%------------------------------------------------------------------------------
% 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/0007_Exponent_QuasiBorel/prob_00504_021934__15343818_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 3136 ( 712 unt;1857 typ;   0 def)
%            Number of atoms       : 2451 (1078 equ;   4 cnn)
%            Maximal formula atoms :   36 (   1 avg)
%            Number of connectives : 11162 ( 167   ~;   5   |;  73   &;9945   @)
%                                         (   0 <=>; 972  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   6 avg)
%            Number of types       :  518 ( 517 usr)
%            Number of type conns  : 5930 (5930   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1347 (1340 usr;  59 con; 0-6 aty)
%            Number of variables   : 3832 ( 706   ^;3080   !;  46   ?;3832   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:10:07.887
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_Eo_M_062_I_Eo_M_Eo_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__b_M_062_Itf__a_Mtf__a_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_Eo_Mt__Real__Oreal_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_M_062_It__Nat__Onat_Mt__Real__Oreal_J_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__QuasiBorel__Oquasi____borel_I_062_I_Eo_M_062_I_Eo_M_Eo_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mtf__b_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__b_J_Mtf__b_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__b_J_Mtf__a_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_It__Real__Oreal_M_Eo_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_It__Nat__Onat_Mtf__b_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_M_062_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_It__Nat__Onat_Mtf__b_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_Mt__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Real__Oreal_M_062_Itf__a_Mtf__a_J_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_It__Nat__Onat_M_Eo_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__Product____Type__Oprod_It__Nat__Onat_M_062_I_Eo_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_I_Eo_M_Eo_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_M_062_I_Eo_M_062_I_Eo_M_Eo_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__b_J_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_M_062_It__Real__Oreal_M_Eo_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_M_062_I_Eo_Mt__Real__Oreal_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_Mtf__a_J_Mtf__a_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__Set__Oset_I_062_It__Nat__Onat_M_062_I_Eo_M_Eo_J_J_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__QuasiBorel__Oquasi____borel_I_062_Itf__a_Mtf__a_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__a_Mtf__a_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__Set__Oset_I_062_Itf__b_M_062_Itf__a_Mtf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_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__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__b_M_062_I_Eo_M_Eo_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_I_Eo_M_Eo_J_J_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__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_I_062_I_Eo_M_062_I_Eo_M_Eo_J_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_tf__a,type,
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% Explicit typings (1340)
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thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Binary__Product__QuasiBorel_Opair__qbs_001_062_I_Eo_M_Eo_J_001_Eo,type,
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thf(sy_c_Binary__Product__QuasiBorel_Opair__qbs_001_062_I_Eo_M_Eo_J_001t__Nat__Onat,type,
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thf(sy_c_Binary__Product__QuasiBorel_Opair__qbs_001_062_I_Eo_M_Eo_J_001t__Real__Oreal,type,
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thf(sy_c_Binary__Product__QuasiBorel_Opair__qbs_001_062_I_Eo_Mt__Real__Oreal_J_001_Eo,type,
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thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_J_001_062_Itf__a_Mtf__a_J,type,
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thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_J_001tf__a,type,
    expone3972300415888346915at_b_a: quasi_8132950864781310810_nat_b > quasi_borel_a > quasi_2328276896549485343at_b_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J,type,
    expone1622171987252446258_b_a_a: quasi_7777979582039567160real_b > quasi_7934145443646537599_b_a_a > quasi_72846094295016550_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J,type,
    expone3019181631264744327at_b_a: quasi_7777979582039567160real_b > quasi_2107853155996167056at_b_a > quasi_1718342455875313023at_b_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001_062_Itf__a_Mtf__a_J,type,
    expone6122776153604691080_b_a_a: quasi_7777979582039567160real_b > quasi_borel_a_a > quasi_1728610414117680828_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J,type,
    expone761354066351259609_b_a_a: quasi_6353345141195568462_b_a_a > quasi_6353345141195568462_b_a_a > quasi_8493789682903986389_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a,type,
    expone790792417327382575_a_a_a: quasi_6353345141195568462_b_a_a > quasi_borel_a > quasi_2547937147209033259_a_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_J_001_062_Itf__a_Mtf__a_J,type,
    expone8080306092094974073_b_a_a: quasi_6341084831132689065_nat_b > quasi_borel_a_a > quasi_9011436741951440429_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_001tf__a,type,
    expone6319500819088444429_b_a_a: quasi_3683678768404249670at_b_a > quasi_borel_a > quasi_2781656761898266373_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Real__Oreal_001_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J,type,
    expone5458538795283732920_b_a_a: quasi_borel_real > quasi_3364728345918372901_b_a_a > quasi_5135531091791863728_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Real__Oreal_001_062_Itf__b_M_062_Itf__a_Mtf__a_J_J,type,
    expone3903273362589924110_b_a_a: quasi_borel_real > quasi_borel_b_a_a > quasi_5545028370205528838_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Real__Oreal_001_Eo,type,
    expone1659820337974317777real_o: quasi_borel_real > quasi_borel_o > quasi_borel_real_o ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001t__Real__Oreal_001tf__a,type,
    expone319700646606540215real_a: quasi_borel_real > quasi_borel_a > quasi_borel_real_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001tf__a_001_062_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_Mtf__a_J,type,
    expone5702679380600246409_a_a_a: quasi_borel_a > quasi_6251275217727459860_a_a_a > quasi_9112982132339698617_a_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001tf__a_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J,type,
    expone5574410363414041987_b_a_a: quasi_borel_a > quasi_6353345141195568462_b_a_a > quasi_3696870048931568255_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001tf__a_001tf__a,type,
    exponent_exp_qbs_a_a: quasi_borel_a > quasi_borel_a > quasi_borel_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001tf__b_001_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J,type,
    expone5294601226626914677_b_a_a: quasi_borel_b > quasi_7934145443646537599_b_a_a > quasi_3364728345918372901_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001tf__b_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_Mtf__a_J,type,
    expone3665158019928849063_b_a_a: quasi_borel_b > quasi_7686113649425967283_b_a_a > quasi_1416857589949220899_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs_001tf__b_001_062_Itf__a_Mtf__a_J,type,
    expone5499329870516987339_b_a_a: quasi_borel_b > quasi_borel_a_a > quasi_borel_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001_062_It__Nat__Onat_Mtf__b_J_001_062_It__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_Mtf__a_J,type,
    expone2444466320815533341_a_a_a: quasi_borel_nat_b > quasi_2547937147209033259_a_a_a > set_re3587539620453870461_a_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001_Eo_001_062_I_Eo_M_Eo_J,type,
    expone8146444206115271801_o_o_o: quasi_borel_o > quasi_borel_o_o > set_real_o_o_o ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001_Eo_001_Eo,type,
    expone9009657249589208708Mx_o_o: quasi_borel_o > quasi_borel_o > set_real_o_o ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001t__Nat__Onat_001tf__b,type,
    expone2173014692789731113_nat_b: quasi_borel_nat > quasi_borel_b > set_real_nat_b ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_Itf__a_Mtf__a_J,type,
    expone1541841343128190326_b_a_a: quasi_7334096685896923387_nat_b > quasi_borel_a_a > set_re127182097576499728_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_J_Mtf__a_J_001tf__a,type,
    expone4687418993921882065_b_a_a: quasi_805850127037308055at_b_a > quasi_borel_a > set_re1180566231054566451_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a,type,
    expone2990877755589709658_a_a_a: quasi_6353345141195568462_b_a_a > quasi_borel_a > set_re1328106051401742268_a_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001t__Real__Oreal_001tf__a,type,
    expone6795215201378684364real_a: quasi_borel_real > quasi_borel_a > set_real_real_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001tf__a_001tf__a,type,
    expone7022597896679715716Mx_a_a: quasi_borel_a > quasi_borel_a > set_real_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001tf__b_001_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J,type,
    expone2961370955173070346_b_a_a: quasi_borel_b > quasi_7934145443646537599_b_a_a > set_re2972810236359023978_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oexp__qbs__Mx_001tf__b_001_062_Itf__a_Mtf__a_J,type,
    expone4990064882076513888_b_a_a: quasi_borel_b > quasi_borel_a_a > set_real_b_a_a ).

thf(sy_c_Exponent__QuasiBorel_Oqbs__eval_001_Eo_001_Eo,type,
    expone8878782944832497578al_o_o: product_prod_o_o_o2 > $o ).

thf(sy_c_Exponent__QuasiBorel_Oqbs__eval_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_Itf__a_Mtf__a_J,type,
    expone8712287761860780188_b_a_a: produc5941874481157875579_nat_b > a > a ).

thf(sy_c_Exponent__QuasiBorel_Oqbs__eval_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a,type,
    expone3941855099027480448_a_a_a: produc7813404874125118754_b_a_a > a ).

thf(sy_c_Exponent__QuasiBorel_Oqbs__eval_001tf__a_001tf__a,type,
    expone7211744221881241258al_a_a: product_prod_a_a_a > a ).

thf(sy_c_Exponent__QuasiBorel_Oqbs__eval_001tf__b_001_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J,type,
    expone6597005973208256868_b_a_a: produc202582562792018919_a_a_b > produc7194034526493574155_nat_b > a > a ).

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

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_Mtf__b_J_001_062_I_Eo_M_Eo_J_001t__Real__Oreal,type,
    comp_nat_b_o_o_real: ( ( nat > b ) > $o > $o ) > ( real > nat > b ) > real > $o > $o ).

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_Mtf__b_J_001_062_It__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_Mtf__a_J_001t__Real__Oreal,type,
    comp_n1554393163112911916a_real: ( ( nat > b ) > produc432120507726958376_b_a_a > a ) > ( real > nat > b ) > real > produc432120507726958376_b_a_a > a ).

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_Mtf__b_J_001_062_Itf__a_Mtf__a_J_001t__Real__Oreal,type,
    comp_nat_b_a_a_real: ( ( nat > b ) > a > a ) > ( real > nat > b ) > real > a > a ).

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_Mtf__b_J_001_062_Itf__b_M_062_Itf__a_Mtf__a_J_J_001t__Real__Oreal,type,
    comp_n7768269484769491838a_real: ( ( nat > b ) > b > a > a ) > ( real > nat > b ) > real > b > a > a ).

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_Mtf__b_J_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    comp_n3851728294588896943_nat_b: ( ( nat > b ) > produc5451071141416689831_nat_b ) > ( produc7194034526493574155_nat_b > nat > b ) > produc7194034526493574155_nat_b > produc5451071141416689831_nat_b ).

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_Mtf__b_J_001tf__b_001t__Real__Oreal,type,
    comp_nat_b_b_real: ( ( nat > b ) > b ) > ( real > nat > b ) > real > b ).

thf(sy_c_Fun_Ocomp_001_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_001t__Product____Type__Oprod_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J,type,
    comp_b3789700955105422685_b_a_a: ( ( b > produc7194034526493574155_nat_b > a > a ) > produc2067044289602187751_b_a_a ) > ( produc432120507726958376_b_a_a > b > produc7194034526493574155_nat_b > a > a ) > produc432120507726958376_b_a_a > produc2067044289602187751_b_a_a ).

thf(sy_c_Fun_Ocomp_001_Eo_001_062_I_Eo_M_Eo_J_001t__Real__Oreal,type,
    comp_o_o_o_real: ( $o > $o > $o ) > ( real > $o ) > real > $o > $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_001_Eo_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_o_real_nat: ( $o > real ) > ( nat > $o ) > nat > real ).

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

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    comp_l1050289791651215098_nat_b: ( list_b > produc7194034526493574155_nat_b ) > ( produc7194034526493574155_nat_b > list_b ) > produc7194034526493574155_nat_b > produc7194034526493574155_nat_b ).

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

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

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    comp_n8051653116556791961_nat_b: ( nat > product_prod_nat_nat ) > ( produc7194034526493574155_nat_b > nat ) > produc7194034526493574155_nat_b > product_prod_nat_nat ).

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

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

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

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_It__Nat__Onat_Mtf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    comp_P3511690182758988141_nat_b: ( produc5451071141416689831_nat_b > nat > b ) > ( produc7194034526493574155_nat_b > produc5451071141416689831_nat_b ) > produc7194034526493574155_nat_b > nat > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_It__Nat__Onat_Mtf__b_J_001t__Real__Oreal,type,
    comp_P5734460479892768782b_real: ( produc5451071141416689831_nat_b > nat > b ) > ( real > produc5451071141416689831_nat_b ) > real > nat > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__b_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mtf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    comp_P272664786628135081_nat_b: ( produc2403698915930647397_b_nat > nat > b ) > ( produc7194034526493574155_nat_b > produc2403698915930647397_b_nat ) > produc7194034526493574155_nat_b > nat > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__b_J_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    comp_P6619190391790938277_nat_b: ( produc2403698915930647397_b_nat > nat ) > ( produc7194034526493574155_nat_b > produc2403698915930647397_b_nat ) > produc7194034526493574155_nat_b > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mtf__b_J_Mtf__a_J_001_062_It__Nat__Onat_Mtf__b_J_001t__Real__Oreal,type,
    comp_P2494481557755200758b_real: ( product_prod_nat_b_a > nat > b ) > ( real > product_prod_nat_b_a ) > real > nat > b ).

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    comp_P6943517474843379220al_nat: ( produc2422161461964618553l_real > real ) > ( nat > produc2422161461964618553l_real ) > nat > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_P3276979369905350645l_real: ( produc2422161461964618553l_real > real ) > ( produc2422161461964618553l_real > produc2422161461964618553l_real ) > produc2422161461964618553l_real > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__a_J_001tf__a_001t__Real__Oreal,type,
    comp_P7874914735527052288a_real: ( product_prod_real_a > a ) > ( real > product_prod_real_a ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001_062_Itf__a_Mtf__a_J_001t__Real__Oreal,type,
    comp_P2278155893825476926a_real: ( product_prod_real_b > a > a ) > ( real > product_prod_real_b ) > real > a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001t__Real__Oreal_001_Eo,type,
    comp_P4534312704559669631real_o: ( product_prod_real_b > real ) > ( $o > product_prod_real_b ) > $o > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_P5368948159003514281al_nat: ( product_prod_real_b > real ) > ( nat > product_prod_real_b ) > nat > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001t__Real__Oreal_001t__Product____Type__Oprod_Itf__b_Mt__Real__Oreal_J,type,
    comp_P8514091664527323699b_real: ( product_prod_real_b > real ) > ( product_prod_b_real > product_prod_real_b ) > product_prod_b_real > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001tf__b_001_Eo,type,
    comp_P7576070003299083396_b_b_o: ( product_prod_real_b > b ) > ( $o > product_prod_real_b ) > $o > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001tf__b_001t__Nat__Onat,type,
    comp_P1903834106658840036_b_nat: ( product_prod_real_b > b ) > ( nat > product_prod_real_b ) > nat > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001tf__b_001t__Product____Type__Oprod_Itf__b_Mt__Real__Oreal_J,type,
    comp_P488263325884094958b_real: ( product_prod_real_b > b ) > ( product_prod_b_real > product_prod_real_b ) > product_prod_b_real > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_It__Nat__Onat_Mtf__b_J_001t__Real__Oreal,type,
    comp_P2831363238483770278b_real: ( product_prod_a_nat_b > nat > b ) > ( real > product_prod_a_nat_b ) > real > nat > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mtf__b_J_J_001tf__a_001t__Real__Oreal,type,
    comp_P6709701325929358526a_real: ( product_prod_a_nat_b > a ) > ( real > product_prod_a_nat_b ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_001t__Product____Type__Oprod_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_Mtf__a_J,type,
    comp_P3970313683510312364_a_a_a: ( produc432120507726958376_b_a_a > b > produc7194034526493574155_nat_b > a > a ) > ( produc202582558488790118_a_a_a > produc432120507726958376_b_a_a ) > produc202582558488790118_a_a_a > b > produc7194034526493574155_nat_b > a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a_001t__Product____Type__Oprod_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_J,type,
    comp_P3156607648531775677at_b_a: ( produc432120507726958376_b_a_a > a ) > ( produc778026099599033348at_b_a > produc432120507726958376_b_a_a ) > produc778026099599033348at_b_a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a_001t__Product____Type__Oprod_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_Mtf__a_J,type,
    comp_P8744978583584356205_a_a_a: ( produc432120507726958376_b_a_a > a ) > ( produc202582558488790118_a_a_a > produc432120507726958376_b_a_a ) > produc202582558488790118_a_a_a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J,type,
    comp_P6428769393572761825at_b_a: ( produc432120507726958376_b_a_a > a ) > ( produc6209191202727158440at_b_a > produc432120507726958376_b_a_a ) > produc6209191202727158440at_b_a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
    comp_P2579465633625509435_nat_b: ( produc432120507726958376_b_a_a > a ) > ( produc3260441645384719156_nat_b > produc432120507726958376_b_a_a ) > produc3260441645384719156_nat_b > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J,type,
    comp_P8974516532822524463_b_a_a: ( produc432120507726958376_b_a_a > a ) > ( produc432120507726958376_b_a_a > produc432120507726958376_b_a_a ) > produc432120507726958376_b_a_a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J,type,
    comp_P1539513916346917903at_b_a: ( produc432120507726958376_b_a_a > a ) > ( produc4883615123598087894at_b_a > produc432120507726958376_b_a_a ) > produc4883615123598087894at_b_a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__a_001tf__a,type,
    comp_P466993921277732229_a_a_a: ( produc432120507726958376_b_a_a > a ) > ( a > produc432120507726958376_b_a_a ) > a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J,type,
    comp_P1156977722522261166_b_a_a: ( product_prod_a_a > a ) > ( produc432120507726958376_b_a_a > product_prod_a_a ) > produc432120507726958376_b_a_a > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a_001t__Real__Oreal,type,
    comp_P7111135413877744038a_real: ( product_prod_a_a > a ) > ( real > product_prod_a_a ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__b_001t__Real__Oreal,type,
    comp_P3483981992211098919b_real: ( product_prod_a_a > b ) > ( real > product_prod_a_a ) > real > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a_001t__Real__Oreal,type,
    comp_P1838655012641556709a_real: ( product_prod_a_b > a ) > ( real > product_prod_a_b ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b_001t__Real__Oreal,type,
    comp_P7434873627829687398b_real: ( product_prod_a_b > b ) > ( real > product_prod_a_b ) > real > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__b_M_062_It__Nat__Onat_Mtf__b_J_J_001_062_It__Nat__Onat_Mtf__b_J_001t__Real__Oreal,type,
    comp_P9026943010836981671b_real: ( product_prod_b_nat_b > nat > b ) > ( real > product_prod_b_nat_b ) > real > nat > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__b_M_062_It__Nat__Onat_Mtf__b_J_J_001tf__b_001t__Real__Oreal,type,
    comp_P447806182887879104b_real: ( product_prod_b_nat_b > b ) > ( real > product_prod_b_nat_b ) > real > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_Mtf__b_J,type,
    comp_P4442744351759092527_b_a_b: ( produc4883615123598087894at_b_a > produc6209191202727158440at_b_a ) > ( produc7743328458780956108_b_a_b > produc4883615123598087894at_b_a ) > produc7743328458780956108_b_a_b > produc6209191202727158440at_b_a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_001tf__b_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_Mtf__b_J,type,
    comp_P4460891510108933714_b_a_b: ( produc4883615123598087894at_b_a > b ) > ( produc7743328458780956108_b_a_b > produc4883615123598087894at_b_a ) > produc7743328458780956108_b_a_b > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__a_001t__Real__Oreal,type,
    comp_P7778034889417652391a_real: ( product_prod_b_a > a ) > ( real > product_prod_b_a ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__b_Mtf__a_J_001tf__b_001t__Real__Oreal,type,
    comp_P4150881467751007272b_real: ( product_prod_b_a > b ) > ( real > product_prod_b_a ) > real > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__a_001t__Real__Oreal,type,
    comp_P2505554488181465062a_real: ( product_prod_b_b > a ) > ( real > product_prod_b_b ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b_001t__Real__Oreal,type,
    comp_P8101773103369595751b_real: ( product_prod_b_b > b ) > ( real > product_prod_b_b ) > real > b ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_062_It__Nat__Onat_Mtf__b_J_001t__Real__Oreal,type,
    comp_real_nat_b_real: ( real > nat > b ) > ( real > real ) > real > nat > b ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_001t__Real__Oreal,type,
    comp_r2587692399004186460a_real: ( real > b > produc7194034526493574155_nat_b > a > a ) > ( real > real ) > real > b > produc7194034526493574155_nat_b > a > a ).

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_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    comp_r6281409797179841921nnreal: ( real > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > real ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

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

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

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_J_001t__Real__Oreal,type,
    comp_r2834535067444373313a_real: ( real > produc778026099599033348at_b_a ) > ( real > real ) > real > produc778026099599033348at_b_a ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    comp_r4825950202331181903t_real: ( real > produc7716430852924023517t_real ) > ( produc7716430852924023517t_real > real ) > produc7716430852924023517t_real > produc7716430852924023517t_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_001t__Real__Oreal,type,
    comp_r4030900867926482106l_real: ( real > produc7716430852924023517t_real ) > ( real > real ) > real > produc7716430852924023517t_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_r97482993418849103l_real: ( real > produc2422161461964618553l_real ) > ( produc2422161461964618553l_real > real ) > produc2422161461964618553l_real > produc2422161461964618553l_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal,type,
    comp_r4464757161866874646l_real: ( real > produc2422161461964618553l_real ) > ( real > real ) > real > produc2422161461964618553l_real ).

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

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

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    comp_r8349034141028359008t_real: ( real > real ) > ( produc7716430852924023517t_real > real ) > produc7716430852924023517t_real > real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_r4460121664545661500l_real: ( real > real ) > ( produc2422161461964618553l_real > real ) > produc2422161461964618553l_real > real ).

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

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

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001t__Nat__Onat,type,
    comp_real_a_nat: ( real > a ) > ( nat > real ) > nat > a ).

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

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

thf(sy_c_Fun_Ocomp_001tf__a_001_062_I_Eo_M_Eo_J_001t__Real__Oreal,type,
    comp_a_o_o_real: ( a > $o > $o ) > ( real > a ) > real > $o > $o ).

thf(sy_c_Fun_Ocomp_001tf__a_001_062_Itf__a_Mtf__a_J_001t__Real__Oreal,type,
    comp_a_a_a_real: ( a > a > a ) > ( real > a ) > real > a > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001_062_Itf__b_M_062_Itf__a_Mtf__a_J_J_001t__Real__Oreal,type,
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thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001t__Real__Oreal,type,
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thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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thf(sy_c_Measure__QuasiBorel__Adjunction_Omeasure__to__qbs_001t__Nat__Onat,type,
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thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mtf__b_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_062_It__Real__Oreal_M_Eo_J_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001_Eo_001t__Real__Oreal,type,
    product_snd_o_real: product_prod_o_real > real ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__b_J,type,
    produc4809910044363821583_nat_b: produc7194034526493574155_nat_b > nat > b ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Real__Oreal,type,
    product_snd_nat_real: produc7716430852924023517t_real > real ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J,type,
    produc6064941407584602266real_b: produc6558228558545702196real_b > product_prod_real_b ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_001tf__a,type,
    produc5227817866736700332at_b_a: produc6209191202727158440at_b_a > a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_001tf__b,type,
    produc4895988176267928656_b_a_b: produc7743328458780956108_b_a_b > b ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_J_001tf__a,type,
    produc7105243515406072463at_b_a: produc1285742814635146737at_b_a > a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    produc5631749817756603418_nat_b: produc3260441645384719156_nat_b > produc7194034526493574155_nat_b ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_001_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J,type,
    produc5086754709204743842_b_a_a: produc1055083314214236062_b_a_a > b > produc7194034526493574155_nat_b > a > a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001_Eo,type,
    product_snd_real_o: product_prod_real_o > $o ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__Real__Oreal,type,
    produc3484788084999411615l_real: produc2422161461964618553l_real > real ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001tf__b,type,
    product_snd_real_b: product_prod_real_b > b ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__b_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
    produc6427044318018601168_nat_b: produc1757427708480497458_nat_b > set_real_nat_b ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_J,type,
    produc6712843908289693712_b_a_a: produc3668604958088222194_b_a_a > set_re2972810236359023978_b_a_a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Product____Type__Oprod_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_J_J_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Product____Type__Oprod_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_J_J_J,type,
    produc1277482548320373382at_b_a: produc7370224728536802976at_b_a > set_re301636585399362053at_b_a ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_Itf__a_J_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
    produc4125011029606581840real_a: produc680249927945959346real_a > set_real_a ).

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

thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001_062_It__Nat__Onat_Mtf__b_J,type,
    product_snd_a_nat_b: product_prod_a_nat_b > nat > b ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J,type,
    produc7355951146076385094_b_a_a: produc432120507726958376_b_a_a > b > produc7194034526493574155_nat_b > a > a ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    produc7959643354479256708_nat_b: produc7306929691551298616_nat_b > produc7194034526493574155_nat_b ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
    product_snd_a_a: product_prod_a_a > a ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__b,type,
    product_snd_a_b: product_prod_a_b > b ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__b_001_062_It__Nat__Onat_Mtf__b_J,type,
    product_snd_b_nat_b: product_prod_b_nat_b > nat > b ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__b_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J,type,
    produc7666751653692995234at_b_a: produc4883615123598087894at_b_a > produc6209191202727158440at_b_a ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__b_001t__Real__Oreal,type,
    product_snd_b_real: product_prod_b_real > real ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__b_001tf__a,type,
    product_snd_b_a: product_prod_b_a > a ).

thf(sy_c_Product__Type_Oprod_Osnd_001tf__b_001tf__b,type,
    product_snd_b_b: product_prod_b_b > b ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_It__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_Mtf__a_J_J,type,
    qbs_Mx6122946776669592201_a_a_a: quasi_8909271866012278712_a_a_a > set_re3587539620453870461_a_a_a ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
    qbs_Mx_o_o_o: quasi_borel_o_o_o > set_real_o_o_o ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_I_Eo_M_Eo_J,type,
    qbs_Mx_o_o: quasi_borel_o_o > set_real_o_o ).

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

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

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Nat__Onat_Mtf__b_J,type,
    qbs_Mx_nat_b: quasi_borel_nat_b > set_real_nat_b ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J,type,
    qbs_Mx442805946153557208_b_a_a: quasi_7934145443646537599_b_a_a > set_re127182097576499728_b_a_a ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_Mtf__a_J,type,
    qbs_Mx506040344577873796_a_a_a: quasi_2547937147209033259_a_a_a > set_re1328106051401742268_a_a_a ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_Itf__a_Mtf__a_J,type,
    qbs_Mx_a_a: quasi_borel_a_a > set_real_a_a ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J,type,
    qbs_Mx703024977670096246_b_a_a: quasi_3364728345918372901_b_a_a > set_re2972810236359023978_b_a_a ).

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

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

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Product____Type__Oprod_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_J,type,
    qbs_Mx5919617464495388621at_b_a: quasi_331246355085487092at_b_a > set_re301636585399362053at_b_a ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    qbs_Mx1531206082039394900_nat_b: quasi_7334096685896923387_nat_b > set_re2747528124059546252_nat_b ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_Mx6884344600386557780t_real: quasi_7702169472194803971t_real > set_re7361189984880485448t_real ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    qbs_Mx2863504943393711152l_real: quasi_5476411728266286559l_real > set_re3939601646622484004l_real ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J,type,
    qbs_Mx2044252502963291921real_b: quasi_7777979582039567160real_b > set_re843112671205889097real_b ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J,type,
    qbs_Mx450739389344589599_b_a_a: quasi_6353345141195568462_b_a_a > set_re8315446236746704915_b_a_a ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J,type,
    qbs_Mx5318780673203594015at_b_a: quasi_3683678768404249670at_b_a > set_re1661134273638700119at_b_a ).

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_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__morphism_001_062_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_M_062_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_Mtf__a_J_J_001_062_It__Product____Type__Oprod_I_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J_J_Mtf__a_J,type,
    qbs_mo7857648355211746802_b_a_a: quasi_6672903365909695221_b_a_a > quasi_1503741678319062615_b_a_a > set_b_703048041285733507_b_a_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_I_Eo_M_Eo_J_001_062_I_Eo_M_Eo_J,type,
    qbs_morphism_o_o_o_o: quasi_borel_o_o > quasi_borel_o_o > set_o_o_o_o2 ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_I_Eo_M_Eo_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    qbs_mo7207216841383384598o_real: quasi_borel_o_o > quasi_borel_o_real > set_o_o_o_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_I_Eo_Mt__Real__Oreal_J_001_062_I_Eo_M_Eo_J,type,
    qbs_mo1176275702449585058al_o_o: quasi_borel_o_real > quasi_borel_o_o > set_o_real_o_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_I_Eo_Mt__Real__Oreal_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    qbs_mo365478700572875064o_real: quasi_borel_o_real > quasi_borel_o_real > set_o_real_o_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_I_Eo_Mt__Real__Oreal_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_mo1345815866363944536t_real: quasi_borel_o_real > quasi_borel_nat_real > set_o_real_nat_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_Itf__a_Mtf__a_J_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J,type,
    qbs_mo6989279613654713658_b_a_a: quasi_2692898989360589641_b_a_a > quasi_7934145443646537599_b_a_a > set_na6516019671708011139_b_a_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Nat__Onat_Mtf__b_J_001_062_I_Eo_M_Eo_J,type,
    qbs_mo3001173925234491475_b_o_o: quasi_borel_nat_b > quasi_borel_o_o > set_nat_b_o_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Nat__Onat_Mtf__b_J_001_062_It__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_Mt__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_J,type,
    qbs_mo3092601099906693971_b_a_a: quasi_borel_nat_b > quasi_8493789682903986389_b_a_a > set_na5164923478861862172_b_a_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Nat__Onat_Mtf__b_J_001_062_It__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_Mtf__a_J,type,
    qbs_mo8618518167108125737_a_a_a: quasi_borel_nat_b > quasi_2547937147209033259_a_a_a > set_na8676686263003397490_a_a_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Nat__Onat_Mtf__b_J_001_062_Itf__a_Mt__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J_J,type,
    qbs_mo7389655384932950653_b_a_a: quasi_borel_nat_b > quasi_3696870048931568255_b_a_a > set_na4279188205053773766_b_a_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Nat__Onat_Mtf__b_J_001_062_Itf__a_Mtf__a_J,type,
    qbs_mo9010864224564201555_b_a_a: quasi_borel_nat_b > quasi_borel_a_a > set_nat_b_a_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Nat__Onat_Mtf__b_J_001_062_Itf__b_M_062_Itf__a_Mtf__a_J_J,type,
    qbs_mo4269413408830428775_b_a_a: quasi_borel_nat_b > quasi_borel_b_a_a > set_nat_b_b_a_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Nat__Onat_Mtf__b_J_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J,type,
    qbs_mo3787342477198169849_nat_b: quasi_borel_nat_b > quasi_7334096685896923387_nat_b > set_na3180526565051942978_nat_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Nat__Onat_Mtf__b_J_001tf__b,type,
    qbs_morphism_nat_b_b: quasi_borel_nat_b > quasi_borel_b > set_nat_b_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mtf__b_J_M_062_Itf__a_Mtf__a_J_J_J,type,
    qbs_mo988760484062759898_b_a_a: quasi_7934145443646537599_b_a_a > quasi_2692898989360589641_b_a_a > set_Pr580766649202342891_b_a_a ).

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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J,type,
    member8909807733118965439at_b_a: produc6209191202727158440at_b_a > set_Pr7000326631795273694at_b_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_J_Mtf__a_J,type,
    member1674185049213098906at_b_a: produc1285742814635146737at_b_a > set_Pr1374702368013165137at_b_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J_Mt__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_J,type,
    member388886286662666717_nat_b: produc3260441645384719156_nat_b > set_Pr196252041547729684_nat_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J,type,
    member7555082154638246367real_b: product_prod_real_b > set_Pr6660261784198275838real_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_M_062_Itf__b_M_062_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_M_062_Itf__a_Mtf__a_J_J_J_J,type,
    member6783937185859681745_b_a_a: produc432120507726958376_b_a_a > set_Pr7640018354816763144_b_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__b_J_J_Mtf__a_J_J,type,
    member8939744840176901101at_b_a: produc4883615123598087894at_b_a > set_Pr4420982220611316236at_b_a > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_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_v_X,type,
    x: quasi_borel_b ).

thf(sy_v_Y,type,
    y: quasi_borel_a ).

thf(sy_v__092_060alpha_062_H____,type,
    alpha: real > nat > b ).

thf(sy_v__092_060alpha_062____,type,
    alpha2: real > nat > b ).

thf(sy_v__092_060beta_0621____,type,
    beta_1: real > b > produc7194034526493574155_nat_b > a > a ).

thf(sy_v__092_060beta_0622____,type,
    beta_2: real > b ).

thf(sy_v__092_060beta_0623____,type,
    beta_3: real > produc7194034526493574155_nat_b ).

thf(sy_v__092_060beta_0624____,type,
    beta_4: real > a ).

thf(sy_v__092_060beta_062____,type,
    beta: real > produc778026099599033348at_b_a ).

thf(sy_v_a____,type,
    a2: real > b ).

thf(sy_v_na____,type,
    na: nat ).

% Relevant facts (1273)
thf(fact_0_hb_I3_J,axiom,
    member_real_b @ beta_2 @ ( qbs_Mx_b @ x ) ).

% hb(3)
thf(fact_1_hb_I5_J,axiom,
    member_real_a @ beta_4 @ ( qbs_Mx_a @ y ) ).

% hb(5)
thf(fact_2_hb_I4_J,axiom,
    ( member8664213473904227437_nat_b @ beta_3
    @ ( qbs_Mx1531206082039394900_nat_b
      @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
        @ ^ [N: nat] :
            ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
            @ ^ [I: nat] : x ) ) ) ) ).

% hb(4)
thf(fact_3_case__prod__conv,axiom,
    ! [F: real > b > a > a,A: real,B: b] :
      ( ( produc7827042608974387084_b_a_a @ F @ ( product_Pair_real_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4_case__prod__conv,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a,A: product_prod_real_b,B: produc7194034526493574155_nat_b] :
      ( ( produc634687561522189117at_b_a @ F @ ( produc1946830598933159916_nat_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5_case__prod__conv,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,A: product_prod_real_b,B: produc7194034526493574155_nat_b] :
      ( ( produc8421786640700244582_b_a_a @ F @ ( produc1946830598933159916_nat_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6_case__prod__conv,axiom,
    ! [F: produc7194034526493574155_nat_b > a > a,A: produc7194034526493574155_nat_b,B: a] :
      ( ( produc2543982510095308599_b_a_a @ F @ ( produc8509509857186669914at_b_a @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_7_case__prod__conv,axiom,
    ! [F: real > b > produc7194034526493574155_nat_b > a > a,A: real,B: b] :
      ( ( produc7016990068875982646_b_a_a @ F @ ( product_Pair_real_b @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_8_case__prod__conv,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,A: b > produc7194034526493574155_nat_b > a > a,B: produc4883615123598087894at_b_a] :
      ( ( produc286368409806729619_b_a_a @ F @ ( produc653446490150804990at_b_a @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_9_case__prod__conv,axiom,
    ! [F: b > produc6209191202727158440at_b_a > a,A: b,B: produc6209191202727158440at_b_a] :
      ( ( produc6302854954412089281_b_a_a @ F @ ( produc992390952543482448at_b_a @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_10_compose__const_I2_J,axiom,
    ! [A: produc2422161461964618553l_real,G: produc2422161461964618553l_real > real] :
      ( ( comp_r97482993418849103l_real
        @ ^ [X: real] : A
        @ G )
      = ( ^ [X: produc2422161461964618553l_real] : A ) ) ).

% compose_const(2)
thf(fact_11_compose__const_I2_J,axiom,
    ! [A: produc7716430852924023517t_real,G: produc7716430852924023517t_real > real] :
      ( ( comp_r4825950202331181903t_real
        @ ^ [X: real] : A
        @ G )
      = ( ^ [X: produc7716430852924023517t_real] : A ) ) ).

% compose_const(2)
thf(fact_12_compose__const_I2_J,axiom,
    ! [A: real,G: real > real] :
      ( ( comp_real_real_real
        @ ^ [X: real] : A
        @ G )
      = ( ^ [X: real] : A ) ) ).

% compose_const(2)
thf(fact_13_compose__const_I2_J,axiom,
    ! [A: $o,G: $o > real] :
      ( ( comp_real_o_o
        @ ^ [X: real] : A
        @ G )
      = ( ^ [X: $o] : A ) ) ).

% compose_const(2)
thf(fact_14_compose__const_I2_J,axiom,
    ! [A: nat,G: nat > real] :
      ( ( comp_real_nat_nat
        @ ^ [X: real] : A
        @ G )
      = ( ^ [X: nat] : A ) ) ).

% compose_const(2)
thf(fact_15_K__record__comp,axiom,
    ! [C: produc2422161461964618553l_real,F: produc2422161461964618553l_real > real] :
      ( ( comp_r97482993418849103l_real
        @ ^ [X: real] : C
        @ F )
      = ( ^ [X: produc2422161461964618553l_real] : C ) ) ).

% K_record_comp
thf(fact_16_K__record__comp,axiom,
    ! [C: produc7716430852924023517t_real,F: produc7716430852924023517t_real > real] :
      ( ( comp_r4825950202331181903t_real
        @ ^ [X: real] : C
        @ F )
      = ( ^ [X: produc7716430852924023517t_real] : C ) ) ).

% K_record_comp
thf(fact_17_K__record__comp,axiom,
    ! [C: real,F: real > real] :
      ( ( comp_real_real_real
        @ ^ [X: real] : C
        @ F )
      = ( ^ [X: real] : C ) ) ).

% K_record_comp
thf(fact_18_K__record__comp,axiom,
    ! [C: $o,F: $o > real] :
      ( ( comp_real_o_o
        @ ^ [X: real] : C
        @ F )
      = ( ^ [X: $o] : C ) ) ).

% K_record_comp
thf(fact_19_K__record__comp,axiom,
    ! [C: nat,F: nat > real] :
      ( ( comp_real_nat_nat
        @ ^ [X: real] : C
        @ F )
      = ( ^ [X: nat] : C ) ) ).

% K_record_comp
thf(fact_20_hb_I2_J,axiom,
    ( member924168396124881715_b_a_a @ beta_1
    @ ( qbs_Mx703024977670096246_b_a_a
      @ ( expone5294601226626914677_b_a_a @ x
        @ ( expone5933728560452245707_b_a_a
          @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
            @ ^ [N: nat] :
                ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                @ ^ [I: nat] : x ) )
          @ ( exponent_exp_qbs_a_a @ y @ y ) ) ) ) ) ).

% hb(2)
thf(fact_21_comp__apply,axiom,
    ( comp_r97482993418849103l_real
    = ( ^ [F2: real > produc2422161461964618553l_real,G2: produc2422161461964618553l_real > real,X: produc2422161461964618553l_real] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_22_comp__apply,axiom,
    ( comp_r4825950202331181903t_real
    = ( ^ [F2: real > produc7716430852924023517t_real,G2: produc7716430852924023517t_real > real,X: produc7716430852924023517t_real] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_23_comp__apply,axiom,
    ( comp_real_real_real
    = ( ^ [F2: real > real,G2: real > real,X: real] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_24_comp__apply,axiom,
    ( comp_real_o_o
    = ( ^ [F2: real > $o,G2: $o > real,X: $o] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_25_comp__apply,axiom,
    ( comp_real_nat_nat
    = ( ^ [F2: real > nat,G2: nat > real,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_26_case__prod__app,axiom,
    ( produc7827042608974387084_b_a_a
    = ( ^ [F2: real > b > a > a,X: product_prod_real_b,Y: a] :
          ( produc2835462372567216023al_b_a
          @ ^ [L: real,R: b] : ( F2 @ L @ R @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_27_case__prod__app,axiom,
    ( produc8421786640700244582_b_a_a
    = ( ^ [F2: product_prod_real_b > produc7194034526493574155_nat_b > a > a,X: produc3260441645384719156_nat_b,Y: a] :
          ( produc634687561522189117at_b_a
          @ ^ [L: product_prod_real_b,R: produc7194034526493574155_nat_b] : ( F2 @ L @ R @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_28_case__prod__app,axiom,
    ( produc7016990068875982646_b_a_a
    = ( ^ [F2: real > b > produc7194034526493574155_nat_b > a > a,X: product_prod_real_b,Y: produc7194034526493574155_nat_b] :
          ( produc7827042608974387084_b_a_a
          @ ^ [L: real,R: b] : ( F2 @ L @ R @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_29_prod_Oinject,axiom,
    ! [X1: produc7194034526493574155_nat_b,X2: a,Y1: produc7194034526493574155_nat_b,Y2: a] :
      ( ( ( produc8509509857186669914at_b_a @ X1 @ X2 )
        = ( produc8509509857186669914at_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_30_prod_Oinject,axiom,
    ! [X1: product_prod_real_b,X2: produc7194034526493574155_nat_b,Y1: product_prod_real_b,Y2: produc7194034526493574155_nat_b] :
      ( ( ( produc1946830598933159916_nat_b @ X1 @ X2 )
        = ( produc1946830598933159916_nat_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_31_prod_Oinject,axiom,
    ! [X1: real,X2: b,Y1: real,Y2: b] :
      ( ( ( product_Pair_real_b @ X1 @ X2 )
        = ( product_Pair_real_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_32_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_33_prod_Oinject,axiom,
    ! [X1: nat,X2: nat > b,Y1: nat,Y2: nat > b] :
      ( ( ( produc5292568363641424317_nat_b @ X1 @ X2 )
        = ( produc5292568363641424317_nat_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_34_prod_Oinject,axiom,
    ! [X1: b > produc7194034526493574155_nat_b > a > a,X2: produc4883615123598087894at_b_a,Y1: b > produc7194034526493574155_nat_b > a > a,Y2: produc4883615123598087894at_b_a] :
      ( ( ( produc653446490150804990at_b_a @ X1 @ X2 )
        = ( produc653446490150804990at_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_35_prod_Oinject,axiom,
    ! [X1: b,X2: produc6209191202727158440at_b_a,Y1: b,Y2: produc6209191202727158440at_b_a] :
      ( ( ( produc992390952543482448at_b_a @ X1 @ X2 )
        = ( produc992390952543482448at_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_36_old_Oprod_Oinject,axiom,
    ! [A: real,B: b,A2: real,B2: b] :
      ( ( ( product_Pair_real_b @ A @ B )
        = ( product_Pair_real_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_37_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_38_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat > b,A2: nat,B2: nat > b] :
      ( ( ( produc5292568363641424317_nat_b @ A @ B )
        = ( produc5292568363641424317_nat_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_39_old_Oprod_Oinject,axiom,
    ! [A: b > produc7194034526493574155_nat_b > a > a,B: produc4883615123598087894at_b_a,A2: b > produc7194034526493574155_nat_b > a > a,B2: produc4883615123598087894at_b_a] :
      ( ( ( produc653446490150804990at_b_a @ A @ B )
        = ( produc653446490150804990at_b_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_40_old_Oprod_Oinject,axiom,
    ! [A: b,B: produc6209191202727158440at_b_a,A2: b,B2: produc6209191202727158440at_b_a] :
      ( ( ( produc992390952543482448at_b_a @ A @ B )
        = ( produc992390952543482448at_b_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_41_old_Oprod_Oinject,axiom,
    ! [A: produc7194034526493574155_nat_b,B: a,A2: produc7194034526493574155_nat_b,B2: a] :
      ( ( ( produc8509509857186669914at_b_a @ A @ B )
        = ( produc8509509857186669914at_b_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_42_old_Oprod_Oinject,axiom,
    ! [A: product_prod_real_b,B: produc7194034526493574155_nat_b,A2: product_prod_real_b,B2: produc7194034526493574155_nat_b] :
      ( ( ( produc1946830598933159916_nat_b @ A @ B )
        = ( produc1946830598933159916_nat_b @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_43_case__prodE2,axiom,
    ! [Q: ( a > a ) > $o,P: product_prod_real_b > produc7194034526493574155_nat_b > a > a,Z: produc3260441645384719156_nat_b] :
      ( ( Q @ ( produc8421786640700244582_b_a_a @ P @ Z ) )
     => ~ ! [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
            ( ( Z
              = ( produc1946830598933159916_nat_b @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_44_case__prodE2,axiom,
    ! [Q: a > $o,P: produc7194034526493574155_nat_b > a > a,Z: produc6209191202727158440at_b_a] :
      ( ( Q @ ( produc2543982510095308599_b_a_a @ P @ Z ) )
     => ~ ! [X3: produc7194034526493574155_nat_b,Y3: a] :
            ( ( Z
              = ( produc8509509857186669914at_b_a @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_45_case__prodE2,axiom,
    ! [Q: ( produc7194034526493574155_nat_b > a > a ) > $o,P: real > b > produc7194034526493574155_nat_b > a > a,Z: product_prod_real_b] :
      ( ( Q @ ( produc7016990068875982646_b_a_a @ P @ Z ) )
     => ~ ! [X3: real,Y3: b] :
            ( ( Z
              = ( product_Pair_real_b @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_46_case__prodE2,axiom,
    ! [Q: a > $o,P: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,Z: produc778026099599033348at_b_a] :
      ( ( Q @ ( produc286368409806729619_b_a_a @ P @ Z ) )
     => ~ ! [X3: b > produc7194034526493574155_nat_b > a > a,Y3: produc4883615123598087894at_b_a] :
            ( ( Z
              = ( produc653446490150804990at_b_a @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_47_case__prodE2,axiom,
    ! [Q: a > $o,P: b > produc6209191202727158440at_b_a > a,Z: produc4883615123598087894at_b_a] :
      ( ( Q @ ( produc6302854954412089281_b_a_a @ P @ Z ) )
     => ~ ! [X3: b,Y3: produc6209191202727158440at_b_a] :
            ( ( Z
              = ( produc992390952543482448at_b_a @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_48_case__prodE2,axiom,
    ! [Q: ( a > a ) > $o,P: real > b > a > a,Z: product_prod_real_b] :
      ( ( Q @ ( produc7827042608974387084_b_a_a @ P @ Z ) )
     => ~ ! [X3: real,Y3: b] :
            ( ( Z
              = ( product_Pair_real_b @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_49_case__prodE2,axiom,
    ! [Q: a > $o,P: product_prod_real_b > produc7194034526493574155_nat_b > a,Z: produc3260441645384719156_nat_b] :
      ( ( Q @ ( produc634687561522189117at_b_a @ P @ Z ) )
     => ~ ! [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
            ( ( Z
              = ( produc1946830598933159916_nat_b @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_50_case__prod__eta,axiom,
    ! [F: produc3260441645384719156_nat_b > a > a] :
      ( ( produc8421786640700244582_b_a_a
        @ ^ [X: product_prod_real_b,Y: produc7194034526493574155_nat_b] : ( F @ ( produc1946830598933159916_nat_b @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_51_case__prod__eta,axiom,
    ! [F: produc6209191202727158440at_b_a > a] :
      ( ( produc2543982510095308599_b_a_a
        @ ^ [X: produc7194034526493574155_nat_b,Y: a] : ( F @ ( produc8509509857186669914at_b_a @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_52_case__prod__eta,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a > a] :
      ( ( produc7016990068875982646_b_a_a
        @ ^ [X: real,Y: b] : ( F @ ( product_Pair_real_b @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_53_case__prod__eta,axiom,
    ! [F: produc778026099599033348at_b_a > a] :
      ( ( produc286368409806729619_b_a_a
        @ ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: produc4883615123598087894at_b_a] : ( F @ ( produc653446490150804990at_b_a @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_54_case__prod__eta,axiom,
    ! [F: produc4883615123598087894at_b_a > a] :
      ( ( produc6302854954412089281_b_a_a
        @ ^ [X: b,Y: produc6209191202727158440at_b_a] : ( F @ ( produc992390952543482448at_b_a @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_55_case__prod__eta,axiom,
    ! [F: product_prod_real_b > a > a] :
      ( ( produc7827042608974387084_b_a_a
        @ ^ [X: real,Y: b] : ( F @ ( product_Pair_real_b @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_56_case__prod__eta,axiom,
    ! [F: produc3260441645384719156_nat_b > a] :
      ( ( produc634687561522189117at_b_a
        @ ^ [X: product_prod_real_b,Y: produc7194034526493574155_nat_b] : ( F @ ( produc1946830598933159916_nat_b @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_57_cond__case__prod__eta,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,G: produc3260441645384719156_nat_b > a > a] :
      ( ! [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc1946830598933159916_nat_b @ X3 @ Y3 ) ) )
     => ( ( produc8421786640700244582_b_a_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_58_cond__case__prod__eta,axiom,
    ! [F: produc7194034526493574155_nat_b > a > a,G: produc6209191202727158440at_b_a > a] :
      ( ! [X3: produc7194034526493574155_nat_b,Y3: a] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc8509509857186669914at_b_a @ X3 @ Y3 ) ) )
     => ( ( produc2543982510095308599_b_a_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_59_cond__case__prod__eta,axiom,
    ! [F: real > b > produc7194034526493574155_nat_b > a > a,G: product_prod_real_b > produc7194034526493574155_nat_b > a > a] :
      ( ! [X3: real,Y3: b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_real_b @ X3 @ Y3 ) ) )
     => ( ( produc7016990068875982646_b_a_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_60_cond__case__prod__eta,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,G: produc778026099599033348at_b_a > a] :
      ( ! [X3: b > produc7194034526493574155_nat_b > a > a,Y3: produc4883615123598087894at_b_a] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc653446490150804990at_b_a @ X3 @ Y3 ) ) )
     => ( ( produc286368409806729619_b_a_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_61_cond__case__prod__eta,axiom,
    ! [F: b > produc6209191202727158440at_b_a > a,G: produc4883615123598087894at_b_a > a] :
      ( ! [X3: b,Y3: produc6209191202727158440at_b_a] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc992390952543482448at_b_a @ X3 @ Y3 ) ) )
     => ( ( produc6302854954412089281_b_a_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_62_cond__case__prod__eta,axiom,
    ! [F: real > b > a > a,G: product_prod_real_b > a > a] :
      ( ! [X3: real,Y3: b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_real_b @ X3 @ Y3 ) ) )
     => ( ( produc7827042608974387084_b_a_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_63_cond__case__prod__eta,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a,G: produc3260441645384719156_nat_b > a] :
      ( ! [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( produc1946830598933159916_nat_b @ X3 @ Y3 ) ) )
     => ( ( produc634687561522189117at_b_a @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_64_qbs__morphism__comp,axiom,
    ! [F: real > real,X4: quasi_borel_real,Y4: quasi_borel_real,G: real > real,Z2: quasi_borel_real] :
      ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X4 @ Y4 ) )
     => ( ( member_real_real @ G @ ( qbs_mo5229770564518008146l_real @ Y4 @ Z2 ) )
       => ( member_real_real @ ( comp_real_real_real @ G @ F ) @ ( qbs_mo5229770564518008146l_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_65_qbs__morphism__comp,axiom,
    ! [F: nat > $o,X4: quasi_borel_nat,Y4: quasi_borel_o,G: $o > real,Z2: quasi_borel_real] :
      ( ( member_nat_o @ F @ ( qbs_morphism_nat_o @ X4 @ Y4 ) )
     => ( ( member_o_real @ G @ ( qbs_morphism_o_real @ Y4 @ Z2 ) )
       => ( member_nat_real @ ( comp_o_real_nat @ G @ F ) @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_66_qbs__morphism__comp,axiom,
    ! [F: $o > nat,X4: quasi_borel_o,Y4: quasi_borel_nat,G: nat > real,Z2: quasi_borel_real] :
      ( ( member_o_nat @ F @ ( qbs_morphism_o_nat @ X4 @ Y4 ) )
     => ( ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z2 ) )
       => ( member_o_real @ ( comp_nat_real_o @ G @ F ) @ ( qbs_morphism_o_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_67_qbs__morphism__comp,axiom,
    ! [F: nat > nat,X4: quasi_borel_nat,Y4: quasi_borel_nat,G: nat > real,Z2: quasi_borel_real] :
      ( ( member_nat_nat @ F @ ( qbs_morphism_nat_nat @ X4 @ Y4 ) )
     => ( ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z2 ) )
       => ( member_nat_real @ ( comp_nat_real_nat @ G @ F ) @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_68_qbs__morphism__comp,axiom,
    ! [F: $o > real,X4: quasi_borel_o,Y4: quasi_borel_real,G: real > real,Z2: quasi_borel_real] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X4 @ Y4 ) )
     => ( ( member_real_real @ G @ ( qbs_mo5229770564518008146l_real @ Y4 @ Z2 ) )
       => ( member_o_real @ ( comp_real_real_o @ G @ F ) @ ( qbs_morphism_o_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_69_qbs__morphism__comp,axiom,
    ! [F: nat > real,X4: quasi_borel_nat,Y4: quasi_borel_real,G: real > nat,Z2: quasi_borel_nat] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
     => ( ( member_real_nat @ G @ ( qbs_mo6567951568834356598al_nat @ Y4 @ Z2 ) )
       => ( member_nat_nat @ ( comp_real_nat_nat @ G @ F ) @ ( qbs_morphism_nat_nat @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_70_qbs__morphism__comp,axiom,
    ! [F: nat > real,X4: quasi_borel_nat,Y4: quasi_borel_real,G: real > real,Z2: quasi_borel_real] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
     => ( ( member_real_real @ G @ ( qbs_mo5229770564518008146l_real @ Y4 @ Z2 ) )
       => ( member_nat_real @ ( comp_real_real_nat @ G @ F ) @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_71_qbs__morphism__comp,axiom,
    ! [F: $o > real,X4: quasi_borel_o,Y4: quasi_borel_real,G: real > $o,Z2: quasi_borel_o] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X4 @ Y4 ) )
     => ( ( member_real_o @ G @ ( qbs_morphism_real_o @ Y4 @ Z2 ) )
       => ( member_o_o @ ( comp_real_o_o @ G @ F ) @ ( qbs_morphism_o_o @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_72_qbs__morphism__comp,axiom,
    ! [F: $o > real,X4: quasi_borel_o,Y4: quasi_borel_real,G: real > a,Z2: quasi_borel_a] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X4 @ Y4 ) )
     => ( ( member_real_a @ G @ ( qbs_morphism_real_a @ Y4 @ Z2 ) )
       => ( member_o_a @ ( comp_real_a_o @ G @ F ) @ ( qbs_morphism_o_a @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_73_qbs__morphism__comp,axiom,
    ! [F: nat > real,X4: quasi_borel_nat,Y4: quasi_borel_real,G: real > a,Z2: quasi_borel_a] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
     => ( ( member_real_a @ G @ ( qbs_morphism_real_a @ Y4 @ Z2 ) )
       => ( member_nat_a @ ( comp_real_a_nat @ G @ F ) @ ( qbs_morphism_nat_a @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_comp
thf(fact_74_iso__tuple__UNIV__I,axiom,
    ! [X5: $o > real] : ( member_o_real @ X5 @ top_top_set_o_real ) ).

% iso_tuple_UNIV_I
thf(fact_75_iso__tuple__UNIV__I,axiom,
    ! [X5: $o > $o] : ( member_o_o @ X5 @ top_top_set_o_o ) ).

% iso_tuple_UNIV_I
thf(fact_76_iso__tuple__UNIV__I,axiom,
    ! [X5: $o > $o > $o] : ( member_o_o_o @ X5 @ top_top_set_o_o_o ) ).

% iso_tuple_UNIV_I
thf(fact_77_iso__tuple__UNIV__I,axiom,
    ! [X5: nat > real] : ( member_nat_real @ X5 @ top_top_set_nat_real ) ).

% iso_tuple_UNIV_I
thf(fact_78_iso__tuple__UNIV__I,axiom,
    ! [X5: ( nat > b ) > produc432120507726958376_b_a_a > a] : ( member1811236362415105851_a_a_a @ X5 @ top_to5015968664104940226_a_a_a ) ).

% iso_tuple_UNIV_I
thf(fact_79_iso__tuple__UNIV__I,axiom,
    ! [X5: produc7716430852924023517t_real] : ( member557208447399453958t_real @ X5 @ top_to4855536200657754381t_real ) ).

% iso_tuple_UNIV_I
thf(fact_80_iso__tuple__UNIV__I,axiom,
    ! [X5: complex] : ( member_complex @ X5 @ top_top_set_complex ) ).

% iso_tuple_UNIV_I
thf(fact_81_iso__tuple__UNIV__I,axiom,
    ! [X5: real] : ( member_real @ X5 @ top_top_set_real ) ).

% iso_tuple_UNIV_I
thf(fact_82_iso__tuple__UNIV__I,axiom,
    ! [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ).

% iso_tuple_UNIV_I
thf(fact_83_iso__tuple__UNIV__I,axiom,
    ! [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_84_mem__case__prodI2,axiom,
    ! [P2: product_prod_real_b,Z: $o > real,C: real > b > set_o_real] :
      ( ! [A3: real,B3: b] :
          ( ( P2
            = ( product_Pair_real_b @ A3 @ B3 ) )
         => ( member_o_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_o_real @ Z @ ( produc266752995426694702o_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_85_mem__case__prodI2,axiom,
    ! [P2: product_prod_real_b,Z: $o > $o,C: real > b > set_o_o] :
      ( ! [A3: real,B3: b] :
          ( ( P2
            = ( product_Pair_real_b @ A3 @ B3 ) )
         => ( member_o_o @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_o_o @ Z @ ( produc1507278367008346818et_o_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_86_mem__case__prodI2,axiom,
    ! [P2: product_prod_real_b,Z: nat > real,C: real > b > set_nat_real] :
      ( ! [A3: real,B3: b] :
          ( ( P2
            = ( product_Pair_real_b @ A3 @ B3 ) )
         => ( member_nat_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat_real @ Z @ ( produc4643918755611922808t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_87_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: $o > real,C: nat > nat > set_o_real] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member_o_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_o_real @ Z @ ( produc2044794759437995847o_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_88_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: $o > $o,C: nat > nat > set_o_o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member_o_o @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_o_o @ Z @ ( produc1059706159101826281et_o_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_89_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: nat > real,C: nat > nat > set_nat_real] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member_nat_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat_real @ Z @ ( produc2304335462895136927t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_90_mem__case__prodI2,axiom,
    ! [P2: product_prod_real_b,Z: $o > $o > $o,C: real > b > set_o_o_o] :
      ( ! [A3: real,B3: b] :
          ( ( P2
            = ( product_Pair_real_b @ A3 @ B3 ) )
         => ( member_o_o_o @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_o_o_o @ Z @ ( produc7579781466765250791_o_o_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_91_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: $o > $o > $o,C: nat > nat > set_o_o_o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member_o_o_o @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_o_o_o @ Z @ ( produc6784641851573448064_o_o_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_92_mem__case__prodI2,axiom,
    ! [P2: produc7194034526493574155_nat_b,Z: $o > real,C: nat > ( nat > b ) > set_o_real] :
      ( ! [A3: nat,B3: nat > b] :
          ( ( P2
            = ( produc5292568363641424317_nat_b @ A3 @ B3 ) )
         => ( member_o_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_o_real @ Z @ ( produc2826515721302849707o_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_93_mem__case__prodI2,axiom,
    ! [P2: produc7194034526493574155_nat_b,Z: $o > $o,C: nat > ( nat > b ) > set_o_o] :
      ( ! [A3: nat,B3: nat > b] :
          ( ( P2
            = ( produc5292568363641424317_nat_b @ A3 @ B3 ) )
         => ( member_o_o @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_o_o @ Z @ ( produc7104104902851696773et_o_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_94_mem__case__prodI,axiom,
    ! [Z: $o > real,C: real > b > set_o_real,A: real,B: b] :
      ( ( member_o_real @ Z @ ( C @ A @ B ) )
     => ( member_o_real @ Z @ ( produc266752995426694702o_real @ C @ ( product_Pair_real_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_95_mem__case__prodI,axiom,
    ! [Z: $o > $o,C: real > b > set_o_o,A: real,B: b] :
      ( ( member_o_o @ Z @ ( C @ A @ B ) )
     => ( member_o_o @ Z @ ( produc1507278367008346818et_o_o @ C @ ( product_Pair_real_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_96_mem__case__prodI,axiom,
    ! [Z: nat > real,C: real > b > set_nat_real,A: real,B: b] :
      ( ( member_nat_real @ Z @ ( C @ A @ B ) )
     => ( member_nat_real @ Z @ ( produc4643918755611922808t_real @ C @ ( product_Pair_real_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_97_mem__case__prodI,axiom,
    ! [Z: $o > real,C: nat > nat > set_o_real,A: nat,B: nat] :
      ( ( member_o_real @ Z @ ( C @ A @ B ) )
     => ( member_o_real @ Z @ ( produc2044794759437995847o_real @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_98_mem__case__prodI,axiom,
    ! [Z: $o > $o,C: nat > nat > set_o_o,A: nat,B: nat] :
      ( ( member_o_o @ Z @ ( C @ A @ B ) )
     => ( member_o_o @ Z @ ( produc1059706159101826281et_o_o @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_99_mem__case__prodI,axiom,
    ! [Z: nat > real,C: nat > nat > set_nat_real,A: nat,B: nat] :
      ( ( member_nat_real @ Z @ ( C @ A @ B ) )
     => ( member_nat_real @ Z @ ( produc2304335462895136927t_real @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_100_mem__case__prodI,axiom,
    ! [Z: $o > $o > $o,C: real > b > set_o_o_o,A: real,B: b] :
      ( ( member_o_o_o @ Z @ ( C @ A @ B ) )
     => ( member_o_o_o @ Z @ ( produc7579781466765250791_o_o_o @ C @ ( product_Pair_real_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_101_mem__case__prodI,axiom,
    ! [Z: $o > $o > $o,C: nat > nat > set_o_o_o,A: nat,B: nat] :
      ( ( member_o_o_o @ Z @ ( C @ A @ B ) )
     => ( member_o_o_o @ Z @ ( produc6784641851573448064_o_o_o @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_102_mem__case__prodI,axiom,
    ! [Z: $o > real,C: nat > ( nat > b ) > set_o_real,A: nat,B: nat > b] :
      ( ( member_o_real @ Z @ ( C @ A @ B ) )
     => ( member_o_real @ Z @ ( produc2826515721302849707o_real @ C @ ( produc5292568363641424317_nat_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_103_mem__case__prodI,axiom,
    ! [Z: $o > $o,C: nat > ( nat > b ) > set_o_o,A: nat,B: nat > b] :
      ( ( member_o_o @ Z @ ( C @ A @ B ) )
     => ( member_o_o @ Z @ ( produc7104104902851696773et_o_o @ C @ ( produc5292568363641424317_nat_b @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_104_case__prodI2,axiom,
    ! [P2: product_prod_real_b,C: real > b > $o] :
      ( ! [A3: real,B3: b] :
          ( ( P2
            = ( product_Pair_real_b @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc5814611744267527345al_b_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_105_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_106_case__prodI2,axiom,
    ! [P2: produc7194034526493574155_nat_b,C: nat > ( nat > b ) > $o] :
      ( ! [A3: nat,B3: nat > b] :
          ( ( P2
            = ( produc5292568363641424317_nat_b @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc7710141471362042158at_b_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_107_case__prodI2,axiom,
    ! [P2: produc778026099599033348at_b_a,C: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > $o] :
      ( ! [A3: b > produc7194034526493574155_nat_b > a > a,B3: produc4883615123598087894at_b_a] :
          ( ( P2
            = ( produc653446490150804990at_b_a @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc6923730833870890157_b_a_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_108_case__prodI2,axiom,
    ! [P2: produc4883615123598087894at_b_a,C: b > produc6209191202727158440at_b_a > $o] :
      ( ! [A3: b,B3: produc6209191202727158440at_b_a] :
          ( ( P2
            = ( produc992390952543482448at_b_a @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc2784492703816452571_b_a_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_109_case__prodI2,axiom,
    ! [P2: produc6209191202727158440at_b_a,C: produc7194034526493574155_nat_b > a > $o] :
      ( ! [A3: produc7194034526493574155_nat_b,B3: a] :
          ( ( P2
            = ( produc8509509857186669914at_b_a @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc5783264217630104657_b_a_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_110_case__prodI2,axiom,
    ! [P2: produc3260441645384719156_nat_b,C: product_prod_real_b > produc7194034526493574155_nat_b > $o] :
      ( ! [A3: product_prod_real_b,B3: produc7194034526493574155_nat_b] :
          ( ( P2
            = ( produc1946830598933159916_nat_b @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc9191534587096267607at_b_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_111_case__prodI,axiom,
    ! [F: real > b > $o,A: real,B: b] :
      ( ( F @ A @ B )
     => ( produc5814611744267527345al_b_o @ F @ ( product_Pair_real_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_112_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_113_case__prodI,axiom,
    ! [F: nat > ( nat > b ) > $o,A: nat,B: nat > b] :
      ( ( F @ A @ B )
     => ( produc7710141471362042158at_b_o @ F @ ( produc5292568363641424317_nat_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_114_case__prodI,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > $o,A: b > produc7194034526493574155_nat_b > a > a,B: produc4883615123598087894at_b_a] :
      ( ( F @ A @ B )
     => ( produc6923730833870890157_b_a_o @ F @ ( produc653446490150804990at_b_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_115_case__prodI,axiom,
    ! [F: b > produc6209191202727158440at_b_a > $o,A: b,B: produc6209191202727158440at_b_a] :
      ( ( F @ A @ B )
     => ( produc2784492703816452571_b_a_o @ F @ ( produc992390952543482448at_b_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_116_case__prodI,axiom,
    ! [F: produc7194034526493574155_nat_b > a > $o,A: produc7194034526493574155_nat_b,B: a] :
      ( ( F @ A @ B )
     => ( produc5783264217630104657_b_a_o @ F @ ( produc8509509857186669914at_b_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_117_case__prodI,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > $o,A: product_prod_real_b,B: produc7194034526493574155_nat_b] :
      ( ( F @ A @ B )
     => ( produc9191534587096267607at_b_o @ F @ ( produc1946830598933159916_nat_b @ A @ B ) ) ) ).

