TPTP Problem File: SLH0935^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Quasi_Borel_Spaces/0003_Binary_CoProduct_QuasiBorel/prob_00520_020734__15237378_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2014 ( 668 unt; 723 typ;   0 def)
%            Number of atoms       : 3664 (1836 equ;   0 cnn)
%            Maximal formula atoms :   36 (   2 avg)
%            Number of connectives : 11093 ( 480   ~;  33   |; 597   &;8883   @)
%                                         (   0 <=>;1100  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   6 avg)
%            Number of types       :  124 ( 123 usr)
%            Number of type conns  : 3226 (3226   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  603 ( 600 usr;  90 con; 0-3 aty)
%            Number of variables   : 3821 ( 684   ^;2914   !; 223   ?;3821   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:09:21.687
%------------------------------------------------------------------------------
% Could-be-implicit typings (123)
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thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mtf__c_J_M_062_It__Real__Oreal_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mtf__c_J_M_062_It__Real__Oreal_Mtf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mtf__c_J_M_062_It__Real__Oreal_Mtf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mtf__b_J_M_062_It__Real__Oreal_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Sum____Type__Osum_Itf__b_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Sum____Type__Osum_Itf__b_Mtf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Sum____Type__Osum_Itf__b_Mtf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Sum____Type__Osum_Itf__a_Mtf__a_J_J_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__b_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__b_J_M_062_It__Real__Oreal_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_062_It__Real__Oreal_Mtf__c_J_M_062_Itf__a_Mtf__b_J_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Real__Oreal_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_It__Real__Oreal_Mtf__c_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_It__Real__Oreal_Mtf__b_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_It__Real__Oreal_Mtf__a_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_It__Real__Oreal_Mtf__c_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_It__Real__Oreal_Mtf__b_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_It__Real__Oreal_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Sigma____Algebra__Omeasure_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Real__Oreal_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_Itf__c_Mtf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_Itf__a_Mtf__b_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_M_062_It__Real__Oreal_Mtf__c_J_J_J,type,
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thf(ty_n_t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_Itf__c_Mtf__b_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_Itf__a_Mtf__b_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__c_Mtf__b_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__a_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_It__Sigma____Algebra__Omeasure_I_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_Itf__c_Mtf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_Itf__a_Mtf__b_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Real__Oreal_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Nat__Onat_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__c_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Sum____Type__Osum_It__Real__Oreal_Mtf__c_J,type,
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thf(ty_n_t__Sum____Type__Osum_It__Real__Oreal_Mtf__a_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__c_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_Itf__c_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_Itf__b_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_Eo_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__c_Mtf__b_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__c_Mtf__a_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__b_Mtf__c_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__b_Mtf__b_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__b_Mtf__a_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__a_Mtf__a_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_I_Eo_J,type,
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thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__c_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__c_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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thf(ty_n_t__Sum____Type__Osum_I_Eo_M_Eo_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_Itf__c_J,type,
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thf(ty_n_t__Set__Oset_Itf__b_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Complex__Ocomplex,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
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thf(ty_n_t__Real__Oreal,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__c,type,
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thf(ty_n_tf__b,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (600)
thf(sy_c_Basic__BNFs_Osetl_001tf__a_001tf__c,type,
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thf(sy_c_Basic__BNFs_Osetr_001tf__a_001tf__c,type,
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thf(sy_c_Basic__BNFs_Osetrp_001tf__a_001tf__c,type,
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thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001tf__a_001tf__c,type,
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thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__a_001tf__a,type,
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thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__a_001tf__b,type,
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thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__a_001tf__c,type,
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thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__b_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__b_001tf__a,type,
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thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__b_001tf__b,type,
    binary3454967616579736122x2_b_b: quasi_borel_b > quasi_borel_b > set_real_Sum_sum_b_b ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__b_001tf__c,type,
    binary3454967616579736123x2_b_c: quasi_borel_b > quasi_borel_c > set_real_Sum_sum_b_c ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__c_001tf__a,type,
    binary667512034607060088x2_c_a: quasi_borel_c > quasi_borel_a > set_real_Sum_sum_c_a ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__c_001tf__b,type,
    binary667512034607060089x2_c_b: quasi_borel_c > quasi_borel_b > set_real_Sum_sum_c_b ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__c_001tf__c,type,
    binary667512034607060090x2_c_c: quasi_borel_c > quasi_borel_c > set_real_Sum_sum_c_c ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx_001tf__a_001tf__a,type,
    binary8286901584692334520Mx_a_a: quasi_borel_a > quasi_borel_a > set_real_Sum_sum_a_a ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx_001tf__a_001tf__b,type,
    binary8286901584692334521Mx_a_b: quasi_borel_a > quasi_borel_b > set_real_Sum_sum_a_b ).

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_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx_001tf__b_001tf__a,type,
    binary5499446002719658487Mx_b_a: quasi_borel_b > quasi_borel_a > set_real_Sum_sum_b_a ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx_001tf__b_001tf__b,type,
    binary5499446002719658488Mx_b_b: quasi_borel_b > quasi_borel_b > set_real_Sum_sum_b_b ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx_001tf__b_001tf__c,type,
    binary5499446002719658489Mx_b_c: quasi_borel_b > quasi_borel_c > set_real_Sum_sum_b_c ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx_001tf__c_001tf__a,type,
    binary2711990420746982454Mx_c_a: quasi_borel_c > quasi_borel_a > set_real_Sum_sum_c_a ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx_001tf__c_001tf__b,type,
    binary2711990420746982455Mx_c_b: quasi_borel_c > quasi_borel_b > set_real_Sum_sum_c_b ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx_001tf__c_001tf__c,type,
    binary2711990420746982456Mx_c_c: quasi_borel_c > quasi_borel_c > set_real_Sum_sum_c_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__Complex__Ocomplex,type,
    borel_1392132677378845456omplex: sigma_3077487657436305159omplex ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Extended____Nonnegative____Real__Oennreal,type,
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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_Complete__Lattices_OSup__class_OSup_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

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

thf(sy_c_Countable__Set_Ocountable_001t__Complex__Ocomplex,type,
    counta5113917769705169331omplex: set_complex > $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_Extended__Nonnegative__Real_Oennreal,type,
    extend7643940197134561352nnreal: real > extend8495563244428889912nnreal ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

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__Real__Oreal_001_Eo,type,
    comp_o_real_o: ( $o > real ) > ( $o > $o ) > $o > real ).

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,
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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__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,
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thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
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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,
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thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
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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_001t__Real__Oreal,type,
    comp_real_a_real: ( real > a ) > ( real > real ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001tf__a,type,
    comp_real_a_a: ( real > a ) > ( a > real ) > a > a ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001tf__c,type,
    comp_real_a_c: ( real > a ) > ( c > real ) > c > a ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__b_001t__Real__Oreal,type,
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thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__b_001tf__a,type,
    comp_real_b_a: ( real > b ) > ( a > real ) > a > b ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__b_001tf__c,type,
    comp_real_b_c: ( real > b ) > ( c > real ) > c > 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__c_001tf__a,type,
    comp_real_c_a: ( real > c ) > ( a > real ) > a > c ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__c_001tf__c,type,
    comp_real_c_c: ( real > c ) > ( c > real ) > c > c ).

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__Real__Oreal_001tf__a,type,
    comp_a_real_a: ( a > real ) > ( a > a ) > a > real ).

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__a_001tf__a,type,
    comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__c,type,
    comp_a_a_c: ( a > a ) > ( c > a ) > c > 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_001t__Sum____Type__Osum_It__Real__Oreal_Mtf__a_J,type,
    comp_a4970644202208605413real_a: ( a > b ) > ( sum_sum_real_a > a ) > sum_sum_real_a > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001t__Sum____Type__Osum_It__Real__Oreal_Mtf__c_J,type,
    comp_a4970644210815063015real_c: ( a > b ) > ( sum_sum_real_c > a ) > sum_sum_real_c > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001t__Sum____Type__Osum_Itf__c_Mt__Real__Oreal_J,type,
    comp_a6422442893462753785c_real: ( a > b ) > ( sum_sum_c_real > a ) > sum_sum_c_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__b,type,
    comp_a_b_b: ( a > b ) > ( b > a ) > b > 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__c_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_a2820443819341408156l_real: ( a > c ) > ( sum_sum_real_real > a ) > sum_sum_real_real > c ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__a,type,
    comp_a_c_a: ( a > c ) > ( a > a ) > a > c ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__c,type,
    comp_a_c_c: ( a > c ) > ( c > a ) > c > c ).

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__Real__Oreal_001tf__c,type,
    comp_b_real_c: ( b > real ) > ( c > b ) > c > real ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001t__Real__Oreal,type,
    comp_b_a_real: ( b > a ) > ( real > b ) > real > a ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__a,type,
    comp_b_a_a: ( b > a ) > ( a > b ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__c,type,
    comp_b_a_c: ( b > a ) > ( c > b ) > c > a ).

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_001t__Sum____Type__Osum_Itf__a_Mtf__a_J,type,
    comp_b_b_Sum_sum_a_a: ( b > b ) > ( sum_sum_a_a > b ) > sum_sum_a_a > 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__b_001tf__b,type,
    comp_b_b_b: ( b > b ) > ( b > b ) > b > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__c,type,
    comp_b_b_c: ( b > b ) > ( c > b ) > c > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001t__Real__Oreal,type,
    comp_b_c_real: ( b > c ) > ( real > b ) > real > c ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__a,type,
    comp_b_c_a: ( b > c ) > ( a > b ) > a > c ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__c,type,
    comp_b_c_c: ( b > c ) > ( c > b ) > c > c ).

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__Real__Oreal_001tf__a,type,
    comp_c_real_a: ( c > real ) > ( a > c ) > a > real ).

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__a_001tf__a,type,
    comp_c_a_a: ( c > a ) > ( a > c ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__c,type,
    comp_c_a_c: ( c > a ) > ( c > c ) > c > 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__b_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_c4637849929068767129l_real: ( c > b ) > ( sum_sum_real_real > c ) > sum_sum_real_real > b ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__a,type,
    comp_c_b_a: ( c > b ) > ( a > c ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__b,type,
    comp_c_b_b: ( c > b ) > ( b > c ) > b > b ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__c,type,
    comp_c_b_c: ( c > b ) > ( c > c ) > c > 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__c_001tf__a,type,
    comp_c_c_a: ( c > c ) > ( a > c ) > a > c ).

thf(sy_c_Fun_Ofcomp_001_Eo_001t__Real__Oreal_001_Eo,type,
    fcomp_o_real_o: ( $o > real ) > ( real > $o ) > $o > $o ).

thf(sy_c_Fun_Ofcomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
    fcomp_nat_real_nat: ( nat > real ) > ( real > nat ) > nat > nat ).

thf(sy_c_Fun_Ofcomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
    fcomp_real_real_real: ( real > real ) > ( real > real ) > real > real ).

thf(sy_c_Fun_Ofcomp_001t__Real__Oreal_001tf__a_001tf__a,type,
    fcomp_real_a_a: ( real > a ) > ( a > a ) > real > a ).

thf(sy_c_Fun_Ofcomp_001t__Real__Oreal_001tf__a_001tf__b,type,
    fcomp_real_a_b: ( real > a ) > ( a > b ) > real > b ).

thf(sy_c_Fun_Ofcomp_001t__Real__Oreal_001tf__a_001tf__c,type,
    fcomp_real_a_c: ( real > a ) > ( a > c ) > real > c ).

thf(sy_c_Fun_Ofcomp_001t__Real__Oreal_001tf__c_001tf__b,type,
    fcomp_real_c_b: ( real > c ) > ( c > b ) > real > b ).

thf(sy_c_Fun_Ofcomp_001tf__a_001tf__a_001tf__b,type,
    fcomp_a_a_b: ( a > a ) > ( a > b ) > a > b ).

thf(sy_c_Fun_Ofcomp_001tf__a_001tf__b_001tf__b,type,
    fcomp_a_b_b: ( a > b ) > ( b > b ) > a > b ).

thf(sy_c_Fun_Ofcomp_001tf__c_001tf__a_001tf__b,type,
    fcomp_c_a_b: ( c > a ) > ( a > b ) > c > b ).

thf(sy_c_FuncSet_OPi_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Real__Oreal_Mtf__c_J,type,
    pi_real_a_real_c: set_real_a > ( ( real > a ) > set_real_c ) > set_real_a_real_c ).

thf(sy_c_FuncSet_OPi_001_062_It__Real__Oreal_Mtf__b_J_001_062_It__Real__Oreal_Mtf__c_J,type,
    pi_real_b_real_c: set_real_b > ( ( real > b ) > set_real_c ) > set_real_b_real_c ).

thf(sy_c_FuncSet_OPi_001_062_It__Real__Oreal_Mtf__c_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    pi_real_c_real_a: set_real_c > ( ( real > c ) > set_real_a ) > set_real_c_real_a ).

thf(sy_c_FuncSet_OPi_001_062_It__Real__Oreal_Mtf__c_J_001_062_It__Real__Oreal_Mtf__b_J,type,
    pi_real_c_real_b: set_real_c > ( ( real > c ) > set_real_b ) > set_real_c_real_b ).

thf(sy_c_FuncSet_OPi_001_062_It__Real__Oreal_Mtf__c_J_001_062_It__Real__Oreal_Mtf__c_J,type,
    pi_real_c_real_c: set_real_c > ( ( real > c ) > set_real_c ) > set_real_c_real_c ).

thf(sy_c_FuncSet_OPi_001_062_It__Real__Oreal_Mtf__c_J_001_062_Itf__a_Mtf__b_J,type,
    pi_real_c_a_b: set_real_c > ( ( real > c ) > set_a_b ) > set_real_c_a_b ).

thf(sy_c_FuncSet_OPi_001_062_It__Real__Oreal_Mtf__c_J_001_062_Itf__c_Mtf__b_J,type,
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thf(sy_c_Measure__Space_OSup__measure_H_001_Eo,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__c_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,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__Complex__Ocomplex_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_M_Eo_J_J,type,
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    top_to2071711978144146653l_real: set_real_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
    top_top_set_real_a: set_real_a ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    top_top_set_real_b: set_real_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__c_J_J,type,
    top_top_set_real_c: set_real_c ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__c_Mtf__b_J_J,type,
    top_top_set_c_b: set_c_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    top_top_set_complex: set_complex ).

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_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_Omap__qbs_001t__Real__Oreal_001t__Real__Oreal,type,
    map_qbs_real_real: ( real > real ) > quasi_borel_real > quasi_borel_real ).

thf(sy_c_QuasiBorel_Omap__qbs_001tf__a_001tf__a,type,
    map_qbs_a_a: ( a > a ) > quasi_borel_a > quasi_borel_a ).

thf(sy_c_QuasiBorel_Omap__qbs_001tf__a_001tf__b,type,
    map_qbs_a_b: ( a > b ) > quasi_borel_a > quasi_borel_b ).

thf(sy_c_QuasiBorel_Omap__qbs_001tf__a_001tf__c,type,
    map_qbs_a_c: ( a > c ) > quasi_borel_a > quasi_borel_c ).

thf(sy_c_QuasiBorel_Omap__qbs_001tf__b_001tf__a,type,
    map_qbs_b_a: ( b > a ) > quasi_borel_b > quasi_borel_a ).

thf(sy_c_QuasiBorel_Omap__qbs_001tf__b_001tf__b,type,
    map_qbs_b_b: ( b > b ) > quasi_borel_b > quasi_borel_b ).

thf(sy_c_QuasiBorel_Omap__qbs_001tf__b_001tf__c,type,
    map_qbs_b_c: ( b > c ) > quasi_borel_b > quasi_borel_c ).

thf(sy_c_QuasiBorel_Omap__qbs_001tf__c_001tf__a,type,
    map_qbs_c_a: ( c > a ) > quasi_borel_c > quasi_borel_a ).

thf(sy_c_QuasiBorel_Omap__qbs_001tf__c_001tf__b,type,
    map_qbs_c_b: ( c > b ) > quasi_borel_c > quasi_borel_b ).

thf(sy_c_QuasiBorel_Omap__qbs_001tf__c_001tf__c,type,
    map_qbs_c_c: ( c > c ) > quasi_borel_c > quasi_borel_c ).

thf(sy_c_QuasiBorel_Omax__quasi__borel_001tf__a,type,
    max_quasi_borel_a: set_a > quasi_borel_a ).

thf(sy_c_QuasiBorel_Omax__quasi__borel_001tf__b,type,
    max_quasi_borel_b: set_b > quasi_borel_b ).

thf(sy_c_QuasiBorel_Omax__quasi__borel_001tf__c,type,
    max_quasi_borel_c: set_c > quasi_borel_c ).

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

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

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Real__Oreal_Mtf__c_J,type,
    qbs_Mx_real_c: quasi_borel_real_c > set_real_real_c ).

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

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

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

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

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

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

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

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

thf(sy_c_QuasiBorel_Oqbs__Mx_001tf__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__closed1_001t__Real__Oreal,type,
    qbs_closed1_real: set_real_real > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed1_001t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
    qbs_cl96855206660534704um_a_c: set_real_Sum_sum_a_c > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed1_001tf__a,type,
    qbs_closed1_a: set_real_a > $o ).

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

thf(sy_c_QuasiBorel_Oqbs__closed1_001tf__c,type,
    qbs_closed1_c: set_real_c > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed3_001t__Real__Oreal,type,
    qbs_closed3_real: set_real_real > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed3_001t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
    qbs_cl7154477953554275762um_a_c: set_real_Sum_sum_a_c > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed3_001tf__a,type,
    qbs_closed3_a: set_real_a > $o ).

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

thf(sy_c_QuasiBorel_Oqbs__closed3_001tf__c,type,
    qbs_closed3_c: set_real_c > $o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__c_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    qbs_mo6326314525454692790real_a: quasi_borel_real_c > quasi_borel_real_a > set_real_c_real_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__c_J_001_062_It__Real__Oreal_Mtf__b_J,type,
    qbs_mo6326314529757921591real_b: quasi_borel_real_c > quasi_borel_real_b > set_real_c_real_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__c_J_001_062_It__Real__Oreal_Mtf__c_J,type,
    qbs_mo6326314534061150392real_c: quasi_borel_real_c > quasi_borel_real_c > set_real_c_real_c ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__c_J_001_062_Itf__a_Mtf__b_J,type,
    qbs_mo3992850854308683129_c_a_b: quasi_borel_real_c > quasi_borel_a_b > set_real_c_a_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__c_J_001_062_Itf__c_Mtf__b_J,type,
    qbs_mo7641311727218106871_c_c_b: quasi_borel_real_c > quasi_borel_c_b > set_real_c_c_b ).

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__Real__Oreal,type,
    qbs_morphism_o_real: quasi_borel_o > quasi_borel_real > set_o_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Nat__Onat_001t__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__Nat__Onat,type,
    qbs_mo6567951568834356598al_nat: quasi_borel_real > quasi_borel_nat > set_real_nat ).

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

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

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001tf__b,type,
    qbs_morphism_real_b: quasi_borel_real > quasi_borel_b > set_real_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001tf__c,type,
    qbs_morphism_real_c: quasi_borel_real > quasi_borel_c > set_real_c ).

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__b_001tf__a,type,
    qbs_morphism_b_a: quasi_borel_b > quasi_borel_a > set_b_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001tf__b,type,
    qbs_morphism_b_b: quasi_borel_b > quasi_borel_b > set_b_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001tf__c,type,
    qbs_morphism_b_c: quasi_borel_b > quasi_borel_c > set_b_c ).

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__space_001_062_It__Real__Oreal_Mtf__a_J,type,
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thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__a_Mtf__b_J,type,
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thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__c_Mtf__b_J,type,
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thf(sy_c_Sigma__Algebra_Osets_001_Eo,type,
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thf(sy_c_Sigma__Algebra_Osets_001t__Complex__Ocomplex,type,
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thf(sy_c_Sigma__Algebra_Osets_001t__Real__Oreal,type,
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thf(sy_c_Sigma__Algebra_Ospace_001_Eo,type,
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thf(sy_c_Sigma__Algebra_Ospace_001t__Complex__Ocomplex,type,
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thf(sy_c_Sigma__Algebra_Ospace_001t__Real__Oreal,type,
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thf(sy_c_StandardBorel_Or01__binary__expansion_H,type,
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thf(sy_c_StandardBorel_Or01__to__r01__r01__fst_H,type,
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thf(sy_c_StandardBorel_Or01__to__r01__r01__snd_H,type,
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thf(sy_c_Starlike_Oorthogonal__comp_001t__Complex__Ocomplex,type,
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thf(sy_c_Starlike_Oorthogonal__comp_001t__Real__Oreal,type,
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thf(sy_c_Sum__Type_OInl_001tf__a_001tf__a,type,
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thf(sy_c_Sum__Type_OInl_001tf__a_001tf__b,type,
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thf(sy_c_Sum__Type_OInl_001tf__a_001tf__c,type,
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thf(sy_c_Sum__Type_OInl_001tf__b_001tf__b,type,
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thf(sy_c_Sum__Type_OInl_001tf__b_001tf__c,type,
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thf(sy_c_Sum__Type_OInl_001tf__c_001tf__a,type,
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thf(sy_c_Sum__Type_OInl_001tf__c_001tf__b,type,
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thf(sy_c_Sum__Type_OInl_001tf__c_001tf__c,type,
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thf(sy_c_Sum__Type_OInr_001t__Extended____Nonnegative____Real__Oennreal_001tf__b,type,
    sum_In2243461111168403281real_b: extend8495563244428889912nnreal > sum_su8602216633299776360nnreal ).

thf(sy_c_Sum__Type_OInr_001tf__a_001tf__a,type,
    sum_Inr_a_a: a > sum_sum_a_a ).

thf(sy_c_Sum__Type_OInr_001tf__a_001tf__b,type,
    sum_Inr_a_b: a > sum_sum_b_a ).

thf(sy_c_Sum__Type_OInr_001tf__a_001tf__c,type,
    sum_Inr_a_c: a > sum_sum_c_a ).

thf(sy_c_Sum__Type_OInr_001tf__b_001tf__a,type,
    sum_Inr_b_a: b > sum_sum_a_b ).

thf(sy_c_Sum__Type_OInr_001tf__b_001tf__b,type,
    sum_Inr_b_b: b > sum_sum_b_b ).

thf(sy_c_Sum__Type_OInr_001tf__b_001tf__c,type,
    sum_Inr_b_c: b > sum_sum_c_b ).

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__c_001tf__b,type,
    sum_Inr_c_b: c > sum_sum_b_c ).

thf(sy_c_Sum__Type_OInr_001tf__c_001tf__c,type,
    sum_Inr_c_c: c > sum_sum_c_c ).

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,
    sum_ca5525272764133257196real_o: ( $o > real ) > ( $o > real ) > sum_sum_o_o > real ).

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_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__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__a_001tf__a,type,
    sum_ca5000516552359814342al_a_a: ( real > a ) > ( a > a ) > sum_sum_real_a > a ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001tf__a_001tf__c,type,
    sum_ca5000516552359814344al_a_c: ( real > a ) > ( c > a ) > sum_sum_real_c > 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__b_001tf__a,type,
    sum_ca2213060970387138309al_b_a: ( real > b ) > ( a > b ) > sum_sum_real_a > b ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001tf__b_001tf__c,type,
    sum_ca2213060970387138311al_b_c: ( real > b ) > ( c > b ) > sum_sum_real_c > 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_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__a_001t__Real__Oreal,type,
    sum_ca2956635407921249476a_real: ( c > a ) > ( real > a ) > sum_sum_c_real > a ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001tf__c_001tf__b_001t__Real__Oreal,type,
    sum_ca8552854023109380165b_real: ( c > b ) > ( real > b ) > sum_sum_c_real > b ).

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

thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: real > real ).

thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: real > real ).

thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: real > real ).

thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: real > real ).

thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > real > real ).

thf(sy_c_Vector__Spaces_Ofinite__dimensional__vector__space_Odimension_001t__Real__Oreal,type,
    vector5117482691322076262n_real: set_real > nat ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__a_J_M_062_It__Real__Oreal_Mtf__c_J_J,type,
    member_real_a_real_c: ( ( real > a ) > real > c ) > set_real_a_real_c > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__b_J_M_062_It__Real__Oreal_Mtf__c_J_J,type,
    member_real_b_real_c: ( ( real > b ) > real > c ) > set_real_b_real_c > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__c_J_M_062_It__Real__Oreal_Mtf__a_J_J,type,
    member_real_c_real_a: ( ( real > c ) > real > a ) > set_real_c_real_a > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__c_J_M_062_It__Real__Oreal_Mtf__b_J_J,type,
    member_real_c_real_b: ( ( real > c ) > real > b ) > set_real_c_real_b > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__c_J_M_062_It__Real__Oreal_Mtf__c_J_J,type,
    member_real_c_real_c: ( ( real > c ) > real > c ) > set_real_c_real_c > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__c_J_M_062_Itf__a_Mtf__b_J_J,type,
    member_real_c_a_b: ( ( real > c ) > a > b ) > set_real_c_a_b > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__c_J_M_062_Itf__c_Mtf__b_J_J,type,
    member_real_c_c_b: ( ( real > c ) > c > b ) > set_real_c_c_b > $o ).

thf(sy_c_member_001_062_I_062_Itf__a_Mtf__b_J_M_062_It__Real__Oreal_Mtf__c_J_J,type,
    member_a_b_real_c: ( ( a > b ) > real > c ) > set_a_b_real_c > $o ).

thf(sy_c_member_001_062_I_062_Itf__c_Mtf__b_J_M_062_It__Real__Oreal_Mtf__c_J_J,type,
    member_c_b_real_c: ( ( c > b ) > real > c ) > set_c_b_real_c > $o ).

thf(sy_c_member_001_062_I_Eo_M_062_It__Real__Oreal_Mtf__c_J_J,type,
    member_o_real_c: ( $o > real > c ) > set_o_real_c > $o ).

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

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

thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_M_062_It__Real__Oreal_Mtf__c_J_J,type,
    member777441237193892147real_c: ( extend8495563244428889912nnreal > real > c ) > set_Ex4970977155637699434real_c > $o ).

thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Real__Oreal_J,type,
    member2874014351250825754l_real: ( extend8495563244428889912nnreal > real ) > set_Ex5658717452565810105l_real > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Real__Oreal_Mtf__c_J_J,type,
    member_nat_real_c: ( nat > real > c ) > set_nat_real_c > $o ).

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

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

thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__a_J_J,type,
    member_real_real_a: ( real > real > a ) > set_real_real_a > $o ).

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

thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__c_J_J,type,
    member_real_real_c: ( real > real > c ) > set_real_real_c > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_062_Itf__a_Mtf__b_J_J,type,
    member_real_a_b: ( real > a > b ) > set_real_a_b > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_062_Itf__c_Mtf__b_J_J,type,
    member_real_c_b: ( real > c > b ) > set_real_c_b > $o ).

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

thf(sy_c_member_001_062_It__Real__Oreal_Mt__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__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_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__b_Mtf__a_J,type,
    member_b_a: ( b > a ) > set_b_a > $o ).

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

thf(sy_c_member_001_062_Itf__b_Mtf__c_J,type,
    member_b_c: ( b > c ) > set_b_c > $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_Eo,type,
    member_o: $o > set_o > $o ).

thf(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > set_complex > $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_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_062_It__Real__Oreal_Mtf__b_J_J,type,
    member_set_real_b: set_real_b > set_set_real_b > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__c_J_J,type,
    member_set_real_c: set_real_c > set_set_real_c > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
    member_set_a_b: set_a_b > set_set_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_Itf__c_Mtf__b_J_J,type,
    member_set_c_b: set_c_b > set_set_c_b > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    member_set_complex: set_complex > set_set_complex > $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__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_v_F____,type,
    f: nat > real > b ).

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

thf(sy_v_S____,type,
    s: set_real ).

thf(sy_v_X,type,
    x: quasi_borel_a ).

thf(sy_v_Y,type,
    y: quasi_borel_c ).

thf(sy_v_Z,type,
    z: quasi_borel_b ).

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,
    f2: a > b ).

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

% Relevant facts (1265)
thf(fact_0_assms_I2_J,axiom,
    member_c_b @ g @ ( qbs_morphism_c_b @ y @ z ) ).

% assms(2)
thf(fact_1_comp__apply,axiom,
    ( comp_b_b_a
    = ( ^ [F: b > b,G: a > b,X: a] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_2_comp__apply,axiom,
    ( comp_a_c_real
    = ( ^ [F: a > c,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_3_comp__apply,axiom,
    ( comp_a_b_c
    = ( ^ [F: a > b,G: c > a,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_4_comp__apply,axiom,
    ( comp_a_b_a
    = ( ^ [F: a > b,G: a > a,X: a] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_5_comp__apply,axiom,
    ( comp_a_a_real
    = ( ^ [F: a > a,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_6_comp__apply,axiom,
    ( comp_real_real_real
    = ( ^ [F: real > real,G: real > real,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_7_comp__apply,axiom,
    ( comp_real_o_o
    = ( ^ [F: real > $o,G: $o > real,X: $o] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_8_comp__apply,axiom,
    ( comp_real_nat_nat
    = ( ^ [F: real > nat,G: nat > real,X: nat] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_9_comp__apply,axiom,
    ( comp_c_b_real
    = ( ^ [F: c > b,G: real > c,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_10_comp__apply,axiom,
    ( comp_a_b_real
    = ( ^ [F: a > b,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_apply
thf(fact_11__092_060open_062f_A_092_060circ_062_A_092_060alpha_0621_A_092_060in_062_Aqbs__Mx_AZ_092_060close_062,axiom,
    member_real_b @ ( comp_a_b_real @ f2 @ alpha_1 ) @ ( qbs_Mx_b @ z ) ).

% \<open>f \<circ> \<alpha>1 \<in> qbs_Mx Z\<close>
thf(fact_12_qbs__eqI,axiom,
    ! [X2: quasi_borel_b,Y: quasi_borel_b] :
      ( ( ( qbs_Mx_b @ X2 )
        = ( qbs_Mx_b @ Y ) )
     => ( X2 = Y ) ) ).

% qbs_eqI
thf(fact_13_qbs__eqI,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X2 )
        = ( qbs_Mx_a @ Y ) )
     => ( X2 = Y ) ) ).

% qbs_eqI
thf(fact_14_qbs__eqI,axiom,
    ! [X2: quasi_borel_c,Y: quasi_borel_c] :
      ( ( ( qbs_Mx_c @ X2 )
        = ( qbs_Mx_c @ Y ) )
     => ( X2 = Y ) ) ).

% qbs_eqI
thf(fact_15_assms_I1_J,axiom,
    member_a_b @ f2 @ ( qbs_morphism_a_b @ x @ z ) ).

% assms(1)
thf(fact_16_fun_Omap__comp,axiom,
    ! [G2: $o > $o,F2: real > $o,V: $o > real] :
      ( ( comp_o_o_o @ G2 @ ( comp_real_o_o @ F2 @ V ) )
      = ( comp_real_o_o @ ( comp_o_o_real @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_17_fun_Omap__comp,axiom,
    ! [G2: nat > nat,F2: real > nat,V: nat > real] :
      ( ( comp_nat_nat_nat @ G2 @ ( comp_real_nat_nat @ F2 @ V ) )
      = ( comp_real_nat_nat @ ( comp_nat_nat_real @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_18_fun_Omap__comp,axiom,
    ! [G2: b > b,F2: c > b,V: real > c] :
      ( ( comp_b_b_real @ G2 @ ( comp_c_b_real @ F2 @ V ) )
      = ( comp_c_b_real @ ( comp_b_b_c @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_19_fun_Omap__comp,axiom,
    ! [G2: b > b,F2: a > b,V: real > a] :
      ( ( comp_b_b_real @ G2 @ ( comp_a_b_real @ F2 @ V ) )
      = ( comp_a_b_real @ ( comp_b_b_a @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_20_fun_Omap__comp,axiom,
    ! [G2: real > real,F2: real > real,V: real > real] :
      ( ( comp_real_real_real @ G2 @ ( comp_real_real_real @ F2 @ V ) )
      = ( comp_real_real_real @ ( comp_real_real_real @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_21_fun_Omap__comp,axiom,
    ! [G2: real > $o,F2: $o > real,V: $o > $o] :
      ( ( comp_real_o_o @ G2 @ ( comp_o_real_o @ F2 @ V ) )
      = ( comp_o_o_o @ ( comp_real_o_o @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_22_fun_Omap__comp,axiom,
    ! [G2: real > $o,F2: real > real,V: $o > real] :
      ( ( comp_real_o_o @ G2 @ ( comp_real_real_o @ F2 @ V ) )
      = ( comp_real_o_o @ ( comp_real_o_real @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_23_fun_Omap__comp,axiom,
    ! [G2: real > nat,F2: nat > real,V: nat > nat] :
      ( ( comp_real_nat_nat @ G2 @ ( comp_nat_real_nat @ F2 @ V ) )
      = ( comp_nat_nat_nat @ ( comp_real_nat_nat @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_24_fun_Omap__comp,axiom,
    ! [G2: real > nat,F2: real > real,V: nat > real] :
      ( ( comp_real_nat_nat @ G2 @ ( comp_real_real_nat @ F2 @ V ) )
      = ( comp_real_nat_nat @ ( comp_real_nat_real @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_25_fun_Omap__comp,axiom,
    ! [G2: c > b,F2: real > c,V: real > real] :
      ( ( comp_c_b_real @ G2 @ ( comp_real_c_real @ F2 @ V ) )
      = ( comp_real_b_real @ ( comp_c_b_real @ G2 @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_26_comp__def,axiom,
    ( comp_b_b_a
    = ( ^ [F: b > b,G: a > b,X: a] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_27_comp__def,axiom,
    ( comp_a_c_real
    = ( ^ [F: a > c,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_28_comp__def,axiom,
    ( comp_a_b_c
    = ( ^ [F: a > b,G: c > a,X: c] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_29_comp__def,axiom,
    ( comp_a_b_a
    = ( ^ [F: a > b,G: a > a,X: a] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_30_comp__def,axiom,
    ( comp_a_a_real
    = ( ^ [F: a > a,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_31_comp__def,axiom,
    ( comp_real_real_real
    = ( ^ [F: real > real,G: real > real,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_32_comp__def,axiom,
    ( comp_real_o_o
    = ( ^ [F: real > $o,G: $o > real,X: $o] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_33_comp__def,axiom,
    ( comp_real_nat_nat
    = ( ^ [F: real > nat,G: nat > real,X: nat] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_34_comp__def,axiom,
    ( comp_c_b_real
    = ( ^ [F: c > b,G: real > c,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_35_comp__def,axiom,
    ( comp_a_b_real
    = ( ^ [F: a > b,G: real > a,X: real] : ( F @ ( G @ X ) ) ) ) ).

% comp_def
thf(fact_36_comp__assoc,axiom,
    ! [F2: real > $o,G2: $o > real,H: $o > $o] :
      ( ( comp_o_o_o @ ( comp_real_o_o @ F2 @ G2 ) @ H )
      = ( comp_real_o_o @ F2 @ ( comp_o_real_o @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_37_comp__assoc,axiom,
    ! [F2: real > nat,G2: nat > real,H: nat > nat] :
      ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F2 @ G2 ) @ H )
      = ( comp_real_nat_nat @ F2 @ ( comp_nat_real_nat @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_38_comp__assoc,axiom,
    ! [F2: c > b,G2: real > c,H: real > real] :
      ( ( comp_real_b_real @ ( comp_c_b_real @ F2 @ G2 ) @ H )
      = ( comp_c_b_real @ F2 @ ( comp_real_c_real @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_39_comp__assoc,axiom,
    ! [F2: a > b,G2: real > a,H: real > real] :
      ( ( comp_real_b_real @ ( comp_a_b_real @ F2 @ G2 ) @ H )
      = ( comp_a_b_real @ F2 @ ( comp_real_a_real @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_40_comp__assoc,axiom,
    ! [F2: real > real,G2: real > real,H: real > real] :
      ( ( comp_real_real_real @ ( comp_real_real_real @ F2 @ G2 ) @ H )
      = ( comp_real_real_real @ F2 @ ( comp_real_real_real @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_41_comp__assoc,axiom,
    ! [F2: $o > $o,G2: real > $o,H: $o > real] :
      ( ( comp_real_o_o @ ( comp_o_o_real @ F2 @ G2 ) @ H )
      = ( comp_o_o_o @ F2 @ ( comp_real_o_o @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_42_comp__assoc,axiom,
    ! [F2: real > $o,G2: real > real,H: $o > real] :
      ( ( comp_real_o_o @ ( comp_real_o_real @ F2 @ G2 ) @ H )
      = ( comp_real_o_o @ F2 @ ( comp_real_real_o @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_43_comp__assoc,axiom,
    ! [F2: nat > nat,G2: real > nat,H: nat > real] :
      ( ( comp_real_nat_nat @ ( comp_nat_nat_real @ F2 @ G2 ) @ H )
      = ( comp_nat_nat_nat @ F2 @ ( comp_real_nat_nat @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_44_comp__assoc,axiom,
    ! [F2: real > nat,G2: real > real,H: nat > real] :
      ( ( comp_real_nat_nat @ ( comp_real_nat_real @ F2 @ G2 ) @ H )
      = ( comp_real_nat_nat @ F2 @ ( comp_real_real_nat @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_45_comp__assoc,axiom,
    ! [F2: b > b,G2: c > b,H: real > c] :
      ( ( comp_c_b_real @ ( comp_b_b_c @ F2 @ G2 ) @ H )
      = ( comp_b_b_real @ F2 @ ( comp_c_b_real @ G2 @ H ) ) ) ).

% comp_assoc
thf(fact_46_comp__eq__dest,axiom,
    ! [A: real > real,B: real > real,C: real > real,D: real > real,V: real] :
      ( ( ( comp_real_real_real @ A @ B )
        = ( comp_real_real_real @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_47_comp__eq__dest,axiom,
    ! [A: real > $o,B: $o > real,C: real > $o,D: $o > real,V: $o] :
      ( ( ( comp_real_o_o @ A @ B )
        = ( comp_real_o_o @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_48_comp__eq__dest,axiom,
    ! [A: real > nat,B: nat > real,C: real > nat,D: nat > real,V: nat] :
      ( ( ( comp_real_nat_nat @ A @ B )
        = ( comp_real_nat_nat @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_49_comp__eq__dest,axiom,
    ! [A: c > b,B: real > c,C: c > b,D: real > c,V: real] :
      ( ( ( comp_c_b_real @ A @ B )
        = ( comp_c_b_real @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_50_comp__eq__dest,axiom,
    ! [A: c > b,B: real > c,C: a > b,D: real > a,V: real] :
      ( ( ( comp_c_b_real @ A @ B )
        = ( comp_a_b_real @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_51_comp__eq__dest,axiom,
    ! [A: a > b,B: real > a,C: c > b,D: real > c,V: real] :
      ( ( ( comp_a_b_real @ A @ B )
        = ( comp_c_b_real @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_52_comp__eq__dest,axiom,
    ! [A: a > b,B: real > a,C: a > b,D: real > a,V: real] :
      ( ( ( comp_a_b_real @ A @ B )
        = ( comp_a_b_real @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_53_comp__eq__dest,axiom,
    ! [A: b > b,B: a > b,C: b > b,D: a > b,V: a] :
      ( ( ( comp_b_b_a @ A @ B )
        = ( comp_b_b_a @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_54_comp__eq__dest,axiom,
    ! [A: b > b,B: a > b,C: a > b,D: a > a,V: a] :
      ( ( ( comp_b_b_a @ A @ B )
        = ( comp_a_b_a @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_55_comp__eq__dest,axiom,
    ! [A: a > c,B: real > a,C: a > c,D: real > a,V: real] :
      ( ( ( comp_a_c_real @ A @ B )
        = ( comp_a_c_real @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_56_comp__eq__elim,axiom,
    ! [A: real > real,B: real > real,C: real > real,D: real > real] :
      ( ( ( comp_real_real_real @ A @ B )
        = ( comp_real_real_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_57_comp__eq__elim,axiom,
    ! [A: real > $o,B: $o > real,C: real > $o,D: $o > real] :
      ( ( ( comp_real_o_o @ A @ B )
        = ( comp_real_o_o @ C @ D ) )
     => ! [V2: $o] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_58_comp__eq__elim,axiom,
    ! [A: real > nat,B: nat > real,C: real > nat,D: nat > real] :
      ( ( ( comp_real_nat_nat @ A @ B )
        = ( comp_real_nat_nat @ C @ D ) )
     => ! [V2: nat] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_59_comp__eq__elim,axiom,
    ! [A: c > b,B: real > c,C: c > b,D: real > c] :
      ( ( ( comp_c_b_real @ A @ B )
        = ( comp_c_b_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_60_comp__eq__elim,axiom,
    ! [A: c > b,B: real > c,C: a > b,D: real > a] :
      ( ( ( comp_c_b_real @ A @ B )
        = ( comp_a_b_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_61_comp__eq__elim,axiom,
    ! [A: a > b,B: real > a,C: c > b,D: real > c] :
      ( ( ( comp_a_b_real @ A @ B )
        = ( comp_c_b_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_62_comp__eq__elim,axiom,
    ! [A: a > b,B: real > a,C: a > b,D: real > a] :
      ( ( ( comp_a_b_real @ A @ B )
        = ( comp_a_b_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_63_comp__eq__elim,axiom,
    ! [A: b > b,B: a > b,C: b > b,D: a > b] :
      ( ( ( comp_b_b_a @ A @ B )
        = ( comp_b_b_a @ C @ D ) )
     => ! [V2: a] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_64_comp__eq__elim,axiom,
    ! [A: b > b,B: a > b,C: a > b,D: a > a] :
      ( ( ( comp_b_b_a @ A @ B )
        = ( comp_a_b_a @ C @ D ) )
     => ! [V2: a] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_65_comp__eq__elim,axiom,
    ! [A: a > c,B: real > a,C: a > c,D: real > a] :
      ( ( ( comp_a_c_real @ A @ B )
        = ( comp_a_c_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_66_comp__cong,axiom,
    ! [F2: real > real,G2: real > real,X3: real,F3: real > real,G3: real > real,X4: real] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_real_real_real @ F2 @ G2 @ X3 )
        = ( comp_real_real_real @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_67_comp__cong,axiom,
    ! [F2: real > $o,G2: $o > real,X3: $o,F3: real > $o,G3: $o > real,X4: $o] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_real_o_o @ F2 @ G2 @ X3 )
        = ( comp_real_o_o @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_68_comp__cong,axiom,
    ! [F2: real > nat,G2: nat > real,X3: nat,F3: real > nat,G3: nat > real,X4: nat] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_real_nat_nat @ F2 @ G2 @ X3 )
        = ( comp_real_nat_nat @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_69_comp__cong,axiom,
    ! [F2: c > b,G2: real > c,X3: real,F3: c > b,G3: real > c,X4: real] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_c_b_real @ F2 @ G2 @ X3 )
        = ( comp_c_b_real @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_70_comp__cong,axiom,
    ! [F2: c > b,G2: real > c,X3: real,F3: a > b,G3: real > a,X4: real] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_c_b_real @ F2 @ G2 @ X3 )
        = ( comp_a_b_real @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_71_comp__cong,axiom,
    ! [F2: c > b,G2: real > c,X3: real,F3: b > b,G3: a > b,X4: a] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_c_b_real @ F2 @ G2 @ X3 )
        = ( comp_b_b_a @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_72_comp__cong,axiom,
    ! [F2: c > b,G2: real > c,X3: real,F3: a > b,G3: c > a,X4: c] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_c_b_real @ F2 @ G2 @ X3 )
        = ( comp_a_b_c @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_73_comp__cong,axiom,
    ! [F2: c > b,G2: real > c,X3: real,F3: a > b,G3: a > a,X4: a] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_c_b_real @ F2 @ G2 @ X3 )
        = ( comp_a_b_a @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_74_comp__cong,axiom,
    ! [F2: a > b,G2: real > a,X3: real,F3: c > b,G3: real > c,X4: real] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_a_b_real @ F2 @ G2 @ X3 )
        = ( comp_c_b_real @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_75_comp__cong,axiom,
    ! [F2: a > b,G2: real > a,X3: real,F3: a > b,G3: real > a,X4: real] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( F3 @ ( G3 @ X4 ) ) )
     => ( ( comp_a_b_real @ F2 @ G2 @ X3 )
        = ( comp_a_b_real @ F3 @ G3 @ X4 ) ) ) ).

