TPTP Problem File: SLH0651^1.p
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%------------------------------------------------------------------------------
% 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 : Universal_Hash_Families/0033_Preliminary_Results/prob_00083_003008__18477988_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1722 ( 496 unt; 443 typ; 0 def)
% Number of atoms : 3815 (1028 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 15098 ( 213 ~; 14 |; 330 &;12832 @)
% ( 0 <=>;1709 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 58 ( 57 usr)
% Number of type conns : 2553 (2553 >; 0 *; 0 +; 0 <<)
% Number of symbols : 389 ( 386 usr; 22 con; 0-5 aty)
% Number of variables : 4351 ( 588 ^;3703 !; 60 ?;4351 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:44:39.018
%------------------------------------------------------------------------------
% Could-be-implicit typings (57)
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__d_J_J,type,
sigma_measure_set_d: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__b_J_J,type,
sigma_measure_set_b: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_Itf__a_J_J,type,
sigma_measure_set_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_062_Itf__b_Mtf__d_J_J,type,
sigma_measure_b_d: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__d_J_J_J,type,
set_set_set_d: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
set_set_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_Itf__b_Mtf__d_J_J_J,type,
set_set_b_d: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
sigma_measure_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mt__Nat__Onat_J_J,type,
set_c_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__c_J_J,type,
set_nat_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
set_o_nat: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__d_J,type,
sigma_measure_d: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__c_J,type,
sigma_measure_c: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__b_J,type,
sigma_measure_b: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__a_J,type,
sigma_measure_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
set_set_d: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
set_set_c: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_Eo_J,type,
sigma_measure_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__d_J_J,type,
set_d_d: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__b_J_J,type,
set_d_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__a_J_J,type,
set_d_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__c_J_J,type,
set_c_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__b_J_J,type,
set_c_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
set_b_d: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__c_J_J,type,
set_b_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
set_b_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
set_b_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__d_J_J,type,
set_a_d: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
set_a_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__c_M_Eo_J_J,type,
set_c_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__b_M_Eo_J_J,type,
set_b_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__d_J_J,type,
set_o_d: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__c_J_J,type,
set_o_c: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__b_J_J,type,
set_o_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
set_o_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__d_J,type,
set_d: $tType ).
thf(ty_n_t__Set__Oset_Itf__c_J,type,
set_c: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__d,type,
d: $tType ).
thf(ty_n_tf__c,type,
c: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (386)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__b_J,type,
comple2307003614231284044_set_b: set_set_b > set_b ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001_Eo_001tf__a,type,
prob_k8052462064657410095_a_o_a: sigma_measure_a > nat > ( $o > sigma_measure_a ) > ( $o > a > a ) > set_o > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001_Eo_001tf__b,type,
prob_k8052462064657410096_a_o_b: sigma_measure_a > nat > ( $o > sigma_measure_b ) > ( $o > a > b ) > set_o > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001_Eo_001tf__d,type,
prob_k8052462064657410098_a_o_d: sigma_measure_a > nat > ( $o > sigma_measure_d ) > ( $o > a > d ) > set_o > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001t__Nat__Onat_001tf__a,type,
prob_k6325968634923510307_nat_a: sigma_measure_a > nat > ( nat > sigma_measure_a ) > ( nat > a > a ) > set_nat > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001t__Nat__Onat_001tf__b,type,
prob_k6325968634923510308_nat_b: sigma_measure_a > nat > ( nat > sigma_measure_b ) > ( nat > a > b ) > set_nat > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001t__Nat__Onat_001tf__d,type,
prob_k6325968634923510310_nat_d: sigma_measure_a > nat > ( nat > sigma_measure_d ) > ( nat > a > d ) > set_nat > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__a_001tf__a,type,
prob_k138169005419483465_a_a_a: sigma_measure_a > nat > ( a > sigma_measure_a ) > ( a > a > a ) > set_a > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__a_001tf__b,type,
prob_k138169005419483466_a_a_b: sigma_measure_a > nat > ( a > sigma_measure_b ) > ( a > a > b ) > set_a > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__a_001tf__d,type,
prob_k138169005419483468_a_a_d: sigma_measure_a > nat > ( a > sigma_measure_d ) > ( a > a > d ) > set_a > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__b_001tf__a,type,
prob_k6574085460301583240_a_b_a: sigma_measure_a > nat > ( b > sigma_measure_a ) > ( b > a > a ) > set_b > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__b_001tf__b,type,
prob_k6574085460301583241_a_b_b: sigma_measure_a > nat > ( b > sigma_measure_b ) > ( b > a > b ) > set_b > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__b_001tf__d,type,
prob_k6574085460301583243_a_b_d: sigma_measure_a > nat > ( b > sigma_measure_d ) > ( b > a > d ) > set_b > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__c_001tf__a,type,
prob_k3786629878328907207_a_c_a: sigma_measure_a > nat > ( c > sigma_measure_a ) > ( c > a > a ) > set_c > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__c_001tf__b,type,
prob_k3786629878328907208_a_c_b: sigma_measure_a > nat > ( c > sigma_measure_b ) > ( c > a > b ) > set_c > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__c_001tf__d,type,
prob_k3786629878328907210_a_c_d: sigma_measure_a > nat > ( c > sigma_measure_d ) > ( c > a > d ) > set_c > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001_Eo_001tf__a,type,
prob_k64534356759028784_b_o_a: sigma_measure_b > nat > ( $o > sigma_measure_a ) > ( $o > b > a ) > set_o > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001_Eo_001tf__b,type,
prob_k64534356759028785_b_o_b: sigma_measure_b > nat > ( $o > sigma_measure_b ) > ( $o > b > b ) > set_o > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001_Eo_001tf__d,type,
prob_k64534356759028787_b_o_d: sigma_measure_b > nat > ( $o > sigma_measure_d ) > ( $o > b > d ) > set_o > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001t__Nat__Onat_001tf__a,type,
prob_k7670433671732998242_nat_a: sigma_measure_b > nat > ( nat > sigma_measure_a ) > ( nat > b > a ) > set_nat > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001t__Nat__Onat_001tf__b,type,
prob_k7670433671732998243_nat_b: sigma_measure_b > nat > ( nat > sigma_measure_b ) > ( nat > b > b ) > set_nat > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001t__Nat__Onat_001tf__d,type,
prob_k7670433671732998245_nat_d: sigma_measure_b > nat > ( nat > sigma_measure_d ) > ( nat > b > d ) > set_nat > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__a_001tf__a,type,
prob_k5734387620607614154_b_a_a: sigma_measure_b > nat > ( a > sigma_measure_a ) > ( a > b > a ) > set_a > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__a_001tf__b,type,
prob_k5734387620607614155_b_a_b: sigma_measure_b > nat > ( a > sigma_measure_b ) > ( a > b > b ) > set_a > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__a_001tf__d,type,
prob_k5734387620607614157_b_a_d: sigma_measure_b > nat > ( a > sigma_measure_d ) > ( a > b > d ) > set_a > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__b_001tf__a,type,
prob_k2946932038634938121_b_b_a: sigma_measure_b > nat > ( b > sigma_measure_a ) > ( b > b > a ) > set_b > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__b_001tf__b,type,
prob_k2946932038634938122_b_b_b: sigma_measure_b > nat > ( b > sigma_measure_b ) > ( b > b > b ) > set_b > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__b_001tf__d,type,
prob_k2946932038634938124_b_b_d: sigma_measure_b > nat > ( b > sigma_measure_d ) > ( b > b > d ) > set_b > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__c_001tf__a,type,
prob_k159476456662262088_b_c_a: sigma_measure_b > nat > ( c > sigma_measure_a ) > ( c > b > a ) > set_c > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__c_001tf__b,type,
prob_k159476456662262089_b_c_b: sigma_measure_b > nat > ( c > sigma_measure_b ) > ( c > b > b ) > set_c > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__b_001tf__c_001tf__d,type,
prob_k159476456662262091_b_c_d: sigma_measure_b > nat > ( c > sigma_measure_d ) > ( c > b > d ) > set_c > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mtf__b_J,type,
finite_finite_a_b: set_a_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__b_Mtf__d_J,type,
finite_finite_b_d: set_b_d > $o ).
thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
finite_finite_o: set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__b_J,type,
finite_finite_set_b: set_set_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
finite_finite_b: set_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__c,type,
finite_finite_c: set_c > $o ).
thf(sy_c_Fun_Ocomp_001tf__b_001tf__d_001tf__a,type,
comp_b_d_a: ( b > d ) > ( a > b ) > a > d ).
thf(sy_c_Giry__Monad_Osubprob__space_001_Eo,type,
giry_subprob_space_o: sigma_measure_o > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001tf__a,type,
giry_subprob_space_a: sigma_measure_a > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001tf__b,type,
giry_subprob_space_b: sigma_measure_b > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001tf__d,type,
giry_subprob_space_d: sigma_measure_d > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__b,type,
if_b: $o > b > b > b ).