% case_prodI
thf(fact_118_hb_I1_J,axiom,
    ( beta
    = ( ^ [R: real] : ( produc653446490150804990at_b_a @ ( beta_1 @ R ) @ ( produc992390952543482448at_b_a @ ( beta_2 @ R ) @ ( produc8509509857186669914at_b_a @ ( beta_3 @ R ) @ ( beta_4 @ R ) ) ) ) ) ) ).

% hb(1)
thf(fact_119_qbs__eqI,axiom,
    ! [X4: quasi_331246355085487092at_b_a,Y4: quasi_331246355085487092at_b_a] :
      ( ( ( qbs_Mx5919617464495388621at_b_a @ X4 )
        = ( qbs_Mx5919617464495388621at_b_a @ Y4 ) )
     => ( X4 = Y4 ) ) ).

% qbs_eqI
thf(fact_120_qbs__eqI,axiom,
    ! [X4: quasi_borel_nat_b,Y4: quasi_borel_nat_b] :
      ( ( ( qbs_Mx_nat_b @ X4 )
        = ( qbs_Mx_nat_b @ Y4 ) )
     => ( X4 = Y4 ) ) ).

% qbs_eqI
thf(fact_121_qbs__eqI,axiom,
    ! [X4: quasi_3364728345918372901_b_a_a,Y4: quasi_3364728345918372901_b_a_a] :
      ( ( ( qbs_Mx703024977670096246_b_a_a @ X4 )
        = ( qbs_Mx703024977670096246_b_a_a @ Y4 ) )
     => ( X4 = Y4 ) ) ).

% qbs_eqI
thf(fact_122_qbs__eqI,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_b] :
      ( ( ( qbs_Mx_b @ X4 )
        = ( qbs_Mx_b @ Y4 ) )
     => ( X4 = Y4 ) ) ).

% qbs_eqI
thf(fact_123_qbs__eqI,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X4 )
        = ( qbs_Mx_a @ Y4 ) )
     => ( X4 = Y4 ) ) ).

% qbs_eqI
thf(fact_124_arg__swap__morphism,axiom,
    ! [F: produc432120507726958376_b_a_a > ( nat > b ) > a,X4: quasi_6353345141195568462_b_a_a,Y4: quasi_borel_nat_b,Z2: quasi_borel_a] :
      ( ( member5820388576109277759at_b_a @ F @ ( qbs_mo4496954763341851693at_b_a @ X4 @ ( expone514113178172992113at_b_a @ Y4 @ Z2 ) ) )
     => ( member1811236362415105851_a_a_a
        @ ^ [Y: nat > b,X: produc432120507726958376_b_a_a] : ( F @ X @ Y )
        @ ( qbs_mo8618518167108125737_a_a_a @ Y4 @ ( expone790792417327382575_a_a_a @ X4 @ Z2 ) ) ) ) ).

% arg_swap_morphism
thf(fact_125_arg__swap__morphism,axiom,
    ! [F: b > b > produc7194034526493574155_nat_b > a > a,X4: quasi_borel_b,Y4: quasi_borel_b,Z2: quasi_7934145443646537599_b_a_a] :
      ( ( member1303456256152176444_b_a_a @ F @ ( qbs_mo522001364118357432_b_a_a @ X4 @ ( expone5294601226626914677_b_a_a @ Y4 @ Z2 ) ) )
     => ( member1303456256152176444_b_a_a
        @ ^ [Y: b,X: b] : ( F @ X @ Y )
        @ ( qbs_mo522001364118357432_b_a_a @ Y4 @ ( expone5294601226626914677_b_a_a @ X4 @ Z2 ) ) ) ) ).

% arg_swap_morphism
thf(fact_126_arg__swap__morphism,axiom,
    ! [F: produc7194034526493574155_nat_b > produc7194034526493574155_nat_b > a > a,X4: quasi_7334096685896923387_nat_b,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a_a] :
      ( ( member5767077552435619664_b_a_a @ F @ ( qbs_mo1970811941495421204_b_a_a @ X4 @ ( expone5933728560452245707_b_a_a @ Y4 @ Z2 ) ) )
     => ( member5767077552435619664_b_a_a
        @ ^ [Y: produc7194034526493574155_nat_b,X: produc7194034526493574155_nat_b] : ( F @ X @ Y )
        @ ( qbs_mo1970811941495421204_b_a_a @ Y4 @ ( expone5933728560452245707_b_a_a @ X4 @ Z2 ) ) ) ) ).

% arg_swap_morphism
thf(fact_127_arg__swap__morphism,axiom,
    ! [F: a > a > a,X4: quasi_borel_a,Y4: quasi_borel_a,Z2: quasi_borel_a] :
      ( ( member_a_a_a @ F @ ( qbs_morphism_a_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) )
     => ( member_a_a_a
        @ ^ [Y: a,X: a] : ( F @ X @ Y )
        @ ( qbs_morphism_a_a_a @ Y4 @ ( exponent_exp_qbs_a_a @ X4 @ Z2 ) ) ) ) ).

% arg_swap_morphism
thf(fact_128_arg__swap__morphism,axiom,
    ! [F: produc432120507726958376_b_a_a > a > a,X4: quasi_6353345141195568462_b_a_a,Y4: quasi_borel_a,Z2: quasi_borel_a] :
      ( ( member1246619165056611495_a_a_a @ F @ ( qbs_mo76743781497322517_a_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) )
     => ( member2150145874327079635_a_a_a
        @ ^ [Y: a,X: produc432120507726958376_b_a_a] : ( F @ X @ Y )
        @ ( qbs_mo4857264843076950849_a_a_a @ Y4 @ ( expone790792417327382575_a_a_a @ X4 @ Z2 ) ) ) ) ).

% arg_swap_morphism
thf(fact_129_arg__swap__morphism,axiom,
    ! [F: ( nat > b ) > produc432120507726958376_b_a_a > a,X4: quasi_borel_nat_b,Y4: quasi_6353345141195568462_b_a_a,Z2: quasi_borel_a] :
      ( ( member1811236362415105851_a_a_a @ F @ ( qbs_mo8618518167108125737_a_a_a @ X4 @ ( expone790792417327382575_a_a_a @ Y4 @ Z2 ) ) )
     => ( member5820388576109277759at_b_a
        @ ^ [Y: produc432120507726958376_b_a_a,X: nat > b] : ( F @ X @ Y )
        @ ( qbs_mo4496954763341851693at_b_a @ Y4 @ ( expone514113178172992113at_b_a @ X4 @ Z2 ) ) ) ) ).

% arg_swap_morphism
thf(fact_130_arg__swap__morphism,axiom,
    ! [F: a > produc432120507726958376_b_a_a > a,X4: quasi_borel_a,Y4: quasi_6353345141195568462_b_a_a,Z2: quasi_borel_a] :
      ( ( member2150145874327079635_a_a_a @ F @ ( qbs_mo4857264843076950849_a_a_a @ X4 @ ( expone790792417327382575_a_a_a @ Y4 @ Z2 ) ) )
     => ( member1246619165056611495_a_a_a
        @ ^ [Y: produc432120507726958376_b_a_a,X: a] : ( F @ X @ Y )
        @ ( qbs_mo76743781497322517_a_a_a @ Y4 @ ( exponent_exp_qbs_a_a @ X4 @ Z2 ) ) ) ) ).

% arg_swap_morphism
thf(fact_131_arg__swap__morphism,axiom,
    ! [F: produc432120507726958376_b_a_a > produc432120507726958376_b_a_a > a,X4: quasi_6353345141195568462_b_a_a,Y4: quasi_6353345141195568462_b_a_a,Z2: quasi_borel_a] :
      ( ( member561746832484703485_a_a_a @ F @ ( qbs_mo5281887857236930283_a_a_a @ X4 @ ( expone790792417327382575_a_a_a @ Y4 @ Z2 ) ) )
     => ( member561746832484703485_a_a_a
        @ ^ [Y: produc432120507726958376_b_a_a,X: produc432120507726958376_b_a_a] : ( F @ X @ Y )
        @ ( qbs_mo5281887857236930283_a_a_a @ Y4 @ ( expone790792417327382575_a_a_a @ X4 @ Z2 ) ) ) ) ).

% arg_swap_morphism
thf(fact_132_arg__swap__morphism,axiom,
    ! [F: $o > $o > $o,X4: quasi_borel_o,Y4: quasi_borel_o,Z2: quasi_borel_o] :
      ( ( member_o_o_o @ F @ ( qbs_morphism_o_o_o @ X4 @ ( exponent_exp_qbs_o_o @ Y4 @ Z2 ) ) )
     => ( member_o_o_o
        @ ^ [Y: $o,X: $o] : ( F @ X @ Y )
        @ ( qbs_morphism_o_o_o @ Y4 @ ( exponent_exp_qbs_o_o @ X4 @ Z2 ) ) ) ) ).

% arg_swap_morphism
thf(fact_133_mem__case__prodE,axiom,
    ! [Z: $o > real,C: real > b > set_o_real,P2: product_prod_real_b] :
      ( ( member_o_real @ Z @ ( produc266752995426694702o_real @ C @ P2 ) )
     => ~ ! [X3: real,Y3: b] :
            ( ( P2
              = ( product_Pair_real_b @ X3 @ Y3 ) )
           => ~ ( member_o_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_134_mem__case__prodE,axiom,
    ! [Z: $o > $o,C: real > b > set_o_o,P2: product_prod_real_b] :
      ( ( member_o_o @ Z @ ( produc1507278367008346818et_o_o @ C @ P2 ) )
     => ~ ! [X3: real,Y3: b] :
            ( ( P2
              = ( product_Pair_real_b @ X3 @ Y3 ) )
           => ~ ( member_o_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_135_mem__case__prodE,axiom,
    ! [Z: nat > real,C: real > b > set_nat_real,P2: product_prod_real_b] :
      ( ( member_nat_real @ Z @ ( produc4643918755611922808t_real @ C @ P2 ) )
     => ~ ! [X3: real,Y3: b] :
            ( ( P2
              = ( product_Pair_real_b @ X3 @ Y3 ) )
           => ~ ( member_nat_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_136_mem__case__prodE,axiom,
    ! [Z: $o > real,C: nat > nat > set_o_real,P2: product_prod_nat_nat] :
      ( ( member_o_real @ Z @ ( produc2044794759437995847o_real @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_o_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_137_mem__case__prodE,axiom,
    ! [Z: $o > $o,C: nat > nat > set_o_o,P2: product_prod_nat_nat] :
      ( ( member_o_o @ Z @ ( produc1059706159101826281et_o_o @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_o_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_138_mem__case__prodE,axiom,
    ! [Z: nat > real,C: nat > nat > set_nat_real,P2: product_prod_nat_nat] :
      ( ( member_nat_real @ Z @ ( produc2304335462895136927t_real @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_nat_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_139_mem__case__prodE,axiom,
    ! [Z: $o > $o > $o,C: real > b > set_o_o_o,P2: product_prod_real_b] :
      ( ( member_o_o_o @ Z @ ( produc7579781466765250791_o_o_o @ C @ P2 ) )
     => ~ ! [X3: real,Y3: b] :
            ( ( P2
              = ( product_Pair_real_b @ X3 @ Y3 ) )
           => ~ ( member_o_o_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_140_mem__case__prodE,axiom,
    ! [Z: $o > $o > $o,C: nat > nat > set_o_o_o,P2: product_prod_nat_nat] :
      ( ( member_o_o_o @ Z @ ( produc6784641851573448064_o_o_o @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_o_o_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_141_mem__case__prodE,axiom,
    ! [Z: $o > real,C: nat > ( nat > b ) > set_o_real,P2: produc7194034526493574155_nat_b] :
      ( ( member_o_real @ Z @ ( produc2826515721302849707o_real @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat > b] :
            ( ( P2
              = ( produc5292568363641424317_nat_b @ X3 @ Y3 ) )
           => ~ ( member_o_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_142_mem__case__prodE,axiom,
    ! [Z: $o > $o,C: nat > ( nat > b ) > set_o_o,P2: produc7194034526493574155_nat_b] :
      ( ( member_o_o @ Z @ ( produc7104104902851696773et_o_o @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat > b] :
            ( ( P2
              = ( produc5292568363641424317_nat_b @ X3 @ Y3 ) )
           => ~ ( member_o_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_143_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_real_b] :
      ( ( produc1702242903178270831real_b @ product_Pair_real_b @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_144_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_145_case__prod__Pair__iden,axiom,
    ! [P2: produc7194034526493574155_nat_b] :
      ( ( produc7949636393531295477_nat_b @ produc5292568363641424317_nat_b @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_146_case__prod__Pair__iden,axiom,
    ! [P2: produc778026099599033348at_b_a] :
      ( ( produc1264538655057408815at_b_a @ produc653446490150804990at_b_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_147_case__prod__Pair__iden,axiom,
    ! [P2: produc4883615123598087894at_b_a] :
      ( ( produc3632542462609453267at_b_a @ produc992390952543482448at_b_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_148_case__prod__Pair__iden,axiom,
    ! [P2: produc6209191202727158440at_b_a] :
      ( ( produc7897861285970668975at_b_a @ produc8509509857186669914at_b_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_149_case__prod__Pair__iden,axiom,
    ! [P2: produc3260441645384719156_nat_b] :
      ( ( produc5543054038463627251_nat_b @ produc1946830598933159916_nat_b @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_150_case__prodE,axiom,
    ! [C: real > b > $o,P2: product_prod_real_b] :
      ( ( produc5814611744267527345al_b_o @ C @ P2 )
     => ~ ! [X3: real,Y3: b] :
            ( ( P2
              = ( product_Pair_real_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_151_case__prodE,axiom,
    ! [C: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P2 )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_152_case__prodE,axiom,
    ! [C: nat > ( nat > b ) > $o,P2: produc7194034526493574155_nat_b] :
      ( ( produc7710141471362042158at_b_o @ C @ P2 )
     => ~ ! [X3: nat,Y3: nat > b] :
            ( ( P2
              = ( produc5292568363641424317_nat_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_153_case__prodE,axiom,
    ! [C: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > $o,P2: produc778026099599033348at_b_a] :
      ( ( produc6923730833870890157_b_a_o @ C @ P2 )
     => ~ ! [X3: b > produc7194034526493574155_nat_b > a > a,Y3: produc4883615123598087894at_b_a] :
            ( ( P2
              = ( produc653446490150804990at_b_a @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_154_case__prodE,axiom,
    ! [C: b > produc6209191202727158440at_b_a > $o,P2: produc4883615123598087894at_b_a] :
      ( ( produc2784492703816452571_b_a_o @ C @ P2 )
     => ~ ! [X3: b,Y3: produc6209191202727158440at_b_a] :
            ( ( P2
              = ( produc992390952543482448at_b_a @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_155_case__prodE,axiom,
    ! [C: produc7194034526493574155_nat_b > a > $o,P2: produc6209191202727158440at_b_a] :
      ( ( produc5783264217630104657_b_a_o @ C @ P2 )
     => ~ ! [X3: produc7194034526493574155_nat_b,Y3: a] :
            ( ( P2
              = ( produc8509509857186669914at_b_a @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_156_case__prodE,axiom,
    ! [C: product_prod_real_b > produc7194034526493574155_nat_b > $o,P2: produc3260441645384719156_nat_b] :
      ( ( produc9191534587096267607at_b_o @ C @ P2 )
     => ~ ! [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
            ( ( P2
              = ( produc1946830598933159916_nat_b @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_157_case__prodD,axiom,
    ! [F: real > b > $o,A: real,B: b] :
      ( ( produc5814611744267527345al_b_o @ F @ ( product_Pair_real_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_158_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_159_case__prodD,axiom,
    ! [F: nat > ( nat > b ) > $o,A: nat,B: nat > b] :
      ( ( produc7710141471362042158at_b_o @ F @ ( produc5292568363641424317_nat_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_160_case__prodD,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > $o,A: b > produc7194034526493574155_nat_b > a > a,B: produc4883615123598087894at_b_a] :
      ( ( produc6923730833870890157_b_a_o @ F @ ( produc653446490150804990at_b_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_161_case__prodD,axiom,
    ! [F: b > produc6209191202727158440at_b_a > $o,A: b,B: produc6209191202727158440at_b_a] :
      ( ( produc2784492703816452571_b_a_o @ F @ ( produc992390952543482448at_b_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_162_case__prodD,axiom,
    ! [F: produc7194034526493574155_nat_b > a > $o,A: produc7194034526493574155_nat_b,B: a] :
      ( ( produc5783264217630104657_b_a_o @ F @ ( produc8509509857186669914at_b_a @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_163_case__prodD,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > $o,A: product_prod_real_b,B: produc7194034526493574155_nat_b] :
      ( ( produc9191534587096267607at_b_o @ F @ ( produc1946830598933159916_nat_b @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_164_exp__qbs__comp__morphism,axiom,
    ! [F: $o > $o > real,W: quasi_borel_o,X4: quasi_borel_o,Y4: quasi_borel_real,G: $o > real > $o,Z2: quasi_borel_o] :
      ( ( member_o_o_real @ F @ ( qbs_mo1138393325155344277o_real @ W @ ( expone150755962933148203o_real @ X4 @ Y4 ) ) )
     => ( ( member_o_real_o @ G @ ( qbs_mo3734336635392818435real_o @ W @ ( expone1659820337974317777real_o @ Y4 @ Z2 ) ) )
       => ( member_o_o_o
          @ ^ [W2: $o] : ( comp_real_o_o @ ( G @ W2 ) @ ( F @ W2 ) )
          @ ( qbs_morphism_o_o_o @ W @ ( exponent_exp_qbs_o_o @ X4 @ Z2 ) ) ) ) ) ).

% exp_qbs_comp_morphism
thf(fact_165_exp__qbs__comp__morphism,axiom,
    ! [F: ( nat > b ) > a > produc432120507726958376_b_a_a,W: quasi_borel_nat_b,X4: quasi_borel_a,Y4: quasi_6353345141195568462_b_a_a,G: ( nat > b ) > produc432120507726958376_b_a_a > a,Z2: quasi_borel_a] :
      ( ( member2960169264137640847_b_a_a @ F @ ( qbs_mo7389655384932950653_b_a_a @ W @ ( expone5574410363414041987_b_a_a @ X4 @ Y4 ) ) )
     => ( ( member1811236362415105851_a_a_a @ G @ ( qbs_mo8618518167108125737_a_a_a @ W @ ( expone790792417327382575_a_a_a @ Y4 @ Z2 ) ) )
       => ( member_nat_b_a_a
          @ ^ [W2: nat > b] : ( comp_P466993921277732229_a_a_a @ ( G @ W2 ) @ ( F @ W2 ) )
          @ ( qbs_mo9010864224564201555_b_a_a @ W @ ( exponent_exp_qbs_a_a @ X4 @ Z2 ) ) ) ) ) ).

% exp_qbs_comp_morphism
thf(fact_166_exp__qbs__comp__morphism,axiom,
    ! [F: ( nat > b ) > produc432120507726958376_b_a_a > produc432120507726958376_b_a_a,W: quasi_borel_nat_b,X4: quasi_6353345141195568462_b_a_a,Y4: quasi_6353345141195568462_b_a_a,G: ( nat > b ) > produc432120507726958376_b_a_a > a,Z2: quasi_borel_a] :
      ( ( member5705524560132844005_b_a_a @ F @ ( qbs_mo3092601099906693971_b_a_a @ W @ ( expone761354066351259609_b_a_a @ X4 @ Y4 ) ) )
     => ( ( member1811236362415105851_a_a_a @ G @ ( qbs_mo8618518167108125737_a_a_a @ W @ ( expone790792417327382575_a_a_a @ Y4 @ Z2 ) ) )
       => ( member1811236362415105851_a_a_a
          @ ^ [W2: nat > b] : ( comp_P8974516532822524463_b_a_a @ ( G @ W2 ) @ ( F @ W2 ) )
          @ ( qbs_mo8618518167108125737_a_a_a @ W @ ( expone790792417327382575_a_a_a @ X4 @ Z2 ) ) ) ) ) ).

% exp_qbs_comp_morphism
thf(fact_167_exp__qbs__comp__morphism,axiom,
    ! [F: ( nat > b ) > produc432120507726958376_b_a_a > a,W: quasi_borel_nat_b,X4: quasi_6353345141195568462_b_a_a,Y4: quasi_borel_a,G: ( nat > b ) > a > a,Z2: quasi_borel_a] :
      ( ( member1811236362415105851_a_a_a @ F @ ( qbs_mo8618518167108125737_a_a_a @ W @ ( expone790792417327382575_a_a_a @ X4 @ Y4 ) ) )
     => ( ( member_nat_b_a_a @ G @ ( qbs_mo9010864224564201555_b_a_a @ W @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) )
       => ( member1811236362415105851_a_a_a
          @ ^ [W2: nat > b] : ( comp_a3817174012071355013_b_a_a @ ( G @ W2 ) @ ( F @ W2 ) )
          @ ( qbs_mo8618518167108125737_a_a_a @ W @ ( expone790792417327382575_a_a_a @ X4 @ Z2 ) ) ) ) ) ).

% exp_qbs_comp_morphism
thf(fact_168_exp__qbs__comp__morphism,axiom,
    ! [F: $o > $o > $o,W: quasi_borel_o,X4: quasi_borel_o,Y4: quasi_borel_o,G: $o > $o > $o,Z2: quasi_borel_o] :
      ( ( member_o_o_o @ F @ ( qbs_morphism_o_o_o @ W @ ( exponent_exp_qbs_o_o @ X4 @ Y4 ) ) )
     => ( ( member_o_o_o @ G @ ( qbs_morphism_o_o_o @ W @ ( exponent_exp_qbs_o_o @ Y4 @ Z2 ) ) )
       => ( member_o_o_o
          @ ^ [W2: $o] : ( comp_o_o_o @ ( G @ W2 ) @ ( F @ W2 ) )
          @ ( qbs_morphism_o_o_o @ W @ ( exponent_exp_qbs_o_o @ X4 @ Z2 ) ) ) ) ) ).

% exp_qbs_comp_morphism
thf(fact_169_Pair__inject,axiom,
    ! [A: real,B: b,A2: real,B2: b] :
      ( ( ( product_Pair_real_b @ A @ B )
        = ( product_Pair_real_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_170_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_171_Pair__inject,axiom,
    ! [A: nat,B: nat > b,A2: nat,B2: nat > b] :
      ( ( ( produc5292568363641424317_nat_b @ A @ B )
        = ( produc5292568363641424317_nat_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_172_Pair__inject,axiom,
    ! [A: b > produc7194034526493574155_nat_b > a > a,B: produc4883615123598087894at_b_a,A2: b > produc7194034526493574155_nat_b > a > a,B2: produc4883615123598087894at_b_a] :
      ( ( ( produc653446490150804990at_b_a @ A @ B )
        = ( produc653446490150804990at_b_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_173_Pair__inject,axiom,
    ! [A: b,B: produc6209191202727158440at_b_a,A2: b,B2: produc6209191202727158440at_b_a] :
      ( ( ( produc992390952543482448at_b_a @ A @ B )
        = ( produc992390952543482448at_b_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_174_Pair__inject,axiom,
    ! [A: produc7194034526493574155_nat_b,B: a,A2: produc7194034526493574155_nat_b,B2: a] :
      ( ( ( produc8509509857186669914at_b_a @ A @ B )
        = ( produc8509509857186669914at_b_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_175_Pair__inject,axiom,
    ! [A: product_prod_real_b,B: produc7194034526493574155_nat_b,A2: product_prod_real_b,B2: produc7194034526493574155_nat_b] :
      ( ( ( produc1946830598933159916_nat_b @ A @ B )
        = ( produc1946830598933159916_nat_b @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_176_prod__cases,axiom,
    ! [P: product_prod_real_b > $o,P2: product_prod_real_b] :
      ( ! [A3: real,B3: b] : ( P @ ( product_Pair_real_b @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_177_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_178_prod__cases,axiom,
    ! [P: produc7194034526493574155_nat_b > $o,P2: produc7194034526493574155_nat_b] :
      ( ! [A3: nat,B3: nat > b] : ( P @ ( produc5292568363641424317_nat_b @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_179_prod__cases,axiom,
    ! [P: produc778026099599033348at_b_a > $o,P2: produc778026099599033348at_b_a] :
      ( ! [A3: b > produc7194034526493574155_nat_b > a > a,B3: produc4883615123598087894at_b_a] : ( P @ ( produc653446490150804990at_b_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_180_prod__cases,axiom,
    ! [P: produc4883615123598087894at_b_a > $o,P2: produc4883615123598087894at_b_a] :
      ( ! [A3: b,B3: produc6209191202727158440at_b_a] : ( P @ ( produc992390952543482448at_b_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_181_prod__cases,axiom,
    ! [P: produc6209191202727158440at_b_a > $o,P2: produc6209191202727158440at_b_a] :
      ( ! [A3: produc7194034526493574155_nat_b,B3: a] : ( P @ ( produc8509509857186669914at_b_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_182_prod__cases,axiom,
    ! [P: produc3260441645384719156_nat_b > $o,P2: produc3260441645384719156_nat_b] :
      ( ! [A3: product_prod_real_b,B3: produc7194034526493574155_nat_b] : ( P @ ( produc1946830598933159916_nat_b @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_183_surj__pair,axiom,
    ! [P2: product_prod_real_b] :
    ? [X3: real,Y3: b] :
      ( P2
      = ( product_Pair_real_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_184_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_185_surj__pair,axiom,
    ! [P2: produc7194034526493574155_nat_b] :
    ? [X3: nat,Y3: nat > b] :
      ( P2
      = ( produc5292568363641424317_nat_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_186_surj__pair,axiom,
    ! [P2: produc778026099599033348at_b_a] :
    ? [X3: b > produc7194034526493574155_nat_b > a > a,Y3: produc4883615123598087894at_b_a] :
      ( P2
      = ( produc653446490150804990at_b_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_187_surj__pair,axiom,
    ! [P2: produc4883615123598087894at_b_a] :
    ? [X3: b,Y3: produc6209191202727158440at_b_a] :
      ( P2
      = ( produc992390952543482448at_b_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_188_surj__pair,axiom,
    ! [P2: produc6209191202727158440at_b_a] :
    ? [X3: produc7194034526493574155_nat_b,Y3: a] :
      ( P2
      = ( produc8509509857186669914at_b_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_189_surj__pair,axiom,
    ! [P2: produc3260441645384719156_nat_b] :
    ? [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
      ( P2
      = ( produc1946830598933159916_nat_b @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_190_old_Oprod_Oexhaust,axiom,
    ! [Y5: product_prod_real_b] :
      ~ ! [A3: real,B3: b] :
          ( Y5
         != ( product_Pair_real_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_191_old_Oprod_Oexhaust,axiom,
    ! [Y5: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y5
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_192_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc7194034526493574155_nat_b] :
      ~ ! [A3: nat,B3: nat > b] :
          ( Y5
         != ( produc5292568363641424317_nat_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_193_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc778026099599033348at_b_a] :
      ~ ! [A3: b > produc7194034526493574155_nat_b > a > a,B3: produc4883615123598087894at_b_a] :
          ( Y5
         != ( produc653446490150804990at_b_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_194_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc4883615123598087894at_b_a] :
      ~ ! [A3: b,B3: produc6209191202727158440at_b_a] :
          ( Y5
         != ( produc992390952543482448at_b_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_195_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc6209191202727158440at_b_a] :
      ~ ! [A3: produc7194034526493574155_nat_b,B3: a] :
          ( Y5
         != ( produc8509509857186669914at_b_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_196_old_Oprod_Oexhaust,axiom,
    ! [Y5: produc3260441645384719156_nat_b] :
      ~ ! [A3: product_prod_real_b,B3: produc7194034526493574155_nat_b] :
          ( Y5
         != ( produc1946830598933159916_nat_b @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_197_rewriteR__comp__comp2,axiom,
    ! [G: $o > real,H: $o > $o,R1: real > real,R2: $o > real,F: real > $o,L2: real > $o] :
      ( ( ( comp_o_real_o @ G @ H )
        = ( comp_real_real_o @ R1 @ R2 ) )
     => ( ( ( comp_real_o_real @ F @ R1 )
          = L2 )
       => ( ( comp_o_o_o @ ( comp_real_o_o @ F @ G ) @ H )
          = ( comp_real_o_o @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_198_rewriteR__comp__comp2,axiom,
    ! [G: nat > real,H: nat > nat,R1: real > real,R2: nat > real,F: real > nat,L2: real > nat] :
      ( ( ( comp_nat_real_nat @ G @ H )
        = ( comp_real_real_nat @ R1 @ R2 ) )
     => ( ( ( comp_real_nat_real @ F @ R1 )
          = L2 )
       => ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F @ G ) @ H )
          = ( comp_real_nat_nat @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_199_rewriteR__comp__comp2,axiom,
    ! [G: real > real,H: $o > real,R1: $o > real,R2: $o > $o,F: real > $o,L2: $o > $o] :
      ( ( ( comp_real_real_o @ G @ H )
        = ( comp_o_real_o @ R1 @ R2 ) )
     => ( ( ( comp_real_o_o @ F @ R1 )
          = L2 )
       => ( ( comp_real_o_o @ ( comp_real_o_real @ F @ G ) @ H )
          = ( comp_o_o_o @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_200_rewriteR__comp__comp2,axiom,
    ! [G: real > real,H: nat > real,R1: nat > real,R2: nat > nat,F: real > nat,L2: nat > nat] :
      ( ( ( comp_real_real_nat @ G @ H )
        = ( comp_nat_real_nat @ R1 @ R2 ) )
     => ( ( ( comp_real_nat_nat @ F @ R1 )
          = L2 )
       => ( ( comp_real_nat_nat @ ( comp_real_nat_real @ F @ G ) @ H )
          = ( comp_nat_nat_nat @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_201_rewriteR__comp__comp2,axiom,
    ! [G: $o > real,H: real > $o,R1: real > real,R2: real > real,F: real > $o,L2: real > $o] :
      ( ( ( comp_o_real_real @ G @ H )
        = ( comp_real_real_real @ R1 @ R2 ) )
     => ( ( ( comp_real_o_real @ F @ R1 )
          = L2 )
       => ( ( comp_o_o_real @ ( comp_real_o_o @ F @ G ) @ H )
          = ( comp_real_o_real @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_202_rewriteR__comp__comp2,axiom,
    ! [G: nat > real,H: real > nat,R1: real > real,R2: real > real,F: real > nat,L2: real > nat] :
      ( ( ( comp_nat_real_real @ G @ H )
        = ( comp_real_real_real @ R1 @ R2 ) )
     => ( ( ( comp_real_nat_real @ F @ R1 )
          = L2 )
       => ( ( comp_nat_nat_real @ ( comp_real_nat_nat @ F @ G ) @ H )
          = ( comp_real_nat_real @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_203_rewriteR__comp__comp2,axiom,
    ! [G: real > real,H: real > real,R1: $o > real,R2: real > $o,F: real > $o,L2: $o > $o] :
      ( ( ( comp_real_real_real @ G @ H )
        = ( comp_o_real_real @ R1 @ R2 ) )
     => ( ( ( comp_real_o_o @ F @ R1 )
          = L2 )
       => ( ( comp_real_o_real @ ( comp_real_o_real @ F @ G ) @ H )
          = ( comp_o_o_real @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_204_rewriteR__comp__comp2,axiom,
    ! [G: real > real,H: real > real,R1: nat > real,R2: real > nat,F: real > nat,L2: nat > nat] :
      ( ( ( comp_real_real_real @ G @ H )
        = ( comp_nat_real_real @ R1 @ R2 ) )
     => ( ( ( comp_real_nat_nat @ F @ R1 )
          = L2 )
       => ( ( comp_real_nat_real @ ( comp_real_nat_real @ F @ G ) @ H )
          = ( comp_nat_nat_real @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_205_rewriteR__comp__comp2,axiom,
    ! [G: real > real,H: real > real,R1: real > real,R2: real > real,F: real > real,L2: real > real] :
      ( ( ( comp_real_real_real @ G @ H )
        = ( comp_real_real_real @ R1 @ R2 ) )
     => ( ( ( comp_real_real_real @ F @ R1 )
          = L2 )
       => ( ( comp_real_real_real @ ( comp_real_real_real @ F @ G ) @ H )
          = ( comp_real_real_real @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_206_rewriteR__comp__comp2,axiom,
    ! [G: real > $o,H: $o > real,R1: real > $o,R2: $o > real,F: $o > $o,L2: real > $o] :
      ( ( ( comp_real_o_o @ G @ H )
        = ( comp_real_o_o @ R1 @ R2 ) )
     => ( ( ( comp_o_o_real @ F @ R1 )
          = L2 )
       => ( ( comp_real_o_o @ ( comp_o_o_real @ F @ G ) @ H )
          = ( comp_real_o_o @ L2 @ R2 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_207_rewriteL__comp__comp2,axiom,
    ! [F: $o > $o,G: real > $o,L1: real > $o,L22: real > real,H: $o > real,R3: $o > real] :
      ( ( ( comp_o_o_real @ F @ G )
        = ( comp_real_o_real @ L1 @ L22 ) )
     => ( ( ( comp_real_real_o @ L22 @ H )
          = R3 )
       => ( ( comp_o_o_o @ F @ ( comp_real_o_o @ G @ H ) )
          = ( comp_real_o_o @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_208_rewriteL__comp__comp2,axiom,
    ! [F: nat > nat,G: real > nat,L1: real > nat,L22: real > real,H: nat > real,R3: nat > real] :
      ( ( ( comp_nat_nat_real @ F @ G )
        = ( comp_real_nat_real @ L1 @ L22 ) )
     => ( ( ( comp_real_real_nat @ L22 @ H )
          = R3 )
       => ( ( comp_nat_nat_nat @ F @ ( comp_real_nat_nat @ G @ H ) )
          = ( comp_real_nat_nat @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_209_rewriteL__comp__comp2,axiom,
    ! [F: real > $o,G: real > real,L1: $o > $o,L22: real > $o,H: $o > real,R3: $o > $o] :
      ( ( ( comp_real_o_real @ F @ G )
        = ( comp_o_o_real @ L1 @ L22 ) )
     => ( ( ( comp_real_o_o @ L22 @ H )
          = R3 )
       => ( ( comp_real_o_o @ F @ ( comp_real_real_o @ G @ H ) )
          = ( comp_o_o_o @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_210_rewriteL__comp__comp2,axiom,
    ! [F: real > nat,G: real > real,L1: nat > nat,L22: real > nat,H: nat > real,R3: nat > nat] :
      ( ( ( comp_real_nat_real @ F @ G )
        = ( comp_nat_nat_real @ L1 @ L22 ) )
     => ( ( ( comp_real_nat_nat @ L22 @ H )
          = R3 )
       => ( ( comp_real_nat_nat @ F @ ( comp_real_real_nat @ G @ H ) )
          = ( comp_nat_nat_nat @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_211_rewriteL__comp__comp2,axiom,
    ! [F: $o > real,G: real > $o,L1: real > real,L22: real > real,H: $o > real,R3: $o > real] :
      ( ( ( comp_o_real_real @ F @ G )
        = ( comp_real_real_real @ L1 @ L22 ) )
     => ( ( ( comp_real_real_o @ L22 @ H )
          = R3 )
       => ( ( comp_o_real_o @ F @ ( comp_real_o_o @ G @ H ) )
          = ( comp_real_real_o @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_212_rewriteL__comp__comp2,axiom,
    ! [F: nat > real,G: real > nat,L1: real > real,L22: real > real,H: nat > real,R3: nat > real] :
      ( ( ( comp_nat_real_real @ F @ G )
        = ( comp_real_real_real @ L1 @ L22 ) )
     => ( ( ( comp_real_real_nat @ L22 @ H )
          = R3 )
       => ( ( comp_nat_real_nat @ F @ ( comp_real_nat_nat @ G @ H ) )
          = ( comp_real_real_nat @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_213_rewriteL__comp__comp2,axiom,
    ! [F: real > real,G: real > real,L1: $o > real,L22: real > $o,H: $o > real,R3: $o > $o] :
      ( ( ( comp_real_real_real @ F @ G )
        = ( comp_o_real_real @ L1 @ L22 ) )
     => ( ( ( comp_real_o_o @ L22 @ H )
          = R3 )
       => ( ( comp_real_real_o @ F @ ( comp_real_real_o @ G @ H ) )
          = ( comp_o_real_o @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_214_rewriteL__comp__comp2,axiom,
    ! [F: real > real,G: real > real,L1: nat > real,L22: real > nat,H: nat > real,R3: nat > nat] :
      ( ( ( comp_real_real_real @ F @ G )
        = ( comp_nat_real_real @ L1 @ L22 ) )
     => ( ( ( comp_real_nat_nat @ L22 @ H )
          = R3 )
       => ( ( comp_real_real_nat @ F @ ( comp_real_real_nat @ G @ H ) )
          = ( comp_nat_real_nat @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_215_rewriteL__comp__comp2,axiom,
    ! [F: real > real,G: real > real,L1: real > real,L22: real > real,H: real > real,R3: real > real] :
      ( ( ( comp_real_real_real @ F @ G )
        = ( comp_real_real_real @ L1 @ L22 ) )
     => ( ( ( comp_real_real_real @ L22 @ H )
          = R3 )
       => ( ( comp_real_real_real @ F @ ( comp_real_real_real @ G @ H ) )
          = ( comp_real_real_real @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_216_rewriteL__comp__comp2,axiom,
    ! [F: real > $o,G: $o > real,L1: real > $o,L22: $o > real,H: $o > $o,R3: $o > real] :
      ( ( ( comp_real_o_o @ F @ G )
        = ( comp_real_o_o @ L1 @ L22 ) )
     => ( ( ( comp_o_real_o @ L22 @ H )
          = R3 )
       => ( ( comp_real_o_o @ F @ ( comp_o_real_o @ G @ H ) )
          = ( comp_real_o_o @ L1 @ R3 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_217_rewriteR__comp__comp,axiom,
    ! [G: $o > real,H: $o > $o,R3: $o > real,F: real > $o] :
      ( ( ( comp_o_real_o @ G @ H )
        = R3 )
     => ( ( comp_o_o_o @ ( comp_real_o_o @ F @ G ) @ H )
        = ( comp_real_o_o @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_218_rewriteR__comp__comp,axiom,
    ! [G: nat > real,H: nat > nat,R3: nat > real,F: real > nat] :
      ( ( ( comp_nat_real_nat @ G @ H )
        = R3 )
     => ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F @ G ) @ H )
        = ( comp_real_nat_nat @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_219_rewriteR__comp__comp,axiom,
    ! [G: real > real,H: $o > real,R3: $o > real,F: real > $o] :
      ( ( ( comp_real_real_o @ G @ H )
        = R3 )
     => ( ( comp_real_o_o @ ( comp_real_o_real @ F @ G ) @ H )
        = ( comp_real_o_o @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_220_rewriteR__comp__comp,axiom,
    ! [G: real > real,H: nat > real,R3: nat > real,F: real > nat] :
      ( ( ( comp_real_real_nat @ G @ H )
        = R3 )
     => ( ( comp_real_nat_nat @ ( comp_real_nat_real @ F @ G ) @ H )
        = ( comp_real_nat_nat @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_221_rewriteR__comp__comp,axiom,
    ! [G: real > real,H: real > real,R3: real > real,F: real > real] :
      ( ( ( comp_real_real_real @ G @ H )
        = R3 )
     => ( ( comp_real_real_real @ ( comp_real_real_real @ F @ G ) @ H )
        = ( comp_real_real_real @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_222_rewriteR__comp__comp,axiom,
    ! [G: real > $o,H: $o > real,R3: $o > $o,F: $o > $o] :
      ( ( ( comp_real_o_o @ G @ H )
        = R3 )
     => ( ( comp_real_o_o @ ( comp_o_o_real @ F @ G ) @ H )
        = ( comp_o_o_o @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_223_rewriteR__comp__comp,axiom,
    ! [G: real > nat,H: nat > real,R3: nat > nat,F: nat > nat] :
      ( ( ( comp_real_nat_nat @ G @ H )
        = R3 )
     => ( ( comp_real_nat_nat @ ( comp_nat_nat_real @ F @ G ) @ H )
        = ( comp_nat_nat_nat @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_224_rewriteR__comp__comp,axiom,
    ! [G: real > real,H: produc2422161461964618553l_real > real,R3: produc2422161461964618553l_real > real,F: real > produc2422161461964618553l_real] :
      ( ( ( comp_r4460121664545661500l_real @ G @ H )
        = R3 )
     => ( ( comp_r97482993418849103l_real @ ( comp_r4464757161866874646l_real @ F @ G ) @ H )
        = ( comp_r97482993418849103l_real @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_225_rewriteR__comp__comp,axiom,
    ! [G: real > real,H: produc7716430852924023517t_real > real,R3: produc7716430852924023517t_real > real,F: real > produc7716430852924023517t_real] :
      ( ( ( comp_r8349034141028359008t_real @ G @ H )
        = R3 )
     => ( ( comp_r4825950202331181903t_real @ ( comp_r4030900867926482106l_real @ F @ G ) @ H )
        = ( comp_r4825950202331181903t_real @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_226_rewriteR__comp__comp,axiom,
    ! [G: produc2422161461964618553l_real > real,H: produc2422161461964618553l_real > produc2422161461964618553l_real,R3: produc2422161461964618553l_real > real,F: real > produc2422161461964618553l_real] :
      ( ( ( comp_P3276979369905350645l_real @ G @ H )
        = R3 )
     => ( ( comp_P4947614994437801686l_real @ ( comp_r97482993418849103l_real @ F @ G ) @ H )
        = ( comp_r97482993418849103l_real @ F @ R3 ) ) ) ).

% rewriteR_comp_comp
thf(fact_227_rewriteL__comp__comp,axiom,
    ! [F: $o > $o,G: real > $o,L2: real > $o,H: $o > real] :
      ( ( ( comp_o_o_real @ F @ G )
        = L2 )
     => ( ( comp_o_o_o @ F @ ( comp_real_o_o @ G @ H ) )
        = ( comp_real_o_o @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_228_rewriteL__comp__comp,axiom,
    ! [F: nat > nat,G: real > nat,L2: real > nat,H: nat > real] :
      ( ( ( comp_nat_nat_real @ F @ G )
        = L2 )
     => ( ( comp_nat_nat_nat @ F @ ( comp_real_nat_nat @ G @ H ) )
        = ( comp_real_nat_nat @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_229_rewriteL__comp__comp,axiom,
    ! [F: real > $o,G: real > real,L2: real > $o,H: $o > real] :
      ( ( ( comp_real_o_real @ F @ G )
        = L2 )
     => ( ( comp_real_o_o @ F @ ( comp_real_real_o @ G @ H ) )
        = ( comp_real_o_o @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_230_rewriteL__comp__comp,axiom,
    ! [F: real > nat,G: real > real,L2: real > nat,H: nat > real] :
      ( ( ( comp_real_nat_real @ F @ G )
        = L2 )
     => ( ( comp_real_nat_nat @ F @ ( comp_real_real_nat @ G @ H ) )
        = ( comp_real_nat_nat @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_231_rewriteL__comp__comp,axiom,
    ! [F: real > real,G: real > real,L2: real > real,H: real > real] :
      ( ( ( comp_real_real_real @ F @ G )
        = L2 )
     => ( ( comp_real_real_real @ F @ ( comp_real_real_real @ G @ H ) )
        = ( comp_real_real_real @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_232_rewriteL__comp__comp,axiom,
    ! [F: real > $o,G: $o > real,L2: $o > $o,H: $o > $o] :
      ( ( ( comp_real_o_o @ F @ G )
        = L2 )
     => ( ( comp_real_o_o @ F @ ( comp_o_real_o @ G @ H ) )
        = ( comp_o_o_o @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_233_rewriteL__comp__comp,axiom,
    ! [F: real > nat,G: nat > real,L2: nat > nat,H: nat > nat] :
      ( ( ( comp_real_nat_nat @ F @ G )
        = L2 )
     => ( ( comp_real_nat_nat @ F @ ( comp_nat_real_nat @ G @ H ) )
        = ( comp_nat_nat_nat @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_234_rewriteL__comp__comp,axiom,
    ! [F: real > produc2422161461964618553l_real,G: real > real,L2: real > produc2422161461964618553l_real,H: produc2422161461964618553l_real > real] :
      ( ( ( comp_r4464757161866874646l_real @ F @ G )
        = L2 )
     => ( ( comp_r97482993418849103l_real @ F @ ( comp_r4460121664545661500l_real @ G @ H ) )
        = ( comp_r97482993418849103l_real @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_235_rewriteL__comp__comp,axiom,
    ! [F: real > produc7716430852924023517t_real,G: real > real,L2: real > produc7716430852924023517t_real,H: produc7716430852924023517t_real > real] :
      ( ( ( comp_r4030900867926482106l_real @ F @ G )
        = L2 )
     => ( ( comp_r4825950202331181903t_real @ F @ ( comp_r8349034141028359008t_real @ G @ H ) )
        = ( comp_r4825950202331181903t_real @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_236_rewriteL__comp__comp,axiom,
    ! [F: produc2422161461964618553l_real > produc2422161461964618553l_real,G: real > produc2422161461964618553l_real,L2: real > produc2422161461964618553l_real,H: produc2422161461964618553l_real > real] :
      ( ( ( comp_P3281614867226563791l_real @ F @ G )
        = L2 )
     => ( ( comp_P4947614994437801686l_real @ F @ ( comp_r97482993418849103l_real @ G @ H ) )
        = ( comp_r97482993418849103l_real @ L2 @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_237_comp__eq__dest__lhs,axiom,
    ! [A: real > produc2422161461964618553l_real,B: produc2422161461964618553l_real > real,C: produc2422161461964618553l_real > produc2422161461964618553l_real,V: produc2422161461964618553l_real] :
      ( ( ( comp_r97482993418849103l_real @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_238_comp__eq__dest__lhs,axiom,
    ! [A: real > produc7716430852924023517t_real,B: produc7716430852924023517t_real > real,C: produc7716430852924023517t_real > produc7716430852924023517t_real,V: produc7716430852924023517t_real] :
      ( ( ( comp_r4825950202331181903t_real @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_239_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_240_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_241_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_242_comp__eq__elim,axiom,
    ! [A: real > produc2422161461964618553l_real,B: produc2422161461964618553l_real > real,C: real > produc2422161461964618553l_real,D: produc2422161461964618553l_real > real] :
      ( ( ( comp_r97482993418849103l_real @ A @ B )
        = ( comp_r97482993418849103l_real @ C @ D ) )
     => ! [V2: produc2422161461964618553l_real] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_243_comp__eq__elim,axiom,
    ! [A: real > produc7716430852924023517t_real,B: produc7716430852924023517t_real > real,C: real > produc7716430852924023517t_real,D: produc7716430852924023517t_real > real] :
      ( ( ( comp_r4825950202331181903t_real @ A @ B )
        = ( comp_r4825950202331181903t_real @ C @ D ) )
     => ! [V2: produc7716430852924023517t_real] :
          ( ( A @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_244_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_245_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_246_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_247_comp__eq__dest,axiom,
    ! [A: real > produc2422161461964618553l_real,B: produc2422161461964618553l_real > real,C: real > produc2422161461964618553l_real,D: produc2422161461964618553l_real > real,V: produc2422161461964618553l_real] :
      ( ( ( comp_r97482993418849103l_real @ A @ B )
        = ( comp_r97482993418849103l_real @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_248_comp__eq__dest,axiom,
    ! [A: real > produc7716430852924023517t_real,B: produc7716430852924023517t_real > real,C: real > produc7716430852924023517t_real,D: produc7716430852924023517t_real > real,V: produc7716430852924023517t_real] :
      ( ( ( comp_r4825950202331181903t_real @ A @ B )
        = ( comp_r4825950202331181903t_real @ C @ D ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_249_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_250_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_251_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_252_comp__assoc,axiom,
    ! [F: real > $o,G: $o > real,H: $o > $o] :
      ( ( comp_o_o_o @ ( comp_real_o_o @ F @ G ) @ H )
      = ( comp_real_o_o @ F @ ( comp_o_real_o @ G @ H ) ) ) ).

% comp_assoc
thf(fact_253_comp__assoc,axiom,
    ! [F: real > nat,G: nat > real,H: nat > nat] :
      ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F @ G ) @ H )
      = ( comp_real_nat_nat @ F @ ( comp_nat_real_nat @ G @ H ) ) ) ).

% comp_assoc
thf(fact_254_comp__assoc,axiom,
    ! [F: real > real,G: real > real,H: real > real] :
      ( ( comp_real_real_real @ ( comp_real_real_real @ F @ G ) @ H )
      = ( comp_real_real_real @ F @ ( comp_real_real_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_255_comp__assoc,axiom,
    ! [F: $o > $o,G: real > $o,H: $o > real] :
      ( ( comp_real_o_o @ ( comp_o_o_real @ F @ G ) @ H )
      = ( comp_o_o_o @ F @ ( comp_real_o_o @ G @ H ) ) ) ).

% comp_assoc
thf(fact_256_comp__assoc,axiom,
    ! [F: real > $o,G: real > real,H: $o > real] :
      ( ( comp_real_o_o @ ( comp_real_o_real @ F @ G ) @ H )
      = ( comp_real_o_o @ F @ ( comp_real_real_o @ G @ H ) ) ) ).

% comp_assoc
thf(fact_257_comp__assoc,axiom,
    ! [F: nat > nat,G: real > nat,H: nat > real] :
      ( ( comp_real_nat_nat @ ( comp_nat_nat_real @ F @ G ) @ H )
      = ( comp_nat_nat_nat @ F @ ( comp_real_nat_nat @ G @ H ) ) ) ).

% comp_assoc
thf(fact_258_comp__assoc,axiom,
    ! [F: real > nat,G: real > real,H: nat > real] :
      ( ( comp_real_nat_nat @ ( comp_real_nat_real @ F @ G ) @ H )
      = ( comp_real_nat_nat @ F @ ( comp_real_real_nat @ G @ H ) ) ) ).

% comp_assoc
thf(fact_259_comp__assoc,axiom,
    ! [F: real > produc2422161461964618553l_real,G: real > real,H: produc2422161461964618553l_real > real] :
      ( ( comp_r97482993418849103l_real @ ( comp_r4464757161866874646l_real @ F @ G ) @ H )
      = ( comp_r97482993418849103l_real @ F @ ( comp_r4460121664545661500l_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_260_comp__assoc,axiom,
    ! [F: real > produc7716430852924023517t_real,G: real > real,H: produc7716430852924023517t_real > real] :
      ( ( comp_r4825950202331181903t_real @ ( comp_r4030900867926482106l_real @ F @ G ) @ H )
      = ( comp_r4825950202331181903t_real @ F @ ( comp_r8349034141028359008t_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_261_comp__assoc,axiom,
    ! [F: real > produc2422161461964618553l_real,G: produc2422161461964618553l_real > real,H: produc2422161461964618553l_real > produc2422161461964618553l_real] :
      ( ( comp_P4947614994437801686l_real @ ( comp_r97482993418849103l_real @ F @ G ) @ H )
      = ( comp_r97482993418849103l_real @ F @ ( comp_P3276979369905350645l_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_262_comp__def,axiom,
    ( comp_r97482993418849103l_real
    = ( ^ [F2: real > produc2422161461964618553l_real,G2: produc2422161461964618553l_real > real,X: produc2422161461964618553l_real] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_263_comp__def,axiom,
    ( comp_r4825950202331181903t_real
    = ( ^ [F2: real > produc7716430852924023517t_real,G2: produc7716430852924023517t_real > real,X: produc7716430852924023517t_real] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_264_comp__def,axiom,
    ( comp_real_real_real
    = ( ^ [F2: real > real,G2: real > real,X: real] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_265_comp__def,axiom,
    ( comp_real_o_o
    = ( ^ [F2: real > $o,G2: $o > real,X: $o] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_266_comp__def,axiom,
    ( comp_real_nat_nat
    = ( ^ [F2: real > nat,G2: nat > real,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_267_mem__Collect__eq,axiom,
    ! [A: $o > real,P: ( $o > real ) > $o] :
      ( ( member_o_real @ A @ ( collect_o_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_268_mem__Collect__eq,axiom,
    ! [A: $o > $o,P: ( $o > $o ) > $o] :
      ( ( member_o_o @ A @ ( collect_o_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_269_mem__Collect__eq,axiom,
    ! [A: $o > $o > $o,P: ( $o > $o > $o ) > $o] :
      ( ( member_o_o_o @ A @ ( collect_o_o_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_270_mem__Collect__eq,axiom,
    ! [A: nat > real,P: ( nat > real ) > $o] :
      ( ( member_nat_real @ A @ ( collect_nat_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_271_mem__Collect__eq,axiom,
    ! [A: ( nat > b ) > produc432120507726958376_b_a_a > a,P: ( ( nat > b ) > produc432120507726958376_b_a_a > a ) > $o] :
      ( ( member1811236362415105851_a_a_a @ A @ ( collec5238665274127055997_a_a_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_272_Collect__mem__eq,axiom,
    ! [A4: set_o_real] :
      ( ( collect_o_real
        @ ^ [X: $o > real] : ( member_o_real @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_273_Collect__mem__eq,axiom,
    ! [A4: set_o_o] :
      ( ( collect_o_o
        @ ^ [X: $o > $o] : ( member_o_o @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_274_Collect__mem__eq,axiom,
    ! [A4: set_o_o_o] :
      ( ( collect_o_o_o
        @ ^ [X: $o > $o > $o] : ( member_o_o_o @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_275_Collect__mem__eq,axiom,
    ! [A4: set_nat_real] :
      ( ( collect_nat_real
        @ ^ [X: nat > real] : ( member_nat_real @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_276_Collect__mem__eq,axiom,
    ! [A4: set_na8676686263003397490_a_a_a] :
      ( ( collec5238665274127055997_a_a_a
        @ ^ [X: ( nat > b ) > produc432120507726958376_b_a_a > a] : ( member1811236362415105851_a_a_a @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_277_qbs__morphismI,axiom,
    ! [X4: quasi_borel_real,F: real > real,Y4: quasi_borel_real] :
      ( ! [Alpha: real > real] :
          ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X4 ) )
         => ( member_real_real @ ( comp_real_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y4 ) ) )
     => ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_278_qbs__morphismI,axiom,
    ! [X4: quasi_borel_o,F: $o > real,Y4: quasi_borel_real] :
      ( ! [Alpha: real > $o] :
          ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X4 ) )
         => ( member_real_real @ ( comp_o_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y4 ) ) )
     => ( member_o_real @ F @ ( qbs_morphism_o_real @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_279_qbs__morphismI,axiom,
    ! [X4: quasi_borel_nat,F: nat > real,Y4: quasi_borel_real] :
      ( ! [Alpha: real > nat] :
          ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X4 ) )
         => ( member_real_real @ ( comp_nat_real_real @ F @ Alpha ) @ ( qbs_Mx_real @ Y4 ) ) )
     => ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_280_qbs__morphismI,axiom,
    ! [X4: quasi_borel_b,F: b > b,Y4: quasi_borel_b] :
      ( ! [Alpha: real > b] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X4 ) )
         => ( member_real_b @ ( comp_b_b_real @ F @ Alpha ) @ ( qbs_Mx_b @ Y4 ) ) )
     => ( member_b_b @ F @ ( qbs_morphism_b_b @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_281_qbs__morphismI,axiom,
    ! [X4: quasi_borel_b,F: b > a,Y4: quasi_borel_a] :
      ( ! [Alpha: real > b] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X4 ) )
         => ( member_real_a @ ( comp_b_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y4 ) ) )
     => ( member_b_a @ F @ ( qbs_morphism_b_a @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_282_qbs__morphismI,axiom,
    ! [X4: quasi_borel_a,F: a > b,Y4: quasi_borel_b] :
      ( ! [Alpha: real > a] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X4 ) )
         => ( member_real_b @ ( comp_a_b_real @ F @ Alpha ) @ ( qbs_Mx_b @ Y4 ) ) )
     => ( member_a_b @ F @ ( qbs_morphism_a_b @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_283_qbs__morphismI,axiom,
    ! [X4: quasi_borel_a,F: a > a,Y4: quasi_borel_a] :
      ( ! [Alpha: real > a] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X4 ) )
         => ( member_real_a @ ( comp_a_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y4 ) ) )
     => ( member_a_a @ F @ ( qbs_morphism_a_a @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_284_qbs__morphismI,axiom,
    ! [X4: quasi_borel_real,F: real > a,Y4: quasi_borel_a] :
      ( ! [Alpha: real > real] :
          ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X4 ) )
         => ( member_real_a @ ( comp_real_a_real @ F @ Alpha ) @ ( qbs_Mx_a @ Y4 ) ) )
     => ( member_real_a @ F @ ( qbs_morphism_real_a @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_285_qbs__morphismI,axiom,
    ! [X4: quasi_borel_o,F: $o > $o,Y4: quasi_borel_o] :
      ( ! [Alpha: real > $o] :
          ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X4 ) )
         => ( member_real_o @ ( comp_o_o_real @ F @ Alpha ) @ ( qbs_Mx_o @ Y4 ) ) )
     => ( member_o_o @ F @ ( qbs_morphism_o_o @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_286_qbs__morphismI,axiom,
    ! [X4: quasi_borel_nat_b,F: ( nat > b ) > b,Y4: quasi_borel_b] :
      ( ! [Alpha: real > nat > b] :
          ( ( member_real_nat_b @ Alpha @ ( qbs_Mx_nat_b @ X4 ) )
         => ( member_real_b @ ( comp_nat_b_b_real @ F @ Alpha ) @ ( qbs_Mx_b @ Y4 ) ) )
     => ( member_nat_b_b @ F @ ( qbs_morphism_nat_b_b @ X4 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_287_qbs__morphismE_I3_J,axiom,
    ! [F: real > real,X4: quasi_borel_real,Y4: quasi_borel_real,Alpha2: real > real] :
      ( ( member_real_real @ F @ ( qbs_mo5229770564518008146l_real @ X4 @ Y4 ) )
     => ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X4 ) )
       => ( member_real_real @ ( comp_real_real_real @ F @ Alpha2 ) @ ( qbs_Mx_real @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_288_qbs__morphismE_I3_J,axiom,
    ! [F: $o > real,X4: quasi_borel_o,Y4: quasi_borel_real,Alpha2: real > $o] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X4 @ Y4 ) )
     => ( ( member_real_o @ Alpha2 @ ( qbs_Mx_o @ X4 ) )
       => ( member_real_real @ ( comp_o_real_real @ F @ Alpha2 ) @ ( qbs_Mx_real @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_289_qbs__morphismE_I3_J,axiom,
    ! [F: nat > real,X4: quasi_borel_nat,Y4: quasi_borel_real,Alpha2: real > nat] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
     => ( ( member_real_nat @ Alpha2 @ ( qbs_Mx_nat @ X4 ) )
       => ( member_real_real @ ( comp_nat_real_real @ F @ Alpha2 ) @ ( qbs_Mx_real @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_290_qbs__morphismE_I3_J,axiom,
    ! [F: b > b,X4: quasi_borel_b,Y4: quasi_borel_b,Alpha2: real > b] :
      ( ( member_b_b @ F @ ( qbs_morphism_b_b @ X4 @ Y4 ) )
     => ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X4 ) )
       => ( member_real_b @ ( comp_b_b_real @ F @ Alpha2 ) @ ( qbs_Mx_b @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_291_qbs__morphismE_I3_J,axiom,
    ! [F: b > a,X4: quasi_borel_b,Y4: quasi_borel_a,Alpha2: real > b] :
      ( ( member_b_a @ F @ ( qbs_morphism_b_a @ X4 @ Y4 ) )
     => ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X4 ) )
       => ( member_real_a @ ( comp_b_a_real @ F @ Alpha2 ) @ ( qbs_Mx_a @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_292_qbs__morphismE_I3_J,axiom,
    ! [F: a > b,X4: quasi_borel_a,Y4: quasi_borel_b,Alpha2: real > a] :
      ( ( member_a_b @ F @ ( qbs_morphism_a_b @ X4 @ Y4 ) )
     => ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X4 ) )
       => ( member_real_b @ ( comp_a_b_real @ F @ Alpha2 ) @ ( qbs_Mx_b @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_293_qbs__morphismE_I3_J,axiom,
    ! [F: a > a,X4: quasi_borel_a,Y4: quasi_borel_a,Alpha2: real > a] :
      ( ( member_a_a @ F @ ( qbs_morphism_a_a @ X4 @ Y4 ) )
     => ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X4 ) )
       => ( member_real_a @ ( comp_a_a_real @ F @ Alpha2 ) @ ( qbs_Mx_a @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_294_qbs__morphismE_I3_J,axiom,
    ! [F: real > a,X4: quasi_borel_real,Y4: quasi_borel_a,Alpha2: real > real] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X4 @ Y4 ) )
     => ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X4 ) )
       => ( member_real_a @ ( comp_real_a_real @ F @ Alpha2 ) @ ( qbs_Mx_a @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_295_qbs__morphismE_I3_J,axiom,
    ! [F: $o > $o,X4: quasi_borel_o,Y4: quasi_borel_o,Alpha2: real > $o] :
      ( ( member_o_o @ F @ ( qbs_morphism_o_o @ X4 @ Y4 ) )
     => ( ( member_real_o @ Alpha2 @ ( qbs_Mx_o @ X4 ) )
       => ( member_real_o @ ( comp_o_o_real @ F @ Alpha2 ) @ ( qbs_Mx_o @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_296_qbs__morphismE_I3_J,axiom,
    ! [F: ( nat > b ) > b,X4: quasi_borel_nat_b,Y4: quasi_borel_b,Alpha2: real > nat > b] :
      ( ( member_nat_b_b @ F @ ( qbs_morphism_nat_b_b @ X4 @ Y4 ) )
     => ( ( member_real_nat_b @ Alpha2 @ ( qbs_Mx_nat_b @ X4 ) )
       => ( member_real_b @ ( comp_nat_b_b_real @ F @ Alpha2 ) @ ( qbs_Mx_b @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_297_compose__const_I1_J,axiom,
    ! [F: real > produc2422161461964618553l_real,A: real] :
      ( ( comp_r97482993418849103l_real @ F
        @ ^ [X: produc2422161461964618553l_real] : A )
      = ( ^ [X: produc2422161461964618553l_real] : ( F @ A ) ) ) ).