% comp_cong
thf(fact_76_comp__eq__dest__lhs,axiom,
    ! [A: real > real,B: real > real,C: real > real,V: real] :
      ( ( ( comp_real_real_real @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_77_comp__eq__dest__lhs,axiom,
    ! [A: real > $o,B: $o > real,C: $o > $o,V: $o] :
      ( ( ( comp_real_o_o @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_78_comp__eq__dest__lhs,axiom,
    ! [A: real > nat,B: nat > real,C: nat > nat,V: nat] :
      ( ( ( comp_real_nat_nat @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_79_comp__eq__dest__lhs,axiom,
    ! [A: c > b,B: real > c,C: real > b,V: real] :
      ( ( ( comp_c_b_real @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_80_comp__eq__dest__lhs,axiom,
    ! [A: a > b,B: real > a,C: real > b,V: real] :
      ( ( ( comp_a_b_real @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_81_comp__eq__dest__lhs,axiom,
    ! [A: b > b,B: a > b,C: a > b,V: a] :
      ( ( ( comp_b_b_a @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_82_comp__eq__dest__lhs,axiom,
    ! [A: a > c,B: real > a,C: real > c,V: real] :
      ( ( ( comp_a_c_real @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_83_comp__eq__dest__lhs,axiom,
    ! [A: a > b,B: c > a,C: c > b,V: c] :
      ( ( ( comp_a_b_c @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_84_comp__eq__dest__lhs,axiom,
    ! [A: a > b,B: a > a,C: a > b,V: a] :
      ( ( ( comp_a_b_a @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_85_comp__eq__dest__lhs,axiom,
    ! [A: a > a,B: real > a,C: real > a,V: real] :
      ( ( ( comp_a_a_real @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_86_qbs__morphism__comp,axiom,
    ! [F2: real > real,X2: quasi_borel_real,Y: quasi_borel_real,G2: real > real,Z: quasi_borel_real] :
      ( ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) )
     => ( ( member_real_real @ G2 @ ( qbs_mo5229770564518008146l_real @ Y @ Z ) )
       => ( member_real_real @ ( comp_real_real_real @ G2 @ F2 ) @ ( qbs_mo5229770564518008146l_real @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_87_qbs__morphism__comp,axiom,
    ! [F2: $o > real,X2: quasi_borel_o,Y: quasi_borel_real,G2: real > $o,Z: quasi_borel_o] :
      ( ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X2 @ Y ) )
     => ( ( member_real_o @ G2 @ ( qbs_morphism_real_o @ Y @ Z ) )
       => ( member_o_o @ ( comp_real_o_o @ G2 @ F2 ) @ ( qbs_morphism_o_o @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_88_qbs__morphism__comp,axiom,
    ! [F2: nat > real,X2: quasi_borel_nat,Y: quasi_borel_real,G2: real > nat,Z: quasi_borel_nat] :
      ( ( member_nat_real @ F2 @ ( qbs_mo2000642995705457910t_real @ X2 @ Y ) )
     => ( ( member_real_nat @ G2 @ ( qbs_mo6567951568834356598al_nat @ Y @ Z ) )
       => ( member_nat_nat @ ( comp_real_nat_nat @ G2 @ F2 ) @ ( qbs_morphism_nat_nat @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_89_qbs__morphism__comp,axiom,
    ! [F2: real > real,X2: quasi_borel_real,Y: quasi_borel_real,G2: real > c,Z: quasi_borel_c] :
      ( ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) )
     => ( ( member_real_c @ G2 @ ( qbs_morphism_real_c @ Y @ Z ) )
       => ( member_real_c @ ( comp_real_c_real @ G2 @ F2 ) @ ( qbs_morphism_real_c @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_90_qbs__morphism__comp,axiom,
    ! [F2: real > real,X2: quasi_borel_real,Y: quasi_borel_real,G2: real > b,Z: quasi_borel_b] :
      ( ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) )
     => ( ( member_real_b @ G2 @ ( qbs_morphism_real_b @ Y @ Z ) )
       => ( member_real_b @ ( comp_real_b_real @ G2 @ F2 ) @ ( qbs_morphism_real_b @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_91_qbs__morphism__comp,axiom,
    ! [F2: real > real,X2: quasi_borel_real,Y: quasi_borel_real,G2: real > a,Z: quasi_borel_a] :
      ( ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) )
     => ( ( member_real_a @ G2 @ ( qbs_morphism_real_a @ Y @ Z ) )
       => ( member_real_a @ ( comp_real_a_real @ G2 @ F2 ) @ ( qbs_morphism_real_a @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_92_qbs__morphism__comp,axiom,
    ! [F2: real > c,X2: quasi_borel_real,Y: quasi_borel_c,G2: c > c,Z: quasi_borel_c] :
      ( ( member_real_c @ F2 @ ( qbs_morphism_real_c @ X2 @ Y ) )
     => ( ( member_c_c @ G2 @ ( qbs_morphism_c_c @ Y @ Z ) )
       => ( member_real_c @ ( comp_c_c_real @ G2 @ F2 ) @ ( qbs_morphism_real_c @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_93_qbs__morphism__comp,axiom,
    ! [F2: real > c,X2: quasi_borel_real,Y: quasi_borel_c,G2: c > a,Z: quasi_borel_a] :
      ( ( member_real_c @ F2 @ ( qbs_morphism_real_c @ X2 @ Y ) )
     => ( ( member_c_a @ G2 @ ( qbs_morphism_c_a @ Y @ Z ) )
       => ( member_real_a @ ( comp_c_a_real @ G2 @ F2 ) @ ( qbs_morphism_real_a @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_94_qbs__morphism__comp,axiom,
    ! [F2: real > b,X2: quasi_borel_real,Y: quasi_borel_b,G2: b > c,Z: quasi_borel_c] :
      ( ( member_real_b @ F2 @ ( qbs_morphism_real_b @ X2 @ Y ) )
     => ( ( member_b_c @ G2 @ ( qbs_morphism_b_c @ Y @ Z ) )
       => ( member_real_c @ ( comp_b_c_real @ G2 @ F2 ) @ ( qbs_morphism_real_c @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_95_qbs__morphism__comp,axiom,
    ! [F2: real > b,X2: quasi_borel_real,Y: quasi_borel_b,G2: b > b,Z: quasi_borel_b] :
      ( ( member_real_b @ F2 @ ( qbs_morphism_real_b @ X2 @ Y ) )
     => ( ( member_b_b @ G2 @ ( qbs_morphism_b_b @ Y @ Z ) )
       => ( member_real_b @ ( comp_b_b_real @ G2 @ F2 ) @ ( qbs_morphism_real_b @ X2 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_96_qbs__morphismI,axiom,
    ! [X2: quasi_borel_real,F2: real > real,Y: quasi_borel_real] :
      ( ! [Alpha: real > real] :
          ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
         => ( member_real_real @ ( comp_real_real_real @ F2 @ Alpha ) @ ( qbs_Mx_real @ Y ) ) )
     => ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_97_qbs__morphismI,axiom,
    ! [X2: quasi_borel_real,F2: real > b,Y: quasi_borel_b] :
      ( ! [Alpha: real > real] :
          ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
         => ( member_real_b @ ( comp_real_b_real @ F2 @ Alpha ) @ ( qbs_Mx_b @ Y ) ) )
     => ( member_real_b @ F2 @ ( qbs_morphism_real_b @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_98_qbs__morphismI,axiom,
    ! [X2: quasi_borel_real,F2: real > a,Y: quasi_borel_a] :
      ( ! [Alpha: real > real] :
          ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
         => ( member_real_a @ ( comp_real_a_real @ F2 @ Alpha ) @ ( qbs_Mx_a @ Y ) ) )
     => ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_99_qbs__morphismI,axiom,
    ! [X2: quasi_borel_real,F2: real > c,Y: quasi_borel_c] :
      ( ! [Alpha: real > real] :
          ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
         => ( member_real_c @ ( comp_real_c_real @ F2 @ Alpha ) @ ( qbs_Mx_c @ Y ) ) )
     => ( member_real_c @ F2 @ ( qbs_morphism_real_c @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_100_qbs__morphismI,axiom,
    ! [X2: quasi_borel_b,F2: b > b,Y: quasi_borel_b] :
      ( ! [Alpha: real > b] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X2 ) )
         => ( member_real_b @ ( comp_b_b_real @ F2 @ Alpha ) @ ( qbs_Mx_b @ Y ) ) )
     => ( member_b_b @ F2 @ ( qbs_morphism_b_b @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_101_qbs__morphismI,axiom,
    ! [X2: quasi_borel_b,F2: b > a,Y: quasi_borel_a] :
      ( ! [Alpha: real > b] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X2 ) )
         => ( member_real_a @ ( comp_b_a_real @ F2 @ Alpha ) @ ( qbs_Mx_a @ Y ) ) )
     => ( member_b_a @ F2 @ ( qbs_morphism_b_a @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_102_qbs__morphismI,axiom,
    ! [X2: quasi_borel_b,F2: b > c,Y: quasi_borel_c] :
      ( ! [Alpha: real > b] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X2 ) )
         => ( member_real_c @ ( comp_b_c_real @ F2 @ Alpha ) @ ( qbs_Mx_c @ Y ) ) )
     => ( member_b_c @ F2 @ ( qbs_morphism_b_c @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_103_qbs__morphismI,axiom,
    ! [X2: quasi_borel_a,F2: a > a,Y: quasi_borel_a] :
      ( ! [Alpha: real > a] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
         => ( member_real_a @ ( comp_a_a_real @ F2 @ Alpha ) @ ( qbs_Mx_a @ Y ) ) )
     => ( member_a_a @ F2 @ ( qbs_morphism_a_a @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_104_qbs__morphismI,axiom,
    ! [X2: quasi_borel_a,F2: a > c,Y: quasi_borel_c] :
      ( ! [Alpha: real > a] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
         => ( member_real_c @ ( comp_a_c_real @ F2 @ Alpha ) @ ( qbs_Mx_c @ Y ) ) )
     => ( member_a_c @ F2 @ ( qbs_morphism_a_c @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_105_qbs__morphismI,axiom,
    ! [X2: quasi_borel_c,F2: c > a,Y: quasi_borel_a] :
      ( ! [Alpha: real > c] :
          ( ( member_real_c @ Alpha @ ( qbs_Mx_c @ X2 ) )
         => ( member_real_a @ ( comp_c_a_real @ F2 @ Alpha ) @ ( qbs_Mx_a @ Y ) ) )
     => ( member_c_a @ F2 @ ( qbs_morphism_c_a @ X2 @ Y ) ) ) ).

% qbs_morphismI
thf(fact_106_qbs__morphismE_I3_J,axiom,
    ! [F2: real > real,X2: quasi_borel_real,Y: quasi_borel_real,Alpha2: real > real] :
      ( ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) )
     => ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X2 ) )
       => ( member_real_real @ ( comp_real_real_real @ F2 @ Alpha2 ) @ ( qbs_Mx_real @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_107_qbs__morphismE_I3_J,axiom,
    ! [F2: real > b,X2: quasi_borel_real,Y: quasi_borel_b,Alpha2: real > real] :
      ( ( member_real_b @ F2 @ ( qbs_morphism_real_b @ X2 @ Y ) )
     => ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X2 ) )
       => ( member_real_b @ ( comp_real_b_real @ F2 @ Alpha2 ) @ ( qbs_Mx_b @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_108_qbs__morphismE_I3_J,axiom,
    ! [F2: real > a,X2: quasi_borel_real,Y: quasi_borel_a,Alpha2: real > real] :
      ( ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X2 @ Y ) )
     => ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X2 ) )
       => ( member_real_a @ ( comp_real_a_real @ F2 @ Alpha2 ) @ ( qbs_Mx_a @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_109_qbs__morphismE_I3_J,axiom,
    ! [F2: real > c,X2: quasi_borel_real,Y: quasi_borel_c,Alpha2: real > real] :
      ( ( member_real_c @ F2 @ ( qbs_morphism_real_c @ X2 @ Y ) )
     => ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X2 ) )
       => ( member_real_c @ ( comp_real_c_real @ F2 @ Alpha2 ) @ ( qbs_Mx_c @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_110_qbs__morphismE_I3_J,axiom,
    ! [F2: b > b,X2: quasi_borel_b,Y: quasi_borel_b,Alpha2: real > b] :
      ( ( member_b_b @ F2 @ ( qbs_morphism_b_b @ X2 @ Y ) )
     => ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X2 ) )
       => ( member_real_b @ ( comp_b_b_real @ F2 @ Alpha2 ) @ ( qbs_Mx_b @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_111_qbs__morphismE_I3_J,axiom,
    ! [F2: b > a,X2: quasi_borel_b,Y: quasi_borel_a,Alpha2: real > b] :
      ( ( member_b_a @ F2 @ ( qbs_morphism_b_a @ X2 @ Y ) )
     => ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X2 ) )
       => ( member_real_a @ ( comp_b_a_real @ F2 @ Alpha2 ) @ ( qbs_Mx_a @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_112_qbs__morphismE_I3_J,axiom,
    ! [F2: b > c,X2: quasi_borel_b,Y: quasi_borel_c,Alpha2: real > b] :
      ( ( member_b_c @ F2 @ ( qbs_morphism_b_c @ X2 @ Y ) )
     => ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X2 ) )
       => ( member_real_c @ ( comp_b_c_real @ F2 @ Alpha2 ) @ ( qbs_Mx_c @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_113_qbs__morphismE_I3_J,axiom,
    ! [F2: a > a,X2: quasi_borel_a,Y: quasi_borel_a,Alpha2: real > a] :
      ( ( member_a_a @ F2 @ ( qbs_morphism_a_a @ X2 @ Y ) )
     => ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X2 ) )
       => ( member_real_a @ ( comp_a_a_real @ F2 @ Alpha2 ) @ ( qbs_Mx_a @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_114_qbs__morphismE_I3_J,axiom,
    ! [F2: a > c,X2: quasi_borel_a,Y: quasi_borel_c,Alpha2: real > a] :
      ( ( member_a_c @ F2 @ ( qbs_morphism_a_c @ X2 @ Y ) )
     => ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X2 ) )
       => ( member_real_c @ ( comp_a_c_real @ F2 @ Alpha2 ) @ ( qbs_Mx_c @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_115_qbs__morphismE_I3_J,axiom,
    ! [F2: c > a,X2: quasi_borel_c,Y: quasi_borel_a,Alpha2: real > c] :
      ( ( member_c_a @ F2 @ ( qbs_morphism_c_a @ X2 @ Y ) )
     => ( ( member_real_c @ Alpha2 @ ( qbs_Mx_c @ X2 ) )
       => ( member_real_a @ ( comp_c_a_real @ F2 @ Alpha2 ) @ ( qbs_Mx_a @ Y ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_116__092_060open_062_092_060alpha_062_A_092_060in_062_Acopair__qbs__Mx_AX_AY_092_060close_062,axiom,
    member2264291325230826761um_a_c @ alpha @ ( binary8286901584692334522Mx_a_c @ x @ y ) ).

% \<open>\<alpha> \<in> copair_qbs_Mx X Y\<close>
thf(fact_117_F__def,axiom,
    ( f
    = ( ^ [I: nat,R: real] : ( if_b @ ( I = zero_zero_nat ) @ ( comp_a_b_real @ f2 @ alpha_1 @ R ) @ ( comp_c_b_real @ g @ alpha_2 @ R ) ) ) ) ).

% F_def
thf(fact_118_h,axiom,
    ( ( member_real_a @ alpha_1 @ ( qbs_Mx_a @ x ) )
    & ( member_real_c @ alpha_2 @ ( qbs_Mx_c @ y ) )
    & ( 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_119_rewriteR__comp__comp2,axiom,
    ! [G2: $o > real,H: $o > $o,R1: real > real,R2: $o > real,F2: real > $o,L: real > $o] :
      ( ( ( comp_o_real_o @ G2 @ H )
        = ( comp_real_real_o @ R1 @ R2 ) )
     => ( ( ( comp_real_o_real @ F2 @ R1 )
          = L )
       => ( ( comp_o_o_o @ ( comp_real_o_o @ F2 @ G2 ) @ H )
          = ( comp_real_o_o @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_120_rewriteR__comp__comp2,axiom,
    ! [G2: nat > real,H: nat > nat,R1: real > real,R2: nat > real,F2: real > nat,L: real > nat] :
      ( ( ( comp_nat_real_nat @ G2 @ H )
        = ( comp_real_real_nat @ R1 @ R2 ) )
     => ( ( ( comp_real_nat_real @ F2 @ R1 )
          = L )
       => ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F2 @ G2 ) @ H )
          = ( comp_real_nat_nat @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_121_rewriteR__comp__comp2,axiom,
    ! [G2: real > c,H: real > real,R1: c > c,R2: real > c,F2: c > b,L: c > b] :
      ( ( ( comp_real_c_real @ G2 @ H )
        = ( comp_c_c_real @ R1 @ R2 ) )
     => ( ( ( comp_c_b_c @ F2 @ R1 )
          = L )
       => ( ( comp_real_b_real @ ( comp_c_b_real @ F2 @ G2 ) @ H )
          = ( comp_c_b_real @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_122_rewriteR__comp__comp2,axiom,
    ! [G2: real > c,H: a > real,R1: b > c,R2: a > b,F2: c > b,L: b > b] :
      ( ( ( comp_real_c_a @ G2 @ H )
        = ( comp_b_c_a @ R1 @ R2 ) )
     => ( ( ( comp_c_b_b @ F2 @ R1 )
          = L )
       => ( ( comp_real_b_a @ ( comp_c_b_real @ F2 @ G2 ) @ H )
          = ( comp_b_b_a @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_123_rewriteR__comp__comp2,axiom,
    ! [G2: real > c,H: c > real,R1: a > c,R2: c > a,F2: c > b,L: a > b] :
      ( ( ( comp_real_c_c @ G2 @ H )
        = ( comp_a_c_c @ R1 @ R2 ) )
     => ( ( ( comp_c_b_a @ F2 @ R1 )
          = L )
       => ( ( comp_real_b_c @ ( comp_c_b_real @ F2 @ G2 ) @ H )
          = ( comp_a_b_c @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_124_rewriteR__comp__comp2,axiom,
    ! [G2: real > c,H: a > real,R1: a > c,R2: a > a,F2: c > b,L: a > b] :
      ( ( ( comp_real_c_a @ G2 @ H )
        = ( comp_a_c_a @ R1 @ R2 ) )
     => ( ( ( comp_c_b_a @ F2 @ R1 )
          = L )
       => ( ( comp_real_b_a @ ( comp_c_b_real @ F2 @ G2 ) @ H )
          = ( comp_a_b_a @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_125_rewriteR__comp__comp2,axiom,
    ! [G2: real > a,H: a > real,R1: b > a,R2: a > b,F2: a > b,L: b > b] :
      ( ( ( comp_real_a_a @ G2 @ H )
        = ( comp_b_a_a @ R1 @ R2 ) )
     => ( ( ( comp_a_b_b @ F2 @ R1 )
          = L )
       => ( ( comp_real_b_a @ ( comp_a_b_real @ F2 @ G2 ) @ H )
          = ( comp_b_b_a @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_126_rewriteR__comp__comp2,axiom,
    ! [G2: c > a,H: a > c,R1: b > a,R2: a > b,F2: a > b,L: b > b] :
      ( ( ( comp_c_a_a @ G2 @ H )
        = ( comp_b_a_a @ R1 @ R2 ) )
     => ( ( ( comp_a_b_b @ F2 @ R1 )
          = L )
       => ( ( comp_c_b_a @ ( comp_a_b_c @ F2 @ G2 ) @ H )
          = ( comp_b_b_a @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_127_rewriteR__comp__comp2,axiom,
    ! [G2: a > a,H: a > a,R1: b > a,R2: a > b,F2: a > b,L: b > b] :
      ( ( ( comp_a_a_a @ G2 @ H )
        = ( comp_b_a_a @ R1 @ R2 ) )
     => ( ( ( comp_a_b_b @ F2 @ R1 )
          = L )
       => ( ( comp_a_b_a @ ( comp_a_b_a @ F2 @ G2 ) @ H )
          = ( comp_b_b_a @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_128_rewriteR__comp__comp2,axiom,
    ! [G2: real > real,H: $o > real,R1: $o > real,R2: $o > $o,F2: real > $o,L: $o > $o] :
      ( ( ( comp_real_real_o @ G2 @ H )
        = ( comp_o_real_o @ R1 @ R2 ) )
     => ( ( ( comp_real_o_o @ F2 @ R1 )
          = L )
       => ( ( comp_real_o_o @ ( comp_real_o_real @ F2 @ G2 ) @ H )
          = ( comp_o_o_o @ L @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_129_rewriteL__comp__comp2,axiom,
    ! [F2: $o > $o,G2: real > $o,L1: real > $o,L2: real > real,H: $o > real,R3: $o > real] :
      ( ( ( comp_o_o_real @ F2 @ G2 )
        = ( comp_real_o_real @ L1 @ L2 ) )
     => ( ( ( comp_real_real_o @ L2 @ H )
          = R3 )
       => ( ( comp_o_o_o @ F2 @ ( comp_real_o_o @ G2 @ H ) )
          = ( comp_real_o_o @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_130_rewriteL__comp__comp2,axiom,
    ! [F2: nat > nat,G2: real > nat,L1: real > nat,L2: real > real,H: nat > real,R3: nat > real] :
      ( ( ( comp_nat_nat_real @ F2 @ G2 )
        = ( comp_real_nat_real @ L1 @ L2 ) )
     => ( ( ( comp_real_real_nat @ L2 @ H )
          = R3 )
       => ( ( comp_nat_nat_nat @ F2 @ ( comp_real_nat_nat @ G2 @ H ) )
          = ( comp_real_nat_nat @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_131_rewriteL__comp__comp2,axiom,
    ! [F2: b > real,G2: c > b,L1: real > real,L2: c > real,H: real > c,R3: real > real] :
      ( ( ( comp_b_real_c @ F2 @ G2 )
        = ( comp_real_real_c @ L1 @ L2 ) )
     => ( ( ( comp_c_real_real @ L2 @ H )
          = R3 )
       => ( ( comp_b_real_real @ F2 @ ( comp_c_b_real @ G2 @ H ) )
          = ( comp_real_real_real @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_132_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G2: c > b,L1: c > b,L2: c > c,H: real > c,R3: real > c] :
      ( ( ( comp_b_b_c @ F2 @ G2 )
        = ( comp_c_b_c @ L1 @ L2 ) )
     => ( ( ( comp_c_c_real @ L2 @ H )
          = R3 )
       => ( ( comp_b_b_real @ F2 @ ( comp_c_b_real @ G2 @ H ) )
          = ( comp_c_b_real @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_133_rewriteL__comp__comp2,axiom,
    ! [F2: b > c,G2: c > b,L1: a > c,L2: c > a,H: real > c,R3: real > a] :
      ( ( ( comp_b_c_c @ F2 @ G2 )
        = ( comp_a_c_c @ L1 @ L2 ) )
     => ( ( ( comp_c_a_real @ L2 @ H )
          = R3 )
       => ( ( comp_b_c_real @ F2 @ ( comp_c_b_real @ G2 @ H ) )
          = ( comp_a_c_real @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_134_rewriteL__comp__comp2,axiom,
    ! [F2: b > a,G2: c > b,L1: a > a,L2: c > a,H: real > c,R3: real > a] :
      ( ( ( comp_b_a_c @ F2 @ G2 )
        = ( comp_a_a_c @ L1 @ L2 ) )
     => ( ( ( comp_c_a_real @ L2 @ H )
          = R3 )
       => ( ( comp_b_a_real @ F2 @ ( comp_c_b_real @ G2 @ H ) )
          = ( comp_a_a_real @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_135_rewriteL__comp__comp2,axiom,
    ! [F2: b > real,G2: a > b,L1: real > real,L2: a > real,H: real > a,R3: real > real] :
      ( ( ( comp_b_real_a @ F2 @ G2 )
        = ( comp_real_real_a @ L1 @ L2 ) )
     => ( ( ( comp_a_real_real @ L2 @ H )
          = R3 )
       => ( ( comp_b_real_real @ F2 @ ( comp_a_b_real @ G2 @ H ) )
          = ( comp_real_real_real @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_136_rewriteL__comp__comp2,axiom,
    ! [F2: c > real,G2: a > c,L1: real > real,L2: a > real,H: real > a,R3: real > real] :
      ( ( ( comp_c_real_a @ F2 @ G2 )
        = ( comp_real_real_a @ L1 @ L2 ) )
     => ( ( ( comp_a_real_real @ L2 @ H )
          = R3 )
       => ( ( comp_c_real_real @ F2 @ ( comp_a_c_real @ G2 @ H ) )
          = ( comp_real_real_real @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_137_rewriteL__comp__comp2,axiom,
    ! [F2: a > real,G2: a > a,L1: real > real,L2: a > real,H: real > a,R3: real > real] :
      ( ( ( comp_a_real_a @ F2 @ G2 )
        = ( comp_real_real_a @ L1 @ L2 ) )
     => ( ( ( comp_a_real_real @ L2 @ H )
          = R3 )
       => ( ( comp_a_real_real @ F2 @ ( comp_a_a_real @ G2 @ H ) )
          = ( comp_real_real_real @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_138_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G2: b > b,L1: a > b,L2: b > a,H: a > b,R3: a > a] :
      ( ( ( comp_b_b_b @ F2 @ G2 )
        = ( comp_a_b_b @ L1 @ L2 ) )
     => ( ( ( comp_b_a_a @ L2 @ H )
          = R3 )
       => ( ( comp_b_b_a @ F2 @ ( comp_b_b_a @ G2 @ H ) )
          = ( comp_a_b_a @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_139_rewriteR__comp__comp,axiom,
    ! [G2: $o > real,H: $o > $o,R3: $o > real,F2: real > $o] :
      ( ( ( comp_o_real_o @ G2 @ H )
        = R3 )
     => ( ( comp_o_o_o @ ( comp_real_o_o @ F2 @ G2 ) @ H )
        = ( comp_real_o_o @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_140_rewriteR__comp__comp,axiom,
    ! [G2: nat > real,H: nat > nat,R3: nat > real,F2: real > nat] :
      ( ( ( comp_nat_real_nat @ G2 @ H )
        = R3 )
     => ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F2 @ G2 ) @ H )
        = ( comp_real_nat_nat @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_141_rewriteR__comp__comp,axiom,
    ! [G2: real > c,H: real > real,R3: real > c,F2: c > b] :
      ( ( ( comp_real_c_real @ G2 @ H )
        = R3 )
     => ( ( comp_real_b_real @ ( comp_c_b_real @ F2 @ G2 ) @ H )
        = ( comp_c_b_real @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_142_rewriteR__comp__comp,axiom,
    ! [G2: real > a,H: real > real,R3: real > a,F2: a > b] :
      ( ( ( comp_real_a_real @ G2 @ H )
        = R3 )
     => ( ( comp_real_b_real @ ( comp_a_b_real @ F2 @ G2 ) @ H )
        = ( comp_a_b_real @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_143_rewriteR__comp__comp,axiom,
    ! [G2: real > a,H: c > real,R3: c > a,F2: a > b] :
      ( ( ( comp_real_a_c @ G2 @ H )
        = R3 )
     => ( ( comp_real_b_c @ ( comp_a_b_real @ F2 @ G2 ) @ H )
        = ( comp_a_b_c @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_144_rewriteR__comp__comp,axiom,
    ! [G2: real > a,H: a > real,R3: a > a,F2: a > b] :
      ( ( ( comp_real_a_a @ G2 @ H )
        = R3 )
     => ( ( comp_real_b_a @ ( comp_a_b_real @ F2 @ G2 ) @ H )
        = ( comp_a_b_a @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_145_rewriteR__comp__comp,axiom,
    ! [G2: real > a,H: real > real,R3: real > a,F2: a > c] :
      ( ( ( comp_real_a_real @ G2 @ H )
        = R3 )
     => ( ( comp_real_c_real @ ( comp_a_c_real @ F2 @ G2 ) @ H )
        = ( comp_a_c_real @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_146_rewriteR__comp__comp,axiom,
    ! [G2: c > a,H: c > c,R3: c > a,F2: a > b] :
      ( ( ( comp_c_a_c @ G2 @ H )
        = R3 )
     => ( ( comp_c_b_c @ ( comp_a_b_c @ F2 @ G2 ) @ H )
        = ( comp_a_b_c @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_147_rewriteR__comp__comp,axiom,
    ! [G2: c > a,H: a > c,R3: a > a,F2: a > b] :
      ( ( ( comp_c_a_a @ G2 @ H )
        = R3 )
     => ( ( comp_c_b_a @ ( comp_a_b_c @ F2 @ G2 ) @ H )
        = ( comp_a_b_a @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_148_rewriteR__comp__comp,axiom,
    ! [G2: real > a,H: real > real,R3: real > a,F2: a > a] :
      ( ( ( comp_real_a_real @ G2 @ H )
        = R3 )
     => ( ( comp_real_a_real @ ( comp_a_a_real @ F2 @ G2 ) @ H )
        = ( comp_a_a_real @ F2 @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_149_rewriteL__comp__comp,axiom,
    ! [F2: $o > $o,G2: real > $o,L: real > $o,H: $o > real] :
      ( ( ( comp_o_o_real @ F2 @ G2 )
        = L )
     => ( ( comp_o_o_o @ F2 @ ( comp_real_o_o @ G2 @ H ) )
        = ( comp_real_o_o @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_150_rewriteL__comp__comp,axiom,
    ! [F2: nat > nat,G2: real > nat,L: real > nat,H: nat > real] :
      ( ( ( comp_nat_nat_real @ F2 @ G2 )
        = L )
     => ( ( comp_nat_nat_nat @ F2 @ ( comp_real_nat_nat @ G2 @ H ) )
        = ( comp_real_nat_nat @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_151_rewriteL__comp__comp,axiom,
    ! [F2: b > b,G2: c > b,L: c > b,H: real > c] :
      ( ( ( comp_b_b_c @ F2 @ G2 )
        = L )
     => ( ( comp_b_b_real @ F2 @ ( comp_c_b_real @ G2 @ H ) )
        = ( comp_c_b_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_152_rewriteL__comp__comp,axiom,
    ! [F2: b > c,G2: a > b,L: a > c,H: real > a] :
      ( ( ( comp_b_c_a @ F2 @ G2 )
        = L )
     => ( ( comp_b_c_real @ F2 @ ( comp_a_b_real @ G2 @ H ) )
        = ( comp_a_c_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_153_rewriteL__comp__comp,axiom,
    ! [F2: b > a,G2: a > b,L: a > a,H: real > a] :
      ( ( ( comp_b_a_a @ F2 @ G2 )
        = L )
     => ( ( comp_b_a_real @ F2 @ ( comp_a_b_real @ G2 @ H ) )
        = ( comp_a_a_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_154_rewriteL__comp__comp,axiom,
    ! [F2: c > c,G2: a > c,L: a > c,H: real > a] :
      ( ( ( comp_c_c_a @ F2 @ G2 )
        = L )
     => ( ( comp_c_c_real @ F2 @ ( comp_a_c_real @ G2 @ H ) )
        = ( comp_a_c_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_155_rewriteL__comp__comp,axiom,
    ! [F2: c > a,G2: a > c,L: a > a,H: real > a] :
      ( ( ( comp_c_a_a @ F2 @ G2 )
        = L )
     => ( ( comp_c_a_real @ F2 @ ( comp_a_c_real @ G2 @ H ) )
        = ( comp_a_a_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_156_rewriteL__comp__comp,axiom,
    ! [F2: real > $o,G2: real > real,L: real > $o,H: $o > real] :
      ( ( ( comp_real_o_real @ F2 @ G2 )
        = L )
     => ( ( comp_real_o_o @ F2 @ ( comp_real_real_o @ G2 @ H ) )
        = ( comp_real_o_o @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_157_rewriteL__comp__comp,axiom,
    ! [F2: real > nat,G2: real > real,L: real > nat,H: nat > real] :
      ( ( ( comp_real_nat_real @ F2 @ G2 )
        = L )
     => ( ( comp_real_nat_nat @ F2 @ ( comp_real_real_nat @ G2 @ H ) )
        = ( comp_real_nat_nat @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_158_rewriteL__comp__comp,axiom,
    ! [F2: c > b,G2: c > c,L: c > b,H: real > c] :
      ( ( ( comp_c_b_c @ F2 @ G2 )
        = L )
     => ( ( comp_c_b_real @ F2 @ ( comp_c_c_real @ G2 @ H ) )
        = ( comp_c_b_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_159_type__copy__map__cong0,axiom,
    ! [M: $o > real,G2: $o > $o,X3: $o,N: real > real,H: $o > real,F2: real > $o] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_o_o_o @ ( comp_real_o_o @ F2 @ M ) @ G2 @ X3 )
        = ( comp_real_o_o @ ( comp_real_o_real @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_160_type__copy__map__cong0,axiom,
    ! [M: nat > real,G2: nat > nat,X3: nat,N: real > real,H: nat > real,F2: real > nat] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F2 @ M ) @ G2 @ X3 )
        = ( comp_real_nat_nat @ ( comp_real_nat_real @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_161_type__copy__map__cong0,axiom,
    ! [M: real > c,G2: real > real,X3: real,N: c > c,H: real > c,F2: c > b] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_real_b_real @ ( comp_c_b_real @ F2 @ M ) @ G2 @ X3 )
        = ( comp_c_b_real @ ( comp_c_b_c @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_162_type__copy__map__cong0,axiom,
    ! [M: real > c,G2: real > real,X3: real,N: a > c,H: real > a,F2: c > b] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_real_b_real @ ( comp_c_b_real @ F2 @ M ) @ G2 @ X3 )
        = ( comp_a_b_real @ ( comp_c_b_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_163_type__copy__map__cong0,axiom,
    ! [M: real > c,G2: a > real,X3: a,N: b > c,H: a > b,F2: c > b] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_real_b_a @ ( comp_c_b_real @ F2 @ M ) @ G2 @ X3 )
        = ( comp_b_b_a @ ( comp_c_b_b @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_164_type__copy__map__cong0,axiom,
    ! [M: real > c,G2: c > real,X3: c,N: a > c,H: c > a,F2: c > b] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_real_b_c @ ( comp_c_b_real @ F2 @ M ) @ G2 @ X3 )
        = ( comp_a_b_c @ ( comp_c_b_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_165_type__copy__map__cong0,axiom,
    ! [M: real > c,G2: a > real,X3: a,N: a > c,H: a > a,F2: c > b] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_real_b_a @ ( comp_c_b_real @ F2 @ M ) @ G2 @ X3 )
        = ( comp_a_b_a @ ( comp_c_b_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_166_type__copy__map__cong0,axiom,
    ! [M: real > a,G2: real > real,X3: real,N: c > a,H: real > c,F2: a > b] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_real_b_real @ ( comp_a_b_real @ F2 @ M ) @ G2 @ X3 )
        = ( comp_c_b_real @ ( comp_a_b_c @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_167_type__copy__map__cong0,axiom,
    ! [M: real > a,G2: real > real,X3: real,N: a > a,H: real > a,F2: a > b] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_real_b_real @ ( comp_a_b_real @ F2 @ M ) @ G2 @ X3 )
        = ( comp_a_b_real @ ( comp_a_b_a @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_168_type__copy__map__cong0,axiom,
    ! [M: real > a,G2: a > real,X3: a,N: b > a,H: a > b,F2: a > b] :
      ( ( ( M @ ( G2 @ X3 ) )
        = ( N @ ( H @ X3 ) ) )
     => ( ( comp_real_b_a @ ( comp_a_b_real @ F2 @ M ) @ G2 @ X3 )
        = ( comp_b_b_a @ ( comp_a_b_b @ F2 @ N ) @ H @ X3 ) ) ) ).

% type_copy_map_cong0
thf(fact_169_function__factors__right,axiom,
    ! [G2: real > real,F2: real > real] :
      ( ( ! [X: real] :
          ? [Y2: real] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: real > real] :
            ( F2
            = ( comp_real_real_real @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_170_function__factors__right,axiom,
    ! [G2: real > $o,F2: $o > $o] :
      ( ( ! [X: $o] :
          ? [Y2: real] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: $o > real] :
            ( F2
            = ( comp_real_o_o @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_171_function__factors__right,axiom,
    ! [G2: real > nat,F2: nat > nat] :
      ( ( ! [X: nat] :
          ? [Y2: real] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: nat > real] :
            ( F2
            = ( comp_real_nat_nat @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_172_function__factors__right,axiom,
    ! [G2: c > b,F2: real > b] :
      ( ( ! [X: real] :
          ? [Y2: c] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: real > c] :
            ( F2
            = ( comp_c_b_real @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_173_function__factors__right,axiom,
    ! [G2: a > b,F2: real > b] :
      ( ( ! [X: real] :
          ? [Y2: a] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: real > a] :
            ( F2
            = ( comp_a_b_real @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_174_function__factors__right,axiom,
    ! [G2: b > b,F2: a > b] :
      ( ( ! [X: a] :
          ? [Y2: b] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: a > b] :
            ( F2
            = ( comp_b_b_a @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_175_function__factors__right,axiom,
    ! [G2: a > c,F2: real > c] :
      ( ( ! [X: real] :
          ? [Y2: a] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: real > a] :
            ( F2
            = ( comp_a_c_real @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_176_function__factors__right,axiom,
    ! [G2: a > b,F2: c > b] :
      ( ( ! [X: c] :
          ? [Y2: a] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: c > a] :
            ( F2
            = ( comp_a_b_c @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_177_function__factors__right,axiom,
    ! [G2: a > b,F2: a > b] :
      ( ( ! [X: a] :
          ? [Y2: a] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: a > a] :
            ( F2
            = ( comp_a_b_a @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_178_function__factors__right,axiom,
    ! [G2: a > a,F2: real > a] :
      ( ( ! [X: real] :
          ? [Y2: a] :
            ( ( G2 @ Y2 )
            = ( F2 @ X ) ) )
      = ( ? [H2: real > a] :
            ( F2
            = ( comp_a_a_real @ G2 @ H2 ) ) ) ) ).

% function_factors_right
thf(fact_179_function__factors__left,axiom,
    ! [G2: real > real,F2: real > real] :
      ( ( ! [X: real,Y2: real] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: real > real] :
            ( F2
            = ( comp_real_real_real @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_180_function__factors__left,axiom,
    ! [G2: $o > real,F2: $o > $o] :
      ( ( ! [X: $o,Y2: $o] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: real > $o] :
            ( F2
            = ( comp_real_o_o @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_181_function__factors__left,axiom,
    ! [G2: nat > real,F2: nat > nat] :
      ( ( ! [X: nat,Y2: nat] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: real > nat] :
            ( F2
            = ( comp_real_nat_nat @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_182_function__factors__left,axiom,
    ! [G2: real > c,F2: real > b] :
      ( ( ! [X: real,Y2: real] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: c > b] :
            ( F2
            = ( comp_c_b_real @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_183_function__factors__left,axiom,
    ! [G2: real > a,F2: real > b] :
      ( ( ! [X: real,Y2: real] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: a > b] :
            ( F2
            = ( comp_a_b_real @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_184_function__factors__left,axiom,
    ! [G2: a > b,F2: a > b] :
      ( ( ! [X: a,Y2: a] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: b > b] :
            ( F2
            = ( comp_b_b_a @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_185_function__factors__left,axiom,
    ! [G2: real > a,F2: real > c] :
      ( ( ! [X: real,Y2: real] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: a > c] :
            ( F2
            = ( comp_a_c_real @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_186_function__factors__left,axiom,
    ! [G2: c > a,F2: c > b] :
      ( ( ! [X: c,Y2: c] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: a > b] :
            ( F2
            = ( comp_a_b_c @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_187_function__factors__left,axiom,
    ! [G2: a > a,F2: a > b] :
      ( ( ! [X: a,Y2: a] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: a > b] :
            ( F2
            = ( comp_a_b_a @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_188_function__factors__left,axiom,
    ! [G2: real > a,F2: real > a] :
      ( ( ! [X: real,Y2: real] :
            ( ( ( G2 @ X )
              = ( G2 @ Y2 ) )
           => ( ( F2 @ X )
              = ( F2 @ Y2 ) ) ) )
      = ( ? [H2: a > a] :
            ( F2
            = ( comp_a_a_real @ H2 @ G2 ) ) ) ) ).

% function_factors_left
thf(fact_189_comp__apply__eq,axiom,
    ! [F2: real > real,G2: real > real,X3: real,H: real > real,K: real > real] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_real_real_real @ F2 @ G2 @ X3 )
        = ( comp_real_real_real @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_190_comp__apply__eq,axiom,
    ! [F2: real > $o,G2: $o > real,X3: $o,H: real > $o,K: $o > real] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_real_o_o @ F2 @ G2 @ X3 )
        = ( comp_real_o_o @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_191_comp__apply__eq,axiom,
    ! [F2: real > nat,G2: nat > real,X3: nat,H: real > nat,K: nat > real] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_real_nat_nat @ F2 @ G2 @ X3 )
        = ( comp_real_nat_nat @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_192_comp__apply__eq,axiom,
    ! [F2: c > b,G2: real > c,X3: real,H: c > b,K: real > c] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_c_b_real @ F2 @ G2 @ X3 )
        = ( comp_c_b_real @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_193_comp__apply__eq,axiom,
    ! [F2: c > b,G2: real > c,X3: real,H: a > b,K: real > a] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_c_b_real @ F2 @ G2 @ X3 )
        = ( comp_a_b_real @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_194_comp__apply__eq,axiom,
    ! [F2: a > b,G2: real > a,X3: real,H: c > b,K: real > c] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_a_b_real @ F2 @ G2 @ X3 )
        = ( comp_c_b_real @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_195_comp__apply__eq,axiom,
    ! [F2: a > b,G2: real > a,X3: real,H: a > b,K: real > a] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_a_b_real @ F2 @ G2 @ X3 )
        = ( comp_a_b_real @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_196_comp__apply__eq,axiom,
    ! [F2: b > b,G2: a > b,X3: a,H: b > b,K: a > b] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_b_b_a @ F2 @ G2 @ X3 )
        = ( comp_b_b_a @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_197_comp__apply__eq,axiom,
    ! [F2: b > b,G2: a > b,X3: a,H: a > b,K: a > a] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_b_b_a @ F2 @ G2 @ X3 )
        = ( comp_a_b_a @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_198_comp__apply__eq,axiom,
    ! [F2: a > c,G2: real > a,X3: real,H: a > c,K: real > a] :
      ( ( ( F2 @ ( G2 @ X3 ) )
        = ( H @ ( K @ X3 ) ) )
     => ( ( comp_a_c_real @ F2 @ G2 @ X3 )
        = ( comp_a_c_real @ H @ K @ X3 ) ) ) ).

% comp_apply_eq
thf(fact_199_fcomp__comp,axiom,
    ( fcomp_real_real_real
    = ( ^ [F: real > real,G: real > real] : ( comp_real_real_real @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_200_fcomp__comp,axiom,
    ( fcomp_o_real_o
    = ( ^ [F: $o > real,G: real > $o] : ( comp_real_o_o @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_201_fcomp__comp,axiom,
    ( fcomp_nat_real_nat
    = ( ^ [F: nat > real,G: real > nat] : ( comp_real_nat_nat @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_202_fcomp__comp,axiom,
    ( fcomp_real_c_b
    = ( ^ [F: real > c,G: c > b] : ( comp_c_b_real @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_203_fcomp__comp,axiom,
    ( fcomp_real_a_b
    = ( ^ [F: real > a,G: a > b] : ( comp_a_b_real @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_204_fcomp__comp,axiom,
    ( fcomp_a_b_b
    = ( ^ [F: a > b,G: b > b] : ( comp_b_b_a @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_205_fcomp__comp,axiom,
    ( fcomp_real_a_c
    = ( ^ [F: real > a,G: a > c] : ( comp_a_c_real @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_206_fcomp__comp,axiom,
    ( fcomp_c_a_b
    = ( ^ [F: c > a,G: a > b] : ( comp_a_b_c @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_207_fcomp__comp,axiom,
    ( fcomp_a_a_b
    = ( ^ [F: a > a,G: a > b] : ( comp_a_b_a @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_208_fcomp__comp,axiom,
    ( fcomp_real_a_a
    = ( ^ [F: real > a,G: a > a] : ( comp_a_a_real @ G @ F ) ) ) ).

% fcomp_comp
thf(fact_209_Inr__Inl__False,axiom,
    ! [X3: c,Y3: a] :
      ( ( sum_Inr_c_a @ X3 )
     != ( sum_Inl_a_c @ Y3 ) ) ).

% Inr_Inl_False
thf(fact_210_Inl__Inr__False,axiom,
    ! [X3: a,Y3: c] :
      ( ( sum_Inl_a_c @ X3 )
     != ( sum_Inr_c_a @ Y3 ) ) ).

% Inl_Inr_False
thf(fact_211__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_AY_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 @ y ) )
          & ( 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 Y \<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_212_fun_Omap__ident,axiom,
    ! [T: real > real] :
      ( ( comp_real_real_real
        @ ^ [X: real] : X
        @ T )
      = T ) ).

% fun.map_ident
thf(fact_213_fun_Omap__ident,axiom,
    ! [T: a > b] :
      ( ( comp_b_b_a
        @ ^ [X: b] : X
        @ T )
      = T ) ).

% fun.map_ident
thf(fact_214_fun_Omap__ident,axiom,
    ! [T: real > a] :
      ( ( comp_a_a_real
        @ ^ [X: a] : X
        @ T )
      = T ) ).

% fun.map_ident
thf(fact_215_P__def,axiom,
    ( p
    = ( ^ [R: real] : ( if_nat @ ( member_real @ R @ s ) @ zero_zero_nat @ one_one_nat ) ) ) ).

% P_def
thf(fact_216_compose__const_I2_J,axiom,
    ! [A: real,G2: real > real] :
      ( ( comp_real_real_real
        @ ^ [X: real] : A
        @ G2 )
      = ( ^ [X: real] : A ) ) ).

% compose_const(2)
thf(fact_217_compose__const_I2_J,axiom,
    ! [A: $o,G2: $o > real] :
      ( ( comp_real_o_o
        @ ^ [X: real] : A
        @ G2 )
      = ( ^ [X: $o] : A ) ) ).

% compose_const(2)
thf(fact_218_compose__const_I2_J,axiom,
    ! [A: nat,G2: nat > real] :
      ( ( comp_real_nat_nat
        @ ^ [X: real] : A
        @ G2 )
      = ( ^ [X: nat] : A ) ) ).

% compose_const(2)
thf(fact_219_compose__const_I2_J,axiom,
    ! [A: b,G2: real > c] :
      ( ( comp_c_b_real
        @ ^ [X: c] : A
        @ G2 )
      = ( ^ [X: real] : A ) ) ).

% compose_const(2)
thf(fact_220_compose__const_I2_J,axiom,
    ! [A: b,G2: real > a] :
      ( ( comp_a_b_real
        @ ^ [X: a] : A
        @ G2 )
      = ( ^ [X: real] : A ) ) ).

% compose_const(2)
thf(fact_221_compose__const_I2_J,axiom,
    ! [A: b,G2: a > b] :
      ( ( comp_b_b_a
        @ ^ [X: b] : A
        @ G2 )
      = ( ^ [X: a] : A ) ) ).

% compose_const(2)
thf(fact_222_compose__const_I2_J,axiom,
    ! [A: c,G2: real > a] :
      ( ( comp_a_c_real
        @ ^ [X: a] : A
        @ G2 )
      = ( ^ [X: real] : A ) ) ).

% compose_const(2)
thf(fact_223_compose__const_I2_J,axiom,
    ! [A: b,G2: c > a] :
      ( ( comp_a_b_c
        @ ^ [X: a] : A
        @ G2 )
      = ( ^ [X: c] : A ) ) ).

% compose_const(2)
thf(fact_224_compose__const_I2_J,axiom,
    ! [A: b,G2: a > a] :
      ( ( comp_a_b_a
        @ ^ [X: a] : A
        @ G2 )
      = ( ^ [X: a] : A ) ) ).

% compose_const(2)
thf(fact_225_compose__const_I2_J,axiom,
    ! [A: a,G2: real > a] :
      ( ( comp_a_a_real
        @ ^ [X: a] : A
        @ G2 )
      = ( ^ [X: real] : A ) ) ).

% compose_const(2)
thf(fact_226_K__record__comp,axiom,
    ! [C: real,F2: real > real] :
      ( ( comp_real_real_real
        @ ^ [X: real] : C
        @ F2 )
      = ( ^ [X: real] : C ) ) ).

% K_record_comp
thf(fact_227_K__record__comp,axiom,
    ! [C: $o,F2: $o > real] :
      ( ( comp_real_o_o
        @ ^ [X: real] : C
        @ F2 )
      = ( ^ [X: $o] : C ) ) ).

% K_record_comp
thf(fact_228_K__record__comp,axiom,
    ! [C: nat,F2: nat > real] :
      ( ( comp_real_nat_nat
        @ ^ [X: real] : C
        @ F2 )
      = ( ^ [X: nat] : C ) ) ).

% K_record_comp
thf(fact_229_K__record__comp,axiom,
    ! [C: b,F2: real > c] :
      ( ( comp_c_b_real
        @ ^ [X: c] : C
        @ F2 )
      = ( ^ [X: real] : C ) ) ).

% K_record_comp
thf(fact_230_K__record__comp,axiom,
    ! [C: b,F2: real > a] :
      ( ( comp_a_b_real
        @ ^ [X: a] : C
        @ F2 )
      = ( ^ [X: real] : C ) ) ).

% K_record_comp
thf(fact_231_K__record__comp,axiom,
    ! [C: b,F2: a > b] :
      ( ( comp_b_b_a
        @ ^ [X: b] : C
        @ F2 )
      = ( ^ [X: a] : C ) ) ).

% K_record_comp
thf(fact_232_K__record__comp,axiom,
    ! [C: c,F2: real > a] :
      ( ( comp_a_c_real
        @ ^ [X: a] : C
        @ F2 )
      = ( ^ [X: real] : C ) ) ).

% K_record_comp
thf(fact_233_K__record__comp,axiom,
    ! [C: b,F2: c > a] :
      ( ( comp_a_b_c
        @ ^ [X: a] : C
        @ F2 )
      = ( ^ [X: c] : C ) ) ).

% K_record_comp
thf(fact_234_K__record__comp,axiom,
    ! [C: b,F2: a > a] :
      ( ( comp_a_b_a
        @ ^ [X: a] : C
        @ F2 )
      = ( ^ [X: a] : C ) ) ).

% K_record_comp
thf(fact_235_K__record__comp,axiom,
    ! [C: a,F2: real > a] :
      ( ( comp_a_a_real
        @ ^ [X: a] : C
        @ F2 )
      = ( ^ [X: real] : C ) ) ).

% K_record_comp
thf(fact_236_old_Osum_Oinject_I1_J,axiom,
    ! [A: a,A2: a] :
      ( ( ( sum_Inl_a_c @ A )
        = ( sum_Inl_a_c @ A2 ) )
      = ( A = A2 ) ) ).

% old.sum.inject(1)
thf(fact_237_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_238_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_239_sum_Oinject_I2_J,axiom,
    ! [X22: c,Y22: c] :
      ( ( ( sum_Inr_c_a @ X22 )
        = ( sum_Inr_c_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% sum.inject(2)
thf(fact_240_mem__Collect__eq,axiom,
    ! [A: real > c,P: ( real > c ) > $o] :
      ( ( member_real_c @ A @ ( collect_real_c @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_241_mem__Collect__eq,axiom,
    ! [A: real > b,P: ( real > b ) > $o] :
      ( ( member_real_b @ A @ ( collect_real_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_242_mem__Collect__eq,axiom,
    ! [A: real > a,P: ( real > a ) > $o] :
      ( ( member_real_a @ A @ ( collect_real_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_243_mem__Collect__eq,axiom,
    ! [A: c > b,P: ( c > b ) > $o] :
      ( ( member_c_b @ A @ ( collect_c_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_244_mem__Collect__eq,axiom,
    ! [A: a > b,P: ( a > b ) > $o] :
      ( ( member_a_b @ A @ ( collect_a_b @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_245_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_246_Collect__mem__eq,axiom,
    ! [A3: set_real_c] :
      ( ( collect_real_c
        @ ^ [X: real > c] : ( member_real_c @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_247_Collect__mem__eq,axiom,
    ! [A3: set_real_b] :
      ( ( collect_real_b
        @ ^ [X: real > b] : ( member_real_b @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_248_Collect__mem__eq,axiom,
    ! [A3: set_real_a] :
      ( ( collect_real_a
        @ ^ [X: real > a] : ( member_real_a @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_249_Collect__mem__eq,axiom,
    ! [A3: set_c_b] :
      ( ( collect_c_b
        @ ^ [X: c > b] : ( member_c_b @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_250_Collect__mem__eq,axiom,
    ! [A3: set_a_b] :
      ( ( collect_a_b
        @ ^ [X: a > b] : ( member_a_b @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_251_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_252_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_253_obj__sumE,axiom,
    ! [S: sum_sum_a_c] :
      ( ! [X5: a] :
          ( S
         != ( sum_Inl_a_c @ X5 ) )
     => ~ ! [X5: c] :
            ( S
           != ( sum_Inr_c_a @ X5 ) ) ) ).