thf(sy_c_If_001tf__d,type,
if_d: $o > d > d > d ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001_Eo,type,
indepe796274259248320173ts_b_o: sigma_measure_b > ( $o > set_b ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001t__Nat__Onat,type,
indepe2786641642957426683_b_nat: sigma_measure_b > ( nat > set_b ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001tf__a,type,
indepe8384762880973921171ts_b_a: sigma_measure_b > ( a > set_b ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001tf__b,type,
indepe8384762880973921172ts_b_b: sigma_measure_b > ( b > set_b ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__b_001tf__c,type,
indepe8384762880973921173ts_b_c: sigma_measure_b > ( c > set_b ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__set_001tf__a,type,
indepe2041756565122539606_set_a: sigma_measure_a > set_set_a > set_set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__set_001tf__b,type,
indepe2041756565122539607_set_b: sigma_measure_b > set_set_b > set_set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__a_Mtf__b_J,type,
indepe1219004915316159480_a_a_b: sigma_measure_a > ( ( a > b ) > set_set_a ) > set_a_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_062_Itf__b_Mtf__d_J,type,
indepe7654921378804716857_a_b_d: sigma_measure_a > ( ( b > d ) > set_set_a ) > set_b_d > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_Eo,type,
indepe7780107833195774214ts_a_o: sigma_measure_a > ( $o > set_set_a ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Nat__Onat,type,
indepe6267730027088848354_a_nat: sigma_measure_a > ( nat > set_set_a ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Set__Oset_Itf__a_J,type,
indepe4967106450811773644_set_a: sigma_measure_a > ( set_a > set_set_a ) > set_set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Set__Oset_Itf__b_J,type,
indepe4967106455115002445_set_b: sigma_measure_a > ( set_b > set_set_a ) > set_set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Set__Oset_Itf__d_J,type,
indepe4967106463721460047_set_d: sigma_measure_a > ( set_d > set_set_a ) > set_set_d > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001tf__a,type,
indepe8927441866673418604ts_a_a: sigma_measure_a > ( a > set_set_a ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001tf__b,type,
indepe8927441866673418605ts_a_b: sigma_measure_a > ( b > set_set_a ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001tf__c,type,
indepe8927441866673418606ts_a_c: sigma_measure_a > ( c > set_set_a ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_062_Itf__a_Mtf__b_J,type,
indepe7072862685212782073_b_a_b: sigma_measure_b > ( ( a > b ) > set_set_b ) > set_a_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_062_Itf__b_Mtf__d_J,type,
indepe4285407111846563642_b_b_d: sigma_measure_b > ( ( b > d ) > set_set_b ) > set_b_d > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001_Eo,type,
indepe4880885433731379909ts_b_o: sigma_measure_b > ( $o > set_set_b ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Nat__Onat,type,
indepe7503174356045242851_b_nat: sigma_measure_b > ( nat > set_set_b ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Set__Oset_Itf__a_J,type,
indepe6311571487621261579_set_a: sigma_measure_b > ( set_a > set_set_b ) > set_set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Set__Oset_Itf__b_J,type,
indepe6311571491924490380_set_b: sigma_measure_b > ( set_b > set_set_b ) > set_set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001t__Set__Oset_Itf__d_J,type,
indepe6311571500530947982_set_d: sigma_measure_b > ( set_d > set_set_b ) > set_set_d > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001tf__a,type,
indepe6139986284700742571ts_b_a: sigma_measure_b > ( a > set_set_b ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001tf__b,type,
indepe6139986284700742572ts_b_b: sigma_measure_b > ( b > set_set_b ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__b_001tf__c,type,
indepe6139986284700742573ts_b_c: sigma_measure_b > ( c > set_set_b ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__d_001_Eo,type,
indepe8305812671657367107ts_d_o: sigma_measure_d > ( $o > set_set_d ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__d_001t__Nat__Onat,type,
indepe750690977103256037_d_nat: sigma_measure_d > ( nat > set_set_d ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__d_001tf__a,type,
indepe565075120755390505ts_d_a: sigma_measure_d > ( a > set_set_d ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__d_001tf__b,type,
indepe565075120755390506ts_d_b: sigma_measure_d > ( b > set_set_d ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__d_001tf__c,type,
indepe565075120755390507ts_d_c: sigma_measure_d > ( c > set_set_d ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__b_001tf__d,type,
indepe8876569649573725966ar_b_d: sigma_measure_b > sigma_measure_d > ( b > d ) > sigma_measure_d > ( b > d ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__a_001tf__c_001tf__d,type,
indepe4851901773132442935_a_c_d: sigma_measure_a > ( c > sigma_measure_d ) > ( c > a > d ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001_Eo_001_Eo,type,
indepe2258988419405707831_b_o_o: sigma_measure_b > ( $o > sigma_measure_o ) > ( $o > b > $o ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001_Eo_001tf__a,type,
indepe4488128152570241565_b_o_a: sigma_measure_b > ( $o > sigma_measure_a ) > ( $o > b > a ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001_Eo_001tf__b,type,
indepe4488128152570241566_b_o_b: sigma_measure_b > ( $o > sigma_measure_b ) > ( $o > b > b ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001_Eo_001tf__c,type,
indepe4488128152570241567_b_o_c: sigma_measure_b > ( $o > sigma_measure_c ) > ( $o > b > c ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001_Eo_001tf__d,type,
indepe4488128152570241568_b_o_d: sigma_measure_b > ( $o > sigma_measure_d ) > ( $o > b > d ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001t__Nat__Onat_001_Eo,type,
indepe3736482776648739791_nat_o: sigma_measure_b > ( nat > sigma_measure_o ) > ( nat > b > $o ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001t__Nat__Onat_001tf__a,type,
indepe4589662937738594229_nat_a: sigma_measure_b > ( nat > sigma_measure_a ) > ( nat > b > a ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001t__Nat__Onat_001tf__b,type,
indepe4589662937738594230_nat_b: sigma_measure_b > ( nat > sigma_measure_b ) > ( nat > b > b ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001t__Nat__Onat_001tf__c,type,
indepe4589662937738594231_nat_c: sigma_measure_b > ( nat > sigma_measure_c ) > ( nat > b > c ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001t__Nat__Onat_001tf__d,type,
indepe4589662937738594232_nat_d: sigma_measure_b > ( nat > sigma_measure_d ) > ( nat > b > d ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__a_001_Eo,type,
indepe4567608309035989329_b_a_o: sigma_measure_b > ( a > sigma_measure_o ) > ( a > b > $o ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__a_001tf__a,type,
indepe6799659515411149879_b_a_a: sigma_measure_b > ( a > sigma_measure_a ) > ( a > b > a ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__a_001tf__b,type,
indepe6799659515411149880_b_a_b: sigma_measure_b > ( a > sigma_measure_b ) > ( a > b > b ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__a_001tf__c,type,
indepe6799659515411149881_b_a_c: sigma_measure_b > ( a > sigma_measure_c ) > ( a > b > c ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__a_001tf__d,type,
indepe6799659515411149882_b_a_d: sigma_measure_b > ( a > sigma_measure_d ) > ( a > b > d ) > set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__b_001_Eo,type,
indepe1668385909571595024_b_b_o: sigma_measure_b > ( b > sigma_measure_o ) > ( b > b > $o ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__b_001tf__a,type,
indepe4012203933438473846_b_b_a: sigma_measure_b > ( b > sigma_measure_a ) > ( b > b > a ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__b_001tf__b,type,
indepe4012203933438473847_b_b_b: sigma_measure_b > ( b > sigma_measure_b ) > ( b > b > b ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__b_001tf__c,type,
indepe4012203933438473848_b_b_c: sigma_measure_b > ( b > sigma_measure_c ) > ( b > b > c ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__b_001tf__d,type,
indepe4012203933438473849_b_b_d: sigma_measure_b > ( b > sigma_measure_d ) > ( b > b > d ) > set_b > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__c_001_Eo,type,
indepe7992535546961976527_b_c_o: sigma_measure_b > ( c > sigma_measure_o ) > ( c > b > $o ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__c_001tf__a,type,
indepe1224748351465797813_b_c_a: sigma_measure_b > ( c > sigma_measure_a ) > ( c > b > a ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__c_001tf__b,type,
indepe1224748351465797814_b_c_b: sigma_measure_b > ( c > sigma_measure_b ) > ( c > b > b ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__c_001tf__c,type,
indepe1224748351465797815_b_c_c: sigma_measure_b > ( c > sigma_measure_c ) > ( c > b > c ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__vars_001tf__b_001tf__c_001tf__d,type,
indepe1224748351465797816_b_c_d: sigma_measure_b > ( c > sigma_measure_d ) > ( c > b > d ) > set_c > $o ).
thf(sy_c_Independent__Family_Oprob__space_Otail__events_001tf__a_001t__Nat__Onat,type,
indepe7538416700049374166_a_nat: sigma_measure_a > ( nat > set_set_a ) > set_set_a ).
thf(sy_c_Independent__Family_Oprob__space_Otail__events_001tf__b_001t__Nat__Onat,type,
indepe8773861029005768663_b_nat: sigma_measure_b > ( nat > set_set_b ) > set_set_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
inf_inf_set_b_d: set_b_d > set_b_d > set_b_d ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
inf_inf_set_set_b: set_set_b > set_set_b > set_set_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
inf_inf_set_set_d: set_set_d > set_set_d > set_set_d ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__c_J,type,
inf_inf_set_c: set_c > set_c > set_c ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__d_J,type,
inf_inf_set_d: set_d > set_d > set_d ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
sup_sup_set_b: set_b > set_b > set_b ).
thf(sy_c_Measure__Space_Odistr_001_Eo_001tf__a,type,
measure_distr_o_a: sigma_measure_o > sigma_measure_a > ( $o > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001_Eo_001tf__b,type,
measure_distr_o_b: sigma_measure_o > sigma_measure_b > ( $o > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001_Eo_001tf__d,type,
measure_distr_o_d: sigma_measure_o > sigma_measure_d > ( $o > d ) > sigma_measure_d ).
thf(sy_c_Measure__Space_Odistr_001t__Nat__Onat_001tf__a,type,
measure_distr_nat_a: sigma_measure_nat > sigma_measure_a > ( nat > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001t__Nat__Onat_001tf__b,type,
measure_distr_nat_b: sigma_measure_nat > sigma_measure_b > ( nat > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001t__Nat__Onat_001tf__d,type,
measure_distr_nat_d: sigma_measure_nat > sigma_measure_d > ( nat > d ) > sigma_measure_d ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001_Eo,type,
measure_distr_a_o: sigma_measure_a > sigma_measure_o > ( a > $o ) > sigma_measure_o ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__a,type,
measure_distr_a_a: sigma_measure_a > sigma_measure_a > ( a > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__b,type,
measure_distr_a_b: sigma_measure_a > sigma_measure_b > ( a > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001tf__a_001tf__d,type,
measure_distr_a_d: sigma_measure_a > sigma_measure_d > ( a > d ) > sigma_measure_d ).
thf(sy_c_Measure__Space_Odistr_001tf__b_001_Eo,type,
measure_distr_b_o: sigma_measure_b > sigma_measure_o > ( b > $o ) > sigma_measure_o ).
thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__a,type,
measure_distr_b_a: sigma_measure_b > sigma_measure_a > ( b > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__b,type,
measure_distr_b_b: sigma_measure_b > sigma_measure_b > ( b > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001tf__b_001tf__d,type,
measure_distr_b_d: sigma_measure_b > sigma_measure_d > ( b > d ) > sigma_measure_d ).
thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__a,type,
measure_distr_c_a: sigma_measure_c > sigma_measure_a > ( c > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__b,type,
measure_distr_c_b: sigma_measure_c > sigma_measure_b > ( c > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001tf__c_001tf__d,type,
measure_distr_c_d: sigma_measure_c > sigma_measure_d > ( c > d ) > sigma_measure_d ).
thf(sy_c_Measure__Space_Odistr_001tf__d_001tf__a,type,
measure_distr_d_a: sigma_measure_d > sigma_measure_a > ( d > a ) > sigma_measure_a ).
thf(sy_c_Measure__Space_Odistr_001tf__d_001tf__b,type,
measure_distr_d_b: sigma_measure_d > sigma_measure_b > ( d > b ) > sigma_measure_b ).
thf(sy_c_Measure__Space_Odistr_001tf__d_001tf__d,type,
measure_distr_d_d: sigma_measure_d > sigma_measure_d > ( d > d ) > sigma_measure_d ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Real__Oreal,type,
measur1776380161843274167a_real: set_set_a > ( set_a > real ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__b_001t__Real__Oreal,type,
measur7372598777031404856b_real: set_set_b > ( set_b > real ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
bot_bot_set_a_b: set_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
bot_bot_set_b_d: set_b_d ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
bot_bot_set_set_b: set_set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
bot_bot_set_set_d: set_set_d ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__c_J,type,
bot_bot_set_c: set_c ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__d_J,type,
bot_bot_set_d: set_d ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
ord_less_eq_set_a_a: set_a_a > set_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
ord_less_eq_set_a_b: set_a_b > set_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__d_J_J,type,
ord_less_eq_set_a_d: set_a_d > set_a_d > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
ord_less_eq_set_b_a: set_b_a > set_b_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
ord_less_eq_set_b_b: set_b_b > set_b_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
ord_less_eq_set_b_d: set_b_d > set_b_d > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__d_Mtf__a_J_J,type,
ord_less_eq_set_d_a: set_d_a > set_d_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__d_Mtf__b_J_J,type,
ord_less_eq_set_d_b: set_d_b > set_d_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__d_Mtf__d_J_J,type,
ord_less_eq_set_d_d: set_d_d > set_d_d > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
ord_le4374716579403074808_set_o: set_set_o > set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__b_J_J_J,type,
ord_le3201067847557142847_set_b: set_set_set_b > set_set_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
ord_le3795704787696855135_set_b: set_set_b > set_set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__c_J_J,type,
ord_le3866738827743201120_set_c: set_set_c > set_set_c > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
ord_le3937772867789547105_set_d: set_set_d > set_set_d > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__c_J,type,
ord_less_eq_set_c: set_c > set_c > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__d_J,type,
ord_less_eq_set_d: set_d > set_d > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Probability__Measure_Oprob__space_001tf__a,type,
probab7247484486040049089pace_a: sigma_measure_a > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001tf__b,type,
probab7247484486040049090pace_b: sigma_measure_b > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001tf__d,type,
probab7247484486040049092pace_d: sigma_measure_d > $o ).
thf(sy_c_Product__Type_Obool_Ocase__bool_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
produc6113963288868236716_set_a: set_set_a > set_set_a > $o > set_set_a ).
thf(sy_c_Product__Type_Obool_Ocase__bool_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
produc6184997328914582701_set_b: set_set_b > set_set_b > $o > set_set_b ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__b_J,type,
collect_a_b: ( ( a > b ) > $o ) > set_a_b ).
thf(sy_c_Set_OCollect_001_062_Itf__b_Mtf__d_J,type,
collect_b_d: ( ( b > d ) > $o ) > set_b_d ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__d_J,type,
collect_set_d: ( set_d > $o ) > set_set_d ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_OCollect_001tf__c,type,
collect_c: ( c > $o ) > set_c ).
thf(sy_c_Set_OCollect_001tf__d,type,
collect_d: ( d > $o ) > set_d ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001tf__a,type,
image_o_a: ( $o > a ) > set_o > set_a ).
thf(sy_c_Set_Oimage_001_Eo_001tf__b,type,
image_o_b: ( $o > b ) > set_o > set_b ).