% compose_const(1)
thf(fact_298_compose__const_I1_J,axiom,
    ! [F: real > produc7716430852924023517t_real,A: real] :
      ( ( comp_r4825950202331181903t_real @ F
        @ ^ [X: produc7716430852924023517t_real] : A )
      = ( ^ [X: produc7716430852924023517t_real] : ( F @ A ) ) ) ).

% compose_const(1)
thf(fact_299_compose__const_I1_J,axiom,
    ! [F: real > real,A: real] :
      ( ( comp_real_real_real @ F
        @ ^ [X: real] : A )
      = ( ^ [X: real] : ( F @ A ) ) ) ).

% compose_const(1)
thf(fact_300_compose__const_I1_J,axiom,
    ! [F: real > $o,A: real] :
      ( ( comp_real_o_o @ F
        @ ^ [X: $o] : A )
      = ( ^ [X: $o] : ( F @ A ) ) ) ).

% compose_const(1)
thf(fact_301_compose__const_I1_J,axiom,
    ! [F: real > nat,A: real] :
      ( ( comp_real_nat_nat @ F
        @ ^ [X: nat] : A )
      = ( ^ [X: nat] : ( F @ A ) ) ) ).

% compose_const(1)
thf(fact_302_prod_Ocase__distrib,axiom,
    ! [H: a > a,F: produc7194034526493574155_nat_b > a > a,Prod: produc6209191202727158440at_b_a] :
      ( ( H @ ( produc2543982510095308599_b_a_a @ F @ Prod ) )
      = ( produc2543982510095308599_b_a_a
        @ ^ [X12: produc7194034526493574155_nat_b,X22: a] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_303_prod_Ocase__distrib,axiom,
    ! [H: ( a > a ) > a > a,F: real > b > a > a,Prod: product_prod_real_b] :
      ( ( H @ ( produc7827042608974387084_b_a_a @ F @ Prod ) )
      = ( produc7827042608974387084_b_a_a
        @ ^ [X12: real,X22: b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_304_prod_Ocase__distrib,axiom,
    ! [H: a > a,F: b > produc6209191202727158440at_b_a > a,Prod: produc4883615123598087894at_b_a] :
      ( ( H @ ( produc6302854954412089281_b_a_a @ F @ Prod ) )
      = ( produc6302854954412089281_b_a_a
        @ ^ [X12: b,X22: produc6209191202727158440at_b_a] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_305_prod_Ocase__distrib,axiom,
    ! [H: a > a,F: product_prod_real_b > produc7194034526493574155_nat_b > a,Prod: produc3260441645384719156_nat_b] :
      ( ( H @ ( produc634687561522189117at_b_a @ F @ Prod ) )
      = ( produc634687561522189117at_b_a
        @ ^ [X12: product_prod_real_b,X22: produc7194034526493574155_nat_b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_306_prod_Ocase__distrib,axiom,
    ! [H: ( a > a ) > a,F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,Prod: produc3260441645384719156_nat_b] :
      ( ( H @ ( produc8421786640700244582_b_a_a @ F @ Prod ) )
      = ( produc634687561522189117at_b_a
        @ ^ [X12: product_prod_real_b,X22: produc7194034526493574155_nat_b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_307_prod_Ocase__distrib,axiom,
    ! [H: a > a > a,F: product_prod_real_b > produc7194034526493574155_nat_b > a,Prod: produc3260441645384719156_nat_b] :
      ( ( H @ ( produc634687561522189117at_b_a @ F @ Prod ) )
      = ( produc8421786640700244582_b_a_a
        @ ^ [X12: product_prod_real_b,X22: produc7194034526493574155_nat_b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_308_prod_Ocase__distrib,axiom,
    ! [H: ( a > a ) > a > a,F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,Prod: produc3260441645384719156_nat_b] :
      ( ( H @ ( produc8421786640700244582_b_a_a @ F @ Prod ) )
      = ( produc8421786640700244582_b_a_a
        @ ^ [X12: product_prod_real_b,X22: produc7194034526493574155_nat_b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_309_prod_Ocase__distrib,axiom,
    ! [H: ( produc7194034526493574155_nat_b > a > a ) > a > a,F: real > b > produc7194034526493574155_nat_b > a > a,Prod: product_prod_real_b] :
      ( ( H @ ( produc7016990068875982646_b_a_a @ F @ Prod ) )
      = ( produc7827042608974387084_b_a_a
        @ ^ [X12: real,X22: b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_310_prod_Ocase__distrib,axiom,
    ! [H: ( a > a ) > produc7194034526493574155_nat_b > a > a,F: real > b > a > a,Prod: product_prod_real_b] :
      ( ( H @ ( produc7827042608974387084_b_a_a @ F @ Prod ) )
      = ( produc7016990068875982646_b_a_a
        @ ^ [X12: real,X22: b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_311_prod_Ocase__distrib,axiom,
    ! [H: ( produc7194034526493574155_nat_b > a > a ) > produc7194034526493574155_nat_b > a > a,F: real > b > produc7194034526493574155_nat_b > a > a,Prod: product_prod_real_b] :
      ( ( H @ ( produc7016990068875982646_b_a_a @ F @ Prod ) )
      = ( produc7016990068875982646_b_a_a
        @ ^ [X12: real,X22: b] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_312_qbs__morphism__ident_H,axiom,
    ! [X4: quasi_borel_o] :
      ( member_o_o
      @ ^ [X: $o] : X
      @ ( qbs_morphism_o_o @ X4 @ X4 ) ) ).

% qbs_morphism_ident'
thf(fact_313_old_Oprod_Ocase,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,X1: product_prod_real_b,X2: produc7194034526493574155_nat_b] :
      ( ( produc8421786640700244582_b_a_a @ F @ ( produc1946830598933159916_nat_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_314_old_Oprod_Ocase,axiom,
    ! [F: produc7194034526493574155_nat_b > a > a,X1: produc7194034526493574155_nat_b,X2: a] :
      ( ( produc2543982510095308599_b_a_a @ F @ ( produc8509509857186669914at_b_a @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_315_old_Oprod_Ocase,axiom,
    ! [F: real > b > produc7194034526493574155_nat_b > a > a,X1: real,X2: b] :
      ( ( produc7016990068875982646_b_a_a @ F @ ( product_Pair_real_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_316_old_Oprod_Ocase,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,X1: b > produc7194034526493574155_nat_b > a > a,X2: produc4883615123598087894at_b_a] :
      ( ( produc286368409806729619_b_a_a @ F @ ( produc653446490150804990at_b_a @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_317_old_Oprod_Ocase,axiom,
    ! [F: b > produc6209191202727158440at_b_a > a,X1: b,X2: produc6209191202727158440at_b_a] :
      ( ( produc6302854954412089281_b_a_a @ F @ ( produc992390952543482448at_b_a @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_318_old_Oprod_Ocase,axiom,
    ! [F: real > b > a > a,X1: real,X2: b] :
      ( ( produc7827042608974387084_b_a_a @ F @ ( product_Pair_real_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_319_old_Oprod_Ocase,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a,X1: product_prod_real_b,X2: produc7194034526493574155_nat_b] :
      ( ( produc634687561522189117at_b_a @ F @ ( produc1946830598933159916_nat_b @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_320_prod__induct4,axiom,
    ! [P: produc778026099599033348at_b_a > $o,X5: produc778026099599033348at_b_a] :
      ( ! [A3: b > produc7194034526493574155_nat_b > a > a,B3: b,C2: produc7194034526493574155_nat_b,D2: a] : ( P @ ( produc653446490150804990at_b_a @ A3 @ ( produc992390952543482448at_b_a @ B3 @ ( produc8509509857186669914at_b_a @ C2 @ D2 ) ) ) )
     => ( P @ X5 ) ) ).

% prod_induct4
thf(fact_321_prod__induct3,axiom,
    ! [P: produc778026099599033348at_b_a > $o,X5: produc778026099599033348at_b_a] :
      ( ! [A3: b > produc7194034526493574155_nat_b > a > a,B3: b,C2: produc6209191202727158440at_b_a] : ( P @ ( produc653446490150804990at_b_a @ A3 @ ( produc992390952543482448at_b_a @ B3 @ C2 ) ) )
     => ( P @ X5 ) ) ).

% prod_induct3
thf(fact_322_prod__induct3,axiom,
    ! [P: produc4883615123598087894at_b_a > $o,X5: produc4883615123598087894at_b_a] :
      ( ! [A3: b,B3: produc7194034526493574155_nat_b,C2: a] : ( P @ ( produc992390952543482448at_b_a @ A3 @ ( produc8509509857186669914at_b_a @ B3 @ C2 ) ) )
     => ( P @ X5 ) ) ).

% prod_induct3
thf(fact_323_prod__induct3,axiom,
    ! [P: produc3260441645384719156_nat_b > $o,X5: produc3260441645384719156_nat_b] :
      ( ! [A3: product_prod_real_b,B3: nat,C2: nat > b] : ( P @ ( produc1946830598933159916_nat_b @ A3 @ ( produc5292568363641424317_nat_b @ B3 @ C2 ) ) )
     => ( P @ X5 ) ) ).

% prod_induct3
thf(fact_324_prod__cases4,axiom,
    ! [Y5: produc778026099599033348at_b_a] :
      ~ ! [A3: b > produc7194034526493574155_nat_b > a > a,B3: b,C2: produc7194034526493574155_nat_b,D2: a] :
          ( Y5
         != ( produc653446490150804990at_b_a @ A3 @ ( produc992390952543482448at_b_a @ B3 @ ( produc8509509857186669914at_b_a @ C2 @ D2 ) ) ) ) ).

% prod_cases4
thf(fact_325_prod__cases3,axiom,
    ! [Y5: produc778026099599033348at_b_a] :
      ~ ! [A3: b > produc7194034526493574155_nat_b > a > a,B3: b,C2: produc6209191202727158440at_b_a] :
          ( Y5
         != ( produc653446490150804990at_b_a @ A3 @ ( produc992390952543482448at_b_a @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_326_prod__cases3,axiom,
    ! [Y5: produc4883615123598087894at_b_a] :
      ~ ! [A3: b,B3: produc7194034526493574155_nat_b,C2: a] :
          ( Y5
         != ( produc992390952543482448at_b_a @ A3 @ ( produc8509509857186669914at_b_a @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_327_prod__cases3,axiom,
    ! [Y5: produc3260441645384719156_nat_b] :
      ~ ! [A3: product_prod_real_b,B3: nat,C2: nat > b] :
          ( Y5
         != ( produc1946830598933159916_nat_b @ A3 @ ( produc5292568363641424317_nat_b @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_328_calculation,axiom,
    ( ( comp_P6318193336758794883a_real
      @ ( produc286368409806729619_b_a_a
        @ ^ [Y: b > produc7194034526493574155_nat_b > a > a] :
            ( produc6302854954412089281_b_a_a
            @ ^ [X12: b] : ( produc2543982510095308599_b_a_a @ ( Y @ X12 ) ) ) )
      @ beta )
    = ( comp_P9131689941513729756a_real @ ( produc5399252211680487944_b_a_a @ ( produc8421786640700244582_b_a_a @ ( produc7016990068875982646_b_a_a @ beta_1 ) ) )
      @ ^ [R: real] : ( produc8437504494124426337at_b_a @ ( produc1946830598933159916_nat_b @ ( product_Pair_real_b @ R @ ( beta_2 @ R ) ) @ ( beta_3 @ R ) ) @ ( beta_4 @ R ) ) ) ) ).

% calculation
thf(fact_329__092_060open_062_092_060exists_062_092_060beta_0621_A_092_060beta_0622_A_092_060beta_0623_A_092_060beta_0624_O_A_092_060beta_062_A_061_A_I_092_060lambda_062r_O_A_I_092_060beta_0621_Ar_M_A_092_060beta_0622_Ar_M_A_092_060beta_0623_Ar_M_A_092_060beta_0624_Ar_J_J_A_092_060and_062_A_092_060beta_0621_A_092_060in_062_Aqbs__Mx_A_IX_A_092_060Rightarrow_062_092_060_094sub_062Q_Alist__of_AX_A_092_060Rightarrow_062_092_060_094sub_062Q_AY_A_092_060Rightarrow_062_092_060_094sub_062Q_AY_J_A_092_060and_062_A_092_060beta_0622_A_092_060in_062_Aqbs__Mx_AX_A_092_060and_062_A_092_060beta_0623_A_092_060in_062_Aqbs__Mx_A_Ilist__of_AX_J_A_092_060and_062_A_092_060beta_0624_A_092_060in_062_Aqbs__Mx_AY_092_060close_062,axiom,
    ? [Beta_1: real > b > produc7194034526493574155_nat_b > a > a,Beta_2: real > b,Beta_3: real > produc7194034526493574155_nat_b,Beta_4: real > a] :
      ( ( beta
        = ( ^ [R: real] : ( produc653446490150804990at_b_a @ ( Beta_1 @ R ) @ ( produc992390952543482448at_b_a @ ( Beta_2 @ R ) @ ( produc8509509857186669914at_b_a @ ( Beta_3 @ R ) @ ( Beta_4 @ R ) ) ) ) ) )
      & ( member924168396124881715_b_a_a @ Beta_1
        @ ( qbs_Mx703024977670096246_b_a_a
          @ ( expone5294601226626914677_b_a_a @ x
            @ ( expone5933728560452245707_b_a_a
              @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
                @ ^ [N: nat] :
                    ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                    @ ^ [I: nat] : x ) )
              @ ( exponent_exp_qbs_a_a @ y @ y ) ) ) ) )
      & ( member_real_b @ Beta_2 @ ( qbs_Mx_b @ x ) )
      & ( member8664213473904227437_nat_b @ Beta_3
        @ ( qbs_Mx1531206082039394900_nat_b
          @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
            @ ^ [N: nat] :
                ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                @ ^ [I: nat] : x ) ) ) )
      & ( member_real_a @ Beta_4 @ ( qbs_Mx_a @ y ) ) ) ).

% \<open>\<exists>\<beta>1 \<beta>2 \<beta>3 \<beta>4. \<beta> = (\<lambda>r. (\<beta>1 r, \<beta>2 r, \<beta>3 r, \<beta>4 r)) \<and> \<beta>1 \<in> qbs_Mx (X \<Rightarrow>\<^sub>Q list_of X \<Rightarrow>\<^sub>Q Y \<Rightarrow>\<^sub>Q Y) \<and> \<beta>2 \<in> qbs_Mx X \<and> \<beta>3 \<in> qbs_Mx (list_of X) \<and> \<beta>4 \<in> qbs_Mx Y\<close>
thf(fact_330__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060beta_0621_A_092_060beta_0622_A_092_060beta_0623_A_092_060beta_0624_O_A_092_060lbrakk_062_092_060beta_062_A_061_A_I_092_060lambda_062r_O_A_I_092_060beta_0621_Ar_M_A_092_060beta_0622_Ar_M_A_092_060beta_0623_Ar_M_A_092_060beta_0624_Ar_J_J_059_A_092_060beta_0621_A_092_060in_062_Aqbs__Mx_A_IX_A_092_060Rightarrow_062_092_060_094sub_062Q_Alist__of_AX_A_092_060Rightarrow_062_092_060_094sub_062Q_AY_A_092_060Rightarrow_062_092_060_094sub_062Q_AY_J_059_A_092_060beta_0622_A_092_060in_062_Aqbs__Mx_AX_059_A_092_060beta_0623_A_092_060in_062_Aqbs__Mx_A_Ilist__of_AX_J_059_A_092_060beta_0624_A_092_060in_062_Aqbs__Mx_AY_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Beta_1: real > b > produc7194034526493574155_nat_b > a > a,Beta_2: real > b,Beta_3: real > produc7194034526493574155_nat_b,Beta_4: real > a] :
        ( ( beta
          = ( ^ [R: real] : ( produc653446490150804990at_b_a @ ( Beta_1 @ R ) @ ( produc992390952543482448at_b_a @ ( Beta_2 @ R ) @ ( produc8509509857186669914at_b_a @ ( Beta_3 @ R ) @ ( Beta_4 @ R ) ) ) ) ) )
       => ( ( member924168396124881715_b_a_a @ Beta_1
            @ ( qbs_Mx703024977670096246_b_a_a
              @ ( expone5294601226626914677_b_a_a @ x
                @ ( expone5933728560452245707_b_a_a
                  @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
                    @ ^ [N: nat] :
                        ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                        @ ^ [I: nat] : x ) )
                  @ ( exponent_exp_qbs_a_a @ y @ y ) ) ) ) )
         => ( ( member_real_b @ Beta_2 @ ( qbs_Mx_b @ x ) )
           => ( ( member8664213473904227437_nat_b @ Beta_3
                @ ( qbs_Mx1531206082039394900_nat_b
                  @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
                    @ ^ [N: nat] :
                        ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                        @ ^ [I: nat] : x ) ) ) )
             => ~ ( member_real_a @ Beta_4 @ ( qbs_Mx_a @ y ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>\<beta>1 \<beta>2 \<beta>3 \<beta>4. \<lbrakk>\<beta> = (\<lambda>r. (\<beta>1 r, \<beta>2 r, \<beta>3 r, \<beta>4 r)); \<beta>1 \<in> qbs_Mx (X \<Rightarrow>\<^sub>Q list_of X \<Rightarrow>\<^sub>Q Y \<Rightarrow>\<^sub>Q Y); \<beta>2 \<in> qbs_Mx X; \<beta>3 \<in> qbs_Mx (list_of X); \<beta>4 \<in> qbs_Mx Y\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_331_ha,axiom,
    member_real_b @ a2 @ ( qbs_Mx_b @ x ) ).

% ha
thf(fact_332_qbs__Mx__is__morphisms,axiom,
    ( qbs_Mx5919617464495388621at_b_a
    = ( qbs_mo225298393485383594at_b_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ).

% qbs_Mx_is_morphisms
thf(fact_333_qbs__Mx__is__morphisms,axiom,
    ( qbs_Mx_nat_b
    = ( qbs_mo3905299618101392192_nat_b @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ).

% qbs_Mx_is_morphisms
thf(fact_334_qbs__Mx__is__morphisms,axiom,
    ( qbs_Mx703024977670096246_b_a_a
    = ( qbs_mo1140304683536899929_b_a_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ).

% qbs_Mx_is_morphisms
thf(fact_335_qbs__Mx__is__morphisms,axiom,
    ( qbs_Mx_b
    = ( qbs_morphism_real_b @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ).

% qbs_Mx_is_morphisms
thf(fact_336_qbs__Mx__is__morphisms,axiom,
    ( qbs_Mx_a
    = ( qbs_morphism_real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) ).

% qbs_Mx_is_morphisms
thf(fact_337_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_a,F: b > a > a,Z2: quasi_borel_a] :
      ( ! [Alpha: real > b,Beta: real > product_prod_real_a] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X4 ) )
         => ( ( member1006887528422719913real_a @ Beta @ ( binary2731563775470309560real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_a @ ( comp_P7874914735527052288a_real @ ( produc5622917954539892056al_a_a @ ( comp_b_a_a_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_a @ Z2 ) ) ) )
     => ( member_b_a_a @ F @ ( qbs_morphism_b_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_338_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_a,F: a > a > a,Z2: quasi_borel_a] :
      ( ! [Alpha: real > a,Beta: real > product_prod_real_a] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X4 ) )
         => ( ( member1006887528422719913real_a @ Beta @ ( binary2731563775470309560real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_a @ ( comp_P7874914735527052288a_real @ ( produc5622917954539892056al_a_a @ ( comp_a_a_a_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_a @ Z2 ) ) ) )
     => ( member_a_a_a @ F @ ( qbs_morphism_a_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_339_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_o,F: b > $o > $o,Z2: quasi_borel_o] :
      ( ! [Alpha: real > b,Beta: real > product_prod_real_o] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X4 ) )
         => ( ( member6586820919158520515real_o @ Beta @ ( binary6447122689459668562real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_o @ ( comp_P4035609806083913856o_real @ ( produc528293406795641944al_o_o @ ( comp_b_o_o_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_o @ Z2 ) ) ) )
     => ( member_b_o_o @ F @ ( qbs_morphism_b_o_o @ X4 @ ( exponent_exp_qbs_o_o @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_340_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_o,F: a > $o > $o,Z2: quasi_borel_o] :
      ( ! [Alpha: real > a,Beta: real > product_prod_real_o] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X4 ) )
         => ( ( member6586820919158520515real_o @ Beta @ ( binary6447122689459668562real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_o @ ( comp_P4035609806083913856o_real @ ( produc528293406795641944al_o_o @ ( comp_a_o_o_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_o @ Z2 ) ) ) )
     => ( member_a_o_o @ F @ ( qbs_morphism_a_o_o @ X4 @ ( exponent_exp_qbs_o_o @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_341_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_o,Y4: quasi_borel_o,F: $o > $o > $o,Z2: quasi_borel_o] :
      ( ! [Alpha: real > $o,Beta: real > product_prod_real_o] :
          ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X4 ) )
         => ( ( member6586820919158520515real_o @ Beta @ ( binary6447122689459668562real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_o @ ( comp_P4035609806083913856o_real @ ( produc528293406795641944al_o_o @ ( comp_o_o_o_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_o @ Z2 ) ) ) )
     => ( member_o_o_o @ F @ ( qbs_morphism_o_o_o @ X4 @ ( exponent_exp_qbs_o_o @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_342_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_nat_b,Y4: quasi_borel_a,F: ( nat > b ) > a > a,Z2: quasi_borel_a] :
      ( ! [Alpha: real > nat > b,Beta: real > product_prod_real_a] :
          ( ( member_real_nat_b @ Alpha @ ( qbs_Mx_nat_b @ X4 ) )
         => ( ( member1006887528422719913real_a @ Beta @ ( binary2731563775470309560real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_a @ ( comp_P7874914735527052288a_real @ ( produc5622917954539892056al_a_a @ ( comp_nat_b_a_a_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_a @ Z2 ) ) ) )
     => ( member_nat_b_a_a @ F @ ( qbs_mo9010864224564201555_b_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_343_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_nat_b,Y4: quasi_borel_o,F: ( nat > b ) > $o > $o,Z2: quasi_borel_o] :
      ( ! [Alpha: real > nat > b,Beta: real > product_prod_real_o] :
          ( ( member_real_nat_b @ Alpha @ ( qbs_Mx_nat_b @ X4 ) )
         => ( ( member6586820919158520515real_o @ Beta @ ( binary6447122689459668562real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_o @ ( comp_P4035609806083913856o_real @ ( produc528293406795641944al_o_o @ ( comp_nat_b_o_o_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_o @ Z2 ) ) ) )
     => ( member_nat_b_o_o @ F @ ( qbs_mo3001173925234491475_b_o_o @ X4 @ ( exponent_exp_qbs_o_o @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_344_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_b,F: b > b > a > a,Z2: quasi_borel_a_a] :
      ( ! [Alpha: real > b,Beta: real > product_prod_real_b] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X4 ) )
         => ( ( member1077921568469065898real_b @ Beta @ ( binary2731563775470309561real_b @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_a_a @ ( comp_P2278155893825476926a_real @ ( produc7827042608974387084_b_a_a @ ( comp_b_b_a_a_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_a_a @ Z2 ) ) ) )
     => ( member_b_b_a_a @ F @ ( qbs_morphism_b_b_a_a @ X4 @ ( expone5499329870516987339_b_a_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_345_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_b,F: a > b > a > a,Z2: quasi_borel_a_a] :
      ( ! [Alpha: real > a,Beta: real > product_prod_real_b] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X4 ) )
         => ( ( member1077921568469065898real_b @ Beta @ ( binary2731563775470309561real_b @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_a_a @ ( comp_P2278155893825476926a_real @ ( produc7827042608974387084_b_a_a @ ( comp_a_b_a_a_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_a_a @ Z2 ) ) ) )
     => ( member_a_b_a_a @ F @ ( qbs_morphism_a_b_a_a @ X4 @ ( expone5499329870516987339_b_a_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_346_qbs__exp__morphismI,axiom,
    ! [X4: quasi_borel_nat_b,Y4: quasi_borel_b,F: ( nat > b ) > b > a > a,Z2: quasi_borel_a_a] :
      ( ! [Alpha: real > nat > b,Beta: real > product_prod_real_b] :
          ( ( member_real_nat_b @ Alpha @ ( qbs_Mx_nat_b @ X4 ) )
         => ( ( member1077921568469065898real_b @ Beta @ ( binary2731563775470309561real_b @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ Y4 ) )
           => ( member_real_a_a @ ( comp_P2278155893825476926a_real @ ( produc7827042608974387084_b_a_a @ ( comp_n7768269484769491838a_real @ F @ Alpha ) ) @ Beta ) @ ( qbs_Mx_a_a @ Z2 ) ) ) )
     => ( member_nat_b_b_a_a @ F @ ( qbs_mo4269413408830428775_b_a_a @ X4 @ ( expone5499329870516987339_b_a_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_exp_morphismI
thf(fact_347_hbq,axiom,
    ( member9091339440257553353_b_a_a @ ( produc5399252211680487944_b_a_a @ ( produc8421786640700244582_b_a_a @ ( produc7016990068875982646_b_a_a @ beta_1 ) ) )
    @ ( qbs_mo7901260496052035013_b_a_a
      @ ( binary4006988140340982583at_b_a
        @ ( binary7297433035349996738_nat_b @ ( binary5762279046363823268real_b @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ x )
          @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
            @ ^ [N: nat] :
                ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                @ ^ [I: nat] : x ) ) )
        @ y )
      @ y ) ) ).

% hbq
thf(fact_348_lessThan__eq__iff,axiom,
    ! [X5: nat,Y5: nat] :
      ( ( ( set_ord_lessThan_nat @ X5 )
        = ( set_ord_lessThan_nat @ Y5 ) )
      = ( X5 = Y5 ) ) ).

% lessThan_eq_iff
thf(fact_349_ini__morphism,axiom,
    ! [J: real,I2: set_real,X4: real > quasi_borel_b] :
      ( ( member_real @ J @ I2 )
     => ( member5108440089302923873real_b @ ( product_Pair_real_b @ J ) @ ( qbs_mo3164329421503706703real_b @ ( X4 @ J ) @ ( coProd2144328941176013730real_b @ I2 @ X4 ) ) ) ) ).

% ini_morphism
thf(fact_350_ini__morphism,axiom,
    ! [J: nat,I2: set_nat,X4: nat > quasi_borel_nat] :
      ( ( member_nat @ J @ I2 )
     => ( member6743353555990375057at_nat @ ( product_Pair_nat_nat @ J ) @ ( qbs_mo3651027331794440503at_nat @ ( X4 @ J ) @ ( coProd2455237193333462417at_nat @ I2 @ X4 ) ) ) ) ).

% ini_morphism
thf(fact_351_ini__morphism,axiom,
    ! [J: b > produc7194034526493574155_nat_b > a > a,I2: set_b_5076249247947527647_b_a_a,X4: ( b > produc7194034526493574155_nat_b > a > a ) > quasi_3683678768404249670at_b_a] :
      ( ( member8306749417536871976_b_a_a @ J @ I2 )
     => ( member9142974548522789008at_b_a @ ( produc653446490150804990at_b_a @ J ) @ ( qbs_mo4893338430385348052at_b_a @ ( X4 @ J ) @ ( coProd5845954756983474342at_b_a @ I2 @ X4 ) ) ) ) ).

% ini_morphism
thf(fact_352_ini__morphism,axiom,
    ! [J: b,I2: set_b,X4: b > quasi_1826430223877327896at_b_a] :
      ( ( member_b @ J @ I2 )
     => ( member2688202504559472564at_b_a @ ( produc992390952543482448at_b_a @ J ) @ ( qbs_mo6862686886989468792at_b_a @ ( X4 @ J ) @ ( coProd8414286804247253240at_b_a @ I2 @ X4 ) ) ) ) ).

% ini_morphism
thf(fact_353_ini__morphism,axiom,
    ! [J: produc7194034526493574155_nat_b,I2: set_Pr2992590090106802881_nat_b,X4: produc7194034526493574155_nat_b > quasi_borel_a] :
      ( ( member2882485241872049954_nat_b @ J @ I2 )
     => ( member4213348123347683520at_b_a @ ( produc8509509857186669914at_b_a @ J ) @ ( qbs_mo388721149857207982at_b_a @ ( X4 @ J ) @ ( coProd7166816528838454274at_b_a @ I2 @ X4 ) ) ) ) ).

% ini_morphism
thf(fact_354_ini__morphism,axiom,
    ! [J: product_prod_real_b,I2: set_Pr6660261784198275838real_b,X4: product_prod_real_b > quasi_7334096685896923387_nat_b] :
      ( ( member7555082154638246367real_b @ J @ I2 )
     => ( member3516147537297476681_nat_b @ ( produc1946830598933159916_nat_b @ J ) @ ( qbs_mo3539708487391302639_nat_b @ ( X4 @ J ) @ ( coProd362979566082915140_nat_b @ I2 @ X4 ) ) ) ) ).

% ini_morphism
thf(fact_355_ini__morphism,axiom,
    ! [J: nat,I2: set_nat,X4: nat > quasi_borel_nat_b] :
      ( ( member_nat @ J @ I2 )
     => ( member3100617602161326603_nat_b @ ( produc5292568363641424317_nat_b @ J ) @ ( qbs_mo3787342477198169849_nat_b @ ( X4 @ J ) @ ( coProd3505932218054916197_nat_b @ I2 @ X4 ) ) ) ) ).

% ini_morphism
thf(fact_356_UNIV__I,axiom,
    ! [X5: $o > real] : ( member_o_real @ X5 @ top_top_set_o_real ) ).

% UNIV_I
thf(fact_357_UNIV__I,axiom,
    ! [X5: $o > $o] : ( member_o_o @ X5 @ top_top_set_o_o ) ).

% UNIV_I
thf(fact_358_UNIV__I,axiom,
    ! [X5: $o > $o > $o] : ( member_o_o_o @ X5 @ top_top_set_o_o_o ) ).

% UNIV_I
thf(fact_359_UNIV__I,axiom,
    ! [X5: nat > real] : ( member_nat_real @ X5 @ top_top_set_nat_real ) ).

% UNIV_I
thf(fact_360_UNIV__I,axiom,
    ! [X5: ( nat > b ) > produc432120507726958376_b_a_a > a] : ( member1811236362415105851_a_a_a @ X5 @ top_to5015968664104940226_a_a_a ) ).

% UNIV_I
thf(fact_361_UNIV__I,axiom,
    ! [X5: produc7716430852924023517t_real] : ( member557208447399453958t_real @ X5 @ top_to4855536200657754381t_real ) ).

% UNIV_I
thf(fact_362_UNIV__I,axiom,
    ! [X5: complex] : ( member_complex @ X5 @ top_top_set_complex ) ).

% UNIV_I
thf(fact_363_UNIV__I,axiom,
    ! [X5: real] : ( member_real @ X5 @ top_top_set_real ) ).

% UNIV_I
thf(fact_364_UNIV__I,axiom,
    ! [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ).

% UNIV_I
thf(fact_365_UNIV__I,axiom,
    ! [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_366_top__set__def,axiom,
    ( top_to4855536200657754381t_real
    = ( collec413939898802455368t_real @ top_to7655735016164553232real_o ) ) ).

% top_set_def
thf(fact_367_top__set__def,axiom,
    ( top_top_set_complex
    = ( collect_complex @ top_top_complex_o ) ) ).

% top_set_def
thf(fact_368_top__set__def,axiom,
    ( top_top_set_real
    = ( collect_real @ top_top_real_o ) ) ).

% top_set_def
thf(fact_369_top__set__def,axiom,
    ( top_top_set_o
    = ( collect_o @ top_top_o_o ) ) ).

% top_set_def
thf(fact_370_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_371_uncurry__preserves__morphisms,axiom,
    ! [F: produc3260441645384719156_nat_b > a > a,X4: quasi_8132950864781310810_nat_b,Y4: quasi_borel_a,Z2: quasi_borel_a] :
      ( ( member7748159150162262195_b_a_a @ F @ ( qbs_mo6578283766602973217_b_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) )
     => ( member9091339440257553353_b_a_a @ ( produc5399252211680487944_b_a_a @ F ) @ ( qbs_mo7901260496052035013_b_a_a @ ( binary4006988140340982583at_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_372_uncurry__preserves__morphisms,axiom,
    ! [F: ( nat > b ) > produc432120507726958376_b_a_a > a,X4: quasi_borel_nat_b,Y4: quasi_6353345141195568462_b_a_a,Z2: quasi_borel_a] :
      ( ( member1811236362415105851_a_a_a @ F @ ( qbs_mo8618518167108125737_a_a_a @ X4 @ ( expone790792417327382575_a_a_a @ Y4 @ Z2 ) ) )
     => ( member1981363422186161233_a_a_a @ ( produc4979302019304372496_a_a_a @ F ) @ ( qbs_mo7915889915729830349_a_a_a @ ( binary1958873873046607663_b_a_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_373_uncurry__preserves__morphisms,axiom,
    ! [F: $o > $o > $o,X4: quasi_borel_o,Y4: quasi_borel_o,Z2: quasi_borel_o] :
      ( ( member_o_o_o @ F @ ( qbs_morphism_o_o_o @ X4 @ ( exponent_exp_qbs_o_o @ Y4 @ Z2 ) ) )
     => ( member5279437561400393645_o_o_o @ ( produc6197397395684419436_o_o_o @ F ) @ ( qbs_mo6404984969470912169_o_o_o @ ( binary9159730317414444269bs_o_o @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_374_uncurry__preserves__morphisms,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,X4: quasi_7777979582039567160real_b,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a_a] :
      ( ( member1306660226910552845_b_a_a @ F @ ( qbs_mo3736047266677648337_b_a_a @ X4 @ ( expone5933728560452245707_b_a_a @ Y4 @ Z2 ) ) )
     => ( member7748159150162262195_b_a_a @ ( produc8421786640700244582_b_a_a @ F ) @ ( qbs_mo6578283766602973217_b_a_a @ ( binary7297433035349996738_nat_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_375_uncurry__preserves__morphisms,axiom,
    ! [F: produc7194034526493574155_nat_b > a > a,X4: quasi_7334096685896923387_nat_b,Y4: quasi_borel_a,Z2: quasi_borel_a] :
      ( ( member970290822420441254_b_a_a @ F @ ( qbs_mo7242095808399995242_b_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) )
     => ( member1447347221985152182_b_a_a @ ( produc2543982510095308599_b_a_a @ F ) @ ( qbs_mo1215615477044746460_b_a_a @ ( binary4113689719790979332at_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_376_uncurry__preserves__morphisms,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a,X4: quasi_7777979582039567160real_b,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a] :
      ( ( member7050793433021125506at_b_a @ F @ ( qbs_mo3341213321107793832at_b_a @ X4 @ ( expone8918854433262535192at_b_a @ Y4 @ Z2 ) ) )
     => ( member7813222213774494534at_b_a @ ( produc634687561522189117at_b_a @ F ) @ ( qbs_mo8230009346921551938at_b_a @ ( binary7297433035349996738_nat_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_377_uncurry__preserves__morphisms,axiom,
    ! [F: real > b > produc7194034526493574155_nat_b > a > a,X4: quasi_borel_real,Y4: quasi_borel_b,Z2: quasi_7934145443646537599_b_a_a] :
      ( ( member924168396124881715_b_a_a @ F @ ( qbs_mo1140304683536899929_b_a_a @ X4 @ ( expone5294601226626914677_b_a_a @ Y4 @ Z2 ) ) )
     => ( member1306660226910552845_b_a_a @ ( produc7016990068875982646_b_a_a @ F ) @ ( qbs_mo3736047266677648337_b_a_a @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_378_uncurry__preserves__morphisms,axiom,
    ! [F: real > b > a > a,X4: quasi_borel_real,Y4: quasi_borel_b,Z2: quasi_borel_a_a] :
      ( ( member_real_b_a_a @ F @ ( qbs_mo6882034808174728367_b_a_a @ X4 @ ( expone5499329870516987339_b_a_a @ Y4 @ Z2 ) ) )
     => ( member9111837334973778019_b_a_a @ ( produc7827042608974387084_b_a_a @ F ) @ ( qbs_mo3896594311416080423_b_a_a @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_379_uncurry__preserves__morphisms,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,X4: quasi_3364728345918372901_b_a_a,Y4: quasi_3683678768404249670at_b_a,Z2: quasi_borel_a] :
      ( ( member4317787861150391324_b_a_a @ F @ ( qbs_mo5202423023110816408_b_a_a @ X4 @ ( expone6319500819088444429_b_a_a @ Y4 @ Z2 ) ) )
     => ( member763877586905102170_b_a_a @ ( produc286368409806729619_b_a_a @ F ) @ ( qbs_mo1240948812564023296_b_a_a @ ( binary8035175780229655208at_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_380_uncurry__preserves__morphisms,axiom,
    ! [F: b > produc6209191202727158440at_b_a > a,X4: quasi_borel_b,Y4: quasi_1826430223877327896at_b_a,Z2: quasi_borel_a] :
      ( ( member5328788390466478410_b_a_a @ F @ ( qbs_mo3491766053686598726_b_a_a @ X4 @ ( expone405746273668894395_b_a_a @ Y4 @ Z2 ) ) )
     => ( member871964287106612744_b_a_a @ ( produc6302854954412089281_b_a_a @ F ) @ ( qbs_mo1164823027738856750_b_a_a @ ( binary3850590118635936762at_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_preserves_morphisms
thf(fact_381_UNIV__witness,axiom,
    ? [X3: $o > real] : ( member_o_real @ X3 @ top_top_set_o_real ) ).

% UNIV_witness
thf(fact_382_UNIV__witness,axiom,
    ? [X3: $o > $o] : ( member_o_o @ X3 @ top_top_set_o_o ) ).

% UNIV_witness
thf(fact_383_UNIV__witness,axiom,
    ? [X3: $o > $o > $o] : ( member_o_o_o @ X3 @ top_top_set_o_o_o ) ).

% UNIV_witness
thf(fact_384_UNIV__witness,axiom,
    ? [X3: nat > real] : ( member_nat_real @ X3 @ top_top_set_nat_real ) ).

% UNIV_witness
thf(fact_385_UNIV__witness,axiom,
    ? [X3: ( nat > b ) > produc432120507726958376_b_a_a > a] : ( member1811236362415105851_a_a_a @ X3 @ top_to5015968664104940226_a_a_a ) ).

% UNIV_witness
thf(fact_386_UNIV__witness,axiom,
    ? [X3: produc7716430852924023517t_real] : ( member557208447399453958t_real @ X3 @ top_to4855536200657754381t_real ) ).

% UNIV_witness
thf(fact_387_UNIV__witness,axiom,
    ? [X3: complex] : ( member_complex @ X3 @ top_top_set_complex ) ).

% UNIV_witness
thf(fact_388_UNIV__witness,axiom,
    ? [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).

% UNIV_witness
thf(fact_389_UNIV__witness,axiom,
    ? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).

% UNIV_witness
thf(fact_390_UNIV__witness,axiom,
    ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_391_UNIV__eq__I,axiom,
    ! [A4: set_o_real] :
      ( ! [X3: $o > real] : ( member_o_real @ X3 @ A4 )
     => ( top_top_set_o_real = A4 ) ) ).

% UNIV_eq_I
thf(fact_392_UNIV__eq__I,axiom,
    ! [A4: set_o_o] :
      ( ! [X3: $o > $o] : ( member_o_o @ X3 @ A4 )
     => ( top_top_set_o_o = A4 ) ) ).

% UNIV_eq_I
thf(fact_393_UNIV__eq__I,axiom,
    ! [A4: set_o_o_o] :
      ( ! [X3: $o > $o > $o] : ( member_o_o_o @ X3 @ A4 )
     => ( top_top_set_o_o_o = A4 ) ) ).

% UNIV_eq_I
thf(fact_394_UNIV__eq__I,axiom,
    ! [A4: set_nat_real] :
      ( ! [X3: nat > real] : ( member_nat_real @ X3 @ A4 )
     => ( top_top_set_nat_real = A4 ) ) ).

% UNIV_eq_I
thf(fact_395_UNIV__eq__I,axiom,
    ! [A4: set_na8676686263003397490_a_a_a] :
      ( ! [X3: ( nat > b ) > produc432120507726958376_b_a_a > a] : ( member1811236362415105851_a_a_a @ X3 @ A4 )
     => ( top_to5015968664104940226_a_a_a = A4 ) ) ).

% UNIV_eq_I
thf(fact_396_UNIV__eq__I,axiom,
    ! [A4: set_Pr320017278500174781t_real] :
      ( ! [X3: produc7716430852924023517t_real] : ( member557208447399453958t_real @ X3 @ A4 )
     => ( top_to4855536200657754381t_real = A4 ) ) ).

% UNIV_eq_I
thf(fact_397_UNIV__eq__I,axiom,
    ! [A4: set_complex] :
      ( ! [X3: complex] : ( member_complex @ X3 @ A4 )
     => ( top_top_set_complex = A4 ) ) ).

% UNIV_eq_I
thf(fact_398_UNIV__eq__I,axiom,
    ! [A4: set_real] :
      ( ! [X3: real] : ( member_real @ X3 @ A4 )
     => ( top_top_set_real = A4 ) ) ).

% UNIV_eq_I
thf(fact_399_UNIV__eq__I,axiom,
    ! [A4: set_o] :
      ( ! [X3: $o] : ( member_o @ X3 @ A4 )
     => ( top_top_set_o = A4 ) ) ).

% UNIV_eq_I
thf(fact_400_UNIV__eq__I,axiom,
    ! [A4: set_nat] :
      ( ! [X3: nat] : ( member_nat @ X3 @ A4 )
     => ( top_top_set_nat = A4 ) ) ).

% UNIV_eq_I
thf(fact_401_UNIV__def,axiom,
    ( top_to4855536200657754381t_real
    = ( collec413939898802455368t_real
      @ ^ [X: produc7716430852924023517t_real] : $true ) ) ).

% UNIV_def
thf(fact_402_UNIV__def,axiom,
    ( top_top_set_complex
    = ( collect_complex
      @ ^ [X: complex] : $true ) ) ).

% UNIV_def
thf(fact_403_UNIV__def,axiom,
    ( top_top_set_real
    = ( collect_real
      @ ^ [X: real] : $true ) ) ).

% UNIV_def
thf(fact_404_UNIV__def,axiom,
    ( top_top_set_o
    = ( collect_o
      @ ^ [X: $o] : $true ) ) ).

% UNIV_def
thf(fact_405_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $true ) ) ).

% UNIV_def
thf(fact_406_uncurry__morphism,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_nat_b,Z2: quasi_borel_a_a] : ( member7836499454956176076_b_a_a @ produc7065089332050921039_b_a_a @ ( qbs_mo6989279613654713658_b_a_a @ ( expone7803431766090835025_b_a_a @ X4 @ ( expone932865709032713778_b_a_a @ Y4 @ Z2 ) ) @ ( expone5933728560452245707_b_a_a @ ( binary7246370547177875303_nat_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_407_uncurry__morphism,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a_a] : ( member3151113425149562966_b_a_a @ produc6838043970507933765_b_a_a @ ( qbs_mo4821322441459649476_b_a_a @ ( expone5294601226626914677_b_a_a @ X4 @ ( expone5933728560452245707_b_a_a @ Y4 @ Z2 ) ) @ ( expone8080306092094974073_b_a_a @ ( binary994859541383405405_nat_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_408_uncurry__morphism,axiom,
    ! [X4: quasi_7777979582039567160real_b,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a_a] : ( member6400930607172261620_b_a_a @ produc8421786640700244582_b_a_a @ ( qbs_mo1950602160588923290_b_a_a @ ( expone1622171987252446258_b_a_a @ X4 @ ( expone5933728560452245707_b_a_a @ Y4 @ Z2 ) ) @ ( expone3210307275161404672_b_a_a @ ( binary7297433035349996738_nat_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_409_uncurry__morphism,axiom,
    ! [X4: quasi_7334096685896923387_nat_b,Y4: quasi_borel_a,Z2: quasi_borel_a] : ( member7400051019931824670_b_a_a @ produc2543982510095308599_b_a_a @ ( qbs_mo3123651541299603524_b_a_a @ ( expone5933728560452245707_b_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) @ ( expone405746273668894395_b_a_a @ ( binary4113689719790979332at_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_410_uncurry__morphism,axiom,
    ! [X4: quasi_7777979582039567160real_b,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a] : ( member6420637408017863330at_b_a @ produc634687561522189117at_b_a @ ( qbs_mo9043266432731987472at_b_a @ ( expone3019181631264744327at_b_a @ X4 @ ( expone8918854433262535192at_b_a @ Y4 @ Z2 ) ) @ ( expone3972300415888346915at_b_a @ ( binary7297433035349996738_nat_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_411_uncurry__morphism,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b,Z2: quasi_7934145443646537599_b_a_a] : ( member1258501860276290714_b_a_a @ produc7016990068875982646_b_a_a @ ( qbs_mo4523754438958348424_b_a_a @ ( expone5458538795283732920_b_a_a @ X4 @ ( expone5294601226626914677_b_a_a @ Y4 @ Z2 ) ) @ ( expone1622171987252446258_b_a_a @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_412_uncurry__morphism,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b,Z2: quasi_borel_a_a] : ( member1114180529494385478_b_a_a @ produc7827042608974387084_b_a_a @ ( qbs_mo4709250047596395060_b_a_a @ ( expone3903273362589924110_b_a_a @ X4 @ ( expone5499329870516987339_b_a_a @ Y4 @ Z2 ) ) @ ( expone6122776153604691080_b_a_a @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_413_uncurry__morphism,axiom,
    ! [X4: quasi_3364728345918372901_b_a_a,Y4: quasi_3683678768404249670at_b_a,Z2: quasi_borel_a] : ( member8535123054158726860_b_a_a @ produc286368409806729619_b_a_a @ ( qbs_mo7857648355211746802_b_a_a @ ( expone7735065173500681337_b_a_a @ X4 @ ( expone6319500819088444429_b_a_a @ Y4 @ Z2 ) ) @ ( expone1118283818123570015_b_a_a @ ( binary8035175780229655208at_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_414_uncurry__morphism,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_1826430223877327896at_b_a,Z2: quasi_borel_a] : ( member6428233467481601228_b_a_a @ produc6302854954412089281_b_a_a @ ( qbs_mo8320231944529395186_b_a_a @ ( expone3665158019928849063_b_a_a @ X4 @ ( expone405746273668894395_b_a_a @ Y4 @ Z2 ) ) @ ( expone6319500819088444429_b_a_a @ ( binary3850590118635936762at_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_415_uncurry__morphism,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_3364728345918372901_b_a_a,Z2: quasi_borel_a] : ( member839204257469859816_a_a_a @ produc8533229574609689361_a_a_a @ ( qbs_mo60665032963471702_a_a_a @ ( expone5702679380600246409_a_a_a @ X4 @ ( expone7391014374654541464_a_a_a @ Y4 @ Z2 ) ) @ ( expone790792417327382575_a_a_a @ ( binary9021634363669778414_b_a_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% uncurry_morphism
thf(fact_416_pair__qbs__Mx,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b] :
      ( ( qbs_Mx2044252502963291921real_b @ ( binary5762279046363823268real_b @ X4 @ Y4 ) )
      = ( binary2731563775470309561real_b @ X4 @ Y4 ) ) ).

% pair_qbs_Mx
thf(fact_417_pair__qbs__Mx,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_real] :
      ( ( qbs_Mx6884344600386557780t_real @ ( binary2938201662863479403t_real @ X4 @ Y4 ) )
      = ( binary1593783336058016534t_real @ X4 @ Y4 ) ) ).

% pair_qbs_Mx
thf(fact_418_pair__qbs__Mx,axiom,
    ! [X4: quasi_3364728345918372901_b_a_a,Y4: quasi_3683678768404249670at_b_a] :
      ( ( qbs_Mx5919617464495388621at_b_a @ ( binary8035175780229655208at_b_a @ X4 @ Y4 ) )
      = ( binary1056969044873907517at_b_a @ X4 @ Y4 ) ) ).

% pair_qbs_Mx
thf(fact_419_pair__qbs__Mx,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_1826430223877327896at_b_a] :
      ( ( qbs_Mx5318780673203594015at_b_a @ ( binary3850590118635936762at_b_a @ X4 @ Y4 ) )
      = ( binary4303792690612612367at_b_a @ X4 @ Y4 ) ) ).

% pair_qbs_Mx
thf(fact_420_pair__qbs__Mx,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_3364728345918372901_b_a_a] :
      ( ( qbs_Mx450739389344589599_b_a_a @ ( binary9021634363669778414_b_a_a @ X4 @ Y4 ) )
      = ( binary3316000448429436441_b_a_a @ X4 @ Y4 ) ) ).

% pair_qbs_Mx
thf(fact_421_qbs__morphism__pair__assoc2,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_nat_b,Z2: quasi_borel_a] :
      ( member7011697144958356034at_b_a
      @ ( produc2970609168315946219at_b_a
        @ ^ [X: nat] :
            ( produc2881758435863576684at_b_a
            @ ^ [Y: nat > b] : ( produc8509509857186669914at_b_a @ ( produc5292568363641424317_nat_b @ X @ Y ) ) ) )
      @ ( qbs_mo252606531986980614at_b_a @ ( binary2038820404064428608at_b_a @ X4 @ ( binary8929016912194566789at_b_a @ Y4 @ Z2 ) ) @ ( binary4113689719790979332at_b_a @ ( binary7246370547177875303_nat_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_assoc2
thf(fact_422_qbs__morphism__pair__assoc2,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b,Z2: quasi_7334096685896923387_nat_b] :
      ( member531737949577091306_nat_b
      @ ( produc5725680957569560545_nat_b
        @ ^ [X: real] :
            ( produc254703461041418580_nat_b
            @ ^ [Y: b] : ( produc1946830598933159916_nat_b @ ( product_Pair_real_b @ X @ Y ) ) ) )
      @ ( qbs_mo4271075201823506790_nat_b @ ( binary2099412680721235540_nat_b @ X4 @ ( binary994859541383405405_nat_b @ Y4 @ Z2 ) ) @ ( binary7297433035349996738_nat_b @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_assoc2
thf(fact_423_qbs__morphism__pair__assoc2,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_real,Z2: quasi_borel_b] :
      ( member128747509785894348real_b
      @ ( produc7961561948777721411real_b
        @ ^ [X: nat] :
            ( produc6309598231035532322real_b
            @ ^ [Y: real] : ( produc3704156752222575865real_b @ ( produc7837566107596912789t_real @ X @ Y ) ) ) )
      @ ( qbs_mo9148312218546427336real_b @ ( binary3428804859926564999real_b @ X4 @ ( binary5762279046363823268real_b @ Y4 @ Z2 ) ) @ ( binary8766389241107001551real_b @ ( binary2938201662863479403t_real @ X4 @ Y4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_assoc2
thf(fact_424_qbs__morphism__pair__assoc2,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( member5982905077879908628t_real
      @ ( produc923536381276967741t_real
        @ ^ [X: nat] :
            ( produc3142128790114867206t_real
            @ ^ [Y: nat] : ( produc904203886716845910t_real @ ( product_Pair_nat_nat @ X @ Y ) ) ) )
      @ ( qbs_mo4277729064851017432t_real @ ( binary5992119153063443550t_real @ X4 @ ( binary2938201662863479403t_real @ Y4 @ Z2 ) ) @ ( binary7654069540790141440t_real @ ( binary6297550310575005455at_nat @ X4 @ Y4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_assoc2
thf(fact_425_qbs__morphism__pair__assoc2,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_3364728345918372901_b_a_a,Z2: quasi_3683678768404249670at_b_a] :
      ( member5017080915215164692at_b_a
      @ ( produc1425556010000434693at_b_a
        @ ^ [X: a] :
            ( produc6522456130969049432at_b_a
            @ ^ [Y: b > produc7194034526493574155_nat_b > a > a] : ( produc7656016108333055527at_b_a @ ( produc3671031927252941592_b_a_a @ X @ Y ) ) ) )
      @ ( qbs_mo7072313174336811864at_b_a @ ( binary5853139439745774421at_b_a @ X4 @ ( binary8035175780229655208at_b_a @ Y4 @ Z2 ) ) @ ( binary112817302848082129at_b_a @ ( binary9021634363669778414_b_a_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_assoc2
thf(fact_426_qbs__morphism__pair__assoc2,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b,Z2: quasi_1826430223877327896at_b_a] :
      ( member6894896908566155178at_b_a
      @ ( produc5023584469290631585at_b_a
        @ ^ [X: real] :
            ( produc295520359679177108at_b_a
            @ ^ [Y: b] : ( produc548985412187518345at_b_a @ ( product_Pair_real_b @ X @ Y ) ) ) )
      @ ( qbs_mo271743898166686886at_b_a @ ( binary552644141153030257at_b_a @ X4 @ ( binary3850590118635936762at_b_a @ Y4 @ Z2 ) ) @ ( binary5341841095258850399at_b_a @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_assoc2
thf(fact_427_qbs__morphism__pair__assoc2,axiom,
    ! [X4: quasi_7334096685896923387_nat_b,Y4: quasi_borel_a,Z2: quasi_3364728345918372901_b_a_a] :
      ( member7381443511799818836_b_a_a
      @ ( produc4471674032456031741_b_a_a
        @ ^ [X: produc7194034526493574155_nat_b] :
            ( produc4676916073658054521_b_a_a
            @ ^ [Y: a] : ( produc1581865879569254142_b_a_a @ ( produc8509509857186669914at_b_a @ X @ Y ) ) ) )
      @ ( qbs_mo8037536863429192728_b_a_a @ ( binary6206370300979919406_b_a_a @ X4 @ ( binary9021634363669778414_b_a_a @ Y4 @ Z2 ) ) @ ( binary6290418155309665704_b_a_a @ ( binary4113689719790979332at_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_assoc2
thf(fact_428_qbs__morphism__pair__assoc2,axiom,
    ! [X4: quasi_3364728345918372901_b_a_a,Y4: quasi_borel_b,Z2: quasi_1826430223877327896at_b_a] :
      ( member730532415096251384at_b_a
      @ ( produc2490192595346053993at_b_a
        @ ^ [X: b > produc7194034526493574155_nat_b > a > a] :
            ( produc7949956296359166907at_b_a
            @ ^ [Y: b] : ( produc8472000655522976568at_b_a @ ( produc6478256549973590487_a_a_b @ X @ Y ) ) ) )
      @ ( qbs_mo9067502892833404092at_b_a @ ( binary8035175780229655208at_b_a @ X4 @ ( binary3850590118635936762at_b_a @ Y4 @ Z2 ) ) @ ( binary6630357908747050978at_b_a @ ( binary2605486949535651501_a_a_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_assoc2
thf(fact_429_qbs__morphism__pair__assoc2,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a] :
      ( member6218801586492877702at_b_a
      @ ( produc7897278118259750775at_b_a
        @ ^ [X: b] :
            ( produc2106565865032386945at_b_a
            @ ^ [Y: produc7194034526493574155_nat_b] : ( produc8850021836743849772at_b_a @ ( produc5390679678779095987_nat_b @ X @ Y ) ) ) )
      @ ( qbs_mo3123876966264954698at_b_a @ ( binary3850590118635936762at_b_a @ X4 @ ( binary4113689719790979332at_b_a @ Y4 @ Z2 ) ) @ ( binary2484848965981528278at_b_a @ ( binary994859541383405405_nat_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_assoc2
thf(fact_430_qbs__morphism__pair__assoc1,axiom,
    ! [X4: quasi_7777979582039567160real_b,Y4: quasi_borel_nat,Z2: quasi_borel_nat_b] :
      ( member1208682416858941148_nat_b
      @ ( produc3796444233586022427_nat_b
        @ ( produc1602700697537262585_nat_b
          @ ^ [X: product_prod_real_b,Y: nat,Z3: nat > b] : ( produc1946830598933159916_nat_b @ X @ ( produc5292568363641424317_nat_b @ Y @ Z3 ) ) ) )
      @ ( qbs_mo4948019669105356632_nat_b @ ( binary4609318140483817114_nat_b @ ( binary1752603940819022343_b_nat @ X4 @ Y4 ) @ Z2 ) @ ( binary7297433035349996738_nat_b @ X4 @ ( binary7246370547177875303_nat_b @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_assoc1
thf(fact_431_qbs__morphism__pair__assoc1,axiom,
    ! [X4: quasi_3364728345918372901_b_a_a,Y4: quasi_borel_b,Z2: quasi_1826430223877327896at_b_a] :
      ( member622410630895769336at_b_a
      @ ( produc2652827125192679017at_b_a
        @ ( produc6073306815516608951at_b_a
          @ ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: b,Z3: produc6209191202727158440at_b_a] : ( produc653446490150804990at_b_a @ X @ ( produc992390952543482448at_b_a @ Y @ Z3 ) ) ) )
      @ ( qbs_mo600803144020306620at_b_a @ ( binary6630357908747050978at_b_a @ ( binary2605486949535651501_a_a_b @ X4 @ Y4 ) @ Z2 ) @ ( binary8035175780229655208at_b_a @ X4 @ ( binary3850590118635936762at_b_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_assoc1
thf(fact_432_qbs__morphism__pair__assoc1,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a] :
      ( member1297401021104003974at_b_a
      @ ( produc7968461916840466351at_b_a
        @ ( produc2676470379460466149at_b_a
          @ ^ [X: b,Y: produc7194034526493574155_nat_b,Z3: a] : ( produc992390952543482448at_b_a @ X @ ( produc8509509857186669914at_b_a @ Y @ Z3 ) ) ) )
      @ ( qbs_mo8530797794480700746at_b_a @ ( binary2484848965981528278at_b_a @ ( binary994859541383405405_nat_b @ X4 @ Y4 ) @ Z2 ) @ ( binary3850590118635936762at_b_a @ X4 @ ( binary4113689719790979332at_b_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_assoc1
thf(fact_433_qbs__morphism__pair__assoc1,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b,Z2: quasi_1826430223877327896at_b_a] :
      ( member6011215547500141902at_b_a
      @ ( produc8389110780133206853at_b_a
        @ ( produc3559213747023088523at_b_a
          @ ^ [X: real,Y: b,Z3: produc6209191202727158440at_b_a] : ( produc4983160494936474011at_b_a @ X @ ( produc992390952543482448at_b_a @ Y @ Z3 ) ) ) )
      @ ( qbs_mo4352086668803648330at_b_a @ ( binary5341841095258850399at_b_a @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) @ ( binary552644141153030257at_b_a @ X4 @ ( binary3850590118635936762at_b_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_assoc1
thf(fact_434_qbs__morphism__pair__assoc1,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_real,Z2: quasi_borel_b] :
      ( member6542205321770129068real_b
      @ ( produc2883147689336301163real_b
        @ ( produc5024734715871732425real_b
          @ ^ [X: nat,Y: real,Z3: b] : ( produc7589944407896915121real_b @ X @ ( product_Pair_real_b @ Y @ Z3 ) ) ) )
      @ ( qbs_mo1130700570939999912real_b @ ( binary8766389241107001551real_b @ ( binary2938201662863479403t_real @ X4 @ Y4 ) @ Z2 ) @ ( binary3428804859926564999real_b @ X4 @ ( binary5762279046363823268real_b @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_assoc1
thf(fact_435_qbs__morphism__pair__assoc1,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_3364728345918372901_b_a_a,Z2: quasi_3683678768404249670at_b_a] :
      ( member8651758763641498388at_b_a
      @ ( produc6457155736653863173at_b_a
        @ ( produc5438551805114339411at_b_a
          @ ^ [X: a,Y: b > produc7194034526493574155_nat_b > a > a,Z3: produc4883615123598087894at_b_a] : ( produc4172966208375972011at_b_a @ X @ ( produc653446490150804990at_b_a @ Y @ Z3 ) ) ) )
      @ ( qbs_mo8818491396796087128at_b_a @ ( binary112817302848082129at_b_a @ ( binary9021634363669778414_b_a_a @ X4 @ Y4 ) @ Z2 ) @ ( binary5853139439745774421at_b_a @ X4 @ ( binary8035175780229655208at_b_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_assoc1
thf(fact_436_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_nat,F: product_prod_nat_nat > b,Z2: quasi_borel_b] :
      ( ! [Alpha: real > nat,Beta: real > nat] :
          ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X4 ) )
         => ( ( member_real_nat @ Beta @ ( qbs_Mx_nat @ Y4 ) )
           => ( member_real_b
              @ ( comp_P4860171283570654797b_real @ F
                @ ^ [R: real] : ( product_Pair_nat_nat @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_b @ Z2 ) ) ) )
     => ( member482440888942232122_nat_b @ F @ ( qbs_mo3342300167638714294_nat_b @ ( binary6297550310575005455at_nat @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_437_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_nat,F: product_prod_nat_nat > a,Z2: quasi_borel_a] :
      ( ! [Alpha: real > nat,Beta: real > nat] :
          ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X4 ) )
         => ( ( member_real_nat @ Beta @ ( qbs_Mx_nat @ Y4 ) )
           => ( member_real_a
              @ ( comp_P8487324705237299916a_real @ F
                @ ^ [R: real] : ( product_Pair_nat_nat @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_a @ Z2 ) ) ) )
     => ( member482440884639003321_nat_a @ F @ ( qbs_mo3342300167638714293_nat_a @ ( binary6297550310575005455at_nat @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_438_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_b,F: product_prod_b_b > b,Z2: quasi_borel_b] :
      ( ! [Alpha: real > b,Beta: real > b] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X4 ) )
         => ( ( member_real_b @ Beta @ ( qbs_Mx_b @ Y4 ) )
           => ( member_real_b
              @ ( comp_P8101773103369595751b_real @ F
                @ ^ [R: real] : ( product_Pair_b_b @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_b @ Z2 ) ) ) )
     => ( member6677139785246673620_b_b_b @ F @ ( qbs_mo7892277270058608336_b_b_b @ ( binary1131512637584519981bs_b_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_439_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_b,F: product_prod_b_b > a,Z2: quasi_borel_a] :
      ( ! [Alpha: real > b,Beta: real > b] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X4 ) )
         => ( ( member_real_b @ Beta @ ( qbs_Mx_b @ Y4 ) )
           => ( member_real_a
              @ ( comp_P2505554488181465062a_real @ F
                @ ^ [R: real] : ( product_Pair_b_b @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_a @ Z2 ) ) ) )
     => ( member6677139780943444819_b_b_a @ F @ ( qbs_mo7892277270058608335_b_b_a @ ( binary1131512637584519981bs_b_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_440_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_a,F: product_prod_b_a > b,Z2: quasi_borel_b] :
      ( ! [Alpha: real > b,Beta: real > a] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X4 ) )
         => ( ( member_real_a @ Beta @ ( qbs_Mx_a @ Y4 ) )
           => ( member_real_b
              @ ( comp_P4150881467751007272b_real @ F
                @ ^ [R: real] : ( product_Pair_b_a @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_b @ Z2 ) ) ) )
     => ( member7312788785851660309_b_a_b @ F @ ( qbs_mo8527926270663595025_b_a_b @ ( binary1131512637584519980bs_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_441_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_a,F: product_prod_b_a > a,Z2: quasi_borel_a] :
      ( ! [Alpha: real > b,Beta: real > a] :
          ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X4 ) )
         => ( ( member_real_a @ Beta @ ( qbs_Mx_a @ Y4 ) )
           => ( member_real_a
              @ ( comp_P7778034889417652391a_real @ F
                @ ^ [R: real] : ( product_Pair_b_a @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_a @ Z2 ) ) ) )
     => ( member7312788781548431508_b_a_a @ F @ ( qbs_mo8527926270663595024_b_a_a @ ( binary1131512637584519980bs_b_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_442_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_b,F: product_prod_a_b > b,Z2: quasi_borel_b] :
      ( ! [Alpha: real > a,Beta: real > b] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X4 ) )
         => ( ( member_real_b @ Beta @ ( qbs_Mx_b @ Y4 ) )
           => ( member_real_b
              @ ( comp_P7434873627829687398b_real @ F
                @ ^ [R: real] : ( product_Pair_a_b @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_b @ Z2 ) ) ) )
     => ( member1080921170058542931_a_b_b @ F @ ( qbs_mo2296058654870477647_a_b_b @ ( binary3918968219557196014bs_a_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_443_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_b,F: product_prod_a_b > a,Z2: quasi_borel_a] :
      ( ! [Alpha: real > a,Beta: real > b] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X4 ) )
         => ( ( member_real_b @ Beta @ ( qbs_Mx_b @ Y4 ) )
           => ( member_real_a
              @ ( comp_P1838655012641556709a_real @ F
                @ ^ [R: real] : ( product_Pair_a_b @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_a @ Z2 ) ) ) )
     => ( member1080921165755314130_a_b_a @ F @ ( qbs_mo2296058654870477646_a_b_a @ ( binary3918968219557196014bs_a_b @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_444_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_a,F: product_prod_a_a > b,Z2: quasi_borel_b] :
      ( ! [Alpha: real > a,Beta: real > a] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X4 ) )
         => ( ( member_real_a @ Beta @ ( qbs_Mx_a @ Y4 ) )
           => ( member_real_b
              @ ( comp_P3483981992211098919b_real @ F
                @ ^ [R: real] : ( product_Pair_a_a @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_b @ Z2 ) ) ) )
     => ( member1716570170663529620_a_a_b @ F @ ( qbs_mo2931707655475464336_a_a_b @ ( binary3918968219557196013bs_a_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_445_pair__qbs__morphismI,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_a,F: product_prod_a_a > a,Z2: quasi_borel_a] :
      ( ! [Alpha: real > a,Beta: real > a] :
          ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X4 ) )
         => ( ( member_real_a @ Beta @ ( qbs_Mx_a @ Y4 ) )
           => ( member_real_a
              @ ( comp_P7111135413877744038a_real @ F
                @ ^ [R: real] : ( product_Pair_a_a @ ( Alpha @ R ) @ ( Beta @ R ) ) )
              @ ( qbs_Mx_a @ Z2 ) ) ) )
     => ( member1716570166360300819_a_a_a @ F @ ( qbs_mo2931707655475464335_a_a_a @ ( binary3918968219557196013bs_a_a @ X4 @ Y4 ) @ Z2 ) ) ) ).