% obj_sumE
thf(fact_254_split__sum__all,axiom,
    ( ( ^ [P2: sum_sum_a_c > $o] :
        ! [X6: sum_sum_a_c] : ( P2 @ X6 ) )
    = ( ^ [P3: sum_sum_a_c > $o] :
          ( ! [X: a] : ( P3 @ ( sum_Inl_a_c @ X ) )
          & ! [X: c] : ( P3 @ ( sum_Inr_c_a @ X ) ) ) ) ) ).

% split_sum_all
thf(fact_255_split__sum__ex,axiom,
    ( ( ^ [P2: sum_sum_a_c > $o] :
        ? [X6: sum_sum_a_c] : ( P2 @ X6 ) )
    = ( ^ [P3: sum_sum_a_c > $o] :
          ( ? [X: a] : ( P3 @ ( sum_Inl_a_c @ X ) )
          | ? [X: c] : ( P3 @ ( sum_Inr_c_a @ X ) ) ) ) ) ).

% split_sum_ex
thf(fact_256_one__reorient,axiom,
    ! [X3: nat] :
      ( ( one_one_nat = X3 )
      = ( X3 = one_one_nat ) ) ).

% one_reorient
thf(fact_257_one__reorient,axiom,
    ! [X3: real] :
      ( ( one_one_real = X3 )
      = ( X3 = one_one_real ) ) ).

% one_reorient
thf(fact_258_one__reorient,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( ( one_on2969667320475766781nnreal = X3 )
      = ( X3 = one_on2969667320475766781nnreal ) ) ).

% one_reorient
thf(fact_259_zero__reorient,axiom,
    ! [X3: nat] :
      ( ( zero_zero_nat = X3 )
      = ( X3 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_260_zero__reorient,axiom,
    ! [X3: real] :
      ( ( zero_zero_real = X3 )
      = ( X3 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_261_zero__reorient,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( ( zero_z7100319975126383169nnreal = X3 )
      = ( X3 = zero_z7100319975126383169nnreal ) ) ).

% zero_reorient
thf(fact_262_Inr__inject,axiom,
    ! [X3: c,Y3: c] :
      ( ( ( sum_Inr_c_a @ X3 )
        = ( sum_Inr_c_a @ Y3 ) )
     => ( X3 = Y3 ) ) ).

% Inr_inject
thf(fact_263_not__arg__cong__Inr,axiom,
    ! [X3: c,Y3: c] :
      ( ( X3 != Y3 )
     => ( ( sum_Inr_c_a @ X3 )
       != ( sum_Inr_c_a @ Y3 ) ) ) ).

% not_arg_cong_Inr
thf(fact_264_Inl__inject,axiom,
    ! [X3: a,Y3: a] :
      ( ( ( sum_Inl_a_c @ X3 )
        = ( sum_Inl_a_c @ Y3 ) )
     => ( X3 = Y3 ) ) ).

% Inl_inject
thf(fact_265_compose__const_I1_J,axiom,
    ! [F2: real > real,A: real] :
      ( ( comp_real_real_real @ F2
        @ ^ [X: real] : A )
      = ( ^ [X: real] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_266_compose__const_I1_J,axiom,
    ! [F2: real > $o,A: real] :
      ( ( comp_real_o_o @ F2
        @ ^ [X: $o] : A )
      = ( ^ [X: $o] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_267_compose__const_I1_J,axiom,
    ! [F2: real > nat,A: real] :
      ( ( comp_real_nat_nat @ F2
        @ ^ [X: nat] : A )
      = ( ^ [X: nat] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_268_compose__const_I1_J,axiom,
    ! [F2: c > b,A: c] :
      ( ( comp_c_b_real @ F2
        @ ^ [X: real] : A )
      = ( ^ [X: real] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_269_compose__const_I1_J,axiom,
    ! [F2: a > b,A: a] :
      ( ( comp_a_b_real @ F2
        @ ^ [X: real] : A )
      = ( ^ [X: real] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_270_compose__const_I1_J,axiom,
    ! [F2: b > b,A: b] :
      ( ( comp_b_b_a @ F2
        @ ^ [X: a] : A )
      = ( ^ [X: a] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_271_compose__const_I1_J,axiom,
    ! [F2: a > c,A: a] :
      ( ( comp_a_c_real @ F2
        @ ^ [X: real] : A )
      = ( ^ [X: real] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_272_compose__const_I1_J,axiom,
    ! [F2: a > b,A: a] :
      ( ( comp_a_b_c @ F2
        @ ^ [X: c] : A )
      = ( ^ [X: c] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_273_compose__const_I1_J,axiom,
    ! [F2: a > b,A: a] :
      ( ( comp_a_b_a @ F2
        @ ^ [X: a] : A )
      = ( ^ [X: a] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_274_compose__const_I1_J,axiom,
    ! [F2: a > a,A: a] :
      ( ( comp_a_a_real @ F2
        @ ^ [X: real] : A )
      = ( ^ [X: real] : ( F2 @ A ) ) ) ).

% compose_const(1)
thf(fact_275_sum_Odistinct_I1_J,axiom,
    ! [X1: a,X22: c] :
      ( ( sum_Inl_a_c @ X1 )
     != ( sum_Inr_c_a @ X22 ) ) ).

% sum.distinct(1)
thf(fact_276_old_Osum_Odistinct_I2_J,axiom,
    ! [B2: c,A: a] :
      ( ( sum_Inr_c_a @ B2 )
     != ( sum_Inl_a_c @ A ) ) ).

% old.sum.distinct(2)
thf(fact_277_old_Osum_Odistinct_I1_J,axiom,
    ! [A: a,B2: c] :
      ( ( sum_Inl_a_c @ A )
     != ( sum_Inr_c_a @ B2 ) ) ).

% old.sum.distinct(1)
thf(fact_278_old_Osum_Oexhaust,axiom,
    ! [Y3: sum_sum_a_c] :
      ( ! [A4: a] :
          ( Y3
         != ( sum_Inl_a_c @ A4 ) )
     => ~ ! [B3: c] :
            ( Y3
           != ( sum_Inr_c_a @ B3 ) ) ) ).

% old.sum.exhaust
thf(fact_279_sumE,axiom,
    ! [S: sum_sum_a_c] :
      ( ! [X5: a] :
          ( S
         != ( sum_Inl_a_c @ X5 ) )
     => ~ ! [Y4: c] :
            ( S
           != ( sum_Inr_c_a @ Y4 ) ) ) ).

% sumE
thf(fact_280_Inr__not__Inl,axiom,
    ! [B: c,A: a] :
      ( ( sum_Inr_c_a @ B )
     != ( sum_Inl_a_c @ A ) ) ).

% Inr_not_Inl
thf(fact_281_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_282_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_283_zero__neq__one,axiom,
    zero_z7100319975126383169nnreal != one_on2969667320475766781nnreal ).

% zero_neq_one
thf(fact_284_copair__qbs__Mx__equiv,axiom,
    binary8286901584692334522Mx_a_c = binary6242423198552412156x2_a_c ).

% copair_qbs_Mx_equiv
thf(fact_285_copair__qbs__Mx,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_c] :
      ( ( qbs_Mx_Sum_sum_a_c @ ( binary8555328655094383375bs_a_c @ X2 @ Y ) )
      = ( binary8286901584692334522Mx_a_c @ X2 @ Y ) ) ).

% copair_qbs_Mx
thf(fact_286_hs,axiom,
    ( ( member_set_real @ s @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
    & ( ( s = bot_bot_set_real )
     => ? [X5: real > a] :
          ( ( member_real_a @ X5 @ ( qbs_Mx_a @ x ) )
          & ( alpha
            = ( ^ [R: real] : ( sum_Inl_a_c @ ( X5 @ R ) ) ) ) ) )
    & ( ( s = top_top_set_real )
     => ? [X5: real > c] :
          ( ( member_real_c @ X5 @ ( qbs_Mx_c @ y ) )
          & ( alpha
            = ( ^ [R: real] : ( sum_Inr_c_a @ ( X5 @ R ) ) ) ) ) )
    & ( ( ( s != bot_bot_set_real )
        & ( s != top_top_set_real ) )
     => ? [X5: real > a] :
          ( ( member_real_a @ X5 @ ( qbs_Mx_a @ x ) )
          & ? [Xa: real > c] :
              ( ( member_real_c @ Xa @ ( qbs_Mx_c @ y ) )
              & ( alpha
                = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ s ) @ ( sum_Inl_a_c @ ( X5 @ R ) ) @ ( sum_Inr_c_a @ ( Xa @ R ) ) ) ) ) ) ) ) ) ).

% hs
thf(fact_287_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_288__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062S_O_AS_A_092_060in_062_Asets_Areal__borel_A_092_060and_062_A_IS_A_061_A_123_125_A_092_060longrightarrow_062_A_I_092_060exists_062_092_060alpha_0621_092_060in_062qbs__Mx_AX_O_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_AInl_A_I_092_060alpha_0621_Ar_J_J_J_J_A_092_060and_062_A_IS_A_061_AUNIV_A_092_060longrightarrow_062_A_I_092_060exists_062_092_060alpha_0622_092_060in_062qbs__Mx_AY_O_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_AInr_A_I_092_060alpha_0622_Ar_J_J_J_J_A_092_060and_062_A_IS_A_092_060noteq_062_A_123_125_A_092_060and_062_AS_A_092_060noteq_062_AUNIV_A_092_060longrightarrow_062_A_I_092_060exists_062_092_060alpha_0621_092_060in_062qbs__Mx_AX_O_A_092_060exists_062_092_060alpha_0622_092_060in_062qbs__Mx_AY_O_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_Aif_Ar_A_092_060in_062_AS_Athen_AInl_A_I_092_060alpha_0621_Ar_J_Aelse_AInr_A_I_092_060alpha_0622_Ar_J_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [S2: set_real] :
        ~ ( ( member_set_real @ S2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
          & ( ( S2 = bot_bot_set_real )
           => ? [X5: real > a] :
                ( ( member_real_a @ X5 @ ( qbs_Mx_a @ x ) )
                & ( alpha
                  = ( ^ [R: real] : ( sum_Inl_a_c @ ( X5 @ R ) ) ) ) ) )
          & ( ( S2 = top_top_set_real )
           => ? [X5: real > c] :
                ( ( member_real_c @ X5 @ ( qbs_Mx_c @ y ) )
                & ( alpha
                  = ( ^ [R: real] : ( sum_Inr_c_a @ ( X5 @ R ) ) ) ) ) )
          & ( ( ( S2 != bot_bot_set_real )
              & ( S2 != top_top_set_real ) )
           => ? [X5: real > a] :
                ( ( member_real_a @ X5 @ ( qbs_Mx_a @ x ) )
                & ? [Xa: real > c] :
                    ( ( member_real_c @ Xa @ ( qbs_Mx_c @ y ) )
                    & ( alpha
                      = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ S2 ) @ ( sum_Inl_a_c @ ( X5 @ R ) ) @ ( sum_Inr_c_a @ ( Xa @ R ) ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>S. S \<in> sets real_borel \<and> (S = {} \<longrightarrow> (\<exists>\<alpha>1\<in>qbs_Mx X. \<alpha> = (\<lambda>r. Inl (\<alpha>1 r)))) \<and> (S = UNIV \<longrightarrow> (\<exists>\<alpha>2\<in>qbs_Mx Y. \<alpha> = (\<lambda>r. Inr (\<alpha>2 r)))) \<and> (S \<noteq> {} \<and> S \<noteq> UNIV \<longrightarrow> (\<exists>\<alpha>1\<in>qbs_Mx X. \<exists>\<alpha>2\<in>qbs_Mx Y. \<alpha> = (\<lambda>r. if r \<in> S then Inl (\<alpha>1 r) else Inr (\<alpha>2 r)))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_289_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_290__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_291__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_292_iso__tuple__UNIV__I,axiom,
    ! [X3: real > c] : ( member_real_c @ X3 @ top_top_set_real_c ) ).

% iso_tuple_UNIV_I
thf(fact_293_iso__tuple__UNIV__I,axiom,
    ! [X3: real > b] : ( member_real_b @ X3 @ top_top_set_real_b ) ).

% iso_tuple_UNIV_I
thf(fact_294_iso__tuple__UNIV__I,axiom,
    ! [X3: real > a] : ( member_real_a @ X3 @ top_top_set_real_a ) ).

% iso_tuple_UNIV_I
thf(fact_295_iso__tuple__UNIV__I,axiom,
    ! [X3: c > b] : ( member_c_b @ X3 @ top_top_set_c_b ) ).

% iso_tuple_UNIV_I
thf(fact_296_iso__tuple__UNIV__I,axiom,
    ! [X3: a > b] : ( member_a_b @ X3 @ top_top_set_a_b ) ).

% iso_tuple_UNIV_I
thf(fact_297_iso__tuple__UNIV__I,axiom,
    ! [X3: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X3 @ top_to7994903218803871134nnreal ) ).

% iso_tuple_UNIV_I
thf(fact_298_iso__tuple__UNIV__I,axiom,
    ! [X3: complex] : ( member_complex @ X3 @ top_top_set_complex ) ).

% iso_tuple_UNIV_I
thf(fact_299_iso__tuple__UNIV__I,axiom,
    ! [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).

% iso_tuple_UNIV_I
thf(fact_300_iso__tuple__UNIV__I,axiom,
    ! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).

% iso_tuple_UNIV_I
thf(fact_301_iso__tuple__UNIV__I,axiom,
    ! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_302_Inr__qbs__morphism,axiom,
    ! [Y: quasi_borel_c,X2: quasi_borel_a] : ( member_c_Sum_sum_a_c @ sum_Inr_c_a @ ( qbs_mo5084992033439934511um_a_c @ Y @ ( binary8555328655094383375bs_a_c @ X2 @ Y ) ) ) ).

% Inr_qbs_morphism
thf(fact_303_Inl__qbs__morphism,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_c] : ( member_a_Sum_sum_a_c @ sum_Inl_a_c @ ( qbs_mo7250741323400969261um_a_c @ X2 @ ( binary8555328655094383375bs_a_c @ X2 @ Y ) ) ) ).

% Inl_qbs_morphism
thf(fact_304_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_305_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec6648975593938027277nnreal
            @ ^ [S3: extend8495563244428889912nnreal] : P )
          = top_to7994903218803871134nnreal ) )
      & ( ~ P
       => ( ( collec6648975593938027277nnreal
            @ ^ [S3: extend8495563244428889912nnreal] : P )
          = bot_bo4854962954004695426nnreal ) ) ) ).

% Collect_const
thf(fact_306_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_complex
            @ ^ [S3: complex] : P )
          = top_top_set_complex ) )
      & ( ~ P
       => ( ( collect_complex
            @ ^ [S3: complex] : P )
          = bot_bot_set_complex ) ) ) ).

% Collect_const
thf(fact_307_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_real
            @ ^ [S3: real] : P )
          = top_top_set_real ) )
      & ( ~ P
       => ( ( collect_real
            @ ^ [S3: real] : P )
          = bot_bot_set_real ) ) ) ).

% Collect_const
thf(fact_308_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_o
            @ ^ [S3: $o] : P )
          = top_top_set_o ) )
      & ( ~ P
       => ( ( collect_o
            @ ^ [S3: $o] : P )
          = bot_bot_set_o ) ) ) ).

% Collect_const
thf(fact_309_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_nat
            @ ^ [S3: nat] : P )
          = top_top_set_nat ) )
      & ( ~ P
       => ( ( collect_nat
            @ ^ [S3: nat] : P )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_310_sets_Oempty__sets,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.empty_sets
thf(fact_311_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_real] : ( member_set_real @ bot_bot_set_real @ ( sigma_sets_real @ M ) ) ).

% sets.empty_sets
thf(fact_312_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_o] : ( member_set_o @ bot_bot_set_o @ ( sigma_sets_o @ M ) ) ).

% sets.empty_sets
thf(fact_313_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_nat] : ( member_set_nat @ bot_bot_set_nat @ ( sigma_sets_nat @ M ) ) ).

% sets.empty_sets
thf(fact_314_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_315_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_316_space__in__borel,axiom,
    member_set_complex @ top_top_set_complex @ ( sigma_sets_complex @ borel_1392132677378845456omplex ) ).

% space_in_borel
thf(fact_317_space__in__borel,axiom,
    member_set_real @ top_top_set_real @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).

% space_in_borel
thf(fact_318_space__in__borel,axiom,
    member_set_nat @ top_top_set_nat @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ).

% space_in_borel
thf(fact_319_space__in__borel,axiom,
    member603777416030116741nnreal @ top_to7994903218803871134nnreal @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ).

% space_in_borel
thf(fact_320_space__in__borel,axiom,
    member_set_o @ top_top_set_o @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ).

% space_in_borel
thf(fact_321_empty__Collect__eq,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ( ( bot_bo4854962954004695426nnreal
        = ( collec6648975593938027277nnreal @ P ) )
      = ( ! [X: extend8495563244428889912nnreal] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_322_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X: real] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_323_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_324_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_325_Collect__empty__eq,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ( ( ( collec6648975593938027277nnreal @ P )
        = bot_bo4854962954004695426nnreal )
      = ( ! [X: extend8495563244428889912nnreal] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_326_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X: real] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_327_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_328_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_329_all__not__in__conv,axiom,
    ! [A3: set_real_c] :
      ( ( ! [X: real > c] :
            ~ ( member_real_c @ X @ A3 ) )
      = ( A3 = bot_bot_set_real_c ) ) ).

% all_not_in_conv
thf(fact_330_all__not__in__conv,axiom,
    ! [A3: set_real_b] :
      ( ( ! [X: real > b] :
            ~ ( member_real_b @ X @ A3 ) )
      = ( A3 = bot_bot_set_real_b ) ) ).

% all_not_in_conv
thf(fact_331_all__not__in__conv,axiom,
    ! [A3: set_real_a] :
      ( ( ! [X: real > a] :
            ~ ( member_real_a @ X @ A3 ) )
      = ( A3 = bot_bot_set_real_a ) ) ).

% all_not_in_conv
thf(fact_332_all__not__in__conv,axiom,
    ! [A3: set_c_b] :
      ( ( ! [X: c > b] :
            ~ ( member_c_b @ X @ A3 ) )
      = ( A3 = bot_bot_set_c_b ) ) ).

% all_not_in_conv
thf(fact_333_all__not__in__conv,axiom,
    ! [A3: set_a_b] :
      ( ( ! [X: a > b] :
            ~ ( member_a_b @ X @ A3 ) )
      = ( A3 = bot_bot_set_a_b ) ) ).

% all_not_in_conv
thf(fact_334_all__not__in__conv,axiom,
    ! [A3: set_Ex3793607809372303086nnreal] :
      ( ( ! [X: extend8495563244428889912nnreal] :
            ~ ( member7908768830364227535nnreal @ X @ A3 ) )
      = ( A3 = bot_bo4854962954004695426nnreal ) ) ).

% all_not_in_conv
thf(fact_335_all__not__in__conv,axiom,
    ! [A3: set_real] :
      ( ( ! [X: real] :
            ~ ( member_real @ X @ A3 ) )
      = ( A3 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_336_all__not__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ! [X: $o] :
            ~ ( member_o @ X @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_337_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_338_empty__iff,axiom,
    ! [C: real > c] :
      ~ ( member_real_c @ C @ bot_bot_set_real_c ) ).

% empty_iff
thf(fact_339_empty__iff,axiom,
    ! [C: real > b] :
      ~ ( member_real_b @ C @ bot_bot_set_real_b ) ).

% empty_iff
thf(fact_340_empty__iff,axiom,
    ! [C: real > a] :
      ~ ( member_real_a @ C @ bot_bot_set_real_a ) ).

% empty_iff
thf(fact_341_empty__iff,axiom,
    ! [C: c > b] :
      ~ ( member_c_b @ C @ bot_bot_set_c_b ) ).

% empty_iff
thf(fact_342_empty__iff,axiom,
    ! [C: a > b] :
      ~ ( member_a_b @ C @ bot_bot_set_a_b ) ).

% empty_iff
thf(fact_343_empty__iff,axiom,
    ! [C: extend8495563244428889912nnreal] :
      ~ ( member7908768830364227535nnreal @ C @ bot_bo4854962954004695426nnreal ) ).

% empty_iff
thf(fact_344_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_345_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_346_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_347_UNIV__I,axiom,
    ! [X3: real > c] : ( member_real_c @ X3 @ top_top_set_real_c ) ).

% UNIV_I
thf(fact_348_UNIV__I,axiom,
    ! [X3: real > b] : ( member_real_b @ X3 @ top_top_set_real_b ) ).

% UNIV_I
thf(fact_349_UNIV__I,axiom,
    ! [X3: real > a] : ( member_real_a @ X3 @ top_top_set_real_a ) ).

% UNIV_I
thf(fact_350_UNIV__I,axiom,
    ! [X3: c > b] : ( member_c_b @ X3 @ top_top_set_c_b ) ).

% UNIV_I
thf(fact_351_UNIV__I,axiom,
    ! [X3: a > b] : ( member_a_b @ X3 @ top_top_set_a_b ) ).

% UNIV_I
thf(fact_352_UNIV__I,axiom,
    ! [X3: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X3 @ top_to7994903218803871134nnreal ) ).

% UNIV_I
thf(fact_353_UNIV__I,axiom,
    ! [X3: complex] : ( member_complex @ X3 @ top_top_set_complex ) ).

% UNIV_I
thf(fact_354_UNIV__I,axiom,
    ! [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).

% UNIV_I
thf(fact_355_UNIV__I,axiom,
    ! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).

% UNIV_I
thf(fact_356_UNIV__I,axiom,
    ! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_357_bot__set__def,axiom,
    ( bot_bo4854962954004695426nnreal
    = ( collec6648975593938027277nnreal @ bot_bo412624608084785539real_o ) ) ).

% bot_set_def
thf(fact_358_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_359_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_360_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_361_UNIV__eq__I,axiom,
    ! [A3: set_real_c] :
      ( ! [X5: real > c] : ( member_real_c @ X5 @ A3 )
     => ( top_top_set_real_c = A3 ) ) ).

% UNIV_eq_I
thf(fact_362_UNIV__eq__I,axiom,
    ! [A3: set_real_b] :
      ( ! [X5: real > b] : ( member_real_b @ X5 @ A3 )
     => ( top_top_set_real_b = A3 ) ) ).

% UNIV_eq_I
thf(fact_363_UNIV__eq__I,axiom,
    ! [A3: set_real_a] :
      ( ! [X5: real > a] : ( member_real_a @ X5 @ A3 )
     => ( top_top_set_real_a = A3 ) ) ).

% UNIV_eq_I
thf(fact_364_UNIV__eq__I,axiom,
    ! [A3: set_c_b] :
      ( ! [X5: c > b] : ( member_c_b @ X5 @ A3 )
     => ( top_top_set_c_b = A3 ) ) ).

% UNIV_eq_I
thf(fact_365_UNIV__eq__I,axiom,
    ! [A3: set_a_b] :
      ( ! [X5: a > b] : ( member_a_b @ X5 @ A3 )
     => ( top_top_set_a_b = A3 ) ) ).

% UNIV_eq_I
thf(fact_366_UNIV__eq__I,axiom,
    ! [A3: set_Ex3793607809372303086nnreal] :
      ( ! [X5: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X5 @ A3 )
     => ( top_to7994903218803871134nnreal = A3 ) ) ).

% UNIV_eq_I
thf(fact_367_UNIV__eq__I,axiom,
    ! [A3: set_complex] :
      ( ! [X5: complex] : ( member_complex @ X5 @ A3 )
     => ( top_top_set_complex = A3 ) ) ).

% UNIV_eq_I
thf(fact_368_UNIV__eq__I,axiom,
    ! [A3: set_real] :
      ( ! [X5: real] : ( member_real @ X5 @ A3 )
     => ( top_top_set_real = A3 ) ) ).

% UNIV_eq_I
thf(fact_369_UNIV__eq__I,axiom,
    ! [A3: set_o] :
      ( ! [X5: $o] : ( member_o @ X5 @ A3 )
     => ( top_top_set_o = A3 ) ) ).

% UNIV_eq_I
thf(fact_370_UNIV__eq__I,axiom,
    ! [A3: set_nat] :
      ( ! [X5: nat] : ( member_nat @ X5 @ A3 )
     => ( top_top_set_nat = A3 ) ) ).

% UNIV_eq_I
thf(fact_371_UNIV__witness,axiom,
    ? [X5: real > c] : ( member_real_c @ X5 @ top_top_set_real_c ) ).

% UNIV_witness
thf(fact_372_UNIV__witness,axiom,
    ? [X5: real > b] : ( member_real_b @ X5 @ top_top_set_real_b ) ).

% UNIV_witness
thf(fact_373_UNIV__witness,axiom,
    ? [X5: real > a] : ( member_real_a @ X5 @ top_top_set_real_a ) ).

% UNIV_witness
thf(fact_374_UNIV__witness,axiom,
    ? [X5: c > b] : ( member_c_b @ X5 @ top_top_set_c_b ) ).

% UNIV_witness
thf(fact_375_UNIV__witness,axiom,
    ? [X5: a > b] : ( member_a_b @ X5 @ top_top_set_a_b ) ).

% UNIV_witness
thf(fact_376_UNIV__witness,axiom,
    ? [X5: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X5 @ top_to7994903218803871134nnreal ) ).

% UNIV_witness
thf(fact_377_UNIV__witness,axiom,
    ? [X5: complex] : ( member_complex @ X5 @ top_top_set_complex ) ).

% UNIV_witness
thf(fact_378_UNIV__witness,axiom,
    ? [X5: real] : ( member_real @ X5 @ top_top_set_real ) ).

% UNIV_witness
thf(fact_379_UNIV__witness,axiom,
    ? [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ).

% UNIV_witness
thf(fact_380_UNIV__witness,axiom,
    ? [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_381_emptyE,axiom,
    ! [A: real > c] :
      ~ ( member_real_c @ A @ bot_bot_set_real_c ) ).

% emptyE
thf(fact_382_emptyE,axiom,
    ! [A: real > b] :
      ~ ( member_real_b @ A @ bot_bot_set_real_b ) ).

% emptyE
thf(fact_383_emptyE,axiom,
    ! [A: real > a] :
      ~ ( member_real_a @ A @ bot_bot_set_real_a ) ).

% emptyE
thf(fact_384_emptyE,axiom,
    ! [A: c > b] :
      ~ ( member_c_b @ A @ bot_bot_set_c_b ) ).

% emptyE
thf(fact_385_emptyE,axiom,
    ! [A: a > b] :
      ~ ( member_a_b @ A @ bot_bot_set_a_b ) ).

% emptyE
thf(fact_386_emptyE,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ~ ( member7908768830364227535nnreal @ A @ bot_bo4854962954004695426nnreal ) ).

% emptyE
thf(fact_387_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_388_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_389_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_390_equals0D,axiom,
    ! [A3: set_real_c,A: real > c] :
      ( ( A3 = bot_bot_set_real_c )
     => ~ ( member_real_c @ A @ A3 ) ) ).

% equals0D
thf(fact_391_equals0D,axiom,
    ! [A3: set_real_b,A: real > b] :
      ( ( A3 = bot_bot_set_real_b )
     => ~ ( member_real_b @ A @ A3 ) ) ).

% equals0D
thf(fact_392_equals0D,axiom,
    ! [A3: set_real_a,A: real > a] :
      ( ( A3 = bot_bot_set_real_a )
     => ~ ( member_real_a @ A @ A3 ) ) ).

% equals0D
thf(fact_393_equals0D,axiom,
    ! [A3: set_c_b,A: c > b] :
      ( ( A3 = bot_bot_set_c_b )
     => ~ ( member_c_b @ A @ A3 ) ) ).

% equals0D
thf(fact_394_equals0D,axiom,
    ! [A3: set_a_b,A: a > b] :
      ( ( A3 = bot_bot_set_a_b )
     => ~ ( member_a_b @ A @ A3 ) ) ).

% equals0D
thf(fact_395_equals0D,axiom,
    ! [A3: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] :
      ( ( A3 = bot_bo4854962954004695426nnreal )
     => ~ ( member7908768830364227535nnreal @ A @ A3 ) ) ).

% equals0D
thf(fact_396_equals0D,axiom,
    ! [A3: set_real,A: real] :
      ( ( A3 = bot_bot_set_real )
     => ~ ( member_real @ A @ A3 ) ) ).

% equals0D
thf(fact_397_equals0D,axiom,
    ! [A3: set_o,A: $o] :
      ( ( A3 = bot_bot_set_o )
     => ~ ( member_o @ A @ A3 ) ) ).

% equals0D
thf(fact_398_equals0D,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A3 ) ) ).

% equals0D
thf(fact_399_equals0I,axiom,
    ! [A3: set_real_c] :
      ( ! [Y4: real > c] :
          ~ ( member_real_c @ Y4 @ A3 )
     => ( A3 = bot_bot_set_real_c ) ) ).

% equals0I
thf(fact_400_equals0I,axiom,
    ! [A3: set_real_b] :
      ( ! [Y4: real > b] :
          ~ ( member_real_b @ Y4 @ A3 )
     => ( A3 = bot_bot_set_real_b ) ) ).

% equals0I
thf(fact_401_equals0I,axiom,
    ! [A3: set_real_a] :
      ( ! [Y4: real > a] :
          ~ ( member_real_a @ Y4 @ A3 )
     => ( A3 = bot_bot_set_real_a ) ) ).

% equals0I
thf(fact_402_equals0I,axiom,
    ! [A3: set_c_b] :
      ( ! [Y4: c > b] :
          ~ ( member_c_b @ Y4 @ A3 )
     => ( A3 = bot_bot_set_c_b ) ) ).

% equals0I
thf(fact_403_equals0I,axiom,
    ! [A3: set_a_b] :
      ( ! [Y4: a > b] :
          ~ ( member_a_b @ Y4 @ A3 )
     => ( A3 = bot_bot_set_a_b ) ) ).

% equals0I
thf(fact_404_equals0I,axiom,
    ! [A3: set_Ex3793607809372303086nnreal] :
      ( ! [Y4: extend8495563244428889912nnreal] :
          ~ ( member7908768830364227535nnreal @ Y4 @ A3 )
     => ( A3 = bot_bo4854962954004695426nnreal ) ) ).

% equals0I
thf(fact_405_equals0I,axiom,
    ! [A3: set_real] :
      ( ! [Y4: real] :
          ~ ( member_real @ Y4 @ A3 )
     => ( A3 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_406_equals0I,axiom,
    ! [A3: set_o] :
      ( ! [Y4: $o] :
          ~ ( member_o @ Y4 @ A3 )
     => ( A3 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_407_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A3 )
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_408_ex__in__conv,axiom,
    ! [A3: set_real_c] :
      ( ( ? [X: real > c] : ( member_real_c @ X @ A3 ) )
      = ( A3 != bot_bot_set_real_c ) ) ).

% ex_in_conv
thf(fact_409_ex__in__conv,axiom,
    ! [A3: set_real_b] :
      ( ( ? [X: real > b] : ( member_real_b @ X @ A3 ) )
      = ( A3 != bot_bot_set_real_b ) ) ).

% ex_in_conv
thf(fact_410_ex__in__conv,axiom,
    ! [A3: set_real_a] :
      ( ( ? [X: real > a] : ( member_real_a @ X @ A3 ) )
      = ( A3 != bot_bot_set_real_a ) ) ).

% ex_in_conv
thf(fact_411_ex__in__conv,axiom,
    ! [A3: set_c_b] :
      ( ( ? [X: c > b] : ( member_c_b @ X @ A3 ) )
      = ( A3 != bot_bot_set_c_b ) ) ).

% ex_in_conv
thf(fact_412_ex__in__conv,axiom,
    ! [A3: set_a_b] :
      ( ( ? [X: a > b] : ( member_a_b @ X @ A3 ) )
      = ( A3 != bot_bot_set_a_b ) ) ).

% ex_in_conv
thf(fact_413_ex__in__conv,axiom,
    ! [A3: set_Ex3793607809372303086nnreal] :
      ( ( ? [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ A3 ) )
      = ( A3 != bot_bo4854962954004695426nnreal ) ) ).

% ex_in_conv
thf(fact_414_ex__in__conv,axiom,
    ! [A3: set_real] :
      ( ( ? [X: real] : ( member_real @ X @ A3 ) )
      = ( A3 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_415_ex__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ? [X: $o] : ( member_o @ X @ A3 ) )
      = ( A3 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_416_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A3 ) )
      = ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_417_UNIV__def,axiom,
    ( top_to7994903218803871134nnreal
    = ( collec6648975593938027277nnreal
      @ ^ [X: extend8495563244428889912nnreal] : $true ) ) ).

% UNIV_def
thf(fact_418_UNIV__def,axiom,
    ( top_top_set_complex
    = ( collect_complex
      @ ^ [X: complex] : $true ) ) ).

% UNIV_def
thf(fact_419_UNIV__def,axiom,
    ( top_top_set_real
    = ( collect_real
      @ ^ [X: real] : $true ) ) ).

% UNIV_def
thf(fact_420_UNIV__def,axiom,
    ( top_top_set_o
    = ( collect_o
      @ ^ [X: $o] : $true ) ) ).

% UNIV_def
thf(fact_421_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $true ) ) ).

% UNIV_def
thf(fact_422_Set_Oempty__def,axiom,
    ( bot_bo4854962954004695426nnreal
    = ( collec6648975593938027277nnreal
      @ ^ [X: extend8495563244428889912nnreal] : $false ) ) ).

% Set.empty_def
thf(fact_423_Set_Oempty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X: real] : $false ) ) ).

% Set.empty_def
thf(fact_424_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X: $o] : $false ) ) ).

% Set.empty_def
thf(fact_425_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_426_empty__not__UNIV,axiom,
    bot_bo4854962954004695426nnreal != top_to7994903218803871134nnreal ).

% empty_not_UNIV
thf(fact_427_empty__not__UNIV,axiom,
    bot_bot_set_complex != top_top_set_complex ).

% empty_not_UNIV
thf(fact_428_empty__not__UNIV,axiom,
    bot_bot_set_real != top_top_set_real ).

% empty_not_UNIV
thf(fact_429_empty__not__UNIV,axiom,
    bot_bot_set_o != top_top_set_o ).

% empty_not_UNIV
thf(fact_430_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_431_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_432_eqb__Mx,axiom,
    ( ( qbs_Mx_b @ empty_quasi_borel_b )
    = bot_bot_set_real_b ) ).

% eqb_Mx
thf(fact_433_eqb__Mx,axiom,
    ( ( qbs_Mx_a @ empty_quasi_borel_a )
    = bot_bot_set_real_a ) ).

% eqb_Mx
thf(fact_434_eqb__Mx,axiom,
    ( ( qbs_Mx_c @ empty_quasi_borel_c )
    = bot_bot_set_real_c ) ).

% eqb_Mx
thf(fact_435_Set_Ois__empty__def,axiom,
    ( is_emp182806100662350310nnreal
    = ( ^ [A5: set_Ex3793607809372303086nnreal] : ( A5 = bot_bo4854962954004695426nnreal ) ) ) ).

% Set.is_empty_def
thf(fact_436_Set_Ois__empty__def,axiom,
    ( is_empty_real
    = ( ^ [A5: set_real] : ( A5 = bot_bot_set_real ) ) ) ).

% Set.is_empty_def
thf(fact_437_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A5: set_o] : ( A5 = bot_bot_set_o ) ) ) ).

% Set.is_empty_def
thf(fact_438_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A5: set_nat] : ( A5 = bot_bot_set_nat ) ) ) ).

% Set.is_empty_def
thf(fact_439_copair__qbs__Mx__def,axiom,
    ( binary5499446002719658488Mx_b_b
    = ( ^ [X7: quasi_borel_b,Y5: quasi_borel_b] :
          ( collec5812062441570659333um_b_b
          @ ^ [G: real > sum_sum_b_b] :
            ? [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
              & ( ( X = bot_bot_set_real )
               => ? [Y2: real > b] :
                    ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inl_b_b @ ( Y2 @ R ) ) ) ) ) )
              & ( ( X = top_top_set_real )
               => ? [Y2: real > b] :
                    ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ Y5 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inr_b_b @ ( Y2 @ R ) ) ) ) ) )
              & ( ( ( X != bot_bot_set_real )
                  & ( X != top_top_set_real ) )
               => ? [Y2: real > b] :
                    ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                    & ? [Z2: real > b] :
                        ( ( member_real_b @ Z2 @ ( qbs_Mx_b @ Y5 ) )
                        & ( G
                          = ( ^ [R: real] : ( if_Sum_sum_b_b @ ( member_real @ R @ X ) @ ( sum_Inl_b_b @ ( Y2 @ R ) ) @ ( sum_Inr_b_b @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx_def
thf(fact_440_copair__qbs__Mx__def,axiom,
    ( binary5499446002719658487Mx_b_a
    = ( ^ [X7: quasi_borel_b,Y5: quasi_borel_a] :
          ( collec5741028401524313348um_b_a
          @ ^ [G: real > sum_sum_b_a] :
            ? [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
              & ( ( X = bot_bot_set_real )
               => ? [Y2: real > b] :
                    ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inl_b_a @ ( Y2 @ R ) ) ) ) ) )
              & ( ( X = top_top_set_real )
               => ? [Y2: real > a] :
                    ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ Y5 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inr_a_b @ ( Y2 @ R ) ) ) ) ) )
              & ( ( ( X != bot_bot_set_real )
                  & ( X != top_top_set_real ) )
               => ? [Y2: real > b] :
                    ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                    & ? [Z2: real > a] :
                        ( ( member_real_a @ Z2 @ ( qbs_Mx_a @ Y5 ) )
                        & ( G
                          = ( ^ [R: real] : ( if_Sum_sum_b_a @ ( member_real @ R @ X ) @ ( sum_Inl_b_a @ ( Y2 @ R ) ) @ ( sum_Inr_a_b @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx_def
thf(fact_441_copair__qbs__Mx__def,axiom,
    ( binary5499446002719658489Mx_b_c
    = ( ^ [X7: quasi_borel_b,Y5: quasi_borel_c] :
          ( collec5883096481617005318um_b_c
          @ ^ [G: real > sum_sum_b_c] :
            ? [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
              & ( ( X = bot_bot_set_real )
               => ? [Y2: real > b] :
                    ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inl_b_c @ ( Y2 @ R ) ) ) ) ) )
              & ( ( X = top_top_set_real )
               => ? [Y2: real > c] :
                    ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ Y5 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inr_c_b @ ( Y2 @ R ) ) ) ) ) )
              & ( ( ( X != bot_bot_set_real )
                  & ( X != top_top_set_real ) )
               => ? [Y2: real > b] :
                    ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                    & ? [Z2: real > c] :
                        ( ( member_real_c @ Z2 @ ( qbs_Mx_c @ Y5 ) )
                        & ( G
                          = ( ^ [R: real] : ( if_Sum_sum_b_c @ ( member_real @ R @ X ) @ ( sum_Inl_b_c @ ( Y2 @ R ) ) @ ( sum_Inr_c_b @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx_def
thf(fact_442_copair__qbs__Mx__def,axiom,
    ( binary8286901584692334521Mx_a_b
    = ( ^ [X7: quasi_borel_a,Y5: quasi_borel_b] :
          ( collec6447711442175646022um_a_b
          @ ^ [G: real > sum_sum_a_b] :
            ? [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
              & ( ( X = bot_bot_set_real )
               => ? [Y2: real > a] :
                    ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ X7 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inl_a_b @ ( Y2 @ R ) ) ) ) ) )
              & ( ( X = top_top_set_real )
               => ? [Y2: real > b] :
                    ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ Y5 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inr_b_a @ ( Y2 @ R ) ) ) ) ) )
              & ( ( ( X != bot_bot_set_real )
                  & ( X != top_top_set_real ) )
               => ? [Y2: real > a] :
                    ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ X7 ) )
                    & ? [Z2: real > b] :
                        ( ( member_real_b @ Z2 @ ( qbs_Mx_b @ Y5 ) )
                        & ( G
                          = ( ^ [R: real] : ( if_Sum_sum_a_b @ ( member_real @ R @ X ) @ ( sum_Inl_a_b @ ( Y2 @ R ) ) @ ( sum_Inr_b_a @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx_def
thf(fact_443_copair__qbs__Mx__def,axiom,
    ( binary8286901584692334520Mx_a_a
    = ( ^ [X7: quasi_borel_a,Y5: quasi_borel_a] :
          ( collec6376677402129300037um_a_a
          @ ^ [G: real > sum_sum_a_a] :
            ? [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
              & ( ( X = bot_bot_set_real )
               => ? [Y2: real > a] :
                    ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ X7 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inl_a_a @ ( Y2 @ R ) ) ) ) ) )
              & ( ( X = top_top_set_real )
               => ? [Y2: real > a] :
                    ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ Y5 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inr_a_a @ ( Y2 @ R ) ) ) ) ) )
              & ( ( ( X != bot_bot_set_real )
                  & ( X != top_top_set_real ) )
               => ? [Y2: real > a] :
                    ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ X7 ) )
                    & ? [Z2: real > a] :
                        ( ( member_real_a @ Z2 @ ( qbs_Mx_a @ Y5 ) )
                        & ( G
                          = ( ^ [R: real] : ( if_Sum_sum_a_a @ ( member_real @ R @ X ) @ ( sum_Inl_a_a @ ( Y2 @ R ) ) @ ( sum_Inr_a_a @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx_def
thf(fact_444_copair__qbs__Mx__def,axiom,
    ( binary2711990420746982455Mx_c_b
    = ( ^ [X7: quasi_borel_c,Y5: quasi_borel_b] :
          ( collec5176413440965672644um_c_b
          @ ^ [G: real > sum_sum_c_b] :
            ? [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
              & ( ( X = bot_bot_set_real )
               => ? [Y2: real > c] :
                    ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ X7 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inl_c_b @ ( Y2 @ R ) ) ) ) ) )
              & ( ( X = top_top_set_real )
               => ? [Y2: real > b] :
                    ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ Y5 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inr_b_c @ ( Y2 @ R ) ) ) ) ) )
              & ( ( ( X != bot_bot_set_real )
                  & ( X != top_top_set_real ) )
               => ? [Y2: real > c] :
                    ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ X7 ) )
                    & ? [Z2: real > b] :
                        ( ( member_real_b @ Z2 @ ( qbs_Mx_b @ Y5 ) )
                        & ( G
                          = ( ^ [R: real] : ( if_Sum_sum_c_b @ ( member_real @ R @ X ) @ ( sum_Inl_c_b @ ( Y2 @ R ) ) @ ( sum_Inr_b_c @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx_def
thf(fact_445_copair__qbs__Mx__def,axiom,
    ( binary2711990420746982454Mx_c_a
    = ( ^ [X7: quasi_borel_c,Y5: quasi_borel_a] :
          ( collec5105379400919326659um_c_a
          @ ^ [G: real > sum_sum_c_a] :
            ? [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
              & ( ( X = bot_bot_set_real )
               => ? [Y2: real > c] :
                    ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ X7 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inl_c_a @ ( Y2 @ R ) ) ) ) ) )
              & ( ( X = top_top_set_real )
               => ? [Y2: real > a] :
                    ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ Y5 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inr_a_c @ ( Y2 @ R ) ) ) ) ) )
              & ( ( ( X != bot_bot_set_real )
                  & ( X != top_top_set_real ) )
               => ? [Y2: real > c] :
                    ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ X7 ) )
                    & ? [Z2: real > a] :
                        ( ( member_real_a @ Z2 @ ( qbs_Mx_a @ Y5 ) )
                        & ( G
                          = ( ^ [R: real] : ( if_Sum_sum_c_a @ ( member_real @ R @ X ) @ ( sum_Inl_c_a @ ( Y2 @ R ) ) @ ( sum_Inr_a_c @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx_def
thf(fact_446_copair__qbs__Mx__def,axiom,
    ( binary2711990420746982456Mx_c_c
    = ( ^ [X7: quasi_borel_c,Y5: quasi_borel_c] :
          ( collec5247447481012018629um_c_c
          @ ^ [G: real > sum_sum_c_c] :
            ? [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
              & ( ( X = bot_bot_set_real )
               => ? [Y2: real > c] :
                    ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ X7 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inl_c_c @ ( Y2 @ R ) ) ) ) ) )
              & ( ( X = top_top_set_real )
               => ? [Y2: real > c] :
                    ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ Y5 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inr_c_c @ ( Y2 @ R ) ) ) ) ) )
              & ( ( ( X != bot_bot_set_real )
                  & ( X != top_top_set_real ) )
               => ? [Y2: real > c] :
                    ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ X7 ) )
                    & ? [Z2: real > c] :
                        ( ( member_real_c @ Z2 @ ( qbs_Mx_c @ Y5 ) )
                        & ( G
                          = ( ^ [R: real] : ( if_Sum_sum_c_c @ ( member_real @ R @ X ) @ ( sum_Inl_c_c @ ( Y2 @ R ) ) @ ( sum_Inr_c_c @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx_def
thf(fact_447_copair__qbs__Mx__def,axiom,
    ( binary8286901584692334522Mx_a_c
    = ( ^ [X7: quasi_borel_a,Y5: quasi_borel_c] :
          ( collec6518745482221992007um_a_c
          @ ^ [G: real > sum_sum_a_c] :
            ? [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
              & ( ( X = bot_bot_set_real )
               => ? [Y2: real > a] :
                    ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ X7 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inl_a_c @ ( Y2 @ R ) ) ) ) ) )
              & ( ( X = top_top_set_real )
               => ? [Y2: real > c] :
                    ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ Y5 ) )
                    & ( G
                      = ( ^ [R: real] : ( sum_Inr_c_a @ ( Y2 @ R ) ) ) ) ) )
              & ( ( ( X != bot_bot_set_real )
                  & ( X != top_top_set_real ) )
               => ? [Y2: real > a] :
                    ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ X7 ) )
                    & ? [Z2: real > c] :
                        ( ( member_real_c @ Z2 @ ( qbs_Mx_c @ Y5 ) )
                        & ( G
                          = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ X ) @ ( sum_Inl_a_c @ ( Y2 @ R ) ) @ ( sum_Inr_c_a @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx_def
thf(fact_448_powr__zero__eq__one,axiom,
    ! [X3: real] :
      ( ( ( X3 = zero_zero_real )
       => ( ( powr_real @ X3 @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X3 != zero_zero_real )
       => ( ( powr_real @ X3 @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_449_sum__set__simps_I3_J,axiom,
    ! [X3: a] :
      ( ( basic_setr_a_c @ ( sum_Inl_a_c @ X3 ) )
      = bot_bot_set_c ) ).

% sum_set_simps(3)
thf(fact_450_eucl__ivals_I1_J,axiom,
    ! [A: real] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X: real] : ( topolo2105956845596822908s_real @ X @ A ) )
      @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% eucl_ivals(1)
thf(fact_451_eucl__ivals_I2_J,axiom,
    ! [A: real] : ( member_set_real @ ( collect_real @ ( topolo2105956845596822908s_real @ A ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% eucl_ivals(2)
thf(fact_452_sum__set__simps_I2_J,axiom,
    ! [X3: c] :
      ( ( basic_setl_a_c @ ( sum_Inr_c_a @ X3 ) )
      = bot_bot_set_a ) ).

% sum_set_simps(2)
thf(fact_453_bex__empty,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ~ ? [X8: extend8495563244428889912nnreal] :
          ( ( member7908768830364227535nnreal @ X8 @ bot_bo4854962954004695426nnreal )
          & ( P @ X8 ) ) ).

% bex_empty
thf(fact_454_bex__empty,axiom,
    ! [P: real > $o] :
      ~ ? [X8: real] :
          ( ( member_real @ X8 @ bot_bot_set_real )
          & ( P @ X8 ) ) ).

% bex_empty
thf(fact_455_bex__empty,axiom,
    ! [P: $o > $o] :
      ~ ? [X8: $o] :
          ( ( member_o @ X8 @ bot_bot_set_o )
          & ( P @ X8 ) ) ).

% bex_empty
thf(fact_456_bex__empty,axiom,
    ! [P: nat > $o] :
      ~ ? [X8: nat] :
          ( ( member_nat @ X8 @ bot_bot_set_nat )
          & ( P @ X8 ) ) ).