thf(sy_c_Set_Oimage_001_Eo_001tf__c,type,
image_o_c: ( $o > c ) > set_o > set_c ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__b_J,type,
image_nat_set_b: ( nat > set_b ) > set_nat > set_set_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__b,type,
image_nat_b: ( nat > b ) > set_nat > set_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__c,type,
image_nat_c: ( nat > c ) > set_nat > set_c ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__b_J,type,
image_set_a_set_b: ( set_a > set_b ) > set_set_a > set_set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__d_J,type,
image_set_a_set_d: ( set_a > set_d ) > set_set_a > set_set_d ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__a_J,type,
image_set_b_set_a: ( set_b > set_a ) > set_set_b > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
image_set_b_set_b: ( set_b > set_b ) > set_set_b > set_set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__d_J_001t__Set__Oset_Itf__a_J,type,
image_set_d_set_a: ( set_d > set_a ) > set_set_d > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__d_J_001t__Set__Oset_Itf__b_J,type,
image_set_d_set_b: ( set_d > set_b ) > set_set_d > set_set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__d_J_001t__Set__Oset_Itf__d_J,type,
image_set_d_set_d: ( set_d > set_d ) > set_set_d > set_set_d ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
image_a_b: ( a > b ) > set_a > set_b ).
thf(sy_c_Set_Oimage_001tf__a_001tf__c,type,
image_a_c: ( a > c ) > set_a > set_c ).
thf(sy_c_Set_Oimage_001tf__a_001tf__d,type,
image_a_d: ( a > d ) > set_a > set_d ).
thf(sy_c_Set_Oimage_001tf__b_001_Eo,type,
image_b_o: ( b > $o ) > set_b > set_o ).
thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
image_b_nat: ( b > nat ) > set_b > set_nat ).
thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
image_b_a: ( b > a ) > set_b > set_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001tf__c,type,
image_b_c: ( b > c ) > set_b > set_c ).
thf(sy_c_Set_Oimage_001tf__b_001tf__d,type,
image_b_d: ( b > d ) > set_b > set_d ).
thf(sy_c_Set_Oimage_001tf__c_001_Eo,type,
image_c_o: ( c > $o ) > set_c > set_o ).
thf(sy_c_Set_Oimage_001tf__c_001t__Nat__Onat,type,
image_c_nat: ( c > nat ) > set_c > set_nat ).
thf(sy_c_Set_Oimage_001tf__c_001tf__a,type,
image_c_a: ( c > a ) > set_c > set_a ).
thf(sy_c_Set_Oimage_001tf__c_001tf__b,type,
image_c_b: ( c > b ) > set_c > set_b ).
thf(sy_c_Set_Oimage_001tf__c_001tf__c,type,
image_c_c: ( c > c ) > set_c > set_c ).
thf(sy_c_Set_Oimage_001tf__d_001tf__a,type,
image_d_a: ( d > a ) > set_d > set_a ).
thf(sy_c_Set_Oimage_001tf__d_001tf__b,type,
image_d_b: ( d > b ) > set_d > set_b ).
thf(sy_c_Set_Oimage_001tf__d_001tf__d,type,
image_d_d: ( d > d ) > set_d > set_d ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001_Eo,type,
vimage_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001t__Nat__Onat,type,
vimage_o_nat: ( $o > nat ) > set_nat > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001tf__a,type,
vimage_o_a: ( $o > a ) > set_a > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001tf__b,type,
vimage_o_b: ( $o > b ) > set_b > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001tf__c,type,
vimage_o_c: ( $o > c ) > set_c > set_o ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_Eo,type,
vimage_nat_o: ( nat > $o ) > set_o > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
vimage_nat_set_a: ( nat > set_a ) > set_set_a > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Set__Oset_Itf__b_J,type,
vimage_nat_set_b: ( nat > set_b ) > set_set_b > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__a,type,
vimage_nat_a: ( nat > a ) > set_a > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__b,type,
vimage_nat_b: ( nat > b ) > set_b > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__c,type,
vimage_nat_c: ( nat > c ) > set_c > set_nat ).
thf(sy_c_Set_Ovimage_001tf__a_001_Eo,type,
vimage_a_o: ( a > $o ) > set_o > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Nat__Onat,type,
vimage_a_nat: ( a > nat ) > set_nat > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__b,type,
vimage_a_b: ( a > b ) > set_b > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__c,type,
vimage_a_c: ( a > c ) > set_c > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__d,type,
vimage_a_d: ( a > d ) > set_d > set_a ).
thf(sy_c_Set_Ovimage_001tf__b_001_Eo,type,
vimage_b_o: ( b > $o ) > set_o > set_b ).
thf(sy_c_Set_Ovimage_001tf__b_001t__Nat__Onat,type,
vimage_b_nat: ( b > nat ) > set_nat > set_b ).
thf(sy_c_Set_Ovimage_001tf__b_001tf__a,type,
vimage_b_a: ( b > a ) > set_a > set_b ).
thf(sy_c_Set_Ovimage_001tf__b_001tf__b,type,
vimage_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Ovimage_001tf__b_001tf__c,type,
vimage_b_c: ( b > c ) > set_c > set_b ).
thf(sy_c_Set_Ovimage_001tf__b_001tf__d,type,
vimage_b_d: ( b > d ) > set_d > set_b ).
thf(sy_c_Set_Ovimage_001tf__c_001_Eo,type,
vimage_c_o: ( c > $o ) > set_o > set_c ).
thf(sy_c_Set_Ovimage_001tf__c_001t__Nat__Onat,type,
vimage_c_nat: ( c > nat ) > set_nat > set_c ).
thf(sy_c_Set_Ovimage_001tf__c_001tf__a,type,
vimage_c_a: ( c > a ) > set_a > set_c ).
thf(sy_c_Set_Ovimage_001tf__c_001tf__b,type,
vimage_c_b: ( c > b ) > set_b > set_c ).
thf(sy_c_Set_Ovimage_001tf__c_001tf__c,type,
vimage_c_c: ( c > c ) > set_c > set_c ).
thf(sy_c_Set_Ovimage_001tf__d_001tf__a,type,
vimage_d_a: ( d > a ) > set_a > set_d ).
thf(sy_c_Set_Ovimage_001tf__d_001tf__b,type,
vimage_d_b: ( d > b ) > set_b > set_d ).
thf(sy_c_Set_Ovimage_001tf__d_001tf__d,type,
vimage_d_d: ( d > d ) > set_d > set_d ).
thf(sy_c_Sigma__Algebra_ODynkin_001tf__a,type,
sigma_Dynkin_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_ODynkin_001tf__b,type,
sigma_Dynkin_b: set_b > set_set_b > set_set_b ).
thf(sy_c_Sigma__Algebra_ODynkin_001tf__d,type,
sigma_Dynkin_d: set_d > set_set_d > set_set_d ).
thf(sy_c_Sigma__Algebra_OInt__stable_001tf__a,type,
sigma_Int_stable_a: set_set_a > $o ).
thf(sy_c_Sigma__Algebra_OInt__stable_001tf__b,type,
sigma_Int_stable_b: set_set_b > $o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001_Eo,type,
sigma_measurable_o_o: sigma_measure_o > sigma_measure_o > set_o_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Nat__Onat,type,
sigma_1999164137574644376_o_nat: sigma_measure_o > sigma_measure_nat > set_o_nat ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001tf__a,type,
sigma_measurable_o_a: sigma_measure_o > sigma_measure_a > set_o_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001tf__b,type,
sigma_measurable_o_b: sigma_measure_o > sigma_measure_b > set_o_b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001tf__c,type,
sigma_measurable_o_c: sigma_measure_o > sigma_measure_c > set_o_c ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001tf__d,type,
sigma_measurable_o_d: sigma_measure_o > sigma_measure_d > set_o_d ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001_Eo,type,
sigma_5101835498682829686_nat_o: sigma_measure_nat > sigma_measure_o > set_nat_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001t__Nat__Onat,type,
sigma_4350458207664084850at_nat: sigma_measure_nat > sigma_measure_nat > set_nat_nat ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001tf__c,type,
sigma_4105081583803843550_nat_c: sigma_measure_nat > sigma_measure_c > set_nat_c ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001_Eo,type,
sigma_measurable_a_o: sigma_measure_a > sigma_measure_o > set_a_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__a,type,
sigma_measurable_a_a: sigma_measure_a > sigma_measure_a > set_a_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__b,type,
sigma_measurable_a_b: sigma_measure_a > sigma_measure_b > set_a_b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__d,type,
sigma_measurable_a_d: sigma_measure_a > sigma_measure_d > set_a_d ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001_Eo,type,
sigma_measurable_b_o: sigma_measure_b > sigma_measure_o > set_b_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__a,type,
sigma_measurable_b_a: sigma_measure_b > sigma_measure_a > set_b_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__b,type,
sigma_measurable_b_b: sigma_measure_b > sigma_measure_b > set_b_b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__c,type,
sigma_measurable_b_c: sigma_measure_b > sigma_measure_c > set_b_c ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__d,type,
sigma_measurable_b_d: sigma_measure_b > sigma_measure_d > set_b_d ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001_Eo,type,
sigma_measurable_c_o: sigma_measure_c > sigma_measure_o > set_c_o ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001t__Nat__Onat,type,
sigma_2544038740538346112_c_nat: sigma_measure_c > sigma_measure_nat > set_c_nat ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001tf__b,type,
sigma_measurable_c_b: sigma_measure_c > sigma_measure_b > set_c_b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001tf__c,type,
sigma_measurable_c_c: sigma_measure_c > sigma_measure_c > set_c_c ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__d_001tf__a,type,
sigma_measurable_d_a: sigma_measure_d > sigma_measure_a > set_d_a ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__d_001tf__b,type,
sigma_measurable_d_b: sigma_measure_d > sigma_measure_b > set_d_b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__d_001tf__d,type,
sigma_measurable_d_d: sigma_measure_d > sigma_measure_d > set_d_d ).
thf(sy_c_Sigma__Algebra_Omeasure_001tf__a,type,
sigma_measure_a2: sigma_measure_a > set_a > real ).
thf(sy_c_Sigma__Algebra_Omeasure_001tf__b,type,
sigma_measure_b2: sigma_measure_b > set_b > real ).
thf(sy_c_Sigma__Algebra_Omeasure_001tf__d,type,
sigma_measure_d2: sigma_measure_d > set_d > real ).
thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__b_Mtf__d_J,type,
sigma_sets_b_d: sigma_measure_b_d > set_set_b_d ).
thf(sy_c_Sigma__Algebra_Osets_001_Eo,type,
sigma_sets_o: sigma_measure_o > set_set_o ).
thf(sy_c_Sigma__Algebra_Osets_001t__Nat__Onat,type,
sigma_sets_nat: sigma_measure_nat > set_set_nat ).
thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_Itf__a_J,type,
sigma_sets_set_a: sigma_measure_set_a > set_set_set_a ).
thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_Itf__b_J,type,
sigma_sets_set_b: sigma_measure_set_b > set_set_set_b ).
thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_Itf__d_J,type,
sigma_sets_set_d: sigma_measure_set_d > set_set_set_d ).
thf(sy_c_Sigma__Algebra_Osets_001tf__a,type,
sigma_sets_a: sigma_measure_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Osets_001tf__b,type,
sigma_sets_b: sigma_measure_b > set_set_b ).
thf(sy_c_Sigma__Algebra_Osets_001tf__c,type,
sigma_sets_c: sigma_measure_c > set_set_c ).
thf(sy_c_Sigma__Algebra_Osets_001tf__d,type,
sigma_sets_d: sigma_measure_d > set_set_d ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001tf__a,type,
sigma_4968961713055010667ebra_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001tf__b,type,
sigma_4968961713055010668ebra_b: set_b > set_set_b > $o ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001tf__a,type,
sigma_sigma_sets_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Osigma__sets_001tf__b,type,
sigma_sigma_sets_b: set_b > set_set_b > set_set_b ).
thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__b_Mtf__d_J,type,
sigma_space_b_d: sigma_measure_b_d > set_b_d ).
thf(sy_c_Sigma__Algebra_Ospace_001_Eo,type,
sigma_space_o: sigma_measure_o > set_o ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Nat__Onat,type,
sigma_space_nat: sigma_measure_nat > set_nat ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__a_J,type,
sigma_space_set_a: sigma_measure_set_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__b_J,type,
sigma_space_set_b: sigma_measure_set_b > set_set_b ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_Itf__d_J,type,
sigma_space_set_d: sigma_measure_set_d > set_set_d ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__b,type,
sigma_space_b: sigma_measure_b > set_b ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__c,type,
sigma_space_c: sigma_measure_c > set_c ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__d,type,
sigma_space_d: sigma_measure_d > set_d ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
member_o_a: ( $o > a ) > set_o_a > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__b_J,type,
member_o_b: ( $o > b ) > set_o_b > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__c_J,type,
member_o_c: ( $o > c ) > set_o_c > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__d_J,type,
member_o_d: ( $o > d ) > set_o_d > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__c_J,type,
member_nat_c: ( nat > c ) > set_nat_c > $o ).