% pair_qbs_morphismI
thf(fact_446_qbs__morphism__tuple,axiom,
    ! [F: $o > real,Z2: quasi_borel_o,X4: quasi_borel_real,G: $o > real,Y4: quasi_borel_real] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ Z2 @ X4 ) )
     => ( ( member_o_real @ G @ ( qbs_morphism_o_real @ Z2 @ Y4 ) )
       => ( member8602590400058854173l_real
          @ ^ [Z3: $o] : ( produc4511245868158468465l_real @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo6076292073755557401l_real @ Z2 @ ( binary210465342804807751l_real @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_447_qbs__morphism__tuple,axiom,
    ! [F: nat > real,Z2: quasi_borel_nat,X4: quasi_borel_real,G: nat > real,Y4: quasi_borel_real] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ Z2 @ X4 ) )
     => ( ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ Z2 @ Y4 ) )
       => ( member4944736856719700937l_real
          @ ^ [Z3: nat] : ( produc4511245868158468465l_real @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo6520021129653552495l_real @ Z2 @ ( binary210465342804807751l_real @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_448_qbs__morphism__tuple,axiom,
    ! [F: $o > real,Z2: quasi_borel_o,X4: quasi_borel_real,G: $o > $o,Y4: quasi_borel_o] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ Z2 @ X4 ) )
     => ( ( member_o_o @ G @ ( qbs_morphism_o_o @ Z2 @ Y4 ) )
       => ( member3923301605023534227real_o
          @ ^ [Z3: $o] : ( product_Pair_real_o @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo6612738326854831105real_o @ Z2 @ ( binary4057238779996960701real_o @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_449_qbs__morphism__tuple,axiom,
    ! [F: real > a,Z2: quasi_borel_real,X4: quasi_borel_a,G: real > a,Y4: quasi_borel_a] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ Z2 @ X4 ) )
     => ( ( member_real_a @ G @ ( qbs_morphism_real_a @ Z2 @ Y4 ) )
       => ( member4570177857406309467od_a_a
          @ ^ [Z3: real] : ( product_Pair_a_a @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo7578050507665182465od_a_a @ Z2 @ ( binary3918968219557196013bs_a_a @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_450_qbs__morphism__tuple,axiom,
    ! [F: $o > $o,Z2: quasi_borel_o,X4: quasi_borel_o,G: $o > real,Y4: quasi_borel_real] :
      ( ( member_o_o @ F @ ( qbs_morphism_o_o @ Z2 @ X4 ) )
     => ( ( member_o_real @ G @ ( qbs_morphism_o_real @ Z2 @ Y4 ) )
       => ( member5560034075029800485o_real
          @ ^ [Z3: $o] : ( product_Pair_o_real @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo4016795016617356947o_real @ Z2 @ ( binary2548174404955791127o_real @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_451_qbs__morphism__tuple,axiom,
    ! [F: $o > $o,Z2: quasi_borel_o,X4: quasi_borel_o,G: $o > $o,Y4: quasi_borel_o] :
      ( ( member_o_o @ F @ ( qbs_morphism_o_o @ Z2 @ X4 ) )
     => ( ( member_o_o @ G @ ( qbs_morphism_o_o @ Z2 @ Y4 ) )
       => ( member3028044298935597963od_o_o
          @ ^ [Z3: $o] : ( product_Pair_o_o @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo2565689321687337991od_o_o @ Z2 @ ( binary9159730317414444269bs_o_o @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_452_qbs__morphism__tuple,axiom,
    ! [F: $o > real,Z2: quasi_borel_o,X4: quasi_borel_real,G: $o > b,Y4: quasi_borel_b] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ Z2 @ X4 ) )
     => ( ( member_o_b @ G @ ( qbs_morphism_o_b @ Z2 @ Y4 ) )
       => ( member5019927097664096890real_b
          @ ^ [Z3: $o] : ( product_Pair_real_b @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo4240396306489266408real_b @ Z2 @ ( binary5762279046363823268real_b @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_453_qbs__morphism__tuple,axiom,
    ! [F: nat > real,Z2: quasi_borel_nat,X4: quasi_borel_real,G: nat > b,Y4: quasi_borel_b] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ Z2 @ X4 ) )
     => ( ( member_nat_b @ G @ ( qbs_morphism_nat_b @ Z2 @ Y4 ) )
       => ( member3863283534846713550real_b
          @ ^ [Z3: nat] : ( product_Pair_real_b @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo2610884262441363474real_b @ Z2 @ ( binary5762279046363823268real_b @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_454_qbs__morphism__tuple,axiom,
    ! [F: $o > nat,Z2: quasi_borel_o,X4: quasi_borel_nat,G: $o > real,Y4: quasi_borel_real] :
      ( ( member_o_nat @ F @ ( qbs_morphism_o_nat @ Z2 @ X4 ) )
     => ( ( member_o_real @ G @ ( qbs_morphism_o_real @ Z2 @ Y4 ) )
       => ( member8739982677887872321t_real
          @ ^ [Z3: $o] : ( produc7837566107596912789t_real @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo928686258813670845t_real @ Z2 @ ( binary2938201662863479403t_real @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_455_qbs__morphism__tuple,axiom,
    ! [F: nat > nat,Z2: quasi_borel_nat,X4: quasi_borel_nat,G: nat > real,Y4: quasi_borel_real] :
      ( ( member_nat_nat @ F @ ( qbs_morphism_nat_nat @ Z2 @ X4 ) )
     => ( ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ Z2 @ Y4 ) )
       => ( member4237364305948650477t_real
          @ ^ [Z3: nat] : ( produc7837566107596912789t_real @ ( F @ Z3 ) @ ( G @ Z3 ) )
          @ ( qbs_mo2896996399151425555t_real @ Z2 @ ( binary2938201662863479403t_real @ X4 @ Y4 ) ) ) ) ) ).

% qbs_morphism_tuple
thf(fact_456_qbs__morphism__pair__swap,axiom,
    ! [F: produc7878349794339525469_nat_b > a,X4: quasi_borel_a,Y4: quasi_8132950864781310810_nat_b,Z2: quasi_borel_a] :
      ( ( member5131579601459065181at_b_a @ F @ ( qbs_mo3941500657253546841at_b_a @ ( binary8437419225524438435_nat_b @ X4 @ Y4 ) @ Z2 ) )
     => ( member9091339440257553353_b_a_a
        @ ( produc5399252211680487944_b_a_a
          @ ^ [X: produc3260441645384719156_nat_b,Y: a] : ( F @ ( produc3644563542453106381_nat_b @ Y @ X ) ) )
        @ ( qbs_mo7901260496052035013_b_a_a @ ( binary4006988140340982583at_b_a @ Y4 @ X4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_swap
thf(fact_457_qbs__morphism__pair__swap,axiom,
    ! [F: produc1285742814635146737at_b_a > a,X4: quasi_8132950864781310810_nat_b,Y4: quasi_borel_a,Z2: quasi_borel_a] :
      ( ( member9091339440257553353_b_a_a @ F @ ( qbs_mo7901260496052035013_b_a_a @ ( binary4006988140340982583at_b_a @ X4 @ Y4 ) @ Z2 ) )
     => ( member5131579601459065181at_b_a
        @ ( produc7385741450531719836at_b_a
          @ ^ [X: a,Y: produc3260441645384719156_nat_b] : ( F @ ( produc8437504494124426337at_b_a @ Y @ X ) ) )
        @ ( qbs_mo3941500657253546841at_b_a @ ( binary8437419225524438435_nat_b @ Y4 @ X4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_swap
thf(fact_458_qbs__morphism__pair__swap,axiom,
    ! [F: produc6558228558545702196real_b > a > a,X4: quasi_7334096685896923387_nat_b,Y4: quasi_7777979582039567160real_b,Z2: quasi_borel_a_a] :
      ( ( member528990613720640179_b_a_a @ F @ ( qbs_mo8582487267016127009_b_a_a @ ( binary7730624625177995586real_b @ X4 @ Y4 ) @ Z2 ) )
     => ( member7748159150162262195_b_a_a
        @ ( produc8421786640700244582_b_a_a
          @ ^ [X: product_prod_real_b,Y: produc7194034526493574155_nat_b] : ( F @ ( produc2380022188761158764real_b @ Y @ X ) ) )
        @ ( qbs_mo6578283766602973217_b_a_a @ ( binary7297433035349996738_nat_b @ Y4 @ X4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_swap
thf(fact_459_qbs__morphism__pair__swap,axiom,
    ! [F: produc7306929691551298616_nat_b > a,X4: quasi_borel_a,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a] :
      ( ( member8479242775280434982at_b_a @ F @ ( qbs_mo8247511030340029260at_b_a @ ( binary6845515207533535708_nat_b @ X4 @ Y4 ) @ Z2 ) )
     => ( member1447347221985152182_b_a_a
        @ ( produc2543982510095308599_b_a_a
          @ ^ [X: produc7194034526493574155_nat_b,Y: a] : ( F @ ( produc2017963308074450482_nat_b @ Y @ X ) ) )
        @ ( qbs_mo1215615477044746460_b_a_a @ ( binary4113689719790979332at_b_a @ Y4 @ X4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_swap
thf(fact_460_qbs__morphism__pair__swap,axiom,
    ! [F: produc6558228558545702196real_b > a,X4: quasi_7334096685896923387_nat_b,Y4: quasi_7777979582039567160real_b,Z2: quasi_borel_a] :
      ( ( member5359709749645695302al_b_a @ F @ ( qbs_mo5776496882792752706al_b_a @ ( binary7730624625177995586real_b @ X4 @ Y4 ) @ Z2 ) )
     => ( member7813222213774494534at_b_a
        @ ( produc634687561522189117at_b_a
          @ ^ [X: product_prod_real_b,Y: produc7194034526493574155_nat_b] : ( F @ ( produc2380022188761158764real_b @ Y @ X ) ) )
        @ ( qbs_mo8230009346921551938at_b_a @ ( binary7297433035349996738_nat_b @ Y4 @ X4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_swap
thf(fact_461_qbs__morphism__pair__swap,axiom,
    ! [F: product_prod_b_real > produc7194034526493574155_nat_b > a > a,X4: quasi_borel_b,Y4: quasi_borel_real,Z2: quasi_7934145443646537599_b_a_a] :
      ( ( member1607198150618329375_b_a_a @ F @ ( qbs_mo4036585190385424867_b_a_a @ ( binary1505106857224228350b_real @ X4 @ Y4 ) @ Z2 ) )
     => ( member1306660226910552845_b_a_a
        @ ( produc7016990068875982646_b_a_a
          @ ^ [X: real,Y: b] : ( F @ ( product_Pair_b_real @ Y @ X ) ) )
        @ ( qbs_mo3736047266677648337_b_a_a @ ( binary5762279046363823268real_b @ Y4 @ X4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_swap
thf(fact_462_qbs__morphism__pair__swap,axiom,
    ! [F: product_prod_b_real > a > a,X4: quasi_borel_b,Y4: quasi_borel_real,Z2: quasi_borel_a_a] :
      ( ( member383449281184013941al_a_a @ F @ ( qbs_mo4391578294481092153al_a_a @ ( binary1505106857224228350b_real @ X4 @ Y4 ) @ Z2 ) )
     => ( member9111837334973778019_b_a_a
        @ ( produc7827042608974387084_b_a_a
          @ ^ [X: real,Y: b] : ( F @ ( product_Pair_b_real @ Y @ X ) ) )
        @ ( qbs_mo3896594311416080423_b_a_a @ ( binary5762279046363823268real_b @ Y4 @ X4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_swap
thf(fact_463_qbs__morphism__pair__swap,axiom,
    ! [F: produc1055083314214236062_b_a_a > a,X4: quasi_3683678768404249670at_b_a,Y4: quasi_3364728345918372901_b_a_a,Z2: quasi_borel_a] :
      ( ( member8912907593070208576_a_a_a @ F @ ( qbs_mo166606781874353894_a_a_a @ ( binary6237893839664861178_b_a_a @ X4 @ Y4 ) @ Z2 ) )
     => ( member763877586905102170_b_a_a
        @ ( produc286368409806729619_b_a_a
          @ ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: produc4883615123598087894at_b_a] : ( F @ ( produc8079536586440786768_b_a_a @ Y @ X ) ) )
        @ ( qbs_mo1240948812564023296_b_a_a @ ( binary8035175780229655208at_b_a @ Y4 @ X4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_swap
thf(fact_464_qbs__morphism__pair__swap,axiom,
    ! [F: produc7743328458780956108_b_a_b > a,X4: quasi_1826430223877327896at_b_a,Y4: quasi_borel_b,Z2: quasi_borel_a] :
      ( ( member3353999721882806418_a_b_a @ F @ ( qbs_mo3646858462515050424_a_b_a @ ( binary1079826641210870184_b_a_b @ X4 @ Y4 ) @ Z2 ) )
     => ( member871964287106612744_b_a_a
        @ ( produc6302854954412089281_b_a_a
          @ ^ [X: b,Y: produc6209191202727158440at_b_a] : ( F @ ( produc7444999511973191678_b_a_b @ Y @ X ) ) )
        @ ( qbs_mo1164823027738856750_b_a_a @ ( binary3850590118635936762at_b_a @ Y4 @ X4 ) @ Z2 ) ) ) ).

% qbs_morphism_pair_swap
thf(fact_465_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_nat] :
      ( member8885076297122219836at_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) )
      @ ( qbs_mo43725994543658552at_nat @ ( binary6297550310575005455at_nat @ X4 @ Y4 ) @ ( binary6297550310575005455at_nat @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_466_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_real] :
      ( member6570605000191323352real_b
      @ ( produc6367669841173908169real_b
        @ ^ [X: b,Y: real] : ( product_Pair_real_b @ Y @ X ) )
      @ ( qbs_mo964639070500085148real_b @ ( binary1505106857224228350b_real @ X4 @ Y4 ) @ ( binary5762279046363823268real_b @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_467_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_nat] :
      ( member4575602350711296188t_real
      @ ( produc3658697984569465395t_real
        @ ^ [X: real,Y: nat] : ( produc7837566107596912789t_real @ Y @ X ) )
      @ ( qbs_mo6219505925795312824t_real @ ( binary7505510235992378091al_nat @ X4 @ Y4 ) @ ( binary2938201662863479403t_real @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_468_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b] :
      ( member4112490518706476760b_real
      @ ( produc6781195011795835521b_real
        @ ^ [X: real,Y: b] : ( product_Pair_b_real @ Y @ X ) )
      @ ( qbs_mo8843654461432590492b_real @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ ( binary1505106857224228350b_real @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_469_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_real] :
      ( member2568373704639994300al_nat
      @ ( produc4028160848036898611al_nat
        @ ^ [X: nat,Y: real] : ( produc3181502643871035669al_nat @ Y @ X ) )
      @ ( qbs_mo6270607485908303288al_nat @ ( binary2938201662863479403t_real @ X4 @ Y4 ) @ ( binary7505510235992378091al_nat @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_470_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_borel_nat_b,Y4: quasi_borel_nat] :
      ( member5929715518534014630_nat_b
      @ ( produc6066811893999682327_nat_b
        @ ^ [X: nat > b,Y: nat] : ( produc5292568363641424317_nat_b @ Y @ X ) )
      @ ( qbs_mo7286046936281110250_nat_b @ ( binary4663048038059423881_b_nat @ X4 @ Y4 ) @ ( binary7246370547177875303_nat_b @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_471_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_7334096685896923387_nat_b] :
      ( member1160318262226876182at_b_a
      @ ( produc8484962418594779783at_b_a
        @ ^ [X: a,Y: produc7194034526493574155_nat_b] : ( produc8509509857186669914at_b_a @ Y @ X ) )
      @ ( qbs_mo3624599686110276570at_b_a @ ( binary6845515207533535708_nat_b @ X4 @ Y4 ) @ ( binary4113689719790979332at_b_a @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_472_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_7334096685896923387_nat_b,Y4: quasi_7777979582039567160real_b] :
      ( member5401647013537137660_nat_b
      @ ( produc8266046411308739443_nat_b
        @ ^ [X: produc7194034526493574155_nat_b,Y: product_prod_real_b] : ( produc1946830598933159916_nat_b @ Y @ X ) )
      @ ( qbs_mo9140984265783553144_nat_b @ ( binary7730624625177995586real_b @ X4 @ Y4 ) @ ( binary7297433035349996738_nat_b @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_473_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_1826430223877327896at_b_a,Y4: quasi_borel_b] :
      ( member1883103383683449560at_b_a
      @ ( produc4594359411059224449at_b_a
        @ ^ [X: produc6209191202727158440at_b_a,Y: b] : ( produc992390952543482448at_b_a @ Y @ X ) )
      @ ( qbs_mo9116500157060146332at_b_a @ ( binary1079826641210870184_b_a_b @ X4 @ Y4 ) @ ( binary3850590118635936762at_b_a @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_474_qbs__morphism__pair__swap_H,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_1826430223877327896at_b_a] :
      ( member7674033304963341016_b_a_b
      @ ( produc6492255797792321481_b_a_b
        @ ^ [X: b,Y: produc6209191202727158440at_b_a] : ( produc7444999511973191678_b_a_b @ Y @ X ) )
      @ ( qbs_mo1718854645797525404_b_a_b @ ( binary3850590118635936762at_b_a @ X4 @ Y4 ) @ ( binary1079826641210870184_b_a_b @ Y4 @ X4 ) ) ) ).

% qbs_morphism_pair_swap'
thf(fact_475_exp__qbs__Mx__def,axiom,
    ( expone2961370955173070346_b_a_a
    = ( ^ [X6: quasi_borel_b,Y6: quasi_7934145443646537599_b_a_a] :
          ( collec8422114425917487221_b_a_a
          @ ^ [G2: real > b > produc7194034526493574155_nat_b > a > a] : ( member1306660226910552845_b_a_a @ ( produc7016990068875982646_b_a_a @ G2 ) @ ( qbs_mo3736047266677648337_b_a_a @ ( binary5762279046363823268real_b @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ X6 ) @ Y6 ) ) ) ) ) ).

% exp_qbs_Mx_def
thf(fact_476_exp__qbs__Mx__def,axiom,
    ( expone4990064882076513888_b_a_a
    = ( ^ [X6: quasi_borel_b,Y6: quasi_borel_a_a] :
          ( collect_real_b_a_a
          @ ^ [G2: real > b > a > a] : ( member9111837334973778019_b_a_a @ ( produc7827042608974387084_b_a_a @ G2 ) @ ( qbs_mo3896594311416080423_b_a_a @ ( binary5762279046363823268real_b @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ X6 ) @ Y6 ) ) ) ) ) ).

% exp_qbs_Mx_def
thf(fact_477__092_060open_062_092_060beta_062_A_092_060in_062_Aqbs__Mx_A_I_IX_A_092_060Rightarrow_062_092_060_094sub_062Q_Alist__of_AX_A_092_060Rightarrow_062_092_060_094sub_062Q_AY_A_092_060Rightarrow_062_092_060_094sub_062Q_AY_J_A_092_060Otimes_062_092_060_094sub_062Q_AX_A_092_060Otimes_062_092_060_094sub_062Q_Alist__of_AX_A_092_060Otimes_062_092_060_094sub_062Q_AY_J_092_060close_062,axiom,
    ( member8314179565955891814at_b_a @ beta
    @ ( qbs_Mx5919617464495388621at_b_a
      @ ( binary8035175780229655208at_b_a
        @ ( expone5294601226626914677_b_a_a @ x
          @ ( expone5933728560452245707_b_a_a
            @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
              @ ^ [N: nat] :
                  ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                  @ ^ [I: nat] : x ) )
            @ ( exponent_exp_qbs_a_a @ y @ y ) ) )
        @ ( binary3850590118635936762at_b_a @ x
          @ ( binary4113689719790979332at_b_a
            @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
              @ ^ [N: nat] :
                  ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                  @ ^ [I: nat] : x ) )
            @ y ) ) ) ) ) ).

% \<open>\<beta> \<in> qbs_Mx ((X \<Rightarrow>\<^sub>Q list_of X \<Rightarrow>\<^sub>Q Y \<Rightarrow>\<^sub>Q Y) \<Otimes>\<^sub>Q X \<Otimes>\<^sub>Q list_of X \<Otimes>\<^sub>Q Y)\<close>
thf(fact_478_exp__qbs__Mx,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_b] :
      ( ( qbs_Mx_nat_b @ ( expone1825828912295380116_nat_b @ X4 @ Y4 ) )
      = ( expone2173014692789731113_nat_b @ X4 @ Y4 ) ) ).

% exp_qbs_Mx
thf(fact_479_exp__qbs__Mx,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_7934145443646537599_b_a_a] :
      ( ( qbs_Mx703024977670096246_b_a_a @ ( expone5294601226626914677_b_a_a @ X4 @ Y4 ) )
      = ( expone2961370955173070346_b_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_Mx
thf(fact_480_exp__qbs__Mx,axiom,
    ! [X4: quasi_7334096685896923387_nat_b,Y4: quasi_borel_a_a] :
      ( ( qbs_Mx442805946153557208_b_a_a @ ( expone5933728560452245707_b_a_a @ X4 @ Y4 ) )
      = ( expone1541841343128190326_b_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_Mx
thf(fact_481_exp__qbs__Mx,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_a] :
      ( ( qbs_Mx_a_a @ ( exponent_exp_qbs_a_a @ X4 @ Y4 ) )
      = ( expone7022597896679715716Mx_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_Mx
thf(fact_482_exp__qbs__Mx,axiom,
    ! [X4: quasi_6353345141195568462_b_a_a,Y4: quasi_borel_a] :
      ( ( qbs_Mx506040344577873796_a_a_a @ ( expone790792417327382575_a_a_a @ X4 @ Y4 ) )
      = ( expone2990877755589709658_a_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_Mx
thf(fact_483_exp__qbs__Mx,axiom,
    ! [X4: quasi_borel_o,Y4: quasi_borel_o] :
      ( ( qbs_Mx_o_o @ ( exponent_exp_qbs_o_o @ X4 @ Y4 ) )
      = ( expone9009657249589208708Mx_o_o @ X4 @ Y4 ) ) ).

% exp_qbs_Mx
thf(fact_484_gcd_Ocases,axiom,
    ! [X5: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( X5
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% gcd.cases
thf(fact_485_qbs__morphism__component__singleton,axiom,
    ! [I3: nat,I2: set_nat,M: nat > quasi_borel_b] :
      ( ( member_nat @ I3 @ I2 )
     => ( member_nat_b_b
        @ ^ [X: nat > b] : ( X @ I3 )
        @ ( qbs_morphism_nat_b_b @ ( product_PiQ_nat_b @ I2 @ M ) @ ( M @ I3 ) ) ) ) ).

% qbs_morphism_component_singleton
thf(fact_486_qbs__eval__morphism,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_7934145443646537599_b_a_a] : ( member1140255728439836176_b_a_a @ expone6597005973208256868_b_a_a @ ( qbs_mo6518393854002529918_b_a_a @ ( binary2605486949535651501_a_a_b @ ( expone5294601226626914677_b_a_a @ X4 @ Y4 ) @ X4 ) @ Y4 ) ) ).

% qbs_eval_morphism
thf(fact_487_qbs__eval__morphism,axiom,
    ! [X4: quasi_7334096685896923387_nat_b,Y4: quasi_borel_a_a] : ( member5525254186805185018_b_a_a @ expone8712287761860780188_b_a_a @ ( qbs_mo5258765042558225128_b_a_a @ ( binary6439553939627338377_nat_b @ ( expone5933728560452245707_b_a_a @ X4 @ Y4 ) @ X4 ) @ Y4 ) ) ).

% qbs_eval_morphism
thf(fact_488_qbs__eval__morphism,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_a] : ( member1898022153095508880_a_a_a @ expone7211744221881241258al_a_a @ ( qbs_mo9170152465216341558_a_a_a @ ( binary5146134066237354538_a_a_a @ ( exponent_exp_qbs_a_a @ X4 @ Y4 ) @ X4 ) @ Y4 ) ) ).

% qbs_eval_morphism
thf(fact_489_qbs__eval__morphism,axiom,
    ! [X4: quasi_6353345141195568462_b_a_a,Y4: quasi_borel_a] : ( member5564030145598096316_a_a_a @ expone3941855099027480448_a_a_a @ ( qbs_mo3967762102501529954_a_a_a @ ( binary4651267417200808190_b_a_a @ ( expone790792417327382575_a_a_a @ X4 @ Y4 ) @ X4 ) @ Y4 ) ) ).

% qbs_eval_morphism
thf(fact_490_qbs__eval__morphism,axiom,
    ! [X4: quasi_borel_o,Y4: quasi_borel_o] : ( member8725249704849985552_o_o_o @ expone8878782944832497578al_o_o @ ( qbs_mo7976596037687805878_o_o_o @ ( binary7336721751802915012_o_o_o @ ( exponent_exp_qbs_o_o @ X4 @ Y4 ) @ X4 ) @ Y4 ) ) ).

% qbs_eval_morphism
thf(fact_491_top__empty__eq,axiom,
    ( top_top_o_real_o
    = ( ^ [X: $o > real] : ( member_o_real @ X @ top_top_set_o_real ) ) ) ).

% top_empty_eq
thf(fact_492_top__empty__eq,axiom,
    ( top_top_o_o_o
    = ( ^ [X: $o > $o] : ( member_o_o @ X @ top_top_set_o_o ) ) ) ).

% top_empty_eq
thf(fact_493_top__empty__eq,axiom,
    ( top_top_o_o_o_o
    = ( ^ [X: $o > $o > $o] : ( member_o_o_o @ X @ top_top_set_o_o_o ) ) ) ).

% top_empty_eq
thf(fact_494_top__empty__eq,axiom,
    ( top_top_nat_real_o
    = ( ^ [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ) ) ).

% top_empty_eq
thf(fact_495_top__empty__eq,axiom,
    ( top_to2609013995415524507_a_a_o
    = ( ^ [X: ( nat > b ) > produc432120507726958376_b_a_a > a] : ( member1811236362415105851_a_a_a @ X @ top_to5015968664104940226_a_a_a ) ) ) ).

% top_empty_eq
thf(fact_496_top__empty__eq,axiom,
    ( top_to7655735016164553232real_o
    = ( ^ [X: produc7716430852924023517t_real] : ( member557208447399453958t_real @ X @ top_to4855536200657754381t_real ) ) ) ).

% top_empty_eq
thf(fact_497_top__empty__eq,axiom,
    ( top_top_complex_o
    = ( ^ [X: complex] : ( member_complex @ X @ top_top_set_complex ) ) ) ).

% top_empty_eq
thf(fact_498_top__empty__eq,axiom,
    ( top_top_real_o
    = ( ^ [X: real] : ( member_real @ X @ top_top_set_real ) ) ) ).

% top_empty_eq
thf(fact_499_top__empty__eq,axiom,
    ( top_top_o_o
    = ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).

% top_empty_eq
thf(fact_500_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_501_coproduct__qbs__Mx,axiom,
    ! [I2: set_b_5076249247947527647_b_a_a,X4: ( b > produc7194034526493574155_nat_b > a > a ) > quasi_3683678768404249670at_b_a] :
      ( ( qbs_Mx5919617464495388621at_b_a @ ( coProd5845954756983474342at_b_a @ I2 @ X4 ) )
      = ( coProd204034348430421307at_b_a @ I2 @ X4 ) ) ).

% coproduct_qbs_Mx
thf(fact_502_coproduct__qbs__Mx,axiom,
    ! [I2: set_nat,X4: nat > quasi_borel_nat_b] :
      ( ( qbs_Mx1531206082039394900_nat_b @ ( coProd3505932218054916197_nat_b @ I2 @ X4 ) )
      = ( coProd6018169600328615290_nat_b @ I2 @ X4 ) ) ).

% coproduct_qbs_Mx
thf(fact_503_split__cong,axiom,
    ! [Q2: produc3260441645384719156_nat_b,F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,G: product_prod_real_b > produc7194034526493574155_nat_b > a > a,P2: produc3260441645384719156_nat_b] :
      ( ! [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
          ( ( ( produc1946830598933159916_nat_b @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc8421786640700244582_b_a_a @ F @ P2 )
          = ( produc8421786640700244582_b_a_a @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_504_split__cong,axiom,
    ! [Q2: produc6209191202727158440at_b_a,F: produc7194034526493574155_nat_b > a > a,G: produc7194034526493574155_nat_b > a > a,P2: produc6209191202727158440at_b_a] :
      ( ! [X3: produc7194034526493574155_nat_b,Y3: a] :
          ( ( ( produc8509509857186669914at_b_a @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc2543982510095308599_b_a_a @ F @ P2 )
          = ( produc2543982510095308599_b_a_a @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_505_split__cong,axiom,
    ! [Q2: product_prod_real_b,F: real > b > produc7194034526493574155_nat_b > a > a,G: real > b > produc7194034526493574155_nat_b > a > a,P2: product_prod_real_b] :
      ( ! [X3: real,Y3: b] :
          ( ( ( product_Pair_real_b @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc7016990068875982646_b_a_a @ F @ P2 )
          = ( produc7016990068875982646_b_a_a @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_506_split__cong,axiom,
    ! [Q2: produc778026099599033348at_b_a,F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,G: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,P2: produc778026099599033348at_b_a] :
      ( ! [X3: b > produc7194034526493574155_nat_b > a > a,Y3: produc4883615123598087894at_b_a] :
          ( ( ( produc653446490150804990at_b_a @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc286368409806729619_b_a_a @ F @ P2 )
          = ( produc286368409806729619_b_a_a @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_507_split__cong,axiom,
    ! [Q2: produc4883615123598087894at_b_a,F: b > produc6209191202727158440at_b_a > a,G: b > produc6209191202727158440at_b_a > a,P2: produc4883615123598087894at_b_a] :
      ( ! [X3: b,Y3: produc6209191202727158440at_b_a] :
          ( ( ( produc992390952543482448at_b_a @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc6302854954412089281_b_a_a @ F @ P2 )
          = ( produc6302854954412089281_b_a_a @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_508_split__cong,axiom,
    ! [Q2: product_prod_real_b,F: real > b > a > a,G: real > b > a > a,P2: product_prod_real_b] :
      ( ! [X3: real,Y3: b] :
          ( ( ( product_Pair_real_b @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc7827042608974387084_b_a_a @ F @ P2 )
          = ( produc7827042608974387084_b_a_a @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_509_split__cong,axiom,
    ! [Q2: produc3260441645384719156_nat_b,F: product_prod_real_b > produc7194034526493574155_nat_b > a,G: product_prod_real_b > produc7194034526493574155_nat_b > a,P2: produc3260441645384719156_nat_b] :
      ( ! [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
          ( ( ( produc1946830598933159916_nat_b @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P2 = Q2 )
       => ( ( produc634687561522189117at_b_a @ F @ P2 )
          = ( produc634687561522189117at_b_a @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_510_top__prod__def,axiom,
    ( top_to5616782037112009081omplex
    = ( produc3790773574474814305omplex @ top_top_set_complex @ top_top_set_complex ) ) ).

% top_prod_def
thf(fact_511_top__prod__def,axiom,
    ( top_to4011229823500134647t_real
    = ( produc3148467163867838047t_real @ top_top_set_complex @ top_top_set_real ) ) ).

% top_prod_def
thf(fact_512_top__prod__def,axiom,
    ( top_to3591092333680416569_set_o
    = ( produc3955722774888272699_set_o @ top_top_set_complex @ top_top_set_o ) ) ).

% top_prod_def
thf(fact_513_top__prod__def,axiom,
    ( top_to5881799373427395739et_nat
    = ( produc451057102532134275et_nat @ top_top_set_complex @ top_top_set_nat ) ) ).

% top_prod_def
thf(fact_514_top__prod__def,axiom,
    ( top_to5687657296277248631omplex
    = ( produc3672341788591977183omplex @ top_top_set_real @ top_top_set_complex ) ) ).

% top_prod_def
thf(fact_515_top__prod__def,axiom,
    ( top_to1822899090350206709t_real
    = ( produc6377492392003646173t_real @ top_top_set_real @ top_top_set_real ) ) ).

% top_prod_def
thf(fact_516_top__prod__def,axiom,
    ( top_to5627709924142012091_set_o
    = ( produc9171573333813886013_set_o @ top_top_set_real @ top_top_set_o ) ) ).

% top_prod_def
thf(fact_517_top__prod__def,axiom,
    ( top_to8233803056352276889et_nat
    = ( produc4432056366243864833et_nat @ top_top_set_real @ top_top_set_nat ) ) ).

% top_prod_def
thf(fact_518_top__prod__def,axiom,
    ( top_to4749799991558211003omplex
    = ( produc3905962715022834821omplex @ top_top_set_o @ top_top_set_complex ) ) ).

% top_prod_def
thf(fact_519_top__prod__def,axiom,
    ( top_to8585549949225041977t_real
    = ( produc2850126452351882627t_real @ top_top_set_o @ top_top_set_real ) ) ).

% top_prod_def
thf(fact_520_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K: real > a,Y4: sigma_measure_a] :
      ( ( member_real_a @ K @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ Y4 ) )
     => ( member_real_a @ K @ ( qbs_morphism_real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6507891955840068946_qbs_a @ Y4 ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_521_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K: real > produc2422161461964618553l_real,Y4: sigma_2308072346491277622l_real] :
      ( ( member9086635009091248365l_real @ K @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ Y4 ) )
     => ( member9086635009091248365l_real @ K @ ( qbs_mo6845038431372961811l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur3029786804310284173l_real @ Y4 ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_522_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K: real > produc7716430852924023517t_real,Y4: sigma_5310753476256395226t_real] :
      ( ( member969486235890772753t_real @ K @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ Y4 ) )
     => ( member969486235890772753t_real @ K @ ( qbs_mo3601394221918002359t_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2011440102552004913t_real @ Y4 ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_523_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K: real > real,Y4: sigma_measure_real] :
      ( ( member_real_real @ K @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ Y4 ) )
     => ( member_real_real @ K @ ( qbs_mo5229770564518008146l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6875533127466166616s_real @ Y4 ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_524_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K: real > $o,Y4: sigma_measure_o] :
      ( ( member_real_o @ K @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ Y4 ) )
     => ( member_real_o @ K @ ( qbs_morphism_real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_525_real_Oqbs__morphism__measurable__intro,axiom,
    ! [K: real > nat,Y4: sigma_measure_nat] :
      ( ( member_real_nat @ K @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ Y4 ) )
     => ( member_real_nat @ K @ ( qbs_mo6567951568834356598al_nat @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) ) ) ).

% real.qbs_morphism_measurable_intro
thf(fact_526__C6_C,axiom,
    ( member8664213473904227437_nat_b
    @ ^ [R: real] : ( produc5292568363641424317_nat_b @ na @ ( alpha @ R ) )
    @ ( qbs_Mx1531206082039394900_nat_b
      @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
        @ ^ [N: nat] :
            ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
            @ ^ [I: nat] : x ) ) ) ) ).

% "6"
thf(fact_527_exp__qbs__def,axiom,
    ( expone319700646606540215real_a
    = ( ^ [X6: quasi_borel_real,Y6: quasi_borel_a] : ( quasi_6377562139329607883real_a @ ( produc8792329689427759714real_a @ ( qbs_morphism_real_a @ X6 @ Y6 ) @ ( expone6795215201378684364real_a @ X6 @ Y6 ) ) ) ) ) ).

% exp_qbs_def
thf(fact_528_exp__qbs__def,axiom,
    ( expone4627587591657327526_b_a_a
    = ( ^ [X6: quasi_805850127037308055at_b_a,Y6: quasi_borel_a] : ( quasi_3966700921869153264_b_a_a @ ( produc2067008855343870260_b_a_a @ ( qbs_mo7901260496052035013_b_a_a @ X6 @ Y6 ) @ ( expone4687418993921882065_b_a_a @ X6 @ Y6 ) ) ) ) ) ).

% exp_qbs_def
thf(fact_529_exp__qbs__def,axiom,
    ( expone6720796868436823688_a_a_a
    = ( ^ [X6: quasi_borel_nat_b,Y6: quasi_2547937147209033259_a_a_a] : ( quasi_4113067884626317524_a_a_a @ ( produc1611076569815217314_a_a_a @ ( qbs_mo8618518167108125737_a_a_a @ X6 @ Y6 ) @ ( expone2444466320815533341_a_a_a @ X6 @ Y6 ) ) ) ) ) ).

% exp_qbs_def
thf(fact_530_exp__qbs__def,axiom,
    ( expone1670929651343127652_o_o_o
    = ( ^ [X6: quasi_borel_o,Y6: quasi_borel_o_o] : ( quasi_2672704784731211056_o_o_o @ ( produc523828491365516066_o_o_o @ ( qbs_morphism_o_o_o @ X6 @ Y6 ) @ ( expone8146444206115271801_o_o_o @ X6 @ Y6 ) ) ) ) ) ).

% exp_qbs_def
thf(fact_531_exp__qbs__def,axiom,
    ( expone5294601226626914677_b_a_a
    = ( ^ [X6: quasi_borel_b,Y6: quasi_7934145443646537599_b_a_a] : ( quasi_4989793359850368449_b_a_a @ ( produc5264547532651997410_b_a_a @ ( qbs_mo1542099550572995862_b_a_a @ X6 @ Y6 ) @ ( expone2961370955173070346_b_a_a @ X6 @ Y6 ) ) ) ) ) ).

% exp_qbs_def
thf(fact_532_exp__qbs__def,axiom,
    ( expone5933728560452245707_b_a_a
    = ( ^ [X6: quasi_7334096685896923387_nat_b,Y6: quasi_borel_a_a] : ( quasi_4498582803870391117_b_a_a @ ( produc3806978686075393262_b_a_a @ ( qbs_mo7242095808399995242_b_a_a @ X6 @ Y6 ) @ ( expone1541841343128190326_b_a_a @ X6 @ Y6 ) ) ) ) ) ).

% exp_qbs_def
thf(fact_533_exp__qbs__def,axiom,
    ( exponent_exp_qbs_a_a
    = ( ^ [X6: quasi_borel_a,Y6: quasi_borel_a] : ( quasi_8994498933364049827el_a_a @ ( produc8176344774097317018al_a_a @ ( qbs_morphism_a_a @ X6 @ Y6 ) @ ( expone7022597896679715716Mx_a_a @ X6 @ Y6 ) ) ) ) ) ).

% exp_qbs_def
thf(fact_534_exp__qbs__def,axiom,
    ( expone790792417327382575_a_a_a
    = ( ^ [X6: quasi_6353345141195568462_b_a_a,Y6: quasi_borel_a] : ( quasi_2958659650385320825_a_a_a @ ( produc9059353793759839558_a_a_a @ ( qbs_mo5048501348360587598_a_a_a @ X6 @ Y6 ) @ ( expone2990877755589709658_a_a_a @ X6 @ Y6 ) ) ) ) ) ).

% exp_qbs_def
thf(fact_535_exp__qbs__def,axiom,
    ( exponent_exp_qbs_o_o
    = ( ^ [X6: quasi_borel_o,Y6: quasi_borel_o] : ( quasi_6990158127852425123el_o_o @ ( produc1980721420443017882al_o_o @ ( qbs_morphism_o_o @ X6 @ Y6 ) @ ( expone9009657249589208708Mx_o_o @ X6 @ Y6 ) ) ) ) ) ).

% exp_qbs_def
thf(fact_536_curry__morphism,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_7334096685896923387_nat_b,Z2: quasi_borel_a_a] : ( member934395255241073122_b_a_a @ produc5532546626253661158_b_a_a @ ( qbs_mo5035750169361120648_b_a_a @ ( expone8080306092094974073_b_a_a @ ( binary994859541383405405_nat_b @ X4 @ Y4 ) @ Z2 ) @ ( expone5294601226626914677_b_a_a @ X4 @ ( expone5933728560452245707_b_a_a @ Y4 @ Z2 ) ) ) ) ).

% curry_morphism
thf(fact_537_curry__morphism,axiom,
    ! [X4: quasi_7334096685896923387_nat_b,Y4: quasi_borel_a,Z2: quasi_borel_a] : ( member4037896789923528630_b_a_a @ produc4332919034239082454_b_a_a @ ( qbs_mo4463654274022106532_b_a_a @ ( expone405746273668894395_b_a_a @ ( binary4113689719790979332at_b_a @ X4 @ Y4 ) @ Z2 ) @ ( expone5933728560452245707_b_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) ) ) ).

% curry_morphism
thf(fact_538_curry__morphism,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_nat_b,Z2: quasi_borel_a_a] : ( member2618646215268313140_b_a_a @ produc75761518356490992_b_a_a @ ( qbs_mo988760484062759898_b_a_a @ ( expone5933728560452245707_b_a_a @ ( binary7246370547177875303_nat_b @ X4 @ Y4 ) @ Z2 ) @ ( expone7803431766090835025_b_a_a @ X4 @ ( expone932865709032713778_b_a_a @ Y4 @ Z2 ) ) ) ) ).

% curry_morphism
thf(fact_539_curry__morphism,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b,Z2: quasi_7934145443646537599_b_a_a] : ( member1723069925862156404_b_a_a @ produc3697779342064585943_b_a_a @ ( qbs_mo3779576572135142810_b_a_a @ ( expone1622171987252446258_b_a_a @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) @ ( expone5458538795283732920_b_a_a @ X4 @ ( expone5294601226626914677_b_a_a @ Y4 @ Z2 ) ) ) ) ).

% curry_morphism
thf(fact_540_curry__morphism,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_3364728345918372901_b_a_a,Z2: quasi_borel_a] : ( member1660163903167422968_a_a_a @ produc7311142274493443890_a_a_a @ ( qbs_mo4033728420622776990_a_a_a @ ( expone790792417327382575_a_a_a @ ( binary9021634363669778414_b_a_a @ X4 @ Y4 ) @ Z2 ) @ ( expone5702679380600246409_a_a_a @ X4 @ ( expone7391014374654541464_a_a_a @ Y4 @ Z2 ) ) ) ) ).

% curry_morphism
thf(fact_541_curryI,axiom,
    ! [F: product_prod_real_b > $o,A: real,B: b] :
      ( ( F @ ( product_Pair_real_b @ A @ B ) )
     => ( produc1170039606973775568al_b_o @ F @ A @ B ) ) ).

% curryI
thf(fact_542_curryI,axiom,
    ! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( produc1310100445399344235_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_543_curryI,axiom,
    ! [F: produc7194034526493574155_nat_b > $o,A: nat,B: nat > b] :
      ( ( F @ ( produc5292568363641424317_nat_b @ A @ B ) )
     => ( produc9135650966359124045at_b_o @ F @ A @ B ) ) ).

% curryI
thf(fact_544_curryI,axiom,
    ! [F: produc778026099599033348at_b_a > $o,A: b > produc7194034526493574155_nat_b > a > a,B: produc4883615123598087894at_b_a] :
      ( ( F @ ( produc653446490150804990at_b_a @ A @ B ) )
     => ( produc490538171352226380_b_a_o @ F @ A @ B ) ) ).

% curryI
thf(fact_545_curryI,axiom,
    ! [F: produc4883615123598087894at_b_a > $o,A: b,B: produc6209191202727158440at_b_a] :
      ( ( F @ ( produc992390952543482448at_b_a @ A @ B ) )
     => ( produc3283929926326352634_b_a_o @ F @ A @ B ) ) ).

% curryI
thf(fact_546_curryI,axiom,
    ! [F: produc6209191202727158440at_b_a > $o,A: produc7194034526493574155_nat_b,B: a] :
      ( ( F @ ( produc8509509857186669914at_b_a @ A @ B ) )
     => ( produc1535869559436854384_b_a_o @ F @ A @ B ) ) ).

% curryI
thf(fact_547_curryI,axiom,
    ! [F: produc3260441645384719156_nat_b > $o,A: product_prod_real_b,B: produc7194034526493574155_nat_b] :
      ( ( F @ ( produc1946830598933159916_nat_b @ A @ B ) )
     => ( produc8029516459795763192at_b_o @ F @ A @ B ) ) ).

% curryI
thf(fact_548_case__prod__curry,axiom,
    ! [F: produc3260441645384719156_nat_b > a > a] :
      ( ( produc8421786640700244582_b_a_a @ ( produc2364549660447693317_b_a_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_549_case__prod__curry,axiom,
    ! [F: produc6209191202727158440at_b_a > a] :
      ( ( produc2543982510095308599_b_a_a @ ( produc4332919034239082454_b_a_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_550_case__prod__curry,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a > a] :
      ( ( produc7016990068875982646_b_a_a @ ( produc3697779342064585943_b_a_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_551_case__prod__curry,axiom,
    ! [F: produc778026099599033348at_b_a > a] :
      ( ( produc286368409806729619_b_a_a @ ( produc4590375859809585074_b_a_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_552_case__prod__curry,axiom,
    ! [F: produc4883615123598087894at_b_a > a] :
      ( ( produc6302854954412089281_b_a_a @ ( produc5925076442738454880_b_a_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_553_case__prod__curry,axiom,
    ! [F: product_prod_real_b > a > a] :
      ( ( produc7827042608974387084_b_a_a @ ( produc3249657668307812653_b_a_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_554_case__prod__curry,axiom,
    ! [F: produc3260441645384719156_nat_b > a] :
      ( ( produc634687561522189117at_b_a @ ( produc8635972298260719454at_b_a @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_555_curry__case__prod,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a > a] :
      ( ( produc2364549660447693317_b_a_a @ ( produc8421786640700244582_b_a_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_556_curry__case__prod,axiom,
    ! [F: produc7194034526493574155_nat_b > a > a] :
      ( ( produc4332919034239082454_b_a_a @ ( produc2543982510095308599_b_a_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_557_curry__case__prod,axiom,
    ! [F: real > b > produc7194034526493574155_nat_b > a > a] :
      ( ( produc3697779342064585943_b_a_a @ ( produc7016990068875982646_b_a_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_558_curry__case__prod,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a] :
      ( ( produc4590375859809585074_b_a_a @ ( produc286368409806729619_b_a_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_559_curry__case__prod,axiom,
    ! [F: b > produc6209191202727158440at_b_a > a] :
      ( ( produc5925076442738454880_b_a_a @ ( produc6302854954412089281_b_a_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_560_curry__case__prod,axiom,
    ! [F: real > b > a > a] :
      ( ( produc3249657668307812653_b_a_a @ ( produc7827042608974387084_b_a_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_561_curry__case__prod,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a] :
      ( ( produc8635972298260719454at_b_a @ ( produc634687561522189117at_b_a @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_562__C1_C,axiom,
    ( member_real_nat_b @ alpha
    @ ( qbs_Mx_nat_b
      @ ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ na )
        @ ^ [I: nat] : x ) ) ) ).

% "1"
thf(fact_563_qbs__Mx__R,axiom,
    ! [X4: sigma_6586933000485258589at_b_a] :
      ( ( qbs_Mx5919617464495388621at_b_a @ ( measur2292414412926267568at_b_a @ X4 ) )
      = ( sigma_3101138000788093954at_b_a @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_564_qbs__Mx__R,axiom,
    ! [X4: sigma_measure_nat_b] :
      ( ( qbs_Mx_nat_b @ ( measur7835506533441958586_nat_b @ X4 ) )
      = ( sigma_6425555513962012904_nat_b @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_565_qbs__Mx__R,axiom,
    ! [X4: sigma_3742508075133793660_b_a_a] :
      ( ( qbs_Mx703024977670096246_b_a_a @ ( measur1881545681227145939_b_a_a @ X4 ) )
      = ( sigma_8612542618026319105_b_a_a @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_566_qbs__Mx__R,axiom,
    ! [X4: sigma_measure_b] :
      ( ( qbs_Mx_b @ ( measur6507891955840068947_qbs_b @ X4 ) )
      = ( sigma_523072396149930113real_b @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_567_qbs__Mx__R,axiom,
    ! [X4: sigma_measure_a] :
      ( ( qbs_Mx_a @ ( measur6507891955840068946_qbs_a @ X4 ) )
      = ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_568_qbs__Mx__R,axiom,
    ! [X4: sigma_2308072346491277622l_real] :
      ( ( qbs_Mx2863504943393711152l_real @ ( measur3029786804310284173l_real @ X4 ) )
      = ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_569_qbs__Mx__R,axiom,
    ! [X4: sigma_5310753476256395226t_real] :
      ( ( qbs_Mx6884344600386557780t_real @ ( measur2011440102552004913t_real @ X4 ) )
      = ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_570_qbs__Mx__R,axiom,
    ! [X4: sigma_measure_real] :
      ( ( qbs_Mx_real @ ( measur6875533127466166616s_real @ X4 ) )
      = ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_571_qbs__Mx__R,axiom,
    ! [X4: sigma_measure_o] :
      ( ( qbs_Mx_o @ ( measur2705496967258476524_qbs_o @ X4 ) )
      = ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_572_qbs__Mx__R,axiom,
    ! [X4: sigma_measure_nat] :
      ( ( qbs_Mx_nat @ ( measur4416158800429964412bs_nat @ X4 ) )
      = ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ X4 ) ) ).

% qbs_Mx_R
thf(fact_573_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X5: produc4471711990508489141at_nat] :
      ~ ! [F3: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
          ( X5
         != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_574_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X5: produc7529714137711950045_nat_b] :
      ~ ! [F3: nat > ( nat > b ) > nat > b,A3: nat,B3: nat,Acc: nat > b] :
          ( X5
         != ( produc7899952307785402255_nat_b @ F3 @ ( produc9037676286296983540_nat_b @ A3 @ ( produc5292568363641424317_nat_b @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_575_curryD,axiom,
    ! [F: product_prod_real_b > $o,A: real,B: b] :
      ( ( produc1170039606973775568al_b_o @ F @ A @ B )
     => ( F @ ( product_Pair_real_b @ A @ B ) ) ) ).

% curryD
thf(fact_576_curryD,axiom,
    ! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryD
thf(fact_577_curryD,axiom,
    ! [F: produc7194034526493574155_nat_b > $o,A: nat,B: nat > b] :
      ( ( produc9135650966359124045at_b_o @ F @ A @ B )
     => ( F @ ( produc5292568363641424317_nat_b @ A @ B ) ) ) ).

% curryD
thf(fact_578_curryD,axiom,
    ! [F: produc778026099599033348at_b_a > $o,A: b > produc7194034526493574155_nat_b > a > a,B: produc4883615123598087894at_b_a] :
      ( ( produc490538171352226380_b_a_o @ F @ A @ B )
     => ( F @ ( produc653446490150804990at_b_a @ A @ B ) ) ) ).

% curryD
thf(fact_579_curryD,axiom,
    ! [F: produc4883615123598087894at_b_a > $o,A: b,B: produc6209191202727158440at_b_a] :
      ( ( produc3283929926326352634_b_a_o @ F @ A @ B )
     => ( F @ ( produc992390952543482448at_b_a @ A @ B ) ) ) ).

% curryD
thf(fact_580_curryD,axiom,
    ! [F: produc6209191202727158440at_b_a > $o,A: produc7194034526493574155_nat_b,B: a] :
      ( ( produc1535869559436854384_b_a_o @ F @ A @ B )
     => ( F @ ( produc8509509857186669914at_b_a @ A @ B ) ) ) ).

% curryD
thf(fact_581_curryD,axiom,
    ! [F: produc3260441645384719156_nat_b > $o,A: product_prod_real_b,B: produc7194034526493574155_nat_b] :
      ( ( produc8029516459795763192at_b_o @ F @ A @ B )
     => ( F @ ( produc1946830598933159916_nat_b @ A @ B ) ) ) ).

% curryD
thf(fact_582_curryE,axiom,
    ! [F: product_prod_real_b > $o,A: real,B: b] :
      ( ( produc1170039606973775568al_b_o @ F @ A @ B )
     => ( F @ ( product_Pair_real_b @ A @ B ) ) ) ).

% curryE
thf(fact_583_curryE,axiom,
    ! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryE
thf(fact_584_curryE,axiom,
    ! [F: produc7194034526493574155_nat_b > $o,A: nat,B: nat > b] :
      ( ( produc9135650966359124045at_b_o @ F @ A @ B )
     => ( F @ ( produc5292568363641424317_nat_b @ A @ B ) ) ) ).

% curryE
thf(fact_585_curryE,axiom,
    ! [F: produc778026099599033348at_b_a > $o,A: b > produc7194034526493574155_nat_b > a > a,B: produc4883615123598087894at_b_a] :
      ( ( produc490538171352226380_b_a_o @ F @ A @ B )
     => ( F @ ( produc653446490150804990at_b_a @ A @ B ) ) ) ).

% curryE
thf(fact_586_curryE,axiom,
    ! [F: produc4883615123598087894at_b_a > $o,A: b,B: produc6209191202727158440at_b_a] :
      ( ( produc3283929926326352634_b_a_o @ F @ A @ B )
     => ( F @ ( produc992390952543482448at_b_a @ A @ B ) ) ) ).

% curryE
thf(fact_587_curryE,axiom,
    ! [F: produc6209191202727158440at_b_a > $o,A: produc7194034526493574155_nat_b,B: a] :
      ( ( produc1535869559436854384_b_a_o @ F @ A @ B )
     => ( F @ ( produc8509509857186669914at_b_a @ A @ B ) ) ) ).

% curryE
thf(fact_588_curryE,axiom,
    ! [F: produc3260441645384719156_nat_b > $o,A: product_prod_real_b,B: produc7194034526493574155_nat_b] :
      ( ( produc8029516459795763192at_b_o @ F @ A @ B )
     => ( F @ ( produc1946830598933159916_nat_b @ A @ B ) ) ) ).

% curryE
thf(fact_589_qbs__closed1__dest,axiom,
    ! [Alpha2: real > real,X4: quasi_borel_real,F: real > real] :
      ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X4 ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_real @ ( comp_real_real_real @ Alpha2 @ F ) @ ( qbs_Mx_real @ X4 ) ) ) ) ).

% qbs_closed1_dest
thf(fact_590_qbs__closed1__dest,axiom,
    ! [Alpha2: real > produc778026099599033348at_b_a,X4: quasi_331246355085487092at_b_a,F: real > real] :
      ( ( member8314179565955891814at_b_a @ Alpha2 @ ( qbs_Mx5919617464495388621at_b_a @ X4 ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member8314179565955891814at_b_a @ ( comp_r2834535067444373313a_real @ Alpha2 @ F ) @ ( qbs_Mx5919617464495388621at_b_a @ X4 ) ) ) ) ).

% qbs_closed1_dest
thf(fact_591_qbs__closed1__dest,axiom,
    ! [Alpha2: real > nat > b,X4: quasi_borel_nat_b,F: real > real] :
      ( ( member_real_nat_b @ Alpha2 @ ( qbs_Mx_nat_b @ X4 ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_nat_b @ ( comp_real_nat_b_real @ Alpha2 @ F ) @ ( qbs_Mx_nat_b @ X4 ) ) ) ) ).

% qbs_closed1_dest
thf(fact_592_qbs__closed1__dest,axiom,
    ! [Alpha2: real > b > produc7194034526493574155_nat_b > a > a,X4: quasi_3364728345918372901_b_a_a,F: real > real] :
      ( ( member924168396124881715_b_a_a @ Alpha2 @ ( qbs_Mx703024977670096246_b_a_a @ X4 ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member924168396124881715_b_a_a @ ( comp_r2587692399004186460a_real @ Alpha2 @ F ) @ ( qbs_Mx703024977670096246_b_a_a @ X4 ) ) ) ) ).

% qbs_closed1_dest
thf(fact_593_qbs__closed1__dest,axiom,
    ! [Alpha2: real > b,X4: quasi_borel_b,F: real > real] :
      ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X4 ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_b @ ( comp_real_b_real @ Alpha2 @ F ) @ ( qbs_Mx_b @ X4 ) ) ) ) ).

% qbs_closed1_dest
thf(fact_594_qbs__closed1__dest,axiom,
    ! [Alpha2: real > a,X4: quasi_borel_a,F: real > real] :
      ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X4 ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
       => ( member_real_a @ ( comp_real_a_real @ Alpha2 @ F ) @ ( qbs_Mx_a @ X4 ) ) ) ) ).