% bex_empty
thf(fact_457_powr__eq__0__iff,axiom,
    ! [W: real,Z3: real] :
      ( ( ( powr_real @ W @ Z3 )
        = zero_zero_real )
      = ( W = zero_zero_real ) ) ).

% powr_eq_0_iff
thf(fact_458_powr__0,axiom,
    ! [Z3: real] :
      ( ( powr_real @ zero_zero_real @ Z3 )
      = zero_zero_real ) ).

% powr_0
thf(fact_459_powr__one__eq__one,axiom,
    ! [A: real] :
      ( ( powr_real @ one_one_real @ A )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_460_Bex__def,axiom,
    ( bex_real_c
    = ( ^ [A5: set_real_c,P3: ( real > c ) > $o] :
        ? [X: real > c] :
          ( ( member_real_c @ X @ A5 )
          & ( P3 @ X ) ) ) ) ).

% Bex_def
thf(fact_461_Bex__def,axiom,
    ( bex_real_b
    = ( ^ [A5: set_real_b,P3: ( real > b ) > $o] :
        ? [X: real > b] :
          ( ( member_real_b @ X @ A5 )
          & ( P3 @ X ) ) ) ) ).

% Bex_def
thf(fact_462_Bex__def,axiom,
    ( bex_real_a
    = ( ^ [A5: set_real_a,P3: ( real > a ) > $o] :
        ? [X: real > a] :
          ( ( member_real_a @ X @ A5 )
          & ( P3 @ X ) ) ) ) ).

% Bex_def
thf(fact_463_Bex__def,axiom,
    ( bex_c_b
    = ( ^ [A5: set_c_b,P3: ( c > b ) > $o] :
        ? [X: c > b] :
          ( ( member_c_b @ X @ A5 )
          & ( P3 @ X ) ) ) ) ).

% Bex_def
thf(fact_464_Bex__def,axiom,
    ( bex_a_b
    = ( ^ [A5: set_a_b,P3: ( a > b ) > $o] :
        ? [X: a > b] :
          ( ( member_a_b @ X @ A5 )
          & ( P3 @ X ) ) ) ) ).

% Bex_def
thf(fact_465_top__set__def,axiom,
    ( top_to7994903218803871134nnreal
    = ( collec6648975593938027277nnreal @ top_to5118619752887738471real_o ) ) ).

% top_set_def
thf(fact_466_top__set__def,axiom,
    ( top_top_set_complex
    = ( collect_complex @ top_top_complex_o ) ) ).

% top_set_def
thf(fact_467_top__set__def,axiom,
    ( top_top_set_real
    = ( collect_real @ top_top_real_o ) ) ).

% top_set_def
thf(fact_468_top__set__def,axiom,
    ( top_top_set_o
    = ( collect_o @ top_top_o_o ) ) ).

% top_set_def
thf(fact_469_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_470_setr_Ocases,axiom,
    ! [A: c,S: sum_sum_a_c] :
      ( ( member_c @ A @ ( basic_setr_a_c @ S ) )
     => ( S
        = ( sum_Inr_c_a @ A ) ) ) ).

% setr.cases
thf(fact_471_setr_Osimps,axiom,
    ! [A: c,S: sum_sum_a_c] :
      ( ( member_c @ A @ ( basic_setr_a_c @ S ) )
      = ( ? [X: c] :
            ( ( A = X )
            & ( S
              = ( sum_Inr_c_a @ X ) ) ) ) ) ).

% setr.simps
thf(fact_472_setr_Ointros,axiom,
    ! [S: sum_sum_a_c,X3: c] :
      ( ( S
        = ( sum_Inr_c_a @ X3 ) )
     => ( member_c @ X3 @ ( basic_setr_a_c @ S ) ) ) ).

% setr.intros
thf(fact_473_setl_Ocases,axiom,
    ! [A: a,S: sum_sum_a_c] :
      ( ( member_a @ A @ ( basic_setl_a_c @ S ) )
     => ( S
        = ( sum_Inl_a_c @ A ) ) ) ).

% setl.cases
thf(fact_474_setl_Osimps,axiom,
    ! [A: a,S: sum_sum_a_c] :
      ( ( member_a @ A @ ( basic_setl_a_c @ S ) )
      = ( ? [X: a] :
            ( ( A = X )
            & ( S
              = ( sum_Inl_a_c @ X ) ) ) ) ) ).

% setl.simps
thf(fact_475_setl_Ointros,axiom,
    ! [S: sum_sum_a_c,X3: a] :
      ( ( S
        = ( sum_Inl_a_c @ X3 ) )
     => ( member_a @ X3 @ ( basic_setl_a_c @ S ) ) ) ).

% setl.intros
thf(fact_476_sets__Ball,axiom,
    ! [I2: set_real_c,A3: ( real > c ) > set_real,M: ( real > c ) > sigma_measure_real,I3: real > c] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ I2 )
         => ( member_set_real @ ( A3 @ X5 ) @ ( sigma_sets_real @ ( M @ X5 ) ) ) )
     => ( ( member_real_c @ I3 @ I2 )
       => ( member_set_real @ ( A3 @ I3 ) @ ( sigma_sets_real @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_477_sets__Ball,axiom,
    ! [I2: set_real_b,A3: ( real > b ) > set_real,M: ( real > b ) > sigma_measure_real,I3: real > b] :
      ( ! [X5: real > b] :
          ( ( member_real_b @ X5 @ I2 )
         => ( member_set_real @ ( A3 @ X5 ) @ ( sigma_sets_real @ ( M @ X5 ) ) ) )
     => ( ( member_real_b @ I3 @ I2 )
       => ( member_set_real @ ( A3 @ I3 ) @ ( sigma_sets_real @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_478_sets__Ball,axiom,
    ! [I2: set_real_a,A3: ( real > a ) > set_real,M: ( real > a ) > sigma_measure_real,I3: real > a] :
      ( ! [X5: real > a] :
          ( ( member_real_a @ X5 @ I2 )
         => ( member_set_real @ ( A3 @ X5 ) @ ( sigma_sets_real @ ( M @ X5 ) ) ) )
     => ( ( member_real_a @ I3 @ I2 )
       => ( member_set_real @ ( A3 @ I3 ) @ ( sigma_sets_real @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_479_sets__Ball,axiom,
    ! [I2: set_c_b,A3: ( c > b ) > set_real,M: ( c > b ) > sigma_measure_real,I3: c > b] :
      ( ! [X5: c > b] :
          ( ( member_c_b @ X5 @ I2 )
         => ( member_set_real @ ( A3 @ X5 ) @ ( sigma_sets_real @ ( M @ X5 ) ) ) )
     => ( ( member_c_b @ I3 @ I2 )
       => ( member_set_real @ ( A3 @ I3 ) @ ( sigma_sets_real @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_480_sets__Ball,axiom,
    ! [I2: set_a_b,A3: ( a > b ) > set_real,M: ( a > b ) > sigma_measure_real,I3: a > b] :
      ( ! [X5: a > b] :
          ( ( member_a_b @ X5 @ I2 )
         => ( member_set_real @ ( A3 @ X5 ) @ ( sigma_sets_real @ ( M @ X5 ) ) ) )
     => ( ( member_a_b @ I3 @ I2 )
       => ( member_set_real @ ( A3 @ I3 ) @ ( sigma_sets_real @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_481_sets__Ball,axiom,
    ! [I2: set_real_c,A3: ( real > c ) > set_nat,M: ( real > c ) > sigma_measure_nat,I3: real > c] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ I2 )
         => ( member_set_nat @ ( A3 @ X5 ) @ ( sigma_sets_nat @ ( M @ X5 ) ) ) )
     => ( ( member_real_c @ I3 @ I2 )
       => ( member_set_nat @ ( A3 @ I3 ) @ ( sigma_sets_nat @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_482_sets__Ball,axiom,
    ! [I2: set_real_b,A3: ( real > b ) > set_nat,M: ( real > b ) > sigma_measure_nat,I3: real > b] :
      ( ! [X5: real > b] :
          ( ( member_real_b @ X5 @ I2 )
         => ( member_set_nat @ ( A3 @ X5 ) @ ( sigma_sets_nat @ ( M @ X5 ) ) ) )
     => ( ( member_real_b @ I3 @ I2 )
       => ( member_set_nat @ ( A3 @ I3 ) @ ( sigma_sets_nat @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_483_sets__Ball,axiom,
    ! [I2: set_real_a,A3: ( real > a ) > set_nat,M: ( real > a ) > sigma_measure_nat,I3: real > a] :
      ( ! [X5: real > a] :
          ( ( member_real_a @ X5 @ I2 )
         => ( member_set_nat @ ( A3 @ X5 ) @ ( sigma_sets_nat @ ( M @ X5 ) ) ) )
     => ( ( member_real_a @ I3 @ I2 )
       => ( member_set_nat @ ( A3 @ I3 ) @ ( sigma_sets_nat @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_484_sets__Ball,axiom,
    ! [I2: set_c_b,A3: ( c > b ) > set_nat,M: ( c > b ) > sigma_measure_nat,I3: c > b] :
      ( ! [X5: c > b] :
          ( ( member_c_b @ X5 @ I2 )
         => ( member_set_nat @ ( A3 @ X5 ) @ ( sigma_sets_nat @ ( M @ X5 ) ) ) )
     => ( ( member_c_b @ I3 @ I2 )
       => ( member_set_nat @ ( A3 @ I3 ) @ ( sigma_sets_nat @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_485_sets__Ball,axiom,
    ! [I2: set_a_b,A3: ( a > b ) > set_nat,M: ( a > b ) > sigma_measure_nat,I3: a > b] :
      ( ! [X5: a > b] :
          ( ( member_a_b @ X5 @ I2 )
         => ( member_set_nat @ ( A3 @ X5 ) @ ( sigma_sets_nat @ ( M @ X5 ) ) ) )
     => ( ( member_a_b @ I3 @ I2 )
       => ( member_set_nat @ ( A3 @ I3 ) @ ( sigma_sets_nat @ ( M @ I3 ) ) ) ) ) ).

% sets_Ball
thf(fact_486_map__qbs__Mx,axiom,
    ! [F2: real > real,X2: quasi_borel_real] :
      ( ( qbs_Mx_real @ ( map_qbs_real_real @ F2 @ X2 ) )
      = ( collect_real_real
        @ ^ [Beta: real > real] :
          ? [X: real > real] :
            ( ( member_real_real @ X @ ( qbs_Mx_real @ X2 ) )
            & ( Beta
              = ( comp_real_real_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_487_map__qbs__Mx,axiom,
    ! [F2: b > b,X2: quasi_borel_b] :
      ( ( qbs_Mx_b @ ( map_qbs_b_b @ F2 @ X2 ) )
      = ( collect_real_b
        @ ^ [Beta: real > b] :
          ? [X: real > b] :
            ( ( member_real_b @ X @ ( qbs_Mx_b @ X2 ) )
            & ( Beta
              = ( comp_b_b_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_488_map__qbs__Mx,axiom,
    ! [F2: a > b,X2: quasi_borel_a] :
      ( ( qbs_Mx_b @ ( map_qbs_a_b @ F2 @ X2 ) )
      = ( collect_real_b
        @ ^ [Beta: real > b] :
          ? [X: real > a] :
            ( ( member_real_a @ X @ ( qbs_Mx_a @ X2 ) )
            & ( Beta
              = ( comp_a_b_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_489_map__qbs__Mx,axiom,
    ! [F2: c > b,X2: quasi_borel_c] :
      ( ( qbs_Mx_b @ ( map_qbs_c_b @ F2 @ X2 ) )
      = ( collect_real_b
        @ ^ [Beta: real > b] :
          ? [X: real > c] :
            ( ( member_real_c @ X @ ( qbs_Mx_c @ X2 ) )
            & ( Beta
              = ( comp_c_b_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_490_map__qbs__Mx,axiom,
    ! [F2: b > a,X2: quasi_borel_b] :
      ( ( qbs_Mx_a @ ( map_qbs_b_a @ F2 @ X2 ) )
      = ( collect_real_a
        @ ^ [Beta: real > a] :
          ? [X: real > b] :
            ( ( member_real_b @ X @ ( qbs_Mx_b @ X2 ) )
            & ( Beta
              = ( comp_b_a_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_491_map__qbs__Mx,axiom,
    ! [F2: a > a,X2: quasi_borel_a] :
      ( ( qbs_Mx_a @ ( map_qbs_a_a @ F2 @ X2 ) )
      = ( collect_real_a
        @ ^ [Beta: real > a] :
          ? [X: real > a] :
            ( ( member_real_a @ X @ ( qbs_Mx_a @ X2 ) )
            & ( Beta
              = ( comp_a_a_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_492_map__qbs__Mx,axiom,
    ! [F2: c > a,X2: quasi_borel_c] :
      ( ( qbs_Mx_a @ ( map_qbs_c_a @ F2 @ X2 ) )
      = ( collect_real_a
        @ ^ [Beta: real > a] :
          ? [X: real > c] :
            ( ( member_real_c @ X @ ( qbs_Mx_c @ X2 ) )
            & ( Beta
              = ( comp_c_a_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_493_map__qbs__Mx,axiom,
    ! [F2: b > c,X2: quasi_borel_b] :
      ( ( qbs_Mx_c @ ( map_qbs_b_c @ F2 @ X2 ) )
      = ( collect_real_c
        @ ^ [Beta: real > c] :
          ? [X: real > b] :
            ( ( member_real_b @ X @ ( qbs_Mx_b @ X2 ) )
            & ( Beta
              = ( comp_b_c_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_494_map__qbs__Mx,axiom,
    ! [F2: a > c,X2: quasi_borel_a] :
      ( ( qbs_Mx_c @ ( map_qbs_a_c @ F2 @ X2 ) )
      = ( collect_real_c
        @ ^ [Beta: real > c] :
          ? [X: real > a] :
            ( ( member_real_a @ X @ ( qbs_Mx_a @ X2 ) )
            & ( Beta
              = ( comp_a_c_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_495_map__qbs__Mx,axiom,
    ! [F2: c > c,X2: quasi_borel_c] :
      ( ( qbs_Mx_c @ ( map_qbs_c_c @ F2 @ X2 ) )
      = ( collect_real_c
        @ ^ [Beta: real > c] :
          ? [X: real > c] :
            ( ( member_real_c @ X @ ( qbs_Mx_c @ X2 ) )
            & ( Beta
              = ( comp_c_c_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_Mx
thf(fact_496_copair__qbs__Mx2__def,axiom,
    ( binary3454967616579736122x2_b_b
    = ( ^ [X7: quasi_borel_b,Y5: quasi_borel_b] :
          ( collec5812062441570659333um_b_b
          @ ^ [G: real > sum_sum_b_b] :
              ( ~ ( ( ( qbs_space_b @ X7 )
                    = bot_bot_set_b )
                  & ( ( qbs_space_b @ Y5 )
                    = bot_bot_set_b ) )
              & ( ~ ( ( ( qbs_space_b @ X7 )
                      = bot_bot_set_b )
                    & ( ( qbs_space_b @ Y5 )
                      = bot_bot_set_b ) )
               => ( ( ( ( ( qbs_space_b @ X7 )
                       != bot_bot_set_b )
                      & ( ( qbs_space_b @ Y5 )
                        = bot_bot_set_b ) )
                   => ? [X: real > b] :
                        ( ( member_real_b @ X @ ( qbs_Mx_b @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_Inl_b_b @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_b @ X7 )
                         != bot_bot_set_b )
                        & ( ( qbs_space_b @ Y5 )
                          = bot_bot_set_b ) )
                   => ( ( ( ( ( qbs_space_b @ X7 )
                            = bot_bot_set_b )
                          & ( ( qbs_space_b @ Y5 )
                           != bot_bot_set_b ) )
                       => ? [X: real > b] :
                            ( ( member_real_b @ X @ ( qbs_Mx_b @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_Inr_b_b @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_b @ X7 )
                              = bot_bot_set_b )
                            & ( ( qbs_space_b @ Y5 )
                             != bot_bot_set_b ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > b] :
                                ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                                & ? [Z2: real > b] :
                                    ( ( member_real_b @ Z2 @ ( qbs_Mx_b @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum_sum_b_b @ ( member_real @ R @ X ) @ ( sum_Inl_b_b @ ( Y2 @ R ) ) @ ( sum_Inr_b_b @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_497_copair__qbs__Mx2__def,axiom,
    ( binary3454967616579736121x2_b_a
    = ( ^ [X7: quasi_borel_b,Y5: quasi_borel_a] :
          ( collec5741028401524313348um_b_a
          @ ^ [G: real > sum_sum_b_a] :
              ( ~ ( ( ( qbs_space_b @ X7 )
                    = bot_bot_set_b )
                  & ( ( qbs_space_a @ Y5 )
                    = bot_bot_set_a ) )
              & ( ~ ( ( ( qbs_space_b @ X7 )
                      = bot_bot_set_b )
                    & ( ( qbs_space_a @ Y5 )
                      = bot_bot_set_a ) )
               => ( ( ( ( ( qbs_space_b @ X7 )
                       != bot_bot_set_b )
                      & ( ( qbs_space_a @ Y5 )
                        = bot_bot_set_a ) )
                   => ? [X: real > b] :
                        ( ( member_real_b @ X @ ( qbs_Mx_b @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_Inl_b_a @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_b @ X7 )
                         != bot_bot_set_b )
                        & ( ( qbs_space_a @ Y5 )
                          = bot_bot_set_a ) )
                   => ( ( ( ( ( qbs_space_b @ X7 )
                            = bot_bot_set_b )
                          & ( ( qbs_space_a @ Y5 )
                           != bot_bot_set_a ) )
                       => ? [X: real > a] :
                            ( ( member_real_a @ X @ ( qbs_Mx_a @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_Inr_a_b @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_b @ X7 )
                              = bot_bot_set_b )
                            & ( ( qbs_space_a @ Y5 )
                             != bot_bot_set_a ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > b] :
                                ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                                & ? [Z2: real > a] :
                                    ( ( member_real_a @ Z2 @ ( qbs_Mx_a @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum_sum_b_a @ ( member_real @ R @ X ) @ ( sum_Inl_b_a @ ( Y2 @ R ) ) @ ( sum_Inr_a_b @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_498_copair__qbs__Mx2__def,axiom,
    ( binary3454967616579736123x2_b_c
    = ( ^ [X7: quasi_borel_b,Y5: quasi_borel_c] :
          ( collec5883096481617005318um_b_c
          @ ^ [G: real > sum_sum_b_c] :
              ( ~ ( ( ( qbs_space_b @ X7 )
                    = bot_bot_set_b )
                  & ( ( qbs_space_c @ Y5 )
                    = bot_bot_set_c ) )
              & ( ~ ( ( ( qbs_space_b @ X7 )
                      = bot_bot_set_b )
                    & ( ( qbs_space_c @ Y5 )
                      = bot_bot_set_c ) )
               => ( ( ( ( ( qbs_space_b @ X7 )
                       != bot_bot_set_b )
                      & ( ( qbs_space_c @ Y5 )
                        = bot_bot_set_c ) )
                   => ? [X: real > b] :
                        ( ( member_real_b @ X @ ( qbs_Mx_b @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_Inl_b_c @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_b @ X7 )
                         != bot_bot_set_b )
                        & ( ( qbs_space_c @ Y5 )
                          = bot_bot_set_c ) )
                   => ( ( ( ( ( qbs_space_b @ X7 )
                            = bot_bot_set_b )
                          & ( ( qbs_space_c @ Y5 )
                           != bot_bot_set_c ) )
                       => ? [X: real > c] :
                            ( ( member_real_c @ X @ ( qbs_Mx_c @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_Inr_c_b @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_b @ X7 )
                              = bot_bot_set_b )
                            & ( ( qbs_space_c @ Y5 )
                             != bot_bot_set_c ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > b] :
                                ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                                & ? [Z2: real > c] :
                                    ( ( member_real_c @ Z2 @ ( qbs_Mx_c @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum_sum_b_c @ ( member_real @ R @ X ) @ ( sum_Inl_b_c @ ( Y2 @ R ) ) @ ( sum_Inr_c_b @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_499_copair__qbs__Mx2__def,axiom,
    ( binary6242423198552412155x2_a_b
    = ( ^ [X7: quasi_borel_a,Y5: quasi_borel_b] :
          ( collec6447711442175646022um_a_b
          @ ^ [G: real > sum_sum_a_b] :
              ( ~ ( ( ( qbs_space_a @ X7 )
                    = bot_bot_set_a )
                  & ( ( qbs_space_b @ Y5 )
                    = bot_bot_set_b ) )
              & ( ~ ( ( ( qbs_space_a @ X7 )
                      = bot_bot_set_a )
                    & ( ( qbs_space_b @ Y5 )
                      = bot_bot_set_b ) )
               => ( ( ( ( ( qbs_space_a @ X7 )
                       != bot_bot_set_a )
                      & ( ( qbs_space_b @ Y5 )
                        = bot_bot_set_b ) )
                   => ? [X: real > a] :
                        ( ( member_real_a @ X @ ( qbs_Mx_a @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_Inl_a_b @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_a @ X7 )
                         != bot_bot_set_a )
                        & ( ( qbs_space_b @ Y5 )
                          = bot_bot_set_b ) )
                   => ( ( ( ( ( qbs_space_a @ X7 )
                            = bot_bot_set_a )
                          & ( ( qbs_space_b @ Y5 )
                           != bot_bot_set_b ) )
                       => ? [X: real > b] :
                            ( ( member_real_b @ X @ ( qbs_Mx_b @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_Inr_b_a @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_a @ X7 )
                              = bot_bot_set_a )
                            & ( ( qbs_space_b @ Y5 )
                             != bot_bot_set_b ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > a] :
                                ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ X7 ) )
                                & ? [Z2: real > b] :
                                    ( ( member_real_b @ Z2 @ ( qbs_Mx_b @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum_sum_a_b @ ( member_real @ R @ X ) @ ( sum_Inl_a_b @ ( Y2 @ R ) ) @ ( sum_Inr_b_a @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_500_copair__qbs__Mx2__def,axiom,
    ( binary6242423198552412154x2_a_a
    = ( ^ [X7: quasi_borel_a,Y5: quasi_borel_a] :
          ( collec6376677402129300037um_a_a
          @ ^ [G: real > sum_sum_a_a] :
              ( ~ ( ( ( qbs_space_a @ X7 )
                    = bot_bot_set_a )
                  & ( ( qbs_space_a @ Y5 )
                    = bot_bot_set_a ) )
              & ( ~ ( ( ( qbs_space_a @ X7 )
                      = bot_bot_set_a )
                    & ( ( qbs_space_a @ Y5 )
                      = bot_bot_set_a ) )
               => ( ( ( ( ( qbs_space_a @ X7 )
                       != bot_bot_set_a )
                      & ( ( qbs_space_a @ Y5 )
                        = bot_bot_set_a ) )
                   => ? [X: real > a] :
                        ( ( member_real_a @ X @ ( qbs_Mx_a @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_Inl_a_a @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_a @ X7 )
                         != bot_bot_set_a )
                        & ( ( qbs_space_a @ Y5 )
                          = bot_bot_set_a ) )
                   => ( ( ( ( ( qbs_space_a @ X7 )
                            = bot_bot_set_a )
                          & ( ( qbs_space_a @ Y5 )
                           != bot_bot_set_a ) )
                       => ? [X: real > a] :
                            ( ( member_real_a @ X @ ( qbs_Mx_a @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_Inr_a_a @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_a @ X7 )
                              = bot_bot_set_a )
                            & ( ( qbs_space_a @ Y5 )
                             != bot_bot_set_a ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > a] :
                                ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ X7 ) )
                                & ? [Z2: real > a] :
                                    ( ( member_real_a @ Z2 @ ( qbs_Mx_a @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum_sum_a_a @ ( member_real @ R @ X ) @ ( sum_Inl_a_a @ ( Y2 @ R ) ) @ ( sum_Inr_a_a @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_501_copair__qbs__Mx2__def,axiom,
    ( binary6242423198552412156x2_a_c
    = ( ^ [X7: quasi_borel_a,Y5: quasi_borel_c] :
          ( collec6518745482221992007um_a_c
          @ ^ [G: real > sum_sum_a_c] :
              ( ~ ( ( ( qbs_space_a @ X7 )
                    = bot_bot_set_a )
                  & ( ( qbs_space_c @ Y5 )
                    = bot_bot_set_c ) )
              & ( ~ ( ( ( qbs_space_a @ X7 )
                      = bot_bot_set_a )
                    & ( ( qbs_space_c @ Y5 )
                      = bot_bot_set_c ) )
               => ( ( ( ( ( qbs_space_a @ X7 )
                       != bot_bot_set_a )
                      & ( ( qbs_space_c @ Y5 )
                        = bot_bot_set_c ) )
                   => ? [X: real > a] :
                        ( ( member_real_a @ X @ ( qbs_Mx_a @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_Inl_a_c @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_a @ X7 )
                         != bot_bot_set_a )
                        & ( ( qbs_space_c @ Y5 )
                          = bot_bot_set_c ) )
                   => ( ( ( ( ( qbs_space_a @ X7 )
                            = bot_bot_set_a )
                          & ( ( qbs_space_c @ Y5 )
                           != bot_bot_set_c ) )
                       => ? [X: real > c] :
                            ( ( member_real_c @ X @ ( qbs_Mx_c @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_Inr_c_a @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_a @ X7 )
                              = bot_bot_set_a )
                            & ( ( qbs_space_c @ Y5 )
                             != bot_bot_set_c ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > a] :
                                ( ( member_real_a @ Y2 @ ( qbs_Mx_a @ X7 ) )
                                & ? [Z2: real > c] :
                                    ( ( member_real_c @ Z2 @ ( qbs_Mx_c @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ X ) @ ( sum_Inl_a_c @ ( Y2 @ R ) ) @ ( sum_Inr_c_a @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_502_copair__qbs__Mx2__def,axiom,
    ( binary667512034607060089x2_c_b
    = ( ^ [X7: quasi_borel_c,Y5: quasi_borel_b] :
          ( collec5176413440965672644um_c_b
          @ ^ [G: real > sum_sum_c_b] :
              ( ~ ( ( ( qbs_space_c @ X7 )
                    = bot_bot_set_c )
                  & ( ( qbs_space_b @ Y5 )
                    = bot_bot_set_b ) )
              & ( ~ ( ( ( qbs_space_c @ X7 )
                      = bot_bot_set_c )
                    & ( ( qbs_space_b @ Y5 )
                      = bot_bot_set_b ) )
               => ( ( ( ( ( qbs_space_c @ X7 )
                       != bot_bot_set_c )
                      & ( ( qbs_space_b @ Y5 )
                        = bot_bot_set_b ) )
                   => ? [X: real > c] :
                        ( ( member_real_c @ X @ ( qbs_Mx_c @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_Inl_c_b @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_c @ X7 )
                         != bot_bot_set_c )
                        & ( ( qbs_space_b @ Y5 )
                          = bot_bot_set_b ) )
                   => ( ( ( ( ( qbs_space_c @ X7 )
                            = bot_bot_set_c )
                          & ( ( qbs_space_b @ Y5 )
                           != bot_bot_set_b ) )
                       => ? [X: real > b] :
                            ( ( member_real_b @ X @ ( qbs_Mx_b @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_Inr_b_c @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_c @ X7 )
                              = bot_bot_set_c )
                            & ( ( qbs_space_b @ Y5 )
                             != bot_bot_set_b ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > c] :
                                ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ X7 ) )
                                & ? [Z2: real > b] :
                                    ( ( member_real_b @ Z2 @ ( qbs_Mx_b @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum_sum_c_b @ ( member_real @ R @ X ) @ ( sum_Inl_c_b @ ( Y2 @ R ) ) @ ( sum_Inr_b_c @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_503_copair__qbs__Mx2__def,axiom,
    ( binary667512034607060088x2_c_a
    = ( ^ [X7: quasi_borel_c,Y5: quasi_borel_a] :
          ( collec5105379400919326659um_c_a
          @ ^ [G: real > sum_sum_c_a] :
              ( ~ ( ( ( qbs_space_c @ X7 )
                    = bot_bot_set_c )
                  & ( ( qbs_space_a @ Y5 )
                    = bot_bot_set_a ) )
              & ( ~ ( ( ( qbs_space_c @ X7 )
                      = bot_bot_set_c )
                    & ( ( qbs_space_a @ Y5 )
                      = bot_bot_set_a ) )
               => ( ( ( ( ( qbs_space_c @ X7 )
                       != bot_bot_set_c )
                      & ( ( qbs_space_a @ Y5 )
                        = bot_bot_set_a ) )
                   => ? [X: real > c] :
                        ( ( member_real_c @ X @ ( qbs_Mx_c @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_Inl_c_a @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_c @ X7 )
                         != bot_bot_set_c )
                        & ( ( qbs_space_a @ Y5 )
                          = bot_bot_set_a ) )
                   => ( ( ( ( ( qbs_space_c @ X7 )
                            = bot_bot_set_c )
                          & ( ( qbs_space_a @ Y5 )
                           != bot_bot_set_a ) )
                       => ? [X: real > a] :
                            ( ( member_real_a @ X @ ( qbs_Mx_a @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_Inr_a_c @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_c @ X7 )
                              = bot_bot_set_c )
                            & ( ( qbs_space_a @ Y5 )
                             != bot_bot_set_a ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > c] :
                                ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ X7 ) )
                                & ? [Z2: real > a] :
                                    ( ( member_real_a @ Z2 @ ( qbs_Mx_a @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum_sum_c_a @ ( member_real @ R @ X ) @ ( sum_Inl_c_a @ ( Y2 @ R ) ) @ ( sum_Inr_a_c @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_504_copair__qbs__Mx2__def,axiom,
    ( binary667512034607060090x2_c_c
    = ( ^ [X7: quasi_borel_c,Y5: quasi_borel_c] :
          ( collec5247447481012018629um_c_c
          @ ^ [G: real > sum_sum_c_c] :
              ( ~ ( ( ( qbs_space_c @ X7 )
                    = bot_bot_set_c )
                  & ( ( qbs_space_c @ Y5 )
                    = bot_bot_set_c ) )
              & ( ~ ( ( ( qbs_space_c @ X7 )
                      = bot_bot_set_c )
                    & ( ( qbs_space_c @ Y5 )
                      = bot_bot_set_c ) )
               => ( ( ( ( ( qbs_space_c @ X7 )
                       != bot_bot_set_c )
                      & ( ( qbs_space_c @ Y5 )
                        = bot_bot_set_c ) )
                   => ? [X: real > c] :
                        ( ( member_real_c @ X @ ( qbs_Mx_c @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_Inl_c_c @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_c @ X7 )
                         != bot_bot_set_c )
                        & ( ( qbs_space_c @ Y5 )
                          = bot_bot_set_c ) )
                   => ( ( ( ( ( qbs_space_c @ X7 )
                            = bot_bot_set_c )
                          & ( ( qbs_space_c @ Y5 )
                           != bot_bot_set_c ) )
                       => ? [X: real > c] :
                            ( ( member_real_c @ X @ ( qbs_Mx_c @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_Inr_c_c @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_c @ X7 )
                              = bot_bot_set_c )
                            & ( ( qbs_space_c @ Y5 )
                             != bot_bot_set_c ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > c] :
                                ( ( member_real_c @ Y2 @ ( qbs_Mx_c @ X7 ) )
                                & ? [Z2: real > c] :
                                    ( ( member_real_c @ Z2 @ ( qbs_Mx_c @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum_sum_c_c @ ( member_real @ R @ X ) @ ( sum_Inl_c_c @ ( Y2 @ R ) ) @ ( sum_Inr_c_c @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_505_copair__qbs__Mx2__def,axiom,
    ( binary4989506101540728637nnreal
    = ( ^ [X7: quasi_borel_b,Y5: quasi_9015997321629101608nnreal] :
          ( collec1455287207176083166nnreal
          @ ^ [G: real > sum_su8602216633299776360nnreal] :
              ( ~ ( ( ( qbs_space_b @ X7 )
                    = bot_bot_set_b )
                  & ( ( qbs_sp175953267596557954nnreal @ Y5 )
                    = bot_bo4854962954004695426nnreal ) )
              & ( ~ ( ( ( qbs_space_b @ X7 )
                      = bot_bot_set_b )
                    & ( ( qbs_sp175953267596557954nnreal @ Y5 )
                      = bot_bo4854962954004695426nnreal ) )
               => ( ( ( ( ( qbs_space_b @ X7 )
                       != bot_bot_set_b )
                      & ( ( qbs_sp175953267596557954nnreal @ Y5 )
                        = bot_bo4854962954004695426nnreal ) )
                   => ? [X: real > b] :
                        ( ( member_real_b @ X @ ( qbs_Mx_b @ X7 ) )
                        & ( G
                          = ( ^ [R: real] : ( sum_In7041624391549913417nnreal @ ( X @ R ) ) ) ) ) )
                  & ( ~ ( ( ( qbs_space_b @ X7 )
                         != bot_bot_set_b )
                        & ( ( qbs_sp175953267596557954nnreal @ Y5 )
                          = bot_bo4854962954004695426nnreal ) )
                   => ( ( ( ( ( qbs_space_b @ X7 )
                            = bot_bot_set_b )
                          & ( ( qbs_sp175953267596557954nnreal @ Y5 )
                           != bot_bo4854962954004695426nnreal ) )
                       => ? [X: real > extend8495563244428889912nnreal] :
                            ( ( member2919562650594848410nnreal @ X @ ( qbs_Mx6523938229262583809nnreal @ Y5 ) )
                            & ( G
                              = ( ^ [R: real] : ( sum_In2243461111168403281real_b @ ( X @ R ) ) ) ) ) )
                      & ( ~ ( ( ( qbs_space_b @ X7 )
                              = bot_bot_set_b )
                            & ( ( qbs_sp175953267596557954nnreal @ Y5 )
                             != bot_bo4854962954004695426nnreal ) )
                       => ? [X: set_real] :
                            ( ( member_set_real @ X @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
                            & ? [Y2: real > b] :
                                ( ( member_real_b @ Y2 @ ( qbs_Mx_b @ X7 ) )
                                & ? [Z2: real > extend8495563244428889912nnreal] :
                                    ( ( member2919562650594848410nnreal @ Z2 @ ( qbs_Mx6523938229262583809nnreal @ Y5 ) )
                                    & ( G
                                      = ( ^ [R: real] : ( if_Sum7506096409499832110nnreal @ ( member_real @ R @ X ) @ ( sum_In7041624391549913417nnreal @ ( Y2 @ R ) ) @ ( sum_In2243461111168403281real_b @ ( Z2 @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% copair_qbs_Mx2_def
thf(fact_506_Collect__empty__eq__bot,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ( ( ( collec6648975593938027277nnreal @ P )
        = bot_bo4854962954004695426nnreal )
      = ( P = bot_bo412624608084785539real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_507_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_508_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_509_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_510_bot__empty__eq,axiom,
    ( bot_bot_real_c_o
    = ( ^ [X: real > c] : ( member_real_c @ X @ bot_bot_set_real_c ) ) ) ).

% bot_empty_eq
thf(fact_511_bot__empty__eq,axiom,
    ( bot_bot_real_b_o
    = ( ^ [X: real > b] : ( member_real_b @ X @ bot_bot_set_real_b ) ) ) ).

% bot_empty_eq
thf(fact_512_bot__empty__eq,axiom,
    ( bot_bot_real_a_o
    = ( ^ [X: real > a] : ( member_real_a @ X @ bot_bot_set_real_a ) ) ) ).

% bot_empty_eq
thf(fact_513_bot__empty__eq,axiom,
    ( bot_bot_c_b_o
    = ( ^ [X: c > b] : ( member_c_b @ X @ bot_bot_set_c_b ) ) ) ).

% bot_empty_eq
thf(fact_514_bot__empty__eq,axiom,
    ( bot_bot_a_b_o
    = ( ^ [X: a > b] : ( member_a_b @ X @ bot_bot_set_a_b ) ) ) ).

% bot_empty_eq
thf(fact_515_bot__empty__eq,axiom,
    ( bot_bo412624608084785539real_o
    = ( ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ bot_bo4854962954004695426nnreal ) ) ) ).

% bot_empty_eq
thf(fact_516_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X: real] : ( member_real @ X @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_517_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_518_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_519_map__qbs__closed1,axiom,
    ! [X2: quasi_borel_real,F2: real > real] :
      ( qbs_closed1_real
      @ ( collect_real_real
        @ ^ [Beta: real > real] :
          ? [X: real > real] :
            ( ( member_real_real @ X @ ( qbs_Mx_real @ X2 ) )
            & ( Beta
              = ( comp_real_real_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed1
thf(fact_520_map__qbs__closed1,axiom,
    ! [X2: quasi_borel_a,F2: a > b] :
      ( qbs_closed1_b
      @ ( collect_real_b
        @ ^ [Beta: real > b] :
          ? [X: real > a] :
            ( ( member_real_a @ X @ ( qbs_Mx_a @ X2 ) )
            & ( Beta
              = ( comp_a_b_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed1
thf(fact_521_map__qbs__closed1,axiom,
    ! [X2: quasi_borel_a,F2: a > c] :
      ( qbs_closed1_c
      @ ( collect_real_c
        @ ^ [Beta: real > c] :
          ? [X: real > a] :
            ( ( member_real_a @ X @ ( qbs_Mx_a @ X2 ) )
            & ( Beta
              = ( comp_a_c_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed1
thf(fact_522_map__qbs__closed1,axiom,
    ! [X2: quasi_borel_a,F2: a > a] :
      ( qbs_closed1_a
      @ ( collect_real_a
        @ ^ [Beta: real > a] :
          ? [X: real > a] :
            ( ( member_real_a @ X @ ( qbs_Mx_a @ X2 ) )
            & ( Beta
              = ( comp_a_a_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed1
thf(fact_523_map__qbs__closed1,axiom,
    ! [X2: quasi_borel_c,F2: c > b] :
      ( qbs_closed1_b
      @ ( collect_real_b
        @ ^ [Beta: real > b] :
          ? [X: real > c] :
            ( ( member_real_c @ X @ ( qbs_Mx_c @ X2 ) )
            & ( Beta
              = ( comp_c_b_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed1
thf(fact_524_map__qbs__closed3,axiom,
    ! [X2: quasi_borel_real,F2: real > real] :
      ( qbs_closed3_real
      @ ( collect_real_real
        @ ^ [Beta: real > real] :
          ? [X: real > real] :
            ( ( member_real_real @ X @ ( qbs_Mx_real @ X2 ) )
            & ( Beta
              = ( comp_real_real_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed3
thf(fact_525_map__qbs__closed3,axiom,
    ! [X2: quasi_borel_a,F2: a > b] :
      ( qbs_closed3_b
      @ ( collect_real_b
        @ ^ [Beta: real > b] :
          ? [X: real > a] :
            ( ( member_real_a @ X @ ( qbs_Mx_a @ X2 ) )
            & ( Beta
              = ( comp_a_b_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed3
thf(fact_526_map__qbs__closed3,axiom,
    ! [X2: quasi_borel_a,F2: a > c] :
      ( qbs_closed3_c
      @ ( collect_real_c
        @ ^ [Beta: real > c] :
          ? [X: real > a] :
            ( ( member_real_a @ X @ ( qbs_Mx_a @ X2 ) )
            & ( Beta
              = ( comp_a_c_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed3
thf(fact_527_map__qbs__closed3,axiom,
    ! [X2: quasi_borel_a,F2: a > a] :
      ( qbs_closed3_a
      @ ( collect_real_a
        @ ^ [Beta: real > a] :
          ? [X: real > a] :
            ( ( member_real_a @ X @ ( qbs_Mx_a @ X2 ) )
            & ( Beta
              = ( comp_a_a_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed3
thf(fact_528_map__qbs__closed3,axiom,
    ! [X2: quasi_borel_c,F2: c > b] :
      ( qbs_closed3_b
      @ ( collect_real_b
        @ ^ [Beta: real > b] :
          ? [X: real > c] :
            ( ( member_real_c @ X @ ( qbs_Mx_c @ X2 ) )
            & ( Beta
              = ( comp_c_b_real @ F2 @ X ) ) ) ) ) ).

% map_qbs_closed3
thf(fact_529_sum__set__simps_I1_J,axiom,
    ! [X3: a] :
      ( ( basic_setl_a_c @ ( sum_Inl_a_c @ X3 ) )
      = ( insert_a @ X3 @ bot_bot_set_a ) ) ).

% sum_set_simps(1)
thf(fact_530_sum__set__simps_I4_J,axiom,
    ! [X3: c] :
      ( ( basic_setr_a_c @ ( sum_Inr_c_a @ X3 ) )
      = ( insert_c @ X3 @ bot_bot_set_c ) ) ).

% sum_set_simps(4)
thf(fact_531_insert__absorb2,axiom,
    ! [X3: real,A3: set_real] :
      ( ( insert_real @ X3 @ ( insert_real @ X3 @ A3 ) )
      = ( insert_real @ X3 @ A3 ) ) ).

% insert_absorb2
thf(fact_532_insert__absorb2,axiom,
    ! [X3: nat,A3: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A3 ) )
      = ( insert_nat @ X3 @ A3 ) ) ).

% insert_absorb2
thf(fact_533_insert__absorb2,axiom,
    ! [X3: $o,A3: set_o] :
      ( ( insert_o @ X3 @ ( insert_o @ X3 @ A3 ) )
      = ( insert_o @ X3 @ A3 ) ) ).

% insert_absorb2
thf(fact_534_insert__absorb2,axiom,
    ! [X3: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ X3 @ ( insert7407984058720857448nnreal @ X3 @ A3 ) )
      = ( insert7407984058720857448nnreal @ X3 @ A3 ) ) ).

% insert_absorb2
thf(fact_535_insert__iff,axiom,
    ! [A: real,B: real,A3: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A3 ) )
      = ( ( A = B )
        | ( member_real @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_536_insert__iff,axiom,
    ! [A: nat,B: nat,A3: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_537_insert__iff,axiom,
    ! [A: $o,B: $o,A3: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A3 ) )
      = ( ( A = B )
        | ( member_o @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_538_insert__iff,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ A3 ) )
      = ( ( A = B )
        | ( member7908768830364227535nnreal @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_539_insert__iff,axiom,
    ! [A: real > c,B: real > c,A3: set_real_c] :
      ( ( member_real_c @ A @ ( insert_real_c @ B @ A3 ) )
      = ( ( A = B )
        | ( member_real_c @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_540_insert__iff,axiom,
    ! [A: real > b,B: real > b,A3: set_real_b] :
      ( ( member_real_b @ A @ ( insert_real_b @ B @ A3 ) )
      = ( ( A = B )
        | ( member_real_b @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_541_insert__iff,axiom,
    ! [A: real > a,B: real > a,A3: set_real_a] :
      ( ( member_real_a @ A @ ( insert_real_a @ B @ A3 ) )
      = ( ( A = B )
        | ( member_real_a @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_542_insert__iff,axiom,
    ! [A: c > b,B: c > b,A3: set_c_b] :
      ( ( member_c_b @ A @ ( insert_c_b @ B @ A3 ) )
      = ( ( A = B )
        | ( member_c_b @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_543_insert__iff,axiom,
    ! [A: a > b,B: a > b,A3: set_a_b] :
      ( ( member_a_b @ A @ ( insert_a_b @ B @ A3 ) )
      = ( ( A = B )
        | ( member_a_b @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_544_insertCI,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( ~ ( member_real @ A @ B4 )
       => ( A = B ) )
     => ( member_real @ A @ ( insert_real @ B @ B4 ) ) ) ).

% insertCI
thf(fact_545_insertCI,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B4 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_546_insertCI,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( ~ ( member_o @ A @ B4 )
       => ( A = B ) )
     => ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).

% insertCI
thf(fact_547_insertCI,axiom,
    ! [A: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( ~ ( member7908768830364227535nnreal @ A @ B4 )
       => ( A = B ) )
     => ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ B4 ) ) ) ).

% insertCI
thf(fact_548_insertCI,axiom,
    ! [A: real > c,B4: set_real_c,B: real > c] :
      ( ( ~ ( member_real_c @ A @ B4 )
       => ( A = B ) )
     => ( member_real_c @ A @ ( insert_real_c @ B @ B4 ) ) ) ).

% insertCI
thf(fact_549_insertCI,axiom,
    ! [A: real > b,B4: set_real_b,B: real > b] :
      ( ( ~ ( member_real_b @ A @ B4 )
       => ( A = B ) )
     => ( member_real_b @ A @ ( insert_real_b @ B @ B4 ) ) ) ).

% insertCI
thf(fact_550_insertCI,axiom,
    ! [A: real > a,B4: set_real_a,B: real > a] :
      ( ( ~ ( member_real_a @ A @ B4 )
       => ( A = B ) )
     => ( member_real_a @ A @ ( insert_real_a @ B @ B4 ) ) ) ).

% insertCI
thf(fact_551_insertCI,axiom,
    ! [A: c > b,B4: set_c_b,B: c > b] :
      ( ( ~ ( member_c_b @ A @ B4 )
       => ( A = B ) )
     => ( member_c_b @ A @ ( insert_c_b @ B @ B4 ) ) ) ).

% insertCI
thf(fact_552_insertCI,axiom,
    ! [A: a > b,B4: set_a_b,B: a > b] :
      ( ( ~ ( member_a_b @ A @ B4 )
       => ( A = B ) )
     => ( member_a_b @ A @ ( insert_a_b @ B @ B4 ) ) ) ).

% insertCI
thf(fact_553_singletonI,axiom,
    ! [A: real > c] : ( member_real_c @ A @ ( insert_real_c @ A @ bot_bot_set_real_c ) ) ).

% singletonI
thf(fact_554_singletonI,axiom,
    ! [A: real > b] : ( member_real_b @ A @ ( insert_real_b @ A @ bot_bot_set_real_b ) ) ).

% singletonI
thf(fact_555_singletonI,axiom,
    ! [A: real > a] : ( member_real_a @ A @ ( insert_real_a @ A @ bot_bot_set_real_a ) ) ).

% singletonI
thf(fact_556_singletonI,axiom,
    ! [A: c > b] : ( member_c_b @ A @ ( insert_c_b @ A @ bot_bot_set_c_b ) ) ).

% singletonI
thf(fact_557_singletonI,axiom,
    ! [A: a > b] : ( member_a_b @ A @ ( insert_a_b @ A @ bot_bot_set_a_b ) ) ).

% singletonI
thf(fact_558_singletonI,axiom,
    ! [A: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) ).

% singletonI
thf(fact_559_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_560_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_561_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_562_singleton__conv2,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( collec6648975593938027277nnreal
        @ ( ^ [Y6: extend8495563244428889912nnreal,Z4: extend8495563244428889912nnreal] : ( Y6 = Z4 )
          @ A ) )
      = ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) ).

% singleton_conv2
thf(fact_563_singleton__conv2,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_564_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_565_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_566_singleton__conv,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( collec6648975593938027277nnreal
        @ ^ [X: extend8495563244428889912nnreal] : ( X = A ) )
      = ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) ).

% singleton_conv
thf(fact_567_singleton__conv,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ^ [X: real] : ( X = A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_568_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X: $o] : ( X = A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_569_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_570_eqb__space,axiom,
    ( ( qbs_sp175953267596557954nnreal @ empty_1788085430566700506nnreal )
    = bot_bo4854962954004695426nnreal ) ).

% eqb_space
thf(fact_571_eqb__space,axiom,
    ( ( qbs_space_real @ empty_1876425439295802446l_real )
    = bot_bot_set_real ) ).

% eqb_space
thf(fact_572_eqb__space,axiom,
    ( ( qbs_space_o @ empty_quasi_borel_o )
    = bot_bot_set_o ) ).

% eqb_space
thf(fact_573_eqb__space,axiom,
    ( ( qbs_space_nat @ empty_8278123436611590770el_nat )
    = bot_bot_set_nat ) ).