thf(sy_c_member_001_062_Itf__a_M_Eo_J,type,
member_a_o: ( a > $o ) > set_a_o > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__b_J,type,
member_a_b: ( a > b ) > set_a_b > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__d_J,type,
member_a_d: ( a > d ) > set_a_d > $o ).
thf(sy_c_member_001_062_Itf__b_M_Eo_J,type,
member_b_o: ( b > $o ) > set_b_o > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__a_J,type,
member_b_a: ( b > a ) > set_b_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
member_b_b: ( b > b ) > set_b_b > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__c_J,type,
member_b_c: ( b > c ) > set_b_c > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__d_J,type,
member_b_d: ( b > d ) > set_b_d > $o ).
thf(sy_c_member_001_062_Itf__c_M_Eo_J,type,
member_c_o: ( c > $o ) > set_c_o > $o ).
thf(sy_c_member_001_062_Itf__c_Mt__Nat__Onat_J,type,
member_c_nat: ( c > nat ) > set_c_nat > $o ).
thf(sy_c_member_001_062_Itf__c_Mtf__b_J,type,
member_c_b: ( c > b ) > set_c_b > $o ).
thf(sy_c_member_001_062_Itf__c_Mtf__c_J,type,
member_c_c: ( c > c ) > set_c_c > $o ).
thf(sy_c_member_001_062_Itf__d_Mtf__a_J,type,
member_d_a: ( d > a ) > set_d_a > $o ).
thf(sy_c_member_001_062_Itf__d_Mtf__b_J,type,
member_d_b: ( d > b ) > set_d_b > $o ).
thf(sy_c_member_001_062_Itf__d_Mtf__d_J,type,
member_d_d: ( d > d ) > set_d_d > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_Itf__b_Mtf__d_J_J,type,
member_set_b_d: set_b_d > set_set_b_d > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
member_set_set_b: set_set_b > set_set_set_b > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__d_J_J,type,
member_set_set_d: set_set_d > set_set_set_d > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__c_J,type,
member_set_c: set_c > set_set_c > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__d_J,type,
member_set_d: set_d > set_set_d > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_c_member_001tf__c,type,
member_c: c > set_c > $o ).
thf(sy_c_member_001tf__d,type,
member_d: d > set_d > $o ).
thf(sy_v_I,type,
i: set_c ).
thf(sy_v_M,type,
m: sigma_measure_a ).
thf(sy_v_M_H,type,
m2: c > sigma_measure_d ).
thf(sy_v_N,type,
n: sigma_measure_b ).
thf(sy_v_X_H,type,
x: c > b > d ).
thf(sy_v_f,type,
f: a > b ).
% Relevant facts (1271)
thf(fact_0_D_Oindep__sets__cong,axiom,
! [I: set_set_b,J: set_set_b,F: set_b > set_set_b,G: set_b > set_set_b] :
( ( I = J )
=> ( ! [I2: set_b] :
( ( member_set_b @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_1_D_Oindep__sets__cong,axiom,
! [I: set_set_d,J: set_set_d,F: set_d > set_set_b,G: set_d > set_set_b] :
( ( I = J )
=> ( ! [I2: set_d] :
( ( member_set_d @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6311571500530947982_set_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe6311571500530947982_set_d @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_2_D_Oindep__sets__cong,axiom,
! [I: set_set_a,J: set_set_a,F: set_a > set_set_b,G: set_a > set_set_b] :
( ( I = J )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_3_D_Oindep__sets__cong,axiom,
! [I: set_b_d,J: set_b_d,F: ( b > d ) > set_set_b,G: ( b > d ) > set_set_b] :
( ( I = J )
=> ( ! [I2: b > d] :
( ( member_b_d @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_4_D_Oindep__sets__cong,axiom,
! [I: set_a_b,J: set_a_b,F: ( a > b ) > set_set_b,G: ( a > b ) > set_set_b] :
( ( I = J )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_5_D_Oindep__sets__cong,axiom,
! [I: set_b,J: set_b,F: b > set_set_b,G: b > set_set_b] :
( ( I = J )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_6_D_Oindep__sets__cong,axiom,
! [I: set_a,J: set_a,F: a > set_set_b,G: a > set_set_b] :
( ( I = J )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_7_D_Oindep__sets__cong,axiom,
! [I: set_c,J: set_c,F: c > set_set_b,G: c > set_set_b] :
( ( I = J )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_8_D_Oindep__sets__cong,axiom,
! [I: set_nat,J: set_nat,F: nat > set_set_b,G: nat > set_set_b] :
( ( I = J )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_9_D_Oindep__sets__cong,axiom,
! [I: set_o,J: set_o,F: $o > set_set_b,G: $o > set_set_b] :
( ( I = J )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_cong
thf(fact_10_prob__space__axioms,axiom,
probab7247484486040049089pace_a @ m ).
% prob_space_axioms
thf(fact_11_assms_I1_J,axiom,
member_a_b @ f @ ( sigma_measurable_a_b @ m @ n ) ).
% assms(1)
thf(fact_12_D_Oindep__sets__Dynkin,axiom,
! [F: set_a > set_set_b,I: set_set_a] :
( ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: set_a] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F @ I3 ) )
@ I ) ) ).
% D.indep_sets_Dynkin
thf(fact_13_D_Oindep__sets__Dynkin,axiom,
! [F: ( b > d ) > set_set_b,I: set_b_d] :
( ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: b > d] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F @ I3 ) )
@ I ) ) ).
% D.indep_sets_Dynkin
thf(fact_14_D_Oindep__sets__Dynkin,axiom,
! [F: ( a > b ) > set_set_b,I: set_a_b] :
( ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: a > b] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F @ I3 ) )
@ I ) ) ).
% D.indep_sets_Dynkin
thf(fact_15_D_Oindep__sets__Dynkin,axiom,
! [F: b > set_set_b,I: set_b] :
( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: b] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F @ I3 ) )
@ I ) ) ).
% D.indep_sets_Dynkin
thf(fact_16_D_Oindep__sets__Dynkin,axiom,
! [F: a > set_set_b,I: set_a] :
( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: a] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F @ I3 ) )
@ I ) ) ).
% D.indep_sets_Dynkin
thf(fact_17_D_Oindep__sets__Dynkin,axiom,
! [F: c > set_set_b,I: set_c] :
( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F @ I3 ) )
@ I ) ) ).
% D.indep_sets_Dynkin
thf(fact_18_D_Oindep__sets__Dynkin,axiom,
! [F: nat > set_set_b,I: set_nat] :
( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: nat] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F @ I3 ) )
@ I ) ) ).
% D.indep_sets_Dynkin
thf(fact_19_D_Oindep__sets__Dynkin,axiom,
! [F: $o > set_set_b,I: set_o] :
( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: $o] : ( sigma_Dynkin_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( F @ I3 ) )
@ I ) ) ).
% D.indep_sets_Dynkin
thf(fact_20_assms_I2_J,axiom,
! [I4: c] :
( ( member_c @ I4 @ i )
=> ( member_b_d @ ( x @ I4 ) @ ( sigma_measurable_b_d @ n @ ( m2 @ I4 ) ) ) ) ).
% assms(2)
thf(fact_21_sets_OInt__space__eq1,axiom,
! [X: set_d,M: sigma_measure_d] :
( ( member_set_d @ X @ ( sigma_sets_d @ M ) )
=> ( ( inf_inf_set_d @ ( sigma_space_d @ M ) @ X )
= X ) ) ).
% sets.Int_space_eq1
thf(fact_22_sets_OInt__space__eq1,axiom,
! [X: set_b,M: sigma_measure_b] :
( ( member_set_b @ X @ ( sigma_sets_b @ M ) )
=> ( ( inf_inf_set_b @ ( sigma_space_b @ M ) @ X )
= X ) ) ).
% sets.Int_space_eq1
thf(fact_23_sets_OInt__space__eq1,axiom,
! [X: set_a,M: sigma_measure_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X )
= X ) ) ).
% sets.Int_space_eq1
thf(fact_24_sets_OInt__space__eq2,axiom,
! [X: set_d,M: sigma_measure_d] :
( ( member_set_d @ X @ ( sigma_sets_d @ M ) )
=> ( ( inf_inf_set_d @ X @ ( sigma_space_d @ M ) )
= X ) ) ).
% sets.Int_space_eq2
thf(fact_25_sets_OInt__space__eq2,axiom,
! [X: set_b,M: sigma_measure_b] :
( ( member_set_b @ X @ ( sigma_sets_b @ M ) )
=> ( ( inf_inf_set_b @ X @ ( sigma_space_b @ M ) )
= X ) ) ).
% sets.Int_space_eq2
thf(fact_26_sets_OInt__space__eq2,axiom,
! [X: set_a,M: sigma_measure_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ X @ ( sigma_space_a @ M ) )
= X ) ) ).
% sets.Int_space_eq2
thf(fact_27_D_Osubprob__not__empty,axiom,
( ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) )
!= bot_bot_set_b ) ).
% D.subprob_not_empty
thf(fact_28_D_Oindep__sets__mono__sets,axiom,
! [F: set_b > set_set_b,I: set_set_b,G: set_b > set_set_b] :
( ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: set_b] :
( ( member_set_b @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_29_D_Oindep__sets__mono__sets,axiom,
! [F: set_d > set_set_b,I: set_set_d,G: set_d > set_set_b] :
( ( indepe6311571500530947982_set_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: set_d] :
( ( member_set_d @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6311571500530947982_set_d @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_30_D_Oindep__sets__mono__sets,axiom,
! [F: set_a > set_set_b,I: set_set_a,G: set_a > set_set_b] :
( ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_31_D_Oindep__sets__mono__sets,axiom,
! [F: ( b > d ) > set_set_b,I: set_b_d,G: ( b > d ) > set_set_b] :
( ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: b > d] :
( ( member_b_d @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_32_D_Oindep__sets__mono__sets,axiom,
! [F: ( a > b ) > set_set_b,I: set_a_b,G: ( a > b ) > set_set_b] :
( ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_33_D_Oindep__sets__mono__sets,axiom,
! [F: b > set_set_b,I: set_b,G: b > set_set_b] :
( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_34_D_Oindep__sets__mono__sets,axiom,
! [F: a > set_set_b,I: set_a,G: a > set_set_b] :
( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_35_D_Oindep__sets__mono__sets,axiom,
! [F: c > set_set_b,I: set_c,G: c > set_set_b] :
( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_36_D_Oindep__sets__mono__sets,axiom,
! [F: nat > set_set_b,I: set_nat,G: nat > set_set_b] :
( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_37_D_Oindep__sets__mono__sets,axiom,
! [F: $o > set_set_b,I: set_o,G: $o > set_set_b] :
( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ G @ I ) ) ) ).
% D.indep_sets_mono_sets
thf(fact_38_D_Oindep__sets__mono__index,axiom,
! [J: set_b_d,I: set_b_d,F: ( b > d ) > set_set_b] :
( ( ord_less_eq_set_b_d @ J @ I )
=> ( ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J ) ) ) ).
% D.indep_sets_mono_index
thf(fact_39_D_Oindep__sets__mono__index,axiom,
! [J: set_a_b,I: set_a_b,F: ( a > b ) > set_set_b] :
( ( ord_less_eq_set_a_b @ J @ I )
=> ( ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J ) ) ) ).
% D.indep_sets_mono_index
thf(fact_40_D_Oindep__sets__mono__index,axiom,
! [J: set_set_b,I: set_set_b,F: set_b > set_set_b] :
( ( ord_le3795704787696855135_set_b @ J @ I )
=> ( ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J ) ) ) ).
% D.indep_sets_mono_index
thf(fact_41_D_Oindep__sets__mono__index,axiom,
! [J: set_set_a,I: set_set_a,F: set_a > set_set_b] :
( ( ord_le3724670747650509150_set_a @ J @ I )
=> ( ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J ) ) ) ).
% D.indep_sets_mono_index
thf(fact_42_D_Oindep__sets__mono__index,axiom,
! [J: set_b,I: set_b,F: b > set_set_b] :
( ( ord_less_eq_set_b @ J @ I )
=> ( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J ) ) ) ).