% qbs_closed1_dest
thf(fact_595_comp__cong,axiom,
    ! [F: real > produc2422161461964618553l_real,G: produc2422161461964618553l_real > real,X5: produc2422161461964618553l_real,F4: real > produc2422161461964618553l_real,G3: produc2422161461964618553l_real > real,X7: produc2422161461964618553l_real] :
      ( ( ( F @ ( G @ X5 ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_r97482993418849103l_real @ F @ G @ X5 )
        = ( comp_r97482993418849103l_real @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_596_comp__cong,axiom,
    ! [F: real > produc7716430852924023517t_real,G: produc7716430852924023517t_real > real,X5: produc7716430852924023517t_real,F4: real > produc7716430852924023517t_real,G3: produc7716430852924023517t_real > real,X7: produc7716430852924023517t_real] :
      ( ( ( F @ ( G @ X5 ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_r4825950202331181903t_real @ F @ G @ X5 )
        = ( comp_r4825950202331181903t_real @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_597_comp__cong,axiom,
    ! [F: real > real,G: real > real,X5: real,F4: real > real,G3: real > real,X7: real] :
      ( ( ( F @ ( G @ X5 ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_real_real_real @ F @ G @ X5 )
        = ( comp_real_real_real @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_598_comp__cong,axiom,
    ! [F: real > $o,G: $o > real,X5: $o,F4: real > $o,G3: $o > real,X7: $o] :
      ( ( ( F @ ( G @ X5 ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_real_o_o @ F @ G @ X5 )
        = ( comp_real_o_o @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_599_comp__cong,axiom,
    ! [F: real > nat,G: nat > real,X5: nat,F4: real > nat,G3: nat > real,X7: nat] :
      ( ( ( F @ ( G @ X5 ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_real_nat_nat @ F @ G @ X5 )
        = ( comp_real_nat_nat @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_600_top__empty__eq2,axiom,
    ( top_top_real_b_o
    = ( ^ [X: real,Y: b] : ( member7555082154638246367real_b @ ( product_Pair_real_b @ X @ Y ) @ top_to2678547386975664558real_b ) ) ) ).

% top_empty_eq2
thf(fact_601_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_602_top__empty__eq2,axiom,
    ( top_top_nat_nat_b_o
    = ( ^ [X: nat,Y: nat > b] : ( member2882485241872049954_nat_b @ ( produc5292568363641424317_nat_b @ X @ Y ) @ top_to1215373225175873905_nat_b ) ) ) ).

% top_empty_eq2
thf(fact_603_top__empty__eq2,axiom,
    ( top_to1453699796639584719_b_a_o
    = ( ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: produc4883615123598087894at_b_a] : ( member3285024325984547227at_b_a @ ( produc653446490150804990at_b_a @ X @ Y ) @ top_to2317269531069164906at_b_a ) ) ) ).

% top_empty_eq2
thf(fact_604_top__empty__eq2,axiom,
    ( top_to4798699689507688061_b_a_o
    = ( ^ [X: b,Y: produc6209191202727158440at_b_a] : ( member8939744840176901101at_b_a @ ( produc992390952543482448at_b_a @ X @ Y ) @ top_to5016788476086710460at_b_a ) ) ) ).

% top_empty_eq2
thf(fact_605_top__empty__eq2,axiom,
    ( top_to1662857348365567193_b_a_o
    = ( ^ [X: produc7194034526493574155_nat_b,Y: a] : ( member8909807733118965439at_b_a @ ( produc8509509857186669914at_b_a @ X @ Y ) @ top_to5877169193438701198at_b_a ) ) ) ).

% top_empty_eq2
thf(fact_606_top__empty__eq2,axiom,
    ( top_to4029256250312543683at_b_o
    = ( ^ [X: product_prod_real_b,Y: produc7194034526493574155_nat_b] : ( member388886286662666717_nat_b @ ( produc1946830598933159916_nat_b @ X @ Y ) @ top_to1051096394934767716_nat_b ) ) ) ).

% top_empty_eq2
thf(fact_607_top__empty__eq2,axiom,
    ( top_top_nat_real_o2
    = ( ^ [X: nat,Y: real] : ( member557208447399453958t_real @ ( produc7837566107596912789t_real @ X @ Y ) @ top_to4855536200657754381t_real ) ) ) ).

% top_empty_eq2
thf(fact_608_pred__equals__eq2,axiom,
    ! [R4: set_Pr6660261784198275838real_b,S: set_Pr6660261784198275838real_b] :
      ( ( ( ^ [X: real,Y: b] : ( member7555082154638246367real_b @ ( product_Pair_real_b @ X @ Y ) @ R4 ) )
        = ( ^ [X: real,Y: b] : ( member7555082154638246367real_b @ ( product_Pair_real_b @ X @ Y ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_609_pred__equals__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R4 ) )
        = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_610_pred__equals__eq2,axiom,
    ! [R4: set_Pr2992590090106802881_nat_b,S: set_Pr2992590090106802881_nat_b] :
      ( ( ( ^ [X: nat,Y: nat > b] : ( member2882485241872049954_nat_b @ ( produc5292568363641424317_nat_b @ X @ Y ) @ R4 ) )
        = ( ^ [X: nat,Y: nat > b] : ( member2882485241872049954_nat_b @ ( produc5292568363641424317_nat_b @ X @ Y ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_611_pred__equals__eq2,axiom,
    ! [R4: set_Pr7964249709903003322at_b_a,S: set_Pr7964249709903003322at_b_a] :
      ( ( ( ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: produc4883615123598087894at_b_a] : ( member3285024325984547227at_b_a @ ( produc653446490150804990at_b_a @ X @ Y ) @ R4 ) )
        = ( ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: produc4883615123598087894at_b_a] : ( member3285024325984547227at_b_a @ ( produc653446490150804990at_b_a @ X @ Y ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_612_pred__equals__eq2,axiom,
    ! [R4: set_Pr4420982220611316236at_b_a,S: set_Pr4420982220611316236at_b_a] :
      ( ( ( ^ [X: b,Y: produc6209191202727158440at_b_a] : ( member8939744840176901101at_b_a @ ( produc992390952543482448at_b_a @ X @ Y ) @ R4 ) )
        = ( ^ [X: b,Y: produc6209191202727158440at_b_a] : ( member8939744840176901101at_b_a @ ( produc992390952543482448at_b_a @ X @ Y ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_613_pred__equals__eq2,axiom,
    ! [R4: set_Pr7000326631795273694at_b_a,S: set_Pr7000326631795273694at_b_a] :
      ( ( ( ^ [X: produc7194034526493574155_nat_b,Y: a] : ( member8909807733118965439at_b_a @ ( produc8509509857186669914at_b_a @ X @ Y ) @ R4 ) )
        = ( ^ [X: produc7194034526493574155_nat_b,Y: a] : ( member8909807733118965439at_b_a @ ( produc8509509857186669914at_b_a @ X @ Y ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_614_pred__equals__eq2,axiom,
    ! [R4: set_Pr196252041547729684_nat_b,S: set_Pr196252041547729684_nat_b] :
      ( ( ( ^ [X: product_prod_real_b,Y: produc7194034526493574155_nat_b] : ( member388886286662666717_nat_b @ ( produc1946830598933159916_nat_b @ X @ Y ) @ R4 ) )
        = ( ^ [X: product_prod_real_b,Y: produc7194034526493574155_nat_b] : ( member388886286662666717_nat_b @ ( produc1946830598933159916_nat_b @ X @ Y ) @ S ) ) )
      = ( R4 = S ) ) ).

% pred_equals_eq2
thf(fact_615_curry__preserves__morphisms,axiom,
    ! [F: produc1285742814635146737at_b_a > a,X4: quasi_8132950864781310810_nat_b,Y4: quasi_borel_a,Z2: quasi_borel_a] :
      ( ( member9091339440257553353_b_a_a @ F @ ( qbs_mo7901260496052035013_b_a_a @ ( binary4006988140340982583at_b_a @ X4 @ Y4 ) @ Z2 ) )
     => ( member7748159150162262195_b_a_a @ ( produc1350391050512725289_b_a_a @ F ) @ ( qbs_mo6578283766602973217_b_a_a @ X4 @ ( exponent_exp_qbs_a_a @ Y4 @ Z2 ) ) ) ) ).

% curry_preserves_morphisms
thf(fact_616_curry__preserves__morphisms,axiom,
    ! [F: produc91719739528564329_b_a_a > a,X4: quasi_borel_nat_b,Y4: quasi_6353345141195568462_b_a_a,Z2: quasi_borel_a] :
      ( ( member1981363422186161233_a_a_a @ F @ ( qbs_mo7915889915729830349_a_a_a @ ( binary1958873873046607663_b_a_a @ X4 @ Y4 ) @ Z2 ) )
     => ( member1811236362415105851_a_a_a @ ( produc1757743712678919857_a_a_a @ F ) @ ( qbs_mo8618518167108125737_a_a_a @ X4 @ ( expone790792417327382575_a_a_a @ Y4 @ Z2 ) ) ) ) ).

% curry_preserves_morphisms
thf(fact_617_curry__preserves__morphisms,axiom,
    ! [F: product_prod_o_o > $o,X4: quasi_borel_o,Y4: quasi_borel_o,Z2: quasi_borel_o] :
      ( ( member5279437561400393645_o_o_o @ F @ ( qbs_mo6404984969470912169_o_o_o @ ( binary9159730317414444269bs_o_o @ X4 @ Y4 ) @ Z2 ) )
     => ( member_o_o_o @ ( product_curry_o_o_o @ F ) @ ( qbs_morphism_o_o_o @ X4 @ ( exponent_exp_qbs_o_o @ Y4 @ Z2 ) ) ) ) ).

% curry_preserves_morphisms
thf(fact_618_curry__preserves__morphisms,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,X4: quasi_borel_real,Y4: quasi_borel_b,Z2: quasi_7934145443646537599_b_a_a] :
      ( ( member1306660226910552845_b_a_a @ F @ ( qbs_mo3736047266677648337_b_a_a @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) )
     => ( member924168396124881715_b_a_a @ ( produc3697779342064585943_b_a_a @ F ) @ ( qbs_mo1140304683536899929_b_a_a @ X4 @ ( expone5294601226626914677_b_a_a @ Y4 @ Z2 ) ) ) ) ).

% curry_preserves_morphisms
thf(fact_619_real_Oqbs__morphism__dest,axiom,
    ! [K: real > a,Y4: sigma_measure_a] :
      ( ( member_real_a @ K @ ( qbs_morphism_real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6507891955840068946_qbs_a @ Y4 ) ) )
     => ( member_real_a @ K @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ Y4 ) ) ) ).

% real.qbs_morphism_dest
thf(fact_620_real_Oqbs__morphism__dest,axiom,
    ! [K: real > produc2422161461964618553l_real,Y4: sigma_2308072346491277622l_real] :
      ( ( member9086635009091248365l_real @ K @ ( qbs_mo6845038431372961811l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur3029786804310284173l_real @ Y4 ) ) )
     => ( member9086635009091248365l_real @ K @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ Y4 ) ) ) ).

% real.qbs_morphism_dest
thf(fact_621_real_Oqbs__morphism__dest,axiom,
    ! [K: real > produc7716430852924023517t_real,Y4: sigma_5310753476256395226t_real] :
      ( ( member969486235890772753t_real @ K @ ( qbs_mo3601394221918002359t_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2011440102552004913t_real @ Y4 ) ) )
     => ( member969486235890772753t_real @ K @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ Y4 ) ) ) ).

% real.qbs_morphism_dest
thf(fact_622_real_Oqbs__morphism__dest,axiom,
    ! [K: real > real,Y4: sigma_measure_real] :
      ( ( member_real_real @ K @ ( qbs_mo5229770564518008146l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6875533127466166616s_real @ Y4 ) ) )
     => ( member_real_real @ K @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ Y4 ) ) ) ).

% real.qbs_morphism_dest
thf(fact_623_real_Oqbs__morphism__dest,axiom,
    ! [K: real > $o,Y4: sigma_measure_o] :
      ( ( member_real_o @ K @ ( qbs_morphism_real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) )
     => ( member_real_o @ K @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ Y4 ) ) ) ).

% real.qbs_morphism_dest
thf(fact_624_real_Oqbs__morphism__dest,axiom,
    ! [K: real > nat,Y4: sigma_measure_nat] :
      ( ( member_real_nat @ K @ ( qbs_mo6567951568834356598al_nat @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) )
     => ( member_real_nat @ K @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ Y4 ) ) ) ).

% real.qbs_morphism_dest
thf(fact_625_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_a] :
      ( ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ Y4 )
      = ( qbs_morphism_real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6507891955840068946_qbs_a @ Y4 ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_626_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_2308072346491277622l_real] :
      ( ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ Y4 )
      = ( qbs_mo6845038431372961811l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur3029786804310284173l_real @ Y4 ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_627_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_5310753476256395226t_real] :
      ( ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ Y4 )
      = ( qbs_mo3601394221918002359t_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2011440102552004913t_real @ Y4 ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_628_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_real] :
      ( ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ Y4 )
      = ( qbs_mo5229770564518008146l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6875533127466166616s_real @ Y4 ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_629_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_o] :
      ( ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ Y4 )
      = ( qbs_morphism_real_o @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_630_real_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_nat] :
      ( ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ Y4 )
      = ( qbs_mo6567951568834356598al_nat @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) ) ).

% real.standard_borel_r_full_faithful
thf(fact_631__C2_C,axiom,
    ! [R3: real] :
      ( member2882485241872049954_nat_b @ ( produc5292568363641424317_nat_b @ na @ ( alpha @ R3 ) )
      @ ( qbs_sp2305461467746415189_nat_b
        @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
          @ ^ [N: nat] :
              ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
              @ ^ [I: nat] : x ) ) ) ) ).

% "2"
thf(fact_632_borel__measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member9086635009091248365l_real
          @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_633_borel__measurable__Pair,axiom,
    ! [F: $o > real,M: sigma_measure_o,G: $o > real] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_o_real @ G @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
       => ( member8602590400058854173l_real
          @ ^ [X: $o] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_2489502367751881329l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_634_borel__measurable__Pair,axiom,
    ! [F: nat > real,M: sigma_measure_nat,G: nat > real] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
       => ( member4944736856719700937l_real
          @ ^ [X: nat] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5893925349391517463l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_635_borel__measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > nat] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member4159480375084375953al_nat
          @ ^ [X: real] : ( produc3181502643871035669al_nat @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_8223622787033862879al_nat @ M @ borel_8300761668267423611al_nat ) ) ) ) ).

% borel_measurable_Pair
thf(fact_636_borel__measurable__Pair,axiom,
    ! [F: $o > real,M: sigma_measure_o,G: $o > nat] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_o_nat @ G @ ( sigma_1999164137574644376_o_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member2706604780226699713al_nat
          @ ^ [X: $o] : ( produc3181502643871035669al_nat @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5929096572201269909al_nat @ M @ borel_8300761668267423611al_nat ) ) ) ) ).

% borel_measurable_Pair
thf(fact_637_borel__measurable__Pair,axiom,
    ! [F: nat > real,M: sigma_measure_nat,G: nat > nat] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_nat_nat @ G @ ( sigma_4350458207664084850at_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member7427358445142253677al_nat
          @ ^ [X: nat] : ( produc3181502643871035669al_nat @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_4775202647280382523al_nat @ M @ borel_8300761668267423611al_nat ) ) ) ) ).

% borel_measurable_Pair
thf(fact_638_borel__measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > extend8495563244428889912nnreal] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member4263462704469908217nnreal
          @ ^ [X: real] : ( produc4778015194254607485nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_1079880282724612679nnreal @ M @ borel_4974289084073311971nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_639_borel__measurable__Pair,axiom,
    ! [F: $o > real,M: sigma_measure_o,G: $o > extend8495563244428889912nnreal] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5265953103328284778nnreal @ G @ ( sigma_6459699357617223168nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member8012821517253025065nnreal
          @ ^ [X: $o] : ( produc4778015194254607485nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_8212496664001311229nnreal @ M @ borel_4974289084073311971nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_640_borel__measurable__Pair,axiom,
    ! [F: nat > real,M: sigma_measure_nat,G: nat > extend8495563244428889912nnreal] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member8283130129095025342nnreal @ G @ ( sigma_6306161311797543642nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member1617991602746293205nnreal
          @ ^ [X: nat] : ( produc4778015194254607485nnreal @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_4192619376674453923nnreal @ M @ borel_4974289084073311971nnreal ) ) ) ) ).

% borel_measurable_Pair
thf(fact_641_borel__measurable__Pair,axiom,
    ! [F: $o > nat,M: sigma_measure_o,G: $o > real] :
      ( ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_o_real @ G @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
       => ( member8739982677887872321t_real
          @ ^ [X: $o] : ( produc7837566107596912789t_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_680772226823374357t_real @ M @ borel_3052437322889528059t_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_642_h,axiom,
    ( member_real_nat_b @ alpha2
    @ ( qbs_Mx_nat_b
      @ ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ ( suc @ na ) )
        @ ^ [I: nat] : x ) ) ) ).

% h
thf(fact_643_ih,axiom,
    ( member1811236362415105851_a_a_a
    @ ^ [X: nat > b,Y: produc432120507726958376_b_a_a] : ( coProd5680178011022504904st_a_b @ ( produc5967371699169076996_b_a_a @ Y ) @ ( produc7355951146076385094_b_a_a @ Y ) @ ( produc5292568363641424317_nat_b @ na @ X ) )
    @ ( qbs_mo8618518167108125737_a_a_a
      @ ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ na )
        @ ^ [I: nat] : x )
      @ ( expone790792417327382575_a_a_a
        @ ( binary9021634363669778414_b_a_a @ y
          @ ( expone5294601226626914677_b_a_a @ x
            @ ( expone5933728560452245707_b_a_a
              @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
                @ ^ [N: nat] :
                    ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                    @ ^ [I: nat] : x ) )
              @ ( exponent_exp_qbs_a_a @ y @ y ) ) ) )
        @ y ) ) ) ).

% ih
thf(fact_644__092_060open_062_I_092_060lambda_062_Ir_M_Ay_J_O_Asnd_Ay_A_Ia_Ar_J_A_In_M_A_092_060alpha_062_H_Ar_J_A_Irec__list_H_A_Ifst_Ay_J_A_Isnd_Ay_J_A_In_M_A_092_060alpha_062_H_Ar_J_J_J_A_061_A_I_092_060lambda_062_Iy_M_Ax1_M_Ax2_M_Ax3_J_O_Ay_Ax1_Ax2_Ax3_J_A_092_060circ_062_A_I_092_060lambda_062_Ir_M_Ay_J_O_A_Isnd_Ay_M_Aa_Ar_M_A_In_M_A_092_060alpha_062_H_Ar_J_M_Arec__list_H_A_Ifst_Ay_J_A_Isnd_Ay_J_A_In_M_A_092_060alpha_062_H_Ar_J_J_J_092_060close_062,axiom,
    ( ( produc2426372561900375854_a_a_a
      @ ^ [R: real,Y: produc432120507726958376_b_a_a] : ( produc7355951146076385094_b_a_a @ Y @ ( a2 @ R ) @ ( produc5292568363641424317_nat_b @ na @ ( alpha @ R ) ) @ ( coProd5680178011022504904st_a_b @ ( produc5967371699169076996_b_a_a @ Y ) @ ( produc7355951146076385094_b_a_a @ Y ) @ ( produc5292568363641424317_nat_b @ na @ ( alpha @ R ) ) ) ) )
    = ( comp_P6156462644269672904_b_a_a
      @ ( produc286368409806729619_b_a_a
        @ ^ [Y: b > produc7194034526493574155_nat_b > a > a] :
            ( produc6302854954412089281_b_a_a
            @ ^ [X12: b] : ( produc2543982510095308599_b_a_a @ ( Y @ X12 ) ) ) )
      @ ( produc3647828067904035796at_b_a
        @ ^ [R: real,Y: produc432120507726958376_b_a_a] : ( produc653446490150804990at_b_a @ ( produc7355951146076385094_b_a_a @ Y ) @ ( produc992390952543482448at_b_a @ ( a2 @ R ) @ ( produc8509509857186669914at_b_a @ ( produc5292568363641424317_nat_b @ na @ ( alpha @ R ) ) @ ( coProd5680178011022504904st_a_b @ ( produc5967371699169076996_b_a_a @ Y ) @ ( produc7355951146076385094_b_a_a @ Y ) @ ( produc5292568363641424317_nat_b @ na @ ( alpha @ R ) ) ) ) ) ) ) ) ) ).

% \<open>(\<lambda>(r, y). snd y (a r) (n, \<alpha>' r) (rec_list' (fst y) (snd y) (n, \<alpha>' r))) = (\<lambda>(y, x1, x2, x3). y x1 x2 x3) \<circ> (\<lambda>(r, y). (snd y, a r, (n, \<alpha>' r), rec_list' (fst y) (snd y) (n, \<alpha>' r)))\<close>
thf(fact_645_measurable__product__then__coordinatewise,axiom,
    ! [F: $o > $o > $o,M: sigma_measure_o,I3: $o] :
      ( ( member_o_o_o @ F @ ( sigma_1874301400886517549_o_o_o @ M @ borel_2241930036368985287el_o_o ) )
     => ( member_o_o
        @ ^ [X: $o] : ( F @ X @ I3 )
        @ ( sigma_measurable_o_o @ M @ borel_5500255247093592246orel_o ) ) ) ).

% measurable_product_then_coordinatewise
thf(fact_646_borel__measurable__const,axiom,
    ! [C: $o > $o,M: sigma_measure_o] :
      ( member_o_o_o
      @ ^ [X: $o] : C
      @ ( sigma_1874301400886517549_o_o_o @ M @ borel_2241930036368985287el_o_o ) ) ).

% borel_measurable_const
thf(fact_647_borel__measurable__const,axiom,
    ! [C: real,M: sigma_measure_real] :
      ( member_real_real
      @ ^ [X: real] : C
      @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_648_borel__measurable__const,axiom,
    ! [C: real,M: sigma_measure_o] :
      ( member_o_real
      @ ^ [X: $o] : C
      @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_649_borel__measurable__const,axiom,
    ! [C: real,M: sigma_measure_nat] :
      ( member_nat_real
      @ ^ [X: nat] : C
      @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_650_borel__measurable__const,axiom,
    ! [C: $o,M: sigma_measure_o] :
      ( member_o_o
      @ ^ [X: $o] : C
      @ ( sigma_measurable_o_o @ M @ borel_5500255247093592246orel_o ) ) ).

% borel_measurable_const
thf(fact_651_borel__measurable__const,axiom,
    ! [C: $o,M: sigma_measure_real] :
      ( member_real_o
      @ ^ [X: real] : C
      @ ( sigma_3939073009482781210real_o @ M @ borel_5500255247093592246orel_o ) ) ).

% borel_measurable_const
thf(fact_652_borel__measurable__const,axiom,
    ! [C: nat,M: sigma_measure_real] :
      ( member_real_nat
      @ ^ [X: real] : C
      @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) ) ).

% borel_measurable_const
thf(fact_653_exp__qbs__correct,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_a] :
      ( ( quasi_347859638729175488real_a @ ( expone319700646606540215real_a @ X4 @ Y4 ) )
      = ( produc8792329689427759714real_a @ ( qbs_morphism_real_a @ X4 @ Y4 ) @ ( expone6795215201378684364real_a @ X4 @ Y4 ) ) ) ).

% exp_qbs_correct
thf(fact_654_exp__qbs__correct,axiom,
    ! [X4: quasi_805850127037308055at_b_a,Y4: quasi_borel_a] :
      ( ( quasi_376539980986154811_b_a_a @ ( expone4627587591657327526_b_a_a @ X4 @ Y4 ) )
      = ( produc2067008855343870260_b_a_a @ ( qbs_mo7901260496052035013_b_a_a @ X4 @ Y4 ) @ ( expone4687418993921882065_b_a_a @ X4 @ Y4 ) ) ) ).

% exp_qbs_correct
thf(fact_655_exp__qbs__correct,axiom,
    ! [X4: quasi_borel_nat_b,Y4: quasi_2547937147209033259_a_a_a] :
      ( ( quasi_2727095950656280649_a_a_a @ ( expone6720796868436823688_a_a_a @ X4 @ Y4 ) )
      = ( produc1611076569815217314_a_a_a @ ( qbs_mo8618518167108125737_a_a_a @ X4 @ Y4 ) @ ( expone2444466320815533341_a_a_a @ X4 @ Y4 ) ) ) ).

% exp_qbs_correct
thf(fact_656_exp__qbs__correct,axiom,
    ! [X4: quasi_borel_o,Y4: quasi_borel_o_o] :
      ( ( quasi_5866374320985554469_o_o_o @ ( expone1670929651343127652_o_o_o @ X4 @ Y4 ) )
      = ( produc523828491365516066_o_o_o @ ( qbs_morphism_o_o_o @ X4 @ Y4 ) @ ( expone8146444206115271801_o_o_o @ X4 @ Y4 ) ) ) ).

% exp_qbs_correct
thf(fact_657_exp__qbs__correct,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_7934145443646537599_b_a_a] :
      ( ( quasi_6770997997730906934_b_a_a @ ( expone5294601226626914677_b_a_a @ X4 @ Y4 ) )
      = ( produc5264547532651997410_b_a_a @ ( qbs_mo1542099550572995862_b_a_a @ X4 @ Y4 ) @ ( expone2961370955173070346_b_a_a @ X4 @ Y4 ) ) ) ).

% exp_qbs_correct
thf(fact_658_exp__qbs__correct,axiom,
    ! [X4: quasi_7334096685896923387_nat_b,Y4: quasi_borel_a_a] :
      ( ( quasi_8461169777907420952_b_a_a @ ( expone5933728560452245707_b_a_a @ X4 @ Y4 ) )
      = ( produc3806978686075393262_b_a_a @ ( qbs_mo7242095808399995242_b_a_a @ X4 @ Y4 ) @ ( expone1541841343128190326_b_a_a @ X4 @ Y4 ) ) ) ).

% exp_qbs_correct
thf(fact_659_exp__qbs__correct,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_a] :
      ( ( quasi_8710943612200540014el_a_a @ ( exponent_exp_qbs_a_a @ X4 @ Y4 ) )
      = ( produc8176344774097317018al_a_a @ ( qbs_morphism_a_a @ X4 @ Y4 ) @ ( expone7022597896679715716Mx_a_a @ X4 @ Y4 ) ) ) ).

% exp_qbs_correct
thf(fact_660_exp__qbs__correct,axiom,
    ! [X4: quasi_6353345141195568462_b_a_a,Y4: quasi_borel_a] :
      ( ( quasi_389993656651803076_a_a_a @ ( expone790792417327382575_a_a_a @ X4 @ Y4 ) )
      = ( produc9059353793759839558_a_a_a @ ( qbs_mo5048501348360587598_a_a_a @ X4 @ Y4 ) @ ( expone2990877755589709658_a_a_a @ X4 @ Y4 ) ) ) ).

% exp_qbs_correct
thf(fact_661_exp__qbs__correct,axiom,
    ! [X4: quasi_borel_o,Y4: quasi_borel_o] :
      ( ( quasi_80751349119011438el_o_o @ ( exponent_exp_qbs_o_o @ X4 @ Y4 ) )
      = ( produc1980721420443017882al_o_o @ ( qbs_morphism_o_o @ X4 @ Y4 ) @ ( expone9009657249589208708Mx_o_o @ X4 @ Y4 ) ) ) ).

% exp_qbs_correct
thf(fact_662_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K: $o > $o > $o,Y4: sigma_measure_o_o] :
      ( ( member_o_o_o @ K @ ( sigma_1874301400886517549_o_o_o @ borel_5500255247093592246orel_o @ Y4 ) )
     => ( member_o_o_o @ K @ ( qbs_morphism_o_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur6002702770210135313bs_o_o @ Y4 ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_663_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K: $o > produc2422161461964618553l_real,Y4: sigma_2308072346491277622l_real] :
      ( ( member8602590400058854173l_real @ K @ ( sigma_2489502367751881329l_real @ borel_5500255247093592246orel_o @ Y4 ) )
     => ( member8602590400058854173l_real @ K @ ( qbs_mo6076292073755557401l_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur3029786804310284173l_real @ Y4 ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_664_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K: $o > produc7716430852924023517t_real,Y4: sigma_5310753476256395226t_real] :
      ( ( member8739982677887872321t_real @ K @ ( sigma_680772226823374357t_real @ borel_5500255247093592246orel_o @ Y4 ) )
     => ( member8739982677887872321t_real @ K @ ( qbs_mo928686258813670845t_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2011440102552004913t_real @ Y4 ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_665_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K: $o > real,Y4: sigma_measure_real] :
      ( ( member_o_real @ K @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ Y4 ) )
     => ( member_o_real @ K @ ( qbs_morphism_o_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur6875533127466166616s_real @ Y4 ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_666_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K: $o > $o,Y4: sigma_measure_o] :
      ( ( member_o_o @ K @ ( sigma_measurable_o_o @ borel_5500255247093592246orel_o @ Y4 ) )
     => ( member_o_o @ K @ ( qbs_morphism_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_667_bool_Oqbs__morphism__measurable__intro,axiom,
    ! [K: $o > nat,Y4: sigma_measure_nat] :
      ( ( member_o_nat @ K @ ( sigma_1999164137574644376_o_nat @ borel_5500255247093592246orel_o @ Y4 ) )
     => ( member_o_nat @ K @ ( qbs_morphism_o_nat @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) ) ) ).

% bool.qbs_morphism_measurable_intro
thf(fact_668_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K: nat > produc2422161461964618553l_real,Y4: sigma_2308072346491277622l_real] :
      ( ( member4944736856719700937l_real @ K @ ( sigma_5893925349391517463l_real @ borel_8449730974584783410el_nat @ Y4 ) )
     => ( member4944736856719700937l_real @ K @ ( qbs_mo6520021129653552495l_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur3029786804310284173l_real @ Y4 ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_669_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K: nat > produc7716430852924023517t_real,Y4: sigma_5310753476256395226t_real] :
      ( ( member4237364305948650477t_real @ K @ ( sigma_8750250338757262779t_real @ borel_8449730974584783410el_nat @ Y4 ) )
     => ( member4237364305948650477t_real @ K @ ( qbs_mo2896996399151425555t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2011440102552004913t_real @ Y4 ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_670_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K: nat > real,Y4: sigma_measure_real] :
      ( ( member_nat_real @ K @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ Y4 ) )
     => ( member_nat_real @ K @ ( qbs_mo2000642995705457910t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur6875533127466166616s_real @ Y4 ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_671_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K: nat > $o,Y4: sigma_measure_o] :
      ( ( member_nat_o @ K @ ( sigma_5101835498682829686_nat_o @ borel_8449730974584783410el_nat @ Y4 ) )
     => ( member_nat_o @ K @ ( qbs_morphism_nat_o @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_672_nat_Oqbs__morphism__measurable__intro,axiom,
    ! [K: nat > nat,Y4: sigma_measure_nat] :
      ( ( member_nat_nat @ K @ ( sigma_4350458207664084850at_nat @ borel_8449730974584783410el_nat @ Y4 ) )
     => ( member_nat_nat @ K @ ( qbs_morphism_nat_nat @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) ) ) ).

% nat.qbs_morphism_measurable_intro
thf(fact_673_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K: extend8495563244428889912nnreal > produc2422161461964618553l_real,Y4: sigma_2308072346491277622l_real] :
      ( ( member7354208599470296673l_real @ K @ ( sigma_1014563338549229999l_real @ borel_6524799422816628122nnreal @ Y4 ) )
     => ( member7354208599470296673l_real @ K @ ( qbs_mo8045053474288007175l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur3029786804310284173l_real @ Y4 ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_674_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K: extend8495563244428889912nnreal > produc7716430852924023517t_real,Y4: sigma_5310753476256395226t_real] :
      ( ( member2197300837325509253t_real @ K @ ( sigma_2672489485171276883t_real @ borel_6524799422816628122nnreal @ Y4 ) )
     => ( member2197300837325509253t_real @ K @ ( qbs_mo3970737951806219947t_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2011440102552004913t_real @ Y4 ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_675_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K: extend8495563244428889912nnreal > real,Y4: sigma_measure_real] :
      ( ( member2874014351250825754l_real @ K @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ Y4 ) )
     => ( member2874014351250825754l_real @ K @ ( qbs_mo8573344932208643166l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur6875533127466166616s_real @ Y4 ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_676_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K: extend8495563244428889912nnreal > $o,Y4: sigma_measure_o] :
      ( ( member8095236870201115968real_o @ K @ ( sigma_6279906219187228174real_o @ borel_6524799422816628122nnreal @ Y4 ) )
     => ( member8095236870201115968real_o @ K @ ( qbs_mo7845494039804091494real_o @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_677_ennreal_Oqbs__morphism__measurable__intro,axiom,
    ! [K: extend8495563244428889912nnreal > nat,Y4: sigma_measure_nat] :
      ( ( member6436672275262627518al_nat @ K @ ( sigma_1856489715609627354al_nat @ borel_6524799422816628122nnreal @ Y4 ) )
     => ( member6436672275262627518al_nat @ K @ ( qbs_mo182749989559755394al_nat @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) ) ) ).

% ennreal.qbs_morphism_measurable_intro
thf(fact_678_a__def,axiom,
    ( a2
    = ( ^ [R: real] : ( alpha2 @ R @ zero_zero_nat ) ) ) ).

% a_def
thf(fact_679__092_060open_062_I_092_060lambda_062x_Ay_O_Arec__list_H_A_Ifst_Ay_J_A_Isnd_Ay_J_A_ISuc_An_M_Ax_J_J_A_092_060circ_062_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_Ay_O_Arec__list_H_A_Ifst_Ay_J_A_Isnd_Ay_J_A_ISuc_An_M_A_092_060alpha_062_Ar_J_J_092_060close_062,axiom,
    ( ( comp_n1554393163112911916a_real
      @ ^ [X: nat > b,Y: produc432120507726958376_b_a_a] : ( coProd5680178011022504904st_a_b @ ( produc5967371699169076996_b_a_a @ Y ) @ ( produc7355951146076385094_b_a_a @ Y ) @ ( produc5292568363641424317_nat_b @ ( suc @ na ) @ X ) )
      @ alpha2 )
    = ( ^ [R: real,Y: produc432120507726958376_b_a_a] : ( coProd5680178011022504904st_a_b @ ( produc5967371699169076996_b_a_a @ Y ) @ ( produc7355951146076385094_b_a_a @ Y ) @ ( produc5292568363641424317_nat_b @ ( suc @ na ) @ ( alpha2 @ R ) ) ) ) ) ).

% \<open>(\<lambda>x y. rec_list' (fst y) (snd y) (Suc n, x)) \<circ> \<alpha> = (\<lambda>r y. rec_list' (fst y) (snd y) (Suc n, \<alpha> r))\<close>
thf(fact_680_prod_Ocollapse,axiom,
    ! [Prod: product_prod_real_b] :
      ( ( product_Pair_real_b @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_681_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_682_prod_Ocollapse,axiom,
    ! [Prod: produc778026099599033348at_b_a] :
      ( ( produc653446490150804990at_b_a @ ( produc7871839370796975378at_b_a @ Prod ) @ ( produc6884036649769537872at_b_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_683_prod_Ocollapse,axiom,
    ! [Prod: produc4883615123598087894at_b_a] :
      ( ( produc992390952543482448at_b_a @ ( produc7380424691241595748at_b_a @ Prod ) @ ( produc7666751653692995234at_b_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_684_prod_Ocollapse,axiom,
    ! [Prod: produc6209191202727158440at_b_a] :
      ( ( produc8509509857186669914at_b_a @ ( produc2385479481072933486at_b_a @ Prod ) @ ( produc5227817866736700332at_b_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_685_prod_Ocollapse,axiom,
    ! [Prod: produc3260441645384719156_nat_b] :
      ( ( produc1946830598933159916_nat_b @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_686_prod_Ocollapse,axiom,
    ! [Prod: produc432120507726958376_b_a_a] :
      ( ( produc3671031927252941592_b_a_a @ ( produc5967371699169076996_b_a_a @ Prod ) @ ( produc7355951146076385094_b_a_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_687_prod_Ocollapse,axiom,
    ! [Prod: produc7194034526493574155_nat_b] :
      ( ( produc5292568363641424317_nat_b @ ( produc3194919583230816977_nat_b @ Prod ) @ ( produc4809910044363821583_nat_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_688__C3_C,axiom,
    ! [R3: real] :
      ( member2882485241872049954_nat_b @ ( produc5292568363641424317_nat_b @ ( suc @ na ) @ ( alpha2 @ R3 ) )
      @ ( qbs_sp2305461467746415189_nat_b
        @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
          @ ^ [N: nat] :
              ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
              @ ^ [I: nat] : x ) ) ) ) ).

% "3"
thf(fact_689__092_060open_062_I_092_060lambda_062r_Ay_O_Arec__list_H_A_Ifst_Ay_J_A_Isnd_Ay_J_A_ISuc_An_M_A_092_060alpha_062_Ar_J_J_A_061_A_I_092_060lambda_062r_Ay_O_Asnd_Ay_A_Ia_Ar_J_A_In_M_A_092_060alpha_062_H_Ar_J_A_Irec__list_H_A_Ifst_Ay_J_A_Isnd_Ay_J_A_In_M_A_092_060alpha_062_H_Ar_J_J_J_092_060close_062,axiom,
    ( ( ^ [R: real,Y: produc432120507726958376_b_a_a] : ( coProd5680178011022504904st_a_b @ ( produc5967371699169076996_b_a_a @ Y ) @ ( produc7355951146076385094_b_a_a @ Y ) @ ( produc5292568363641424317_nat_b @ ( suc @ na ) @ ( alpha2 @ R ) ) ) )
    = ( ^ [R: real,Y: produc432120507726958376_b_a_a] : ( produc7355951146076385094_b_a_a @ Y @ ( a2 @ R ) @ ( produc5292568363641424317_nat_b @ na @ ( alpha @ R ) ) @ ( coProd5680178011022504904st_a_b @ ( produc5967371699169076996_b_a_a @ Y ) @ ( produc7355951146076385094_b_a_a @ Y ) @ ( produc5292568363641424317_nat_b @ na @ ( alpha @ R ) ) ) ) ) ) ).

% \<open>(\<lambda>r y. rec_list' (fst y) (snd y) (Suc n, \<alpha> r)) = (\<lambda>r y. snd y (a r) (n, \<alpha>' r) (rec_list' (fst y) (snd y) (n, \<alpha>' r)))\<close>
thf(fact_690_fst__o__paired,axiom,
    ! [F: produc7194034526493574155_nat_b > a > a,G: produc7194034526493574155_nat_b > a > b > produc7194034526493574155_nat_b > a > a] :
      ( ( comp_P6428769393572761825at_b_a @ produc5967371699169076996_b_a_a
        @ ( produc4186246733524169441_b_a_a
          @ ^ [X: produc7194034526493574155_nat_b,Y: a] : ( produc3671031927252941592_b_a_a @ ( F @ X @ Y ) @ ( G @ X @ Y ) ) ) )
      = ( produc2543982510095308599_b_a_a @ F ) ) ).

% fst_o_paired
thf(fact_691_fst__o__paired,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,G: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > b > produc7194034526493574155_nat_b > a > a] :
      ( ( comp_P3156607648531775677at_b_a @ produc5967371699169076996_b_a_a
        @ ( produc1431721540671598397_b_a_a
          @ ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: produc4883615123598087894at_b_a] : ( produc3671031927252941592_b_a_a @ ( F @ X @ Y ) @ ( G @ X @ Y ) ) ) )
      = ( produc286368409806729619_b_a_a @ F ) ) ).

% fst_o_paired
thf(fact_692_fst__o__paired,axiom,
    ! [F: b > produc6209191202727158440at_b_a > a,G: b > produc6209191202727158440at_b_a > b > produc7194034526493574155_nat_b > a > a] :
      ( ( comp_P1539513916346917903at_b_a @ produc5967371699169076996_b_a_a
        @ ( produc4190251302972714603_b_a_a
          @ ^ [X: b,Y: produc6209191202727158440at_b_a] : ( produc3671031927252941592_b_a_a @ ( F @ X @ Y ) @ ( G @ X @ Y ) ) ) )
      = ( produc6302854954412089281_b_a_a @ F ) ) ).

% fst_o_paired
thf(fact_693_fst__o__paired,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a,G: product_prod_real_b > produc7194034526493574155_nat_b > b > produc7194034526493574155_nat_b > a > a] :
      ( ( comp_P2579465633625509435_nat_b @ produc5967371699169076996_b_a_a
        @ ( produc2714732900805866471_b_a_a
          @ ^ [X: product_prod_real_b,Y: produc7194034526493574155_nat_b] : ( produc3671031927252941592_b_a_a @ ( F @ X @ Y ) @ ( G @ X @ Y ) ) ) )
      = ( produc634687561522189117at_b_a @ F ) ) ).

% fst_o_paired
thf(fact_694_snd__o__paired,axiom,
    ! [F: produc7194034526493574155_nat_b > a > produc7194034526493574155_nat_b,G: produc7194034526493574155_nat_b > a > a] :
      ( ( comp_P3164597315454541147at_b_a @ produc5227817866736700332at_b_a
        @ ( produc7897861285970668975at_b_a
          @ ^ [X: produc7194034526493574155_nat_b,Y: a] : ( produc8509509857186669914at_b_a @ ( F @ X @ Y ) @ ( G @ X @ Y ) ) ) )
      = ( produc2543982510095308599_b_a_a @ G ) ) ).

% snd_o_paired
thf(fact_695_snd__o__paired,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > produc7194034526493574155_nat_b,G: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a] :
      ( ( comp_P3028257441997127223at_b_a @ produc5227817866736700332at_b_a
        @ ( produc605504560143574611at_b_a
          @ ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: produc4883615123598087894at_b_a] : ( produc8509509857186669914at_b_a @ ( F @ X @ Y ) @ ( G @ X @ Y ) ) ) )
      = ( produc286368409806729619_b_a_a @ G ) ) ).

% snd_o_paired
thf(fact_696_snd__o__paired,axiom,
    ! [F: b > produc6209191202727158440at_b_a > produc7194034526493574155_nat_b,G: b > produc6209191202727158440at_b_a > a] :
      ( ( comp_P4766272437332485769at_b_a @ produc5227817866736700332at_b_a
        @ ( produc8965135185937645989at_b_a
          @ ^ [X: b,Y: produc6209191202727158440at_b_a] : ( produc8509509857186669914at_b_a @ ( F @ X @ Y ) @ ( G @ X @ Y ) ) ) )
      = ( produc6302854954412089281_b_a_a @ G ) ) ).

% snd_o_paired
thf(fact_697_snd__o__paired,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > produc7194034526493574155_nat_b,G: product_prod_real_b > produc7194034526493574155_nat_b > a] :
      ( ( comp_P7717571827227254337_nat_b @ produc5227817866736700332at_b_a
        @ ( produc6929322651383056681at_b_a
          @ ^ [X: product_prod_real_b,Y: produc7194034526493574155_nat_b] : ( produc8509509857186669914at_b_a @ ( F @ X @ Y ) @ ( G @ X @ Y ) ) ) )
      = ( produc634687561522189117at_b_a @ G ) ) ).

% snd_o_paired
thf(fact_698__C4_C,axiom,
    ! [R3: real] :
      ( ( produc5292568363641424317_nat_b @ na @ ( alpha @ R3 ) )
      = ( coProd8554194001436178208tail_b @ ( produc5292568363641424317_nat_b @ ( suc @ na ) @ ( alpha2 @ R3 ) ) ) ) ).

% "4"
thf(fact_699_nat__qbs__morphism,axiom,
    ! [F: nat > real,Y4: quasi_borel_real] :
      ( ! [N2: nat] : ( member_real @ ( F @ N2 ) @ ( qbs_space_real @ Y4 ) )
     => ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ Y4 ) ) ) ).

% nat_qbs_morphism
thf(fact_700_nat__qbs__morphism,axiom,
    ! [F: nat > $o > real,Y4: quasi_borel_o_real] :
      ( ! [N2: nat] : ( member_o_real @ ( F @ N2 ) @ ( qbs_space_o_real @ Y4 ) )
     => ( member_nat_o_real @ F @ ( qbs_mo6546791585729861355o_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ Y4 ) ) ) ).

% nat_qbs_morphism
thf(fact_701_nat__qbs__morphism,axiom,
    ! [F: nat > $o > $o,Y4: quasi_borel_o_o] :
      ( ! [N2: nat] : ( member_o_o @ ( F @ N2 ) @ ( qbs_space_o_o @ Y4 ) )
     => ( member_nat_o_o @ F @ ( qbs_morphism_nat_o_o @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ Y4 ) ) ) ).

% nat_qbs_morphism
thf(fact_702_nat__qbs__morphism,axiom,
    ! [F: nat > $o > $o > $o,Y4: quasi_borel_o_o_o] :
      ( ! [N2: nat] : ( member_o_o_o @ ( F @ N2 ) @ ( qbs_space_o_o_o @ Y4 ) )
     => ( member_nat_o_o_o @ F @ ( qbs_mo777698081677429284_o_o_o @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ Y4 ) ) ) ).

% nat_qbs_morphism
thf(fact_703_nat__qbs__morphism,axiom,
    ! [F: nat > nat > real,Y4: quasi_borel_nat_real] :
      ( ! [N2: nat] : ( member_nat_real @ ( F @ N2 ) @ ( qbs_space_nat_real @ Y4 ) )
     => ( member_nat_nat_real @ F @ ( qbs_mo3397633600088470373t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ Y4 ) ) ) ).

% nat_qbs_morphism
thf(fact_704_nat__qbs__morphism,axiom,
    ! [F: nat > ( nat > b ) > produc432120507726958376_b_a_a > a,Y4: quasi_8909271866012278712_a_a_a] :
      ( ! [N2: nat] : ( member1811236362415105851_a_a_a @ ( F @ N2 ) @ ( qbs_sp8333110983225252808_a_a_a @ Y4 ) )
     => ( member1855690233985635746_a_a_a @ F @ ( qbs_mo5963866097022083016_a_a_a @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ Y4 ) ) ) ).

% nat_qbs_morphism
thf(fact_705_nat__qbs__morphism,axiom,
    ! [F: nat > produc7194034526493574155_nat_b,Y4: quasi_7334096685896923387_nat_b] :
      ( ! [N2: nat] : ( member2882485241872049954_nat_b @ ( F @ N2 ) @ ( qbs_sp2305461467746415189_nat_b @ Y4 ) )
     => ( member7258681010406981265_nat_b @ F @ ( qbs_mo3020559592811146837_nat_b @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ Y4 ) ) ) ).

% nat_qbs_morphism
thf(fact_706_bool__qbs__morphism,axiom,
    ! [F: $o > real,Y4: quasi_borel_real] :
      ( ! [B3: $o] : ( member_real @ ( F @ B3 ) @ ( qbs_space_real @ Y4 ) )
     => ( member_o_real @ F @ ( qbs_morphism_o_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ Y4 ) ) ) ).

% bool_qbs_morphism
thf(fact_707_bool__qbs__morphism,axiom,
    ! [F: $o > $o > real,Y4: quasi_borel_o_real] :
      ( ! [B3: $o] : ( member_o_real @ ( F @ B3 ) @ ( qbs_space_o_real @ Y4 ) )
     => ( member_o_o_real @ F @ ( qbs_mo1138393325155344277o_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ Y4 ) ) ) ).

% bool_qbs_morphism
thf(fact_708_bool__qbs__morphism,axiom,
    ! [F: $o > $o > $o > $o,Y4: quasi_borel_o_o_o] :
      ( ! [B3: $o] : ( member_o_o_o @ ( F @ B3 ) @ ( qbs_space_o_o_o @ Y4 ) )
     => ( member_o_o_o_o2 @ F @ ( qbs_morphism_o_o_o_o2 @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ Y4 ) ) ) ).

% bool_qbs_morphism
thf(fact_709_bool__qbs__morphism,axiom,
    ! [F: $o > nat > real,Y4: quasi_borel_nat_real] :
      ( ! [B3: $o] : ( member_nat_real @ ( F @ B3 ) @ ( qbs_space_nat_real @ Y4 ) )
     => ( member_o_nat_real @ F @ ( qbs_mo370417502978959675t_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ Y4 ) ) ) ).

% bool_qbs_morphism
thf(fact_710_bool__qbs__morphism,axiom,
    ! [F: $o > ( nat > b ) > produc432120507726958376_b_a_a > a,Y4: quasi_8909271866012278712_a_a_a] :
      ( ! [B3: $o] : ( member1811236362415105851_a_a_a @ ( F @ B3 ) @ ( qbs_sp8333110983225252808_a_a_a @ Y4 ) )
     => ( member8988714699417754870_a_a_a @ F @ ( qbs_mo5967126606998171762_a_a_a @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ Y4 ) ) ) ).

% bool_qbs_morphism
thf(fact_711_bool__qbs__morphism,axiom,
    ! [F: $o > produc7194034526493574155_nat_b,Y4: quasi_7334096685896923387_nat_b] :
      ( ! [B3: $o] : ( member2882485241872049954_nat_b @ ( F @ B3 ) @ ( qbs_sp2305461467746415189_nat_b @ Y4 ) )
     => ( member1237389059852570173_nat_b @ F @ ( qbs_mo3134286859365214507_nat_b @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ Y4 ) ) ) ).

% bool_qbs_morphism
thf(fact_712_bool__qbs__morphism,axiom,
    ! [F: $o > $o > $o,Y4: quasi_borel_o_o] :
      ( ! [B3: $o] : ( member_o_o @ ( F @ B3 ) @ ( qbs_space_o_o @ Y4 ) )
     => ( member_o_o_o @ F @ ( qbs_morphism_o_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ Y4 ) ) ) ).

% bool_qbs_morphism
thf(fact_713_bool__qbs__morphism,axiom,
    ! [F: $o > $o,Y4: quasi_borel_o] :
      ( ! [B3: $o] : ( member_o @ ( F @ B3 ) @ ( qbs_space_o @ Y4 ) )
     => ( member_o_o @ F @ ( qbs_morphism_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ Y4 ) ) ) ).

% bool_qbs_morphism
thf(fact_714_surjective__pairing,axiom,
    ! [T: product_prod_real_b] :
      ( T
      = ( product_Pair_real_b @ ( product_fst_real_b @ T ) @ ( product_snd_real_b @ T ) ) ) ).

% surjective_pairing
thf(fact_715_surjective__pairing,axiom,
    ! [T: product_prod_nat_nat] :
      ( T
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_716_surjective__pairing,axiom,
    ! [T: produc778026099599033348at_b_a] :
      ( T
      = ( produc653446490150804990at_b_a @ ( produc7871839370796975378at_b_a @ T ) @ ( produc6884036649769537872at_b_a @ T ) ) ) ).

% surjective_pairing
thf(fact_717_surjective__pairing,axiom,
    ! [T: produc4883615123598087894at_b_a] :
      ( T
      = ( produc992390952543482448at_b_a @ ( produc7380424691241595748at_b_a @ T ) @ ( produc7666751653692995234at_b_a @ T ) ) ) ).

% surjective_pairing
thf(fact_718_surjective__pairing,axiom,
    ! [T: produc6209191202727158440at_b_a] :
      ( T
      = ( produc8509509857186669914at_b_a @ ( produc2385479481072933486at_b_a @ T ) @ ( produc5227817866736700332at_b_a @ T ) ) ) ).

% surjective_pairing
thf(fact_719_surjective__pairing,axiom,
    ! [T: produc3260441645384719156_nat_b] :
      ( T
      = ( produc1946830598933159916_nat_b @ ( produc4243170370849295320_nat_b @ T ) @ ( produc5631749817756603418_nat_b @ T ) ) ) ).

% surjective_pairing
thf(fact_720_surjective__pairing,axiom,
    ! [T: produc432120507726958376_b_a_a] :
      ( T
      = ( produc3671031927252941592_b_a_a @ ( produc5967371699169076996_b_a_a @ T ) @ ( produc7355951146076385094_b_a_a @ T ) ) ) ).

% surjective_pairing
thf(fact_721_surjective__pairing,axiom,
    ! [T: produc7194034526493574155_nat_b] :
      ( T
      = ( produc5292568363641424317_nat_b @ ( produc3194919583230816977_nat_b @ T ) @ ( produc4809910044363821583_nat_b @ T ) ) ) ).

% surjective_pairing
thf(fact_722_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_real_b] :
      ( Prod
      = ( product_Pair_real_b @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_723_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( Prod
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_724_prod_Oexhaust__sel,axiom,
    ! [Prod: produc778026099599033348at_b_a] :
      ( Prod
      = ( produc653446490150804990at_b_a @ ( produc7871839370796975378at_b_a @ Prod ) @ ( produc6884036649769537872at_b_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_725_prod_Oexhaust__sel,axiom,
    ! [Prod: produc4883615123598087894at_b_a] :
      ( Prod
      = ( produc992390952543482448at_b_a @ ( produc7380424691241595748at_b_a @ Prod ) @ ( produc7666751653692995234at_b_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_726_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6209191202727158440at_b_a] :
      ( Prod
      = ( produc8509509857186669914at_b_a @ ( produc2385479481072933486at_b_a @ Prod ) @ ( produc5227817866736700332at_b_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_727_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3260441645384719156_nat_b] :
      ( Prod
      = ( produc1946830598933159916_nat_b @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_728_prod_Oexhaust__sel,axiom,
    ! [Prod: produc432120507726958376_b_a_a] :
      ( Prod
      = ( produc3671031927252941592_b_a_a @ ( produc5967371699169076996_b_a_a @ Prod ) @ ( produc7355951146076385094_b_a_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_729_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7194034526493574155_nat_b] :
      ( Prod
      = ( produc5292568363641424317_nat_b @ ( produc3194919583230816977_nat_b @ Prod ) @ ( produc4809910044363821583_nat_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_730_prod__eq__iff,axiom,
    ( ( ^ [Y7: produc432120507726958376_b_a_a,Z4: produc432120507726958376_b_a_a] : ( Y7 = Z4 ) )
    = ( ^ [S2: produc432120507726958376_b_a_a,T2: produc432120507726958376_b_a_a] :
          ( ( ( produc5967371699169076996_b_a_a @ S2 )
            = ( produc5967371699169076996_b_a_a @ T2 ) )
          & ( ( produc7355951146076385094_b_a_a @ S2 )
            = ( produc7355951146076385094_b_a_a @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_731_prod__eq__iff,axiom,
    ( ( ^ [Y7: produc7194034526493574155_nat_b,Z4: produc7194034526493574155_nat_b] : ( Y7 = Z4 ) )
    = ( ^ [S2: produc7194034526493574155_nat_b,T2: produc7194034526493574155_nat_b] :
          ( ( ( produc3194919583230816977_nat_b @ S2 )
            = ( produc3194919583230816977_nat_b @ T2 ) )
          & ( ( produc4809910044363821583_nat_b @ S2 )
            = ( produc4809910044363821583_nat_b @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_732_prod__eqI,axiom,
    ! [P2: produc432120507726958376_b_a_a,Q2: produc432120507726958376_b_a_a] :
      ( ( ( produc5967371699169076996_b_a_a @ P2 )
        = ( produc5967371699169076996_b_a_a @ Q2 ) )
     => ( ( ( produc7355951146076385094_b_a_a @ P2 )
          = ( produc7355951146076385094_b_a_a @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_733_prod__eqI,axiom,
    ! [P2: produc7194034526493574155_nat_b,Q2: produc7194034526493574155_nat_b] :
      ( ( ( produc3194919583230816977_nat_b @ P2 )
        = ( produc3194919583230816977_nat_b @ Q2 ) )
     => ( ( ( produc4809910044363821583_nat_b @ P2 )
          = ( produc4809910044363821583_nat_b @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_734_prod_Oexpand,axiom,
    ! [Prod: produc432120507726958376_b_a_a,Prod2: produc432120507726958376_b_a_a] :
      ( ( ( ( produc5967371699169076996_b_a_a @ Prod )
          = ( produc5967371699169076996_b_a_a @ Prod2 ) )
        & ( ( produc7355951146076385094_b_a_a @ Prod )
          = ( produc7355951146076385094_b_a_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_735_prod_Oexpand,axiom,
    ! [Prod: produc7194034526493574155_nat_b,Prod2: produc7194034526493574155_nat_b] :
      ( ( ( ( produc3194919583230816977_nat_b @ Prod )
          = ( produc3194919583230816977_nat_b @ Prod2 ) )
        & ( ( produc4809910044363821583_nat_b @ Prod )
          = ( produc4809910044363821583_nat_b @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_736_Product__Type_OCollect__case__prodD,axiom,
    ! [X5: produc432120507726958376_b_a_a,A4: a > ( b > produc7194034526493574155_nat_b > a > a ) > $o] :
      ( ( member6783937185859681745_b_a_a @ X5 @ ( collec3932359806813859091_b_a_a @ ( produc1569100873952324907_a_a_o @ A4 ) ) )
     => ( A4 @ ( produc5967371699169076996_b_a_a @ X5 ) @ ( produc7355951146076385094_b_a_a @ X5 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_737_Product__Type_OCollect__case__prodD,axiom,
    ! [X5: produc7194034526493574155_nat_b,A4: nat > ( nat > b ) > $o] :
      ( ( member2882485241872049954_nat_b @ X5 @ ( collec7375898155600944992_nat_b @ ( produc7710141471362042158at_b_o @ A4 ) ) )
     => ( A4 @ ( produc3194919583230816977_nat_b @ X5 ) @ ( produc4809910044363821583_nat_b @ X5 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_738_split__beta,axiom,
    ( produc8421786640700244582_b_a_a
    = ( ^ [F2: product_prod_real_b > produc7194034526493574155_nat_b > a > a,Prod3: produc3260441645384719156_nat_b] : ( F2 @ ( produc4243170370849295320_nat_b @ Prod3 ) @ ( produc5631749817756603418_nat_b @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_739_split__beta,axiom,
    ( produc2543982510095308599_b_a_a
    = ( ^ [F2: produc7194034526493574155_nat_b > a > a,Prod3: produc6209191202727158440at_b_a] : ( F2 @ ( produc2385479481072933486at_b_a @ Prod3 ) @ ( produc5227817866736700332at_b_a @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_740_split__beta,axiom,
    ( produc7016990068875982646_b_a_a
    = ( ^ [F2: real > b > produc7194034526493574155_nat_b > a > a,Prod3: product_prod_real_b] : ( F2 @ ( product_fst_real_b @ Prod3 ) @ ( product_snd_real_b @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_741_split__beta,axiom,
    ( produc286368409806729619_b_a_a
    = ( ^ [F2: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,Prod3: produc778026099599033348at_b_a] : ( F2 @ ( produc7871839370796975378at_b_a @ Prod3 ) @ ( produc6884036649769537872at_b_a @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_742_split__beta,axiom,
    ( produc6302854954412089281_b_a_a
    = ( ^ [F2: b > produc6209191202727158440at_b_a > a,Prod3: produc4883615123598087894at_b_a] : ( F2 @ ( produc7380424691241595748at_b_a @ Prod3 ) @ ( produc7666751653692995234at_b_a @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_743_split__beta,axiom,
    ( produc7827042608974387084_b_a_a
    = ( ^ [F2: real > b > a > a,Prod3: product_prod_real_b] : ( F2 @ ( product_fst_real_b @ Prod3 ) @ ( product_snd_real_b @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_744_split__beta,axiom,
    ( produc634687561522189117at_b_a
    = ( ^ [F2: product_prod_real_b > produc7194034526493574155_nat_b > a,Prod3: produc3260441645384719156_nat_b] : ( F2 @ ( produc4243170370849295320_nat_b @ Prod3 ) @ ( produc5631749817756603418_nat_b @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_745_case__prod__beta,axiom,
    ( produc8421786640700244582_b_a_a
    = ( ^ [F2: product_prod_real_b > produc7194034526493574155_nat_b > a > a,P3: produc3260441645384719156_nat_b] : ( F2 @ ( produc4243170370849295320_nat_b @ P3 ) @ ( produc5631749817756603418_nat_b @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_746_case__prod__beta,axiom,
    ( produc2543982510095308599_b_a_a
    = ( ^ [F2: produc7194034526493574155_nat_b > a > a,P3: produc6209191202727158440at_b_a] : ( F2 @ ( produc2385479481072933486at_b_a @ P3 ) @ ( produc5227817866736700332at_b_a @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_747_case__prod__beta,axiom,
    ( produc7016990068875982646_b_a_a
    = ( ^ [F2: real > b > produc7194034526493574155_nat_b > a > a,P3: product_prod_real_b] : ( F2 @ ( product_fst_real_b @ P3 ) @ ( product_snd_real_b @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_748_case__prod__beta,axiom,
    ( produc286368409806729619_b_a_a
    = ( ^ [F2: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,P3: produc778026099599033348at_b_a] : ( F2 @ ( produc7871839370796975378at_b_a @ P3 ) @ ( produc6884036649769537872at_b_a @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_749_case__prod__beta,axiom,
    ( produc6302854954412089281_b_a_a
    = ( ^ [F2: b > produc6209191202727158440at_b_a > a,P3: produc4883615123598087894at_b_a] : ( F2 @ ( produc7380424691241595748at_b_a @ P3 ) @ ( produc7666751653692995234at_b_a @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_750_case__prod__beta,axiom,
    ( produc7827042608974387084_b_a_a
    = ( ^ [F2: real > b > a > a,P3: product_prod_real_b] : ( F2 @ ( product_fst_real_b @ P3 ) @ ( product_snd_real_b @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_751_case__prod__beta,axiom,
    ( produc634687561522189117at_b_a
    = ( ^ [F2: product_prod_real_b > produc7194034526493574155_nat_b > a,P3: produc3260441645384719156_nat_b] : ( F2 @ ( produc4243170370849295320_nat_b @ P3 ) @ ( produc5631749817756603418_nat_b @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_752_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_o_o] :
      ( ( sigma_1874301400886517549_o_o_o @ borel_5500255247093592246orel_o @ Y4 )
      = ( qbs_morphism_o_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur6002702770210135313bs_o_o @ Y4 ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_753_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_2308072346491277622l_real] :
      ( ( sigma_2489502367751881329l_real @ borel_5500255247093592246orel_o @ Y4 )
      = ( qbs_mo6076292073755557401l_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur3029786804310284173l_real @ Y4 ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_754_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_5310753476256395226t_real] :
      ( ( sigma_680772226823374357t_real @ borel_5500255247093592246orel_o @ Y4 )
      = ( qbs_mo928686258813670845t_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2011440102552004913t_real @ Y4 ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_755_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_real] :
      ( ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ Y4 )
      = ( qbs_morphism_o_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur6875533127466166616s_real @ Y4 ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_756_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_o] :
      ( ( sigma_measurable_o_o @ borel_5500255247093592246orel_o @ Y4 )
      = ( qbs_morphism_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_757_bool_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_nat] :
      ( ( sigma_1999164137574644376_o_nat @ borel_5500255247093592246orel_o @ Y4 )
      = ( qbs_morphism_o_nat @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) ) ).