% eqb_space
thf(fact_574_mk__disjoint__insert,axiom,
    ! [A: real,A3: set_real] :
      ( ( member_real @ A @ A3 )
     => ? [B5: set_real] :
          ( ( A3
            = ( insert_real @ A @ B5 ) )
          & ~ ( member_real @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_575_mk__disjoint__insert,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ? [B5: set_nat] :
          ( ( A3
            = ( insert_nat @ A @ B5 ) )
          & ~ ( member_nat @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_576_mk__disjoint__insert,axiom,
    ! [A: $o,A3: set_o] :
      ( ( member_o @ A @ A3 )
     => ? [B5: set_o] :
          ( ( A3
            = ( insert_o @ A @ B5 ) )
          & ~ ( member_o @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_577_mk__disjoint__insert,axiom,
    ! [A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ A3 )
     => ? [B5: set_Ex3793607809372303086nnreal] :
          ( ( A3
            = ( insert7407984058720857448nnreal @ A @ B5 ) )
          & ~ ( member7908768830364227535nnreal @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_578_mk__disjoint__insert,axiom,
    ! [A: real > c,A3: set_real_c] :
      ( ( member_real_c @ A @ A3 )
     => ? [B5: set_real_c] :
          ( ( A3
            = ( insert_real_c @ A @ B5 ) )
          & ~ ( member_real_c @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_579_mk__disjoint__insert,axiom,
    ! [A: real > b,A3: set_real_b] :
      ( ( member_real_b @ A @ A3 )
     => ? [B5: set_real_b] :
          ( ( A3
            = ( insert_real_b @ A @ B5 ) )
          & ~ ( member_real_b @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_580_mk__disjoint__insert,axiom,
    ! [A: real > a,A3: set_real_a] :
      ( ( member_real_a @ A @ A3 )
     => ? [B5: set_real_a] :
          ( ( A3
            = ( insert_real_a @ A @ B5 ) )
          & ~ ( member_real_a @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_581_mk__disjoint__insert,axiom,
    ! [A: c > b,A3: set_c_b] :
      ( ( member_c_b @ A @ A3 )
     => ? [B5: set_c_b] :
          ( ( A3
            = ( insert_c_b @ A @ B5 ) )
          & ~ ( member_c_b @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_582_mk__disjoint__insert,axiom,
    ! [A: a > b,A3: set_a_b] :
      ( ( member_a_b @ A @ A3 )
     => ? [B5: set_a_b] :
          ( ( A3
            = ( insert_a_b @ A @ B5 ) )
          & ~ ( member_a_b @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_583_insert__commute,axiom,
    ! [X3: real,Y3: real,A3: set_real] :
      ( ( insert_real @ X3 @ ( insert_real @ Y3 @ A3 ) )
      = ( insert_real @ Y3 @ ( insert_real @ X3 @ A3 ) ) ) ).

% insert_commute
thf(fact_584_insert__commute,axiom,
    ! [X3: nat,Y3: nat,A3: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ Y3 @ A3 ) )
      = ( insert_nat @ Y3 @ ( insert_nat @ X3 @ A3 ) ) ) ).

% insert_commute
thf(fact_585_insert__commute,axiom,
    ! [X3: $o,Y3: $o,A3: set_o] :
      ( ( insert_o @ X3 @ ( insert_o @ Y3 @ A3 ) )
      = ( insert_o @ Y3 @ ( insert_o @ X3 @ A3 ) ) ) ).

% insert_commute
thf(fact_586_insert__commute,axiom,
    ! [X3: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ X3 @ ( insert7407984058720857448nnreal @ Y3 @ A3 ) )
      = ( insert7407984058720857448nnreal @ Y3 @ ( insert7407984058720857448nnreal @ X3 @ A3 ) ) ) ).

% insert_commute
thf(fact_587_insert__eq__iff,axiom,
    ! [A: real,A3: set_real,B: real,B4: set_real] :
      ( ~ ( member_real @ A @ A3 )
     => ( ~ ( member_real @ B @ B4 )
       => ( ( ( insert_real @ A @ A3 )
            = ( insert_real @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_real] :
                  ( ( A3
                    = ( insert_real @ B @ C2 ) )
                  & ~ ( member_real @ B @ C2 )
                  & ( B4
                    = ( insert_real @ A @ C2 ) )
                  & ~ ( member_real @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_588_insert__eq__iff,axiom,
    ! [A: nat,A3: set_nat,B: nat,B4: set_nat] :
      ( ~ ( member_nat @ A @ A3 )
     => ( ~ ( member_nat @ B @ B4 )
       => ( ( ( insert_nat @ A @ A3 )
            = ( insert_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_nat] :
                  ( ( A3
                    = ( insert_nat @ B @ C2 ) )
                  & ~ ( member_nat @ B @ C2 )
                  & ( B4
                    = ( insert_nat @ A @ C2 ) )
                  & ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_589_insert__eq__iff,axiom,
    ! [A: $o,A3: set_o,B: $o,B4: set_o] :
      ( ~ ( member_o @ A @ A3 )
     => ( ~ ( member_o @ B @ B4 )
       => ( ( ( insert_o @ A @ A3 )
            = ( insert_o @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A = ~ B )
             => ? [C2: set_o] :
                  ( ( A3
                    = ( insert_o @ B @ C2 ) )
                  & ~ ( member_o @ B @ C2 )
                  & ( B4
                    = ( insert_o @ A @ C2 ) )
                  & ~ ( member_o @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_590_insert__eq__iff,axiom,
    ! [A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ A @ A3 )
     => ( ~ ( member7908768830364227535nnreal @ B @ B4 )
       => ( ( ( insert7407984058720857448nnreal @ A @ A3 )
            = ( insert7407984058720857448nnreal @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_Ex3793607809372303086nnreal] :
                  ( ( A3
                    = ( insert7407984058720857448nnreal @ B @ C2 ) )
                  & ~ ( member7908768830364227535nnreal @ B @ C2 )
                  & ( B4
                    = ( insert7407984058720857448nnreal @ A @ C2 ) )
                  & ~ ( member7908768830364227535nnreal @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_591_insert__eq__iff,axiom,
    ! [A: real > c,A3: set_real_c,B: real > c,B4: set_real_c] :
      ( ~ ( member_real_c @ A @ A3 )
     => ( ~ ( member_real_c @ B @ B4 )
       => ( ( ( insert_real_c @ A @ A3 )
            = ( insert_real_c @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_real_c] :
                  ( ( A3
                    = ( insert_real_c @ B @ C2 ) )
                  & ~ ( member_real_c @ B @ C2 )
                  & ( B4
                    = ( insert_real_c @ A @ C2 ) )
                  & ~ ( member_real_c @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_592_insert__eq__iff,axiom,
    ! [A: real > b,A3: set_real_b,B: real > b,B4: set_real_b] :
      ( ~ ( member_real_b @ A @ A3 )
     => ( ~ ( member_real_b @ B @ B4 )
       => ( ( ( insert_real_b @ A @ A3 )
            = ( insert_real_b @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_real_b] :
                  ( ( A3
                    = ( insert_real_b @ B @ C2 ) )
                  & ~ ( member_real_b @ B @ C2 )
                  & ( B4
                    = ( insert_real_b @ A @ C2 ) )
                  & ~ ( member_real_b @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_593_insert__eq__iff,axiom,
    ! [A: real > a,A3: set_real_a,B: real > a,B4: set_real_a] :
      ( ~ ( member_real_a @ A @ A3 )
     => ( ~ ( member_real_a @ B @ B4 )
       => ( ( ( insert_real_a @ A @ A3 )
            = ( insert_real_a @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_real_a] :
                  ( ( A3
                    = ( insert_real_a @ B @ C2 ) )
                  & ~ ( member_real_a @ B @ C2 )
                  & ( B4
                    = ( insert_real_a @ A @ C2 ) )
                  & ~ ( member_real_a @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_594_insert__eq__iff,axiom,
    ! [A: c > b,A3: set_c_b,B: c > b,B4: set_c_b] :
      ( ~ ( member_c_b @ A @ A3 )
     => ( ~ ( member_c_b @ B @ B4 )
       => ( ( ( insert_c_b @ A @ A3 )
            = ( insert_c_b @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_c_b] :
                  ( ( A3
                    = ( insert_c_b @ B @ C2 ) )
                  & ~ ( member_c_b @ B @ C2 )
                  & ( B4
                    = ( insert_c_b @ A @ C2 ) )
                  & ~ ( member_c_b @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_595_insert__eq__iff,axiom,
    ! [A: a > b,A3: set_a_b,B: a > b,B4: set_a_b] :
      ( ~ ( member_a_b @ A @ A3 )
     => ( ~ ( member_a_b @ B @ B4 )
       => ( ( ( insert_a_b @ A @ A3 )
            = ( insert_a_b @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A3 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_a_b] :
                  ( ( A3
                    = ( insert_a_b @ B @ C2 ) )
                  & ~ ( member_a_b @ B @ C2 )
                  & ( B4
                    = ( insert_a_b @ A @ C2 ) )
                  & ~ ( member_a_b @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_596_insert__absorb,axiom,
    ! [A: real,A3: set_real] :
      ( ( member_real @ A @ A3 )
     => ( ( insert_real @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_597_insert__absorb,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( insert_nat @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_598_insert__absorb,axiom,
    ! [A: $o,A3: set_o] :
      ( ( member_o @ A @ A3 )
     => ( ( insert_o @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_599_insert__absorb,axiom,
    ! [A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ A3 )
     => ( ( insert7407984058720857448nnreal @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_600_insert__absorb,axiom,
    ! [A: real > c,A3: set_real_c] :
      ( ( member_real_c @ A @ A3 )
     => ( ( insert_real_c @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_601_insert__absorb,axiom,
    ! [A: real > b,A3: set_real_b] :
      ( ( member_real_b @ A @ A3 )
     => ( ( insert_real_b @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_602_insert__absorb,axiom,
    ! [A: real > a,A3: set_real_a] :
      ( ( member_real_a @ A @ A3 )
     => ( ( insert_real_a @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_603_insert__absorb,axiom,
    ! [A: c > b,A3: set_c_b] :
      ( ( member_c_b @ A @ A3 )
     => ( ( insert_c_b @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_604_insert__absorb,axiom,
    ! [A: a > b,A3: set_a_b] :
      ( ( member_a_b @ A @ A3 )
     => ( ( insert_a_b @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_605_insert__ident,axiom,
    ! [X3: real,A3: set_real,B4: set_real] :
      ( ~ ( member_real @ X3 @ A3 )
     => ( ~ ( member_real @ X3 @ B4 )
       => ( ( ( insert_real @ X3 @ A3 )
            = ( insert_real @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_606_insert__ident,axiom,
    ! [X3: nat,A3: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X3 @ A3 )
     => ( ~ ( member_nat @ X3 @ B4 )
       => ( ( ( insert_nat @ X3 @ A3 )
            = ( insert_nat @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_607_insert__ident,axiom,
    ! [X3: $o,A3: set_o,B4: set_o] :
      ( ~ ( member_o @ X3 @ A3 )
     => ( ~ ( member_o @ X3 @ B4 )
       => ( ( ( insert_o @ X3 @ A3 )
            = ( insert_o @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_608_insert__ident,axiom,
    ! [X3: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ X3 @ A3 )
     => ( ~ ( member7908768830364227535nnreal @ X3 @ B4 )
       => ( ( ( insert7407984058720857448nnreal @ X3 @ A3 )
            = ( insert7407984058720857448nnreal @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_609_insert__ident,axiom,
    ! [X3: real > c,A3: set_real_c,B4: set_real_c] :
      ( ~ ( member_real_c @ X3 @ A3 )
     => ( ~ ( member_real_c @ X3 @ B4 )
       => ( ( ( insert_real_c @ X3 @ A3 )
            = ( insert_real_c @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_610_insert__ident,axiom,
    ! [X3: real > b,A3: set_real_b,B4: set_real_b] :
      ( ~ ( member_real_b @ X3 @ A3 )
     => ( ~ ( member_real_b @ X3 @ B4 )
       => ( ( ( insert_real_b @ X3 @ A3 )
            = ( insert_real_b @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_611_insert__ident,axiom,
    ! [X3: real > a,A3: set_real_a,B4: set_real_a] :
      ( ~ ( member_real_a @ X3 @ A3 )
     => ( ~ ( member_real_a @ X3 @ B4 )
       => ( ( ( insert_real_a @ X3 @ A3 )
            = ( insert_real_a @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_612_insert__ident,axiom,
    ! [X3: c > b,A3: set_c_b,B4: set_c_b] :
      ( ~ ( member_c_b @ X3 @ A3 )
     => ( ~ ( member_c_b @ X3 @ B4 )
       => ( ( ( insert_c_b @ X3 @ A3 )
            = ( insert_c_b @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_613_insert__ident,axiom,
    ! [X3: a > b,A3: set_a_b,B4: set_a_b] :
      ( ~ ( member_a_b @ X3 @ A3 )
     => ( ~ ( member_a_b @ X3 @ B4 )
       => ( ( ( insert_a_b @ X3 @ A3 )
            = ( insert_a_b @ X3 @ B4 ) )
          = ( A3 = B4 ) ) ) ) ).

% insert_ident
thf(fact_614_Set_Oset__insert,axiom,
    ! [X3: real,A3: set_real] :
      ( ( member_real @ X3 @ A3 )
     => ~ ! [B5: set_real] :
            ( ( A3
              = ( insert_real @ X3 @ B5 ) )
           => ( member_real @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_615_Set_Oset__insert,axiom,
    ! [X3: nat,A3: set_nat] :
      ( ( member_nat @ X3 @ A3 )
     => ~ ! [B5: set_nat] :
            ( ( A3
              = ( insert_nat @ X3 @ B5 ) )
           => ( member_nat @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_616_Set_Oset__insert,axiom,
    ! [X3: $o,A3: set_o] :
      ( ( member_o @ X3 @ A3 )
     => ~ ! [B5: set_o] :
            ( ( A3
              = ( insert_o @ X3 @ B5 ) )
           => ( member_o @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_617_Set_Oset__insert,axiom,
    ! [X3: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ X3 @ A3 )
     => ~ ! [B5: set_Ex3793607809372303086nnreal] :
            ( ( A3
              = ( insert7407984058720857448nnreal @ X3 @ B5 ) )
           => ( member7908768830364227535nnreal @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_618_Set_Oset__insert,axiom,
    ! [X3: real > c,A3: set_real_c] :
      ( ( member_real_c @ X3 @ A3 )
     => ~ ! [B5: set_real_c] :
            ( ( A3
              = ( insert_real_c @ X3 @ B5 ) )
           => ( member_real_c @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_619_Set_Oset__insert,axiom,
    ! [X3: real > b,A3: set_real_b] :
      ( ( member_real_b @ X3 @ A3 )
     => ~ ! [B5: set_real_b] :
            ( ( A3
              = ( insert_real_b @ X3 @ B5 ) )
           => ( member_real_b @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_620_Set_Oset__insert,axiom,
    ! [X3: real > a,A3: set_real_a] :
      ( ( member_real_a @ X3 @ A3 )
     => ~ ! [B5: set_real_a] :
            ( ( A3
              = ( insert_real_a @ X3 @ B5 ) )
           => ( member_real_a @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_621_Set_Oset__insert,axiom,
    ! [X3: c > b,A3: set_c_b] :
      ( ( member_c_b @ X3 @ A3 )
     => ~ ! [B5: set_c_b] :
            ( ( A3
              = ( insert_c_b @ X3 @ B5 ) )
           => ( member_c_b @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_622_Set_Oset__insert,axiom,
    ! [X3: a > b,A3: set_a_b] :
      ( ( member_a_b @ X3 @ A3 )
     => ~ ! [B5: set_a_b] :
            ( ( A3
              = ( insert_a_b @ X3 @ B5 ) )
           => ( member_a_b @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_623_insertI2,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( member_real @ A @ B4 )
     => ( member_real @ A @ ( insert_real @ B @ B4 ) ) ) ).

% insertI2
thf(fact_624_insertI2,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( member_nat @ A @ B4 )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_625_insertI2,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( member_o @ A @ B4 )
     => ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).

% insertI2
thf(fact_626_insertI2,axiom,
    ! [A: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ A @ B4 )
     => ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ B4 ) ) ) ).

% insertI2
thf(fact_627_insertI2,axiom,
    ! [A: real > c,B4: set_real_c,B: real > c] :
      ( ( member_real_c @ A @ B4 )
     => ( member_real_c @ A @ ( insert_real_c @ B @ B4 ) ) ) ).

% insertI2
thf(fact_628_insertI2,axiom,
    ! [A: real > b,B4: set_real_b,B: real > b] :
      ( ( member_real_b @ A @ B4 )
     => ( member_real_b @ A @ ( insert_real_b @ B @ B4 ) ) ) ).

% insertI2
thf(fact_629_insertI2,axiom,
    ! [A: real > a,B4: set_real_a,B: real > a] :
      ( ( member_real_a @ A @ B4 )
     => ( member_real_a @ A @ ( insert_real_a @ B @ B4 ) ) ) ).

% insertI2
thf(fact_630_insertI2,axiom,
    ! [A: c > b,B4: set_c_b,B: c > b] :
      ( ( member_c_b @ A @ B4 )
     => ( member_c_b @ A @ ( insert_c_b @ B @ B4 ) ) ) ).

% insertI2
thf(fact_631_insertI2,axiom,
    ! [A: a > b,B4: set_a_b,B: a > b] :
      ( ( member_a_b @ A @ B4 )
     => ( member_a_b @ A @ ( insert_a_b @ B @ B4 ) ) ) ).

% insertI2
thf(fact_632_insertI1,axiom,
    ! [A: real,B4: set_real] : ( member_real @ A @ ( insert_real @ A @ B4 ) ) ).

% insertI1
thf(fact_633_insertI1,axiom,
    ! [A: nat,B4: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_634_insertI1,axiom,
    ! [A: $o,B4: set_o] : ( member_o @ A @ ( insert_o @ A @ B4 ) ) ).

% insertI1
thf(fact_635_insertI1,axiom,
    ! [A: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] : ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ A @ B4 ) ) ).

% insertI1
thf(fact_636_insertI1,axiom,
    ! [A: real > c,B4: set_real_c] : ( member_real_c @ A @ ( insert_real_c @ A @ B4 ) ) ).

% insertI1
thf(fact_637_insertI1,axiom,
    ! [A: real > b,B4: set_real_b] : ( member_real_b @ A @ ( insert_real_b @ A @ B4 ) ) ).

% insertI1
thf(fact_638_insertI1,axiom,
    ! [A: real > a,B4: set_real_a] : ( member_real_a @ A @ ( insert_real_a @ A @ B4 ) ) ).

% insertI1
thf(fact_639_insertI1,axiom,
    ! [A: c > b,B4: set_c_b] : ( member_c_b @ A @ ( insert_c_b @ A @ B4 ) ) ).

% insertI1
thf(fact_640_insertI1,axiom,
    ! [A: a > b,B4: set_a_b] : ( member_a_b @ A @ ( insert_a_b @ A @ B4 ) ) ).

% insertI1
thf(fact_641_insertE,axiom,
    ! [A: real,B: real,A3: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A3 ) )
     => ( ( A != B )
       => ( member_real @ A @ A3 ) ) ) ).

% insertE
thf(fact_642_insertE,axiom,
    ! [A: nat,B: nat,A3: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
     => ( ( A != B )
       => ( member_nat @ A @ A3 ) ) ) ).

% insertE
thf(fact_643_insertE,axiom,
    ! [A: $o,B: $o,A3: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A3 ) )
     => ( ( A = ~ B )
       => ( member_o @ A @ A3 ) ) ) ).

% insertE
thf(fact_644_insertE,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ A3 ) )
     => ( ( A != B )
       => ( member7908768830364227535nnreal @ A @ A3 ) ) ) ).

% insertE
thf(fact_645_insertE,axiom,
    ! [A: real > c,B: real > c,A3: set_real_c] :
      ( ( member_real_c @ A @ ( insert_real_c @ B @ A3 ) )
     => ( ( A != B )
       => ( member_real_c @ A @ A3 ) ) ) ).

% insertE
thf(fact_646_insertE,axiom,
    ! [A: real > b,B: real > b,A3: set_real_b] :
      ( ( member_real_b @ A @ ( insert_real_b @ B @ A3 ) )
     => ( ( A != B )
       => ( member_real_b @ A @ A3 ) ) ) ).

% insertE
thf(fact_647_insertE,axiom,
    ! [A: real > a,B: real > a,A3: set_real_a] :
      ( ( member_real_a @ A @ ( insert_real_a @ B @ A3 ) )
     => ( ( A != B )
       => ( member_real_a @ A @ A3 ) ) ) ).

% insertE
thf(fact_648_insertE,axiom,
    ! [A: c > b,B: c > b,A3: set_c_b] :
      ( ( member_c_b @ A @ ( insert_c_b @ B @ A3 ) )
     => ( ( A != B )
       => ( member_c_b @ A @ A3 ) ) ) ).

% insertE
thf(fact_649_insertE,axiom,
    ! [A: a > b,B: a > b,A3: set_a_b] :
      ( ( member_a_b @ A @ ( insert_a_b @ B @ A3 ) )
     => ( ( A != B )
       => ( member_a_b @ A @ A3 ) ) ) ).

% insertE
thf(fact_650_insert__Collect,axiom,
    ! [A: real,P: real > $o] :
      ( ( insert_real @ A @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U: real] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_651_insert__Collect,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( insert_o @ A @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U: $o] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_652_insert__Collect,axiom,
    ! [A: extend8495563244428889912nnreal,P: extend8495563244428889912nnreal > $o] :
      ( ( insert7407984058720857448nnreal @ A @ ( collec6648975593938027277nnreal @ P ) )
      = ( collec6648975593938027277nnreal
        @ ^ [U: extend8495563244428889912nnreal] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_653_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_654_insert__compr,axiom,
    ( insert_real
    = ( ^ [A6: real,B6: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( X = A6 )
              | ( member_real @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_655_insert__compr,axiom,
    ( insert_o
    = ( ^ [A6: $o,B6: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( X = A6 )
              | ( member_o @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_656_insert__compr,axiom,
    ( insert7407984058720857448nnreal
    = ( ^ [A6: extend8495563244428889912nnreal,B6: set_Ex3793607809372303086nnreal] :
          ( collec6648975593938027277nnreal
          @ ^ [X: extend8495563244428889912nnreal] :
              ( ( X = A6 )
              | ( member7908768830364227535nnreal @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_657_insert__compr,axiom,
    ( insert_real_c
    = ( ^ [A6: real > c,B6: set_real_c] :
          ( collect_real_c
          @ ^ [X: real > c] :
              ( ( X = A6 )
              | ( member_real_c @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_658_insert__compr,axiom,
    ( insert_real_b
    = ( ^ [A6: real > b,B6: set_real_b] :
          ( collect_real_b
          @ ^ [X: real > b] :
              ( ( X = A6 )
              | ( member_real_b @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_659_insert__compr,axiom,
    ( insert_real_a
    = ( ^ [A6: real > a,B6: set_real_a] :
          ( collect_real_a
          @ ^ [X: real > a] :
              ( ( X = A6 )
              | ( member_real_a @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_660_insert__compr,axiom,
    ( insert_c_b
    = ( ^ [A6: c > b,B6: set_c_b] :
          ( collect_c_b
          @ ^ [X: c > b] :
              ( ( X = A6 )
              | ( member_c_b @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_661_insert__compr,axiom,
    ( insert_a_b
    = ( ^ [A6: a > b,B6: set_a_b] :
          ( collect_a_b
          @ ^ [X: a > b] :
              ( ( X = A6 )
              | ( member_a_b @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_662_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A6: nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A6 )
              | ( member_nat @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_663_qbs__space__eq__Mx,axiom,
    ! [X2: quasi_borel_b,Y: quasi_borel_b] :
      ( ( ( qbs_Mx_b @ X2 )
        = ( qbs_Mx_b @ Y ) )
     => ( ( qbs_space_b @ X2 )
        = ( qbs_space_b @ Y ) ) ) ).

% qbs_space_eq_Mx
thf(fact_664_qbs__space__eq__Mx,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X2 )
        = ( qbs_Mx_a @ Y ) )
     => ( ( qbs_space_a @ X2 )
        = ( qbs_space_a @ Y ) ) ) ).

% qbs_space_eq_Mx
thf(fact_665_qbs__space__eq__Mx,axiom,
    ! [X2: quasi_borel_c,Y: quasi_borel_c] :
      ( ( ( qbs_Mx_c @ X2 )
        = ( qbs_Mx_c @ Y ) )
     => ( ( qbs_space_c @ X2 )
        = ( qbs_space_c @ Y ) ) ) ).

% qbs_space_eq_Mx
thf(fact_666_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > real > c,X2: quasi_borel_real_c,R3: real] :
      ( ( member_real_real_c @ Alpha2 @ ( qbs_Mx_real_c @ X2 ) )
     => ( member_real_c @ ( Alpha2 @ R3 ) @ ( qbs_space_real_c @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_667_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > real > b,X2: quasi_borel_real_b,R3: real] :
      ( ( member_real_real_b @ Alpha2 @ ( qbs_Mx_real_b @ X2 ) )
     => ( member_real_b @ ( Alpha2 @ R3 ) @ ( qbs_space_real_b @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_668_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > real > a,X2: quasi_borel_real_a,R3: real] :
      ( ( member_real_real_a @ Alpha2 @ ( qbs_Mx_real_a @ X2 ) )
     => ( member_real_a @ ( Alpha2 @ R3 ) @ ( qbs_space_real_a @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_669_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > c > b,X2: quasi_borel_c_b,R3: real] :
      ( ( member_real_c_b @ Alpha2 @ ( qbs_Mx_c_b @ X2 ) )
     => ( member_c_b @ ( Alpha2 @ R3 ) @ ( qbs_space_c_b @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_670_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > a > b,X2: quasi_borel_a_b,R3: real] :
      ( ( member_real_a_b @ Alpha2 @ ( qbs_Mx_a_b @ X2 ) )
     => ( member_a_b @ ( Alpha2 @ R3 ) @ ( qbs_space_a_b @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_671_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > b,X2: quasi_borel_b,R3: real] :
      ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X2 ) )
     => ( member_b @ ( Alpha2 @ R3 ) @ ( qbs_space_b @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_672_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > a,X2: quasi_borel_a,R3: real] :
      ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X2 ) )
     => ( member_a @ ( Alpha2 @ R3 ) @ ( qbs_space_a @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_673_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > c,X2: quasi_borel_c,R3: real] :
      ( ( member_real_c @ Alpha2 @ ( qbs_Mx_c @ X2 ) )
     => ( member_c @ ( Alpha2 @ R3 ) @ ( qbs_space_c @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_674_insert__UNIV,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( ( insert7407984058720857448nnreal @ X3 @ top_to7994903218803871134nnreal )
      = top_to7994903218803871134nnreal ) ).

% insert_UNIV
thf(fact_675_insert__UNIV,axiom,
    ! [X3: complex] :
      ( ( insert_complex @ X3 @ top_top_set_complex )
      = top_top_set_complex ) ).

% insert_UNIV
thf(fact_676_insert__UNIV,axiom,
    ! [X3: real] :
      ( ( insert_real @ X3 @ top_top_set_real )
      = top_top_set_real ) ).

% insert_UNIV
thf(fact_677_insert__UNIV,axiom,
    ! [X3: $o] :
      ( ( insert_o @ X3 @ top_top_set_o )
      = top_top_set_o ) ).

% insert_UNIV
thf(fact_678_insert__UNIV,axiom,
    ! [X3: nat] :
      ( ( insert_nat @ X3 @ top_top_set_nat )
      = top_top_set_nat ) ).

% insert_UNIV
thf(fact_679_singletonD,axiom,
    ! [B: real > c,A: real > c] :
      ( ( member_real_c @ B @ ( insert_real_c @ A @ bot_bot_set_real_c ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_680_singletonD,axiom,
    ! [B: real > b,A: real > b] :
      ( ( member_real_b @ B @ ( insert_real_b @ A @ bot_bot_set_real_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_681_singletonD,axiom,
    ! [B: real > a,A: real > a] :
      ( ( member_real_a @ B @ ( insert_real_a @ A @ bot_bot_set_real_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_682_singletonD,axiom,
    ! [B: c > b,A: c > b] :
      ( ( member_c_b @ B @ ( insert_c_b @ A @ bot_bot_set_c_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_683_singletonD,axiom,
    ! [B: a > b,A: a > b] :
      ( ( member_a_b @ B @ ( insert_a_b @ A @ bot_bot_set_a_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_684_singletonD,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ B @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_685_singletonD,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_686_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_687_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_688_singleton__iff,axiom,
    ! [B: real > c,A: real > c] :
      ( ( member_real_c @ B @ ( insert_real_c @ A @ bot_bot_set_real_c ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_689_singleton__iff,axiom,
    ! [B: real > b,A: real > b] :
      ( ( member_real_b @ B @ ( insert_real_b @ A @ bot_bot_set_real_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_690_singleton__iff,axiom,
    ! [B: real > a,A: real > a] :
      ( ( member_real_a @ B @ ( insert_real_a @ A @ bot_bot_set_real_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_691_singleton__iff,axiom,
    ! [B: c > b,A: c > b] :
      ( ( member_c_b @ B @ ( insert_c_b @ A @ bot_bot_set_c_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_692_singleton__iff,axiom,
    ! [B: a > b,A: a > b] :
      ( ( member_a_b @ B @ ( insert_a_b @ A @ bot_bot_set_a_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_693_singleton__iff,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ B @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_694_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_695_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_696_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_697_doubleton__eq__iff,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
      ( ( ( insert7407984058720857448nnreal @ A @ ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) )
        = ( insert7407984058720857448nnreal @ C @ ( insert7407984058720857448nnreal @ D @ bot_bo4854962954004695426nnreal ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_698_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_699_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_700_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_701_insert__not__empty,axiom,
    ! [A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ A @ A3 )
     != bot_bo4854962954004695426nnreal ) ).

% insert_not_empty
thf(fact_702_insert__not__empty,axiom,
    ! [A: real,A3: set_real] :
      ( ( insert_real @ A @ A3 )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_703_insert__not__empty,axiom,
    ! [A: $o,A3: set_o] :
      ( ( insert_o @ A @ A3 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_704_insert__not__empty,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( insert_nat @ A @ A3 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_705_singleton__inject,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal )
        = ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_706_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( ( insert_real @ A @ bot_bot_set_real )
        = ( insert_real @ B @ bot_bot_set_real ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_707_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_708_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_709_qbs__morphismE_I2_J,axiom,
    ! [F2: real > c,X2: quasi_borel_real,Y: quasi_borel_c,X3: real] :
      ( ( member_real_c @ F2 @ ( qbs_morphism_real_c @ X2 @ Y ) )
     => ( ( member_real @ X3 @ ( qbs_space_real @ X2 ) )
       => ( member_c @ ( F2 @ X3 ) @ ( qbs_space_c @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_710_qbs__morphismE_I2_J,axiom,
    ! [F2: real > b,X2: quasi_borel_real,Y: quasi_borel_b,X3: real] :
      ( ( member_real_b @ F2 @ ( qbs_morphism_real_b @ X2 @ Y ) )
     => ( ( member_real @ X3 @ ( qbs_space_real @ X2 ) )
       => ( member_b @ ( F2 @ X3 ) @ ( qbs_space_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_711_qbs__morphismE_I2_J,axiom,
    ! [F2: real > a,X2: quasi_borel_real,Y: quasi_borel_a,X3: real] :
      ( ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X2 @ Y ) )
     => ( ( member_real @ X3 @ ( qbs_space_real @ X2 ) )
       => ( member_a @ ( F2 @ X3 ) @ ( qbs_space_a @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_712_qbs__morphismE_I2_J,axiom,
    ! [F2: c > b,X2: quasi_borel_c,Y: quasi_borel_b,X3: c] :
      ( ( member_c_b @ F2 @ ( qbs_morphism_c_b @ X2 @ Y ) )
     => ( ( member_c @ X3 @ ( qbs_space_c @ X2 ) )
       => ( member_b @ ( F2 @ X3 ) @ ( qbs_space_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_713_qbs__morphismE_I2_J,axiom,
    ! [F2: a > b,X2: quasi_borel_a,Y: quasi_borel_b,X3: a] :
      ( ( member_a_b @ F2 @ ( qbs_morphism_a_b @ X2 @ Y ) )
     => ( ( member_a @ X3 @ ( qbs_space_a @ X2 ) )
       => ( member_b @ ( F2 @ X3 ) @ ( qbs_space_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_714_qbs__morphismE_I2_J,axiom,
    ! [F2: ( real > c ) > real > c,X2: quasi_borel_real_c,Y: quasi_borel_real_c,X3: real > c] :
      ( ( member_real_c_real_c @ F2 @ ( qbs_mo6326314534061150392real_c @ X2 @ Y ) )
     => ( ( member_real_c @ X3 @ ( qbs_space_real_c @ X2 ) )
       => ( member_real_c @ ( F2 @ X3 ) @ ( qbs_space_real_c @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_715_qbs__morphismE_I2_J,axiom,
    ! [F2: ( real > c ) > real > b,X2: quasi_borel_real_c,Y: quasi_borel_real_b,X3: real > c] :
      ( ( member_real_c_real_b @ F2 @ ( qbs_mo6326314529757921591real_b @ X2 @ Y ) )
     => ( ( member_real_c @ X3 @ ( qbs_space_real_c @ X2 ) )
       => ( member_real_b @ ( F2 @ X3 ) @ ( qbs_space_real_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_716_qbs__morphismE_I2_J,axiom,
    ! [F2: ( real > c ) > real > a,X2: quasi_borel_real_c,Y: quasi_borel_real_a,X3: real > c] :
      ( ( member_real_c_real_a @ F2 @ ( qbs_mo6326314525454692790real_a @ X2 @ Y ) )
     => ( ( member_real_c @ X3 @ ( qbs_space_real_c @ X2 ) )
       => ( member_real_a @ ( F2 @ X3 ) @ ( qbs_space_real_a @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_717_qbs__morphismE_I2_J,axiom,
    ! [F2: ( real > c ) > c > b,X2: quasi_borel_real_c,Y: quasi_borel_c_b,X3: real > c] :
      ( ( member_real_c_c_b @ F2 @ ( qbs_mo7641311727218106871_c_c_b @ X2 @ Y ) )
     => ( ( member_real_c @ X3 @ ( qbs_space_real_c @ X2 ) )
       => ( member_c_b @ ( F2 @ X3 ) @ ( qbs_space_c_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_718_qbs__morphismE_I2_J,axiom,
    ! [F2: ( real > c ) > a > b,X2: quasi_borel_real_c,Y: quasi_borel_a_b,X3: real > c] :
      ( ( member_real_c_a_b @ F2 @ ( qbs_mo3992850854308683129_c_a_b @ X2 @ Y ) )
     => ( ( member_real_c @ X3 @ ( qbs_space_real_c @ X2 ) )
       => ( member_a_b @ ( F2 @ X3 ) @ ( qbs_space_a_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_719_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_real,F2: real > c,G2: real > c,Y: quasi_borel_c] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( qbs_space_real @ X2 ) )
         => ( ( F2 @ X5 )
            = ( G2 @ X5 ) ) )
     => ( ( member_real_c @ F2 @ ( qbs_morphism_real_c @ X2 @ Y ) )
       => ( member_real_c @ G2 @ ( qbs_morphism_real_c @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_720_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_real,F2: real > b,G2: real > b,Y: quasi_borel_b] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( qbs_space_real @ X2 ) )
         => ( ( F2 @ X5 )
            = ( G2 @ X5 ) ) )
     => ( ( member_real_b @ F2 @ ( qbs_morphism_real_b @ X2 @ Y ) )
       => ( member_real_b @ G2 @ ( qbs_morphism_real_b @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_721_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_real,F2: real > a,G2: real > a,Y: quasi_borel_a] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( qbs_space_real @ X2 ) )
         => ( ( F2 @ X5 )
            = ( G2 @ X5 ) ) )
     => ( ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X2 @ Y ) )
       => ( member_real_a @ G2 @ ( qbs_morphism_real_a @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_722_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_c,F2: c > b,G2: c > b,Y: quasi_borel_b] :
      ( ! [X5: c] :
          ( ( member_c @ X5 @ ( qbs_space_c @ X2 ) )
         => ( ( F2 @ X5 )
            = ( G2 @ X5 ) ) )
     => ( ( member_c_b @ F2 @ ( qbs_morphism_c_b @ X2 @ Y ) )
       => ( member_c_b @ G2 @ ( qbs_morphism_c_b @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_723_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_a,F2: a > b,G2: a > b,Y: quasi_borel_b] :
      ( ! [X5: a] :
          ( ( member_a @ X5 @ ( qbs_space_a @ X2 ) )
         => ( ( F2 @ X5 )
            = ( G2 @ X5 ) ) )
     => ( ( member_a_b @ F2 @ ( qbs_morphism_a_b @ X2 @ Y ) )
       => ( member_a_b @ G2 @ ( qbs_morphism_a_b @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_724_qbs__closed2__dest,axiom,
    ! [X3: real > c,X2: quasi_borel_real_c] :
      ( ( member_real_c @ X3 @ ( qbs_space_real_c @ X2 ) )
     => ( member_real_real_c
        @ ^ [R: real] : X3
        @ ( qbs_Mx_real_c @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_725_qbs__closed2__dest,axiom,
    ! [X3: real > b,X2: quasi_borel_real_b] :
      ( ( member_real_b @ X3 @ ( qbs_space_real_b @ X2 ) )
     => ( member_real_real_b
        @ ^ [R: real] : X3
        @ ( qbs_Mx_real_b @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_726_qbs__closed2__dest,axiom,
    ! [X3: real > a,X2: quasi_borel_real_a] :
      ( ( member_real_a @ X3 @ ( qbs_space_real_a @ X2 ) )
     => ( member_real_real_a
        @ ^ [R: real] : X3
        @ ( qbs_Mx_real_a @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_727_qbs__closed2__dest,axiom,
    ! [X3: c > b,X2: quasi_borel_c_b] :
      ( ( member_c_b @ X3 @ ( qbs_space_c_b @ X2 ) )
     => ( member_real_c_b
        @ ^ [R: real] : X3
        @ ( qbs_Mx_c_b @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_728_qbs__closed2__dest,axiom,
    ! [X3: a > b,X2: quasi_borel_a_b] :
      ( ( member_a_b @ X3 @ ( qbs_space_a_b @ X2 ) )
     => ( member_real_a_b
        @ ^ [R: real] : X3
        @ ( qbs_Mx_a_b @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_729_qbs__closed2__dest,axiom,
    ! [X3: b,X2: quasi_borel_b] :
      ( ( member_b @ X3 @ ( qbs_space_b @ X2 ) )
     => ( member_real_b
        @ ^ [R: real] : X3
        @ ( qbs_Mx_b @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_730_qbs__closed2__dest,axiom,
    ! [X3: a,X2: quasi_borel_a] :
      ( ( member_a @ X3 @ ( qbs_space_a @ X2 ) )
     => ( member_real_a
        @ ^ [R: real] : X3
        @ ( qbs_Mx_a @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_731_qbs__closed2__dest,axiom,
    ! [X3: c,X2: quasi_borel_c] :
      ( ( member_c @ X3 @ ( qbs_space_c @ X2 ) )
     => ( member_real_c
        @ ^ [R: real] : X3
        @ ( qbs_Mx_c @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_732_qbs__morphism__const,axiom,
    ! [Y3: c,Y: quasi_borel_c,X2: quasi_borel_real] :
      ( ( member_c @ Y3 @ ( qbs_space_c @ Y ) )
     => ( member_real_c
        @ ^ [Uu: real] : Y3
        @ ( qbs_morphism_real_c @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_733_qbs__morphism__const,axiom,
    ! [Y3: b,Y: quasi_borel_b,X2: quasi_borel_real] :
      ( ( member_b @ Y3 @ ( qbs_space_b @ Y ) )
     => ( member_real_b
        @ ^ [Uu: real] : Y3
        @ ( qbs_morphism_real_b @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_734_qbs__morphism__const,axiom,
    ! [Y3: a,Y: quasi_borel_a,X2: quasi_borel_real] :
      ( ( member_a @ Y3 @ ( qbs_space_a @ Y ) )
     => ( member_real_a
        @ ^ [Uu: real] : Y3
        @ ( qbs_morphism_real_a @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_735_qbs__morphism__const,axiom,
    ! [Y3: b,Y: quasi_borel_b,X2: quasi_borel_c] :
      ( ( member_b @ Y3 @ ( qbs_space_b @ Y ) )
     => ( member_c_b
        @ ^ [Uu: c] : Y3
        @ ( qbs_morphism_c_b @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_736_qbs__morphism__const,axiom,
    ! [Y3: b,Y: quasi_borel_b,X2: quasi_borel_a] :
      ( ( member_b @ Y3 @ ( qbs_space_b @ Y ) )
     => ( member_a_b
        @ ^ [Uu: a] : Y3
        @ ( qbs_morphism_a_b @ X2 @ Y ) ) ) ).

% qbs_morphism_const
thf(fact_737_Collect__conv__if2,axiom,
    ! [P: extend8495563244428889912nnreal > $o,A: extend8495563244428889912nnreal] :
      ( ( ( P @ A )
       => ( ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo4854962954004695426nnreal ) ) ) ).

% Collect_conv_if2
thf(fact_738_Collect__conv__if2,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_739_Collect__conv__if2,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_740_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_741_Collect__conv__if,axiom,
    ! [P: extend8495563244428889912nnreal > $o,A: extend8495563244428889912nnreal] :
      ( ( ( P @ A )
       => ( ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo4854962954004695426nnreal ) ) ) ).

% Collect_conv_if
thf(fact_742_Collect__conv__if,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_743_Collect__conv__if,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_744_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_745_sets_Oinsert__in__sets,axiom,
    ! [X3: extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal @ A3 @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X3 @ A3 ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_746_sets_Oinsert__in__sets,axiom,
    ! [X3: real,M: sigma_measure_real,A3: set_real] :
      ( ( member_set_real @ ( insert_real @ X3 @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real @ A3 @ ( sigma_sets_real @ M ) )
       => ( member_set_real @ ( insert_real @ X3 @ A3 ) @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_747_sets_Oinsert__in__sets,axiom,
    ! [X3: $o,M: sigma_measure_o,A3: set_o] :
      ( ( member_set_o @ ( insert_o @ X3 @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o @ A3 @ ( sigma_sets_o @ M ) )
       => ( member_set_o @ ( insert_o @ X3 @ A3 ) @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_748_sets_Oinsert__in__sets,axiom,
    ! [X3: nat,M: sigma_measure_nat,A3: set_nat] :
      ( ( member_set_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat @ A3 @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat @ ( insert_nat @ X3 @ A3 ) @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_749_borel__singleton,axiom,
    ! [A3: set_real,X3: real] :
      ( ( member_set_real @ A3 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( member_set_real @ ( insert_real @ X3 @ A3 ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% borel_singleton
thf(fact_750_borel__singleton,axiom,
    ! [A3: set_nat,X3: nat] :
      ( ( member_set_nat @ A3 @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) )
     => ( member_set_nat @ ( insert_nat @ X3 @ A3 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% borel_singleton
thf(fact_751_borel__singleton,axiom,
    ! [A3: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( member603777416030116741nnreal @ A3 @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
     => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X3 @ A3 ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).

% borel_singleton
thf(fact_752_borel__singleton,axiom,
    ! [A3: set_o,X3: $o] :
      ( ( member_set_o @ A3 @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) )
     => ( member_set_o @ ( insert_o @ X3 @ A3 ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% borel_singleton
thf(fact_753_empty__quasi__borel__iff,axiom,
    ! [X2: quasi_9015997321629101608nnreal] :
      ( ( ( qbs_sp175953267596557954nnreal @ X2 )
        = bot_bo4854962954004695426nnreal )
      = ( X2 = empty_1788085430566700506nnreal ) ) ).

% empty_quasi_borel_iff
thf(fact_754_empty__quasi__borel__iff,axiom,
    ! [X2: quasi_borel_real] :
      ( ( ( qbs_space_real @ X2 )
        = bot_bot_set_real )
      = ( X2 = empty_1876425439295802446l_real ) ) ).

% empty_quasi_borel_iff
thf(fact_755_empty__quasi__borel__iff,axiom,
    ! [X2: quasi_borel_o] :
      ( ( ( qbs_space_o @ X2 )
        = bot_bot_set_o )
      = ( X2 = empty_quasi_borel_o ) ) ).

% empty_quasi_borel_iff
thf(fact_756_empty__quasi__borel__iff,axiom,
    ! [X2: quasi_borel_nat] :
      ( ( ( qbs_space_nat @ X2 )
        = bot_bot_set_nat )
      = ( X2 = empty_8278123436611590770el_nat ) ) ).

% empty_quasi_borel_iff
thf(fact_757_copair__qbs__closed3,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_c] : ( qbs_cl7154477953554275762um_a_c @ ( binary8286901584692334522Mx_a_c @ X2 @ Y ) ) ).

% copair_qbs_closed3
thf(fact_758_copair__qbs__closed1,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_c] : ( qbs_cl96855206660534704um_a_c @ ( binary8286901584692334522Mx_a_c @ X2 @ Y ) ) ).

% copair_qbs_closed1
thf(fact_759_setrp_Ointros,axiom,
    ! [S: sum_sum_a_c,X3: c] :
      ( ( S
        = ( sum_Inr_c_a @ X3 ) )
     => ( basic_setrp_a_c @ S @ X3 ) ) ).

% setrp.intros
thf(fact_760_setrp_Osimps,axiom,
    ( basic_setrp_a_c
    = ( ^ [S3: sum_sum_a_c,A6: c] :
        ? [X: c] :
          ( ( A6 = X )
          & ( S3
            = ( sum_Inr_c_a @ X ) ) ) ) ) ).

% setrp.simps
thf(fact_761_setrp_Ocases,axiom,
    ! [S: sum_sum_a_c,A: c] :
      ( ( basic_setrp_a_c @ S @ A )
     => ( S
        = ( sum_Inr_c_a @ A ) ) ) ).

% setrp.cases
thf(fact_762_top__empty__eq,axiom,
    ( top_top_real_c_o
    = ( ^ [X: real > c] : ( member_real_c @ X @ top_top_set_real_c ) ) ) ).

% top_empty_eq
thf(fact_763_top__empty__eq,axiom,
    ( top_top_real_b_o
    = ( ^ [X: real > b] : ( member_real_b @ X @ top_top_set_real_b ) ) ) ).

% top_empty_eq
thf(fact_764_top__empty__eq,axiom,
    ( top_top_real_a_o
    = ( ^ [X: real > a] : ( member_real_a @ X @ top_top_set_real_a ) ) ) ).

% top_empty_eq
thf(fact_765_top__empty__eq,axiom,
    ( top_top_c_b_o
    = ( ^ [X: c > b] : ( member_c_b @ X @ top_top_set_c_b ) ) ) ).

% top_empty_eq
thf(fact_766_top__empty__eq,axiom,
    ( top_top_a_b_o
    = ( ^ [X: a > b] : ( member_a_b @ X @ top_top_set_a_b ) ) ) ).

% top_empty_eq
thf(fact_767_top__empty__eq,axiom,
    ( top_to5118619752887738471real_o
    = ( ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ top_to7994903218803871134nnreal ) ) ) ).

% top_empty_eq
thf(fact_768_top__empty__eq,axiom,
    ( top_top_complex_o
    = ( ^ [X: complex] : ( member_complex @ X @ top_top_set_complex ) ) ) ).

% top_empty_eq
thf(fact_769_top__empty__eq,axiom,
    ( top_top_real_o
    = ( ^ [X: real] : ( member_real @ X @ top_top_set_real ) ) ) ).

% top_empty_eq
thf(fact_770_top__empty__eq,axiom,
    ( top_top_o_o
    = ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).

% top_empty_eq
thf(fact_771_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_772_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_b] :
      ( ( ( qbs_space_b @ X2 )
        = bot_bot_set_b )
      = ( ( qbs_Mx_b @ X2 )
        = bot_bot_set_real_b ) ) ).

% qbs_empty_equiv
thf(fact_773_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_a] :
      ( ( ( qbs_space_a @ X2 )
        = bot_bot_set_a )
      = ( ( qbs_Mx_a @ X2 )
        = bot_bot_set_real_a ) ) ).

% qbs_empty_equiv
thf(fact_774_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_c] :
      ( ( ( qbs_space_c @ X2 )
        = bot_bot_set_c )
      = ( ( qbs_Mx_c @ X2 )
        = bot_bot_set_real_c ) ) ).

% qbs_empty_equiv
thf(fact_775_qbs__empty__equiv,axiom,
    ! [X2: quasi_9015997321629101608nnreal] :
      ( ( ( qbs_sp175953267596557954nnreal @ X2 )
        = bot_bo4854962954004695426nnreal )
      = ( ( qbs_Mx6523938229262583809nnreal @ X2 )
        = bot_bo6037503491064675021nnreal ) ) ).

% qbs_empty_equiv
thf(fact_776_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_real] :
      ( ( ( qbs_space_real @ X2 )
        = bot_bot_set_real )
      = ( ( qbs_Mx_real @ X2 )
        = bot_bo6767488733719836353l_real ) ) ).

% qbs_empty_equiv
thf(fact_777_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_o] :
      ( ( ( qbs_space_o @ X2 )
        = bot_bot_set_o )
      = ( ( qbs_Mx_o @ X2 )
        = bot_bot_set_real_o ) ) ).