% D.indep_sets_mono_index
thf(fact_43_D_Oindep__sets__mono__index,axiom,
! [J: set_a,I: set_a,F: a > set_set_b] :
( ( ord_less_eq_set_a @ J @ I )
=> ( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J ) ) ) ).
% D.indep_sets_mono_index
thf(fact_44_D_Oindep__sets__mono__index,axiom,
! [J: set_c,I: set_c,F: c > set_set_b] :
( ( ord_less_eq_set_c @ J @ I )
=> ( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J ) ) ) ).
% D.indep_sets_mono_index
thf(fact_45_D_Oindep__sets__mono__index,axiom,
! [J: set_nat,I: set_nat,F: nat > set_set_b] :
( ( ord_less_eq_set_nat @ J @ I )
=> ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J ) ) ) ).
% D.indep_sets_mono_index
thf(fact_46_D_Oindep__sets__mono__index,axiom,
! [J: set_o,I: set_o,F: $o > set_set_b] :
( ( ord_less_eq_set_o @ J @ I )
=> ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J ) ) ) ).
% D.indep_sets_mono_index
thf(fact_47_vimage__Int,axiom,
! [F2: b > b,A: set_b,B: set_b] :
( ( vimage_b_b @ F2 @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_b @ ( vimage_b_b @ F2 @ A ) @ ( vimage_b_b @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_48_vimage__Int,axiom,
! [F2: a > b,A: set_b,B: set_b] :
( ( vimage_a_b @ F2 @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_a @ ( vimage_a_b @ F2 @ A ) @ ( vimage_a_b @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_49_vimage__Int,axiom,
! [F2: d > b,A: set_b,B: set_b] :
( ( vimage_d_b @ F2 @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_d @ ( vimage_d_b @ F2 @ A ) @ ( vimage_d_b @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_50_vimage__Int,axiom,
! [F2: b > a,A: set_a,B: set_a] :
( ( vimage_b_a @ F2 @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_b @ ( vimage_b_a @ F2 @ A ) @ ( vimage_b_a @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_51_vimage__Int,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( vimage_a_a @ F2 @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ ( vimage_a_a @ F2 @ A ) @ ( vimage_a_a @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_52_vimage__Int,axiom,
! [F2: d > a,A: set_a,B: set_a] :
( ( vimage_d_a @ F2 @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_d @ ( vimage_d_a @ F2 @ A ) @ ( vimage_d_a @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_53_vimage__Int,axiom,
! [F2: b > d,A: set_d,B: set_d] :
( ( vimage_b_d @ F2 @ ( inf_inf_set_d @ A @ B ) )
= ( inf_inf_set_b @ ( vimage_b_d @ F2 @ A ) @ ( vimage_b_d @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_54_vimage__Int,axiom,
! [F2: a > d,A: set_d,B: set_d] :
( ( vimage_a_d @ F2 @ ( inf_inf_set_d @ A @ B ) )
= ( inf_inf_set_a @ ( vimage_a_d @ F2 @ A ) @ ( vimage_a_d @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_55_vimage__Int,axiom,
! [F2: d > d,A: set_d,B: set_d] :
( ( vimage_d_d @ F2 @ ( inf_inf_set_d @ A @ B ) )
= ( inf_inf_set_d @ ( vimage_d_d @ F2 @ A ) @ ( vimage_d_d @ F2 @ B ) ) ) ).
% vimage_Int
thf(fact_56_space__distr,axiom,
! [M: sigma_measure_a,N: sigma_measure_b,F2: a > b] :
( ( sigma_space_b @ ( measure_distr_a_b @ M @ N @ F2 ) )
= ( sigma_space_b @ N ) ) ).
% space_distr
thf(fact_57_sets__distr,axiom,
! [M: sigma_measure_a,N: sigma_measure_b,F2: a > b] :
( ( sigma_sets_b @ ( measure_distr_a_b @ M @ N @ F2 ) )
= ( sigma_sets_b @ N ) ) ).
% sets_distr
thf(fact_58_sets_Otop,axiom,
! [M: sigma_measure_b] : ( member_set_b @ ( sigma_space_b @ M ) @ ( sigma_sets_b @ M ) ) ).
% sets.top
thf(fact_59_sets_Otop,axiom,
! [M: sigma_measure_a] : ( member_set_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.top
thf(fact_60_sets_Otop,axiom,
! [M: sigma_measure_d] : ( member_set_d @ ( sigma_space_d @ M ) @ ( sigma_sets_d @ M ) ) ).
% sets.top
thf(fact_61_sets_OInt,axiom,
! [A2: set_b,M: sigma_measure_b,B2: set_b] :
( ( member_set_b @ A2 @ ( sigma_sets_b @ M ) )
=> ( ( member_set_b @ B2 @ ( sigma_sets_b @ M ) )
=> ( member_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( sigma_sets_b @ M ) ) ) ) ).
% sets.Int
thf(fact_62_sets_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Int
thf(fact_63_sets_OInt,axiom,
! [A2: set_d,M: sigma_measure_d,B2: set_d] :
( ( member_set_d @ A2 @ ( sigma_sets_d @ M ) )
=> ( ( member_set_d @ B2 @ ( sigma_sets_d @ M ) )
=> ( member_set_d @ ( inf_inf_set_d @ A2 @ B2 ) @ ( sigma_sets_d @ M ) ) ) ) ).
% sets.Int
thf(fact_64_D_Osubprob__space__axioms,axiom,
giry_subprob_space_b @ ( measure_distr_a_b @ m @ n @ f ) ).
% D.subprob_space_axioms
thf(fact_65_subsetI,axiom,
! [A: set_b_d,B: set_b_d] :
( ! [X2: b > d] :
( ( member_b_d @ X2 @ A )
=> ( member_b_d @ X2 @ B ) )
=> ( ord_less_eq_set_b_d @ A @ B ) ) ).
% subsetI
thf(fact_66_subsetI,axiom,
! [A: set_a_b,B: set_a_b] :
( ! [X2: a > b] :
( ( member_a_b @ X2 @ A )
=> ( member_a_b @ X2 @ B ) )
=> ( ord_less_eq_set_a_b @ A @ B ) ) ).
% subsetI
thf(fact_67_subsetI,axiom,
! [A: set_set_d,B: set_set_d] :
( ! [X2: set_d] :
( ( member_set_d @ X2 @ A )
=> ( member_set_d @ X2 @ B ) )
=> ( ord_le3937772867789547105_set_d @ A @ B ) ) ).
% subsetI
thf(fact_68_subsetI,axiom,
! [A: set_set_b,B: set_set_b] :
( ! [X2: set_b] :
( ( member_set_b @ X2 @ A )
=> ( member_set_b @ X2 @ B ) )
=> ( ord_le3795704787696855135_set_b @ A @ B ) ) ).
% subsetI
thf(fact_69_subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% subsetI
thf(fact_70_subsetI,axiom,
! [A: set_b,B: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( member_b @ X2 @ B ) )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% subsetI
thf(fact_71_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_72_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member_o @ X2 @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_73_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_74_subsetI,axiom,
! [A: set_c,B: set_c] :
( ! [X2: c] :
( ( member_c @ X2 @ A )
=> ( member_c @ X2 @ B ) )
=> ( ord_less_eq_set_c @ A @ B ) ) ).
% subsetI
thf(fact_75_subset__antisym,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_76_subset__antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_77_subset__antisym,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_78_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_79_subset__antisym,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_80_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_81_subset__antisym,axiom,
! [A: set_c,B: set_c] :
( ( ord_less_eq_set_c @ A @ B )
=> ( ( ord_less_eq_set_c @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_82_empty__iff,axiom,
! [C: b > d] :
~ ( member_b_d @ C @ bot_bot_set_b_d ) ).
% empty_iff
thf(fact_83_empty__iff,axiom,
! [C: a > b] :
~ ( member_a_b @ C @ bot_bot_set_a_b ) ).
% empty_iff
thf(fact_84_empty__iff,axiom,
! [C: c] :
~ ( member_c @ C @ bot_bot_set_c ) ).
% empty_iff
thf(fact_85_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_86_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_87_empty__iff,axiom,
! [C: set_b] :
~ ( member_set_b @ C @ bot_bot_set_set_b ) ).
% empty_iff
thf(fact_88_empty__iff,axiom,
! [C: set_d] :
~ ( member_set_d @ C @ bot_bot_set_set_d ) ).
% empty_iff
thf(fact_89_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_90_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_91_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_92_all__not__in__conv,axiom,
! [A: set_b_d] :
( ( ! [X3: b > d] :
~ ( member_b_d @ X3 @ A ) )
= ( A = bot_bot_set_b_d ) ) ).
% all_not_in_conv
thf(fact_93_all__not__in__conv,axiom,
! [A: set_a_b] :
( ( ! [X3: a > b] :
~ ( member_a_b @ X3 @ A ) )
= ( A = bot_bot_set_a_b ) ) ).
% all_not_in_conv
thf(fact_94_all__not__in__conv,axiom,
! [A: set_c] :
( ( ! [X3: c] :
~ ( member_c @ X3 @ A ) )
= ( A = bot_bot_set_c ) ) ).
% all_not_in_conv
thf(fact_95_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_96_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_97_all__not__in__conv,axiom,
! [A: set_set_b] :
( ( ! [X3: set_b] :
~ ( member_set_b @ X3 @ A ) )
= ( A = bot_bot_set_set_b ) ) ).
% all_not_in_conv
thf(fact_98_all__not__in__conv,axiom,
! [A: set_set_d] :
( ( ! [X3: set_d] :
~ ( member_set_d @ X3 @ A ) )
= ( A = bot_bot_set_set_d ) ) ).
% all_not_in_conv
thf(fact_99_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X3: b] :
~ ( member_b @ X3 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_100_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_101_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X3: $o] :
~ ( member_o @ X3 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_102_Collect__empty__eq,axiom,
! [P: set_b > $o] :
( ( ( collect_set_b @ P )
= bot_bot_set_set_b )
= ( ! [X3: set_b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_103_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_104_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_105_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_106_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_107_empty__Collect__eq,axiom,
! [P: set_b > $o] :
( ( bot_bot_set_set_b
= ( collect_set_b @ P ) )
= ( ! [X3: set_b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_108_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_109_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_110_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_111_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_112_Int__iff,axiom,
! [C: c,A: set_c,B: set_c] :
( ( member_c @ C @ ( inf_inf_set_c @ A @ B ) )
= ( ( member_c @ C @ A )
& ( member_c @ C @ B ) ) ) ).
% Int_iff
thf(fact_113_Int__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
& ( member_o @ C @ B ) ) ) ).
% Int_iff
thf(fact_114_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_115_Int__iff,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
= ( ( member_b @ C @ A )
& ( member_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_116_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_117_Int__iff,axiom,
! [C: d,A: set_d,B: set_d] :
( ( member_d @ C @ ( inf_inf_set_d @ A @ B ) )
= ( ( member_d @ C @ A )
& ( member_d @ C @ B ) ) ) ).
% Int_iff
thf(fact_118_Int__iff,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_119_Int__iff,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
= ( ( member_set_b @ C @ A )
& ( member_set_b @ C @ B ) ) ) ).
% Int_iff
thf(fact_120_Int__iff,axiom,
! [C: set_d,A: set_set_d,B: set_set_d] :
( ( member_set_d @ C @ ( inf_inf_set_set_d @ A @ B ) )
= ( ( member_set_d @ C @ A )
& ( member_set_d @ C @ B ) ) ) ).
% Int_iff
thf(fact_121_Int__iff,axiom,
! [C: b > d,A: set_b_d,B: set_b_d] :
( ( member_b_d @ C @ ( inf_inf_set_b_d @ A @ B ) )
= ( ( member_b_d @ C @ A )
& ( member_b_d @ C @ B ) ) ) ).
% Int_iff
thf(fact_122_IntI,axiom,
! [C: c,A: set_c,B: set_c] :
( ( member_c @ C @ A )
=> ( ( member_c @ C @ B )
=> ( member_c @ C @ ( inf_inf_set_c @ A @ B ) ) ) ) ).
% IntI
thf(fact_123_IntI,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( ( member_o @ C @ B )
=> ( member_o @ C @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% IntI
thf(fact_124_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_125_IntI,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ A )
=> ( ( member_b @ C @ B )
=> ( member_b @ C @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_126_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_127_IntI,axiom,
! [C: d,A: set_d,B: set_d] :
( ( member_d @ C @ A )
=> ( ( member_d @ C @ B )
=> ( member_d @ C @ ( inf_inf_set_d @ A @ B ) ) ) ) ).