% bool.standard_borel_r_full_faithful
thf(fact_758_bool_Oqbs__morphism__dest,axiom,
    ! [K: $o > $o > $o,Y4: sigma_measure_o_o] :
      ( ( member_o_o_o @ K @ ( qbs_morphism_o_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur6002702770210135313bs_o_o @ Y4 ) ) )
     => ( member_o_o_o @ K @ ( sigma_1874301400886517549_o_o_o @ borel_5500255247093592246orel_o @ Y4 ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_759_bool_Oqbs__morphism__dest,axiom,
    ! [K: $o > produc2422161461964618553l_real,Y4: sigma_2308072346491277622l_real] :
      ( ( member8602590400058854173l_real @ K @ ( qbs_mo6076292073755557401l_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur3029786804310284173l_real @ Y4 ) ) )
     => ( member8602590400058854173l_real @ K @ ( sigma_2489502367751881329l_real @ borel_5500255247093592246orel_o @ Y4 ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_760_bool_Oqbs__morphism__dest,axiom,
    ! [K: $o > produc7716430852924023517t_real,Y4: sigma_5310753476256395226t_real] :
      ( ( member8739982677887872321t_real @ K @ ( qbs_mo928686258813670845t_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2011440102552004913t_real @ Y4 ) ) )
     => ( member8739982677887872321t_real @ K @ ( sigma_680772226823374357t_real @ borel_5500255247093592246orel_o @ Y4 ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_761_bool_Oqbs__morphism__dest,axiom,
    ! [K: $o > real,Y4: sigma_measure_real] :
      ( ( member_o_real @ K @ ( qbs_morphism_o_real @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur6875533127466166616s_real @ Y4 ) ) )
     => ( member_o_real @ K @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ Y4 ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_762_bool_Oqbs__morphism__dest,axiom,
    ! [K: $o > $o,Y4: sigma_measure_o] :
      ( ( member_o_o @ K @ ( qbs_morphism_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) )
     => ( member_o_o @ K @ ( sigma_measurable_o_o @ borel_5500255247093592246orel_o @ Y4 ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_763_bool_Oqbs__morphism__dest,axiom,
    ! [K: $o > nat,Y4: sigma_measure_nat] :
      ( ( member_o_nat @ K @ ( qbs_morphism_o_nat @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) )
     => ( member_o_nat @ K @ ( sigma_1999164137574644376_o_nat @ borel_5500255247093592246orel_o @ Y4 ) ) ) ).

% bool.qbs_morphism_dest
thf(fact_764_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_2308072346491277622l_real] :
      ( ( sigma_5893925349391517463l_real @ borel_8449730974584783410el_nat @ Y4 )
      = ( qbs_mo6520021129653552495l_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur3029786804310284173l_real @ Y4 ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_765_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_5310753476256395226t_real] :
      ( ( sigma_8750250338757262779t_real @ borel_8449730974584783410el_nat @ Y4 )
      = ( qbs_mo2896996399151425555t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2011440102552004913t_real @ Y4 ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_766_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_real] :
      ( ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ Y4 )
      = ( qbs_mo2000642995705457910t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur6875533127466166616s_real @ Y4 ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_767_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_o] :
      ( ( sigma_5101835498682829686_nat_o @ borel_8449730974584783410el_nat @ Y4 )
      = ( qbs_morphism_nat_o @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_768_nat_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_nat] :
      ( ( sigma_4350458207664084850at_nat @ borel_8449730974584783410el_nat @ Y4 )
      = ( qbs_morphism_nat_nat @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) ) ).

% nat.standard_borel_r_full_faithful
thf(fact_769_nat_Oqbs__morphism__dest,axiom,
    ! [K: nat > produc2422161461964618553l_real,Y4: sigma_2308072346491277622l_real] :
      ( ( member4944736856719700937l_real @ K @ ( qbs_mo6520021129653552495l_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur3029786804310284173l_real @ Y4 ) ) )
     => ( member4944736856719700937l_real @ K @ ( sigma_5893925349391517463l_real @ borel_8449730974584783410el_nat @ Y4 ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_770_nat_Oqbs__morphism__dest,axiom,
    ! [K: nat > produc7716430852924023517t_real,Y4: sigma_5310753476256395226t_real] :
      ( ( member4237364305948650477t_real @ K @ ( qbs_mo2896996399151425555t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2011440102552004913t_real @ Y4 ) ) )
     => ( member4237364305948650477t_real @ K @ ( sigma_8750250338757262779t_real @ borel_8449730974584783410el_nat @ Y4 ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_771_nat_Oqbs__morphism__dest,axiom,
    ! [K: nat > real,Y4: sigma_measure_real] :
      ( ( member_nat_real @ K @ ( qbs_mo2000642995705457910t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur6875533127466166616s_real @ Y4 ) ) )
     => ( member_nat_real @ K @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ Y4 ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_772_nat_Oqbs__morphism__dest,axiom,
    ! [K: nat > $o,Y4: sigma_measure_o] :
      ( ( member_nat_o @ K @ ( qbs_morphism_nat_o @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) )
     => ( member_nat_o @ K @ ( sigma_5101835498682829686_nat_o @ borel_8449730974584783410el_nat @ Y4 ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_773_nat_Oqbs__morphism__dest,axiom,
    ! [K: nat > nat,Y4: sigma_measure_nat] :
      ( ( member_nat_nat @ K @ ( qbs_morphism_nat_nat @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) )
     => ( member_nat_nat @ K @ ( sigma_4350458207664084850at_nat @ borel_8449730974584783410el_nat @ Y4 ) ) ) ).

% nat.qbs_morphism_dest
thf(fact_774_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_2308072346491277622l_real] :
      ( ( sigma_1014563338549229999l_real @ borel_6524799422816628122nnreal @ Y4 )
      = ( qbs_mo8045053474288007175l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur3029786804310284173l_real @ Y4 ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_775_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_5310753476256395226t_real] :
      ( ( sigma_2672489485171276883t_real @ borel_6524799422816628122nnreal @ Y4 )
      = ( qbs_mo3970737951806219947t_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2011440102552004913t_real @ Y4 ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_776_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_real] :
      ( ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ Y4 )
      = ( qbs_mo8573344932208643166l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur6875533127466166616s_real @ Y4 ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_777_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_o] :
      ( ( sigma_6279906219187228174real_o @ borel_6524799422816628122nnreal @ Y4 )
      = ( qbs_mo7845494039804091494real_o @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_778_ennreal_Ostandard__borel__r__full__faithful,axiom,
    ! [Y4: sigma_measure_nat] :
      ( ( sigma_1856489715609627354al_nat @ borel_6524799422816628122nnreal @ Y4 )
      = ( qbs_mo182749989559755394al_nat @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) ) ).

% ennreal.standard_borel_r_full_faithful
thf(fact_779_ennreal_Oqbs__morphism__dest,axiom,
    ! [K: extend8495563244428889912nnreal > produc2422161461964618553l_real,Y4: sigma_2308072346491277622l_real] :
      ( ( member7354208599470296673l_real @ K @ ( qbs_mo8045053474288007175l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur3029786804310284173l_real @ Y4 ) ) )
     => ( member7354208599470296673l_real @ K @ ( sigma_1014563338549229999l_real @ borel_6524799422816628122nnreal @ Y4 ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_780_ennreal_Oqbs__morphism__dest,axiom,
    ! [K: extend8495563244428889912nnreal > produc7716430852924023517t_real,Y4: sigma_5310753476256395226t_real] :
      ( ( member2197300837325509253t_real @ K @ ( qbs_mo3970737951806219947t_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2011440102552004913t_real @ Y4 ) ) )
     => ( member2197300837325509253t_real @ K @ ( sigma_2672489485171276883t_real @ borel_6524799422816628122nnreal @ Y4 ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_781_ennreal_Oqbs__morphism__dest,axiom,
    ! [K: extend8495563244428889912nnreal > real,Y4: sigma_measure_real] :
      ( ( member2874014351250825754l_real @ K @ ( qbs_mo8573344932208643166l_real @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur6875533127466166616s_real @ Y4 ) ) )
     => ( member2874014351250825754l_real @ K @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ Y4 ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_782_ennreal_Oqbs__morphism__dest,axiom,
    ! [K: extend8495563244428889912nnreal > $o,Y4: sigma_measure_o] :
      ( ( member8095236870201115968real_o @ K @ ( qbs_mo7845494039804091494real_o @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur2705496967258476524_qbs_o @ Y4 ) ) )
     => ( member8095236870201115968real_o @ K @ ( sigma_6279906219187228174real_o @ borel_6524799422816628122nnreal @ Y4 ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_783_ennreal_Oqbs__morphism__dest,axiom,
    ! [K: extend8495563244428889912nnreal > nat,Y4: sigma_measure_nat] :
      ( ( member6436672275262627518al_nat @ K @ ( qbs_mo182749989559755394al_nat @ ( measur2642298986910087140nnreal @ borel_6524799422816628122nnreal ) @ ( measur4416158800429964412bs_nat @ Y4 ) ) )
     => ( member6436672275262627518al_nat @ K @ ( sigma_1856489715609627354al_nat @ borel_6524799422816628122nnreal @ Y4 ) ) ) ).

% ennreal.qbs_morphism_dest
thf(fact_784_case__prod__unfold,axiom,
    ( produc8421786640700244582_b_a_a
    = ( ^ [C3: product_prod_real_b > produc7194034526493574155_nat_b > a > a,P3: produc3260441645384719156_nat_b] : ( C3 @ ( produc4243170370849295320_nat_b @ P3 ) @ ( produc5631749817756603418_nat_b @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_785_case__prod__unfold,axiom,
    ( produc2543982510095308599_b_a_a
    = ( ^ [C3: produc7194034526493574155_nat_b > a > a,P3: produc6209191202727158440at_b_a] : ( C3 @ ( produc2385479481072933486at_b_a @ P3 ) @ ( produc5227817866736700332at_b_a @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_786_case__prod__unfold,axiom,
    ( produc7016990068875982646_b_a_a
    = ( ^ [C3: real > b > produc7194034526493574155_nat_b > a > a,P3: product_prod_real_b] : ( C3 @ ( product_fst_real_b @ P3 ) @ ( product_snd_real_b @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_787_case__prod__unfold,axiom,
    ( produc286368409806729619_b_a_a
    = ( ^ [C3: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,P3: produc778026099599033348at_b_a] : ( C3 @ ( produc7871839370796975378at_b_a @ P3 ) @ ( produc6884036649769537872at_b_a @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_788_case__prod__unfold,axiom,
    ( produc6302854954412089281_b_a_a
    = ( ^ [C3: b > produc6209191202727158440at_b_a > a,P3: produc4883615123598087894at_b_a] : ( C3 @ ( produc7380424691241595748at_b_a @ P3 ) @ ( produc7666751653692995234at_b_a @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_789_case__prod__unfold,axiom,
    ( produc7827042608974387084_b_a_a
    = ( ^ [C3: real > b > a > a,P3: product_prod_real_b] : ( C3 @ ( product_fst_real_b @ P3 ) @ ( product_snd_real_b @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_790_case__prod__unfold,axiom,
    ( produc634687561522189117at_b_a
    = ( ^ [C3: product_prod_real_b > produc7194034526493574155_nat_b > a,P3: produc3260441645384719156_nat_b] : ( C3 @ ( produc4243170370849295320_nat_b @ P3 ) @ ( produc5631749817756603418_nat_b @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_791_case__prod__beta_H,axiom,
    ( produc8421786640700244582_b_a_a
    = ( ^ [F2: product_prod_real_b > produc7194034526493574155_nat_b > a > a,X: produc3260441645384719156_nat_b] : ( F2 @ ( produc4243170370849295320_nat_b @ X ) @ ( produc5631749817756603418_nat_b @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_792_case__prod__beta_H,axiom,
    ( produc2543982510095308599_b_a_a
    = ( ^ [F2: produc7194034526493574155_nat_b > a > a,X: produc6209191202727158440at_b_a] : ( F2 @ ( produc2385479481072933486at_b_a @ X ) @ ( produc5227817866736700332at_b_a @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_793_case__prod__beta_H,axiom,
    ( produc7016990068875982646_b_a_a
    = ( ^ [F2: real > b > produc7194034526493574155_nat_b > a > a,X: product_prod_real_b] : ( F2 @ ( product_fst_real_b @ X ) @ ( product_snd_real_b @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_794_case__prod__beta_H,axiom,
    ( produc286368409806729619_b_a_a
    = ( ^ [F2: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,X: produc778026099599033348at_b_a] : ( F2 @ ( produc7871839370796975378at_b_a @ X ) @ ( produc6884036649769537872at_b_a @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_795_case__prod__beta_H,axiom,
    ( produc6302854954412089281_b_a_a
    = ( ^ [F2: b > produc6209191202727158440at_b_a > a,X: produc4883615123598087894at_b_a] : ( F2 @ ( produc7380424691241595748at_b_a @ X ) @ ( produc7666751653692995234at_b_a @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_796_case__prod__beta_H,axiom,
    ( produc7827042608974387084_b_a_a
    = ( ^ [F2: real > b > a > a,X: product_prod_real_b] : ( F2 @ ( product_fst_real_b @ X ) @ ( product_snd_real_b @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_797_case__prod__beta_H,axiom,
    ( produc634687561522189117at_b_a
    = ( ^ [F2: product_prod_real_b > produc7194034526493574155_nat_b > a,X: produc3260441645384719156_nat_b] : ( F2 @ ( produc4243170370849295320_nat_b @ X ) @ ( produc5631749817756603418_nat_b @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_798_prod_Osplit__sel__asm,axiom,
    ! [P: ( a > a ) > $o,F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,Prod: produc3260441645384719156_nat_b] :
      ( ( P @ ( produc8421786640700244582_b_a_a @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc1946830598933159916_nat_b @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_799_prod_Osplit__sel__asm,axiom,
    ! [P: a > $o,F: produc7194034526493574155_nat_b > a > a,Prod: produc6209191202727158440at_b_a] :
      ( ( P @ ( produc2543982510095308599_b_a_a @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc8509509857186669914at_b_a @ ( produc2385479481072933486at_b_a @ Prod ) @ ( produc5227817866736700332at_b_a @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc2385479481072933486at_b_a @ Prod ) @ ( produc5227817866736700332at_b_a @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_800_prod_Osplit__sel__asm,axiom,
    ! [P: ( produc7194034526493574155_nat_b > a > a ) > $o,F: real > b > produc7194034526493574155_nat_b > a > a,Prod: product_prod_real_b] :
      ( ( P @ ( produc7016990068875982646_b_a_a @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_real_b @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_801_prod_Osplit__sel__asm,axiom,
    ! [P: a > $o,F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,Prod: produc778026099599033348at_b_a] :
      ( ( P @ ( produc286368409806729619_b_a_a @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc653446490150804990at_b_a @ ( produc7871839370796975378at_b_a @ Prod ) @ ( produc6884036649769537872at_b_a @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc7871839370796975378at_b_a @ Prod ) @ ( produc6884036649769537872at_b_a @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_802_prod_Osplit__sel__asm,axiom,
    ! [P: a > $o,F: b > produc6209191202727158440at_b_a > a,Prod: produc4883615123598087894at_b_a] :
      ( ( P @ ( produc6302854954412089281_b_a_a @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc992390952543482448at_b_a @ ( produc7380424691241595748at_b_a @ Prod ) @ ( produc7666751653692995234at_b_a @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc7380424691241595748at_b_a @ Prod ) @ ( produc7666751653692995234at_b_a @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_803_prod_Osplit__sel__asm,axiom,
    ! [P: ( a > a ) > $o,F: real > b > a > a,Prod: product_prod_real_b] :
      ( ( P @ ( produc7827042608974387084_b_a_a @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_real_b @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_804_prod_Osplit__sel__asm,axiom,
    ! [P: a > $o,F: product_prod_real_b > produc7194034526493574155_nat_b > a,Prod: produc3260441645384719156_nat_b] :
      ( ( P @ ( produc634687561522189117at_b_a @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc1946830598933159916_nat_b @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) ) )
            & ~ ( P @ ( F @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_805_prod_Osplit__sel,axiom,
    ! [P: ( a > a ) > $o,F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,Prod: produc3260441645384719156_nat_b] :
      ( ( P @ ( produc8421786640700244582_b_a_a @ F @ Prod ) )
      = ( ( Prod
          = ( produc1946830598933159916_nat_b @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) ) )
       => ( P @ ( F @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_806_prod_Osplit__sel,axiom,
    ! [P: a > $o,F: produc7194034526493574155_nat_b > a > a,Prod: produc6209191202727158440at_b_a] :
      ( ( P @ ( produc2543982510095308599_b_a_a @ F @ Prod ) )
      = ( ( Prod
          = ( produc8509509857186669914at_b_a @ ( produc2385479481072933486at_b_a @ Prod ) @ ( produc5227817866736700332at_b_a @ Prod ) ) )
       => ( P @ ( F @ ( produc2385479481072933486at_b_a @ Prod ) @ ( produc5227817866736700332at_b_a @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_807_prod_Osplit__sel,axiom,
    ! [P: ( produc7194034526493574155_nat_b > a > a ) > $o,F: real > b > produc7194034526493574155_nat_b > a > a,Prod: product_prod_real_b] :
      ( ( P @ ( produc7016990068875982646_b_a_a @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_real_b @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_808_prod_Osplit__sel,axiom,
    ! [P: a > $o,F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,Prod: produc778026099599033348at_b_a] :
      ( ( P @ ( produc286368409806729619_b_a_a @ F @ Prod ) )
      = ( ( Prod
          = ( produc653446490150804990at_b_a @ ( produc7871839370796975378at_b_a @ Prod ) @ ( produc6884036649769537872at_b_a @ Prod ) ) )
       => ( P @ ( F @ ( produc7871839370796975378at_b_a @ Prod ) @ ( produc6884036649769537872at_b_a @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_809_prod_Osplit__sel,axiom,
    ! [P: a > $o,F: b > produc6209191202727158440at_b_a > a,Prod: produc4883615123598087894at_b_a] :
      ( ( P @ ( produc6302854954412089281_b_a_a @ F @ Prod ) )
      = ( ( Prod
          = ( produc992390952543482448at_b_a @ ( produc7380424691241595748at_b_a @ Prod ) @ ( produc7666751653692995234at_b_a @ Prod ) ) )
       => ( P @ ( F @ ( produc7380424691241595748at_b_a @ Prod ) @ ( produc7666751653692995234at_b_a @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_810_prod_Osplit__sel,axiom,
    ! [P: ( a > a ) > $o,F: real > b > a > a,Prod: product_prod_real_b] :
      ( ( P @ ( produc7827042608974387084_b_a_a @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_real_b @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_real_b @ Prod ) @ ( product_snd_real_b @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_811_prod_Osplit__sel,axiom,
    ! [P: a > $o,F: product_prod_real_b > produc7194034526493574155_nat_b > a,Prod: produc3260441645384719156_nat_b] :
      ( ( P @ ( produc634687561522189117at_b_a @ F @ Prod ) )
      = ( ( Prod
          = ( produc1946830598933159916_nat_b @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) ) )
       => ( P @ ( F @ ( produc4243170370849295320_nat_b @ Prod ) @ ( produc5631749817756603418_nat_b @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_812_snd__conv,axiom,
    ! [X1: real,X2: b] :
      ( ( product_snd_real_b @ ( product_Pair_real_b @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_813_snd__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_814_snd__conv,axiom,
    ! [X1: b > produc7194034526493574155_nat_b > a > a,X2: produc4883615123598087894at_b_a] :
      ( ( produc6884036649769537872at_b_a @ ( produc653446490150804990at_b_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_815_snd__conv,axiom,
    ! [X1: b,X2: produc6209191202727158440at_b_a] :
      ( ( produc7666751653692995234at_b_a @ ( produc992390952543482448at_b_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_816_snd__conv,axiom,
    ! [X1: produc7194034526493574155_nat_b,X2: a] :
      ( ( produc5227817866736700332at_b_a @ ( produc8509509857186669914at_b_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_817_snd__conv,axiom,
    ! [X1: product_prod_real_b,X2: produc7194034526493574155_nat_b] :
      ( ( produc5631749817756603418_nat_b @ ( produc1946830598933159916_nat_b @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_818_snd__conv,axiom,
    ! [X1: a,X2: b > produc7194034526493574155_nat_b > a > a] :
      ( ( produc7355951146076385094_b_a_a @ ( produc3671031927252941592_b_a_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_819_snd__conv,axiom,
    ! [X1: nat,X2: nat > b] :
      ( ( produc4809910044363821583_nat_b @ ( produc5292568363641424317_nat_b @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_820_snd__eqD,axiom,
    ! [X5: real,Y5: b,A: b] :
      ( ( ( product_snd_real_b @ ( product_Pair_real_b @ X5 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_821_snd__eqD,axiom,
    ! [X5: nat,Y5: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_822_snd__eqD,axiom,
    ! [X5: b > produc7194034526493574155_nat_b > a > a,Y5: produc4883615123598087894at_b_a,A: produc4883615123598087894at_b_a] :
      ( ( ( produc6884036649769537872at_b_a @ ( produc653446490150804990at_b_a @ X5 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_823_snd__eqD,axiom,
    ! [X5: b,Y5: produc6209191202727158440at_b_a,A: produc6209191202727158440at_b_a] :
      ( ( ( produc7666751653692995234at_b_a @ ( produc992390952543482448at_b_a @ X5 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_824_snd__eqD,axiom,
    ! [X5: produc7194034526493574155_nat_b,Y5: a,A: a] :
      ( ( ( produc5227817866736700332at_b_a @ ( produc8509509857186669914at_b_a @ X5 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_825_snd__eqD,axiom,
    ! [X5: product_prod_real_b,Y5: produc7194034526493574155_nat_b,A: produc7194034526493574155_nat_b] :
      ( ( ( produc5631749817756603418_nat_b @ ( produc1946830598933159916_nat_b @ X5 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_826_snd__eqD,axiom,
    ! [X5: a,Y5: b > produc7194034526493574155_nat_b > a > a,A: b > produc7194034526493574155_nat_b > a > a] :
      ( ( ( produc7355951146076385094_b_a_a @ ( produc3671031927252941592_b_a_a @ X5 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_827_snd__eqD,axiom,
    ! [X5: nat,Y5: nat > b,A: nat > b] :
      ( ( ( produc4809910044363821583_nat_b @ ( produc5292568363641424317_nat_b @ X5 @ Y5 ) )
        = A )
     => ( Y5 = A ) ) ).

% snd_eqD
thf(fact_828_fst__conv,axiom,
    ! [X1: real,X2: b] :
      ( ( product_fst_real_b @ ( product_Pair_real_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_829_fst__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_830_fst__conv,axiom,
    ! [X1: b > produc7194034526493574155_nat_b > a > a,X2: produc4883615123598087894at_b_a] :
      ( ( produc7871839370796975378at_b_a @ ( produc653446490150804990at_b_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_831_fst__conv,axiom,
    ! [X1: b,X2: produc6209191202727158440at_b_a] :
      ( ( produc7380424691241595748at_b_a @ ( produc992390952543482448at_b_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_832_fst__conv,axiom,
    ! [X1: produc7194034526493574155_nat_b,X2: a] :
      ( ( produc2385479481072933486at_b_a @ ( produc8509509857186669914at_b_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_833_fst__conv,axiom,
    ! [X1: product_prod_real_b,X2: produc7194034526493574155_nat_b] :
      ( ( produc4243170370849295320_nat_b @ ( produc1946830598933159916_nat_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_834_fst__conv,axiom,
    ! [X1: a,X2: b > produc7194034526493574155_nat_b > a > a] :
      ( ( produc5967371699169076996_b_a_a @ ( produc3671031927252941592_b_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_835_fst__conv,axiom,
    ! [X1: nat,X2: nat > b] :
      ( ( produc3194919583230816977_nat_b @ ( produc5292568363641424317_nat_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_836_fst__eqD,axiom,
    ! [X5: real,Y5: b,A: real] :
      ( ( ( product_fst_real_b @ ( product_Pair_real_b @ X5 @ Y5 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_837_fst__eqD,axiom,
    ! [X5: nat,Y5: nat,A: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_838_fst__eqD,axiom,
    ! [X5: b > produc7194034526493574155_nat_b > a > a,Y5: produc4883615123598087894at_b_a,A: b > produc7194034526493574155_nat_b > a > a] :
      ( ( ( produc7871839370796975378at_b_a @ ( produc653446490150804990at_b_a @ X5 @ Y5 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_839_fst__eqD,axiom,
    ! [X5: b,Y5: produc6209191202727158440at_b_a,A: b] :
      ( ( ( produc7380424691241595748at_b_a @ ( produc992390952543482448at_b_a @ X5 @ Y5 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_840_fst__eqD,axiom,
    ! [X5: produc7194034526493574155_nat_b,Y5: a,A: produc7194034526493574155_nat_b] :
      ( ( ( produc2385479481072933486at_b_a @ ( produc8509509857186669914at_b_a @ X5 @ Y5 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_841_fst__eqD,axiom,
    ! [X5: product_prod_real_b,Y5: produc7194034526493574155_nat_b,A: product_prod_real_b] :
      ( ( ( produc4243170370849295320_nat_b @ ( produc1946830598933159916_nat_b @ X5 @ Y5 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_842_fst__eqD,axiom,
    ! [X5: a,Y5: b > produc7194034526493574155_nat_b > a > a,A: a] :
      ( ( ( produc5967371699169076996_b_a_a @ ( produc3671031927252941592_b_a_a @ X5 @ Y5 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_843_fst__eqD,axiom,
    ! [X5: nat,Y5: nat > b,A: nat] :
      ( ( ( produc3194919583230816977_nat_b @ ( produc5292568363641424317_nat_b @ X5 @ Y5 ) )
        = A )
     => ( X5 = A ) ) ).

% fst_eqD
thf(fact_844_qbs__morphism__cong,axiom,
    ! [X4: quasi_borel_o,F: $o > real,G: $o > real,Y4: quasi_borel_real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( qbs_space_o @ X4 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X4 @ Y4 ) )
       => ( member_o_real @ G @ ( qbs_morphism_o_real @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_845_qbs__morphism__cong,axiom,
    ! [X4: quasi_borel_nat,F: nat > real,G: nat > real,Y4: quasi_borel_real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( qbs_space_nat @ X4 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
       => ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_846_qbs__morphism__cong,axiom,
    ! [X4: quasi_borel_real,F: real > a,G: real > a,Y4: quasi_borel_a] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( qbs_space_real @ X4 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X4 @ Y4 ) )
       => ( member_real_a @ G @ ( qbs_morphism_real_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_847_qbs__morphism__cong,axiom,
    ! [X4: quasi_805850127037308055at_b_a,F: produc1285742814635146737at_b_a > a,G: produc1285742814635146737at_b_a > a,Y4: quasi_borel_a] :
      ( ! [X3: produc1285742814635146737at_b_a] :
          ( ( member1674185049213098906at_b_a @ X3 @ ( qbs_sp8457271469619332135at_b_a @ X4 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( member9091339440257553353_b_a_a @ F @ ( qbs_mo7901260496052035013_b_a_a @ X4 @ Y4 ) )
       => ( member9091339440257553353_b_a_a @ G @ ( qbs_mo7901260496052035013_b_a_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_848_qbs__morphism__cong,axiom,
    ! [X4: quasi_borel_nat_b,F: ( nat > b ) > produc432120507726958376_b_a_a > a,G: ( nat > b ) > produc432120507726958376_b_a_a > a,Y4: quasi_2547937147209033259_a_a_a] :
      ( ! [X3: nat > b] :
          ( ( member_nat_b @ X3 @ ( qbs_space_nat_b @ X4 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( member1811236362415105851_a_a_a @ F @ ( qbs_mo8618518167108125737_a_a_a @ X4 @ Y4 ) )
       => ( member1811236362415105851_a_a_a @ G @ ( qbs_mo8618518167108125737_a_a_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_849_qbs__morphism__cong,axiom,
    ! [X4: quasi_borel_o,F: $o > $o > $o,G: $o > $o > $o,Y4: quasi_borel_o_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( qbs_space_o @ X4 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( member_o_o_o @ F @ ( qbs_morphism_o_o_o @ X4 @ Y4 ) )
       => ( member_o_o_o @ G @ ( qbs_morphism_o_o_o @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_850_qbs__morphism__cong,axiom,
    ! [X4: quasi_borel_o,F: $o > $o,G: $o > $o,Y4: quasi_borel_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( qbs_space_o @ X4 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( member_o_o @ F @ ( qbs_morphism_o_o @ X4 @ Y4 ) )
       => ( member_o_o @ G @ ( qbs_morphism_o_o @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_851_qbs__morphismE_I2_J,axiom,
    ! [F: $o > real,X4: quasi_borel_o,Y4: quasi_borel_real,X5: $o] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X4 @ Y4 ) )
     => ( ( member_o @ X5 @ ( qbs_space_o @ X4 ) )
       => ( member_real @ ( F @ X5 ) @ ( qbs_space_real @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_852_qbs__morphismE_I2_J,axiom,
    ! [F: nat > real,X4: quasi_borel_nat,Y4: quasi_borel_real,X5: nat] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
     => ( ( member_nat @ X5 @ ( qbs_space_nat @ X4 ) )
       => ( member_real @ ( F @ X5 ) @ ( qbs_space_real @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_853_qbs__morphismE_I2_J,axiom,
    ! [F: real > a,X4: quasi_borel_real,Y4: quasi_borel_a,X5: real] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X4 @ Y4 ) )
     => ( ( member_real @ X5 @ ( qbs_space_real @ X4 ) )
       => ( member_a @ ( F @ X5 ) @ ( qbs_space_a @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_854_qbs__morphismE_I2_J,axiom,
    ! [F: $o > $o,X4: quasi_borel_o,Y4: quasi_borel_o,X5: $o] :
      ( ( member_o_o @ F @ ( qbs_morphism_o_o @ X4 @ Y4 ) )
     => ( ( member_o @ X5 @ ( qbs_space_o @ X4 ) )
       => ( member_o @ ( F @ X5 ) @ ( qbs_space_o @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_855_qbs__morphismE_I2_J,axiom,
    ! [F: $o > $o > $o,X4: quasi_borel_o,Y4: quasi_borel_o_o,X5: $o] :
      ( ( member_o_o_o @ F @ ( qbs_morphism_o_o_o @ X4 @ Y4 ) )
     => ( ( member_o @ X5 @ ( qbs_space_o @ X4 ) )
       => ( member_o_o @ ( F @ X5 ) @ ( qbs_space_o_o @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_856_qbs__morphismE_I2_J,axiom,
    ! [F: ( $o > real ) > $o > real,X4: quasi_borel_o_real,Y4: quasi_borel_o_real,X5: $o > real] :
      ( ( member_o_real_o_real @ F @ ( qbs_mo365478700572875064o_real @ X4 @ Y4 ) )
     => ( ( member_o_real @ X5 @ ( qbs_space_o_real @ X4 ) )
       => ( member_o_real @ ( F @ X5 ) @ ( qbs_space_o_real @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_857_qbs__morphismE_I2_J,axiom,
    ! [F: ( $o > real ) > $o > $o,X4: quasi_borel_o_real,Y4: quasi_borel_o_o,X5: $o > real] :
      ( ( member_o_real_o_o @ F @ ( qbs_mo1176275702449585058al_o_o @ X4 @ Y4 ) )
     => ( ( member_o_real @ X5 @ ( qbs_space_o_real @ X4 ) )
       => ( member_o_o @ ( F @ X5 ) @ ( qbs_space_o_o @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_858_qbs__morphismE_I2_J,axiom,
    ! [F: ( $o > real ) > nat > real,X4: quasi_borel_o_real,Y4: quasi_borel_nat_real,X5: $o > real] :
      ( ( member1678482115722949866t_real @ F @ ( qbs_mo1345815866363944536t_real @ X4 @ Y4 ) )
     => ( ( member_o_real @ X5 @ ( qbs_space_o_real @ X4 ) )
       => ( member_nat_real @ ( F @ X5 ) @ ( qbs_space_nat_real @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_859_qbs__morphismE_I2_J,axiom,
    ! [F: ( $o > $o ) > $o > real,X4: quasi_borel_o_o,Y4: quasi_borel_o_real,X5: $o > $o] :
      ( ( member_o_o_o_real @ F @ ( qbs_mo7207216841383384598o_real @ X4 @ Y4 ) )
     => ( ( member_o_o @ X5 @ ( qbs_space_o_o @ X4 ) )
       => ( member_o_real @ ( F @ X5 ) @ ( qbs_space_o_real @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_860_qbs__morphismE_I2_J,axiom,
    ! [F: ( $o > $o ) > $o > $o,X4: quasi_borel_o_o,Y4: quasi_borel_o_o,X5: $o > $o] :
      ( ( member_o_o_o_o @ F @ ( qbs_morphism_o_o_o_o @ X4 @ Y4 ) )
     => ( ( member_o_o @ X5 @ ( qbs_space_o_o @ X4 ) )
       => ( member_o_o @ ( F @ X5 ) @ ( qbs_space_o_o @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_861_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > b,X4: quasi_borel_b,R3: real] :
      ( ( member_real_b @ Alpha2 @ ( qbs_Mx_b @ X4 ) )
     => ( member_b @ ( Alpha2 @ R3 ) @ ( qbs_space_b @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_862_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > a,X4: quasi_borel_a,R3: real] :
      ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X4 ) )
     => ( member_a @ ( Alpha2 @ R3 ) @ ( qbs_space_a @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_863_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > $o > real,X4: quasi_borel_o_real,R3: real] :
      ( ( member_real_o_real @ Alpha2 @ ( qbs_Mx_o_real @ X4 ) )
     => ( member_o_real @ ( Alpha2 @ R3 ) @ ( qbs_space_o_real @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_864_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > $o > $o,X4: quasi_borel_o_o,R3: real] :
      ( ( member_real_o_o @ Alpha2 @ ( qbs_Mx_o_o @ X4 ) )
     => ( member_o_o @ ( Alpha2 @ R3 ) @ ( qbs_space_o_o @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_865_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > nat > real,X4: quasi_borel_nat_real,R3: real] :
      ( ( member_real_nat_real @ Alpha2 @ ( qbs_Mx_nat_real @ X4 ) )
     => ( member_nat_real @ ( Alpha2 @ R3 ) @ ( qbs_space_nat_real @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_866_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > nat > b,X4: quasi_borel_nat_b,R3: real] :
      ( ( member_real_nat_b @ Alpha2 @ ( qbs_Mx_nat_b @ X4 ) )
     => ( member_nat_b @ ( Alpha2 @ R3 ) @ ( qbs_space_nat_b @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_867_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > $o > $o > $o,X4: quasi_borel_o_o_o,R3: real] :
      ( ( member_real_o_o_o @ Alpha2 @ ( qbs_Mx_o_o_o @ X4 ) )
     => ( member_o_o_o @ ( Alpha2 @ R3 ) @ ( qbs_space_o_o_o @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_868_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > produc7194034526493574155_nat_b,X4: quasi_7334096685896923387_nat_b,R3: real] :
      ( ( member8664213473904227437_nat_b @ Alpha2 @ ( qbs_Mx1531206082039394900_nat_b @ X4 ) )
     => ( member2882485241872049954_nat_b @ ( Alpha2 @ R3 ) @ ( qbs_sp2305461467746415189_nat_b @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_869_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > b > produc7194034526493574155_nat_b > a > a,X4: quasi_3364728345918372901_b_a_a,R3: real] :
      ( ( member924168396124881715_b_a_a @ Alpha2 @ ( qbs_Mx703024977670096246_b_a_a @ X4 ) )
     => ( member8306749417536871976_b_a_a @ ( Alpha2 @ R3 ) @ ( qbs_sp4382579563652706485_b_a_a @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_870_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha2: real > ( nat > b ) > produc432120507726958376_b_a_a > a,X4: quasi_8909271866012278712_a_a_a,R3: real] :
      ( ( member2371902028876673734_a_a_a @ Alpha2 @ ( qbs_Mx6122946776669592201_a_a_a @ X4 ) )
     => ( member1811236362415105851_a_a_a @ ( Alpha2 @ R3 ) @ ( qbs_sp8333110983225252808_a_a_a @ X4 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_871_qbs__space__eq__Mx,axiom,
    ! [X4: quasi_331246355085487092at_b_a,Y4: quasi_331246355085487092at_b_a] :
      ( ( ( qbs_Mx5919617464495388621at_b_a @ X4 )
        = ( qbs_Mx5919617464495388621at_b_a @ Y4 ) )
     => ( ( qbs_sp4001621062360912206at_b_a @ X4 )
        = ( qbs_sp4001621062360912206at_b_a @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_872_qbs__space__eq__Mx,axiom,
    ! [X4: quasi_borel_nat_b,Y4: quasi_borel_nat_b] :
      ( ( ( qbs_Mx_nat_b @ X4 )
        = ( qbs_Mx_nat_b @ Y4 ) )
     => ( ( qbs_space_nat_b @ X4 )
        = ( qbs_space_nat_b @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_873_qbs__space__eq__Mx,axiom,
    ! [X4: quasi_3364728345918372901_b_a_a,Y4: quasi_3364728345918372901_b_a_a] :
      ( ( ( qbs_Mx703024977670096246_b_a_a @ X4 )
        = ( qbs_Mx703024977670096246_b_a_a @ Y4 ) )
     => ( ( qbs_sp4382579563652706485_b_a_a @ X4 )
        = ( qbs_sp4382579563652706485_b_a_a @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_874_qbs__space__eq__Mx,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_borel_b] :
      ( ( ( qbs_Mx_b @ X4 )
        = ( qbs_Mx_b @ Y4 ) )
     => ( ( qbs_space_b @ X4 )
        = ( qbs_space_b @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_875_qbs__space__eq__Mx,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X4 )
        = ( qbs_Mx_a @ Y4 ) )
     => ( ( qbs_space_a @ X4 )
        = ( qbs_space_a @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_876_qbs__space__eq__Mx,axiom,
    ! [X4: quasi_7334096685896923387_nat_b,Y4: quasi_7334096685896923387_nat_b] :
      ( ( ( qbs_Mx1531206082039394900_nat_b @ X4 )
        = ( qbs_Mx1531206082039394900_nat_b @ Y4 ) )
     => ( ( qbs_sp2305461467746415189_nat_b @ X4 )
        = ( qbs_sp2305461467746415189_nat_b @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_877_snd__def,axiom,
    ( produc7355951146076385094_b_a_a
    = ( produc6460842837556059154_b_a_a
      @ ^ [X12: a,X22: b > produc7194034526493574155_nat_b > a > a] : X22 ) ) ).

% snd_def
thf(fact_878_snd__def,axiom,
    ( produc4809910044363821583_nat_b
    = ( produc8787918764284747772_nat_b
      @ ^ [X12: nat,X22: nat > b] : X22 ) ) ).

% snd_def
thf(fact_879_snd__def,axiom,
    ( produc5227817866736700332at_b_a
    = ( produc2543982510095308599_b_a_a
      @ ^ [X12: produc7194034526493574155_nat_b,X22: a] : X22 ) ) ).

% snd_def
thf(fact_880_qbs__morphism__const,axiom,
    ! [Y5: real,Y4: quasi_borel_real,X4: quasi_borel_o] :
      ( ( member_real @ Y5 @ ( qbs_space_real @ Y4 ) )
     => ( member_o_real
        @ ^ [Uu: $o] : Y5
        @ ( qbs_morphism_o_real @ X4 @ Y4 ) ) ) ).

% qbs_morphism_const
thf(fact_881_qbs__morphism__const,axiom,
    ! [Y5: real,Y4: quasi_borel_real,X4: quasi_borel_nat] :
      ( ( member_real @ Y5 @ ( qbs_space_real @ Y4 ) )
     => ( member_nat_real
        @ ^ [Uu: nat] : Y5
        @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) ) ) ).

% qbs_morphism_const
thf(fact_882_qbs__morphism__const,axiom,
    ! [Y5: a,Y4: quasi_borel_a,X4: quasi_borel_real] :
      ( ( member_a @ Y5 @ ( qbs_space_a @ Y4 ) )
     => ( member_real_a
        @ ^ [Uu: real] : Y5
        @ ( qbs_morphism_real_a @ X4 @ Y4 ) ) ) ).

% qbs_morphism_const
thf(fact_883_qbs__morphism__const,axiom,
    ! [Y5: a,Y4: quasi_borel_a,X4: quasi_805850127037308055at_b_a] :
      ( ( member_a @ Y5 @ ( qbs_space_a @ Y4 ) )
     => ( member9091339440257553353_b_a_a
        @ ^ [Uu: produc1285742814635146737at_b_a] : Y5
        @ ( qbs_mo7901260496052035013_b_a_a @ X4 @ Y4 ) ) ) ).

% qbs_morphism_const
thf(fact_884_qbs__morphism__const,axiom,
    ! [Y5: produc432120507726958376_b_a_a > a,Y4: quasi_2547937147209033259_a_a_a,X4: quasi_borel_nat_b] :
      ( ( member4631714215213530194_a_a_a @ Y5 @ ( qbs_sp9089677150445993861_a_a_a @ Y4 ) )
     => ( member1811236362415105851_a_a_a
        @ ^ [Uu: nat > b] : Y5
        @ ( qbs_mo8618518167108125737_a_a_a @ X4 @ Y4 ) ) ) ).

% qbs_morphism_const
thf(fact_885_qbs__morphism__const,axiom,
    ! [Y5: $o > $o,Y4: quasi_borel_o_o,X4: quasi_borel_o] :
      ( ( member_o_o @ Y5 @ ( qbs_space_o_o @ Y4 ) )
     => ( member_o_o_o
        @ ^ [Uu: $o] : Y5
        @ ( qbs_morphism_o_o_o @ X4 @ Y4 ) ) ) ).

% qbs_morphism_const
thf(fact_886_qbs__morphism__const,axiom,
    ! [Y5: $o,Y4: quasi_borel_o,X4: quasi_borel_o] :
      ( ( member_o @ Y5 @ ( qbs_space_o @ Y4 ) )
     => ( member_o_o
        @ ^ [Uu: $o] : Y5
        @ ( qbs_morphism_o_o @ X4 @ Y4 ) ) ) ).

% qbs_morphism_const
thf(fact_887_qbs__closed2__dest,axiom,
    ! [X5: b,X4: quasi_borel_b] :
      ( ( member_b @ X5 @ ( qbs_space_b @ X4 ) )
     => ( member_real_b
        @ ^ [R: real] : X5
        @ ( qbs_Mx_b @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_888_qbs__closed2__dest,axiom,
    ! [X5: a,X4: quasi_borel_a] :
      ( ( member_a @ X5 @ ( qbs_space_a @ X4 ) )
     => ( member_real_a
        @ ^ [R: real] : X5
        @ ( qbs_Mx_a @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_889_qbs__closed2__dest,axiom,
    ! [X5: $o > real,X4: quasi_borel_o_real] :
      ( ( member_o_real @ X5 @ ( qbs_space_o_real @ X4 ) )
     => ( member_real_o_real
        @ ^ [R: real] : X5
        @ ( qbs_Mx_o_real @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_890_qbs__closed2__dest,axiom,
    ! [X5: $o > $o,X4: quasi_borel_o_o] :
      ( ( member_o_o @ X5 @ ( qbs_space_o_o @ X4 ) )
     => ( member_real_o_o
        @ ^ [R: real] : X5
        @ ( qbs_Mx_o_o @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_891_qbs__closed2__dest,axiom,
    ! [X5: nat > real,X4: quasi_borel_nat_real] :
      ( ( member_nat_real @ X5 @ ( qbs_space_nat_real @ X4 ) )
     => ( member_real_nat_real
        @ ^ [R: real] : X5
        @ ( qbs_Mx_nat_real @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_892_qbs__closed2__dest,axiom,
    ! [X5: nat > b,X4: quasi_borel_nat_b] :
      ( ( member_nat_b @ X5 @ ( qbs_space_nat_b @ X4 ) )
     => ( member_real_nat_b
        @ ^ [R: real] : X5
        @ ( qbs_Mx_nat_b @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_893_qbs__closed2__dest,axiom,
    ! [X5: $o > $o > $o,X4: quasi_borel_o_o_o] :
      ( ( member_o_o_o @ X5 @ ( qbs_space_o_o_o @ X4 ) )
     => ( member_real_o_o_o
        @ ^ [R: real] : X5
        @ ( qbs_Mx_o_o_o @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_894_qbs__closed2__dest,axiom,
    ! [X5: produc7194034526493574155_nat_b,X4: quasi_7334096685896923387_nat_b] :
      ( ( member2882485241872049954_nat_b @ X5 @ ( qbs_sp2305461467746415189_nat_b @ X4 ) )
     => ( member8664213473904227437_nat_b
        @ ^ [R: real] : X5
        @ ( qbs_Mx1531206082039394900_nat_b @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_895_qbs__closed2__dest,axiom,
    ! [X5: b > produc7194034526493574155_nat_b > a > a,X4: quasi_3364728345918372901_b_a_a] :
      ( ( member8306749417536871976_b_a_a @ X5 @ ( qbs_sp4382579563652706485_b_a_a @ X4 ) )
     => ( member924168396124881715_b_a_a
        @ ^ [R: real] : X5
        @ ( qbs_Mx703024977670096246_b_a_a @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_896_qbs__closed2__dest,axiom,
    ! [X5: ( nat > b ) > produc432120507726958376_b_a_a > a,X4: quasi_8909271866012278712_a_a_a] :
      ( ( member1811236362415105851_a_a_a @ X5 @ ( qbs_sp8333110983225252808_a_a_a @ X4 ) )
     => ( member2371902028876673734_a_a_a
        @ ^ [R: real] : X5
        @ ( qbs_Mx6122946776669592201_a_a_a @ X4 ) ) ) ).

% qbs_closed2_dest
thf(fact_897_fst__def,axiom,
    ( produc5967371699169076996_b_a_a
    = ( produc8533229574609689361_a_a_a
      @ ^ [X12: a,X22: b > produc7194034526493574155_nat_b > a > a] : X12 ) ) ).

% fst_def
thf(fact_898_fst__def,axiom,
    ( produc3194919583230816977_nat_b
    = ( produc5914306107723085754_b_nat
      @ ^ [X12: nat,X22: nat > b] : X12 ) ) ).

% fst_def
thf(fact_899_pair__qbs__Mx__def,axiom,
    ( binary3024450868691050456Mx_b_b
    = ( ^ [X6: quasi_borel_b,Y6: quasi_borel_b] :
          ( collec944675128910930013od_b_b
          @ ^ [F2: real > product_prod_b_b] :
              ( ( member_real_b @ ( comp_P8101773103369595751b_real @ product_fst_b_b @ F2 ) @ ( qbs_Mx_b @ X6 ) )
              & ( member_real_b @ ( comp_P8101773103369595751b_real @ product_snd_b_b @ F2 ) @ ( qbs_Mx_b @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_900_pair__qbs__Mx__def,axiom,
    ( binary3024450868691050455Mx_b_a
    = ( ^ [X6: quasi_borel_b,Y6: quasi_borel_a] :
          ( collec873641088864584028od_b_a
          @ ^ [F2: real > product_prod_b_a] :
              ( ( member_real_b @ ( comp_P4150881467751007272b_real @ product_fst_b_a @ F2 ) @ ( qbs_Mx_b @ X6 ) )
              & ( member_real_a @ ( comp_P7778034889417652391a_real @ product_snd_b_a @ F2 ) @ ( qbs_Mx_a @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_901_pair__qbs__Mx__def,axiom,
    ( binary5811906450663726489Mx_a_b
    = ( ^ [X6: quasi_borel_a,Y6: quasi_borel_b] :
          ( collec1580324129515916702od_a_b
          @ ^ [F2: real > product_prod_a_b] :
              ( ( member_real_a @ ( comp_P1838655012641556709a_real @ product_fst_a_b @ F2 ) @ ( qbs_Mx_a @ X6 ) )
              & ( member_real_b @ ( comp_P7434873627829687398b_real @ product_snd_a_b @ F2 ) @ ( qbs_Mx_b @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_902_pair__qbs__Mx__def,axiom,
    ( binary5811906450663726488Mx_a_a
    = ( ^ [X6: quasi_borel_a,Y6: quasi_borel_a] :
          ( collec1509290089469570717od_a_a
          @ ^ [F2: real > product_prod_a_a] :
              ( ( member_real_a @ ( comp_P7111135413877744038a_real @ product_fst_a_a @ F2 ) @ ( qbs_Mx_a @ X6 ) )
              & ( member_real_a @ ( comp_P7111135413877744038a_real @ product_snd_a_a @ F2 ) @ ( qbs_Mx_a @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_903_pair__qbs__Mx__def,axiom,
    ( binary8007162529914129073at_b_b
    = ( ^ [X6: quasi_borel_nat_b,Y6: quasi_borel_b] :
          ( collec1478500567074120246at_b_b
          @ ^ [F2: real > product_prod_nat_b_b] :
              ( ( member_real_nat_b @ ( comp_P8709072139942781365b_real @ product_fst_nat_b_b @ F2 ) @ ( qbs_Mx_nat_b @ X6 ) )
              & ( member_real_b @ ( comp_P7469846031817660622b_real @ product_snd_nat_b_b @ F2 ) @ ( qbs_Mx_b @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_904_pair__qbs__Mx__def,axiom,
    ( binary8007162529914129072at_b_a
    = ( ^ [X6: quasi_borel_nat_b,Y6: quasi_borel_a] :
          ( collec1407466527027774261at_b_a
          @ ^ [F2: real > product_prod_nat_b_a] :
              ( ( member_real_nat_b @ ( comp_P2494481557755200758b_real @ product_fst_nat_b_a @ F2 ) @ ( qbs_Mx_nat_b @ X6 ) )
              & ( member_real_a @ ( comp_P7146107817865717262a_real @ product_snd_nat_b_a @ F2 ) @ ( qbs_Mx_a @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_905_pair__qbs__Mx__def,axiom,
    ( binary302611147984943551_nat_b
    = ( ^ [X6: quasi_borel_b,Y6: quasi_borel_nat_b] :
          ( collec8659260748451152068_nat_b
          @ ^ [F2: real > product_prod_b_nat_b] :
              ( ( member_real_b @ ( comp_P447806182887879104b_real @ product_fst_b_nat_b @ F2 ) @ ( qbs_Mx_b @ X6 ) )
              & ( member_real_nat_b @ ( comp_P9026943010836981671b_real @ product_snd_b_nat_b @ F2 ) @ ( qbs_Mx_nat_b @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_906_pair__qbs__Mx__def,axiom,
    ( binary3388367389974938112_nat_b
    = ( ^ [X6: quasi_borel_a,Y6: quasi_borel_nat_b] :
          ( collec4458043773282809989_nat_b
          @ ^ [F2: real > product_prod_a_nat_b] :
              ( ( member_real_a @ ( comp_P6709701325929358526a_real @ product_fst_a_nat_b @ F2 ) @ ( qbs_Mx_a @ X6 ) )
              & ( member_real_nat_b @ ( comp_P2831363238483770278b_real @ product_snd_a_nat_b @ F2 ) @ ( qbs_Mx_nat_b @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_907_pair__qbs__Mx__def,axiom,
    ( binary8267100837097976444_nat_b
    = ( ^ [X6: quasi_borel_nat,Y6: quasi_borel_nat_b] :
          ( collec500969705129490603_nat_b
          @ ^ [F2: real > produc7194034526493574155_nat_b] :
              ( ( member_real_nat @ ( comp_P2417870527327321324t_real @ produc3194919583230816977_nat_b @ F2 ) @ ( qbs_Mx_nat @ X6 ) )
              & ( member_real_nat_b @ ( comp_P5830504534140287012b_real @ produc4809910044363821583_nat_b @ F2 ) @ ( qbs_Mx_nat_b @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_908_pair__qbs__Mx__def,axiom,
    ( binary5772008827079125784_nat_b
    = ( ^ [X6: quasi_borel_nat_b,Y6: quasi_borel_nat_b] :
          ( collec1372824333772097821_nat_b
          @ ^ [F2: real > produc5451071141416689831_nat_b] :
              ( ( member_real_nat_b @ ( comp_P5734460479892768782b_real @ produc2218887877279390083_nat_b @ F2 ) @ ( qbs_Mx_nat_b @ X6 ) )
              & ( member_real_nat_b @ ( comp_P5734460479892768782b_real @ produc1773315609376244421_nat_b @ F2 ) @ ( qbs_Mx_nat_b @ Y6 ) ) ) ) ) ) ).

% pair_qbs_Mx_def
thf(fact_909_qbs__morphism__pair__iff,axiom,
    ! [F: $o > produc2422161461964618553l_real,X4: quasi_borel_o,Y4: quasi_borel_real,Z2: quasi_borel_real] :
      ( ( member8602590400058854173l_real @ F @ ( qbs_mo6076292073755557401l_real @ X4 @ ( binary210465342804807751l_real @ Y4 @ Z2 ) ) )
      = ( ( member_o_real @ ( comp_P7245199795817254996real_o @ produc5828954698716094813l_real @ F ) @ ( qbs_morphism_o_real @ X4 @ Y4 ) )
        & ( member_o_real @ ( comp_P7245199795817254996real_o @ produc3484788084999411615l_real @ F ) @ ( qbs_morphism_o_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_910_qbs__morphism__pair__iff,axiom,
    ! [F: nat > produc2422161461964618553l_real,X4: quasi_borel_nat,Y4: quasi_borel_real,Z2: quasi_borel_real] :
      ( ( member4944736856719700937l_real @ F @ ( qbs_mo6520021129653552495l_real @ X4 @ ( binary210465342804807751l_real @ Y4 @ Z2 ) ) )
      = ( ( member_nat_real @ ( comp_P6943517474843379220al_nat @ produc5828954698716094813l_real @ F ) @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
        & ( member_nat_real @ ( comp_P6943517474843379220al_nat @ produc3484788084999411615l_real @ F ) @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_911_qbs__morphism__pair__iff,axiom,
    ! [F: $o > product_prod_real_o,X4: quasi_borel_o,Y4: quasi_borel_real,Z2: quasi_borel_o] :
      ( ( member3923301605023534227real_o @ F @ ( qbs_mo6612738326854831105real_o @ X4 @ ( binary4057238779996960701real_o @ Y4 @ Z2 ) ) )
      = ( ( member_o_real @ ( comp_P5544674181125083430real_o @ product_fst_real_o @ F ) @ ( qbs_morphism_o_real @ X4 @ Y4 ) )
        & ( member_o_o @ ( comp_P6996295418169922244_o_o_o @ product_snd_real_o @ F ) @ ( qbs_morphism_o_o @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_912_qbs__morphism__pair__iff,axiom,
    ! [F: real > product_prod_a_a,X4: quasi_borel_real,Y4: quasi_borel_a,Z2: quasi_borel_a] :
      ( ( member4570177857406309467od_a_a @ F @ ( qbs_mo7578050507665182465od_a_a @ X4 @ ( binary3918968219557196013bs_a_a @ Y4 @ Z2 ) ) )
      = ( ( member_real_a @ ( comp_P7111135413877744038a_real @ product_fst_a_a @ F ) @ ( qbs_morphism_real_a @ X4 @ Y4 ) )
        & ( member_real_a @ ( comp_P7111135413877744038a_real @ product_snd_a_a @ F ) @ ( qbs_morphism_real_a @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_913_qbs__morphism__pair__iff,axiom,
    ! [F: $o > product_prod_o_real,X4: quasi_borel_o,Y4: quasi_borel_o,Z2: quasi_borel_real] :
      ( ( member5560034075029800485o_real @ F @ ( qbs_mo4016795016617356947o_real @ X4 @ ( binary2548174404955791127o_real @ Y4 @ Z2 ) ) )
      = ( ( member_o_o @ ( comp_P3594550797714445910al_o_o @ product_fst_o_real @ F ) @ ( qbs_morphism_o_o @ X4 @ Y4 ) )
        & ( member_o_real @ ( comp_P2467800417234829076real_o @ product_snd_o_real @ F ) @ ( qbs_morphism_o_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_914_qbs__morphism__pair__iff,axiom,
    ! [F: $o > product_prod_o_o,X4: quasi_borel_o,Y4: quasi_borel_o,Z2: quasi_borel_o] :
      ( ( member3028044298935597963od_o_o @ F @ ( qbs_mo2565689321687337991od_o_o @ X4 @ ( binary9159730317414444269bs_o_o @ Y4 @ Z2 ) ) )
      = ( ( member_o_o @ ( comp_P2137996956111167876_o_o_o @ product_fst_o_o @ F ) @ ( qbs_morphism_o_o @ X4 @ Y4 ) )
        & ( member_o_o @ ( comp_P2137996956111167876_o_o_o @ product_snd_o_o @ F ) @ ( qbs_morphism_o_o @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_915_qbs__morphism__pair__iff,axiom,
    ! [F: $o > product_prod_real_b,X4: quasi_borel_o,Y4: quasi_borel_real,Z2: quasi_borel_b] :
      ( ( member5019927097664096890real_b @ F @ ( qbs_mo4240396306489266408real_b @ X4 @ ( binary5762279046363823268real_b @ Y4 @ Z2 ) ) )
      = ( ( member_o_real @ ( comp_P4534312704559669631real_o @ product_fst_real_b @ F ) @ ( qbs_morphism_o_real @ X4 @ Y4 ) )
        & ( member_o_b @ ( comp_P7576070003299083396_b_b_o @ product_snd_real_b @ F ) @ ( qbs_morphism_o_b @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_916_qbs__morphism__pair__iff,axiom,
    ! [F: nat > product_prod_real_b,X4: quasi_borel_nat,Y4: quasi_borel_real,Z2: quasi_borel_b] :
      ( ( member3863283534846713550real_b @ F @ ( qbs_mo2610884262441363474real_b @ X4 @ ( binary5762279046363823268real_b @ Y4 @ Z2 ) ) )
      = ( ( member_nat_real @ ( comp_P5368948159003514281al_nat @ product_fst_real_b @ F ) @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
        & ( member_nat_b @ ( comp_P1903834106658840036_b_nat @ product_snd_real_b @ F ) @ ( qbs_morphism_nat_b @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_917_qbs__morphism__pair__iff,axiom,
    ! [F: $o > produc7716430852924023517t_real,X4: quasi_borel_o,Y4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member8739982677887872321t_real @ F @ ( qbs_mo928686258813670845t_real @ X4 @ ( binary2938201662863479403t_real @ Y4 @ Z2 ) ) )
      = ( ( member_o_nat @ ( comp_P7117026637527641996_nat_o @ product_fst_nat_real @ F ) @ ( qbs_morphism_o_nat @ X4 @ Y4 ) )
        & ( member_o_real @ ( comp_P5209122424551656240real_o @ product_snd_nat_real @ F ) @ ( qbs_morphism_o_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_918_qbs__morphism__pair__iff,axiom,
    ! [F: nat > produc7716430852924023517t_real,X4: quasi_borel_nat,Y4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member4237364305948650477t_real @ F @ ( qbs_mo2896996399151425555t_real @ X4 @ ( binary2938201662863479403t_real @ Y4 @ Z2 ) ) )
      = ( ( member_nat_nat @ ( comp_P3616034732471067868at_nat @ product_fst_nat_real @ F ) @ ( qbs_morphism_nat_nat @ X4 @ Y4 ) )
        & ( member_nat_real @ ( comp_P2638273056294992440al_nat @ product_snd_nat_real @ F ) @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_pair_iff
thf(fact_919_snd__qbs__morphism,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_nat_b] : ( member3507851424016974971_nat_b @ produc4809910044363821583_nat_b @ ( qbs_mo8783080198091093025_nat_b @ ( binary7246370547177875303_nat_b @ X4 @ Y4 ) @ Y4 ) ) ).

% snd_qbs_morphism
thf(fact_920_snd__qbs__morphism,axiom,
    ! [X4: quasi_8132950864781310810_nat_b,Y4: quasi_borel_a] : ( member9091339440257553353_b_a_a @ produc7105243515406072463at_b_a @ ( qbs_mo7901260496052035013_b_a_a @ ( binary4006988140340982583at_b_a @ X4 @ Y4 ) @ Y4 ) ) ).

% snd_qbs_morphism
thf(fact_921_snd__qbs__morphism,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b] : ( member5457248586424104727al_b_b @ product_snd_real_b @ ( qbs_mo4743646238736226749al_b_b @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Y4 ) ) ).

% snd_qbs_morphism
thf(fact_922_snd__qbs__morphism,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_real] : ( member4539693599630147619l_real @ product_snd_nat_real @ ( qbs_mo8506632985670901265l_real @ ( binary2938201662863479403t_real @ X4 @ Y4 ) @ Y4 ) ) ).

% snd_qbs_morphism
thf(fact_923_snd__qbs__morphism,axiom,
    ! [X4: quasi_3364728345918372901_b_a_a,Y4: quasi_3683678768404249670at_b_a] : ( member849528351786989072at_b_a @ produc6884036649769537872at_b_a @ ( qbs_mo404491099253798484at_b_a @ ( binary8035175780229655208at_b_a @ X4 @ Y4 ) @ Y4 ) ) ).

% snd_qbs_morphism
thf(fact_924_snd__qbs__morphism,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_1826430223877327896at_b_a] : ( member506078014504387252at_b_a @ produc7666751653692995234at_b_a @ ( qbs_mo2050090585163134072at_b_a @ ( binary3850590118635936762at_b_a @ X4 @ Y4 ) @ Y4 ) ) ).

% snd_qbs_morphism
thf(fact_925_snd__qbs__morphism,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_3364728345918372901_b_a_a] : ( member7720416752616689619_b_a_a @ produc7355951146076385094_b_a_a @ ( qbs_mo187615740505320655_b_a_a @ ( binary9021634363669778414_b_a_a @ X4 @ Y4 ) @ Y4 ) ) ).

% snd_qbs_morphism
thf(fact_926_fst__qbs__morphism,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_nat_b] : ( member7852119857225428113_b_nat @ produc3194919583230816977_nat_b @ ( qbs_mo1186143403256568021_b_nat @ ( binary7246370547177875303_nat_b @ X4 @ Y4 ) @ X4 ) ) ).

% fst_qbs_morphism
thf(fact_927_fst__qbs__morphism,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_b] : ( member5160407386997391786b_real @ product_fst_real_b @ ( qbs_mo7149229302404819822b_real @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ X4 ) ) ).

% fst_qbs_morphism
thf(fact_928_fst__qbs__morphism,axiom,
    ! [X4: quasi_borel_nat,Y4: quasi_borel_real] : ( member3149932044224082247al_nat @ product_fst_nat_real @ ( qbs_mo2463579765191384501al_nat @ ( binary2938201662863479403t_real @ X4 @ Y4 ) @ X4 ) ) ).