% qbs_empty_equiv
thf(fact_778_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_nat] :
      ( ( ( qbs_space_nat @ X2 )
        = bot_bot_set_nat )
      = ( ( qbs_Mx_nat @ X2 )
        = bot_bot_set_real_nat ) ) ).

% qbs_empty_equiv
thf(fact_779_sets__bot,axiom,
    ( ( sigma_5465916536984168985nnreal @ bot_bo1740529460517930749nnreal )
    = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) ) ).

% sets_bot
thf(fact_780_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_781_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_782_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_783_vector__space__over__itself_Ozero__not__in__Basis,axiom,
    ~ ( member_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) ).

% vector_space_over_itself.zero_not_in_Basis
thf(fact_784_set__one,axiom,
    ( one_on449101747310943155nnreal
    = ( insert7407984058720857448nnreal @ one_on2969667320475766781nnreal @ bot_bo4854962954004695426nnreal ) ) ).

% set_one
thf(fact_785_set__one,axiom,
    ( one_one_set_real
    = ( insert_real @ one_one_real @ bot_bot_set_real ) ) ).

% set_one
thf(fact_786_set__one,axiom,
    ( one_one_set_nat
    = ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ).

% set_one
thf(fact_787_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( top_to7994903218803871134nnreal
     != ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) ).

% perfect_space_class.UNIV_not_singleton
thf(fact_788_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [X3: complex] :
      ( top_top_set_complex
     != ( insert_complex @ X3 @ bot_bot_set_complex ) ) ).

% perfect_space_class.UNIV_not_singleton
thf(fact_789_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [X3: real] :
      ( top_top_set_real
     != ( insert_real @ X3 @ bot_bot_set_real ) ) ).

% perfect_space_class.UNIV_not_singleton
thf(fact_790_set__zero,axiom,
    ( zero_z81811184781112823nnreal
    = ( insert7407984058720857448nnreal @ zero_z7100319975126383169nnreal @ bot_bo4854962954004695426nnreal ) ) ).

% set_zero
thf(fact_791_set__zero,axiom,
    ( zero_zero_set_real
    = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).

% set_zero
thf(fact_792_set__zero,axiom,
    ( zero_zero_set_nat
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% set_zero
thf(fact_793_is__singletonI,axiom,
    ! [X3: extend8495563244428889912nnreal] : ( is_sin3654761921782142788nnreal @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) ).

% is_singletonI
thf(fact_794_is__singletonI,axiom,
    ! [X3: real] : ( is_singleton_real @ ( insert_real @ X3 @ bot_bot_set_real ) ) ).

% is_singletonI
thf(fact_795_is__singletonI,axiom,
    ! [X3: $o] : ( is_singleton_o @ ( insert_o @ X3 @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_796_is__singletonI,axiom,
    ! [X3: nat] : ( is_singleton_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_797_sets__eq__bot,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) )
      = ( M = bot_bo1740529460517930749nnreal ) ) ).

% sets_eq_bot
thf(fact_798_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_799_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_800_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_801_sets__eq__bot2,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal )
        = ( sigma_5465916536984168985nnreal @ M ) )
      = ( M = bot_bo1740529460517930749nnreal ) ) ).

% sets_eq_bot2
thf(fact_802_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_803_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_804_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_805_is__singletonI_H,axiom,
    ! [A3: set_real_c] :
      ( ( A3 != bot_bot_set_real_c )
     => ( ! [X5: real > c,Y4: real > c] :
            ( ( member_real_c @ X5 @ A3 )
           => ( ( member_real_c @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_real_c @ A3 ) ) ) ).

% is_singletonI'
thf(fact_806_is__singletonI_H,axiom,
    ! [A3: set_real_b] :
      ( ( A3 != bot_bot_set_real_b )
     => ( ! [X5: real > b,Y4: real > b] :
            ( ( member_real_b @ X5 @ A3 )
           => ( ( member_real_b @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_real_b @ A3 ) ) ) ).

% is_singletonI'
thf(fact_807_is__singletonI_H,axiom,
    ! [A3: set_real_a] :
      ( ( A3 != bot_bot_set_real_a )
     => ( ! [X5: real > a,Y4: real > a] :
            ( ( member_real_a @ X5 @ A3 )
           => ( ( member_real_a @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_real_a @ A3 ) ) ) ).

% is_singletonI'
thf(fact_808_is__singletonI_H,axiom,
    ! [A3: set_c_b] :
      ( ( A3 != bot_bot_set_c_b )
     => ( ! [X5: c > b,Y4: c > b] :
            ( ( member_c_b @ X5 @ A3 )
           => ( ( member_c_b @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_c_b @ A3 ) ) ) ).

% is_singletonI'
thf(fact_809_is__singletonI_H,axiom,
    ! [A3: set_a_b] :
      ( ( A3 != bot_bot_set_a_b )
     => ( ! [X5: a > b,Y4: a > b] :
            ( ( member_a_b @ X5 @ A3 )
           => ( ( member_a_b @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_a_b @ A3 ) ) ) ).

% is_singletonI'
thf(fact_810_is__singletonI_H,axiom,
    ! [A3: set_Ex3793607809372303086nnreal] :
      ( ( A3 != bot_bo4854962954004695426nnreal )
     => ( ! [X5: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X5 @ A3 )
           => ( ( member7908768830364227535nnreal @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_sin3654761921782142788nnreal @ A3 ) ) ) ).

% is_singletonI'
thf(fact_811_is__singletonI_H,axiom,
    ! [A3: set_real] :
      ( ( A3 != bot_bot_set_real )
     => ( ! [X5: real,Y4: real] :
            ( ( member_real @ X5 @ A3 )
           => ( ( member_real @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_real @ A3 ) ) ) ).

% is_singletonI'
thf(fact_812_is__singletonI_H,axiom,
    ! [A3: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X5: $o,Y4: $o] :
            ( ( member_o @ X5 @ A3 )
           => ( ( member_o @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_o @ A3 ) ) ) ).

% is_singletonI'
thf(fact_813_is__singletonI_H,axiom,
    ! [A3: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X5: nat,Y4: nat] :
            ( ( member_nat @ X5 @ A3 )
           => ( ( member_nat @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_nat @ A3 ) ) ) ).

% is_singletonI'
thf(fact_814_is__singletonE,axiom,
    ! [A3: set_Ex3793607809372303086nnreal] :
      ( ( is_sin3654761921782142788nnreal @ A3 )
     => ~ ! [X5: extend8495563244428889912nnreal] :
            ( A3
           != ( insert7407984058720857448nnreal @ X5 @ bot_bo4854962954004695426nnreal ) ) ) ).

% is_singletonE
thf(fact_815_is__singletonE,axiom,
    ! [A3: set_real] :
      ( ( is_singleton_real @ A3 )
     => ~ ! [X5: real] :
            ( A3
           != ( insert_real @ X5 @ bot_bot_set_real ) ) ) ).

% is_singletonE
thf(fact_816_is__singletonE,axiom,
    ! [A3: set_o] :
      ( ( is_singleton_o @ A3 )
     => ~ ! [X5: $o] :
            ( A3
           != ( insert_o @ X5 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_817_is__singletonE,axiom,
    ! [A3: set_nat] :
      ( ( is_singleton_nat @ A3 )
     => ~ ! [X5: nat] :
            ( A3
           != ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_818_is__singleton__def,axiom,
    ( is_sin3654761921782142788nnreal
    = ( ^ [A5: set_Ex3793607809372303086nnreal] :
        ? [X: extend8495563244428889912nnreal] :
          ( A5
          = ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% is_singleton_def
thf(fact_819_is__singleton__def,axiom,
    ( is_singleton_real
    = ( ^ [A5: set_real] :
        ? [X: real] :
          ( A5
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% is_singleton_def
thf(fact_820_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
        ? [X: $o] :
          ( A5
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_821_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
        ? [X: nat] :
          ( A5
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_822_dimension__eq__1,axiom,
    ( ( vector5117482691322076262n_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) )
    = one_one_nat ) ).

% dimension_eq_1
thf(fact_823_is__singleton__the__elem,axiom,
    ( is_sin3654761921782142788nnreal
    = ( ^ [A5: set_Ex3793607809372303086nnreal] :
          ( A5
          = ( insert7407984058720857448nnreal @ ( the_el3795950934141317635nnreal @ A5 ) @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% is_singleton_the_elem
thf(fact_824_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_825_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_826_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_827_the__elem__eq,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( ( the_el3795950934141317635nnreal @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) )
      = X3 ) ).

% the_elem_eq
thf(fact_828_the__elem__eq,axiom,
    ! [X3: real] :
      ( ( the_elem_real @ ( insert_real @ X3 @ bot_bot_set_real ) )
      = X3 ) ).

% the_elem_eq
thf(fact_829_the__elem__eq,axiom,
    ! [X3: $o] :
      ( ( the_elem_o @ ( insert_o @ X3 @ bot_bot_set_o ) )
      = X3 ) ).

% the_elem_eq
thf(fact_830_the__elem__eq,axiom,
    ! [X3: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
      = X3 ) ).

% the_elem_eq
thf(fact_831_orthogonal__comp__null,axiom,
    ( ( orthog7815041348650528364omplex @ ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) )
    = top_top_set_complex ) ).

% orthogonal_comp_null
thf(fact_832_orthogonal__comp__null,axiom,
    ( ( orthogonal_comp_real @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) )
    = top_top_set_real ) ).

% orthogonal_comp_null
thf(fact_833_orthogonal__comp__UNIV,axiom,
    ( ( orthog7815041348650528364omplex @ top_top_set_complex )
    = ( insert_complex @ zero_zero_complex @ bot_bot_set_complex ) ) ).

% orthogonal_comp_UNIV
thf(fact_834_orthogonal__comp__UNIV,axiom,
    ( ( orthogonal_comp_real @ top_top_set_real )
    = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).

% orthogonal_comp_UNIV
thf(fact_835_max__qbs__Mx,axiom,
    ! [X2: set_b] :
      ( ( qbs_Mx_b @ ( max_quasi_borel_b @ X2 ) )
      = ( pi_real_b @ top_top_set_real
        @ ^ [Uu: real] : X2 ) ) ).

% max_qbs_Mx
thf(fact_836_max__qbs__Mx,axiom,
    ! [X2: set_a] :
      ( ( qbs_Mx_a @ ( max_quasi_borel_a @ X2 ) )
      = ( pi_real_a @ top_top_set_real
        @ ^ [Uu: real] : X2 ) ) ).

% max_qbs_Mx
thf(fact_837_max__qbs__Mx,axiom,
    ! [X2: set_c] :
      ( ( qbs_Mx_c @ ( max_quasi_borel_c @ X2 ) )
      = ( pi_real_c @ top_top_set_real
        @ ^ [Uu: real] : X2 ) ) ).

% max_qbs_Mx
thf(fact_838_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_839_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_840_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_841_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_842_Pi__I,axiom,
    ! [A3: set_real,F2: real > c,B4: real > set_c] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_c @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c @ F2 @ ( pi_real_c @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_843_Pi__I,axiom,
    ! [A3: set_real,F2: real > b,B4: real > set_b] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_b @ F2 @ ( pi_real_b @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_844_Pi__I,axiom,
    ! [A3: set_real,F2: real > a,B4: real > set_a] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_a @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_a @ F2 @ ( pi_real_a @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_845_Pi__I,axiom,
    ! [A3: set_c,F2: c > b,B4: c > set_b] :
      ( ! [X5: c] :
          ( ( member_c @ X5 @ A3 )
         => ( member_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_c_b @ F2 @ ( pi_c_b @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_846_Pi__I,axiom,
    ! [A3: set_a,F2: a > b,B4: a > set_b] :
      ( ! [X5: a] :
          ( ( member_a @ X5 @ A3 )
         => ( member_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_a_b @ F2 @ ( pi_a_b @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_847_Pi__I,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > real > c,B4: ( real > c ) > set_real_c] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_real_c @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_real_c @ F2 @ ( pi_real_c_real_c @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_848_Pi__I,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > real > b,B4: ( real > c ) > set_real_b] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_real_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_real_b @ F2 @ ( pi_real_c_real_b @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_849_Pi__I,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > real > a,B4: ( real > c ) > set_real_a] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_real_a @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_real_a @ F2 @ ( pi_real_c_real_a @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_850_Pi__I,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > c > b,B4: ( real > c ) > set_c_b] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_c_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_c_b @ F2 @ ( pi_real_c_c_b @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_851_Pi__I,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > a > b,B4: ( real > c ) > set_a_b] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_a_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_a_b @ F2 @ ( pi_real_c_a_b @ A3 @ B4 ) ) ) ).

% Pi_I
thf(fact_852_sets_Otop,axiom,
    ! [M: sigma_measure_real] : ( member_set_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).

% sets.top
thf(fact_853_sets_Otop,axiom,
    ! [M: sigma_measure_nat] : ( member_set_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) ).

% sets.top
thf(fact_854_sets_Otop,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( member603777416030116741nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.top
thf(fact_855_sets_Otop,axiom,
    ! [M: sigma_measure_o] : ( member_set_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) ).

% sets.top
thf(fact_856_Pi__split__insert__domain,axiom,
    ! [X3: real > c,I3: real,I2: set_real,X2: real > set_c] :
      ( ( member_real_c @ X3 @ ( pi_real_c @ ( insert_real @ I3 @ I2 ) @ X2 ) )
      = ( ( member_real_c @ X3 @ ( pi_real_c @ I2 @ X2 ) )
        & ( member_c @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_857_Pi__split__insert__domain,axiom,
    ! [X3: real > b,I3: real,I2: set_real,X2: real > set_b] :
      ( ( member_real_b @ X3 @ ( pi_real_b @ ( insert_real @ I3 @ I2 ) @ X2 ) )
      = ( ( member_real_b @ X3 @ ( pi_real_b @ I2 @ X2 ) )
        & ( member_b @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_858_Pi__split__insert__domain,axiom,
    ! [X3: real > a,I3: real,I2: set_real,X2: real > set_a] :
      ( ( member_real_a @ X3 @ ( pi_real_a @ ( insert_real @ I3 @ I2 ) @ X2 ) )
      = ( ( member_real_a @ X3 @ ( pi_real_a @ I2 @ X2 ) )
        & ( member_a @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_859_Pi__split__insert__domain,axiom,
    ! [X3: c > b,I3: c,I2: set_c,X2: c > set_b] :
      ( ( member_c_b @ X3 @ ( pi_c_b @ ( insert_c @ I3 @ I2 ) @ X2 ) )
      = ( ( member_c_b @ X3 @ ( pi_c_b @ I2 @ X2 ) )
        & ( member_b @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_860_Pi__split__insert__domain,axiom,
    ! [X3: a > b,I3: a,I2: set_a,X2: a > set_b] :
      ( ( member_a_b @ X3 @ ( pi_a_b @ ( insert_a @ I3 @ I2 ) @ X2 ) )
      = ( ( member_a_b @ X3 @ ( pi_a_b @ I2 @ X2 ) )
        & ( member_b @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_861_Pi__split__insert__domain,axiom,
    ! [X3: real > real > c,I3: real,I2: set_real,X2: real > set_real_c] :
      ( ( member_real_real_c @ X3 @ ( pi_real_real_c @ ( insert_real @ I3 @ I2 ) @ X2 ) )
      = ( ( member_real_real_c @ X3 @ ( pi_real_real_c @ I2 @ X2 ) )
        & ( member_real_c @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_862_Pi__split__insert__domain,axiom,
    ! [X3: nat > real > c,I3: nat,I2: set_nat,X2: nat > set_real_c] :
      ( ( member_nat_real_c @ X3 @ ( pi_nat_real_c @ ( insert_nat @ I3 @ I2 ) @ X2 ) )
      = ( ( member_nat_real_c @ X3 @ ( pi_nat_real_c @ I2 @ X2 ) )
        & ( member_real_c @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_863_Pi__split__insert__domain,axiom,
    ! [X3: $o > real > c,I3: $o,I2: set_o,X2: $o > set_real_c] :
      ( ( member_o_real_c @ X3 @ ( pi_o_real_c @ ( insert_o @ I3 @ I2 ) @ X2 ) )
      = ( ( member_o_real_c @ X3 @ ( pi_o_real_c @ I2 @ X2 ) )
        & ( member_real_c @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_864_Pi__split__insert__domain,axiom,
    ! [X3: extend8495563244428889912nnreal > real > c,I3: extend8495563244428889912nnreal,I2: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal > set_real_c] :
      ( ( member777441237193892147real_c @ X3 @ ( pi_Ext4284999243647831750real_c @ ( insert7407984058720857448nnreal @ I3 @ I2 ) @ X2 ) )
      = ( ( member777441237193892147real_c @ X3 @ ( pi_Ext4284999243647831750real_c @ I2 @ X2 ) )
        & ( member_real_c @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_865_Pi__split__insert__domain,axiom,
    ! [X3: real > real > b,I3: real,I2: set_real,X2: real > set_real_b] :
      ( ( member_real_real_b @ X3 @ ( pi_real_real_b @ ( insert_real @ I3 @ I2 ) @ X2 ) )
      = ( ( member_real_real_b @ X3 @ ( pi_real_real_b @ I2 @ X2 ) )
        & ( member_real_b @ ( X3 @ I3 ) @ ( X2 @ I3 ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_866_space__borel,axiom,
    ( ( sigma_space_complex @ borel_1392132677378845456omplex )
    = top_top_set_complex ) ).

% space_borel
thf(fact_867_space__borel,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% space_borel
thf(fact_868_space__borel,axiom,
    ( ( sigma_space_nat @ borel_8449730974584783410el_nat )
    = top_top_set_nat ) ).

% space_borel
thf(fact_869_space__borel,axiom,
    ( ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal )
    = top_to7994903218803871134nnreal ) ).

% space_borel
thf(fact_870_space__borel,axiom,
    ( ( sigma_space_o @ borel_5500255247093592246orel_o )
    = top_top_set_o ) ).

% space_borel
thf(fact_871_space__bot,axiom,
    ( ( sigma_3147302497200244656nnreal @ bot_bo1740529460517930749nnreal )
    = bot_bo4854962954004695426nnreal ) ).

% space_bot
thf(fact_872_space__bot,axiom,
    ( ( sigma_space_real @ bot_bo5982154664989874033e_real )
    = bot_bot_set_real ) ).

% space_bot
thf(fact_873_space__bot,axiom,
    ( ( sigma_space_o @ bot_bo5758314138661044393sure_o )
    = bot_bot_set_o ) ).

% space_bot
thf(fact_874_space__bot,axiom,
    ( ( sigma_space_nat @ bot_bo6718502177978453909re_nat )
    = bot_bot_set_nat ) ).

% space_bot
thf(fact_875_funcsetI,axiom,
    ! [A3: set_real,F2: real > c,B4: set_c] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_c @ ( F2 @ X5 ) @ B4 ) )
     => ( member_real_c @ F2
        @ ( pi_real_c @ A3
          @ ^ [Uu: real] : B4 ) ) ) ).

% funcsetI
thf(fact_876_funcsetI,axiom,
    ! [A3: set_real,F2: real > b,B4: set_b] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_b @ ( F2 @ X5 ) @ B4 ) )
     => ( member_real_b @ F2
        @ ( pi_real_b @ A3
          @ ^ [Uu: real] : B4 ) ) ) ).

% funcsetI
thf(fact_877_funcsetI,axiom,
    ! [A3: set_real,F2: real > a,B4: set_a] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_a @ ( F2 @ X5 ) @ B4 ) )
     => ( member_real_a @ F2
        @ ( pi_real_a @ A3
          @ ^ [Uu: real] : B4 ) ) ) ).

% funcsetI
thf(fact_878_funcsetI,axiom,
    ! [A3: set_c,F2: c > b,B4: set_b] :
      ( ! [X5: c] :
          ( ( member_c @ X5 @ A3 )
         => ( member_b @ ( F2 @ X5 ) @ B4 ) )
     => ( member_c_b @ F2
        @ ( pi_c_b @ A3
          @ ^ [Uu: c] : B4 ) ) ) ).

% funcsetI
thf(fact_879_funcsetI,axiom,
    ! [A3: set_a,F2: a > b,B4: set_b] :
      ( ! [X5: a] :
          ( ( member_a @ X5 @ A3 )
         => ( member_b @ ( F2 @ X5 ) @ B4 ) )
     => ( member_a_b @ F2
        @ ( pi_a_b @ A3
          @ ^ [Uu: a] : B4 ) ) ) ).

% funcsetI
thf(fact_880_funcsetI,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > real > c,B4: set_real_c] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_real_c @ ( F2 @ X5 ) @ B4 ) )
     => ( member_real_c_real_c @ F2
        @ ( pi_real_c_real_c @ A3
          @ ^ [Uu: real > c] : B4 ) ) ) ).

% funcsetI
thf(fact_881_funcsetI,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > real > b,B4: set_real_b] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_real_b @ ( F2 @ X5 ) @ B4 ) )
     => ( member_real_c_real_b @ F2
        @ ( pi_real_c_real_b @ A3
          @ ^ [Uu: real > c] : B4 ) ) ) ).

% funcsetI
thf(fact_882_funcsetI,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > real > a,B4: set_real_a] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_real_a @ ( F2 @ X5 ) @ B4 ) )
     => ( member_real_c_real_a @ F2
        @ ( pi_real_c_real_a @ A3
          @ ^ [Uu: real > c] : B4 ) ) ) ).

% funcsetI
thf(fact_883_funcsetI,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > c > b,B4: set_c_b] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_c_b @ ( F2 @ X5 ) @ B4 ) )
     => ( member_real_c_c_b @ F2
        @ ( pi_real_c_c_b @ A3
          @ ^ [Uu: real > c] : B4 ) ) ) ).

% funcsetI
thf(fact_884_funcsetI,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > a > b,B4: set_a_b] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_a_b @ ( F2 @ X5 ) @ B4 ) )
     => ( member_real_c_a_b @ F2
        @ ( pi_real_c_a_b @ A3
          @ ^ [Uu: real > c] : B4 ) ) ) ).

% funcsetI
thf(fact_885_funcset__mem,axiom,
    ! [F2: real > c,A3: set_real,B4: set_c,X3: real] :
      ( ( member_real_c @ F2
        @ ( pi_real_c @ A3
          @ ^ [Uu: real] : B4 ) )
     => ( ( member_real @ X3 @ A3 )
       => ( member_c @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_886_funcset__mem,axiom,
    ! [F2: real > b,A3: set_real,B4: set_b,X3: real] :
      ( ( member_real_b @ F2
        @ ( pi_real_b @ A3
          @ ^ [Uu: real] : B4 ) )
     => ( ( member_real @ X3 @ A3 )
       => ( member_b @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_887_funcset__mem,axiom,
    ! [F2: real > a,A3: set_real,B4: set_a,X3: real] :
      ( ( member_real_a @ F2
        @ ( pi_real_a @ A3
          @ ^ [Uu: real] : B4 ) )
     => ( ( member_real @ X3 @ A3 )
       => ( member_a @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_888_funcset__mem,axiom,
    ! [F2: c > b,A3: set_c,B4: set_b,X3: c] :
      ( ( member_c_b @ F2
        @ ( pi_c_b @ A3
          @ ^ [Uu: c] : B4 ) )
     => ( ( member_c @ X3 @ A3 )
       => ( member_b @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_889_funcset__mem,axiom,
    ! [F2: a > b,A3: set_a,B4: set_b,X3: a] :
      ( ( member_a_b @ F2
        @ ( pi_a_b @ A3
          @ ^ [Uu: a] : B4 ) )
     => ( ( member_a @ X3 @ A3 )
       => ( member_b @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_890_funcset__mem,axiom,
    ! [F2: ( real > c ) > real > c,A3: set_real_c,B4: set_real_c,X3: real > c] :
      ( ( member_real_c_real_c @ F2
        @ ( pi_real_c_real_c @ A3
          @ ^ [Uu: real > c] : B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_real_c @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_891_funcset__mem,axiom,
    ! [F2: ( real > c ) > real > b,A3: set_real_c,B4: set_real_b,X3: real > c] :
      ( ( member_real_c_real_b @ F2
        @ ( pi_real_c_real_b @ A3
          @ ^ [Uu: real > c] : B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_real_b @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_892_funcset__mem,axiom,
    ! [F2: ( real > c ) > real > a,A3: set_real_c,B4: set_real_a,X3: real > c] :
      ( ( member_real_c_real_a @ F2
        @ ( pi_real_c_real_a @ A3
          @ ^ [Uu: real > c] : B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_real_a @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_893_funcset__mem,axiom,
    ! [F2: ( real > c ) > c > b,A3: set_real_c,B4: set_c_b,X3: real > c] :
      ( ( member_real_c_c_b @ F2
        @ ( pi_real_c_c_b @ A3
          @ ^ [Uu: real > c] : B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_c_b @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_894_funcset__mem,axiom,
    ! [F2: ( real > c ) > a > b,A3: set_real_c,B4: set_a_b,X3: real > c] :
      ( ( member_real_c_a_b @ F2
        @ ( pi_real_c_a_b @ A3
          @ ^ [Uu: real > c] : B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_a_b @ ( F2 @ X3 ) @ B4 ) ) ) ).

% funcset_mem
thf(fact_895_Pi__cong,axiom,
    ! [A3: set_real,F2: real > c,G2: real > c,B4: real > set_c] :
      ( ! [W2: real] :
          ( ( member_real @ W2 @ A3 )
         => ( ( F2 @ W2 )
            = ( G2 @ W2 ) ) )
     => ( ( member_real_c @ F2 @ ( pi_real_c @ A3 @ B4 ) )
        = ( member_real_c @ G2 @ ( pi_real_c @ A3 @ B4 ) ) ) ) ).

% Pi_cong
thf(fact_896_Pi__cong,axiom,
    ! [A3: set_real,F2: real > b,G2: real > b,B4: real > set_b] :
      ( ! [W2: real] :
          ( ( member_real @ W2 @ A3 )
         => ( ( F2 @ W2 )
            = ( G2 @ W2 ) ) )
     => ( ( member_real_b @ F2 @ ( pi_real_b @ A3 @ B4 ) )
        = ( member_real_b @ G2 @ ( pi_real_b @ A3 @ B4 ) ) ) ) ).

% Pi_cong
thf(fact_897_Pi__cong,axiom,
    ! [A3: set_real,F2: real > a,G2: real > a,B4: real > set_a] :
      ( ! [W2: real] :
          ( ( member_real @ W2 @ A3 )
         => ( ( F2 @ W2 )
            = ( G2 @ W2 ) ) )
     => ( ( member_real_a @ F2 @ ( pi_real_a @ A3 @ B4 ) )
        = ( member_real_a @ G2 @ ( pi_real_a @ A3 @ B4 ) ) ) ) ).

% Pi_cong
thf(fact_898_Pi__cong,axiom,
    ! [A3: set_c,F2: c > b,G2: c > b,B4: c > set_b] :
      ( ! [W2: c] :
          ( ( member_c @ W2 @ A3 )
         => ( ( F2 @ W2 )
            = ( G2 @ W2 ) ) )
     => ( ( member_c_b @ F2 @ ( pi_c_b @ A3 @ B4 ) )
        = ( member_c_b @ G2 @ ( pi_c_b @ A3 @ B4 ) ) ) ) ).

% Pi_cong
thf(fact_899_Pi__cong,axiom,
    ! [A3: set_a,F2: a > b,G2: a > b,B4: a > set_b] :
      ( ! [W2: a] :
          ( ( member_a @ W2 @ A3 )
         => ( ( F2 @ W2 )
            = ( G2 @ W2 ) ) )
     => ( ( member_a_b @ F2 @ ( pi_a_b @ A3 @ B4 ) )
        = ( member_a_b @ G2 @ ( pi_a_b @ A3 @ B4 ) ) ) ) ).

% Pi_cong
thf(fact_900_Pi__mem,axiom,
    ! [F2: real > c,A3: set_real,B4: real > set_c,X3: real] :
      ( ( member_real_c @ F2 @ ( pi_real_c @ A3 @ B4 ) )
     => ( ( member_real @ X3 @ A3 )
       => ( member_c @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_901_Pi__mem,axiom,
    ! [F2: real > b,A3: set_real,B4: real > set_b,X3: real] :
      ( ( member_real_b @ F2 @ ( pi_real_b @ A3 @ B4 ) )
     => ( ( member_real @ X3 @ A3 )
       => ( member_b @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_902_Pi__mem,axiom,
    ! [F2: real > a,A3: set_real,B4: real > set_a,X3: real] :
      ( ( member_real_a @ F2 @ ( pi_real_a @ A3 @ B4 ) )
     => ( ( member_real @ X3 @ A3 )
       => ( member_a @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_903_Pi__mem,axiom,
    ! [F2: c > b,A3: set_c,B4: c > set_b,X3: c] :
      ( ( member_c_b @ F2 @ ( pi_c_b @ A3 @ B4 ) )
     => ( ( member_c @ X3 @ A3 )
       => ( member_b @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_904_Pi__mem,axiom,
    ! [F2: a > b,A3: set_a,B4: a > set_b,X3: a] :
      ( ( member_a_b @ F2 @ ( pi_a_b @ A3 @ B4 ) )
     => ( ( member_a @ X3 @ A3 )
       => ( member_b @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_905_Pi__mem,axiom,
    ! [F2: ( real > c ) > real > c,A3: set_real_c,B4: ( real > c ) > set_real_c,X3: real > c] :
      ( ( member_real_c_real_c @ F2 @ ( pi_real_c_real_c @ A3 @ B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_real_c @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_906_Pi__mem,axiom,
    ! [F2: ( real > c ) > real > b,A3: set_real_c,B4: ( real > c ) > set_real_b,X3: real > c] :
      ( ( member_real_c_real_b @ F2 @ ( pi_real_c_real_b @ A3 @ B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_real_b @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_907_Pi__mem,axiom,
    ! [F2: ( real > c ) > real > a,A3: set_real_c,B4: ( real > c ) > set_real_a,X3: real > c] :
      ( ( member_real_c_real_a @ F2 @ ( pi_real_c_real_a @ A3 @ B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_real_a @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_908_Pi__mem,axiom,
    ! [F2: ( real > c ) > c > b,A3: set_real_c,B4: ( real > c ) > set_c_b,X3: real > c] :
      ( ( member_real_c_c_b @ F2 @ ( pi_real_c_c_b @ A3 @ B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_c_b @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_909_Pi__mem,axiom,
    ! [F2: ( real > c ) > a > b,A3: set_real_c,B4: ( real > c ) > set_a_b,X3: real > c] :
      ( ( member_real_c_a_b @ F2 @ ( pi_real_c_a_b @ A3 @ B4 ) )
     => ( ( member_real_c @ X3 @ A3 )
       => ( member_a_b @ ( F2 @ X3 ) @ ( B4 @ X3 ) ) ) ) ).

% Pi_mem
thf(fact_910_Pi__iff,axiom,
    ! [F2: real > c,I2: set_real,X2: real > set_c] :
      ( ( member_real_c @ F2 @ ( pi_real_c @ I2 @ X2 ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ I2 )
           => ( member_c @ ( F2 @ X ) @ ( X2 @ X ) ) ) ) ) ).

% Pi_iff
thf(fact_911_Pi__iff,axiom,
    ! [F2: real > b,I2: set_real,X2: real > set_b] :
      ( ( member_real_b @ F2 @ ( pi_real_b @ I2 @ X2 ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ I2 )
           => ( member_b @ ( F2 @ X ) @ ( X2 @ X ) ) ) ) ) ).

% Pi_iff
thf(fact_912_Pi__iff,axiom,
    ! [F2: real > a,I2: set_real,X2: real > set_a] :
      ( ( member_real_a @ F2 @ ( pi_real_a @ I2 @ X2 ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ I2 )
           => ( member_a @ ( F2 @ X ) @ ( X2 @ X ) ) ) ) ) ).

% Pi_iff
thf(fact_913_Pi__iff,axiom,
    ! [F2: c > b,I2: set_c,X2: c > set_b] :
      ( ( member_c_b @ F2 @ ( pi_c_b @ I2 @ X2 ) )
      = ( ! [X: c] :
            ( ( member_c @ X @ I2 )
           => ( member_b @ ( F2 @ X ) @ ( X2 @ X ) ) ) ) ) ).

% Pi_iff
thf(fact_914_Pi__iff,axiom,
    ! [F2: a > b,I2: set_a,X2: a > set_b] :
      ( ( member_a_b @ F2 @ ( pi_a_b @ I2 @ X2 ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ I2 )
           => ( member_b @ ( F2 @ X ) @ ( X2 @ X ) ) ) ) ) ).

% Pi_iff
thf(fact_915_Pi__I_H,axiom,
    ! [A3: set_real,F2: real > c,B4: real > set_c] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_c @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c @ F2 @ ( pi_real_c @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_916_Pi__I_H,axiom,
    ! [A3: set_real,F2: real > b,B4: real > set_b] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_b @ F2 @ ( pi_real_b @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_917_Pi__I_H,axiom,
    ! [A3: set_real,F2: real > a,B4: real > set_a] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_a @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_a @ F2 @ ( pi_real_a @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_918_Pi__I_H,axiom,
    ! [A3: set_c,F2: c > b,B4: c > set_b] :
      ( ! [X5: c] :
          ( ( member_c @ X5 @ A3 )
         => ( member_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_c_b @ F2 @ ( pi_c_b @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_919_Pi__I_H,axiom,
    ! [A3: set_a,F2: a > b,B4: a > set_b] :
      ( ! [X5: a] :
          ( ( member_a @ X5 @ A3 )
         => ( member_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_a_b @ F2 @ ( pi_a_b @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_920_Pi__I_H,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > real > c,B4: ( real > c ) > set_real_c] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_real_c @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_real_c @ F2 @ ( pi_real_c_real_c @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_921_Pi__I_H,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > real > b,B4: ( real > c ) > set_real_b] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_real_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_real_b @ F2 @ ( pi_real_c_real_b @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_922_Pi__I_H,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > real > a,B4: ( real > c ) > set_real_a] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_real_a @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_real_a @ F2 @ ( pi_real_c_real_a @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_923_Pi__I_H,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > c > b,B4: ( real > c ) > set_c_b] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_c_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_c_b @ F2 @ ( pi_real_c_c_b @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_924_Pi__I_H,axiom,
    ! [A3: set_real_c,F2: ( real > c ) > a > b,B4: ( real > c ) > set_a_b] :
      ( ! [X5: real > c] :
          ( ( member_real_c @ X5 @ A3 )
         => ( member_a_b @ ( F2 @ X5 ) @ ( B4 @ X5 ) ) )
     => ( member_real_c_a_b @ F2 @ ( pi_real_c_a_b @ A3 @ B4 ) ) ) ).

% Pi_I'
thf(fact_925_PiE,axiom,
    ! [F2: real > c,A3: set_real,B4: real > set_c,X3: real] :
      ( ( member_real_c @ F2 @ ( pi_real_c @ A3 @ B4 ) )
     => ( ~ ( member_c @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_real @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_926_PiE,axiom,
    ! [F2: real > b,A3: set_real,B4: real > set_b,X3: real] :
      ( ( member_real_b @ F2 @ ( pi_real_b @ A3 @ B4 ) )
     => ( ~ ( member_b @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_real @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_927_PiE,axiom,
    ! [F2: real > a,A3: set_real,B4: real > set_a,X3: real] :
      ( ( member_real_a @ F2 @ ( pi_real_a @ A3 @ B4 ) )
     => ( ~ ( member_a @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_real @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_928_PiE,axiom,
    ! [F2: c > b,A3: set_c,B4: c > set_b,X3: c] :
      ( ( member_c_b @ F2 @ ( pi_c_b @ A3 @ B4 ) )
     => ( ~ ( member_b @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_c @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_929_PiE,axiom,
    ! [F2: a > b,A3: set_a,B4: a > set_b,X3: a] :
      ( ( member_a_b @ F2 @ ( pi_a_b @ A3 @ B4 ) )
     => ( ~ ( member_b @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_a @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_930_PiE,axiom,
    ! [F2: ( real > c ) > real > c,A3: set_real_c,B4: ( real > c ) > set_real_c,X3: real > c] :
      ( ( member_real_c_real_c @ F2 @ ( pi_real_c_real_c @ A3 @ B4 ) )
     => ( ~ ( member_real_c @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_real_c @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_931_PiE,axiom,
    ! [F2: ( real > b ) > real > c,A3: set_real_b,B4: ( real > b ) > set_real_c,X3: real > b] :
      ( ( member_real_b_real_c @ F2 @ ( pi_real_b_real_c @ A3 @ B4 ) )
     => ( ~ ( member_real_c @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_real_b @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_932_PiE,axiom,
    ! [F2: ( real > a ) > real > c,A3: set_real_a,B4: ( real > a ) > set_real_c,X3: real > a] :
      ( ( member_real_a_real_c @ F2 @ ( pi_real_a_real_c @ A3 @ B4 ) )
     => ( ~ ( member_real_c @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_real_a @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_933_PiE,axiom,
    ! [F2: ( c > b ) > real > c,A3: set_c_b,B4: ( c > b ) > set_real_c,X3: c > b] :
      ( ( member_c_b_real_c @ F2 @ ( pi_c_b_real_c @ A3 @ B4 ) )
     => ( ~ ( member_real_c @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_c_b @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_934_PiE,axiom,
    ! [F2: ( a > b ) > real > c,A3: set_a_b,B4: ( a > b ) > set_real_c,X3: a > b] :
      ( ( member_a_b_real_c @ F2 @ ( pi_a_b_real_c @ A3 @ B4 ) )
     => ( ~ ( member_real_c @ ( F2 @ X3 ) @ ( B4 @ X3 ) )
       => ~ ( member_a_b @ X3 @ A3 ) ) ) ).

% PiE
thf(fact_935_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_real,M2: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M2 ) )
     => ( ( sigma_space_real @ M )
        = ( sigma_space_real @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_936_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_nat,M2: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ M2 ) )
     => ( ( sigma_space_nat @ M )
        = ( sigma_space_nat @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_937_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( sigma_5465916536984168985nnreal @ M2 ) )
     => ( ( sigma_3147302497200244656nnreal @ M )
        = ( sigma_3147302497200244656nnreal @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_938_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_o,M2: sigma_measure_o] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ M2 ) )
     => ( ( sigma_space_o @ M )
        = ( sigma_space_o @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_939_o__case__sum,axiom,
    ! [H: real > real,F2: real > real,G2: real > real] :
      ( ( comp_r5151246396438109300l_real @ H @ ( sum_ca8732840427581260704l_real @ F2 @ G2 ) )
      = ( sum_ca8732840427581260704l_real @ ( comp_real_real_real @ H @ F2 ) @ ( comp_real_real_real @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_940_o__case__sum,axiom,
    ! [H: real > $o,F2: $o > real,G2: $o > real] :
      ( ( comp_r962287379166727918um_o_o @ H @ ( sum_ca5525272764133257196real_o @ F2 @ G2 ) )
      = ( sum_case_sum_o_o_o @ ( comp_real_o_o @ H @ F2 ) @ ( comp_real_o_o @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_941_o__case__sum,axiom,
    ! [H: real > nat,F2: nat > real,G2: nat > real] :
      ( ( comp_r6376950801160420448at_nat @ H @ ( sum_ca8334624595930125032al_nat @ F2 @ G2 ) )
      = ( sum_ca6763686470577984908at_nat @ ( comp_real_nat_nat @ H @ F2 ) @ ( comp_real_nat_nat @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_942_o__case__sum,axiom,
    ! [H: c > b,F2: real > c,G2: real > c] :
      ( ( comp_c4637849929068767129l_real @ H @ ( sum_ca5660074461753380326c_real @ F2 @ G2 ) )
      = ( sum_ca63855846565249637b_real @ ( comp_c_b_real @ H @ F2 ) @ ( comp_c_b_real @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_943_o__case__sum,axiom,
    ! [H: a > b,F2: real > a,G2: real > a] :
      ( ( comp_a5455185540716242459l_real @ H @ ( sum_ca3691009268231894756a_real @ F2 @ G2 ) )
      = ( sum_ca63855846565249637b_real @ ( comp_a_b_real @ H @ F2 ) @ ( comp_a_b_real @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_944_o__case__sum,axiom,
    ! [H: a > b,F2: real > a,G2: c > a] :
      ( ( comp_a4970644210815063015real_c @ H @ ( sum_ca5000516552359814344al_a_c @ F2 @ G2 ) )
      = ( sum_ca2213060970387138311al_b_c @ ( comp_a_b_real @ H @ F2 ) @ ( comp_a_b_c @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_945_o__case__sum,axiom,
    ! [H: a > b,F2: real > a,G2: a > a] :
      ( ( comp_a4970644202208605413real_a @ H @ ( sum_ca5000516552359814342al_a_a @ F2 @ G2 ) )
      = ( sum_ca2213060970387138309al_b_a @ ( comp_a_b_real @ H @ F2 ) @ ( comp_a_b_a @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_946_o__case__sum,axiom,
    ! [H: b > b,F2: a > b,G2: a > b] :
      ( ( comp_b_b_Sum_sum_a_a @ H @ ( sum_case_sum_a_b_a @ F2 @ G2 ) )
      = ( sum_case_sum_a_b_a @ ( comp_b_b_a @ H @ F2 ) @ ( comp_b_b_a @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_947_o__case__sum,axiom,
    ! [H: a > c,F2: real > a,G2: real > a] :
      ( ( comp_a2820443819341408156l_real @ H @ ( sum_ca3691009268231894756a_real @ F2 @ G2 ) )
      = ( sum_ca5660074461753380326c_real @ ( comp_a_c_real @ H @ F2 ) @ ( comp_a_c_real @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_948_o__case__sum,axiom,
    ! [H: a > b,F2: c > a,G2: real > a] :
      ( ( comp_a6422442893462753785c_real @ H @ ( sum_ca2956635407921249476a_real @ F2 @ G2 ) )
      = ( sum_ca8552854023109380165b_real @ ( comp_a_b_c @ H @ F2 ) @ ( comp_a_b_real @ H @ G2 ) ) ) ).

% o_case_sum
thf(fact_949_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_real_c,P: $o] :
      ( member_set_real_c
      @ ( collect_real_c
        @ ^ [X: real > c] :
            ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
            & P ) )
      @ ( sigma_sets_real_c @ M ) ) ).

% sets.sets_Collect_const
thf(fact_950_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_real_b,P: $o] :
      ( member_set_real_b
      @ ( collect_real_b
        @ ^ [X: real > b] :
            ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
            & P ) )
      @ ( sigma_sets_real_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_951_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_real_a,P: $o] :
      ( member_set_real_a
      @ ( collect_real_a
        @ ^ [X: real > a] :
            ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
            & P ) )
      @ ( sigma_sets_real_a @ M ) ) ).

% sets.sets_Collect_const
thf(fact_952_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_c_b,P: $o] :
      ( member_set_c_b
      @ ( collect_c_b
        @ ^ [X: c > b] :
            ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
            & P ) )
      @ ( sigma_sets_c_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_953_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_a_b,P: $o] :
      ( member_set_a_b
      @ ( collect_a_b
        @ ^ [X: a > b] :
            ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
            & P ) )
      @ ( sigma_sets_a_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_954_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_real,P: $o] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ( sigma_space_real @ M ) )
            & P ) )
      @ ( sigma_sets_real @ M ) ) ).

% sets.sets_Collect_const
thf(fact_955_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_nat,P: $o] :
      ( member_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
            & P ) )
      @ ( sigma_sets_nat @ M ) ) ).

% sets.sets_Collect_const
thf(fact_956_sets_Osets__Collect__const,axiom,
    ! [M: sigma_7234349610311085201nnreal,P: $o] :
      ( member603777416030116741nnreal
      @ ( collec6648975593938027277nnreal
        @ ^ [X: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
            & P ) )
      @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.sets_Collect_const
thf(fact_957_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_o,P: $o] :
      ( member_set_o
      @ ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( sigma_space_o @ M ) )
            & P ) )
      @ ( sigma_sets_o @ M ) ) ).