% IntI
thf(fact_128_IntI,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_129_IntI,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ A )
=> ( ( member_set_b @ C @ B )
=> ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) ) ) ) ).
% IntI
thf(fact_130_IntI,axiom,
! [C: set_d,A: set_set_d,B: set_set_d] :
( ( member_set_d @ C @ A )
=> ( ( member_set_d @ C @ B )
=> ( member_set_d @ C @ ( inf_inf_set_set_d @ A @ B ) ) ) ) ).
% IntI
thf(fact_131_IntI,axiom,
! [C: b > d,A: set_b_d,B: set_b_d] :
( ( member_b_d @ C @ A )
=> ( ( member_b_d @ C @ B )
=> ( member_b_d @ C @ ( inf_inf_set_b_d @ A @ B ) ) ) ) ).
% IntI
thf(fact_132_prob__space__distr,axiom,
! [F2: a > a,M2: sigma_measure_a] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ m @ M2 ) )
=> ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ m @ M2 @ F2 ) ) ) ).
% prob_space_distr
thf(fact_133_prob__space__distr,axiom,
! [F2: a > b,M2: sigma_measure_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ m @ M2 ) )
=> ( probab7247484486040049090pace_b @ ( measure_distr_a_b @ m @ M2 @ F2 ) ) ) ).
% prob_space_distr
thf(fact_134_vimage__eq,axiom,
! [A2: c,F2: c > c,B: set_c] :
( ( member_c @ A2 @ ( vimage_c_c @ F2 @ B ) )
= ( member_c @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_135_vimage__eq,axiom,
! [A2: c,F2: c > b,B: set_b] :
( ( member_c @ A2 @ ( vimage_c_b @ F2 @ B ) )
= ( member_b @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_136_vimage__eq,axiom,
! [A2: c,F2: c > a,B: set_a] :
( ( member_c @ A2 @ ( vimage_c_a @ F2 @ B ) )
= ( member_a @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_137_vimage__eq,axiom,
! [A2: c,F2: c > $o,B: set_o] :
( ( member_c @ A2 @ ( vimage_c_o @ F2 @ B ) )
= ( member_o @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_138_vimage__eq,axiom,
! [A2: c,F2: c > nat,B: set_nat] :
( ( member_c @ A2 @ ( vimage_c_nat @ F2 @ B ) )
= ( member_nat @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_139_vimage__eq,axiom,
! [A2: b,F2: b > c,B: set_c] :
( ( member_b @ A2 @ ( vimage_b_c @ F2 @ B ) )
= ( member_c @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_140_vimage__eq,axiom,
! [A2: b,F2: b > b,B: set_b] :
( ( member_b @ A2 @ ( vimage_b_b @ F2 @ B ) )
= ( member_b @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_141_vimage__eq,axiom,
! [A2: b,F2: b > a,B: set_a] :
( ( member_b @ A2 @ ( vimage_b_a @ F2 @ B ) )
= ( member_a @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_142_vimage__eq,axiom,
! [A2: b,F2: b > $o,B: set_o] :
( ( member_b @ A2 @ ( vimage_b_o @ F2 @ B ) )
= ( member_o @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_143_vimage__eq,axiom,
! [A2: b,F2: b > nat,B: set_nat] :
( ( member_b @ A2 @ ( vimage_b_nat @ F2 @ B ) )
= ( member_nat @ ( F2 @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_144_vimageI,axiom,
! [F2: c > c,A2: c,B2: c,B: set_c] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_c @ B2 @ B )
=> ( member_c @ A2 @ ( vimage_c_c @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_145_vimageI,axiom,
! [F2: b > c,A2: b,B2: c,B: set_c] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_c @ B2 @ B )
=> ( member_b @ A2 @ ( vimage_b_c @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_146_vimageI,axiom,
! [F2: a > c,A2: a,B2: c,B: set_c] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_c @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_c @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_147_vimageI,axiom,
! [F2: $o > c,A2: $o,B2: c,B: set_c] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_c @ B2 @ B )
=> ( member_o @ A2 @ ( vimage_o_c @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_148_vimageI,axiom,
! [F2: nat > c,A2: nat,B2: c,B: set_c] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_c @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_c @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_149_vimageI,axiom,
! [F2: c > b,A2: c,B2: b,B: set_b] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_b @ B2 @ B )
=> ( member_c @ A2 @ ( vimage_c_b @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_150_vimageI,axiom,
! [F2: b > b,A2: b,B2: b,B: set_b] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_b @ B2 @ B )
=> ( member_b @ A2 @ ( vimage_b_b @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_151_vimageI,axiom,
! [F2: a > b,A2: a,B2: b,B: set_b] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_b @ B2 @ B )
=> ( member_a @ A2 @ ( vimage_a_b @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_152_vimageI,axiom,
! [F2: $o > b,A2: $o,B2: b,B: set_b] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_b @ B2 @ B )
=> ( member_o @ A2 @ ( vimage_o_b @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_153_vimageI,axiom,
! [F2: nat > b,A2: nat,B2: b,B: set_b] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_b @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_b @ F2 @ B ) ) ) ) ).
% vimageI
thf(fact_154_vimage__Collect__eq,axiom,
! [F2: b > d,P: d > $o] :
( ( vimage_b_d @ F2 @ ( collect_d @ P ) )
= ( collect_b
@ ^ [Y: b] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_155_vimage__Collect__eq,axiom,
! [F2: a > a,P: a > $o] :
( ( vimage_a_a @ F2 @ ( collect_a @ P ) )
= ( collect_a
@ ^ [Y: a] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_156_vimage__Collect__eq,axiom,
! [F2: b > a,P: a > $o] :
( ( vimage_b_a @ F2 @ ( collect_a @ P ) )
= ( collect_b
@ ^ [Y: b] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_157_vimage__Collect__eq,axiom,
! [F2: nat > a,P: a > $o] :
( ( vimage_nat_a @ F2 @ ( collect_a @ P ) )
= ( collect_nat
@ ^ [Y: nat] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_158_vimage__Collect__eq,axiom,
! [F2: a > b,P: b > $o] :
( ( vimage_a_b @ F2 @ ( collect_b @ P ) )
= ( collect_a
@ ^ [Y: a] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_159_vimage__Collect__eq,axiom,
! [F2: b > b,P: b > $o] :
( ( vimage_b_b @ F2 @ ( collect_b @ P ) )
= ( collect_b
@ ^ [Y: b] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_160_vimage__Collect__eq,axiom,
! [F2: nat > b,P: b > $o] :
( ( vimage_nat_b @ F2 @ ( collect_b @ P ) )
= ( collect_nat
@ ^ [Y: nat] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_161_vimage__Collect__eq,axiom,
! [F2: a > nat,P: nat > $o] :
( ( vimage_a_nat @ F2 @ ( collect_nat @ P ) )
= ( collect_a
@ ^ [Y: a] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_162_vimage__Collect__eq,axiom,
! [F2: b > nat,P: nat > $o] :
( ( vimage_b_nat @ F2 @ ( collect_nat @ P ) )
= ( collect_b
@ ^ [Y: b] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_163_vimage__Collect__eq,axiom,
! [F2: nat > nat,P: nat > $o] :
( ( vimage_nat_nat @ F2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [Y: nat] : ( P @ ( F2 @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_164_Dynkin__Basic,axiom,
! [A: set_a,M: set_set_a,Omega: set_a] :
( ( member_set_a @ A @ M )
=> ( member_set_a @ A @ ( sigma_Dynkin_a @ Omega @ M ) ) ) ).
% Dynkin_Basic
thf(fact_165_Dynkin__Basic,axiom,
! [A: set_d,M: set_set_d,Omega: set_d] :
( ( member_set_d @ A @ M )
=> ( member_set_d @ A @ ( sigma_Dynkin_d @ Omega @ M ) ) ) ).
% Dynkin_Basic
thf(fact_166_Dynkin__Basic,axiom,
! [A: set_b,M: set_set_b,Omega: set_b] :
( ( member_set_b @ A @ M )
=> ( member_set_b @ A @ ( sigma_Dynkin_b @ Omega @ M ) ) ) ).
% Dynkin_Basic
thf(fact_167_subprob__space__distr,axiom,
! [F2: a > d,M2: sigma_measure_d] :
( ( member_a_d @ F2 @ ( sigma_measurable_a_d @ m @ M2 ) )
=> ( ( ( sigma_space_d @ M2 )
!= bot_bot_set_d )
=> ( giry_subprob_space_d @ ( measure_distr_a_d @ m @ M2 @ F2 ) ) ) ) ).
% subprob_space_distr
thf(fact_168_subprob__space__distr,axiom,
! [F2: a > $o,M2: sigma_measure_o] :
( ( member_a_o @ F2 @ ( sigma_measurable_a_o @ m @ M2 ) )
=> ( ( ( sigma_space_o @ M2 )
!= bot_bot_set_o )
=> ( giry_subprob_space_o @ ( measure_distr_a_o @ m @ M2 @ F2 ) ) ) ) ).
% subprob_space_distr
thf(fact_169_subprob__space__distr,axiom,
! [F2: a > a,M2: sigma_measure_a] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ m @ M2 ) )
=> ( ( ( sigma_space_a @ M2 )
!= bot_bot_set_a )
=> ( giry_subprob_space_a @ ( measure_distr_a_a @ m @ M2 @ F2 ) ) ) ) ).
% subprob_space_distr
thf(fact_170_subprob__space__distr,axiom,
! [F2: a > b,M2: sigma_measure_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ m @ M2 ) )
=> ( ( ( sigma_space_b @ M2 )
!= bot_bot_set_b )
=> ( giry_subprob_space_b @ ( measure_distr_a_b @ m @ M2 @ F2 ) ) ) ) ).
% subprob_space_distr
thf(fact_171_D_Oprob__space__distr,axiom,
! [F2: b > d,M2: sigma_measure_d] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M2 ) )
=> ( probab7247484486040049092pace_d @ ( measure_distr_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ F2 ) ) ) ).
% D.prob_space_distr
thf(fact_172_D_Oprob__space__distr,axiom,
! [F2: b > a,M2: sigma_measure_a] :
( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 ) )
=> ( probab7247484486040049089pace_a @ ( measure_distr_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ F2 ) ) ) ).
% D.prob_space_distr
thf(fact_173_D_Oprob__space__distr,axiom,
! [F2: b > b,M2: sigma_measure_b] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 ) )
=> ( probab7247484486040049090pace_b @ ( measure_distr_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ F2 ) ) ) ).
% D.prob_space_distr
thf(fact_174_subset__empty,axiom,
! [A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ bot_bot_set_set_b )
= ( A = bot_bot_set_set_b ) ) ).
% subset_empty
thf(fact_175_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_176_subset__empty,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_177_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_178_subset__empty,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_179_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_180_subset__empty,axiom,
! [A: set_c] :
( ( ord_less_eq_set_c @ A @ bot_bot_set_c )
= ( A = bot_bot_set_c ) ) ).
% subset_empty
thf(fact_181_empty__subsetI,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ bot_bot_set_set_b @ A ) ).
% empty_subsetI
thf(fact_182_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_183_empty__subsetI,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% empty_subsetI
thf(fact_184_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_185_empty__subsetI,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% empty_subsetI
thf(fact_186_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_187_empty__subsetI,axiom,
! [A: set_c] : ( ord_less_eq_set_c @ bot_bot_set_c @ A ) ).
% empty_subsetI
thf(fact_188_sets_Oempty__sets,axiom,
! [M: sigma_measure_d] : ( member_set_d @ bot_bot_set_d @ ( sigma_sets_d @ M ) ) ).
% sets.empty_sets
thf(fact_189_sets_Oempty__sets,axiom,
! [M: sigma_measure_b] : ( member_set_b @ bot_bot_set_b @ ( sigma_sets_b @ M ) ) ).
% sets.empty_sets
thf(fact_190_sets_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sets_a @ M ) ) ).
% sets.empty_sets
thf(fact_191_sets_Oempty__sets,axiom,
! [M: sigma_measure_o] : ( member_set_o @ bot_bot_set_o @ ( sigma_sets_o @ M ) ) ).