% fst_qbs_morphism
thf(fact_929_fst__qbs__morphism,axiom,
    ! [X4: quasi_3364728345918372901_b_a_a,Y4: quasi_3683678768404249670at_b_a] : ( member7779032602282684123_b_a_a @ produc7871839370796975378at_b_a @ ( qbs_mo272395263599635585_b_a_a @ ( binary8035175780229655208at_b_a @ X4 @ Y4 ) @ X4 ) ) ).

% fst_qbs_morphism
thf(fact_930_fst__qbs__morphism,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_1826430223877327896at_b_a] : ( member871964291409841545_b_a_b @ produc7380424691241595748at_b_a @ ( qbs_mo1164823027738856751_b_a_b @ ( binary3850590118635936762at_b_a @ X4 @ Y4 ) @ X4 ) ) ).

% fst_qbs_morphism
thf(fact_931_fst__qbs__morphism,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_3364728345918372901_b_a_a] : ( member4631714215213530194_a_a_a @ produc5967371699169076996_b_a_a @ ( qbs_mo5048501348360587598_a_a_a @ ( binary9021634363669778414_b_a_a @ X4 @ Y4 ) @ X4 ) ) ).

% fst_qbs_morphism
thf(fact_932_qbs__closed3__dest_H,axiom,
    ! [P: real > nat,Fi: nat > real > produc778026099599033348at_b_a,X4: quasi_331246355085487092at_b_a] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( ! [I4: nat] : ( member8314179565955891814at_b_a @ ( Fi @ I4 ) @ ( qbs_Mx5919617464495388621at_b_a @ X4 ) )
       => ( member8314179565955891814at_b_a
          @ ^ [R: real] : ( Fi @ ( P @ R ) @ R )
          @ ( qbs_Mx5919617464495388621at_b_a @ X4 ) ) ) ) ).

% qbs_closed3_dest'
thf(fact_933_qbs__closed3__dest_H,axiom,
    ! [P: real > nat,Fi: nat > real > nat > b,X4: quasi_borel_nat_b] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( ! [I4: nat] : ( member_real_nat_b @ ( Fi @ I4 ) @ ( qbs_Mx_nat_b @ X4 ) )
       => ( member_real_nat_b
          @ ^ [R: real] : ( Fi @ ( P @ R ) @ R )
          @ ( qbs_Mx_nat_b @ X4 ) ) ) ) ).

% qbs_closed3_dest'
thf(fact_934_qbs__closed3__dest_H,axiom,
    ! [P: real > nat,Fi: nat > real > b > produc7194034526493574155_nat_b > a > a,X4: quasi_3364728345918372901_b_a_a] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( ! [I4: nat] : ( member924168396124881715_b_a_a @ ( Fi @ I4 ) @ ( qbs_Mx703024977670096246_b_a_a @ X4 ) )
       => ( member924168396124881715_b_a_a
          @ ^ [R: real] : ( Fi @ ( P @ R ) @ R )
          @ ( qbs_Mx703024977670096246_b_a_a @ X4 ) ) ) ) ).

% qbs_closed3_dest'
thf(fact_935_qbs__closed3__dest_H,axiom,
    ! [P: real > nat,Fi: nat > real > b,X4: quasi_borel_b] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( ! [I4: nat] : ( member_real_b @ ( Fi @ I4 ) @ ( qbs_Mx_b @ X4 ) )
       => ( member_real_b
          @ ^ [R: real] : ( Fi @ ( P @ R ) @ R )
          @ ( qbs_Mx_b @ X4 ) ) ) ) ).

% qbs_closed3_dest'
thf(fact_936_qbs__closed3__dest_H,axiom,
    ! [P: real > nat,Fi: nat > real > a,X4: quasi_borel_a] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( ! [I4: nat] : ( member_real_a @ ( Fi @ I4 ) @ ( qbs_Mx_a @ X4 ) )
       => ( member_real_a
          @ ^ [R: real] : ( Fi @ ( P @ R ) @ R )
          @ ( qbs_Mx_a @ X4 ) ) ) ) ).

% qbs_closed3_dest'
thf(fact_937_qbs__morphism__Pair1,axiom,
    ! [X5: nat,X4: quasi_borel_nat,Y4: quasi_borel_nat] :
      ( ( member_nat @ X5 @ ( qbs_space_nat @ X4 ) )
     => ( member6743353555990375057at_nat @ ( product_Pair_nat_nat @ X5 ) @ ( qbs_mo3651027331794440503at_nat @ Y4 @ ( binary6297550310575005455at_nat @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair1
thf(fact_938_qbs__morphism__Pair1,axiom,
    ! [X5: nat,X4: quasi_borel_nat,Y4: quasi_borel_nat_b] :
      ( ( member_nat @ X5 @ ( qbs_space_nat @ X4 ) )
     => ( member3100617602161326603_nat_b @ ( produc5292568363641424317_nat_b @ X5 ) @ ( qbs_mo3787342477198169849_nat_b @ Y4 @ ( binary7246370547177875303_nat_b @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair1
thf(fact_939_qbs__morphism__Pair1,axiom,
    ! [X5: product_prod_real_b,X4: quasi_7777979582039567160real_b,Y4: quasi_7334096685896923387_nat_b] :
      ( ( member7555082154638246367real_b @ X5 @ ( qbs_sp4150966586872015890real_b @ X4 ) )
     => ( member3516147537297476681_nat_b @ ( produc1946830598933159916_nat_b @ X5 ) @ ( qbs_mo3539708487391302639_nat_b @ Y4 @ ( binary7297433035349996738_nat_b @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair1
thf(fact_940_qbs__morphism__Pair1,axiom,
    ! [X5: produc7194034526493574155_nat_b,X4: quasi_7334096685896923387_nat_b,Y4: quasi_borel_a] :
      ( ( member2882485241872049954_nat_b @ X5 @ ( qbs_sp2305461467746415189_nat_b @ X4 ) )
     => ( member4213348123347683520at_b_a @ ( produc8509509857186669914at_b_a @ X5 ) @ ( qbs_mo388721149857207982at_b_a @ Y4 @ ( binary4113689719790979332at_b_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair1
thf(fact_941_qbs__morphism__Pair1,axiom,
    ! [X5: real,X4: quasi_borel_real,Y4: quasi_borel_b] :
      ( ( member_real @ X5 @ ( qbs_space_real @ X4 ) )
     => ( member5108440089302923873real_b @ ( product_Pair_real_b @ X5 ) @ ( qbs_mo3164329421503706703real_b @ Y4 @ ( binary5762279046363823268real_b @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair1
thf(fact_942_qbs__morphism__Pair1,axiom,
    ! [X5: nat,X4: quasi_borel_nat,Y4: quasi_borel_real] :
      ( ( member_nat @ X5 @ ( qbs_space_nat @ X4 ) )
     => ( member969486235890772753t_real @ ( produc7837566107596912789t_real @ X5 ) @ ( qbs_mo3601394221918002359t_real @ Y4 @ ( binary2938201662863479403t_real @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair1
thf(fact_943_qbs__morphism__Pair1,axiom,
    ! [X5: b > produc7194034526493574155_nat_b > a > a,X4: quasi_3364728345918372901_b_a_a,Y4: quasi_3683678768404249670at_b_a] :
      ( ( member8306749417536871976_b_a_a @ X5 @ ( qbs_sp4382579563652706485_b_a_a @ X4 ) )
     => ( member9142974548522789008at_b_a @ ( produc653446490150804990at_b_a @ X5 ) @ ( qbs_mo4893338430385348052at_b_a @ Y4 @ ( binary8035175780229655208at_b_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair1
thf(fact_944_qbs__morphism__Pair1,axiom,
    ! [X5: b,X4: quasi_borel_b,Y4: quasi_1826430223877327896at_b_a] :
      ( ( member_b @ X5 @ ( qbs_space_b @ X4 ) )
     => ( member2688202504559472564at_b_a @ ( produc992390952543482448at_b_a @ X5 ) @ ( qbs_mo6862686886989468792at_b_a @ Y4 @ ( binary3850590118635936762at_b_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair1
thf(fact_945_qbs__morphism__Pair1,axiom,
    ! [X5: a,X4: quasi_borel_a,Y4: quasi_3364728345918372901_b_a_a] :
      ( ( member_a @ X5 @ ( qbs_space_a @ X4 ) )
     => ( member6989227910922024165_b_a_a @ ( produc3671031927252941592_b_a_a @ X5 ) @ ( qbs_mo8505670199031306849_b_a_a @ Y4 @ ( binary9021634363669778414_b_a_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair1
thf(fact_946_qbs__morphism__Pair2,axiom,
    ! [Y5: nat,Y4: quasi_borel_nat,X4: quasi_borel_nat] :
      ( ( member_nat @ Y5 @ ( qbs_space_nat @ Y4 ) )
     => ( member6743353555990375057at_nat
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ Y5 )
        @ ( qbs_mo3651027331794440503at_nat @ X4 @ ( binary6297550310575005455at_nat @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair2
thf(fact_947_qbs__morphism__Pair2,axiom,
    ! [Y5: nat > b,Y4: quasi_borel_nat_b,X4: quasi_borel_nat] :
      ( ( member_nat_b @ Y5 @ ( qbs_space_nat_b @ Y4 ) )
     => ( member7258681010406981265_nat_b
        @ ^ [X: nat] : ( produc5292568363641424317_nat_b @ X @ Y5 )
        @ ( qbs_mo3020559592811146837_nat_b @ X4 @ ( binary7246370547177875303_nat_b @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair2
thf(fact_948_qbs__morphism__Pair2,axiom,
    ! [Y5: a,Y4: quasi_borel_a,X4: quasi_7334096685896923387_nat_b] :
      ( ( member_a @ Y5 @ ( qbs_space_a @ Y4 ) )
     => ( member2603177927859825257at_b_a
        @ ^ [X: produc7194034526493574155_nat_b] : ( produc8509509857186669914at_b_a @ X @ Y5 )
        @ ( qbs_mo8559266479951684141at_b_a @ X4 @ ( binary4113689719790979332at_b_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair2
thf(fact_949_qbs__morphism__Pair2,axiom,
    ! [Y5: produc7194034526493574155_nat_b,Y4: quasi_7334096685896923387_nat_b,X4: quasi_7777979582039567160real_b] :
      ( ( member2882485241872049954_nat_b @ Y5 @ ( qbs_sp2305461467746415189_nat_b @ Y4 ) )
     => ( member1339377625142798924_nat_b
        @ ^ [X: product_prod_real_b] : ( produc1946830598933159916_nat_b @ X @ Y5 )
        @ ( qbs_mo6514647438471240434_nat_b @ X4 @ ( binary7297433035349996738_nat_b @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair2
thf(fact_950_qbs__morphism__Pair2,axiom,
    ! [Y5: b,Y4: quasi_borel_b,X4: quasi_borel_real] :
      ( ( member_b @ Y5 @ ( qbs_space_b @ Y4 ) )
     => ( member1077921568469065898real_b
        @ ^ [X: real] : ( product_Pair_real_b @ X @ Y5 )
        @ ( qbs_mo1948987424667538542real_b @ X4 @ ( binary5762279046363823268real_b @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair2
thf(fact_951_qbs__morphism__Pair2,axiom,
    ! [Y5: real,Y4: quasi_borel_real,X4: quasi_borel_nat] :
      ( ( member_real @ Y5 @ ( qbs_space_real @ Y4 ) )
     => ( member4237364305948650477t_real
        @ ^ [X: nat] : ( produc7837566107596912789t_real @ X @ Y5 )
        @ ( qbs_mo2896996399151425555t_real @ X4 @ ( binary2938201662863479403t_real @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair2
thf(fact_952_qbs__morphism__Pair2,axiom,
    ! [Y5: produc4883615123598087894at_b_a,Y4: quasi_3683678768404249670at_b_a,X4: quasi_3364728345918372901_b_a_a] :
      ( ( member8939744840176901101at_b_a @ Y5 @ ( qbs_sp8005886530717159712at_b_a @ Y4 ) )
     => ( member980455446143721501at_b_a
        @ ^ [X: b > produc7194034526493574155_nat_b > a > a] : ( produc653446490150804990at_b_a @ X @ Y5 )
        @ ( qbs_mo966930433891195147at_b_a @ X4 @ ( binary8035175780229655208at_b_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair2
thf(fact_953_qbs__morphism__Pair2,axiom,
    ! [Y5: produc6209191202727158440at_b_a,Y4: quasi_1826430223877327896at_b_a,X4: quasi_borel_b] :
      ( ( member8909807733118965439at_b_a @ Y5 @ ( qbs_sp2434242944003618546at_b_a @ Y4 ) )
     => ( member7606463673362447471at_b_a
        @ ^ [X: b] : ( produc992390952543482448at_b_a @ X @ Y5 )
        @ ( qbs_mo6936898997215871069at_b_a @ X4 @ ( binary3850590118635936762at_b_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair2
thf(fact_954_qbs__morphism__Pair2,axiom,
    ! [Y5: b > produc7194034526493574155_nat_b > a > a,Y4: quasi_3364728345918372901_b_a_a,X4: quasi_borel_a] :
      ( ( member8306749417536871976_b_a_a @ Y5 @ ( qbs_sp4382579563652706485_b_a_a @ Y4 ) )
     => ( member3402851433038355110_b_a_a
        @ ^ [X: a] : ( produc3671031927252941592_b_a_a @ X @ Y5 )
        @ ( qbs_mo608747257592471202_b_a_a @ X4 @ ( binary9021634363669778414_b_a_a @ X4 @ Y4 ) ) ) ) ).

% qbs_morphism_Pair2
thf(fact_955_qbs__morphism__snd_H_H,axiom,
    ! [F: a > a,X4: quasi_borel_a,Y4: quasi_borel_a,Z2: quasi_8132950864781310810_nat_b] :
      ( ( member_a_a @ F @ ( qbs_morphism_a_a @ X4 @ Y4 ) )
     => ( member9091339440257553353_b_a_a
        @ ^ [K2: produc1285742814635146737at_b_a] : ( F @ ( produc7105243515406072463at_b_a @ K2 ) )
        @ ( qbs_mo7901260496052035013_b_a_a @ ( binary4006988140340982583at_b_a @ Z2 @ X4 ) @ Y4 ) ) ) ).

% qbs_morphism_snd''
thf(fact_956_qbs__morphism__snd_H_H,axiom,
    ! [F: ( nat > b ) > produc432120507726958376_b_a_a > a,X4: quasi_borel_nat_b,Y4: quasi_2547937147209033259_a_a_a,Z2: quasi_borel_nat] :
      ( ( member1811236362415105851_a_a_a @ F @ ( qbs_mo8618518167108125737_a_a_a @ X4 @ Y4 ) )
     => ( member7268114585821029756_a_a_a
        @ ^ [K2: produc7194034526493574155_nat_b] : ( F @ ( produc4809910044363821583_nat_b @ K2 ) )
        @ ( qbs_mo6729265999899219392_a_a_a @ ( binary7246370547177875303_nat_b @ Z2 @ X4 ) @ Y4 ) ) ) ).

% qbs_morphism_snd''
thf(fact_957_qbs__morphism__snd_H_H,axiom,
    ! [F: real > a,X4: quasi_borel_real,Y4: quasi_borel_a,Z2: quasi_borel_nat] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X4 @ Y4 ) )
     => ( member7970359591612582109real_a
        @ ^ [K2: produc7716430852924023517t_real] : ( F @ ( product_snd_nat_real @ K2 ) )
        @ ( qbs_mo7948468643621800025real_a @ ( binary2938201662863479403t_real @ Z2 @ X4 ) @ Y4 ) ) ) ).

% qbs_morphism_snd''
thf(fact_958_qbs__morphism__fst_H_H,axiom,
    ! [F: nat > real,X4: quasi_borel_nat,Y4: quasi_borel_real,Z2: quasi_borel_nat_b] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
     => ( member8270543924707358317b_real
        @ ^ [K2: produc7194034526493574155_nat_b] : ( F @ ( produc3194919583230816977_nat_b @ K2 ) )
        @ ( qbs_mo2982017945243388209b_real @ ( binary7246370547177875303_nat_b @ X4 @ Z2 ) @ Y4 ) ) ) ).

% qbs_morphism_fst''
thf(fact_959_qbs__morphism__fst_H_H,axiom,
    ! [F: produc3260441645384719156_nat_b > a,X4: quasi_8132950864781310810_nat_b,Y4: quasi_borel_a,Z2: quasi_borel_a] :
      ( ( member7813222213774494534at_b_a @ F @ ( qbs_mo8230009346921551938at_b_a @ X4 @ Y4 ) )
     => ( member9091339440257553353_b_a_a
        @ ^ [K2: produc1285742814635146737at_b_a] : ( F @ ( produc168530464060425293at_b_a @ K2 ) )
        @ ( qbs_mo7901260496052035013_b_a_a @ ( binary4006988140340982583at_b_a @ X4 @ Z2 ) @ Y4 ) ) ) ).

% qbs_morphism_fst''
thf(fact_960_qbs__morphism__fst_H_H,axiom,
    ! [F: real > a,X4: quasi_borel_real,Y4: quasi_borel_a,Z2: quasi_borel_b] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X4 @ Y4 ) )
     => ( member5457248582120875926al_b_a
        @ ^ [K2: product_prod_real_b] : ( F @ ( product_fst_real_b @ K2 ) )
        @ ( qbs_mo4743646238736226748al_b_a @ ( binary5762279046363823268real_b @ X4 @ Z2 ) @ Y4 ) ) ) ).

% qbs_morphism_fst''
thf(fact_961_qbs__morphism__fst_H_H,axiom,
    ! [F: nat > real,X4: quasi_borel_nat,Y4: quasi_borel_real,Z2: quasi_borel_real] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X4 @ Y4 ) )
     => ( member4539693599630147619l_real
        @ ^ [K2: produc7716430852924023517t_real] : ( F @ ( product_fst_nat_real @ K2 ) )
        @ ( qbs_mo8506632985670901265l_real @ ( binary2938201662863479403t_real @ X4 @ Z2 ) @ Y4 ) ) ) ).

% qbs_morphism_fst''
thf(fact_962_qbs__morphism__Pair2_H,axiom,
    ! [Y5: nat,Y4: quasi_borel_nat,F: product_prod_nat_nat > real,X4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_nat @ Y5 @ ( qbs_space_nat @ Y4 ) )
     => ( ( member4564283293661824327t_real @ F @ ( qbs_mo4558946786878123445t_real @ ( binary6297550310575005455at_nat @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [X: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y5 ) )
          @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_963_qbs__morphism__Pair2_H,axiom,
    ! [Y5: b,Y4: quasi_borel_b,F: product_prod_real_b > a,X4: quasi_borel_real,Z2: quasi_borel_a] :
      ( ( member_b @ Y5 @ ( qbs_space_b @ Y4 ) )
     => ( ( member5457248582120875926al_b_a @ F @ ( qbs_mo4743646238736226748al_b_a @ ( binary5762279046363823268real_b @ X4 @ Y4 ) @ Z2 ) )
       => ( member_real_a
          @ ^ [X: real] : ( F @ ( product_Pair_real_b @ X @ Y5 ) )
          @ ( qbs_morphism_real_a @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_964_qbs__morphism__Pair2_H,axiom,
    ! [Y5: real,Y4: quasi_borel_real,F: produc7716430852924023517t_real > real,X4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_real @ Y5 @ ( qbs_space_real @ Y4 ) )
     => ( ( member4539693599630147619l_real @ F @ ( qbs_mo8506632985670901265l_real @ ( binary2938201662863479403t_real @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [X: nat] : ( F @ ( produc7837566107596912789t_real @ X @ Y5 ) )
          @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_965_qbs__morphism__Pair2_H,axiom,
    ! [Y5: $o > real,Y4: quasi_borel_o_real,F: produc5742938386009301956o_real > real,X4: quasi_borel_o,Z2: quasi_borel_real] :
      ( ( member_o_real @ Y5 @ ( qbs_space_o_real @ Y4 ) )
     => ( ( member2756556309132193674l_real @ F @ ( qbs_mo5985082283978848376l_real @ ( binary3642445751346049034o_real @ X4 @ Y4 ) @ Z2 ) )
       => ( member_o_real
          @ ^ [X: $o] : ( F @ ( produc2807985352718036020o_real @ X @ Y5 ) )
          @ ( qbs_morphism_o_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_966_qbs__morphism__Pair2_H,axiom,
    ! [Y5: $o > real,Y4: quasi_borel_o_real,F: produc280003423409713882o_real > real,X4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_o_real @ Y5 @ ( qbs_space_o_real @ Y4 ) )
     => ( ( member6445645701922451900l_real @ F @ ( qbs_mo1157119722458481792l_real @ ( binary5684049858718027190o_real @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [X: nat] : ( F @ ( produc3730247675181576204o_real @ X @ Y5 ) )
          @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_967_qbs__morphism__Pair2_H,axiom,
    ! [Y5: $o > $o,Y4: quasi_borel_o_o,F: product_prod_o_o_o > real,X4: quasi_borel_o,Z2: quasi_borel_real] :
      ( ( member_o_o @ Y5 @ ( qbs_space_o_o @ Y4 ) )
     => ( ( member351977418977672334o_real @ F @ ( qbs_mo2340799334385100370o_real @ ( binary7113508051100410704_o_o_o @ X4 @ Y4 ) @ Z2 ) )
       => ( member_o_real
          @ ^ [X: $o] : ( F @ ( product_Pair_o_o_o2 @ X @ Y5 ) )
          @ ( qbs_morphism_o_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_968_qbs__morphism__Pair2_H,axiom,
    ! [Y5: $o > $o,Y4: quasi_borel_o_o,F: product_prod_nat_o_o > real,X4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_o_o @ Y5 @ ( qbs_space_o_o @ Y4 ) )
     => ( ( member6568351589859881180o_real @ F @ ( qbs_mo6718410125248496714o_real @ ( binary6585809602380229796at_o_o @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [X: nat] : ( F @ ( product_Pair_nat_o_o @ X @ Y5 ) )
          @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_969_qbs__morphism__Pair2_H,axiom,
    ! [Y5: nat > real,Y4: quasi_borel_nat_real,F: produc6359219807994557794t_real > real,X4: quasi_borel_o,Z2: quasi_borel_real] :
      ( ( member_nat_real @ Y5 @ ( qbs_space_nat_real @ Y4 ) )
     => ( ( member8304547364788473924l_real @ F @ ( qbs_mo3016021385324503816l_real @ ( binary8731047812821901318t_real @ X4 @ Y4 ) @ Z2 ) )
       => ( member_o_real
          @ ^ [X: $o] : ( F @ ( produc6777245629285450332t_real @ X @ Y5 ) )
          @ ( qbs_morphism_o_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_970_qbs__morphism__Pair2_H,axiom,
    ! [Y5: nat > real,Y4: quasi_borel_nat_real,F: produc9032034819326515148t_real > real,X4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_nat_real @ Y5 @ ( qbs_space_nat_real @ Y4 ) )
     => ( ( member8955425030672039570l_real @ F @ ( qbs_mo7647184989296994560l_real @ ( binary7981337641200944474t_real @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [X: nat] : ( F @ ( produc2678254361277162628t_real @ X @ Y5 ) )
          @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_971_qbs__morphism__Pair2_H,axiom,
    ! [Y5: nat > b,Y4: quasi_borel_nat_b,F: produc7194034526493574155_nat_b > real,X4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_nat_b @ Y5 @ ( qbs_space_nat_b @ Y4 ) )
     => ( ( member8270543924707358317b_real @ F @ ( qbs_mo2982017945243388209b_real @ ( binary7246370547177875303_nat_b @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [X: nat] : ( F @ ( produc5292568363641424317_nat_b @ X @ Y5 ) )
          @ ( qbs_mo2000642995705457910t_real @ X4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair2'
thf(fact_972_qbs__morphism__Pair1_H,axiom,
    ! [X5: nat,X4: quasi_borel_nat,F: product_prod_nat_nat > real,Y4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_nat @ X5 @ ( qbs_space_nat @ X4 ) )
     => ( ( member4564283293661824327t_real @ F @ ( qbs_mo4558946786878123445t_real @ ( binary6297550310575005455at_nat @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [Y: nat] : ( F @ ( product_Pair_nat_nat @ X5 @ Y ) )
          @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_973_qbs__morphism__Pair1_H,axiom,
    ! [X5: nat,X4: quasi_borel_nat,F: produc7716430852924023517t_real > a,Y4: quasi_borel_real,Z2: quasi_borel_a] :
      ( ( member_nat @ X5 @ ( qbs_space_nat @ X4 ) )
     => ( ( member7970359591612582109real_a @ F @ ( qbs_mo7948468643621800025real_a @ ( binary2938201662863479403t_real @ X4 @ Y4 ) @ Z2 ) )
       => ( member_real_a
          @ ^ [Y: real] : ( F @ ( produc7837566107596912789t_real @ X5 @ Y ) )
          @ ( qbs_morphism_real_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_974_qbs__morphism__Pair1_H,axiom,
    ! [X5: $o > real,X4: quasi_borel_o_real,F: produc1282461823455724746real_o > real,Y4: quasi_borel_o,Z2: quasi_borel_real] :
      ( ( member_o_real @ X5 @ ( qbs_space_o_real @ X4 ) )
     => ( ( member2138021612121904528o_real @ F @ ( qbs_mo5366547586968559230o_real @ ( binary4854780749520403088real_o @ X4 @ Y4 ) @ Z2 ) )
       => ( member_o_real
          @ ^ [Y: $o] : ( F @ ( produc4020320350892390074real_o @ X5 @ Y ) )
          @ ( qbs_morphism_o_real @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_975_qbs__morphism__Pair1_H,axiom,
    ! [X5: $o > real,X4: quasi_borel_o_real,F: produc3906027766007223732al_nat > real,Y4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_o_real @ X5 @ ( qbs_space_o_real @ X4 ) )
     => ( ( member8195127911244750742t_real @ F @ ( qbs_mo2906601931780780634t_real @ ( binary6165376888136000216al_nat @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [Y: nat] : ( F @ ( produc4211574704599549230al_nat @ X5 @ Y ) )
          @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_976_qbs__morphism__Pair1_H,axiom,
    ! [X5: $o > $o,X4: quasi_borel_o_o,F: product_prod_o_o_o2 > real,Y4: quasi_borel_o,Z2: quasi_borel_real] :
      ( ( member_o_o @ X5 @ ( qbs_space_o_o @ X4 ) )
     => ( ( member6128203064704897866o_real @ F @ ( qbs_mo8117024980112325902o_real @ ( binary7336721751802915012_o_o_o @ X4 @ Y4 ) @ Z2 ) )
       => ( member_o_real
          @ ^ [Y: $o] : ( F @ ( product_Pair_o_o_o @ X5 @ Y ) )
          @ ( qbs_morphism_o_real @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_977_qbs__morphism__Pair1_H,axiom,
    ! [X5: $o > $o,X4: quasi_borel_o_o,F: product_prod_o_o_nat > real,Y4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_o_o @ X5 @ ( qbs_space_o_o @ X4 ) )
     => ( ( member6881674728302740572t_real @ F @ ( qbs_mo7031733263691356106t_real @ ( binary1055580586269223332_o_nat @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [Y: nat] : ( F @ ( product_Pair_o_o_nat @ X5 @ Y ) )
          @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_978_qbs__morphism__Pair1_H,axiom,
    ! [X5: nat > real,X4: quasi_borel_nat_real,F: produc455489999581664562real_o > real,Y4: quasi_borel_o,Z2: quasi_borel_real] :
      ( ( member_nat_real @ X5 @ ( qbs_space_nat_real @ X4 ) )
     => ( ( member8323372967221233684o_real @ F @ ( qbs_mo3034846987757263576o_real @ ( binary2308975958763133326real_o @ X4 @ Y4 ) @ Z2 ) )
       => ( member_o_real
          @ ^ [Y: $o] : ( F @ ( produc355173775226682340real_o @ X5 @ Y ) )
          @ ( qbs_morphism_o_real @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_979_qbs__morphism__Pair1_H,axiom,
    ! [X5: nat > real,X4: quasi_borel_nat_real,F: produc6623316681580375884al_nat > real,Y4: quasi_borel_nat,Z2: quasi_borel_real] :
      ( ( member_nat_real @ X5 @ ( qbs_space_nat_real @ X4 ) )
     => ( ( member1551507034919228946t_real @ F @ ( qbs_mo243266993544183936t_real @ ( binary5460140620756238170al_nat @ X4 @ Y4 ) @ Z2 ) )
       => ( member_nat_real
          @ ^ [Y: nat] : ( F @ ( produc157057340832456324al_nat @ X5 @ Y ) )
          @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_980_qbs__morphism__Pair1_H,axiom,
    ! [X5: $o > real,X4: quasi_borel_o_real,F: produc2537719793687985680l_real > a,Y4: quasi_borel_real,Z2: quasi_borel_a] :
      ( ( member_o_real @ X5 @ ( qbs_space_o_real @ X4 ) )
     => ( ( member3619123087330862158real_a @ F @ ( qbs_mo4109026080723182324real_a @ ( binary6186829326267171252l_real @ X4 @ Y4 ) @ Z2 ) )
       => ( member_real_a
          @ ^ [Y: real] : ( F @ ( produc4108164088747874570l_real @ X5 @ Y ) )
          @ ( qbs_morphism_real_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_981_qbs__morphism__Pair1_H,axiom,
    ! [X5: $o > $o,X4: quasi_borel_o_o,F: produc3878078923543209970o_real > a,Y4: quasi_borel_real,Z2: quasi_borel_a] :
      ( ( member_o_o @ X5 @ ( qbs_space_o_o @ X4 ) )
     => ( ( member2298721579218887944real_a @ F @ ( qbs_mo5667206408012003716real_a @ ( binary8537276784804192896o_real @ X4 @ Y4 ) @ Z2 ) )
       => ( member_real_a
          @ ^ [Y: real] : ( F @ ( produc7702816386176179882o_real @ X5 @ Y ) )
          @ ( qbs_morphism_real_a @ Y4 @ Z2 ) ) ) ) ).

% qbs_morphism_Pair1'
thf(fact_982_case__prod__const,axiom,
    ! [C: a > a] :
      ( ( produc8421786640700244582_b_a_a
        @ ^ [I: product_prod_real_b,J2: produc7194034526493574155_nat_b] : C )
      = ( ^ [Uu: produc3260441645384719156_nat_b] : C ) ) ).

% case_prod_const
thf(fact_983_case__prod__const,axiom,
    ! [C: a] :
      ( ( produc2543982510095308599_b_a_a
        @ ^ [I: produc7194034526493574155_nat_b,J2: a] : C )
      = ( ^ [Uu: produc6209191202727158440at_b_a] : C ) ) ).

% case_prod_const
thf(fact_984_case__prod__const,axiom,
    ! [C: produc7194034526493574155_nat_b > a > a] :
      ( ( produc7016990068875982646_b_a_a
        @ ^ [I: real,J2: b] : C )
      = ( ^ [Uu: product_prod_real_b] : C ) ) ).

% case_prod_const
thf(fact_985_case__prod__const,axiom,
    ! [C: a] :
      ( ( produc286368409806729619_b_a_a
        @ ^ [I: b > produc7194034526493574155_nat_b > a > a,J2: produc4883615123598087894at_b_a] : C )
      = ( ^ [Uu: produc778026099599033348at_b_a] : C ) ) ).

% case_prod_const
thf(fact_986_case__prod__const,axiom,
    ! [C: a] :
      ( ( produc6302854954412089281_b_a_a
        @ ^ [I: b,J2: produc6209191202727158440at_b_a] : C )
      = ( ^ [Uu: produc4883615123598087894at_b_a] : C ) ) ).

% case_prod_const
thf(fact_987_case__prod__const,axiom,
    ! [C: a > a] :
      ( ( produc7827042608974387084_b_a_a
        @ ^ [I: real,J2: b] : C )
      = ( ^ [Uu: product_prod_real_b] : C ) ) ).

% case_prod_const
thf(fact_988_case__prod__const,axiom,
    ! [C: a] :
      ( ( produc634687561522189117at_b_a
        @ ^ [I: product_prod_real_b,J2: produc7194034526493574155_nat_b] : C )
      = ( ^ [Uu: produc3260441645384719156_nat_b] : C ) ) ).

% case_prod_const
thf(fact_989__092_060alpha_062_H__def,axiom,
    ( alpha
    = ( ^ [R: real] : ( produc4809910044363821583_nat_b @ ( coProd8554194001436178208tail_b @ ( produc5292568363641424317_nat_b @ ( suc @ na ) @ ( alpha2 @ R ) ) ) ) ) ) ).

% \<alpha>'_def
thf(fact_990_snd__fst__flip,axiom,
    ( product_snd_b_real
    = ( comp_P8514091664527323699b_real @ product_fst_real_b
      @ ( produc6367669841173908169real_b
        @ ^ [X: b,Y: real] : ( product_Pair_real_b @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_991_snd__fst__flip,axiom,
    ( product_snd_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_992_snd__fst__flip,axiom,
    ( produc5086754709204743842_b_a_a
    = ( comp_P3916664037200898678_b_a_a @ produc7871839370796975378at_b_a
      @ ( produc493878141207180929at_b_a
        @ ^ [X: produc4883615123598087894at_b_a,Y: b > produc7194034526493574155_nat_b > a > a] : ( produc653446490150804990at_b_a @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_993_snd__fst__flip,axiom,
    ( produc4895988176267928656_b_a_b
    = ( comp_P4460891510108933714_b_a_b @ produc7380424691241595748at_b_a
      @ ( produc4594359411059224449at_b_a
        @ ^ [X: produc6209191202727158440at_b_a,Y: b] : ( produc992390952543482448at_b_a @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_994_snd__fst__flip,axiom,
    ( produc7959643354479256708_nat_b
    = ( comp_P9041367404280096848_nat_b @ produc2385479481072933486at_b_a
      @ ( produc8484962418594779783at_b_a
        @ ^ [X: a,Y: produc7194034526493574155_nat_b] : ( produc8509509857186669914at_b_a @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_995_snd__fst__flip,axiom,
    ( produc6064941407584602266real_b
    = ( comp_P2632048606743913113real_b @ produc4243170370849295320_nat_b
      @ ( produc8266046411308739443_nat_b
        @ ^ [X: produc7194034526493574155_nat_b,Y: product_prod_real_b] : ( produc1946830598933159916_nat_b @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_996_snd__fst__flip,axiom,
    ( produc939803731942258180_a_a_a
    = ( comp_P8744978583584356205_a_a_a @ produc5967371699169076996_b_a_a
      @ ( produc5515788090559159677_b_a_a
        @ ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: a] : ( produc3671031927252941592_b_a_a @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_997_snd__fst__flip,axiom,
    ( produc2226587535245370161_b_nat
    = ( comp_P4321060880961368485_b_nat @ produc3194919583230816977_nat_b
      @ ( produc6066811893999682327_nat_b
        @ ^ [X: nat > b,Y: nat] : ( produc5292568363641424317_nat_b @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_998_snd__fst__flip,axiom,
    ( produc7355951146076385094_b_a_a
    = ( comp_P2588703832359241388_b_a_a @ produc8774596321889725890_a_a_a
      @ ( produc834709304643892985_a_a_a
        @ ^ [X: a,Y: b > produc7194034526493574155_nat_b > a > a] : ( produc6478256549973590486_a_a_a @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_999_snd__fst__flip,axiom,
    ( produc4809910044363821583_nat_b
    = ( comp_P272664786628135081_nat_b @ produc611597074112365555_b_nat
      @ ( produc3159300782968368719_b_nat
        @ ^ [X: nat,Y: nat > b] : ( produc2709245854522972895_b_nat @ Y @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_1000_fst__snd__flip,axiom,
    ( product_fst_b_real
    = ( comp_P488263325884094958b_real @ product_snd_real_b
      @ ( produc6367669841173908169real_b
        @ ^ [X: b,Y: real] : ( product_Pair_real_b @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1001_fst__snd__flip,axiom,
    ( product_fst_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1002_fst__snd__flip,axiom,
    ( produc6074557430232181348_b_a_a
    = ( comp_P477566498786489949_b_a_a @ produc6884036649769537872at_b_a
      @ ( produc493878141207180929at_b_a
        @ ^ [X: produc4883615123598087894at_b_a,Y: b > produc7194034526493574155_nat_b > a > a] : ( produc653446490150804990at_b_a @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1003_fst__snd__flip,axiom,
    ( produc4609661213816529170_b_a_b
    = ( comp_P4442744351759092527_b_a_b @ produc7666751653692995234at_b_a
      @ ( produc4594359411059224449at_b_a
        @ ^ [X: produc6209191202727158440at_b_a,Y: b] : ( produc992390952543482448at_b_a @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1004_fst__snd__flip,axiom,
    ( produc5117304968815489862_nat_b
    = ( comp_P4262335804278681323_nat_b @ produc5227817866736700332at_b_a
      @ ( produc8484962418594779783at_b_a
        @ ^ [X: a,Y: produc7194034526493574155_nat_b] : ( produc8509509857186669914at_b_a @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1005_fst__snd__flip,axiom,
    ( produc4676361960677294168real_b
    = ( comp_P1267499494818674838real_b @ produc5631749817756603418_nat_b
      @ ( produc8266046411308739443_nat_b
        @ ^ [X: produc7194034526493574155_nat_b,Y: product_prod_real_b] : ( produc1946830598933159916_nat_b @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1006_fst__snd__flip,axiom,
    ( produc5967371699169076996_b_a_a
    = ( comp_P8837201170851432813_b_a_a @ produc939803731942258180_a_a_a
      @ ( produc834709304643892985_a_a_a
        @ ^ [X: a,Y: b > produc7194034526493574155_nat_b > a > a] : ( produc6478256549973590486_a_a_a @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1007_fst__snd__flip,axiom,
    ( produc3194919583230816977_nat_b
    = ( comp_P6619190391790938277_nat_b @ produc2226587535245370161_b_nat
      @ ( produc3159300782968368719_b_nat
        @ ^ [X: nat,Y: nat > b] : ( produc2709245854522972895_b_nat @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1008_fst__snd__flip,axiom,
    ( produc8774596321889725890_a_a_a
    = ( comp_P3970313683510312364_a_a_a @ produc7355951146076385094_b_a_a
      @ ( produc5515788090559159677_b_a_a
        @ ^ [X: b > produc7194034526493574155_nat_b > a > a,Y: a] : ( produc3671031927252941592_b_a_a @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1009_fst__snd__flip,axiom,
    ( produc611597074112365555_b_nat
    = ( comp_P5478636132350303453_b_nat @ produc4809910044363821583_nat_b
      @ ( produc6066811893999682327_nat_b
        @ ^ [X: nat > b,Y: nat] : ( produc5292568363641424317_nat_b @ Y @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_1010_fst__diag__snd,axiom,
    ( ( comp_P3276979369905350645l_real @ produc5828954698716094813l_real
      @ ( comp_r97482993418849103l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ X @ X )
        @ produc3484788084999411615l_real ) )
    = produc3484788084999411615l_real ) ).

% fst_diag_snd
thf(fact_1011_fst__diag__snd,axiom,
    ( ( comp_P9159556223120822829_b_a_a @ produc7930298036489509699_b_a_a
      @ ( comp_b3789700955105422685_b_a_a
        @ ^ [X: b > produc7194034526493574155_nat_b > a > a] : ( produc5775047331815311703_b_a_a @ X @ X )
        @ produc7355951146076385094_b_a_a ) )
    = produc7355951146076385094_b_a_a ) ).

% fst_diag_snd
thf(fact_1012_fst__diag__snd,axiom,
    ( ( comp_P3511690182758988141_nat_b @ produc2218887877279390083_nat_b
      @ ( comp_n3851728294588896943_nat_b
        @ ^ [X: nat > b] : ( produc1832820645975422871_nat_b @ X @ X )
        @ produc4809910044363821583_nat_b ) )
    = produc4809910044363821583_nat_b ) ).

% fst_diag_snd
thf(fact_1013_snd__diag__fst,axiom,
    ( ( comp_P3276979369905350645l_real @ produc3484788084999411615l_real
      @ ( comp_r97482993418849103l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ X @ X )
        @ produc5828954698716094813l_real ) )
    = produc5828954698716094813l_real ) ).

% snd_diag_fst
thf(fact_1014_snd__diag__fst,axiom,
    ( ( comp_P1156977722522261166_b_a_a @ product_snd_a_a
      @ ( comp_a5577347777374339356_b_a_a
        @ ^ [X: a] : ( product_Pair_a_a @ X @ X )
        @ produc5967371699169076996_b_a_a ) )
    = produc5967371699169076996_b_a_a ) ).

% snd_diag_fst
thf(fact_1015_snd__diag__fst,axiom,
    ( ( comp_P5143406238261914043_nat_b @ product_snd_nat_nat
      @ ( comp_n8051653116556791961_nat_b
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
        @ produc3194919583230816977_nat_b ) )
    = produc3194919583230816977_nat_b ) ).

% snd_diag_fst
thf(fact_1016_measurable__split__replace,axiom,
    ! [F: $o > nat > ( nat > b ) > $o,G: $o > produc7194034526493574155_nat_b,M: sigma_measure_o,N3: sigma_measure_o] :
      ( ( member_o_o
        @ ^ [X: $o] : ( F @ X @ ( produc3194919583230816977_nat_b @ ( G @ X ) ) @ ( produc4809910044363821583_nat_b @ ( G @ X ) ) )
        @ ( sigma_measurable_o_o @ M @ N3 ) )
     => ( member_o_o
        @ ^ [X: $o] : ( produc7710141471362042158at_b_o @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_measurable_o_o @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1017_measurable__split__replace,axiom,
    ! [F: real > nat > ( nat > b ) > real,G: real > produc7194034526493574155_nat_b,M: sigma_measure_real,N3: sigma_measure_real] :
      ( ( member_real_real
        @ ^ [X: real] : ( F @ X @ ( produc3194919583230816977_nat_b @ ( G @ X ) ) @ ( produc4809910044363821583_nat_b @ ( G @ X ) ) )
        @ ( sigma_5267869275261027754l_real @ M @ N3 ) )
     => ( member_real_real
        @ ^ [X: real] : ( produc1876314459100775830b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1018_measurable__split__replace,axiom,
    ! [F: real > nat > ( nat > b ) > $o,G: real > produc7194034526493574155_nat_b,M: sigma_measure_real,N3: sigma_measure_o] :
      ( ( member_real_o
        @ ^ [X: real] : ( F @ X @ ( produc3194919583230816977_nat_b @ ( G @ X ) ) @ ( produc4809910044363821583_nat_b @ ( G @ X ) ) )
        @ ( sigma_3939073009482781210real_o @ M @ N3 ) )
     => ( member_real_o
        @ ^ [X: real] : ( produc7710141471362042158at_b_o @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_3939073009482781210real_o @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1019_measurable__split__replace,axiom,
    ! [F: real > nat > ( nat > b ) > nat,G: real > produc7194034526493574155_nat_b,M: sigma_measure_real,N3: sigma_measure_nat] :
      ( ( member_real_nat
        @ ^ [X: real] : ( F @ X @ ( produc3194919583230816977_nat_b @ ( G @ X ) ) @ ( produc4809910044363821583_nat_b @ ( G @ X ) ) )
        @ ( sigma_6315060578831106510al_nat @ M @ N3 ) )
     => ( member_real_nat
        @ ^ [X: real] : ( produc5914306107723085754_b_nat @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_6315060578831106510al_nat @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1020_measurable__split__replace,axiom,
    ! [F: $o > nat > ( nat > b ) > real,G: $o > produc7194034526493574155_nat_b,M: sigma_measure_o,N3: sigma_measure_real] :
      ( ( member_o_real
        @ ^ [X: $o] : ( F @ X @ ( produc3194919583230816977_nat_b @ ( G @ X ) ) @ ( produc4809910044363821583_nat_b @ ( G @ X ) ) )
        @ ( sigma_2430008634441611636o_real @ M @ N3 ) )
     => ( member_o_real
        @ ^ [X: $o] : ( produc1876314459100775830b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_2430008634441611636o_real @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1021_measurable__split__replace,axiom,
    ! [F: nat > nat > ( nat > b ) > real,G: nat > produc7194034526493574155_nat_b,M: sigma_measure_nat,N3: sigma_measure_real] :
      ( ( member_nat_real
        @ ^ [X: nat] : ( F @ X @ ( produc3194919583230816977_nat_b @ ( G @ X ) ) @ ( produc4809910044363821583_nat_b @ ( G @ X ) ) )
        @ ( sigma_1747752005702207822t_real @ M @ N3 ) )
     => ( member_nat_real
        @ ^ [X: nat] : ( produc1876314459100775830b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1747752005702207822t_real @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1022_measurable__split__replace,axiom,
    ! [F: $o > nat > ( nat > b ) > $o > $o,G: $o > produc7194034526493574155_nat_b,M: sigma_measure_o,N3: sigma_measure_o_o] :
      ( ( member_o_o_o
        @ ^ [X: $o] : ( F @ X @ ( produc3194919583230816977_nat_b @ ( G @ X ) ) @ ( produc4809910044363821583_nat_b @ ( G @ X ) ) )
        @ ( sigma_1874301400886517549_o_o_o @ M @ N3 ) )
     => ( member_o_o_o
        @ ^ [X: $o] : ( produc204931096969673295_b_o_o @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1874301400886517549_o_o_o @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1023_measurable__split__replace,axiom,
    ! [F: $o > a > ( b > produc7194034526493574155_nat_b > a > a ) > $o,G: $o > produc432120507726958376_b_a_a,M: sigma_measure_o,N3: sigma_measure_o] :
      ( ( member_o_o
        @ ^ [X: $o] : ( F @ X @ ( produc5967371699169076996_b_a_a @ ( G @ X ) ) @ ( produc7355951146076385094_b_a_a @ ( G @ X ) ) )
        @ ( sigma_measurable_o_o @ M @ N3 ) )
     => ( member_o_o
        @ ^ [X: $o] : ( produc1569100873952324907_a_a_o @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_measurable_o_o @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1024_measurable__split__replace,axiom,
    ! [F: real > a > ( b > produc7194034526493574155_nat_b > a > a ) > real,G: real > produc432120507726958376_b_a_a,M: sigma_measure_real,N3: sigma_measure_real] :
      ( ( member_real_real
        @ ^ [X: real] : ( F @ X @ ( produc5967371699169076996_b_a_a @ ( G @ X ) ) @ ( produc7355951146076385094_b_a_a @ ( G @ X ) ) )
        @ ( sigma_5267869275261027754l_real @ M @ N3 ) )
     => ( member_real_real
        @ ^ [X: real] : ( produc1440120111801149785a_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1025_measurable__split__replace,axiom,
    ! [F: real > a > ( b > produc7194034526493574155_nat_b > a > a ) > $o,G: real > produc432120507726958376_b_a_a,M: sigma_measure_real,N3: sigma_measure_o] :
      ( ( member_real_o
        @ ^ [X: real] : ( F @ X @ ( produc5967371699169076996_b_a_a @ ( G @ X ) ) @ ( produc7355951146076385094_b_a_a @ ( G @ X ) ) )
        @ ( sigma_3939073009482781210real_o @ M @ N3 ) )
     => ( member_real_o
        @ ^ [X: real] : ( produc1569100873952324907_a_a_o @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_3939073009482781210real_o @ M @ N3 ) ) ) ).

% measurable_split_replace
thf(fact_1026_from__list__in__list__of_H,axiom,
    ! [L2: list_P3644420460460130531t_real] :
      ( member2668624195008970752t_real @ ( coProd6724821034314123748t_real @ L2 )
      @ ( qbs_sp5813997553491121203t_real
        @ ( coProd4740449861057989187t_real @ top_top_set_nat
          @ ^ [N: nat] :
              ( produc7198240415595337743t_real @ ( set_ord_lessThan_nat @ N )
              @ ^ [I: nat] : ( quasi_5340063075079884575t_real @ ( produc2405820699097220962t_real @ top_to4855536200657754381t_real @ top_to4102014873413542296t_real ) ) ) ) ) ) ).

% from_list_in_list_of'
thf(fact_1027_from__list__in__list__of_H,axiom,
    ! [L2: list_complex] :
      ( member147869362354246647omplex @ ( coProd1929593393765550311omplex @ L2 )
      @ ( qbs_sp4672747147356854276omplex
        @ ( coProd4521580844970066270omplex @ top_top_set_nat
          @ ^ [N: nat] :
              ( produc4391159522165087484omplex @ ( set_ord_lessThan_nat @ N )
              @ ^ [I: nat] : ( quasi_5875727614544941548omplex @ ( produc5917795231952384940omplex @ top_top_set_complex @ top_to3254105281827402847omplex ) ) ) ) ) ) ).

% from_list_in_list_of'
thf(fact_1028_from__list__in__list__of_H,axiom,
    ! [L2: list_real] :
      ( member1761831456504113269t_real @ ( coProd42620066860459109t_real @ L2 )
      @ ( qbs_sp7060840734277079042t_real
        @ ( coProd2933363516889264092t_real @ top_top_set_nat
          @ ^ [N: nat] :
              ( product_PiQ_nat_real @ ( set_ord_lessThan_nat @ N )
              @ ^ [I: nat] : ( quasi_6150793623345104106l_real @ ( produc3470952364605845544l_real @ top_top_set_real @ top_to2071711978144146653l_real ) ) ) ) ) ) ).

% from_list_in_list_of'
thf(fact_1029_from__list__in__list__of_H,axiom,
    ! [L2: list_o] :
      ( member2870736337157897531_nat_o @ ( coProd2271143566531055903list_o @ L2 )
      @ ( qbs_sp7669535023837000174_nat_o
        @ ( coProd2185642157359440382_nat_o @ top_top_set_nat
          @ ^ [N: nat] :
              ( product_PiQ_nat_o @ ( set_ord_lessThan_nat @ N )
              @ ^ [I: nat] : ( quasi_6672779326458340058orel_o @ ( produc2133903995702845986real_o @ top_top_set_o @ top_top_set_real_o ) ) ) ) ) ) ).

% from_list_in_list_of'
thf(fact_1030_from__list__in__list__of_H,axiom,
    ! [L2: list_nat] :
      ( member8336108448496249625at_nat @ ( coProd6427887730212605449st_nat @ L2 )
      @ ( qbs_sp6217465386016753958at_nat
        @ ( coProd5302874912193113472at_nat @ top_top_set_nat
          @ ^ [N: nat] :
              ( product_PiQ_nat_nat @ ( set_ord_lessThan_nat @ N )
              @ ^ [I: nat] : ( quasi_3391071144330191118el_nat @ ( produc3843710909396041968al_nat @ top_top_set_nat @ top_top_set_real_nat ) ) ) ) ) ) ).

% from_list_in_list_of'
thf(fact_1031_from__list__in__list__of_H,axiom,
    ! [L2: list_b] :
      ( member2882485241872049954_nat_b @ ( coProd4444644356563764486list_b @ L2 )
      @ ( qbs_sp2305461467746415189_nat_b
        @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
          @ ^ [N: nat] :
              ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
              @ ^ [I: nat] : ( quasi_2002468295286565185orel_b @ ( produc5260422477894090978real_b @ top_top_set_b @ top_top_set_real_b ) ) ) ) ) ) ).

% from_list_in_list_of'
thf(fact_1032_is__borel__def,axiom,
    ( borel_is_borel_o_o_o
    = ( ^ [F2: $o > $o > $o,M2: sigma_measure_o] : ( member_o_o_o @ F2 @ ( sigma_1874301400886517549_o_o_o @ M2 @ borel_2241930036368985287el_o_o ) ) ) ) ).

% is_borel_def
thf(fact_1033_is__borel__def,axiom,
    ( borel_236569967776329622l_real
    = ( ^ [F2: real > real,M2: sigma_measure_real] : ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% is_borel_def
thf(fact_1034_is__borel__def,axiom,
    ( borel_2269360593130276488o_real
    = ( ^ [F2: $o > real,M2: sigma_measure_o] : ( member_o_real @ F2 @ ( sigma_2430008634441611636o_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% is_borel_def
thf(fact_1035_is__borel__def,axiom,
    ( borel_9213571707143006522t_real
    = ( ^ [F2: nat > real,M2: sigma_measure_nat] : ( member_nat_real @ F2 @ ( sigma_1747752005702207822t_real @ M2 @ borel_5078946678739801102l_real ) ) ) ) ).

% is_borel_def
thf(fact_1036_is__borel__def,axiom,
    ( borel_is_borel_o_o
    = ( ^ [F2: $o > $o,M2: sigma_measure_o] : ( member_o_o @ F2 @ ( sigma_measurable_o_o @ M2 @ borel_5500255247093592246orel_o ) ) ) ) ).

% is_borel_def
thf(fact_1037_is__borel__def,axiom,
    ( borel_3778424968171446062real_o
    = ( ^ [F2: real > $o,M2: sigma_measure_real] : ( member_real_o @ F2 @ ( sigma_3939073009482781210real_o @ M2 @ borel_5500255247093592246orel_o ) ) ) ) ).

% is_borel_def
thf(fact_1038_is__borel__def,axiom,
    ( borel_4557508243417129402al_nat
    = ( ^ [F2: real > nat,M2: sigma_measure_real] : ( member_real_nat @ F2 @ ( sigma_6315060578831106510al_nat @ M2 @ borel_8449730974584783410el_nat ) ) ) ) ).

% is_borel_def
thf(fact_1039_snd__diag__snd,axiom,
    ( ( comp_P3276979369905350645l_real @ produc3484788084999411615l_real
      @ ( comp_r97482993418849103l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ X @ X )
        @ produc3484788084999411615l_real ) )
    = produc3484788084999411615l_real ) ).

% snd_diag_snd
thf(fact_1040_snd__diag__snd,axiom,
    ( ( comp_P9159556223120822829_b_a_a @ produc7732007854759685253_b_a_a
      @ ( comp_b3789700955105422685_b_a_a
        @ ^ [X: b > produc7194034526493574155_nat_b > a > a] : ( produc5775047331815311703_b_a_a @ X @ X )
        @ produc7355951146076385094_b_a_a ) )
    = produc7355951146076385094_b_a_a ) ).

% snd_diag_snd
thf(fact_1041_snd__diag__snd,axiom,
    ( ( comp_P3511690182758988141_nat_b @ produc1773315609376244421_nat_b
      @ ( comp_n3851728294588896943_nat_b
        @ ^ [X: nat > b] : ( produc1832820645975422871_nat_b @ X @ X )
        @ produc4809910044363821583_nat_b ) )
    = produc4809910044363821583_nat_b ) ).

% snd_diag_snd
thf(fact_1042_exp__qbs__space,axiom,
    ! [X4: quasi_borel_real,Y4: quasi_borel_a] :
      ( ( qbs_space_real_a @ ( expone319700646606540215real_a @ X4 @ Y4 ) )
      = ( qbs_morphism_real_a @ X4 @ Y4 ) ) ).

% exp_qbs_space
thf(fact_1043_exp__qbs__space,axiom,
    ! [X4: quasi_805850127037308055at_b_a,Y4: quasi_borel_a] :
      ( ( qbs_sp499793078126619644_b_a_a @ ( expone4627587591657327526_b_a_a @ X4 @ Y4 ) )
      = ( qbs_mo7901260496052035013_b_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_space
thf(fact_1044_exp__qbs__space,axiom,
    ! [X4: quasi_borel_nat_b,Y4: quasi_2547937147209033259_a_a_a] :
      ( ( qbs_sp8333110983225252808_a_a_a @ ( expone6720796868436823688_a_a_a @ X4 @ Y4 ) )
      = ( qbs_mo8618518167108125737_a_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_space
thf(fact_1045_exp__qbs__space,axiom,
    ! [X4: quasi_borel_o,Y4: quasi_borel_o_o] :
      ( ( qbs_space_o_o_o @ ( expone1670929651343127652_o_o_o @ X4 @ Y4 ) )
      = ( qbs_morphism_o_o_o @ X4 @ Y4 ) ) ).

% exp_qbs_space
thf(fact_1046_exp__qbs__space,axiom,
    ! [X4: quasi_borel_b,Y4: quasi_7934145443646537599_b_a_a] :
      ( ( qbs_sp4382579563652706485_b_a_a @ ( expone5294601226626914677_b_a_a @ X4 @ Y4 ) )
      = ( qbs_mo1542099550572995862_b_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_space
thf(fact_1047_exp__qbs__space,axiom,
    ! [X4: quasi_7334096685896923387_nat_b,Y4: quasi_borel_a_a] :
      ( ( qbs_sp7017503477097497945_b_a_a @ ( expone5933728560452245707_b_a_a @ X4 @ Y4 ) )
      = ( qbs_mo7242095808399995242_b_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_space
thf(fact_1048_exp__qbs__space,axiom,
    ! [X4: quasi_borel_a,Y4: quasi_borel_a] :
      ( ( qbs_space_a_a @ ( exponent_exp_qbs_a_a @ X4 @ Y4 ) )
      = ( qbs_morphism_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_space
thf(fact_1049_exp__qbs__space,axiom,
    ! [X4: quasi_6353345141195568462_b_a_a,Y4: quasi_borel_a] :
      ( ( qbs_sp9089677150445993861_a_a_a @ ( expone790792417327382575_a_a_a @ X4 @ Y4 ) )
      = ( qbs_mo5048501348360587598_a_a_a @ X4 @ Y4 ) ) ).

% exp_qbs_space
thf(fact_1050_exp__qbs__space,axiom,
    ! [X4: quasi_borel_o,Y4: quasi_borel_o] :
      ( ( qbs_space_o_o @ ( exponent_exp_qbs_o_o @ X4 @ Y4 ) )
      = ( qbs_morphism_o_o @ X4 @ Y4 ) ) ).

% exp_qbs_space
thf(fact_1051_or__qbs__morphism,axiom,
    member_o_o_o @ (|) @ ( qbs_morphism_o_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( exponent_exp_qbs_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) ) ) ).

% or_qbs_morphism
thf(fact_1052_and__qbs__morphism,axiom,
    member_o_o_o @ (&) @ ( qbs_morphism_o_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( exponent_exp_qbs_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) ) ) ).

% and_qbs_morphism
thf(fact_1053_not__qbs__morphism,axiom,
    member_o_o @ (~) @ ( qbs_morphism_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) ) ).

% not_qbs_morphism
thf(fact_1054_implies__qbs__morphism,axiom,
    member_o_o_o @ (=>) @ ( qbs_morphism_o_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( exponent_exp_qbs_o_o @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) @ ( measur2705496967258476524_qbs_o @ borel_5500255247093592246orel_o ) ) ) ).

% implies_qbs_morphism
thf(fact_1055_qbs__Mx__def,axiom,
    ( qbs_Mx5919617464495388621at_b_a
    = ( ^ [X6: quasi_331246355085487092at_b_a] : ( produc1277482548320373382at_b_a @ ( quasi_3612281633575004685at_b_a @ X6 ) ) ) ) ).

% qbs_Mx_def
thf(fact_1056_qbs__Mx__def,axiom,
    ( qbs_Mx_nat_b
    = ( ^ [X6: quasi_borel_nat_b] : ( produc6427044318018601168_nat_b @ ( quasi_4957602499819279645_nat_b @ X6 ) ) ) ) ).

% qbs_Mx_def
thf(fact_1057_qbs__Mx__def,axiom,
    ( qbs_Mx703024977670096246_b_a_a
    = ( ^ [X6: quasi_3364728345918372901_b_a_a] : ( produc6712843908289693712_b_a_a @ ( quasi_6770997997730906934_b_a_a @ X6 ) ) ) ) ).

% qbs_Mx_def
thf(fact_1058_qbs__Mx__def,axiom,
    ( qbs_Mx_b
    = ( ^ [X6: quasi_borel_b] : ( produc1187263614985732624real_b @ ( quasi_4958298574314430518orel_b @ X6 ) ) ) ) ).

% qbs_Mx_def
thf(fact_1059_qbs__Mx__def,axiom,
    ( qbs_Mx_a
    = ( ^ [X6: quasi_borel_a] : ( produc4125011029606581840real_a @ ( quasi_4958298574314430517orel_a @ X6 ) ) ) ) ).

% qbs_Mx_def
thf(fact_1060_qbs__space__def,axiom,
    ( qbs_sp2305461467746415189_nat_b
    = ( ^ [X6: quasi_7334096685896923387_nat_b] : ( produc6119163444293898962_nat_b @ ( quasi_2328903794196513940_nat_b @ X6 ) ) ) ) ).

% qbs_space_def
thf(fact_1061_to__list_H_Ocases,axiom,
    ! [X5: produc7194034526493574155_nat_b] :
      ( ! [Uu2: nat > b] :
          ( X5
         != ( produc5292568363641424317_nat_b @ zero_zero_nat @ Uu2 ) )
     => ~ ! [N2: nat,F3: nat > b] :
            ( X5
           != ( produc5292568363641424317_nat_b @ ( suc @ N2 ) @ F3 ) ) ) ).

% to_list'.cases
thf(fact_1062_comp__apply__eq,axiom,
    ! [F: real > produc2422161461964618553l_real,G: produc2422161461964618553l_real > real,X5: produc2422161461964618553l_real,H: real > produc2422161461964618553l_real,K: produc2422161461964618553l_real > real] :
      ( ( ( F @ ( G @ X5 ) )
        = ( H @ ( K @ X5 ) ) )
     => ( ( comp_r97482993418849103l_real @ F @ G @ X5 )
        = ( comp_r97482993418849103l_real @ H @ K @ X5 ) ) ) ).

% comp_apply_eq
thf(fact_1063_comp__apply__eq,axiom,
    ! [F: real > produc7716430852924023517t_real,G: produc7716430852924023517t_real > real,X5: produc7716430852924023517t_real,H: real > produc7716430852924023517t_real,K: produc7716430852924023517t_real > real] :
      ( ( ( F @ ( G @ X5 ) )
        = ( H @ ( K @ X5 ) ) )
     => ( ( comp_r4825950202331181903t_real @ F @ G @ X5 )
        = ( comp_r4825950202331181903t_real @ H @ K @ X5 ) ) ) ).

% comp_apply_eq
thf(fact_1064_comp__apply__eq,axiom,
    ! [F: real > real,G: real > real,X5: real,H: real > real,K: real > real] :
      ( ( ( F @ ( G @ X5 ) )
        = ( H @ ( K @ X5 ) ) )
     => ( ( comp_real_real_real @ F @ G @ X5 )
        = ( comp_real_real_real @ H @ K @ X5 ) ) ) ).