% sets.sets_Collect_const
thf(fact_958_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_real_c,P: ( real > c ) > $o,Q: ( real > c ) > $o] :
      ( ( member_set_real_c
        @ ( collect_real_c
          @ ^ [X: real > c] :
              ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_c @ M ) )
     => ( ( member_set_real_c
          @ ( collect_real_c
            @ ^ [X: real > c] :
                ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_c @ M ) )
       => ( member_set_real_c
          @ ( collect_real_c
            @ ^ [X: real > c] :
                ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_real_c @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_959_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_real_b,P: ( real > b ) > $o,Q: ( real > b ) > $o] :
      ( ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X: real > b] :
              ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_b @ M ) )
     => ( ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X: real > b] :
                ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_b @ M ) )
       => ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X: real > b] :
                ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_real_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_960_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_real_a,P: ( real > a ) > $o,Q: ( real > a ) > $o] :
      ( ( member_set_real_a
        @ ( collect_real_a
          @ ^ [X: real > a] :
              ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_a @ M ) )
     => ( ( member_set_real_a
          @ ( collect_real_a
            @ ^ [X: real > a] :
                ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_a @ M ) )
       => ( member_set_real_a
          @ ( collect_real_a
            @ ^ [X: real > a] :
                ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_real_a @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_961_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_c_b,P: ( c > b ) > $o,Q: ( c > b ) > $o] :
      ( ( member_set_c_b
        @ ( collect_c_b
          @ ^ [X: c > b] :
              ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_c_b @ M ) )
     => ( ( member_set_c_b
          @ ( collect_c_b
            @ ^ [X: c > b] :
                ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_c_b @ M ) )
       => ( member_set_c_b
          @ ( collect_c_b
            @ ^ [X: c > b] :
                ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_c_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_962_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_a_b,P: ( a > b ) > $o,Q: ( a > b ) > $o] :
      ( ( member_set_a_b
        @ ( collect_a_b
          @ ^ [X: a > b] :
              ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_a_b @ M ) )
     => ( ( member_set_a_b
          @ ( collect_a_b
            @ ^ [X: a > b] :
                ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_a_b @ M ) )
       => ( member_set_a_b
          @ ( collect_a_b
            @ ^ [X: a > b] :
                ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_a_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_963_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real @ M ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_964_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
      ( ( member_set_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_965_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_7234349610311085201nnreal,P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
      ( ( member603777416030116741nnreal
        @ ( collec6648975593938027277nnreal
          @ ^ [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( member603777416030116741nnreal
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_966_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
      ( ( member_set_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ ( sigma_space_o @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o
          @ ( collect_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ ( sigma_space_o @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_o @ M ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ ( sigma_space_o @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_967_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_real_c,P: ( real > c ) > $o,Q: ( real > c ) > $o] :
      ( ( member_set_real_c
        @ ( collect_real_c
          @ ^ [X: real > c] :
              ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_c @ M ) )
     => ( ( member_set_real_c
          @ ( collect_real_c
            @ ^ [X: real > c] :
                ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_c @ M ) )
       => ( member_set_real_c
          @ ( collect_real_c
            @ ^ [X: real > c] :
                ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_real_c @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_968_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_real_b,P: ( real > b ) > $o,Q: ( real > b ) > $o] :
      ( ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X: real > b] :
              ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_b @ M ) )
     => ( ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X: real > b] :
                ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_b @ M ) )
       => ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X: real > b] :
                ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_real_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_969_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_real_a,P: ( real > a ) > $o,Q: ( real > a ) > $o] :
      ( ( member_set_real_a
        @ ( collect_real_a
          @ ^ [X: real > a] :
              ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_a @ M ) )
     => ( ( member_set_real_a
          @ ( collect_real_a
            @ ^ [X: real > a] :
                ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_a @ M ) )
       => ( member_set_real_a
          @ ( collect_real_a
            @ ^ [X: real > a] :
                ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_real_a @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_970_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_c_b,P: ( c > b ) > $o,Q: ( c > b ) > $o] :
      ( ( member_set_c_b
        @ ( collect_c_b
          @ ^ [X: c > b] :
              ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_c_b @ M ) )
     => ( ( member_set_c_b
          @ ( collect_c_b
            @ ^ [X: c > b] :
                ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_c_b @ M ) )
       => ( member_set_c_b
          @ ( collect_c_b
            @ ^ [X: c > b] :
                ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_c_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_971_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_a_b,P: ( a > b ) > $o,Q: ( a > b ) > $o] :
      ( ( member_set_a_b
        @ ( collect_a_b
          @ ^ [X: a > b] :
              ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_a_b @ M ) )
     => ( ( member_set_a_b
          @ ( collect_a_b
            @ ^ [X: a > b] :
                ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_a_b @ M ) )
       => ( member_set_a_b
          @ ( collect_a_b
            @ ^ [X: a > b] :
                ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_a_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_972_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real @ M ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_973_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
      ( ( member_set_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_974_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_7234349610311085201nnreal,P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
      ( ( member603777416030116741nnreal
        @ ( collec6648975593938027277nnreal
          @ ^ [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( member603777416030116741nnreal
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_975_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
      ( ( member_set_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ ( sigma_space_o @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o
          @ ( collect_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ ( sigma_space_o @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_o @ M ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ ( sigma_space_o @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_976_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_real_c,P: ( real > c ) > $o] :
      ( ( member_set_real_c
        @ ( collect_real_c
          @ ^ [X: real > c] :
              ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_c @ M ) )
     => ( member_set_real_c
        @ ( collect_real_c
          @ ^ [X: real > c] :
              ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_real_c @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_977_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_real_b,P: ( real > b ) > $o] :
      ( ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X: real > b] :
              ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_b @ M ) )
     => ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X: real > b] :
              ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_real_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_978_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_real_a,P: ( real > a ) > $o] :
      ( ( member_set_real_a
        @ ( collect_real_a
          @ ^ [X: real > a] :
              ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_a @ M ) )
     => ( member_set_real_a
        @ ( collect_real_a
          @ ^ [X: real > a] :
              ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_real_a @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_979_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_c_b,P: ( c > b ) > $o] :
      ( ( member_set_c_b
        @ ( collect_c_b
          @ ^ [X: c > b] :
              ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_c_b @ M ) )
     => ( member_set_c_b
        @ ( collect_c_b
          @ ^ [X: c > b] :
              ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_c_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_980_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_a_b,P: ( a > b ) > $o] :
      ( ( member_set_a_b
        @ ( collect_a_b
          @ ^ [X: a > b] :
              ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_a_b @ M ) )
     => ( member_set_a_b
        @ ( collect_a_b
          @ ^ [X: a > b] :
              ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_a_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_981_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_real,P: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_982_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_nat,P: nat > $o] :
      ( ( member_set_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_nat @ M ) )
     => ( member_set_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_nat @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_983_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_7234349610311085201nnreal,P: extend8495563244428889912nnreal > $o] :
      ( ( member603777416030116741nnreal
        @ ( collec6648975593938027277nnreal
          @ ^ [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( member603777416030116741nnreal
        @ ( collec6648975593938027277nnreal
          @ ^ [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_984_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_o,P: $o > $o] :
      ( ( member_set_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ ( sigma_space_o @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_o @ M ) )
     => ( member_set_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ ( sigma_space_o @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_o @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_985_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_real_c,P: ( real > c ) > $o,Q: ( real > c ) > $o] :
      ( ( member_set_real_c
        @ ( collect_real_c
          @ ^ [X: real > c] :
              ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_c @ M ) )
     => ( ( member_set_real_c
          @ ( collect_real_c
            @ ^ [X: real > c] :
                ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_c @ M ) )
       => ( member_set_real_c
          @ ( collect_real_c
            @ ^ [X: real > c] :
                ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_real_c @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_986_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_real_b,P: ( real > b ) > $o,Q: ( real > b ) > $o] :
      ( ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X: real > b] :
              ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_b @ M ) )
     => ( ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X: real > b] :
                ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_b @ M ) )
       => ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X: real > b] :
                ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_real_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_987_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_real_a,P: ( real > a ) > $o,Q: ( real > a ) > $o] :
      ( ( member_set_real_a
        @ ( collect_real_a
          @ ^ [X: real > a] :
              ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_a @ M ) )
     => ( ( member_set_real_a
          @ ( collect_real_a
            @ ^ [X: real > a] :
                ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_a @ M ) )
       => ( member_set_real_a
          @ ( collect_real_a
            @ ^ [X: real > a] :
                ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_real_a @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_988_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_c_b,P: ( c > b ) > $o,Q: ( c > b ) > $o] :
      ( ( member_set_c_b
        @ ( collect_c_b
          @ ^ [X: c > b] :
              ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_c_b @ M ) )
     => ( ( member_set_c_b
          @ ( collect_c_b
            @ ^ [X: c > b] :
                ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_c_b @ M ) )
       => ( member_set_c_b
          @ ( collect_c_b
            @ ^ [X: c > b] :
                ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_c_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_989_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_a_b,P: ( a > b ) > $o,Q: ( a > b ) > $o] :
      ( ( member_set_a_b
        @ ( collect_a_b
          @ ^ [X: a > b] :
              ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_a_b @ M ) )
     => ( ( member_set_a_b
          @ ( collect_a_b
            @ ^ [X: a > b] :
                ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_a_b @ M ) )
       => ( member_set_a_b
          @ ( collect_a_b
            @ ^ [X: a > b] :
                ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_a_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_990_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real @ M ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_991_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
      ( ( member_set_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_992_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_7234349610311085201nnreal,P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
      ( ( member603777416030116741nnreal
        @ ( collec6648975593938027277nnreal
          @ ^ [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( member603777416030116741nnreal
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_993_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
      ( ( member_set_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ ( sigma_space_o @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o
          @ ( collect_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ ( sigma_space_o @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_o @ M ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ ( sigma_space_o @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_994_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_real_c,Pb: $o] :
      ( member_set_real_c
      @ ( collect_real_c
        @ ^ [X: real > c] :
            ( ( member_real_c @ X @ ( sigma_space_real_c @ M ) )
            & Pb ) )
      @ ( sigma_sets_real_c @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_995_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_real_b,Pb: $o] :
      ( member_set_real_b
      @ ( collect_real_b
        @ ^ [X: real > b] :
            ( ( member_real_b @ X @ ( sigma_space_real_b @ M ) )
            & Pb ) )
      @ ( sigma_sets_real_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_996_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_real_a,Pb: $o] :
      ( member_set_real_a
      @ ( collect_real_a
        @ ^ [X: real > a] :
            ( ( member_real_a @ X @ ( sigma_space_real_a @ M ) )
            & Pb ) )
      @ ( sigma_sets_real_a @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_997_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_c_b,Pb: $o] :
      ( member_set_c_b
      @ ( collect_c_b
        @ ^ [X: c > b] :
            ( ( member_c_b @ X @ ( sigma_space_c_b @ M ) )
            & Pb ) )
      @ ( sigma_sets_c_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_998_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_a_b,Pb: $o] :
      ( member_set_a_b
      @ ( collect_a_b
        @ ^ [X: a > b] :
            ( ( member_a_b @ X @ ( sigma_space_a_b @ M ) )
            & Pb ) )
      @ ( sigma_sets_a_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_999_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_real,Pb: $o] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ( sigma_space_real @ M ) )
            & Pb ) )
      @ ( sigma_sets_real @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1000_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_nat,Pb: $o] :
      ( member_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
            & Pb ) )
      @ ( sigma_sets_nat @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1001_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_7234349610311085201nnreal,Pb: $o] :
      ( member603777416030116741nnreal
      @ ( collec6648975593938027277nnreal
        @ ^ [X: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
            & Pb ) )
      @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1002_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_o,Pb: $o] :
      ( member_set_o
      @ ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( sigma_space_o @ M ) )
            & Pb ) )
      @ ( sigma_sets_o @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1003_funcset__to__empty__iff,axiom,
    ! [A3: set_Ex3793607809372303086nnreal] :
      ( ( ( A3 = bot_bo4854962954004695426nnreal )
       => ( ( pi_Ext6719389991241943869nnreal @ A3
            @ ^ [Uu: extend8495563244428889912nnreal] : bot_bo4854962954004695426nnreal )
          = top_to7542815285589172853nnreal ) )
      & ( ( A3 != bot_bo4854962954004695426nnreal )
       => ( ( pi_Ext6719389991241943869nnreal @ A3
            @ ^ [Uu: extend8495563244428889912nnreal] : bot_bo4854962954004695426nnreal )
          = bot_bo7180108423516133465nnreal ) ) ) ).

% funcset_to_empty_iff
thf(fact_1004_funcset__to__empty__iff,axiom,
    ! [A3: set_real] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( pi_rea7198910874028739761nnreal @ A3
            @ ^ [Uu: real] : bot_bo4854962954004695426nnreal )
          = top_to315565310957491945nnreal ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( pi_rea7198910874028739761nnreal @ A3
            @ ^ [Uu: real] : bot_bo4854962954004695426nnreal )
          = bot_bo6037503491064675021nnreal ) ) ) ).

% funcset_to_empty_iff
thf(fact_1005_funcset__to__empty__iff,axiom,
    ! [A3: set_o] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( pi_o_E3657889280065143877nnreal @ A3
            @ ^ [Uu: $o] : bot_bo4854962954004695426nnreal )
          = top_to8359376978561864305nnreal ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( pi_o_E3657889280065143877nnreal @ A3
            @ ^ [Uu: $o] : bot_bo4854962954004695426nnreal )
          = bot_bo8089301180103932813nnreal ) ) ) ).

% funcset_to_empty_iff
thf(fact_1006_funcset__to__empty__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( pi_nat6594671786139111381nnreal @ A3
            @ ^ [Uu: nat] : bot_bo4854962954004695426nnreal )
          = top_to1572151773405874957nnreal ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( pi_nat6594671786139111381nnreal @ A3
            @ ^ [Uu: nat] : bot_bo4854962954004695426nnreal )
          = bot_bo2612803927560216817nnreal ) ) ) ).

% funcset_to_empty_iff
thf(fact_1007_funcset__to__empty__iff,axiom,
    ! [A3: set_Ex3793607809372303086nnreal] :
      ( ( ( A3 = bot_bo4854962954004695426nnreal )
       => ( ( pi_Ext5231164604578195505l_real @ A3
            @ ^ [Uu: extend8495563244428889912nnreal] : bot_bot_set_real )
          = top_to645609954874935913l_real ) )
      & ( ( A3 != bot_bo4854962954004695426nnreal )
       => ( ( pi_Ext5231164604578195505l_real @ A3
            @ ^ [Uu: extend8495563244428889912nnreal] : bot_bot_set_real )
          = bot_bo6367548134982118989l_real ) ) ) ).

% funcset_to_empty_iff
thf(fact_1008_funcset__to__empty__iff,axiom,
    ! [A3: set_real] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( pi_real_real @ A3
            @ ^ [Uu: real] : bot_bot_set_real )
          = top_to2071711978144146653l_real ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( pi_real_real @ A3
            @ ^ [Uu: real] : bot_bot_set_real )
          = bot_bo6767488733719836353l_real ) ) ) ).

% funcset_to_empty_iff
thf(fact_1009_funcset__to__empty__iff,axiom,
    ! [A3: set_o] :
      ( ( ( A3 = bot_bot_set_o )
       => ( ( pi_o_real @ A3
            @ ^ [Uu: $o] : bot_bot_set_real )
          = top_top_set_o_real ) )
      & ( ( A3 != bot_bot_set_o )
       => ( ( pi_o_real @ A3
            @ ^ [Uu: $o] : bot_bot_set_real )
          = bot_bot_set_o_real ) ) ) ).

% funcset_to_empty_iff
thf(fact_1010_funcset__to__empty__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( pi_nat_real @ A3
            @ ^ [Uu: nat] : bot_bot_set_real )
          = top_top_set_nat_real ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( pi_nat_real @ A3
            @ ^ [Uu: nat] : bot_bot_set_real )
          = bot_bot_set_nat_real ) ) ) ).

% funcset_to_empty_iff
thf(fact_1011_funcset__to__empty__iff,axiom,
    ! [A3: set_Ex3793607809372303086nnreal] :
      ( ( ( A3 = bot_bo4854962954004695426nnreal )
       => ( ( pi_Ext3478096141635148883real_o @ A3
            @ ^ [Uu: extend8495563244428889912nnreal] : bot_bot_set_o )
          = top_to4606021423082044487real_o ) )
      & ( ( A3 != bot_bo4854962954004695426nnreal )
       => ( ( pi_Ext3478096141635148883real_o @ A3
            @ ^ [Uu: extend8495563244428889912nnreal] : bot_bot_set_o )
          = bot_bo4335945624624112995real_o ) ) ) ).

% funcset_to_empty_iff
thf(fact_1012_funcset__to__empty__iff,axiom,
    ! [A3: set_real] :
      ( ( ( A3 = bot_bot_set_real )
       => ( ( pi_real_o @ A3
            @ ^ [Uu: real] : bot_bot_set_o )
          = top_top_set_real_o ) )
      & ( ( A3 != bot_bot_set_real )
       => ( ( pi_real_o @ A3
            @ ^ [Uu: real] : bot_bot_set_o )
          = bot_bot_set_real_o ) ) ) ).

% funcset_to_empty_iff
thf(fact_1013_qbs__morphismE_I1_J,axiom,
    ! [F2: real > c,X2: quasi_borel_real,Y: quasi_borel_c] :
      ( ( member_real_c @ F2 @ ( qbs_morphism_real_c @ X2 @ Y ) )
     => ( member_real_c @ F2
        @ ( pi_real_c @ ( qbs_space_real @ X2 )
          @ ^ [Uu: real] : ( qbs_space_c @ Y ) ) ) ) ).

% qbs_morphismE(1)
thf(fact_1014_qbs__morphismE_I1_J,axiom,
    ! [F2: real > b,X2: quasi_borel_real,Y: quasi_borel_b] :
      ( ( member_real_b @ F2 @ ( qbs_morphism_real_b @ X2 @ Y ) )
     => ( member_real_b @ F2
        @ ( pi_real_b @ ( qbs_space_real @ X2 )
          @ ^ [Uu: real] : ( qbs_space_b @ Y ) ) ) ) ).

% qbs_morphismE(1)
thf(fact_1015_qbs__morphismE_I1_J,axiom,
    ! [F2: real > a,X2: quasi_borel_real,Y: quasi_borel_a] :
      ( ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X2 @ Y ) )
     => ( member_real_a @ F2
        @ ( pi_real_a @ ( qbs_space_real @ X2 )
          @ ^ [Uu: real] : ( qbs_space_a @ Y ) ) ) ) ).

% qbs_morphismE(1)
thf(fact_1016_qbs__morphismE_I1_J,axiom,
    ! [F2: c > b,X2: quasi_borel_c,Y: quasi_borel_b] :
      ( ( member_c_b @ F2 @ ( qbs_morphism_c_b @ X2 @ Y ) )
     => ( member_c_b @ F2
        @ ( pi_c_b @ ( qbs_space_c @ X2 )
          @ ^ [Uu: c] : ( qbs_space_b @ Y ) ) ) ) ).

% qbs_morphismE(1)
thf(fact_1017_qbs__morphismE_I1_J,axiom,
    ! [F2: a > b,X2: quasi_borel_a,Y: quasi_borel_b] :
      ( ( member_a_b @ F2 @ ( qbs_morphism_a_b @ X2 @ Y ) )
     => ( member_a_b @ F2
        @ ( pi_a_b @ ( qbs_space_a @ X2 )
          @ ^ [Uu: a] : ( qbs_space_b @ Y ) ) ) ) ).

% qbs_morphismE(1)
thf(fact_1018_space__empty__eq__bot,axiom,
    ! [A: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ A )
        = bot_bo4854962954004695426nnreal )
      = ( A = bot_bo1740529460517930749nnreal ) ) ).

% space_empty_eq_bot
thf(fact_1019_space__empty__eq__bot,axiom,
    ! [A: sigma_measure_real] :
      ( ( ( sigma_space_real @ A )
        = bot_bot_set_real )
      = ( A = bot_bo5982154664989874033e_real ) ) ).

% space_empty_eq_bot
thf(fact_1020_space__empty__eq__bot,axiom,
    ! [A: sigma_measure_o] :
      ( ( ( sigma_space_o @ A )
        = bot_bot_set_o )
      = ( A = bot_bo5758314138661044393sure_o ) ) ).

% space_empty_eq_bot
thf(fact_1021_space__empty__eq__bot,axiom,
    ! [A: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ A )
        = bot_bot_set_nat )
      = ( A = bot_bo6718502177978453909re_nat ) ) ).

% space_empty_eq_bot
thf(fact_1022_qbs__Mx__to__X_I1_J,axiom,
    ! [Alpha2: real > b,X2: quasi_borel_b] :
      ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X2 ) )
     => ( member_real_b @ Alpha2
        @ ( pi_real_b @ top_top_set_real
          @ ^ [Uu: real] : ( qbs_space_b @ X2 ) ) ) ) ).

% qbs_Mx_to_X(1)
thf(fact_1023_qbs__Mx__to__X_I1_J,axiom,
    ! [Alpha2: real > a,X2: quasi_borel_a] :
      ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X2 ) )
     => ( member_real_a @ Alpha2
        @ ( pi_real_a @ top_top_set_real
          @ ^ [Uu: real] : ( qbs_space_a @ X2 ) ) ) ) ).

% qbs_Mx_to_X(1)
thf(fact_1024_qbs__Mx__to__X_I1_J,axiom,
    ! [Alpha2: real > c,X2: quasi_borel_c] :
      ( ( member_real_c @ Alpha2 @ ( qbs_Mx_c @ X2 ) )
     => ( member_real_c @ Alpha2
        @ ( pi_real_c @ top_top_set_real
          @ ^ [Uu: real] : ( qbs_space_c @ X2 ) ) ) ) ).

% qbs_Mx_to_X(1)
thf(fact_1025_real_Osingleton__sets,axiom,
    ! [X3: real] :
      ( ( member_real @ X3 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( member_set_real @ ( insert_real @ X3 @ bot_bot_set_real ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% real.singleton_sets
thf(fact_1026_real_Ospace__UNIV,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% real.space_UNIV
thf(fact_1027_sub__qbs__closed_I1_J,axiom,
    ! [X2: quasi_borel_b,U2: set_b] :
      ( qbs_closed1_b
      @ ( collect_real_b
        @ ^ [F: real > b] :
            ( ( member_real_b @ F
              @ ( pi_real_b @ top_top_set_real
                @ ^ [Uu: real] : ( inf_inf_set_b @ ( qbs_space_b @ X2 ) @ U2 ) ) )
            & ( member_real_b @ F @ ( qbs_Mx_b @ X2 ) ) ) ) ) ).

% sub_qbs_closed(1)
thf(fact_1028_sub__qbs__closed_I1_J,axiom,
    ! [X2: quasi_borel_a,U2: set_a] :
      ( qbs_closed1_a
      @ ( collect_real_a
        @ ^ [F: real > a] :
            ( ( member_real_a @ F
              @ ( pi_real_a @ top_top_set_real
                @ ^ [Uu: real] : ( inf_inf_set_a @ ( qbs_space_a @ X2 ) @ U2 ) ) )
            & ( member_real_a @ F @ ( qbs_Mx_a @ X2 ) ) ) ) ) ).

% sub_qbs_closed(1)
thf(fact_1029_sub__qbs__closed_I1_J,axiom,
    ! [X2: quasi_borel_c,U2: set_c] :
      ( qbs_closed1_c
      @ ( collect_real_c
        @ ^ [F: real > c] :
            ( ( member_real_c @ F
              @ ( pi_real_c @ top_top_set_real
                @ ^ [Uu: real] : ( inf_inf_set_c @ ( qbs_space_c @ X2 ) @ U2 ) ) )
            & ( member_real_c @ F @ ( qbs_Mx_c @ X2 ) ) ) ) ) ).

% sub_qbs_closed(1)
thf(fact_1030_sub__qbs__closed_I3_J,axiom,
    ! [X2: quasi_borel_b,U2: set_b] :
      ( qbs_closed3_b
      @ ( collect_real_b
        @ ^ [F: real > b] :
            ( ( member_real_b @ F
              @ ( pi_real_b @ top_top_set_real
                @ ^ [Uu: real] : ( inf_inf_set_b @ ( qbs_space_b @ X2 ) @ U2 ) ) )
            & ( member_real_b @ F @ ( qbs_Mx_b @ X2 ) ) ) ) ) ).

% sub_qbs_closed(3)
thf(fact_1031_sub__qbs__closed_I3_J,axiom,
    ! [X2: quasi_borel_a,U2: set_a] :
      ( qbs_closed3_a
      @ ( collect_real_a
        @ ^ [F: real > a] :
            ( ( member_real_a @ F
              @ ( pi_real_a @ top_top_set_real
                @ ^ [Uu: real] : ( inf_inf_set_a @ ( qbs_space_a @ X2 ) @ U2 ) ) )
            & ( member_real_a @ F @ ( qbs_Mx_a @ X2 ) ) ) ) ) ).

% sub_qbs_closed(3)
thf(fact_1032_sub__qbs__closed_I3_J,axiom,
    ! [X2: quasi_borel_c,U2: set_c] :
      ( qbs_closed3_c
      @ ( collect_real_c
        @ ^ [F: real > c] :
            ( ( member_real_c @ F
              @ ( pi_real_c @ top_top_set_real
                @ ^ [Uu: real] : ( inf_inf_set_c @ ( qbs_space_c @ X2 ) @ U2 ) ) )
            & ( member_real_c @ F @ ( qbs_Mx_c @ X2 ) ) ) ) ) ).

% sub_qbs_closed(3)
thf(fact_1033_space__sup__measure_H,axiom,
    ! [B4: sigma_measure_real,A3: sigma_measure_real] :
      ( ( ( sigma_sets_real @ B4 )
        = ( sigma_sets_real @ A3 ) )
     => ( ( sigma_space_real @ ( measur2147279183506585690e_real @ A3 @ B4 ) )
        = ( sigma_space_real @ A3 ) ) ) ).

% space_sup_measure'
thf(fact_1034_space__sup__measure_H,axiom,
    ! [B4: sigma_measure_nat,A3: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ B4 )
        = ( sigma_sets_nat @ A3 ) )
     => ( ( sigma_space_nat @ ( measur876423496291765374re_nat @ A3 @ B4 ) )
        = ( sigma_space_nat @ A3 ) ) ) ).

% space_sup_measure'
thf(fact_1035_space__sup__measure_H,axiom,
    ! [B4: sigma_7234349610311085201nnreal,A3: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ B4 )
        = ( sigma_5465916536984168985nnreal @ A3 ) )
     => ( ( sigma_3147302497200244656nnreal @ ( measur4473656680840910822nnreal @ A3 @ B4 ) )
        = ( sigma_3147302497200244656nnreal @ A3 ) ) ) ).

% space_sup_measure'
thf(fact_1036_space__sup__measure_H,axiom,
    ! [B4: sigma_measure_o,A3: sigma_measure_o] :
      ( ( ( sigma_sets_o @ B4 )
        = ( sigma_sets_o @ A3 ) )
     => ( ( sigma_space_o @ ( measur4529518739368704874sure_o @ A3 @ B4 ) )
        = ( sigma_space_o @ A3 ) ) ) ).

% space_sup_measure'
thf(fact_1037_qbs__morphism__def,axiom,
    ( qbs_mo5229770564518008146l_real
    = ( ^ [X7: quasi_borel_real,Y5: quasi_borel_real] :
          ( collect_real_real
          @ ^ [F: real > real] :
              ( ( member_real_real @ F
                @ ( pi_real_real @ ( qbs_space_real @ X7 )
                  @ ^ [Uu: real] : ( qbs_space_real @ Y5 ) ) )
              & ! [X: real > real] :
                  ( ( member_real_real @ X @ ( qbs_Mx_real @ X7 ) )
                 => ( member_real_real @ ( comp_real_real_real @ F @ X ) @ ( qbs_Mx_real @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1038_qbs__morphism__def,axiom,
    ( qbs_morphism_real_b
    = ( ^ [X7: quasi_borel_real,Y5: quasi_borel_b] :
          ( collect_real_b
          @ ^ [F: real > b] :
              ( ( member_real_b @ F
                @ ( pi_real_b @ ( qbs_space_real @ X7 )
                  @ ^ [Uu: real] : ( qbs_space_b @ Y5 ) ) )
              & ! [X: real > real] :
                  ( ( member_real_real @ X @ ( qbs_Mx_real @ X7 ) )
                 => ( member_real_b @ ( comp_real_b_real @ F @ X ) @ ( qbs_Mx_b @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1039_qbs__morphism__def,axiom,
    ( qbs_morphism_real_a
    = ( ^ [X7: quasi_borel_real,Y5: quasi_borel_a] :
          ( collect_real_a
          @ ^ [F: real > a] :
              ( ( member_real_a @ F
                @ ( pi_real_a @ ( qbs_space_real @ X7 )
                  @ ^ [Uu: real] : ( qbs_space_a @ Y5 ) ) )
              & ! [X: real > real] :
                  ( ( member_real_real @ X @ ( qbs_Mx_real @ X7 ) )
                 => ( member_real_a @ ( comp_real_a_real @ F @ X ) @ ( qbs_Mx_a @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1040_qbs__morphism__def,axiom,
    ( qbs_morphism_real_c
    = ( ^ [X7: quasi_borel_real,Y5: quasi_borel_c] :
          ( collect_real_c
          @ ^ [F: real > c] :
              ( ( member_real_c @ F
                @ ( pi_real_c @ ( qbs_space_real @ X7 )
                  @ ^ [Uu: real] : ( qbs_space_c @ Y5 ) ) )
              & ! [X: real > real] :
                  ( ( member_real_real @ X @ ( qbs_Mx_real @ X7 ) )
                 => ( member_real_c @ ( comp_real_c_real @ F @ X ) @ ( qbs_Mx_c @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1041_qbs__morphism__def,axiom,
    ( qbs_morphism_b_b
    = ( ^ [X7: quasi_borel_b,Y5: quasi_borel_b] :
          ( collect_b_b
          @ ^ [F: b > b] :
              ( ( member_b_b @ F
                @ ( pi_b_b @ ( qbs_space_b @ X7 )
                  @ ^ [Uu: b] : ( qbs_space_b @ Y5 ) ) )
              & ! [X: real > b] :
                  ( ( member_real_b @ X @ ( qbs_Mx_b @ X7 ) )
                 => ( member_real_b @ ( comp_b_b_real @ F @ X ) @ ( qbs_Mx_b @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1042_qbs__morphism__def,axiom,
    ( qbs_morphism_b_a
    = ( ^ [X7: quasi_borel_b,Y5: quasi_borel_a] :
          ( collect_b_a
          @ ^ [F: b > a] :
              ( ( member_b_a @ F
                @ ( pi_b_a @ ( qbs_space_b @ X7 )
                  @ ^ [Uu: b] : ( qbs_space_a @ Y5 ) ) )
              & ! [X: real > b] :
                  ( ( member_real_b @ X @ ( qbs_Mx_b @ X7 ) )
                 => ( member_real_a @ ( comp_b_a_real @ F @ X ) @ ( qbs_Mx_a @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1043_qbs__morphism__def,axiom,
    ( qbs_morphism_b_c
    = ( ^ [X7: quasi_borel_b,Y5: quasi_borel_c] :
          ( collect_b_c
          @ ^ [F: b > c] :
              ( ( member_b_c @ F
                @ ( pi_b_c @ ( qbs_space_b @ X7 )
                  @ ^ [Uu: b] : ( qbs_space_c @ Y5 ) ) )
              & ! [X: real > b] :
                  ( ( member_real_b @ X @ ( qbs_Mx_b @ X7 ) )
                 => ( member_real_c @ ( comp_b_c_real @ F @ X ) @ ( qbs_Mx_c @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1044_qbs__morphism__def,axiom,
    ( qbs_morphism_a_a
    = ( ^ [X7: quasi_borel_a,Y5: quasi_borel_a] :
          ( collect_a_a
          @ ^ [F: a > a] :
              ( ( member_a_a @ F
                @ ( pi_a_a @ ( qbs_space_a @ X7 )
                  @ ^ [Uu: a] : ( qbs_space_a @ Y5 ) ) )
              & ! [X: real > a] :
                  ( ( member_real_a @ X @ ( qbs_Mx_a @ X7 ) )
                 => ( member_real_a @ ( comp_a_a_real @ F @ X ) @ ( qbs_Mx_a @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1045_qbs__morphism__def,axiom,
    ( qbs_morphism_a_c
    = ( ^ [X7: quasi_borel_a,Y5: quasi_borel_c] :
          ( collect_a_c
          @ ^ [F: a > c] :
              ( ( member_a_c @ F
                @ ( pi_a_c @ ( qbs_space_a @ X7 )
                  @ ^ [Uu: a] : ( qbs_space_c @ Y5 ) ) )
              & ! [X: real > a] :
                  ( ( member_real_a @ X @ ( qbs_Mx_a @ X7 ) )
                 => ( member_real_c @ ( comp_a_c_real @ F @ X ) @ ( qbs_Mx_c @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1046_qbs__morphism__def,axiom,
    ( qbs_morphism_c_a
    = ( ^ [X7: quasi_borel_c,Y5: quasi_borel_a] :
          ( collect_c_a
          @ ^ [F: c > a] :
              ( ( member_c_a @ F
                @ ( pi_c_a @ ( qbs_space_c @ X7 )
                  @ ^ [Uu: c] : ( qbs_space_a @ Y5 ) ) )
              & ! [X: real > c] :
                  ( ( member_real_c @ X @ ( qbs_Mx_c @ X7 ) )
                 => ( member_real_a @ ( comp_c_a_real @ F @ X ) @ ( qbs_Mx_a @ Y5 ) ) ) ) ) ) ) ).

% qbs_morphism_def
thf(fact_1047_space__Sup__measure_H,axiom,
    ! [M: set_Si6059263944882162789e_real,A3: sigma_measure_real] :
      ( ! [M3: sigma_measure_real] :
          ( ( member4553183543495551918e_real @ M3 @ M )
         => ( ( sigma_sets_real @ M3 )
            = ( sigma_sets_real @ A3 ) ) )
     => ( ( M != bot_bo5686449298802467025e_real )
       => ( ( sigma_space_real @ ( measur8657758558638653562e_real @ M ) )
          = ( sigma_space_real @ A3 ) ) ) ) ).

% space_Sup_measure'
thf(fact_1048_space__Sup__measure_H,axiom,
    ! [M: set_Si3048223896905877257re_nat,A3: sigma_measure_nat] :
      ( ! [M3: sigma_measure_nat] :
          ( ( member4416920341759242834re_nat @ M3 @ M )
         => ( ( sigma_sets_nat @ M3 )
            = ( sigma_sets_nat @ A3 ) ) )
     => ( ( M != bot_bo8872222457363190133re_nat )
       => ( ( sigma_space_nat @ ( measur3575099672463795358re_nat @ M ) )
          = ( sigma_space_nat @ A3 ) ) ) ) ).

% space_Sup_measure'
thf(fact_1049_space__Sup__measure_H,axiom,
    ! [M: set_Si97717610131227249nnreal,A3: sigma_7234349610311085201nnreal] :
      ( ! [M3: sigma_7234349610311085201nnreal] :
          ( ( member6261374078160781754nnreal @ M3 @ M )
         => ( ( sigma_5465916536984168985nnreal @ M3 )
            = ( sigma_5465916536984168985nnreal @ A3 ) ) )
     => ( ( M != bot_bo8227844048696536285nnreal )
       => ( ( sigma_3147302497200244656nnreal @ ( measur1651139276328235014nnreal @ M ) )
          = ( sigma_3147302497200244656nnreal @ A3 ) ) ) ) ).

% space_Sup_measure'
thf(fact_1050_space__Sup__measure_H,axiom,
    ! [M: set_Sigma_measure_o,A3: sigma_measure_o] :
      ( ! [M3: sigma_measure_o] :
          ( ( member1844656263901471916sure_o @ M3 @ M )
         => ( ( sigma_sets_o @ M3 )
            = ( sigma_sets_o @ A3 ) ) )
     => ( ( M != bot_bo7838039659004643295sure_o )
       => ( ( sigma_space_o @ ( measur1214336222341667658sure_o @ M ) )
          = ( sigma_space_o @ A3 ) ) ) ) ).

% space_Sup_measure'
thf(fact_1051_IntI,axiom,
    ! [C: real > c,A3: set_real_c,B4: set_real_c] :
      ( ( member_real_c @ C @ A3 )
     => ( ( member_real_c @ C @ B4 )
       => ( member_real_c @ C @ ( inf_inf_set_real_c @ A3 @ B4 ) ) ) ) ).

% IntI
thf(fact_1052_IntI,axiom,
    ! [C: real > b,A3: set_real_b,B4: set_real_b] :
      ( ( member_real_b @ C @ A3 )
     => ( ( member_real_b @ C @ B4 )
       => ( member_real_b @ C @ ( inf_inf_set_real_b @ A3 @ B4 ) ) ) ) ).

% IntI
thf(fact_1053_IntI,axiom,
    ! [C: real > a,A3: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ A3 )
     => ( ( member_real_a @ C @ B4 )
       => ( member_real_a @ C @ ( inf_inf_set_real_a @ A3 @ B4 ) ) ) ) ).

% IntI
thf(fact_1054_IntI,axiom,
    ! [C: c > b,A3: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ A3 )
     => ( ( member_c_b @ C @ B4 )
       => ( member_c_b @ C @ ( inf_inf_set_c_b @ A3 @ B4 ) ) ) ) ).

% IntI
thf(fact_1055_IntI,axiom,
    ! [C: a > b,A3: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ A3 )
     => ( ( member_a_b @ C @ B4 )
       => ( member_a_b @ C @ ( inf_inf_set_a_b @ A3 @ B4 ) ) ) ) ).

% IntI
thf(fact_1056_Int__iff,axiom,
    ! [C: real > c,A3: set_real_c,B4: set_real_c] :
      ( ( member_real_c @ C @ ( inf_inf_set_real_c @ A3 @ B4 ) )
      = ( ( member_real_c @ C @ A3 )
        & ( member_real_c @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_1057_Int__iff,axiom,
    ! [C: real > b,A3: set_real_b,B4: set_real_b] :
      ( ( member_real_b @ C @ ( inf_inf_set_real_b @ A3 @ B4 ) )
      = ( ( member_real_b @ C @ A3 )
        & ( member_real_b @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_1058_Int__iff,axiom,
    ! [C: real > a,A3: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ ( inf_inf_set_real_a @ A3 @ B4 ) )
      = ( ( member_real_a @ C @ A3 )
        & ( member_real_a @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_1059_Int__iff,axiom,
    ! [C: c > b,A3: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ ( inf_inf_set_c_b @ A3 @ B4 ) )
      = ( ( member_c_b @ C @ A3 )
        & ( member_c_b @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_1060_Int__iff,axiom,
    ! [C: a > b,A3: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ ( inf_inf_set_a_b @ A3 @ B4 ) )
      = ( ( member_a_b @ C @ A3 )
        & ( member_a_b @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_1061_Int__UNIV,axiom,
    ! [A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( ( inf_in3368558534146122112nnreal @ A3 @ B4 )
        = top_to7994903218803871134nnreal )
      = ( ( A3 = top_to7994903218803871134nnreal )
        & ( B4 = top_to7994903218803871134nnreal ) ) ) ).

% Int_UNIV
thf(fact_1062_Int__UNIV,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( ( ( inf_inf_set_complex @ A3 @ B4 )
        = top_top_set_complex )
      = ( ( A3 = top_top_set_complex )
        & ( B4 = top_top_set_complex ) ) ) ).

% Int_UNIV
thf(fact_1063_Int__UNIV,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( ( inf_inf_set_real @ A3 @ B4 )
        = top_top_set_real )
      = ( ( A3 = top_top_set_real )
        & ( B4 = top_top_set_real ) ) ) ).

% Int_UNIV
thf(fact_1064_Int__UNIV,axiom,
    ! [A3: set_o,B4: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B4 )
        = top_top_set_o )
      = ( ( A3 = top_top_set_o )
        & ( B4 = top_top_set_o ) ) ) ).

% Int_UNIV
thf(fact_1065_Int__UNIV,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B4 )
        = top_top_set_nat )
      = ( ( A3 = top_top_set_nat )
        & ( B4 = top_top_set_nat ) ) ) ).

% Int_UNIV
thf(fact_1066_Set_Oball__empty,axiom,
    ! [P: extend8495563244428889912nnreal > $o,X8: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ X8 @ bot_bo4854962954004695426nnreal )
     => ( P @ X8 ) ) ).

% Set.ball_empty
thf(fact_1067_Set_Oball__empty,axiom,
    ! [P: real > $o,X8: real] :
      ( ( member_real @ X8 @ bot_bot_set_real )
     => ( P @ X8 ) ) ).

% Set.ball_empty
thf(fact_1068_Set_Oball__empty,axiom,
    ! [P: $o > $o,X8: $o] :
      ( ( member_o @ X8 @ bot_bot_set_o )
     => ( P @ X8 ) ) ).

% Set.ball_empty
thf(fact_1069_Set_Oball__empty,axiom,
    ! [P: nat > $o,X8: nat] :
      ( ( member_nat @ X8 @ bot_bot_set_nat )
     => ( P @ X8 ) ) ).

% Set.ball_empty
thf(fact_1070_Int__insert__left__if0,axiom,
    ! [A: real,C3: set_real,B4: set_real] :
      ( ~ ( member_real @ A @ C3 )
     => ( ( inf_inf_set_real @ ( insert_real @ A @ B4 ) @ C3 )
        = ( inf_inf_set_real @ B4 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1071_Int__insert__left__if0,axiom,
    ! [A: nat,C3: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ A @ C3 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C3 )
        = ( inf_inf_set_nat @ B4 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1072_Int__insert__left__if0,axiom,
    ! [A: $o,C3: set_o,B4: set_o] :
      ( ~ ( member_o @ A @ C3 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C3 )
        = ( inf_inf_set_o @ B4 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1073_Int__insert__left__if0,axiom,
    ! [A: extend8495563244428889912nnreal,C3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ A @ C3 )
     => ( ( inf_in3368558534146122112nnreal @ ( insert7407984058720857448nnreal @ A @ B4 ) @ C3 )
        = ( inf_in3368558534146122112nnreal @ B4 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1074_Int__insert__left__if0,axiom,
    ! [A: real > c,C3: set_real_c,B4: set_real_c] :
      ( ~ ( member_real_c @ A @ C3 )
     => ( ( inf_inf_set_real_c @ ( insert_real_c @ A @ B4 ) @ C3 )
        = ( inf_inf_set_real_c @ B4 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1075_Int__insert__left__if0,axiom,
    ! [A: real > b,C3: set_real_b,B4: set_real_b] :
      ( ~ ( member_real_b @ A @ C3 )
     => ( ( inf_inf_set_real_b @ ( insert_real_b @ A @ B4 ) @ C3 )
        = ( inf_inf_set_real_b @ B4 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1076_Int__insert__left__if0,axiom,
    ! [A: real > a,C3: set_real_a,B4: set_real_a] :
      ( ~ ( member_real_a @ A @ C3 )
     => ( ( inf_inf_set_real_a @ ( insert_real_a @ A @ B4 ) @ C3 )
        = ( inf_inf_set_real_a @ B4 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1077_Int__insert__left__if0,axiom,
    ! [A: c > b,C3: set_c_b,B4: set_c_b] :
      ( ~ ( member_c_b @ A @ C3 )
     => ( ( inf_inf_set_c_b @ ( insert_c_b @ A @ B4 ) @ C3 )
        = ( inf_inf_set_c_b @ B4 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1078_Int__insert__left__if0,axiom,
    ! [A: a > b,C3: set_a_b,B4: set_a_b] :
      ( ~ ( member_a_b @ A @ C3 )
     => ( ( inf_inf_set_a_b @ ( insert_a_b @ A @ B4 ) @ C3 )
        = ( inf_inf_set_a_b @ B4 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1079_Int__insert__left__if1,axiom,
    ! [A: real,C3: set_real,B4: set_real] :
      ( ( member_real @ A @ C3 )
     => ( ( inf_inf_set_real @ ( insert_real @ A @ B4 ) @ C3 )
        = ( insert_real @ A @ ( inf_inf_set_real @ B4 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1080_Int__insert__left__if1,axiom,
    ! [A: nat,C3: set_nat,B4: set_nat] :
      ( ( member_nat @ A @ C3 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C3 )
        = ( insert_nat @ A @ ( inf_inf_set_nat @ B4 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1081_Int__insert__left__if1,axiom,
    ! [A: $o,C3: set_o,B4: set_o] :
      ( ( member_o @ A @ C3 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C3 )
        = ( insert_o @ A @ ( inf_inf_set_o @ B4 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1082_Int__insert__left__if1,axiom,
    ! [A: extend8495563244428889912nnreal,C3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ C3 )
     => ( ( inf_in3368558534146122112nnreal @ ( insert7407984058720857448nnreal @ A @ B4 ) @ C3 )
        = ( insert7407984058720857448nnreal @ A @ ( inf_in3368558534146122112nnreal @ B4 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1083_Int__insert__left__if1,axiom,
    ! [A: real > c,C3: set_real_c,B4: set_real_c] :
      ( ( member_real_c @ A @ C3 )
     => ( ( inf_inf_set_real_c @ ( insert_real_c @ A @ B4 ) @ C3 )
        = ( insert_real_c @ A @ ( inf_inf_set_real_c @ B4 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1084_Int__insert__left__if1,axiom,
    ! [A: real > b,C3: set_real_b,B4: set_real_b] :
      ( ( member_real_b @ A @ C3 )
     => ( ( inf_inf_set_real_b @ ( insert_real_b @ A @ B4 ) @ C3 )
        = ( insert_real_b @ A @ ( inf_inf_set_real_b @ B4 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1085_Int__insert__left__if1,axiom,
    ! [A: real > a,C3: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ A @ C3 )
     => ( ( inf_inf_set_real_a @ ( insert_real_a @ A @ B4 ) @ C3 )
        = ( insert_real_a @ A @ ( inf_inf_set_real_a @ B4 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1086_Int__insert__left__if1,axiom,
    ! [A: c > b,C3: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ A @ C3 )
     => ( ( inf_inf_set_c_b @ ( insert_c_b @ A @ B4 ) @ C3 )
        = ( insert_c_b @ A @ ( inf_inf_set_c_b @ B4 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1087_Int__insert__left__if1,axiom,
    ! [A: a > b,C3: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ A @ C3 )
     => ( ( inf_inf_set_a_b @ ( insert_a_b @ A @ B4 ) @ C3 )
        = ( insert_a_b @ A @ ( inf_inf_set_a_b @ B4 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1088_insert__inter__insert,axiom,
    ! [A: real,A3: set_real,B4: set_real] :
      ( ( inf_inf_set_real @ ( insert_real @ A @ A3 ) @ ( insert_real @ A @ B4 ) )
      = ( insert_real @ A @ ( inf_inf_set_real @ A3 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_1089_insert__inter__insert,axiom,
    ! [A: nat,A3: set_nat,B4: set_nat] :
      ( ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ ( insert_nat @ A @ B4 ) )
      = ( insert_nat @ A @ ( inf_inf_set_nat @ A3 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_1090_insert__inter__insert,axiom,
    ! [A: $o,A3: set_o,B4: set_o] :
      ( ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ ( insert_o @ A @ B4 ) )
      = ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_1091_insert__inter__insert,axiom,
    ! [A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( inf_in3368558534146122112nnreal @ ( insert7407984058720857448nnreal @ A @ A3 ) @ ( insert7407984058720857448nnreal @ A @ B4 ) )
      = ( insert7407984058720857448nnreal @ A @ ( inf_in3368558534146122112nnreal @ A3 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_1092_Int__insert__right__if0,axiom,
    ! [A: real,A3: set_real,B4: set_real] :
      ( ~ ( member_real @ A @ A3 )
     => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A @ B4 ) )
        = ( inf_inf_set_real @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1093_Int__insert__right__if0,axiom,
    ! [A: nat,A3: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ A @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A @ B4 ) )
        = ( inf_inf_set_nat @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1094_Int__insert__right__if0,axiom,
    ! [A: $o,A3: set_o,B4: set_o] :
      ( ~ ( member_o @ A @ A3 )
     => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B4 ) )
        = ( inf_inf_set_o @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1095_Int__insert__right__if0,axiom,
    ! [A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ A @ A3 )
     => ( ( inf_in3368558534146122112nnreal @ A3 @ ( insert7407984058720857448nnreal @ A @ B4 ) )
        = ( inf_in3368558534146122112nnreal @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1096_Int__insert__right__if0,axiom,
    ! [A: real > c,A3: set_real_c,B4: set_real_c] :
      ( ~ ( member_real_c @ A @ A3 )
     => ( ( inf_inf_set_real_c @ A3 @ ( insert_real_c @ A @ B4 ) )
        = ( inf_inf_set_real_c @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1097_Int__insert__right__if0,axiom,
    ! [A: real > b,A3: set_real_b,B4: set_real_b] :
      ( ~ ( member_real_b @ A @ A3 )
     => ( ( inf_inf_set_real_b @ A3 @ ( insert_real_b @ A @ B4 ) )
        = ( inf_inf_set_real_b @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1098_Int__insert__right__if0,axiom,
    ! [A: real > a,A3: set_real_a,B4: set_real_a] :
      ( ~ ( member_real_a @ A @ A3 )
     => ( ( inf_inf_set_real_a @ A3 @ ( insert_real_a @ A @ B4 ) )
        = ( inf_inf_set_real_a @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1099_Int__insert__right__if0,axiom,
    ! [A: c > b,A3: set_c_b,B4: set_c_b] :
      ( ~ ( member_c_b @ A @ A3 )
     => ( ( inf_inf_set_c_b @ A3 @ ( insert_c_b @ A @ B4 ) )
        = ( inf_inf_set_c_b @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1100_Int__insert__right__if0,axiom,
    ! [A: a > b,A3: set_a_b,B4: set_a_b] :
      ( ~ ( member_a_b @ A @ A3 )
     => ( ( inf_inf_set_a_b @ A3 @ ( insert_a_b @ A @ B4 ) )
        = ( inf_inf_set_a_b @ A3 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_1101_Int__insert__right__if1,axiom,
    ! [A: real,A3: set_real,B4: set_real] :
      ( ( member_real @ A @ A3 )
     => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A @ B4 ) )
        = ( insert_real @ A @ ( inf_inf_set_real @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1102_Int__insert__right__if1,axiom,
    ! [A: nat,A3: set_nat,B4: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A @ B4 ) )
        = ( insert_nat @ A @ ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1103_Int__insert__right__if1,axiom,
    ! [A: $o,A3: set_o,B4: set_o] :
      ( ( member_o @ A @ A3 )
     => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B4 ) )
        = ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1104_Int__insert__right__if1,axiom,
    ! [A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ A3 )
     => ( ( inf_in3368558534146122112nnreal @ A3 @ ( insert7407984058720857448nnreal @ A @ B4 ) )
        = ( insert7407984058720857448nnreal @ A @ ( inf_in3368558534146122112nnreal @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1105_Int__insert__right__if1,axiom,
    ! [A: real > c,A3: set_real_c,B4: set_real_c] :
      ( ( member_real_c @ A @ A3 )
     => ( ( inf_inf_set_real_c @ A3 @ ( insert_real_c @ A @ B4 ) )
        = ( insert_real_c @ A @ ( inf_inf_set_real_c @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1106_Int__insert__right__if1,axiom,
    ! [A: real > b,A3: set_real_b,B4: set_real_b] :
      ( ( member_real_b @ A @ A3 )
     => ( ( inf_inf_set_real_b @ A3 @ ( insert_real_b @ A @ B4 ) )
        = ( insert_real_b @ A @ ( inf_inf_set_real_b @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1107_Int__insert__right__if1,axiom,
    ! [A: real > a,A3: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ A @ A3 )
     => ( ( inf_inf_set_real_a @ A3 @ ( insert_real_a @ A @ B4 ) )
        = ( insert_real_a @ A @ ( inf_inf_set_real_a @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1108_Int__insert__right__if1,axiom,
    ! [A: c > b,A3: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ A @ A3 )
     => ( ( inf_inf_set_c_b @ A3 @ ( insert_c_b @ A @ B4 ) )
        = ( insert_c_b @ A @ ( inf_inf_set_c_b @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1109_Int__insert__right__if1,axiom,
    ! [A: a > b,A3: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ A @ A3 )
     => ( ( inf_inf_set_a_b @ A3 @ ( insert_a_b @ A @ B4 ) )
        = ( insert_a_b @ A @ ( inf_inf_set_a_b @ A3 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1110_sets_OInt,axiom,
    ! [A: set_real,M: sigma_measure_real,B: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ M ) )
       => ( member_set_real @ ( inf_inf_set_real @ A @ B ) @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.Int
thf(fact_1111_sets_OInt,axiom,
    ! [A: set_nat,M: sigma_measure_nat,B: set_nat] :
      ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat @ B @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.Int
thf(fact_1112_sets_OInt,axiom,
    ! [A: set_o,M: sigma_measure_o,B: set_o] :
      ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o @ B @ ( sigma_sets_o @ M ) )
       => ( member_set_o @ ( inf_inf_set_o @ A @ B ) @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.Int
thf(fact_1113_sets_OInt,axiom,
    ! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal @ B @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ A @ B ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.Int
thf(fact_1114_sets__sup__measure_H,axiom,
    ! [B4: sigma_measure_real,A3: sigma_measure_real] :
      ( ( ( sigma_sets_real @ B4 )
        = ( sigma_sets_real @ A3 ) )
     => ( ( sigma_sets_real @ ( measur2147279183506585690e_real @ A3 @ B4 ) )
        = ( sigma_sets_real @ A3 ) ) ) ).