% sets.empty_sets
thf(fact_192_Int__subset__iff,axiom,
! [C2: set_d,A: set_d,B: set_d] :
( ( ord_less_eq_set_d @ C2 @ ( inf_inf_set_d @ A @ B ) )
= ( ( ord_less_eq_set_d @ C2 @ A )
& ( ord_less_eq_set_d @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_193_Int__subset__iff,axiom,
! [C2: set_set_b,A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ C2 @ ( inf_inf_set_set_b @ A @ B ) )
= ( ( ord_le3795704787696855135_set_b @ C2 @ A )
& ( ord_le3795704787696855135_set_b @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_194_Int__subset__iff,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) )
= ( ( ord_le3724670747650509150_set_a @ C2 @ A )
& ( ord_le3724670747650509150_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_195_Int__subset__iff,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B ) )
= ( ( ord_less_eq_set_b @ C2 @ A )
& ( ord_less_eq_set_b @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_196_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_197_Int__subset__iff,axiom,
! [C2: set_o,A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
= ( ( ord_less_eq_set_o @ C2 @ A )
& ( ord_less_eq_set_o @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_198_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_199_Int__subset__iff,axiom,
! [C2: set_c,A: set_c,B: set_c] :
( ( ord_less_eq_set_c @ C2 @ ( inf_inf_set_c @ A @ B ) )
= ( ( ord_less_eq_set_c @ C2 @ A )
& ( ord_less_eq_set_c @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_200_measurable__distr__eq2,axiom,
! [Mg: sigma_measure_a,Mg2: sigma_measure_a,Ng: sigma_measure_b,G2: a > b] :
( ( sigma_measurable_a_b @ Mg @ ( measure_distr_a_b @ Mg2 @ Ng @ G2 ) )
= ( sigma_measurable_a_b @ Mg @ Ng ) ) ).
% measurable_distr_eq2
thf(fact_201_measurable__distr__eq1,axiom,
! [Mf: sigma_measure_a,Nf: sigma_measure_b,F2: a > b,Mf2: sigma_measure_d] :
( ( sigma_measurable_b_d @ ( measure_distr_a_b @ Mf @ Nf @ F2 ) @ Mf2 )
= ( sigma_measurable_b_d @ Nf @ Mf2 ) ) ).
% measurable_distr_eq1
thf(fact_202_vimage__empty,axiom,
! [F2: b > d] :
( ( vimage_b_d @ F2 @ bot_bot_set_d )
= bot_bot_set_b ) ).
% vimage_empty
thf(fact_203_vimage__empty,axiom,
! [F2: b > b] :
( ( vimage_b_b @ F2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% vimage_empty
thf(fact_204_vimage__empty,axiom,
! [F2: a > b] :
( ( vimage_a_b @ F2 @ bot_bot_set_b )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_205_vimage__empty,axiom,
! [F2: $o > b] :
( ( vimage_o_b @ F2 @ bot_bot_set_b )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_206_vimage__empty,axiom,
! [F2: b > a] :
( ( vimage_b_a @ F2 @ bot_bot_set_a )
= bot_bot_set_b ) ).
% vimage_empty
thf(fact_207_vimage__empty,axiom,
! [F2: a > a] :
( ( vimage_a_a @ F2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_208_vimage__empty,axiom,
! [F2: $o > a] :
( ( vimage_o_a @ F2 @ bot_bot_set_a )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_209_vimage__empty,axiom,
! [F2: b > $o] :
( ( vimage_b_o @ F2 @ bot_bot_set_o )
= bot_bot_set_b ) ).
% vimage_empty
thf(fact_210_vimage__empty,axiom,
! [F2: a > $o] :
( ( vimage_a_o @ F2 @ bot_bot_set_o )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_211_vimage__empty,axiom,
! [F2: $o > $o] :
( ( vimage_o_o @ F2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_212_D_Oindep__sets__mono,axiom,
! [F: set_d > set_set_b,I: set_set_d,J: set_set_d,G: set_d > set_set_b] :
( ( indepe6311571500530947982_set_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_le3937772867789547105_set_d @ J @ I )
=> ( ! [I2: set_d] :
( ( member_set_d @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6311571500530947982_set_d @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_213_D_Oindep__sets__mono,axiom,
! [F: set_b > set_set_b,I: set_set_b,J: set_set_b,G: set_b > set_set_b] :
( ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_le3795704787696855135_set_b @ J @ I )
=> ( ! [I2: set_b] :
( ( member_set_b @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_214_D_Oindep__sets__mono,axiom,
! [F: c > set_set_b,I: set_c,J: set_c,G: c > set_set_b] :
( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_less_eq_set_c @ J @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_215_D_Oindep__sets__mono,axiom,
! [F: nat > set_set_b,I: set_nat,J: set_nat,G: nat > set_set_b] :
( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_less_eq_set_nat @ J @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_216_D_Oindep__sets__mono,axiom,
! [F: $o > set_set_b,I: set_o,J: set_o,G: $o > set_set_b] :
( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_less_eq_set_o @ J @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_217_D_Oindep__sets__mono,axiom,
! [F: set_a > set_set_b,I: set_set_a,J: set_set_a,G: set_a > set_set_b] :
( ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_le3724670747650509150_set_a @ J @ I )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_218_D_Oindep__sets__mono,axiom,
! [F: ( b > d ) > set_set_b,I: set_b_d,J: set_b_d,G: ( b > d ) > set_set_b] :
( ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_less_eq_set_b_d @ J @ I )
=> ( ! [I2: b > d] :
( ( member_b_d @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_219_D_Oindep__sets__mono,axiom,
! [F: ( a > b ) > set_set_b,I: set_a_b,J: set_a_b,G: ( a > b ) > set_set_b] :
( ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_less_eq_set_a_b @ J @ I )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_220_D_Oindep__sets__mono,axiom,
! [F: b > set_set_b,I: set_b,J: set_b,G: b > set_set_b] :
( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_less_eq_set_b @ J @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_221_D_Oindep__sets__mono,axiom,
! [F: a > set_set_b,I: set_a,J: set_a,G: a > set_set_b] :
( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
=> ( ( ord_less_eq_set_a @ J @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ J )
=> ( ord_le3795704787696855135_set_b @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ G @ J ) ) ) ) ).
% D.indep_sets_mono
thf(fact_222_D_Osubprob__space__distr,axiom,
! [F2: b > $o,M2: sigma_measure_o] :
( ( member_b_o @ F2 @ ( sigma_measurable_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ M2 ) )
=> ( ( ( sigma_space_o @ M2 )
!= bot_bot_set_o )
=> ( giry_subprob_space_o @ ( measure_distr_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ F2 ) ) ) ) ).
% D.subprob_space_distr
thf(fact_223_D_Osubprob__space__distr,axiom,
! [F2: b > b,M2: sigma_measure_b] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 ) )
=> ( ( ( sigma_space_b @ M2 )
!= bot_bot_set_b )
=> ( giry_subprob_space_b @ ( measure_distr_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ F2 ) ) ) ) ).
% D.subprob_space_distr
thf(fact_224_D_Osubprob__space__distr,axiom,
! [F2: b > a,M2: sigma_measure_a] :
( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 ) )
=> ( ( ( sigma_space_a @ M2 )
!= bot_bot_set_a )
=> ( giry_subprob_space_a @ ( measure_distr_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ F2 ) ) ) ) ).
% D.subprob_space_distr
thf(fact_225_D_Osubprob__space__distr,axiom,
! [F2: b > d,M2: sigma_measure_d] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M2 ) )
=> ( ( ( sigma_space_d @ M2 )
!= bot_bot_set_d )
=> ( giry_subprob_space_d @ ( measure_distr_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ F2 ) ) ) ) ).
% D.subprob_space_distr
thf(fact_226_tail__events__sets,axiom,
! [A: nat > set_set_a] :
( ! [I2: nat] : ( ord_le3724670747650509150_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ m ) )
=> ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ m @ A ) @ ( sigma_sets_a @ m ) ) ) ).
% tail_events_sets
thf(fact_227_D_Otail__events__sets,axiom,
! [A: nat > set_set_b] :
( ! [I2: nat] : ( ord_le3795704787696855135_set_b @ ( A @ I2 ) @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ord_le3795704787696855135_set_b @ ( indepe8773861029005768663_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) ) ).
% D.tail_events_sets
thf(fact_228_emptyE,axiom,
! [A2: b > d] :
~ ( member_b_d @ A2 @ bot_bot_set_b_d ) ).
% emptyE
thf(fact_229_emptyE,axiom,
! [A2: a > b] :
~ ( member_a_b @ A2 @ bot_bot_set_a_b ) ).
% emptyE
thf(fact_230_emptyE,axiom,
! [A2: c] :
~ ( member_c @ A2 @ bot_bot_set_c ) ).
% emptyE
thf(fact_231_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_232_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_233_emptyE,axiom,
! [A2: set_b] :
~ ( member_set_b @ A2 @ bot_bot_set_set_b ) ).
% emptyE
thf(fact_234_emptyE,axiom,
! [A2: set_d] :
~ ( member_set_d @ A2 @ bot_bot_set_set_d ) ).
% emptyE
thf(fact_235_emptyE,axiom,
! [A2: b] :
~ ( member_b @ A2 @ bot_bot_set_b ) ).
% emptyE
thf(fact_236_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_237_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_238_mem__Collect__eq,axiom,
! [A2: b > d,P: ( b > d ) > $o] :
( ( member_b_d @ A2 @ ( collect_b_d @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_239_mem__Collect__eq,axiom,
! [A2: a > b,P: ( a > b ) > $o] :
( ( member_a_b @ A2 @ ( collect_a_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_240_mem__Collect__eq,axiom,
! [A2: c,P: c > $o] :
( ( member_c @ A2 @ ( collect_c @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_241_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_242_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_243_mem__Collect__eq,axiom,
! [A2: set_d,P: set_d > $o] :
( ( member_set_d @ A2 @ ( collect_set_d @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_244_mem__Collect__eq,axiom,
! [A2: set_b,P: set_b > $o] :
( ( member_set_b @ A2 @ ( collect_set_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_245_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_246_mem__Collect__eq,axiom,
! [A2: b,P: b > $o] :
( ( member_b @ A2 @ ( collect_b @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_247_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_248_Collect__mem__eq,axiom,
! [A: set_b_d] :
( ( collect_b_d
@ ^ [X3: b > d] : ( member_b_d @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_249_Collect__mem__eq,axiom,
! [A: set_a_b] :
( ( collect_a_b
@ ^ [X3: a > b] : ( member_a_b @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_250_Collect__mem__eq,axiom,
! [A: set_c] :
( ( collect_c
@ ^ [X3: c] : ( member_c @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_251_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X3: $o] : ( member_o @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_252_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_253_Collect__mem__eq,axiom,
! [A: set_set_d] :
( ( collect_set_d
@ ^ [X3: set_d] : ( member_set_d @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_254_Collect__mem__eq,axiom,
! [A: set_set_b] :
( ( collect_set_b
@ ^ [X3: set_b] : ( member_set_b @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_255_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_256_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_257_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_258_Collect__cong,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ! [X2: set_b] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_b @ P )
= ( collect_set_b @ Q ) ) ) ).
% Collect_cong
thf(fact_259_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_260_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X2: b] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_261_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_262_in__mono,axiom,
! [A: set_b_d,B: set_b_d,X: b > d] :
( ( ord_less_eq_set_b_d @ A @ B )
=> ( ( member_b_d @ X @ A )
=> ( member_b_d @ X @ B ) ) ) ).
% in_mono
thf(fact_263_in__mono,axiom,
! [A: set_a_b,B: set_a_b,X: a > b] :
( ( ord_less_eq_set_a_b @ A @ B )
=> ( ( member_a_b @ X @ A )
=> ( member_a_b @ X @ B ) ) ) ).
% in_mono
thf(fact_264_in__mono,axiom,
! [A: set_set_d,B: set_set_d,X: set_d] :
( ( ord_le3937772867789547105_set_d @ A @ B )
=> ( ( member_set_d @ X @ A )
=> ( member_set_d @ X @ B ) ) ) ).
% in_mono
thf(fact_265_in__mono,axiom,
! [A: set_set_b,B: set_set_b,X: set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( member_set_b @ X @ A )
=> ( member_set_b @ X @ B ) ) ) ).
% in_mono
thf(fact_266_in__mono,axiom,
! [A: set_set_a,B: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B ) ) ) ).
% in_mono
thf(fact_267_in__mono,axiom,
! [A: set_b,B: set_b,X: b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_b @ X @ A )
=> ( member_b @ X @ B ) ) ) ).