% comp_apply_eq
thf(fact_1065_comp__apply__eq,axiom,
    ! [F: real > $o,G: $o > real,X5: $o,H: real > $o,K: $o > real] :
      ( ( ( F @ ( G @ X5 ) )
        = ( H @ ( K @ X5 ) ) )
     => ( ( comp_real_o_o @ F @ G @ X5 )
        = ( comp_real_o_o @ H @ K @ X5 ) ) ) ).

% comp_apply_eq
thf(fact_1066_comp__apply__eq,axiom,
    ! [F: real > nat,G: nat > real,X5: nat,H: real > nat,K: nat > real] :
      ( ( ( F @ ( G @ X5 ) )
        = ( H @ ( K @ X5 ) ) )
     => ( ( comp_real_nat_nat @ F @ G @ X5 )
        = ( comp_real_nat_nat @ H @ K @ X5 ) ) ) ).

% comp_apply_eq
thf(fact_1067_fstI,axiom,
    ! [X5: product_prod_real_b,Y5: real,Z: b] :
      ( ( X5
        = ( product_Pair_real_b @ Y5 @ Z ) )
     => ( ( product_fst_real_b @ X5 )
        = Y5 ) ) ).

% fstI
thf(fact_1068_fstI,axiom,
    ! [X5: product_prod_nat_nat,Y5: nat,Z: nat] :
      ( ( X5
        = ( product_Pair_nat_nat @ Y5 @ Z ) )
     => ( ( product_fst_nat_nat @ X5 )
        = Y5 ) ) ).

% fstI
thf(fact_1069_fstI,axiom,
    ! [X5: produc778026099599033348at_b_a,Y5: b > produc7194034526493574155_nat_b > a > a,Z: produc4883615123598087894at_b_a] :
      ( ( X5
        = ( produc653446490150804990at_b_a @ Y5 @ Z ) )
     => ( ( produc7871839370796975378at_b_a @ X5 )
        = Y5 ) ) ).

% fstI
thf(fact_1070_fstI,axiom,
    ! [X5: produc4883615123598087894at_b_a,Y5: b,Z: produc6209191202727158440at_b_a] :
      ( ( X5
        = ( produc992390952543482448at_b_a @ Y5 @ Z ) )
     => ( ( produc7380424691241595748at_b_a @ X5 )
        = Y5 ) ) ).

% fstI
thf(fact_1071_fstI,axiom,
    ! [X5: produc6209191202727158440at_b_a,Y5: produc7194034526493574155_nat_b,Z: a] :
      ( ( X5
        = ( produc8509509857186669914at_b_a @ Y5 @ Z ) )
     => ( ( produc2385479481072933486at_b_a @ X5 )
        = Y5 ) ) ).

% fstI
thf(fact_1072_fstI,axiom,
    ! [X5: produc3260441645384719156_nat_b,Y5: product_prod_real_b,Z: produc7194034526493574155_nat_b] :
      ( ( X5
        = ( produc1946830598933159916_nat_b @ Y5 @ Z ) )
     => ( ( produc4243170370849295320_nat_b @ X5 )
        = Y5 ) ) ).

% fstI
thf(fact_1073_fstI,axiom,
    ! [X5: produc432120507726958376_b_a_a,Y5: a,Z: b > produc7194034526493574155_nat_b > a > a] :
      ( ( X5
        = ( produc3671031927252941592_b_a_a @ Y5 @ Z ) )
     => ( ( produc5967371699169076996_b_a_a @ X5 )
        = Y5 ) ) ).

% fstI
thf(fact_1074_fstI,axiom,
    ! [X5: produc7194034526493574155_nat_b,Y5: nat,Z: nat > b] :
      ( ( X5
        = ( produc5292568363641424317_nat_b @ Y5 @ Z ) )
     => ( ( produc3194919583230816977_nat_b @ X5 )
        = Y5 ) ) ).

% fstI
thf(fact_1075_sndI,axiom,
    ! [X5: product_prod_real_b,Y5: real,Z: b] :
      ( ( X5
        = ( product_Pair_real_b @ Y5 @ Z ) )
     => ( ( product_snd_real_b @ X5 )
        = Z ) ) ).

% sndI
thf(fact_1076_sndI,axiom,
    ! [X5: product_prod_nat_nat,Y5: nat,Z: nat] :
      ( ( X5
        = ( product_Pair_nat_nat @ Y5 @ Z ) )
     => ( ( product_snd_nat_nat @ X5 )
        = Z ) ) ).

% sndI
thf(fact_1077_sndI,axiom,
    ! [X5: produc778026099599033348at_b_a,Y5: b > produc7194034526493574155_nat_b > a > a,Z: produc4883615123598087894at_b_a] :
      ( ( X5
        = ( produc653446490150804990at_b_a @ Y5 @ Z ) )
     => ( ( produc6884036649769537872at_b_a @ X5 )
        = Z ) ) ).

% sndI
thf(fact_1078_sndI,axiom,
    ! [X5: produc4883615123598087894at_b_a,Y5: b,Z: produc6209191202727158440at_b_a] :
      ( ( X5
        = ( produc992390952543482448at_b_a @ Y5 @ Z ) )
     => ( ( produc7666751653692995234at_b_a @ X5 )
        = Z ) ) ).

% sndI
thf(fact_1079_sndI,axiom,
    ! [X5: produc6209191202727158440at_b_a,Y5: produc7194034526493574155_nat_b,Z: a] :
      ( ( X5
        = ( produc8509509857186669914at_b_a @ Y5 @ Z ) )
     => ( ( produc5227817866736700332at_b_a @ X5 )
        = Z ) ) ).

% sndI
thf(fact_1080_sndI,axiom,
    ! [X5: produc3260441645384719156_nat_b,Y5: product_prod_real_b,Z: produc7194034526493574155_nat_b] :
      ( ( X5
        = ( produc1946830598933159916_nat_b @ Y5 @ Z ) )
     => ( ( produc5631749817756603418_nat_b @ X5 )
        = Z ) ) ).

% sndI
thf(fact_1081_sndI,axiom,
    ! [X5: produc432120507726958376_b_a_a,Y5: a,Z: b > produc7194034526493574155_nat_b > a > a] :
      ( ( X5
        = ( produc3671031927252941592_b_a_a @ Y5 @ Z ) )
     => ( ( produc7355951146076385094_b_a_a @ X5 )
        = Z ) ) ).

% sndI
thf(fact_1082_sndI,axiom,
    ! [X5: produc7194034526493574155_nat_b,Y5: nat,Z: nat > b] :
      ( ( X5
        = ( produc5292568363641424317_nat_b @ Y5 @ Z ) )
     => ( ( produc4809910044363821583_nat_b @ X5 )
        = Z ) ) ).

% sndI
thf(fact_1083_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: real > b > $o,X5: real,Y5: b,A: product_prod_real_b] :
      ( ( P @ X5 @ Y5 )
     => ( ( A
          = ( product_Pair_real_b @ X5 @ Y5 ) )
       => ( P @ ( product_fst_real_b @ A ) @ ( product_snd_real_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1084_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: nat > nat > $o,X5: nat,Y5: nat,A: product_prod_nat_nat] :
      ( ( P @ X5 @ Y5 )
     => ( ( A
          = ( product_Pair_nat_nat @ X5 @ Y5 ) )
       => ( P @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1085_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > $o,X5: b > produc7194034526493574155_nat_b > a > a,Y5: produc4883615123598087894at_b_a,A: produc778026099599033348at_b_a] :
      ( ( P @ X5 @ Y5 )
     => ( ( A
          = ( produc653446490150804990at_b_a @ X5 @ Y5 ) )
       => ( P @ ( produc7871839370796975378at_b_a @ A ) @ ( produc6884036649769537872at_b_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1086_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: b > produc6209191202727158440at_b_a > $o,X5: b,Y5: produc6209191202727158440at_b_a,A: produc4883615123598087894at_b_a] :
      ( ( P @ X5 @ Y5 )
     => ( ( A
          = ( produc992390952543482448at_b_a @ X5 @ Y5 ) )
       => ( P @ ( produc7380424691241595748at_b_a @ A ) @ ( produc7666751653692995234at_b_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1087_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: produc7194034526493574155_nat_b > a > $o,X5: produc7194034526493574155_nat_b,Y5: a,A: produc6209191202727158440at_b_a] :
      ( ( P @ X5 @ Y5 )
     => ( ( A
          = ( produc8509509857186669914at_b_a @ X5 @ Y5 ) )
       => ( P @ ( produc2385479481072933486at_b_a @ A ) @ ( produc5227817866736700332at_b_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1088_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: product_prod_real_b > produc7194034526493574155_nat_b > $o,X5: product_prod_real_b,Y5: produc7194034526493574155_nat_b,A: produc3260441645384719156_nat_b] :
      ( ( P @ X5 @ Y5 )
     => ( ( A
          = ( produc1946830598933159916_nat_b @ X5 @ Y5 ) )
       => ( P @ ( produc4243170370849295320_nat_b @ A ) @ ( produc5631749817756603418_nat_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1089_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: a > ( b > produc7194034526493574155_nat_b > a > a ) > $o,X5: a,Y5: b > produc7194034526493574155_nat_b > a > a,A: produc432120507726958376_b_a_a] :
      ( ( P @ X5 @ Y5 )
     => ( ( A
          = ( produc3671031927252941592_b_a_a @ X5 @ Y5 ) )
       => ( P @ ( produc5967371699169076996_b_a_a @ A ) @ ( produc7355951146076385094_b_a_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1090_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: nat > ( nat > b ) > $o,X5: nat,Y5: nat > b,A: produc7194034526493574155_nat_b] :
      ( ( P @ X5 @ Y5 )
     => ( ( A
          = ( produc5292568363641424317_nat_b @ X5 @ Y5 ) )
       => ( P @ ( produc3194919583230816977_nat_b @ A ) @ ( produc4809910044363821583_nat_b @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_1091_measurable__split__conv,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a > a,A4: sigma_1533348360751875505_nat_b,B4: sigma_measure_a_a] :
      ( ( member7748159150162262195_b_a_a @ ( produc8421786640700244582_b_a_a @ F ) @ ( sigma_2494036265332152777_b_a_a @ A4 @ B4 ) )
      = ( member7748159150162262195_b_a_a
        @ ^ [X: produc3260441645384719156_nat_b] : ( F @ ( produc4243170370849295320_nat_b @ X ) @ ( produc5631749817756603418_nat_b @ X ) )
        @ ( sigma_2494036265332152777_b_a_a @ A4 @ B4 ) ) ) ).

% measurable_split_conv
thf(fact_1092_measurable__split__conv,axiom,
    ! [F: produc7194034526493574155_nat_b > a > a,A4: sigma_6723678977712692993at_b_a,B4: sigma_measure_a] :
      ( ( member1447347221985152182_b_a_a @ ( produc2543982510095308599_b_a_a @ F ) @ ( sigma_902110273089948292_b_a_a @ A4 @ B4 ) )
      = ( member1447347221985152182_b_a_a
        @ ^ [X: produc6209191202727158440at_b_a] : ( F @ ( produc2385479481072933486at_b_a @ X ) @ ( produc5227817866736700332at_b_a @ X ) )
        @ ( sigma_902110273089948292_b_a_a @ A4 @ B4 ) ) ) ).

% measurable_split_conv
thf(fact_1093_measurable__split__conv,axiom,
    ! [F: real > b > produc7194034526493574155_nat_b > a > a,A4: sigma_4741609642398120993real_b,B4: sigma_6671653343297732584_b_a_a] :
      ( ( member1306660226910552845_b_a_a @ ( produc7016990068875982646_b_a_a @ F ) @ ( sigma_1481486135004880937_b_a_a @ A4 @ B4 ) )
      = ( member1306660226910552845_b_a_a
        @ ^ [X: product_prod_real_b] : ( F @ ( product_fst_real_b @ X ) @ ( product_snd_real_b @ X ) )
        @ ( sigma_1481486135004880937_b_a_a @ A4 @ B4 ) ) ) ).

% measurable_split_conv
thf(fact_1094_measurable__split__conv,axiom,
    ! [F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a,A4: sigma_6586933000485258589at_b_a,B4: sigma_measure_a] :
      ( ( member763877586905102170_b_a_a @ ( produc286368409806729619_b_a_a @ F ) @ ( sigma_4910332140527951272_b_a_a @ A4 @ B4 ) )
      = ( member763877586905102170_b_a_a
        @ ^ [X: produc778026099599033348at_b_a] : ( F @ ( produc7871839370796975378at_b_a @ X ) @ ( produc6884036649769537872at_b_a @ X ) )
        @ ( sigma_4910332140527951272_b_a_a @ A4 @ B4 ) ) ) ).

% measurable_split_conv
thf(fact_1095_measurable__split__conv,axiom,
    ! [F: b > produc6209191202727158440at_b_a > a,A4: sigma_7305014782616092207at_b_a,B4: sigma_measure_a] :
      ( ( member871964287106612744_b_a_a @ ( produc6302854954412089281_b_a_a @ F ) @ ( sigma_5453631462185342678_b_a_a @ A4 @ B4 ) )
      = ( member871964287106612744_b_a_a
        @ ^ [X: produc4883615123598087894at_b_a] : ( F @ ( produc7380424691241595748at_b_a @ X ) @ ( produc7666751653692995234at_b_a @ X ) )
        @ ( sigma_5453631462185342678_b_a_a @ A4 @ B4 ) ) ) ).

% measurable_split_conv
thf(fact_1096_measurable__split__conv,axiom,
    ! [F: real > b > a > a,A4: sigma_4741609642398120993real_b,B4: sigma_measure_a_a] :
      ( ( member9111837334973778019_b_a_a @ ( produc7827042608974387084_b_a_a @ F ) @ ( sigma_2523884250415250559_b_a_a @ A4 @ B4 ) )
      = ( member9111837334973778019_b_a_a
        @ ^ [X: product_prod_real_b] : ( F @ ( product_fst_real_b @ X ) @ ( product_snd_real_b @ X ) )
        @ ( sigma_2523884250415250559_b_a_a @ A4 @ B4 ) ) ) ).

% measurable_split_conv
thf(fact_1097_measurable__split__conv,axiom,
    ! [F: product_prod_real_b > produc7194034526493574155_nat_b > a,A4: sigma_1533348360751875505_nat_b,B4: sigma_measure_a] :
      ( ( member7813222213774494534at_b_a @ ( produc634687561522189117at_b_a @ F ) @ ( sigma_8999254761160657050at_b_a @ A4 @ B4 ) )
      = ( member7813222213774494534at_b_a
        @ ^ [X: produc3260441645384719156_nat_b] : ( F @ ( produc4243170370849295320_nat_b @ X ) @ ( produc5631749817756603418_nat_b @ X ) )
        @ ( sigma_8999254761160657050at_b_a @ A4 @ B4 ) ) ) ).

% measurable_split_conv
thf(fact_1098_fst__diag__fst,axiom,
    ( ( comp_P3276979369905350645l_real @ produc5828954698716094813l_real
      @ ( comp_r97482993418849103l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ X @ X )
        @ produc5828954698716094813l_real ) )
    = produc5828954698716094813l_real ) ).

% fst_diag_fst
thf(fact_1099_fst__diag__fst,axiom,
    ( ( comp_P1156977722522261166_b_a_a @ product_fst_a_a
      @ ( comp_a5577347777374339356_b_a_a
        @ ^ [X: a] : ( product_Pair_a_a @ X @ X )
        @ produc5967371699169076996_b_a_a ) )
    = produc5967371699169076996_b_a_a ) ).

% fst_diag_fst
thf(fact_1100_fst__diag__fst,axiom,
    ( ( comp_P5143406238261914043_nat_b @ product_fst_nat_nat
      @ ( comp_n8051653116556791961_nat_b
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
        @ produc3194919583230816977_nat_b ) )
    = produc3194919583230816977_nat_b ) ).

% fst_diag_fst
thf(fact_1101_to__list__from__list__ident,axiom,
    ! [L2: produc7194034526493574155_nat_b,X4: quasi_borel_b] :
      ( ( member2882485241872049954_nat_b @ L2
        @ ( qbs_sp2305461467746415189_nat_b
          @ ( coProd3505932218054916197_nat_b @ top_top_set_nat
            @ ^ [N: nat] :
                ( product_PiQ_nat_b @ ( set_ord_lessThan_nat @ N )
                @ ^ [I: nat] : X4 ) ) ) )
     => ( ( comp_l1050289791651215098_nat_b @ coProd4444644356563764486list_b @ coProduct_to_list_b @ L2 )
        = L2 ) ) ).

% to_list_from_list_ident
thf(fact_1102_exE__realizer,axiom,
    ! [P: produc7194034526493574155_nat_b > product_prod_real_b > $o,P2: produc3260441645384719156_nat_b,Q: ( a > a ) > $o,F: product_prod_real_b > produc7194034526493574155_nat_b > a > a] :
      ( ( P @ ( produc5631749817756603418_nat_b @ P2 ) @ ( produc4243170370849295320_nat_b @ P2 ) )
     => ( ! [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc8421786640700244582_b_a_a @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_1103_exE__realizer,axiom,
    ! [P: a > produc7194034526493574155_nat_b > $o,P2: produc6209191202727158440at_b_a,Q: a > $o,F: produc7194034526493574155_nat_b > a > a] :
      ( ( P @ ( produc5227817866736700332at_b_a @ P2 ) @ ( produc2385479481072933486at_b_a @ P2 ) )
     => ( ! [X3: produc7194034526493574155_nat_b,Y3: a] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc2543982510095308599_b_a_a @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_1104_exE__realizer,axiom,
    ! [P: b > real > $o,P2: product_prod_real_b,Q: ( produc7194034526493574155_nat_b > a > a ) > $o,F: real > b > produc7194034526493574155_nat_b > a > a] :
      ( ( P @ ( product_snd_real_b @ P2 ) @ ( product_fst_real_b @ P2 ) )
     => ( ! [X3: real,Y3: b] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc7016990068875982646_b_a_a @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_1105_exE__realizer,axiom,
    ! [P: produc4883615123598087894at_b_a > ( b > produc7194034526493574155_nat_b > a > a ) > $o,P2: produc778026099599033348at_b_a,Q: a > $o,F: ( b > produc7194034526493574155_nat_b > a > a ) > produc4883615123598087894at_b_a > a] :
      ( ( P @ ( produc6884036649769537872at_b_a @ P2 ) @ ( produc7871839370796975378at_b_a @ P2 ) )
     => ( ! [X3: b > produc7194034526493574155_nat_b > a > a,Y3: produc4883615123598087894at_b_a] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc286368409806729619_b_a_a @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_1106_exE__realizer,axiom,
    ! [P: produc6209191202727158440at_b_a > b > $o,P2: produc4883615123598087894at_b_a,Q: a > $o,F: b > produc6209191202727158440at_b_a > a] :
      ( ( P @ ( produc7666751653692995234at_b_a @ P2 ) @ ( produc7380424691241595748at_b_a @ P2 ) )
     => ( ! [X3: b,Y3: produc6209191202727158440at_b_a] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc6302854954412089281_b_a_a @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_1107_exE__realizer,axiom,
    ! [P: b > real > $o,P2: product_prod_real_b,Q: ( a > a ) > $o,F: real > b > a > a] :
      ( ( P @ ( product_snd_real_b @ P2 ) @ ( product_fst_real_b @ P2 ) )
     => ( ! [X3: real,Y3: b] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc7827042608974387084_b_a_a @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_1108_exE__realizer,axiom,
    ! [P: produc7194034526493574155_nat_b > product_prod_real_b > $o,P2: produc3260441645384719156_nat_b,Q: a > $o,F: product_prod_real_b > produc7194034526493574155_nat_b > a] :
      ( ( P @ ( produc5631749817756603418_nat_b @ P2 ) @ ( produc4243170370849295320_nat_b @ P2 ) )
     => ( ! [X3: product_prod_real_b,Y3: produc7194034526493574155_nat_b] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc634687561522189117at_b_a @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_1109__C5_C,axiom,
    ! [R3: real] :
      ( ( produc5292568363641424317_nat_b @ ( suc @ na ) @ ( alpha2 @ R3 ) )
      = ( produc5292568363641424317_nat_b @ ( suc @ ( produc3194919583230816977_nat_b @ ( produc5292568363641424317_nat_b @ na @ ( alpha @ R3 ) ) ) )
        @ ^ [N: nat] : ( if_b @ ( N = zero_zero_nat ) @ ( a2 @ R3 ) @ ( produc4809910044363821583_nat_b @ ( produc5292568363641424317_nat_b @ na @ ( alpha @ R3 ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% "5"
thf(fact_1110_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q: nat > $o,Q2: nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1111_conjI__realizer,axiom,
    ! [P: ( b > produc7194034526493574155_nat_b > a > a ) > $o,P2: b > produc7194034526493574155_nat_b > a > a,Q: produc4883615123598087894at_b_a > $o,Q2: produc4883615123598087894at_b_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc7871839370796975378at_b_a @ ( produc653446490150804990at_b_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc6884036649769537872at_b_a @ ( produc653446490150804990at_b_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1112_conjI__realizer,axiom,
    ! [P: b > $o,P2: b,Q: produc6209191202727158440at_b_a > $o,Q2: produc6209191202727158440at_b_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc7380424691241595748at_b_a @ ( produc992390952543482448at_b_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7666751653692995234at_b_a @ ( produc992390952543482448at_b_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1113_conjI__realizer,axiom,
    ! [P: produc7194034526493574155_nat_b > $o,P2: produc7194034526493574155_nat_b,Q: a > $o,Q2: a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc2385479481072933486at_b_a @ ( produc8509509857186669914at_b_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc5227817866736700332at_b_a @ ( produc8509509857186669914at_b_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1114_conjI__realizer,axiom,
    ! [P: product_prod_real_b > $o,P2: product_prod_real_b,Q: produc7194034526493574155_nat_b > $o,Q2: produc7194034526493574155_nat_b] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc4243170370849295320_nat_b @ ( produc1946830598933159916_nat_b @ P2 @ Q2 ) ) )
          & ( Q @ ( produc5631749817756603418_nat_b @ ( produc1946830598933159916_nat_b @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1115_conjI__realizer,axiom,
    ! [P: a > $o,P2: a,Q: ( b > produc7194034526493574155_nat_b > a > a ) > $o,Q2: b > produc7194034526493574155_nat_b > a > a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc5967371699169076996_b_a_a @ ( produc3671031927252941592_b_a_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7355951146076385094_b_a_a @ ( produc3671031927252941592_b_a_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1116_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q: ( nat > b ) > $o,Q2: nat > b] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc3194919583230816977_nat_b @ ( produc5292568363641424317_nat_b @ P2 @ Q2 ) ) )
          & ( Q @ ( produc4809910044363821583_nat_b @ ( produc5292568363641424317_nat_b @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_1117_diff__Suc__1,axiom,
    ! [N4: nat] :
      ( ( minus_minus_nat @ ( suc @ N4 ) @ one_one_nat )
      = N4 ) ).

% diff_Suc_1
thf(fact_1118_Suc__diff__diff,axiom,
    ! [M3: nat,N4: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N4 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N4 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1119_diff__Suc__Suc,axiom,
    ! [M3: nat,N4: nat] :
      ( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N4 ) )
      = ( minus_minus_nat @ M3 @ N4 ) ) ).

% diff_Suc_Suc
thf(fact_1120_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1121_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_1122_n__not__Suc__n,axiom,
    ! [N4: nat] :
      ( N4
     != ( suc @ N4 ) ) ).

% n_not_Suc_n
thf(fact_1123_Suc__inject,axiom,
    ! [X5: nat,Y5: nat] :
      ( ( ( suc @ X5 )
        = ( suc @ Y5 ) )
     => ( X5 = Y5 ) ) ).

% Suc_inject
thf(fact_1124_not0__implies__Suc,axiom,
    ! [N4: nat] :
      ( ( N4 != zero_zero_nat )
     => ? [M4: nat] :
          ( N4
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_1125_Zero__not__Suc,axiom,
    ! [M3: nat] :
      ( zero_zero_nat
     != ( suc @ M3 ) ) ).

% Zero_not_Suc
thf(fact_1126_Zero__neq__Suc,axiom,
    ! [M3: nat] :
      ( zero_zero_nat
     != ( suc @ M3 ) ) ).

% Zero_neq_Suc
thf(fact_1127_Suc__neq__Zero,axiom,
    ! [M3: nat] :
      ( ( suc @ M3 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1128_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1129_diff__induct,axiom,
    ! [P: nat > nat > $o,M3: nat,N4: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P @ X3 @ Y3 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P @ M3 @ N4 ) ) ) ) ).

% diff_induct
thf(fact_1130_nat__induct,axiom,
    ! [P: nat > $o,N4: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N4 ) ) ) ).

% nat_induct
thf(fact_1131_old_Onat_Oexhaust,axiom,
    ! [Y5: nat] :
      ( ( Y5 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y5
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1132_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1133_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1134_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1135_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_1136_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I3: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).

% zero_induct_lemma
thf(fact_1137_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1138_diff__Suc__eq__diff__pred,axiom,
    ! [M3: nat,N4: nat] :
      ( ( minus_minus_nat @ M3 @ ( suc @ N4 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1139_ennreal__diff__self,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ( ( A != top_to1496364449551166952nnreal )
     => ( ( minus_8429688780609304081nnreal @ A @ A )
        = zero_z7100319975126383169nnreal ) ) ).

% ennreal_diff_self
thf(fact_1140_ennreal__top__minus,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ( minus_8429688780609304081nnreal @ top_to1496364449551166952nnreal @ X5 )
      = top_to1496364449551166952nnreal ) ).

% ennreal_top_minus
thf(fact_1141_ennreal__minus__eq__top,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( minus_8429688780609304081nnreal @ A @ B )
        = top_to1496364449551166952nnreal )
      = ( A = top_to1496364449551166952nnreal ) ) ).

% ennreal_minus_eq_top
thf(fact_1142_Suc__le__mono,axiom,
    ! [N4: nat,M3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N4 ) @ ( suc @ M3 ) )
      = ( ord_less_eq_nat @ N4 @ M3 ) ) ).

% Suc_le_mono
thf(fact_1143_ennreal__minus__cancel,axiom,
    ! [C: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( C != top_to1496364449551166952nnreal )
     => ( ( ord_le3935885782089961368nnreal @ A @ C )
       => ( ( ord_le3935885782089961368nnreal @ B @ C )
         => ( ( ( minus_8429688780609304081nnreal @ C @ A )
              = ( minus_8429688780609304081nnreal @ C @ B ) )
           => ( A = B ) ) ) ) ) ).

% ennreal_minus_cancel
thf(fact_1144_ennreal__minus__cancel__iff,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ( minus_8429688780609304081nnreal @ A @ B )
        = ( minus_8429688780609304081nnreal @ A @ C ) )
      = ( ( B = C )
        | ( ( ord_le3935885782089961368nnreal @ A @ B )
          & ( ord_le3935885782089961368nnreal @ A @ C ) )
        | ( A = top_to1496364449551166952nnreal ) ) ) ).

% ennreal_minus_cancel_iff
thf(fact_1145_neq__top__trans,axiom,
    ! [Y5: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal] :
      ( ( Y5 != top_to1496364449551166952nnreal )
     => ( ( ord_le3935885782089961368nnreal @ X5 @ Y5 )
       => ( X5 != top_to1496364449551166952nnreal ) ) ) ).

% neq_top_trans
thf(fact_1146_bounded__Max__nat,axiom,
    ! [P: nat > $o,X5: nat,M: nat] :
      ( ( P @ X5 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X8: nat] :
                    ( ( P @ X8 )
                   => ( ord_less_eq_nat @ X8 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1147_ennreal__zero__neq__top,axiom,
    zero_z7100319975126383169nnreal != top_to1496364449551166952nnreal ).

% ennreal_zero_neq_top
thf(fact_1148_Suc__leD,axiom,
    ! [M3: nat,N4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N4 )
     => ( ord_less_eq_nat @ M3 @ N4 ) ) ).

% Suc_leD
thf(fact_1149_le__SucE,axiom,
    ! [M3: nat,N4: nat] :
      ( ( ord_less_eq_nat @ M3 @ ( suc @ N4 ) )
     => ( ~ ( ord_less_eq_nat @ M3 @ N4 )
       => ( M3
          = ( suc @ N4 ) ) ) ) ).

% le_SucE
thf(fact_1150_le__SucI,axiom,
    ! [M3: nat,N4: nat] :
      ( ( ord_less_eq_nat @ M3 @ N4 )
     => ( ord_less_eq_nat @ M3 @ ( suc @ N4 ) ) ) ).

% le_SucI
thf(fact_1151_Suc__le__D,axiom,
    ! [N4: nat,M5: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N4 ) @ M5 )
     => ? [M4: nat] :
          ( M5
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_1152_le__Suc__eq,axiom,
    ! [M3: nat,N4: nat] :
      ( ( ord_less_eq_nat @ M3 @ ( suc @ N4 ) )
      = ( ( ord_less_eq_nat @ M3 @ N4 )
        | ( M3
          = ( suc @ N4 ) ) ) ) ).

% le_Suc_eq
thf(fact_1153_Suc__n__not__le__n,axiom,
    ! [N4: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N4 ) @ N4 ) ).

% Suc_n_not_le_n
thf(fact_1154_not__less__eq__eq,axiom,
    ! [M3: nat,N4: nat] :
      ( ( ~ ( ord_less_eq_nat @ M3 @ N4 ) )
      = ( ord_less_eq_nat @ ( suc @ N4 ) @ M3 ) ) ).

% not_less_eq_eq
thf(fact_1155_full__nat__induct,axiom,
    ! [P: nat > $o,N4: nat] :
      ( ! [N2: nat] :
          ( ! [M6: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M6 ) @ N2 )
             => ( P @ M6 ) )
         => ( P @ N2 ) )
     => ( P @ N4 ) ) ).

% full_nat_induct
thf(fact_1156_nat__induct__at__least,axiom,
    ! [M3: nat,N4: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M3 @ N4 )
     => ( ( P @ M3 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N4 ) ) ) ) ).

% nat_induct_at_least
thf(fact_1157_transitive__stepwise__le,axiom,
    ! [M3: nat,N4: nat,R4: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M3 @ N4 )
     => ( ! [X3: nat] : ( R4 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z5: nat] :
              ( ( R4 @ X3 @ Y3 )
             => ( ( R4 @ Y3 @ Z5 )
               => ( R4 @ X3 @ Z5 ) ) )
         => ( ! [N2: nat] : ( R4 @ N2 @ ( suc @ N2 ) )
           => ( R4 @ M3 @ N4 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1158_Suc__diff__le,axiom,
    ! [N4: nat,M3: nat] :
      ( ( ord_less_eq_nat @ N4 @ M3 )
     => ( ( minus_minus_nat @ ( suc @ M3 ) @ N4 )
        = ( suc @ ( minus_minus_nat @ M3 @ N4 ) ) ) ) ).

% Suc_diff_le
thf(fact_1159_minus__top__ennreal,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ( ( X5 = top_to1496364449551166952nnreal )
       => ( ( minus_8429688780609304081nnreal @ X5 @ top_to1496364449551166952nnreal )
          = top_to1496364449551166952nnreal ) )
      & ( ( X5 != top_to1496364449551166952nnreal )
       => ( ( minus_8429688780609304081nnreal @ X5 @ top_to1496364449551166952nnreal )
          = zero_z7100319975126383169nnreal ) ) ) ).

% minus_top_ennreal
thf(fact_1160_nat_Oexist__fg,axiom,
    ? [X3: nat > real] :
      ( ( member_nat_real @ X3 @ ( 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 @ X3 @ Xb )
                = Xb ) ) ) ) ).

% nat.exist_fg
thf(fact_1161_nat__complete__induction,axiom,
    ! [P: nat > $o,N4: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ! [M6: nat] :
                ( ( ord_less_eq_nat @ M6 @ N2 )
               => ( P @ M6 ) )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N4 ) ) ) ).

% nat_complete_induction
thf(fact_1162_real_Oexist__fg,axiom,
    ? [X3: real > real] :
      ( ( member_real_real @ X3 @ ( 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 @ X3 @ Xb )
                = Xb ) ) ) ) ).

% real.exist_fg
thf(fact_1163_real_Ospace__UNIV,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% real.space_UNIV
thf(fact_1164_ennreal_Ospace__UNIV,axiom,
    ( ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal )
    = top_to7994903218803871134nnreal ) ).

% ennreal.space_UNIV
thf(fact_1165_bool_Ospace__UNIV,axiom,
    ( ( sigma_space_o @ borel_5500255247093592246orel_o )
    = top_top_set_o ) ).

% bool.space_UNIV
thf(fact_1166_r01__binary__expansion_H_H_Ocases,axiom,
    ! [X5: produc3741383161447143261al_nat] :
      ( ! [R5: real] :
          ( X5
         != ( produc3181502643871035669al_nat @ R5 @ zero_zero_nat ) )
     => ~ ! [R5: real,N2: nat] :
            ( X5
           != ( produc3181502643871035669al_nat @ R5 @ ( suc @ N2 ) ) ) ) ).

% r01_binary_expansion''.cases
thf(fact_1167_list__decode_Ocases,axiom,
    ! [X5: nat] :
      ( ( X5 != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X5
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_1168_ennreal_Oexist__fg,axiom,
    ? [X3: extend8495563244428889912nnreal > real] :
      ( ( member2874014351250825754l_real @ X3 @ ( 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 @ X3 @ Xb )
                = Xb ) ) ) ) ).

% ennreal.exist_fg
thf(fact_1169_nat_Ospace__UNIV,axiom,
    ( ( sigma_space_nat @ borel_8449730974584783410el_nat )
    = top_top_set_nat ) ).

% nat.space_UNIV
thf(fact_1170_bool_Oexist__fg,axiom,
    ? [X3: $o > real] :
      ( ( member_o_real @ X3 @ ( 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 @ X3 @ Xb )
                = Xb ) ) ) ) ).

% bool.exist_fg
thf(fact_1171_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1172_ennreal__minus__top,axiom,
    ! [A: real] :
      ( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ A ) @ top_to1496364449551166952nnreal )
      = zero_z7100319975126383169nnreal ) ).

% ennreal_minus_top
thf(fact_1173_ennreal3__cases,axiom,
    ! [X5: extend8495563244428889912nnreal,Xa2: extend8495563244428889912nnreal,Xaa: extend8495563244428889912nnreal] :
      ( ! [R5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R5 )
         => ( ( X5
              = ( extend7643940197134561352nnreal @ R5 ) )
           => ! [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 ) ) ) ) ) ) )
     => ( ! [R5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ R5 )
           => ( ( X5
                = ( extend7643940197134561352nnreal @ R5 ) )
             => ! [Ra: real] :
                  ( ( ord_less_eq_real @ zero_zero_real @ Ra )
                 => ( ( Xa2
                      = ( extend7643940197134561352nnreal @ Ra ) )
                   => ( Xaa != top_to1496364449551166952nnreal ) ) ) ) )
       => ( ! [R5: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ R5 )
             => ( ( X5
                  = ( extend7643940197134561352nnreal @ R5 ) )
               => ( ( Xa2 = top_to1496364449551166952nnreal )
                 => ! [Ra: real] :
                      ( ( ord_less_eq_real @ zero_zero_real @ Ra )
                     => ( Xaa
                       != ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
         => ( ! [R5: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ R5 )
               => ( ( X5
                    = ( extend7643940197134561352nnreal @ R5 ) )
                 => ( ( Xa2 = top_to1496364449551166952nnreal )
                   => ( Xaa != top_to1496364449551166952nnreal ) ) ) )
           => ( ( ( X5 = top_to1496364449551166952nnreal )
               => ! [R5: real] :
                    ( ( ord_less_eq_real @ zero_zero_real @ R5 )
                   => ( ( Xa2
                        = ( extend7643940197134561352nnreal @ R5 ) )
                     => ! [Ra: real] :
                          ( ( ord_less_eq_real @ zero_zero_real @ Ra )
                         => ( Xaa
                           != ( extend7643940197134561352nnreal @ Ra ) ) ) ) ) )
             => ( ( ( X5 = top_to1496364449551166952nnreal )
                 => ! [R5: real] :
                      ( ( ord_less_eq_real @ zero_zero_real @ R5 )
                     => ( ( Xa2
                          = ( extend7643940197134561352nnreal @ R5 ) )
                       => ( Xaa != top_to1496364449551166952nnreal ) ) ) )
               => ( ( ( X5 = top_to1496364449551166952nnreal )
                   => ( ( Xa2 = top_to1496364449551166952nnreal )
                     => ! [R5: real] :
                          ( ( ord_less_eq_real @ zero_zero_real @ R5 )
                         => ( Xaa
                           != ( extend7643940197134561352nnreal @ R5 ) ) ) ) )
                 => ~ ( ( X5 = top_to1496364449551166952nnreal )
                     => ( ( Xa2 = top_to1496364449551166952nnreal )
                       => ( Xaa != top_to1496364449551166952nnreal ) ) ) ) ) ) ) ) ) ) ).

% ennreal3_cases
thf(fact_1174_ennreal2__cases,axiom,
    ! [X5: extend8495563244428889912nnreal,Xa2: extend8495563244428889912nnreal] :
      ( ! [R5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R5 )
         => ( ( X5
              = ( extend7643940197134561352nnreal @ R5 ) )
           => ! [Ra: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ Ra )
               => ( Xa2
                 != ( extend7643940197134561352nnreal @ Ra ) ) ) ) )
     => ( ! [R5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ R5 )
           => ( ( X5
                = ( extend7643940197134561352nnreal @ R5 ) )
             => ( Xa2 != top_to1496364449551166952nnreal ) ) )
       => ( ( ( X5 = top_to1496364449551166952nnreal )
           => ! [R5: real] :
                ( ( ord_less_eq_real @ zero_zero_real @ R5 )
               => ( Xa2
                 != ( extend7643940197134561352nnreal @ R5 ) ) ) )
         => ~ ( ( X5 = top_to1496364449551166952nnreal )
             => ( Xa2 != top_to1496364449551166952nnreal ) ) ) ) ) ).

% ennreal2_cases
thf(fact_1175_ennreal__cases,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ! [R5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ R5 )
         => ( X5
           != ( extend7643940197134561352nnreal @ R5 ) ) )
     => ( X5 = top_to1496364449551166952nnreal ) ) ).

% ennreal_cases
thf(fact_1176_top__neq__ennreal,axiom,
    ! [R3: real] :
      ( top_to1496364449551166952nnreal
     != ( extend7643940197134561352nnreal @ R3 ) ) ).

% top_neq_ennreal
thf(fact_1177_measurable__ennreal,axiom,
    member2919562650594848410nnreal @ extend7643940197134561352nnreal @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).

% measurable_ennreal
thf(fact_1178_r01__to__r01__r01__snd__measurable,axiom,
    member_real_real @ r01_to_r01_r01_snd @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% r01_to_r01_r01_snd_measurable
thf(fact_1179_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_1180_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_1181_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_1182_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_1183_r01__to__r01__r01__fst__measurable,axiom,
    member_real_real @ r01_to_r01_r01_fst @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% r01_to_r01_r01_fst_measurable
thf(fact_1184_ennreal_Ostandard__borel__sets,axiom,
    ! [Y4: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal )
        = ( sigma_5465916536984168985nnreal @ Y4 ) )
     => ( standa6705281362538419621nnreal @ Y4 ) ) ).

% ennreal.standard_borel_sets
thf(fact_1185_real_Ostandard__borel__sets,axiom,
    ! [Y4: sigma_measure_real] :
      ( ( ( sigma_sets_real @ borel_5078946678739801102l_real )
        = ( sigma_sets_real @ Y4 ) )
     => ( standard_borel_real @ Y4 ) ) ).

% real.standard_borel_sets
thf(fact_1186_real_Ostandard__borel__axioms,axiom,
    standard_borel_real @ borel_5078946678739801102l_real ).

% real.standard_borel_axioms
thf(fact_1187_ennreal_Ostandard__borel__axioms,axiom,
    standa6705281362538419621nnreal @ borel_6524799422816628122nnreal ).

% ennreal.standard_borel_axioms
thf(fact_1188_nat_Ostandard__borel__axioms,axiom,
    standard_borel_nat @ borel_8449730974584783410el_nat ).

% nat.standard_borel_axioms
thf(fact_1189_nat_Ostandard__borel__sets,axiom,
    ! [Y4: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ borel_8449730974584783410el_nat )
        = ( sigma_sets_nat @ Y4 ) )
     => ( standard_borel_nat @ Y4 ) ) ).

% nat.standard_borel_sets
thf(fact_1190_bool_Ostandard__borel__axioms,axiom,
    standard_borel_o @ borel_5500255247093592246orel_o ).

% bool.standard_borel_axioms
thf(fact_1191_bool_Ostandard__borel__sets,axiom,
    ! [Y4: sigma_measure_o] :
      ( ( ( sigma_sets_o @ borel_5500255247093592246orel_o )
        = ( sigma_sets_o @ Y4 ) )
     => ( standard_borel_o @ Y4 ) ) ).

% bool.standard_borel_sets
thf(fact_1192_r01__to__r01__r01__def,axiom,
    ( r01_to_r01_r01
    = ( ^ [R: real] : ( produc4511245868158468465l_real @ ( r01_to_r01_r01_fst @ R ) @ ( r01_to_r01_r01_snd @ R ) ) ) ) ).

% r01_to_r01_r01_def
thf(fact_1193_real_Og__meas,axiom,
    member_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% real.g_meas
thf(fact_1194_ennreal_Og__meas,axiom,
    member2919562650594848410nnreal @ ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal ) @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).

% ennreal.g_meas
thf(fact_1195_nat_Og__meas,axiom,
    member_real_nat @ ( standard_g_nat @ borel_8449730974584783410el_nat ) @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) ).

% nat.g_meas
thf(fact_1196_bool_Og__meas,axiom,
    member_real_o @ ( standard_g_o @ borel_5500255247093592246orel_o ) @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ borel_5500255247093592246orel_o ) ).

% bool.g_meas
thf(fact_1197_bool_Ogf__comp__id_I1_J,axiom,
    ! [X5: $o] :
      ( ( member_o @ X5 @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
     => ( ( comp_real_o_o @ ( standard_g_o @ borel_5500255247093592246orel_o ) @ ( standard_f_o @ borel_5500255247093592246orel_o ) @ X5 )
        = X5 ) ) ).

% bool.gf_comp_id(1)
thf(fact_1198_nat_Ogf__comp__id_I1_J,axiom,
    ! [X5: nat] :
      ( ( member_nat @ X5 @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
     => ( ( comp_real_nat_nat @ ( standard_g_nat @ borel_8449730974584783410el_nat ) @ ( standard_f_nat @ borel_8449730974584783410el_nat ) @ X5 )
        = X5 ) ) ).

% nat.gf_comp_id(1)
thf(fact_1199_nat_Ogf__comp__id_H_I2_J,axiom,
    ! [X5: nat] :
      ( ( standard_g_nat @ borel_8449730974584783410el_nat @ ( standard_f_nat @ borel_8449730974584783410el_nat @ X5 ) )
      = X5 ) ).

% nat.gf_comp_id'(2)
thf(fact_1200_bool_Ogf__comp__id_H_I2_J,axiom,
    ! [X5: $o] :
      ( ( standard_g_o @ borel_5500255247093592246orel_o @ ( standard_f_o @ borel_5500255247093592246orel_o @ X5 ) )
      = X5 ) ).

% bool.gf_comp_id'(2)
thf(fact_1201_nat_Ogf__comp__id_I2_J,axiom,
    ! [X5: nat] :
      ( ( member_nat @ X5 @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
     => ( ( standard_g_nat @ borel_8449730974584783410el_nat @ ( standard_f_nat @ borel_8449730974584783410el_nat @ X5 ) )
        = X5 ) ) ).

% nat.gf_comp_id(2)
thf(fact_1202_bool_Ogf__comp__id_I2_J,axiom,
    ! [X5: $o] :
      ( ( member_o @ X5 @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
     => ( ( standard_g_o @ borel_5500255247093592246orel_o @ ( standard_f_o @ borel_5500255247093592246orel_o @ X5 ) )
        = X5 ) ) ).

% bool.gf_comp_id(2)
thf(fact_1203_nat_Of__meas,axiom,
    member_nat_real @ ( standard_f_nat @ borel_8449730974584783410el_nat ) @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ).

% nat.f_meas
thf(fact_1204_bool_Of__meas,axiom,
    member_o_real @ ( standard_f_o @ borel_5500255247093592246orel_o ) @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ borel_5078946678739801102l_real ) ).

% bool.f_meas
thf(fact_1205_ennreal_Ogf__comp__id_I1_J,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ X5 @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
     => ( ( comp_r6281409797179841921nnreal @ ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal ) @ ( standa4501783974915749827nnreal @ borel_6524799422816628122nnreal ) @ X5 )
        = X5 ) ) ).

% ennreal.gf_comp_id(1)
thf(fact_1206_real_Ogf__comp__id_I1_J,axiom,
    ! [X5: real] :
      ( ( member_real @ X5 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( ( comp_real_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( standard_f_real @ borel_5078946678739801102l_real ) @ X5 )
        = X5 ) ) ).

% real.gf_comp_id(1)
thf(fact_1207_ennreal_Ogf__comp__id_I2_J,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ X5 @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
     => ( ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal @ ( standa4501783974915749827nnreal @ borel_6524799422816628122nnreal @ X5 ) )
        = X5 ) ) ).

% ennreal.gf_comp_id(2)
thf(fact_1208_real_Ogf__comp__id_I2_J,axiom,
    ! [X5: real] :
      ( ( member_real @ X5 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( ( standard_g_real @ borel_5078946678739801102l_real @ ( standard_f_real @ borel_5078946678739801102l_real @ X5 ) )
        = X5 ) ) ).

% real.gf_comp_id(2)
thf(fact_1209_real_Ogf__comp__id_H_I2_J,axiom,
    ! [X5: real] :
      ( ( standard_g_real @ borel_5078946678739801102l_real @ ( standard_f_real @ borel_5078946678739801102l_real @ X5 ) )
      = X5 ) ).

% real.gf_comp_id'(2)
thf(fact_1210_ennreal_Ogf__comp__id_H_I2_J,axiom,
    ! [X5: extend8495563244428889912nnreal] :
      ( ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal @ ( standa4501783974915749827nnreal @ borel_6524799422816628122nnreal @ X5 ) )
      = X5 ) ).

% ennreal.gf_comp_id'(2)
thf(fact_1211_real_Of__meas,axiom,
    member_real_real @ ( standard_f_real @ borel_5078946678739801102l_real ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% real.f_meas
thf(fact_1212_ennreal_Of__meas,axiom,
    member2874014351250825754l_real @ ( standa4501783974915749827nnreal @ borel_6524799422816628122nnreal ) @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ borel_5078946678739801102l_real ) ).

% ennreal.f_meas
thf(fact_1213_bool_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_real_o_o @ ( standard_g_o @ borel_5500255247093592246orel_o ) @ ( standard_f_o @ borel_5500255247093592246orel_o ) )
    = id_o ) ).

% bool.gf_comp_id'(1)
thf(fact_1214_nat_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_real_nat_nat @ ( standard_g_nat @ borel_8449730974584783410el_nat ) @ ( standard_f_nat @ borel_8449730974584783410el_nat ) )
    = id_nat ) ).

% nat.gf_comp_id'(1)
thf(fact_1215_ennreal_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_r6281409797179841921nnreal @ ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal ) @ ( standa4501783974915749827nnreal @ borel_6524799422816628122nnreal ) )
    = id_Ext8331313139072774535nnreal ) ).

% ennreal.gf_comp_id'(1)
thf(fact_1216_real_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_real_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( standard_f_real @ borel_5078946678739801102l_real ) )
    = id_real ) ).

% real.gf_comp_id'(1)
thf(fact_1217_real_Ostandard__borel__space__UNIV__axioms,axiom,
    standa1306199911732814765V_real @ borel_5078946678739801102l_real ).

% real.standard_borel_space_UNIV_axioms
thf(fact_1218_ennreal_Ostandard__borel__space__UNIV__axioms,axiom,
    standa6183118758442165433nnreal @ borel_6524799422816628122nnreal ).

% ennreal.standard_borel_space_UNIV_axioms
thf(fact_1219_nat_Ostandard__borel__space__UNIV__axioms,axiom,
    standa39772052079309137IV_nat @ borel_8449730974584783410el_nat ).

% nat.standard_borel_space_UNIV_axioms
thf(fact_1220_bool_Ostandard__borel__space__UNIV__axioms,axiom,
    standa6982250985768520663UNIV_o @ borel_5500255247093592246orel_o ).

% bool.standard_borel_space_UNIV_axioms
thf(fact_1221_nat__real_Ol__r__r__sets,axiom,
    ( ( sigma_2363344875859395388t_real @ ( measur6583539744506187129t_real @ ( binary2938201662863479403t_real @ ( measur4416158800429964412bs_nat @ borel_8449730974584783410el_nat ) @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) )
    = ( sigma_2363344875859395388t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) ) ).

% nat_real.l_r_r_sets
thf(fact_1222_nat__real_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_r4825950202331181903t_real @ ( standa3432675769617534289t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) @ ( standa2191550596119666194t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) )
    = id_Pro272570185591972174t_real ) ).

% nat_real.gf_comp_id'(1)
thf(fact_1223_nat__real_Ogf__comp__id_I1_J,axiom,
    ! [X5: produc7716430852924023517t_real] :
      ( ( member557208447399453958t_real @ X5 @ ( sigma_2177939267068080229t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) )
     => ( ( comp_r4825950202331181903t_real @ ( standa3432675769617534289t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) @ ( standa2191550596119666194t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) @ X5 )
        = X5 ) ) ).

% nat_real.gf_comp_id(1)
thf(fact_1224_nat__real_Ogf__comp__id_I2_J,axiom,
    ! [X5: produc7716430852924023517t_real] :
      ( ( member557208447399453958t_real @ X5 @ ( sigma_2177939267068080229t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) )
     => ( ( standa3432675769617534289t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) @ ( standa2191550596119666194t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) @ X5 ) )
        = X5 ) ) ).

% nat_real.gf_comp_id(2)
thf(fact_1225_nat__real_Ogf__comp__id_H_I2_J,axiom,
    ! [X5: produc7716430852924023517t_real] :
      ( ( standa3432675769617534289t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) @ ( standa2191550596119666194t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) @ X5 ) )
      = X5 ) ).

% nat_real.gf_comp_id'(2)
thf(fact_1226_nat__real_Ostandard__borel__lr__sets__ident,axiom,
    ( ( sigma_2363344875859395388t_real @ ( measur6583539744506187129t_real @ ( measur2011440102552004913t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) ) )
    = ( sigma_2363344875859395388t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) ) ).

% nat_real.standard_borel_lr_sets_ident
thf(fact_1227_nat__real_Ostandard__borel__sets,axiom,
    ! [Y4: sigma_5310753476256395226t_real] :
      ( ( ( sigma_2363344875859395388t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) )
        = ( sigma_2363344875859395388t_real @ Y4 ) )
     => ( standa1520499236374798000t_real @ Y4 ) ) ).

% nat_real.standard_borel_sets
thf(fact_1228_nat__real_Ostandard__borel__axioms,axiom,
    standa1520499236374798000t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ).

% nat_real.standard_borel_axioms
thf(fact_1229_nat__real_Oexist__fg,axiom,
    ? [X3: produc7716430852924023517t_real > real] :
      ( ( member4539693599630147619l_real @ X3 @ ( sigma_7880537205408866233l_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > produc7716430852924023517t_real] :
          ( ( member969486235890772753t_real @ Xa @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) )
          & ! [Xb: produc7716430852924023517t_real] :
              ( ( member557208447399453958t_real @ Xb @ ( sigma_2177939267068080229t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) )
             => ( ( comp_r4825950202331181903t_real @ Xa @ X3 @ Xb )
                = Xb ) ) ) ) ).

% nat_real.exist_fg
thf(fact_1230_nat__real_Ospace__UNIV,axiom,
    ( ( sigma_2177939267068080229t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) )
    = top_to4855536200657754381t_real ) ).

% nat_real.space_UNIV
thf(fact_1231_nat__real_Og__meas,axiom,
    member969486235890772753t_real @ ( standa3432675769617534289t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) @ ( sigma_2975298441655967327t_real @ borel_5078946678739801102l_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) ).

% nat_real.g_meas
thf(fact_1232_nat__real_Of__meas,axiom,
    member4539693599630147619l_real @ ( standa2191550596119666194t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) @ ( sigma_7880537205408866233l_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) @ borel_5078946678739801102l_real ) ).

% nat_real.f_meas
thf(fact_1233_nat__real_Ostandard__borel__space__UNIV__axioms,axiom,
    standa1649057328583495836t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ).

% nat_real.standard_borel_space_UNIV_axioms
thf(fact_1234_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K2: nat,M7: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M7 @ K2 ) @ ( product_Pair_nat_nat @ M7 @ ( minus_minus_nat @ K2 @ M7 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M7 @ ( suc @ K2 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_1235_real__real_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_r97482993418849103l_real @ ( standa5011908621721856941l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) @ ( standa1305434478922034158l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) )
    = id_Pro8459329387447996714l_real ) ).

% real_real.gf_comp_id'(1)
thf(fact_1236_real__real_Ogf__comp__id_I1_J,axiom,
    ! [X5: produc2422161461964618553l_real] :
      ( ( member7849222048561428706l_real @ X5 @ ( sigma_2519298556039103681l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) )
     => ( ( comp_r97482993418849103l_real @ ( standa5011908621721856941l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) @ ( standa1305434478922034158l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) @ X5 )
        = X5 ) ) ).

% real_real.gf_comp_id(1)
thf(fact_1237_real__real_Ostandard__borel__lr__sets__ident,axiom,
    ( ( sigma_1543982426361076760l_real @ ( measur6658399249024522197l_real @ ( measur3029786804310284173l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) )
    = ( sigma_1543982426361076760l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) ).

% real_real.standard_borel_lr_sets_ident
thf(fact_1238_real__real_Ogf__comp__id_H_I2_J,axiom,
    ! [X5: produc2422161461964618553l_real] :
      ( ( standa5011908621721856941l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) @ ( standa1305434478922034158l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) @ X5 ) )
      = X5 ) ).

% real_real.gf_comp_id'(2)
thf(fact_1239_real__real_Ogf__comp__id_I2_J,axiom,
    ! [X5: produc2422161461964618553l_real] :
      ( ( member7849222048561428706l_real @ X5 @ ( sigma_2519298556039103681l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) )
     => ( ( standa5011908621721856941l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) @ ( standa1305434478922034158l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) @ X5 ) )
        = X5 ) ) ).

% real_real.gf_comp_id(2)
thf(fact_1240_real__real_Ol__r__r__sets,axiom,
    ( ( sigma_1543982426361076760l_real @ ( measur6658399249024522197l_real @ ( binary210465342804807751l_real @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) ) ) )
    = ( sigma_1543982426361076760l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) ).

% real_real.l_r_r_sets
thf(fact_1241_real__real_Oexist__fg,axiom,
    ? [X3: produc2422161461964618553l_real > real] :
      ( ( member6699615393305559423l_real @ X3 @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > produc2422161461964618553l_real] :
          ( ( member9086635009091248365l_real @ Xa @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) )
          & ! [Xb: produc2422161461964618553l_real] :
              ( ( member7849222048561428706l_real @ Xb @ ( sigma_2519298556039103681l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) )
             => ( ( comp_r97482993418849103l_real @ Xa @ X3 @ Xb )
                = Xb ) ) ) ) ).

% real_real.exist_fg
thf(fact_1242_real__real_Ostandard__borel__space__UNIV__axioms,axiom,
    standa8482556277449587064l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% real_real.standard_borel_space_UNIV_axioms
thf(fact_1243_real__real_Og__meas,axiom,
    member9086635009091248365l_real @ ( standa5011908621721856941l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ).

% real_real.g_meas
thf(fact_1244_real__real_Ostandard__borel__axioms,axiom,
    standa8333227632594042252l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% real_real.standard_borel_axioms
thf(fact_1245_real__real_Ostandard__borel__sets,axiom,
    ! [Y4: sigma_2308072346491277622l_real] :
      ( ( ( sigma_1543982426361076760l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
        = ( sigma_1543982426361076760l_real @ Y4 ) )
     => ( standa8333227632594042252l_real @ Y4 ) ) ).

% real_real.standard_borel_sets
thf(fact_1246_real__real_Of__meas,axiom,
    member6699615393305559423l_real @ ( standa1305434478922034158l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) @ borel_5078946678739801102l_real ) ).

% real_real.f_meas
thf(fact_1247_real__real_Ospace__UNIV,axiom,
    ( ( sigma_2519298556039103681l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
    = top_to1175844992842649833l_real ) ).

% real_real.space_UNIV
thf(fact_1248_prod__decode__aux_Ocases,axiom,
    ! [X5: product_prod_nat_nat] :
      ~ ! [K3: nat,M4: nat] :
          ( X5
         != ( product_Pair_nat_nat @ K3 @ M4 ) ) ).

% prod_decode_aux.cases
thf(fact_1249_r01__to__r01__r01__measurable,axiom,
    member9086635009091248365l_real @ r01_to_r01_r01 @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ).

% r01_to_r01_r01_measurable
thf(fact_1250_real__real_Opair__standard__borel__axioms,axiom,
    pair_s454119610699386988l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ).

% real_real.pair_standard_borel_axioms
thf(fact_1251_nat__real_Opair__standard__borel__axioms,axiom,
    pair_s8264832550775477520t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ).

% nat_real.pair_standard_borel_axioms
thf(fact_1252_prod__decode__aux_Oelims,axiom,
    ! [X5: nat,Xa2: nat,Y5: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X5 @ Xa2 )
        = Y5 )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X5 )
         => ( Y5
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X5 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X5 )
         => ( Y5
            = ( nat_prod_decode_aux @ ( suc @ X5 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X5 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_1253_nat__real_Og__surj_H,axiom,
    ( ( image_917144268913371882t_real @ ( standa3432675769617534289t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) @ top_top_set_real )
    = top_to4855536200657754381t_real ) ).

% nat_real.g_surj'
thf(fact_1254_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).

% zero_notin_Suc_image
thf(fact_1255_real_Og__surj_H,axiom,
    ( ( image_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ top_top_set_real )
    = top_top_set_real ) ).

% real.g_surj'
thf(fact_1256_nat_Og__surj_H,axiom,
    ( ( image_real_nat @ ( standard_g_nat @ borel_8449730974584783410el_nat ) @ top_top_set_real )
    = top_top_set_nat ) ).

% nat.g_surj'
thf(fact_1257_bool_Og__surj_H,axiom,
    ( ( image_real_o @ ( standard_g_o @ borel_5500255247093592246orel_o ) @ top_top_set_real )
    = top_top_set_o ) ).

% bool.g_surj'
thf(fact_1258_ennreal_Og__surj_H,axiom,
    ( ( image_7616191137145695467nnreal @ ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal ) @ top_top_set_real )
    = top_to7994903218803871134nnreal ) ).

% ennreal.g_surj'
thf(fact_1259_real__real_Og__surj_H,axiom,
    ( ( image_4241449277970500806l_real @ ( standa5011908621721856941l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) @ top_top_set_real )
    = top_to1175844992842649833l_real ) ).

% real_real.g_surj'
thf(fact_1260_r01__r01__to__r01__measurable,axiom,
    member6699615393305559423l_real @ r01_r01_to_r01 @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) @ borel_5078946678739801102l_real ) ).

% r01_r01_to_r01_measurable
thf(fact_1261_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_1262_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_1263_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_1264_real__real_Ocountable__space__discrete,axiom,
    ( ( counta6472190628005395188l_real @ ( sigma_2519298556039103681l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) )
   => ( ( sigma_1543982426361076760l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
      = ( sigma_1543982426361076760l_real @ ( sigma_7544814672420218578l_real @ ( sigma_2519298556039103681l_real @ ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) ) ) ) ).

% real_real.countable_space_discrete
thf(fact_1265_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_1266_nat__real_Ocountable__space__discrete,axiom,
    ( ( counta6100287699616567320t_real @ ( sigma_2177939267068080229t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) )
   => ( ( sigma_2363344875859395388t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) )
      = ( sigma_2363344875859395388t_real @ ( sigma_8348511806855285494t_real @ ( sigma_2177939267068080229t_real @ ( binary2880700947547503686t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ) ) ) ) ) ).

% nat_real.countable_space_discrete
thf(fact_1267_uncountable__UNIV__real,axiom,
    ~ ( counta7319604579010473777e_real @ top_top_set_real ) ).

% uncountable_UNIV_real
thf(fact_1268_complex__non__denum,axiom,
    ~ ? [F5: nat > complex] :
        ( ( image_nat_complex @ F5 @ top_top_set_nat )
        = top_top_set_complex ) ).

% complex_non_denum
thf(fact_1269_real__non__denum,axiom,
    ~ ? [F5: nat > real] :
        ( ( image_nat_real @ F5 @ top_top_set_nat )
        = top_top_set_real ) ).

% real_non_denum
thf(fact_1270_uncountable__UNIV__complex,axiom,
    ~ ( counta5113917769705169331omplex @ top_top_set_complex ) ).

% uncountable_UNIV_complex
thf(fact_1271_Least__Suc2,axiom,
    ! [P: nat > $o,N4: nat,Q: nat > $o,M3: nat] :
      ( ( P @ N4 )
     => ( ( Q @ M3 )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K3: nat] :
                ( ( P @ ( suc @ K3 ) )
                = ( Q @ K3 ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_1272_Least__Suc,axiom,
    ! [P: nat > $o,N4: nat] :
      ( ( P @ N4 )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M7: nat] : ( P @ ( suc @ M7 ) ) ) ) ) ) ) ).

% Least_Suc

% Helper facts (5)
thf(help_If_2_1_If_001tf__b_T,axiom,
    ! [X5: b,Y5: b] :
      ( ( if_b @ $false @ X5 @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001tf__b_T,axiom,
    ! [X5: b,Y5: b] :
      ( ( if_b @ $true @ X5 @ Y5 )
      = X5 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X5 @ Y5 )
      = Y5 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X5 @ Y5 )
      = X5 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( member_real_a
    @ ( comp_P9131689941513729756a_real @ ( produc5399252211680487944_b_a_a @ ( produc8421786640700244582_b_a_a @ ( produc7016990068875982646_b_a_a @ beta_1 ) ) )
      @ ^ [R: real] : ( produc8437504494124426337at_b_a @ ( produc1946830598933159916_nat_b @ ( product_Pair_real_b @ R @ ( beta_2 @ R ) ) @ ( beta_3 @ R ) ) @ ( beta_4 @ R ) ) )
    @ ( qbs_morphism_real_a @ ( measur6875533127466166616s_real @ borel_5078946678739801102l_real ) @ y ) ) ).

%------------------------------------------------------------------------------