% sets_sup_measure'
thf(fact_1115_sets__sup__measure_H,axiom,
    ! [B4: sigma_measure_nat,A3: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ B4 )
        = ( sigma_sets_nat @ A3 ) )
     => ( ( sigma_sets_nat @ ( measur876423496291765374re_nat @ A3 @ B4 ) )
        = ( sigma_sets_nat @ A3 ) ) ) ).

% sets_sup_measure'
thf(fact_1116_sets__sup__measure_H,axiom,
    ! [B4: sigma_measure_o,A3: sigma_measure_o] :
      ( ( ( sigma_sets_o @ B4 )
        = ( sigma_sets_o @ A3 ) )
     => ( ( sigma_sets_o @ ( measur4529518739368704874sure_o @ A3 @ B4 ) )
        = ( sigma_sets_o @ A3 ) ) ) ).

% sets_sup_measure'
thf(fact_1117_sets__sup__measure_H,axiom,
    ! [B4: sigma_7234349610311085201nnreal,A3: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ B4 )
        = ( sigma_5465916536984168985nnreal @ A3 ) )
     => ( ( sigma_5465916536984168985nnreal @ ( measur4473656680840910822nnreal @ A3 @ B4 ) )
        = ( sigma_5465916536984168985nnreal @ A3 ) ) ) ).

% sets_sup_measure'
thf(fact_1118_insert__disjoint_I1_J,axiom,
    ! [A: real > c,A3: set_real_c,B4: set_real_c] :
      ( ( ( inf_inf_set_real_c @ ( insert_real_c @ A @ A3 ) @ B4 )
        = bot_bot_set_real_c )
      = ( ~ ( member_real_c @ A @ B4 )
        & ( ( inf_inf_set_real_c @ A3 @ B4 )
          = bot_bot_set_real_c ) ) ) ).

% insert_disjoint(1)
thf(fact_1119_insert__disjoint_I1_J,axiom,
    ! [A: real > b,A3: set_real_b,B4: set_real_b] :
      ( ( ( inf_inf_set_real_b @ ( insert_real_b @ A @ A3 ) @ B4 )
        = bot_bot_set_real_b )
      = ( ~ ( member_real_b @ A @ B4 )
        & ( ( inf_inf_set_real_b @ A3 @ B4 )
          = bot_bot_set_real_b ) ) ) ).

% insert_disjoint(1)
thf(fact_1120_insert__disjoint_I1_J,axiom,
    ! [A: real > a,A3: set_real_a,B4: set_real_a] :
      ( ( ( inf_inf_set_real_a @ ( insert_real_a @ A @ A3 ) @ B4 )
        = bot_bot_set_real_a )
      = ( ~ ( member_real_a @ A @ B4 )
        & ( ( inf_inf_set_real_a @ A3 @ B4 )
          = bot_bot_set_real_a ) ) ) ).

% insert_disjoint(1)
thf(fact_1121_insert__disjoint_I1_J,axiom,
    ! [A: c > b,A3: set_c_b,B4: set_c_b] :
      ( ( ( inf_inf_set_c_b @ ( insert_c_b @ A @ A3 ) @ B4 )
        = bot_bot_set_c_b )
      = ( ~ ( member_c_b @ A @ B4 )
        & ( ( inf_inf_set_c_b @ A3 @ B4 )
          = bot_bot_set_c_b ) ) ) ).

% insert_disjoint(1)
thf(fact_1122_insert__disjoint_I1_J,axiom,
    ! [A: a > b,A3: set_a_b,B4: set_a_b] :
      ( ( ( inf_inf_set_a_b @ ( insert_a_b @ A @ A3 ) @ B4 )
        = bot_bot_set_a_b )
      = ( ~ ( member_a_b @ A @ B4 )
        & ( ( inf_inf_set_a_b @ A3 @ B4 )
          = bot_bot_set_a_b ) ) ) ).

% insert_disjoint(1)
thf(fact_1123_insert__disjoint_I1_J,axiom,
    ! [A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( ( inf_in3368558534146122112nnreal @ ( insert7407984058720857448nnreal @ A @ A3 ) @ B4 )
        = bot_bo4854962954004695426nnreal )
      = ( ~ ( member7908768830364227535nnreal @ A @ B4 )
        & ( ( inf_in3368558534146122112nnreal @ A3 @ B4 )
          = bot_bo4854962954004695426nnreal ) ) ) ).

% insert_disjoint(1)
thf(fact_1124_insert__disjoint_I1_J,axiom,
    ! [A: real,A3: set_real,B4: set_real] :
      ( ( ( inf_inf_set_real @ ( insert_real @ A @ A3 ) @ B4 )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A @ B4 )
        & ( ( inf_inf_set_real @ A3 @ B4 )
          = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
thf(fact_1125_insert__disjoint_I1_J,axiom,
    ! [A: $o,A3: set_o,B4: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ B4 )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B4 )
        & ( ( inf_inf_set_o @ A3 @ B4 )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_1126_insert__disjoint_I1_J,axiom,
    ! [A: nat,A3: set_nat,B4: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ B4 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B4 )
        & ( ( inf_inf_set_nat @ A3 @ B4 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_1127_insert__disjoint_I2_J,axiom,
    ! [A: real > c,A3: set_real_c,B4: set_real_c] :
      ( ( bot_bot_set_real_c
        = ( inf_inf_set_real_c @ ( insert_real_c @ A @ A3 ) @ B4 ) )
      = ( ~ ( member_real_c @ A @ B4 )
        & ( bot_bot_set_real_c
          = ( inf_inf_set_real_c @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1128_insert__disjoint_I2_J,axiom,
    ! [A: real > b,A3: set_real_b,B4: set_real_b] :
      ( ( bot_bot_set_real_b
        = ( inf_inf_set_real_b @ ( insert_real_b @ A @ A3 ) @ B4 ) )
      = ( ~ ( member_real_b @ A @ B4 )
        & ( bot_bot_set_real_b
          = ( inf_inf_set_real_b @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1129_insert__disjoint_I2_J,axiom,
    ! [A: real > a,A3: set_real_a,B4: set_real_a] :
      ( ( bot_bot_set_real_a
        = ( inf_inf_set_real_a @ ( insert_real_a @ A @ A3 ) @ B4 ) )
      = ( ~ ( member_real_a @ A @ B4 )
        & ( bot_bot_set_real_a
          = ( inf_inf_set_real_a @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1130_insert__disjoint_I2_J,axiom,
    ! [A: c > b,A3: set_c_b,B4: set_c_b] :
      ( ( bot_bot_set_c_b
        = ( inf_inf_set_c_b @ ( insert_c_b @ A @ A3 ) @ B4 ) )
      = ( ~ ( member_c_b @ A @ B4 )
        & ( bot_bot_set_c_b
          = ( inf_inf_set_c_b @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1131_insert__disjoint_I2_J,axiom,
    ! [A: a > b,A3: set_a_b,B4: set_a_b] :
      ( ( bot_bot_set_a_b
        = ( inf_inf_set_a_b @ ( insert_a_b @ A @ A3 ) @ B4 ) )
      = ( ~ ( member_a_b @ A @ B4 )
        & ( bot_bot_set_a_b
          = ( inf_inf_set_a_b @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1132_insert__disjoint_I2_J,axiom,
    ! [A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( bot_bo4854962954004695426nnreal
        = ( inf_in3368558534146122112nnreal @ ( insert7407984058720857448nnreal @ A @ A3 ) @ B4 ) )
      = ( ~ ( member7908768830364227535nnreal @ A @ B4 )
        & ( bot_bo4854962954004695426nnreal
          = ( inf_in3368558534146122112nnreal @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1133_insert__disjoint_I2_J,axiom,
    ! [A: real,A3: set_real,B4: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ ( insert_real @ A @ A3 ) @ B4 ) )
      = ( ~ ( member_real @ A @ B4 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1134_insert__disjoint_I2_J,axiom,
    ! [A: $o,A3: set_o,B4: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ B4 ) )
      = ( ~ ( member_o @ A @ B4 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1135_insert__disjoint_I2_J,axiom,
    ! [A: nat,A3: set_nat,B4: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ B4 ) )
      = ( ~ ( member_nat @ A @ B4 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1136_disjoint__insert_I1_J,axiom,
    ! [B4: set_real_c,A: real > c,A3: set_real_c] :
      ( ( ( inf_inf_set_real_c @ B4 @ ( insert_real_c @ A @ A3 ) )
        = bot_bot_set_real_c )
      = ( ~ ( member_real_c @ A @ B4 )
        & ( ( inf_inf_set_real_c @ B4 @ A3 )
          = bot_bot_set_real_c ) ) ) ).

% disjoint_insert(1)
thf(fact_1137_disjoint__insert_I1_J,axiom,
    ! [B4: set_real_b,A: real > b,A3: set_real_b] :
      ( ( ( inf_inf_set_real_b @ B4 @ ( insert_real_b @ A @ A3 ) )
        = bot_bot_set_real_b )
      = ( ~ ( member_real_b @ A @ B4 )
        & ( ( inf_inf_set_real_b @ B4 @ A3 )
          = bot_bot_set_real_b ) ) ) ).

% disjoint_insert(1)
thf(fact_1138_disjoint__insert_I1_J,axiom,
    ! [B4: set_real_a,A: real > a,A3: set_real_a] :
      ( ( ( inf_inf_set_real_a @ B4 @ ( insert_real_a @ A @ A3 ) )
        = bot_bot_set_real_a )
      = ( ~ ( member_real_a @ A @ B4 )
        & ( ( inf_inf_set_real_a @ B4 @ A3 )
          = bot_bot_set_real_a ) ) ) ).

% disjoint_insert(1)
thf(fact_1139_disjoint__insert_I1_J,axiom,
    ! [B4: set_c_b,A: c > b,A3: set_c_b] :
      ( ( ( inf_inf_set_c_b @ B4 @ ( insert_c_b @ A @ A3 ) )
        = bot_bot_set_c_b )
      = ( ~ ( member_c_b @ A @ B4 )
        & ( ( inf_inf_set_c_b @ B4 @ A3 )
          = bot_bot_set_c_b ) ) ) ).

% disjoint_insert(1)
thf(fact_1140_disjoint__insert_I1_J,axiom,
    ! [B4: set_a_b,A: a > b,A3: set_a_b] :
      ( ( ( inf_inf_set_a_b @ B4 @ ( insert_a_b @ A @ A3 ) )
        = bot_bot_set_a_b )
      = ( ~ ( member_a_b @ A @ B4 )
        & ( ( inf_inf_set_a_b @ B4 @ A3 )
          = bot_bot_set_a_b ) ) ) ).

% disjoint_insert(1)
thf(fact_1141_disjoint__insert_I1_J,axiom,
    ! [B4: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal,A3: set_Ex3793607809372303086nnreal] :
      ( ( ( inf_in3368558534146122112nnreal @ B4 @ ( insert7407984058720857448nnreal @ A @ A3 ) )
        = bot_bo4854962954004695426nnreal )
      = ( ~ ( member7908768830364227535nnreal @ A @ B4 )
        & ( ( inf_in3368558534146122112nnreal @ B4 @ A3 )
          = bot_bo4854962954004695426nnreal ) ) ) ).

% disjoint_insert(1)
thf(fact_1142_disjoint__insert_I1_J,axiom,
    ! [B4: set_real,A: real,A3: set_real] :
      ( ( ( inf_inf_set_real @ B4 @ ( insert_real @ A @ A3 ) )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A @ B4 )
        & ( ( inf_inf_set_real @ B4 @ A3 )
          = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
thf(fact_1143_disjoint__insert_I1_J,axiom,
    ! [B4: set_o,A: $o,A3: set_o] :
      ( ( ( inf_inf_set_o @ B4 @ ( insert_o @ A @ A3 ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B4 )
        & ( ( inf_inf_set_o @ B4 @ A3 )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_1144_disjoint__insert_I1_J,axiom,
    ! [B4: set_nat,A: nat,A3: set_nat] :
      ( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A @ A3 ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B4 )
        & ( ( inf_inf_set_nat @ B4 @ A3 )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1145_disjoint__insert_I2_J,axiom,
    ! [A3: set_real_c,B: real > c,B4: set_real_c] :
      ( ( bot_bot_set_real_c
        = ( inf_inf_set_real_c @ A3 @ ( insert_real_c @ B @ B4 ) ) )
      = ( ~ ( member_real_c @ B @ A3 )
        & ( bot_bot_set_real_c
          = ( inf_inf_set_real_c @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1146_disjoint__insert_I2_J,axiom,
    ! [A3: set_real_b,B: real > b,B4: set_real_b] :
      ( ( bot_bot_set_real_b
        = ( inf_inf_set_real_b @ A3 @ ( insert_real_b @ B @ B4 ) ) )
      = ( ~ ( member_real_b @ B @ A3 )
        & ( bot_bot_set_real_b
          = ( inf_inf_set_real_b @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1147_disjoint__insert_I2_J,axiom,
    ! [A3: set_real_a,B: real > a,B4: set_real_a] :
      ( ( bot_bot_set_real_a
        = ( inf_inf_set_real_a @ A3 @ ( insert_real_a @ B @ B4 ) ) )
      = ( ~ ( member_real_a @ B @ A3 )
        & ( bot_bot_set_real_a
          = ( inf_inf_set_real_a @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1148_disjoint__insert_I2_J,axiom,
    ! [A3: set_c_b,B: c > b,B4: set_c_b] :
      ( ( bot_bot_set_c_b
        = ( inf_inf_set_c_b @ A3 @ ( insert_c_b @ B @ B4 ) ) )
      = ( ~ ( member_c_b @ B @ A3 )
        & ( bot_bot_set_c_b
          = ( inf_inf_set_c_b @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1149_disjoint__insert_I2_J,axiom,
    ! [A3: set_a_b,B: a > b,B4: set_a_b] :
      ( ( bot_bot_set_a_b
        = ( inf_inf_set_a_b @ A3 @ ( insert_a_b @ B @ B4 ) ) )
      = ( ~ ( member_a_b @ B @ A3 )
        & ( bot_bot_set_a_b
          = ( inf_inf_set_a_b @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1150_disjoint__insert_I2_J,axiom,
    ! [A3: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( bot_bo4854962954004695426nnreal
        = ( inf_in3368558534146122112nnreal @ A3 @ ( insert7407984058720857448nnreal @ B @ B4 ) ) )
      = ( ~ ( member7908768830364227535nnreal @ B @ A3 )
        & ( bot_bo4854962954004695426nnreal
          = ( inf_in3368558534146122112nnreal @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1151_disjoint__insert_I2_J,axiom,
    ! [A3: set_real,B: real,B4: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ A3 @ ( insert_real @ B @ B4 ) ) )
      = ( ~ ( member_real @ B @ A3 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1152_disjoint__insert_I2_J,axiom,
    ! [A3: set_o,B: $o,B4: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A3 @ ( insert_o @ B @ B4 ) ) )
      = ( ~ ( member_o @ B @ A3 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1153_disjoint__insert_I2_J,axiom,
    ! [A3: set_nat,B: nat,B4: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A3 @ ( insert_nat @ B @ B4 ) ) )
      = ( ~ ( member_nat @ B @ A3 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1154_sets_OInt__space__eq2,axiom,
    ! [X3: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ X3 @ ( sigma_sets_real @ M ) )
     => ( ( inf_inf_set_real @ X3 @ ( sigma_space_real @ M ) )
        = X3 ) ) ).

% sets.Int_space_eq2
thf(fact_1155_sets_OInt__space__eq2,axiom,
    ! [X3: set_nat,M: sigma_measure_nat] :
      ( ( member_set_nat @ X3 @ ( sigma_sets_nat @ M ) )
     => ( ( inf_inf_set_nat @ X3 @ ( sigma_space_nat @ M ) )
        = X3 ) ) ).

% sets.Int_space_eq2
thf(fact_1156_sets_OInt__space__eq2,axiom,
    ! [X3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( member603777416030116741nnreal @ X3 @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( inf_in3368558534146122112nnreal @ X3 @ ( sigma_3147302497200244656nnreal @ M ) )
        = X3 ) ) ).

% sets.Int_space_eq2
thf(fact_1157_sets_OInt__space__eq2,axiom,
    ! [X3: set_o,M: sigma_measure_o] :
      ( ( member_set_o @ X3 @ ( sigma_sets_o @ M ) )
     => ( ( inf_inf_set_o @ X3 @ ( sigma_space_o @ M ) )
        = X3 ) ) ).

% sets.Int_space_eq2
thf(fact_1158_sets_OInt__space__eq1,axiom,
    ! [X3: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ X3 @ ( sigma_sets_real @ M ) )
     => ( ( inf_inf_set_real @ ( sigma_space_real @ M ) @ X3 )
        = X3 ) ) ).

% sets.Int_space_eq1
thf(fact_1159_sets_OInt__space__eq1,axiom,
    ! [X3: set_nat,M: sigma_measure_nat] :
      ( ( member_set_nat @ X3 @ ( sigma_sets_nat @ M ) )
     => ( ( inf_inf_set_nat @ ( sigma_space_nat @ M ) @ X3 )
        = X3 ) ) ).

% sets.Int_space_eq1
thf(fact_1160_sets_OInt__space__eq1,axiom,
    ! [X3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( member603777416030116741nnreal @ X3 @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( inf_in3368558534146122112nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ X3 )
        = X3 ) ) ).

% sets.Int_space_eq1
thf(fact_1161_sets_OInt__space__eq1,axiom,
    ! [X3: set_o,M: sigma_measure_o] :
      ( ( member_set_o @ X3 @ ( sigma_sets_o @ M ) )
     => ( ( inf_inf_set_o @ ( sigma_space_o @ M ) @ X3 )
        = X3 ) ) ).

% sets.Int_space_eq1
thf(fact_1162_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1163_Int__Collect,axiom,
    ! [X3: real > c,A3: set_real_c,P: ( real > c ) > $o] :
      ( ( member_real_c @ X3 @ ( inf_inf_set_real_c @ A3 @ ( collect_real_c @ P ) ) )
      = ( ( member_real_c @ X3 @ A3 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1164_Int__Collect,axiom,
    ! [X3: real > b,A3: set_real_b,P: ( real > b ) > $o] :
      ( ( member_real_b @ X3 @ ( inf_inf_set_real_b @ A3 @ ( collect_real_b @ P ) ) )
      = ( ( member_real_b @ X3 @ A3 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1165_Int__Collect,axiom,
    ! [X3: real > a,A3: set_real_a,P: ( real > a ) > $o] :
      ( ( member_real_a @ X3 @ ( inf_inf_set_real_a @ A3 @ ( collect_real_a @ P ) ) )
      = ( ( member_real_a @ X3 @ A3 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1166_Int__Collect,axiom,
    ! [X3: c > b,A3: set_c_b,P: ( c > b ) > $o] :
      ( ( member_c_b @ X3 @ ( inf_inf_set_c_b @ A3 @ ( collect_c_b @ P ) ) )
      = ( ( member_c_b @ X3 @ A3 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1167_Int__Collect,axiom,
    ! [X3: a > b,A3: set_a_b,P: ( a > b ) > $o] :
      ( ( member_a_b @ X3 @ ( inf_inf_set_a_b @ A3 @ ( collect_a_b @ P ) ) )
      = ( ( member_a_b @ X3 @ A3 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1168_Int__Collect,axiom,
    ! [X3: nat,A3: set_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X3 @ A3 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_1169_Int__def,axiom,
    ( inf_inf_set_real_c
    = ( ^ [A5: set_real_c,B6: set_real_c] :
          ( collect_real_c
          @ ^ [X: real > c] :
              ( ( member_real_c @ X @ A5 )
              & ( member_real_c @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1170_Int__def,axiom,
    ( inf_inf_set_real_b
    = ( ^ [A5: set_real_b,B6: set_real_b] :
          ( collect_real_b
          @ ^ [X: real > b] :
              ( ( member_real_b @ X @ A5 )
              & ( member_real_b @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1171_Int__def,axiom,
    ( inf_inf_set_real_a
    = ( ^ [A5: set_real_a,B6: set_real_a] :
          ( collect_real_a
          @ ^ [X: real > a] :
              ( ( member_real_a @ X @ A5 )
              & ( member_real_a @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1172_Int__def,axiom,
    ( inf_inf_set_c_b
    = ( ^ [A5: set_c_b,B6: set_c_b] :
          ( collect_c_b
          @ ^ [X: c > b] :
              ( ( member_c_b @ X @ A5 )
              & ( member_c_b @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1173_Int__def,axiom,
    ( inf_inf_set_a_b
    = ( ^ [A5: set_a_b,B6: set_a_b] :
          ( collect_a_b
          @ ^ [X: a > b] :
              ( ( member_a_b @ X @ A5 )
              & ( member_a_b @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1174_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A5: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A5 )
              & ( member_nat @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1175_IntE,axiom,
    ! [C: real > c,A3: set_real_c,B4: set_real_c] :
      ( ( member_real_c @ C @ ( inf_inf_set_real_c @ A3 @ B4 ) )
     => ~ ( ( member_real_c @ C @ A3 )
         => ~ ( member_real_c @ C @ B4 ) ) ) ).

% IntE
thf(fact_1176_IntE,axiom,
    ! [C: real > b,A3: set_real_b,B4: set_real_b] :
      ( ( member_real_b @ C @ ( inf_inf_set_real_b @ A3 @ B4 ) )
     => ~ ( ( member_real_b @ C @ A3 )
         => ~ ( member_real_b @ C @ B4 ) ) ) ).

% IntE
thf(fact_1177_IntE,axiom,
    ! [C: real > a,A3: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ ( inf_inf_set_real_a @ A3 @ B4 ) )
     => ~ ( ( member_real_a @ C @ A3 )
         => ~ ( member_real_a @ C @ B4 ) ) ) ).

% IntE
thf(fact_1178_IntE,axiom,
    ! [C: c > b,A3: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ ( inf_inf_set_c_b @ A3 @ B4 ) )
     => ~ ( ( member_c_b @ C @ A3 )
         => ~ ( member_c_b @ C @ B4 ) ) ) ).

% IntE
thf(fact_1179_IntE,axiom,
    ! [C: a > b,A3: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ ( inf_inf_set_a_b @ A3 @ B4 ) )
     => ~ ( ( member_a_b @ C @ A3 )
         => ~ ( member_a_b @ C @ B4 ) ) ) ).

% IntE
thf(fact_1180_IntD1,axiom,
    ! [C: real > c,A3: set_real_c,B4: set_real_c] :
      ( ( member_real_c @ C @ ( inf_inf_set_real_c @ A3 @ B4 ) )
     => ( member_real_c @ C @ A3 ) ) ).

% IntD1
thf(fact_1181_IntD1,axiom,
    ! [C: real > b,A3: set_real_b,B4: set_real_b] :
      ( ( member_real_b @ C @ ( inf_inf_set_real_b @ A3 @ B4 ) )
     => ( member_real_b @ C @ A3 ) ) ).

% IntD1
thf(fact_1182_IntD1,axiom,
    ! [C: real > a,A3: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ ( inf_inf_set_real_a @ A3 @ B4 ) )
     => ( member_real_a @ C @ A3 ) ) ).

% IntD1
thf(fact_1183_IntD1,axiom,
    ! [C: c > b,A3: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ ( inf_inf_set_c_b @ A3 @ B4 ) )
     => ( member_c_b @ C @ A3 ) ) ).

% IntD1
thf(fact_1184_IntD1,axiom,
    ! [C: a > b,A3: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ ( inf_inf_set_a_b @ A3 @ B4 ) )
     => ( member_a_b @ C @ A3 ) ) ).

% IntD1
thf(fact_1185_IntD2,axiom,
    ! [C: real > c,A3: set_real_c,B4: set_real_c] :
      ( ( member_real_c @ C @ ( inf_inf_set_real_c @ A3 @ B4 ) )
     => ( member_real_c @ C @ B4 ) ) ).

% IntD2
thf(fact_1186_IntD2,axiom,
    ! [C: real > b,A3: set_real_b,B4: set_real_b] :
      ( ( member_real_b @ C @ ( inf_inf_set_real_b @ A3 @ B4 ) )
     => ( member_real_b @ C @ B4 ) ) ).

% IntD2
thf(fact_1187_IntD2,axiom,
    ! [C: real > a,A3: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ ( inf_inf_set_real_a @ A3 @ B4 ) )
     => ( member_real_a @ C @ B4 ) ) ).

% IntD2
thf(fact_1188_IntD2,axiom,
    ! [C: c > b,A3: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ ( inf_inf_set_c_b @ A3 @ B4 ) )
     => ( member_c_b @ C @ B4 ) ) ).

% IntD2
thf(fact_1189_IntD2,axiom,
    ! [C: a > b,A3: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ ( inf_inf_set_a_b @ A3 @ B4 ) )
     => ( member_a_b @ C @ B4 ) ) ).

% IntD2
thf(fact_1190_Ball__def,axiom,
    ( ball_real_b
    = ( ^ [A5: set_real_b,P3: ( real > b ) > $o] :
        ! [X: real > b] :
          ( ( member_real_b @ X @ A5 )
         => ( P3 @ X ) ) ) ) ).

% Ball_def
thf(fact_1191_Ball__def,axiom,
    ( ball_real_a
    = ( ^ [A5: set_real_a,P3: ( real > a ) > $o] :
        ! [X: real > a] :
          ( ( member_real_a @ X @ A5 )
         => ( P3 @ X ) ) ) ) ).

% Ball_def
thf(fact_1192_Ball__def,axiom,
    ( ball_c_b
    = ( ^ [A5: set_c_b,P3: ( c > b ) > $o] :
        ! [X: c > b] :
          ( ( member_c_b @ X @ A5 )
         => ( P3 @ X ) ) ) ) ).

% Ball_def
thf(fact_1193_Ball__def,axiom,
    ( ball_a_b
    = ( ^ [A5: set_a_b,P3: ( a > b ) > $o] :
        ! [X: a > b] :
          ( ( member_a_b @ X @ A5 )
         => ( P3 @ X ) ) ) ) ).

% Ball_def
thf(fact_1194_r01__to__r01__r01__snd_Hin01,axiom,
    ! [R3: real,N2: nat] : ( member_nat @ ( r01_to_r01_r01_snd @ R3 @ N2 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% r01_to_r01_r01_snd'in01
thf(fact_1195_r01__to__r01__r01__fst_Hin01,axiom,
    ! [R3: real,N2: nat] : ( member_nat @ ( r01_to_r01_r01_fst @ R3 @ N2 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% r01_to_r01_r01_fst'in01
thf(fact_1196_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_1197_nat_Osingleton__sets,axiom,
    ! [X3: nat] :
      ( ( member_nat @ X3 @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
     => ( member_set_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% nat.singleton_sets
thf(fact_1198_real01__binary__expansion_H__0or1,axiom,
    ! [R3: real,N2: nat] : ( member_nat @ ( r01_binary_expansion @ R3 @ N2 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% real01_binary_expansion'_0or1
thf(fact_1199_nat_Ospace__UNIV,axiom,
    ( ( sigma_space_nat @ borel_8449730974584783410el_nat )
    = top_top_set_nat ) ).

% nat.space_UNIV
thf(fact_1200_measurable__separate,axiom,
    ! [P: real > nat,I3: nat] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( member_set_real @ ( vimage_real_nat @ P @ ( insert_nat @ I3 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% measurable_separate
thf(fact_1201_separate__measurable,axiom,
    ! [P: real > nat] :
      ( ! [I4: nat] : ( member_set_real @ ( vimage_real_nat @ P @ ( insert_nat @ I4 @ 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_1202_ennreal_Oexist__fg,axiom,
    ? [X5: extend8495563244428889912nnreal > real] :
      ( ( member2874014351250825754l_real @ X5 @ ( 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 @ X5 @ Xb )
                = Xb ) ) ) ) ).

% ennreal.exist_fg
thf(fact_1203_bool_Oexist__fg,axiom,
    ? [X5: $o > real] :
      ( ( member_o_real @ X5 @ ( 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 @ X5 @ Xb )
                = Xb ) ) ) ) ).

% bool.exist_fg
thf(fact_1204_real_Oexist__fg,axiom,
    ? [X5: real > real] :
      ( ( member_real_real @ X5 @ ( 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 @ X5 @ Xb )
                = Xb ) ) ) ) ).

% real.exist_fg
thf(fact_1205_r01__binary__expansion_H__lt0,axiom,
    ! [R3: real] :
      ( ( ord_less_eq_real @ R3 @ zero_zero_real )
      = ( ! [N3: nat] :
            ( ( r01_binary_expansion @ R3 @ N3 )
            = zero_zero_nat ) ) ) ).

% r01_binary_expansion'_lt0
thf(fact_1206_r01__binary__expansion_H__gt1,axiom,
    ! [R3: real] :
      ( ( ord_less_eq_real @ one_one_real @ R3 )
      = ( ! [N3: nat] :
            ( ( r01_binary_expansion @ R3 @ N3 )
            = one_one_nat ) ) ) ).

% r01_binary_expansion'_gt1
thf(fact_1207_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 ) )
       => ( ! [R4: real] : ( member_real @ ( F2 @ R4 ) @ ( 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_1208_nat_Oexist__fg,axiom,
    ? [X5: nat > real] :
      ( ( member_nat_real @ X5 @ ( 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 @ X5 @ Xb )
                = Xb ) ) ) ) ).

% nat.exist_fg
thf(fact_1209_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1210_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_1211_powr__nonneg__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X3 ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_1212_powr__one,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ one_one_real )
        = X3 ) ) ).

% powr_one
thf(fact_1213_powr__one__gt__zero__iff,axiom,
    ! [X3: real] :
      ( ( ( powr_real @ X3 @ one_one_real )
        = X3 )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% powr_one_gt_zero_iff
thf(fact_1214_powr__mono,axiom,
    ! [A: real,B: real,X3: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X3 )
       => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ) ).

% powr_mono
thf(fact_1215_powr__le1,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ X3 @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_1216_ln__ge__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ one_one_real @ X3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).

% ln_ge_zero
thf(fact_1217_powr__mono2,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ X3 @ Y3 )
         => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_1218_powr__ge__pzero,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X3 @ Y3 ) ) ).

% powr_ge_pzero
thf(fact_1219_powr__mono__both,axiom,
    ! [A: real,B: real,X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X3 )
         => ( ( ord_less_eq_real @ X3 @ Y3 )
           => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_1220_ge__one__powr__ge__zero,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X3 @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_1221_bool_Osingleton__sets,axiom,
    ! [X3: $o] :
      ( ( member_o @ X3 @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
     => ( member_set_o @ ( insert_o @ X3 @ bot_bot_set_o ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% bool.singleton_sets
thf(fact_1222_ennreal_Osingleton__sets,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ X3 @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
     => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).

% ennreal.singleton_sets
thf(fact_1223_powr__powr__swap,axiom,
    ! [X3: real,A: real,B: real] :
      ( ( powr_real @ ( powr_real @ X3 @ A ) @ B )
      = ( powr_real @ ( powr_real @ X3 @ B ) @ A ) ) ).

% powr_powr_swap
thf(fact_1224_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_1225_ennreal_Ospace__UNIV,axiom,
    ( ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal )
    = top_to7994903218803871134nnreal ) ).

% ennreal.space_UNIV
thf(fact_1226_bool_Ospace__UNIV,axiom,
    ( ( sigma_space_o @ borel_5500255247093592246orel_o )
    = top_top_set_o ) ).

% bool.space_UNIV
thf(fact_1227_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1228_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_1229_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1230_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1231_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1232_kuhn__labelling__lemma_H,axiom,
    ! [P: ( nat > real ) > $o,F2: ( nat > real ) > nat > real,Q: nat > $o] :
      ( ! [X5: nat > real] :
          ( ( P @ X5 )
         => ( P @ ( F2 @ X5 ) ) )
     => ( ! [X5: nat > real] :
            ( ( P @ X5 )
           => ! [I4: nat] :
                ( ( Q @ I4 )
               => ( ( ord_less_eq_real @ zero_zero_real @ ( X5 @ I4 ) )
                  & ( ord_less_eq_real @ ( X5 @ I4 ) @ one_one_real ) ) ) )
       => ? [L3: ( nat > real ) > nat > nat] :
            ( ! [X8: nat > real,I5: nat] : ( ord_less_eq_nat @ ( L3 @ X8 @ I5 ) @ one_one_nat )
            & ! [X8: nat > real,I5: nat] :
                ( ( ( P @ X8 )
                  & ( Q @ I5 )
                  & ( ( X8 @ I5 )
                    = zero_zero_real ) )
               => ( ( L3 @ X8 @ I5 )
                  = zero_zero_nat ) )
            & ! [X8: nat > real,I5: nat] :
                ( ( ( P @ X8 )
                  & ( Q @ I5 )
                  & ( ( X8 @ I5 )
                    = one_one_real ) )
               => ( ( L3 @ X8 @ I5 )
                  = one_one_nat ) )
            & ! [X8: nat > real,I5: nat] :
                ( ( ( P @ X8 )
                  & ( Q @ I5 )
                  & ( ( L3 @ X8 @ I5 )
                    = zero_zero_nat ) )
               => ( ord_less_eq_real @ ( X8 @ I5 ) @ ( F2 @ X8 @ I5 ) ) )
            & ! [X8: nat > real,I5: nat] :
                ( ( ( P @ X8 )
                  & ( Q @ I5 )
                  & ( ( L3 @ X8 @ I5 )
                    = one_one_nat ) )
               => ( ord_less_eq_real @ ( F2 @ X8 @ I5 ) @ ( X8 @ I5 ) ) ) ) ) ) ).

% kuhn_labelling_lemma'
thf(fact_1233_biexp01__well__formedE,axiom,
    ! [A: nat > nat] :
      ( ( biexp01_well_formed @ A )
     => ( ! [N4: nat] : ( member_nat @ ( A @ N4 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
        & ! [N4: nat] :
          ? [M3: nat] :
            ( ( ord_less_eq_nat @ N4 @ M3 )
            & ( ( A @ M3 )
              = zero_zero_nat ) ) ) ) ).

% biexp01_well_formedE
thf(fact_1234_biexp01__well__formedI,axiom,
    ! [A: nat > nat] :
      ( ! [N5: nat] : ( member_nat @ ( A @ N5 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
     => ( ! [N5: nat] :
          ? [M4: nat] :
            ( ( ord_less_eq_nat @ N5 @ M4 )
            & ( ( A @ M4 )
              = zero_zero_nat ) )
       => ( biexp01_well_formed @ A ) ) ) ).

% biexp01_well_formedI
thf(fact_1235_biexp01__well__formed__def,axiom,
    ( biexp01_well_formed
    = ( ^ [A6: nat > nat] :
          ( ! [N3: nat] : ( member_nat @ ( A6 @ N3 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
          & ! [N3: nat] :
            ? [M5: nat] :
              ( ( ord_less_eq_nat @ N3 @ M5 )
              & ( ( A6 @ M5 )
                = zero_zero_nat ) ) ) ) ) ).

% biexp01_well_formed_def
thf(fact_1236_measurable__ennreal,axiom,
    member2919562650594848410nnreal @ extend7643940197134561352nnreal @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).

% measurable_ennreal
thf(fact_1237_ennreal3__cases,axiom,
    ! [X3: extend8495563244428889912nnreal,Xa2: extend8495563244428889912nnreal,Xaa: extend8495563244428889912nnreal] :
      ( ! [R4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R4 )
         => ( ( X3
              = ( extend7643940197134561352nnreal @ R4 ) )
           => ! [Ra: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ Ra )
               => ( ( Xa2
                    = ( extend7643940197134561352nnreal @ Ra ) )
                 => ! [Raa: real] :
                      ( ( ord_less_eq_real @ zero_zero_real @ Raa )
                     => ( Xaa
                       != ( extend7643940197134561352nnreal @ Raa ) ) ) ) ) ) )
     => ( ! [R4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ R4 )
           => ( ( X3
                = ( extend7643940197134561352nnreal @ R4 ) )
             => ! [Ra: real] :
                  ( ( ord_less_eq_real @ zero_zero_real @ Ra )
                 => ( ( Xa2
                      = ( extend7643940197134561352nnreal @ Ra ) )
                   => ( Xaa != top_to1496364449551166952nnreal ) ) ) ) )
       => ( ! [R4: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ R4 )
             => ( ( X3
                  = ( extend7643940197134561352nnreal @ R4 ) )
               => ( ( Xa2 = top_to1496364449551166952nnreal )
                 => ! [Ra: real] :
                      ( ( ord_less_eq_real @ zero_zero_real @ Ra )
                     => ( Xaa
                       != ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
         => ( ! [R4: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ R4 )
               => ( ( X3
                    = ( extend7643940197134561352nnreal @ R4 ) )
                 => ( ( Xa2 = top_to1496364449551166952nnreal )
                   => ( Xaa != top_to1496364449551166952nnreal ) ) ) )
           => ( ( ( X3 = top_to1496364449551166952nnreal )
               => ! [R4: real] :
                    ( ( ord_less_eq_real @ zero_zero_real @ R4 )
                   => ( ( Xa2
                        = ( extend7643940197134561352nnreal @ R4 ) )
                     => ! [Ra: real] :
                          ( ( ord_less_eq_real @ zero_zero_real @ Ra )
                         => ( Xaa
                           != ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
             => ( ( ( X3 = top_to1496364449551166952nnreal )
                 => ! [R4: real] :
                      ( ( ord_less_eq_real @ zero_zero_real @ R4 )
                     => ( ( Xa2
                          = ( extend7643940197134561352nnreal @ R4 ) )
                       => ( Xaa != top_to1496364449551166952nnreal ) ) ) )
               => ( ( ( X3 = top_to1496364449551166952nnreal )
                   => ( ( Xa2 = top_to1496364449551166952nnreal )
                     => ! [R4: real] :
                          ( ( ord_less_eq_real @ zero_zero_real @ R4 )
                         => ( Xaa
                           != ( extend7643940197134561352nnreal @ R4 ) ) ) ) )
                 => ~ ( ( X3 = top_to1496364449551166952nnreal )
                     => ( ( Xa2 = top_to1496364449551166952nnreal )
                       => ( Xaa != top_to1496364449551166952nnreal ) ) ) ) ) ) ) ) ) ) ).

% ennreal3_cases
thf(fact_1238_ennreal2__cases,axiom,
    ! [X3: extend8495563244428889912nnreal,Xa2: extend8495563244428889912nnreal] :
      ( ! [R4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R4 )
         => ( ( X3
              = ( extend7643940197134561352nnreal @ R4 ) )
           => ! [Ra: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ Ra )
               => ( Xa2
                 != ( extend7643940197134561352nnreal @ Ra ) ) ) ) )
     => ( ! [R4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ R4 )
           => ( ( X3
                = ( extend7643940197134561352nnreal @ R4 ) )
             => ( Xa2 != top_to1496364449551166952nnreal ) ) )
       => ( ( ( X3 = top_to1496364449551166952nnreal )
           => ! [R4: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ R4 )
               => ( Xa2
                 != ( extend7643940197134561352nnreal @ R4 ) ) ) )
         => ~ ( ( X3 = top_to1496364449551166952nnreal )
             => ( Xa2 != top_to1496364449551166952nnreal ) ) ) ) ) ).

% ennreal2_cases
thf(fact_1239_ennreal__zero__neq__top,axiom,
    zero_z7100319975126383169nnreal != top_to1496364449551166952nnreal ).

% ennreal_zero_neq_top
thf(fact_1240_ennreal__top__neq__one,axiom,
    top_to1496364449551166952nnreal != one_on2969667320475766781nnreal ).

% ennreal_top_neq_one
thf(fact_1241_neq__top__trans,axiom,
    ! [Y3: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] :
      ( ( Y3 != top_to1496364449551166952nnreal )
     => ( ( ord_le3935885782089961368nnreal @ X3 @ Y3 )
       => ( X3 != top_to1496364449551166952nnreal ) ) ) ).

% neq_top_trans
thf(fact_1242_top__neq__ennreal,axiom,
    ! [R3: real] :
      ( top_to1496364449551166952nnreal
     != ( extend7643940197134561352nnreal @ R3 ) ) ).

% top_neq_ennreal
thf(fact_1243_ennreal__cases,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( ! [R4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R4 )
         => ( X3
           != ( extend7643940197134561352nnreal @ R4 ) ) )
     => ( X3 = top_to1496364449551166952nnreal ) ) ).

% ennreal_cases
thf(fact_1244_bot__ennreal,axiom,
    bot_bo841427958541957580nnreal = zero_z7100319975126383169nnreal ).

% bot_ennreal
thf(fact_1245_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_1246_ennreal_Ocountable__space__discrete,axiom,
    ( ( counta8439243037236335165nnreal @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
   => ( ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal )
      = ( sigma_5465916536984168985nnreal @ ( sigma_7204664791115113951nnreal @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) ) ) ) ) ).

% ennreal.countable_space_discrete
thf(fact_1247_nat_Ocountable__space__discrete,axiom,
    ( ( counta1168086296615599829le_nat @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
   => ( ( sigma_sets_nat @ borel_8449730974584783410el_nat )
      = ( sigma_sets_nat @ ( sigma_7685844798829912695ce_nat @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) ) ) ) ) ).

% nat.countable_space_discrete
thf(fact_1248_bool_Ocountable__space__discrete,axiom,
    ( ( counta5976203206615340371able_o @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
   => ( ( sigma_sets_o @ borel_5500255247093592246orel_o )
      = ( sigma_sets_o @ ( sigma_count_space_o @ ( sigma_space_o @ borel_5500255247093592246orel_o ) ) ) ) ) ).

% bool.countable_space_discrete
thf(fact_1249_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1250_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_1251_finite__less__ub,axiom,
    ! [F2: nat > nat,U3: nat] :
      ( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F2 @ N5 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ U3 ) ) ) ) ).

% finite_less_ub
thf(fact_1252_ln__neg__is__const,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ln_ln_real @ X3 )
        = ( the_real
          @ ^ [X: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_1253_uncountable__UNIV__real,axiom,
    ~ ( counta7319604579010473777e_real @ top_top_set_real ) ).

% uncountable_UNIV_real
thf(fact_1254_complex__non__denum,axiom,
    ~ ? [F4: nat > complex] :
        ( ( image_nat_complex @ F4 @ top_top_set_nat )
        = top_top_set_complex ) ).

% complex_non_denum
thf(fact_1255_real__non__denum,axiom,
    ~ ? [F4: nat > real] :
        ( ( image_nat_real @ F4 @ top_top_set_nat )
        = top_top_set_real ) ).

% real_non_denum
thf(fact_1256_uncountable__UNIV__complex,axiom,
    ~ ( counta5113917769705169331omplex @ top_top_set_complex ) ).

% uncountable_UNIV_complex
thf(fact_1257_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_1258_real_Ostandard__borel__lr__sets__ident,axiom,
    ( ( sigma_sets_real @ ( measur1733462625046462224e_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) )
    = ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% real.standard_borel_lr_sets_ident
thf(fact_1259_ennreal_Ostandard__borel__lr__sets__ident,axiom,
    ( ( sigma_5465916536984168985nnreal @ ( measur7384687747506661788nnreal @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) ) )
    = ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ).

% ennreal.standard_borel_lr_sets_ident
thf(fact_1260_nat_Ostandard__borel__lr__sets__ident,axiom,
    ( ( sigma_sets_nat @ ( measur7418878410283781684re_nat @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) ) )
    = ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ).

% nat.standard_borel_lr_sets_ident
thf(fact_1261_bool_Ostandard__borel__lr__sets__ident,axiom,
    ( ( sigma_sets_o @ ( measur2926627334652526644sure_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) ) )
    = ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ).

% bool.standard_borel_lr_sets_ident
thf(fact_1262_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_1263_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_1264_ennreal__Sup,axiom,
    ! [A3: set_real] :
      ( ( ( comple6814414086264997003nnreal @ ( image_7616191137145695467nnreal @ extend7643940197134561352nnreal @ A3 ) )
       != top_to1496364449551166952nnreal )
     => ( ( A3 != bot_bot_set_real )
       => ( ( extend7643940197134561352nnreal @ ( comple1385675409528146559p_real @ A3 ) )
          = ( comple6814414086264997003nnreal @ ( image_7616191137145695467nnreal @ extend7643940197134561352nnreal @ A3 ) ) ) ) ) ).

% ennreal_Sup

% Helper facts (25)
thf(help_If_2_1_If_001tf__b_T,axiom,
    ! [X3: b,Y3: b] :
      ( ( if_b @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001tf__b_T,axiom,
    ! [X3: b,Y3: b] :
      ( ( if_b @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__a_J_T,axiom,
    ! [X3: sum_sum_a_a,Y3: sum_sum_a_a] :
      ( ( if_Sum_sum_a_a @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__a_J_T,axiom,
    ! [X3: sum_sum_a_a,Y3: sum_sum_a_a] :
      ( ( if_Sum_sum_a_a @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_T,axiom,
    ! [X3: sum_sum_a_b,Y3: sum_sum_a_b] :
      ( ( if_Sum_sum_a_b @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_T,axiom,
    ! [X3: sum_sum_a_b,Y3: sum_sum_a_b] :
      ( ( if_Sum_sum_a_b @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_T,axiom,
    ! [X3: sum_sum_a_c,Y3: sum_sum_a_c] :
      ( ( if_Sum_sum_a_c @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_T,axiom,
    ! [X3: sum_sum_a_c,Y3: sum_sum_a_c] :
      ( ( if_Sum_sum_a_c @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__b_Mtf__a_J_T,axiom,
    ! [X3: sum_sum_b_a,Y3: sum_sum_b_a] :
      ( ( if_Sum_sum_b_a @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__b_Mtf__a_J_T,axiom,
    ! [X3: sum_sum_b_a,Y3: sum_sum_b_a] :
      ( ( if_Sum_sum_b_a @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__b_Mtf__b_J_T,axiom,
    ! [X3: sum_sum_b_b,Y3: sum_sum_b_b] :
      ( ( if_Sum_sum_b_b @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__b_Mtf__b_J_T,axiom,
    ! [X3: sum_sum_b_b,Y3: sum_sum_b_b] :
      ( ( if_Sum_sum_b_b @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__b_Mtf__c_J_T,axiom,
    ! [X3: sum_sum_b_c,Y3: sum_sum_b_c] :
      ( ( if_Sum_sum_b_c @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__b_Mtf__c_J_T,axiom,
    ! [X3: sum_sum_b_c,Y3: sum_sum_b_c] :
      ( ( if_Sum_sum_b_c @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__c_Mtf__a_J_T,axiom,
    ! [X3: sum_sum_c_a,Y3: sum_sum_c_a] :
      ( ( if_Sum_sum_c_a @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__c_Mtf__a_J_T,axiom,
    ! [X3: sum_sum_c_a,Y3: sum_sum_c_a] :
      ( ( if_Sum_sum_c_a @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__c_Mtf__b_J_T,axiom,
    ! [X3: sum_sum_c_b,Y3: sum_sum_c_b] :
      ( ( if_Sum_sum_c_b @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__c_Mtf__b_J_T,axiom,
    ! [X3: sum_sum_c_b,Y3: sum_sum_c_b] :
      ( ( if_Sum_sum_c_b @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__c_Mtf__c_J_T,axiom,
    ! [X3: sum_sum_c_c,Y3: sum_sum_c_c] :
      ( ( if_Sum_sum_c_c @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__c_Mtf__c_J_T,axiom,
    ! [X3: sum_sum_c_c,Y3: sum_sum_c_c] :
      ( ( if_Sum_sum_c_c @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_3_1_If_001t__Sum____Type__Osum_Itf__b_Mt__Extended____Nonnegative____Real__Oennreal_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__b_Mt__Extended____Nonnegative____Real__Oennreal_J_T,axiom,
    ! [X3: sum_su8602216633299776360nnreal,Y3: sum_su8602216633299776360nnreal] :
      ( ( if_Sum7506096409499832110nnreal @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__b_Mt__Extended____Nonnegative____Real__Oennreal_J_T,axiom,
    ! [X3: sum_su8602216633299776360nnreal,Y3: sum_su8602216633299776360nnreal] :
      ( ( if_Sum7506096409499832110nnreal @ $true @ X3 @ Y3 )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    member_real_b @ ( comp_c_b_real @ g @ alpha_2 ) @ ( qbs_Mx_b @ z ) ).

%------------------------------------------------------------------------------