% in_mono
thf(fact_268_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_269_in__mono,axiom,
! [A: set_o,B: set_o,X: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X @ A )
=> ( member_o @ X @ B ) ) ) ).
% in_mono
thf(fact_270_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_271_in__mono,axiom,
! [A: set_c,B: set_c,X: c] :
( ( ord_less_eq_set_c @ A @ B )
=> ( ( member_c @ X @ A )
=> ( member_c @ X @ B ) ) ) ).
% in_mono
thf(fact_272_subsetD,axiom,
! [A: set_b_d,B: set_b_d,C: b > d] :
( ( ord_less_eq_set_b_d @ A @ B )
=> ( ( member_b_d @ C @ A )
=> ( member_b_d @ C @ B ) ) ) ).
% subsetD
thf(fact_273_subsetD,axiom,
! [A: set_a_b,B: set_a_b,C: a > b] :
( ( ord_less_eq_set_a_b @ A @ B )
=> ( ( member_a_b @ C @ A )
=> ( member_a_b @ C @ B ) ) ) ).
% subsetD
thf(fact_274_subsetD,axiom,
! [A: set_set_d,B: set_set_d,C: set_d] :
( ( ord_le3937772867789547105_set_d @ A @ B )
=> ( ( member_set_d @ C @ A )
=> ( member_set_d @ C @ B ) ) ) ).
% subsetD
thf(fact_275_subsetD,axiom,
! [A: set_set_b,B: set_set_b,C: set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( member_set_b @ C @ A )
=> ( member_set_b @ C @ B ) ) ) ).
% subsetD
thf(fact_276_subsetD,axiom,
! [A: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_277_subsetD,axiom,
! [A: set_b,B: set_b,C: b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_b @ C @ A )
=> ( member_b @ C @ B ) ) ) ).
% subsetD
thf(fact_278_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_279_subsetD,axiom,
! [A: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_280_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_281_subsetD,axiom,
! [A: set_c,B: set_c,C: c] :
( ( ord_less_eq_set_c @ A @ B )
=> ( ( member_c @ C @ A )
=> ( member_c @ C @ B ) ) ) ).
% subsetD
thf(fact_282_equals0D,axiom,
! [A: set_b_d,A2: b > d] :
( ( A = bot_bot_set_b_d )
=> ~ ( member_b_d @ A2 @ A ) ) ).
% equals0D
thf(fact_283_equals0D,axiom,
! [A: set_a_b,A2: a > b] :
( ( A = bot_bot_set_a_b )
=> ~ ( member_a_b @ A2 @ A ) ) ).
% equals0D
thf(fact_284_equals0D,axiom,
! [A: set_c,A2: c] :
( ( A = bot_bot_set_c )
=> ~ ( member_c @ A2 @ A ) ) ).
% equals0D
thf(fact_285_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_286_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_287_equals0D,axiom,
! [A: set_set_b,A2: set_b] :
( ( A = bot_bot_set_set_b )
=> ~ ( member_set_b @ A2 @ A ) ) ).
% equals0D
thf(fact_288_equals0D,axiom,
! [A: set_set_d,A2: set_d] :
( ( A = bot_bot_set_set_d )
=> ~ ( member_set_d @ A2 @ A ) ) ).
% equals0D
thf(fact_289_equals0D,axiom,
! [A: set_b,A2: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A2 @ A ) ) ).
% equals0D
thf(fact_290_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_291_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_292_equals0I,axiom,
! [A: set_b_d] :
( ! [Y2: b > d] :
~ ( member_b_d @ Y2 @ A )
=> ( A = bot_bot_set_b_d ) ) ).
% equals0I
thf(fact_293_equals0I,axiom,
! [A: set_a_b] :
( ! [Y2: a > b] :
~ ( member_a_b @ Y2 @ A )
=> ( A = bot_bot_set_a_b ) ) ).
% equals0I
thf(fact_294_equals0I,axiom,
! [A: set_c] :
( ! [Y2: c] :
~ ( member_c @ Y2 @ A )
=> ( A = bot_bot_set_c ) ) ).
% equals0I
thf(fact_295_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_296_equals0I,axiom,
! [A: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_297_equals0I,axiom,
! [A: set_set_b] :
( ! [Y2: set_b] :
~ ( member_set_b @ Y2 @ A )
=> ( A = bot_bot_set_set_b ) ) ).
% equals0I
thf(fact_298_equals0I,axiom,
! [A: set_set_d] :
( ! [Y2: set_d] :
~ ( member_set_d @ Y2 @ A )
=> ( A = bot_bot_set_set_d ) ) ).
% equals0I
thf(fact_299_equals0I,axiom,
! [A: set_b] :
( ! [Y2: b] :
~ ( member_b @ Y2 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_300_equals0I,axiom,
! [A: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_301_equals0I,axiom,
! [A: set_o] :
( ! [Y2: $o] :
~ ( member_o @ Y2 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_302_empty__def,axiom,
( bot_bot_set_set_b
= ( collect_set_b
@ ^ [X3: set_b] : $false ) ) ).
% empty_def
thf(fact_303_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_304_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X3: b] : $false ) ) ).
% empty_def
thf(fact_305_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_306_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X3: $o] : $false ) ) ).
% empty_def
thf(fact_307_equalityE,axiom,
! [A: set_set_b,B: set_set_b] :
( ( A = B )
=> ~ ( ( ord_le3795704787696855135_set_b @ A @ B )
=> ~ ( ord_le3795704787696855135_set_b @ B @ A ) ) ) ).
% equalityE
thf(fact_308_equalityE,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_309_equalityE,axiom,
! [A: set_b,B: set_b] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_b @ A @ B )
=> ~ ( ord_less_eq_set_b @ B @ A ) ) ) ).
% equalityE
thf(fact_310_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_311_equalityE,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_o @ A @ B )
=> ~ ( ord_less_eq_set_o @ B @ A ) ) ) ).
% equalityE
thf(fact_312_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_313_equalityE,axiom,
! [A: set_c,B: set_c] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_c @ A @ B )
=> ~ ( ord_less_eq_set_c @ B @ A ) ) ) ).
% equalityE
thf(fact_314_subset__eq,axiom,
( ord_less_eq_set_b_d
= ( ^ [A3: set_b_d,B3: set_b_d] :
! [X3: b > d] :
( ( member_b_d @ X3 @ A3 )
=> ( member_b_d @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_315_subset__eq,axiom,
( ord_less_eq_set_a_b
= ( ^ [A3: set_a_b,B3: set_a_b] :
! [X3: a > b] :
( ( member_a_b @ X3 @ A3 )
=> ( member_a_b @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_316_subset__eq,axiom,
( ord_le3937772867789547105_set_d
= ( ^ [A3: set_set_d,B3: set_set_d] :
! [X3: set_d] :
( ( member_set_d @ X3 @ A3 )
=> ( member_set_d @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_317_subset__eq,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A3: set_set_b,B3: set_set_b] :
! [X3: set_b] :
( ( member_set_b @ X3 @ A3 )
=> ( member_set_b @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_318_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_319_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( member_b @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_320_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_321_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [X3: $o] :
( ( member_o @ X3 @ A3 )
=> ( member_o @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_322_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_323_subset__eq,axiom,
( ord_less_eq_set_c
= ( ^ [A3: set_c,B3: set_c] :
! [X3: c] :
( ( member_c @ X3 @ A3 )
=> ( member_c @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_324_equalityD1,axiom,
! [A: set_set_b,B: set_set_b] :
( ( A = B )
=> ( ord_le3795704787696855135_set_b @ A @ B ) ) ).
% equalityD1
thf(fact_325_equalityD1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_326_equalityD1,axiom,
! [A: set_b,B: set_b] :
( ( A = B )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% equalityD1
thf(fact_327_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_328_equalityD1,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% equalityD1
thf(fact_329_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_330_equalityD1,axiom,
! [A: set_c,B: set_c] :
( ( A = B )
=> ( ord_less_eq_set_c @ A @ B ) ) ).
% equalityD1
thf(fact_331_equalityD2,axiom,
! [A: set_set_b,B: set_set_b] :
( ( A = B )
=> ( ord_le3795704787696855135_set_b @ B @ A ) ) ).
% equalityD2
thf(fact_332_equalityD2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( A = B )
=> ( ord_le3724670747650509150_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_333_equalityD2,axiom,
! [A: set_b,B: set_b] :
( ( A = B )
=> ( ord_less_eq_set_b @ B @ A ) ) ).
% equalityD2
thf(fact_334_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_335_equalityD2,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ( ord_less_eq_set_o @ B @ A ) ) ).
% equalityD2
thf(fact_336_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_337_equalityD2,axiom,
! [A: set_c,B: set_c] :
( ( A = B )
=> ( ord_less_eq_set_c @ B @ A ) ) ).
% equalityD2
thf(fact_338_ex__in__conv,axiom,
! [A: set_b_d] :
( ( ? [X3: b > d] : ( member_b_d @ X3 @ A ) )
= ( A != bot_bot_set_b_d ) ) ).
% ex_in_conv
thf(fact_339_ex__in__conv,axiom,
! [A: set_a_b] :
( ( ? [X3: a > b] : ( member_a_b @ X3 @ A ) )
= ( A != bot_bot_set_a_b ) ) ).
% ex_in_conv
thf(fact_340_ex__in__conv,axiom,
! [A: set_c] :
( ( ? [X3: c] : ( member_c @ X3 @ A ) )
= ( A != bot_bot_set_c ) ) ).
% ex_in_conv
thf(fact_341_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_342_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_343_ex__in__conv,axiom,
! [A: set_set_b] :
( ( ? [X3: set_b] : ( member_set_b @ X3 @ A ) )
= ( A != bot_bot_set_set_b ) ) ).
% ex_in_conv
thf(fact_344_ex__in__conv,axiom,
! [A: set_set_d] :
( ( ? [X3: set_d] : ( member_set_d @ X3 @ A ) )
= ( A != bot_bot_set_set_d ) ) ).
% ex_in_conv
thf(fact_345_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X3: b] : ( member_b @ X3 @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_346_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_347_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X3: $o] : ( member_o @ X3 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_348_subset__iff,axiom,
( ord_less_eq_set_b_d
= ( ^ [A3: set_b_d,B3: set_b_d] :
! [T: b > d] :
( ( member_b_d @ T @ A3 )
=> ( member_b_d @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_349_subset__iff,axiom,
( ord_less_eq_set_a_b
= ( ^ [A3: set_a_b,B3: set_a_b] :
! [T: a > b] :
( ( member_a_b @ T @ A3 )
=> ( member_a_b @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_350_subset__iff,axiom,
( ord_le3937772867789547105_set_d
= ( ^ [A3: set_set_d,B3: set_set_d] :
! [T: set_d] :
( ( member_set_d @ T @ A3 )
=> ( member_set_d @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_351_subset__iff,axiom,
( ord_le3795704787696855135_set_b
= ( ^ [A3: set_set_b,B3: set_set_b] :
! [T: set_b] :
( ( member_set_b @ T @ A3 )
=> ( member_set_b @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_352_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_353_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
! [T: b] :
( ( member_b @ T @ A3 )
=> ( member_b @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_354_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_355_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_356_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_357_subset__iff,axiom,
( ord_less_eq_set_c
= ( ^ [A3: set_c,B3: set_c] :
! [T: c] :
( ( member_c @ T @ A3 )
=> ( member_c @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_358_subset__refl,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ A @ A ) ).
% subset_refl
thf(fact_359_subset__refl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% subset_refl
thf(fact_360_subset__refl,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% subset_refl
thf(fact_361_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_362_subset__refl,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ A ) ).
% subset_refl
thf(fact_363_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_364_subset__refl,axiom,
! [A: set_c] : ( ord_less_eq_set_c @ A @ A ) ).
% subset_refl
thf(fact_365_Collect__mono,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ! [X2: set_b] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).
% Collect_mono
thf(fact_366_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_367_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X2: b] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_368_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_369_Collect__mono,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X2: $o] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_370_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_371_Collect__mono,axiom,
! [P: c > $o,Q: c > $o] :
( ! [X2: c] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_c @ ( collect_c @ P ) @ ( collect_c @ Q ) ) ) ).
% Collect_mono
thf(fact_372_subset__trans,axiom,
! [A: set_set_b,B: set_set_b,C2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( ord_le3795704787696855135_set_b @ B @ C2 )
=> ( ord_le3795704787696855135_set_b @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_373_subset__trans,axiom,
! [A: set_set_a,B: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C2 )
=> ( ord_le3724670747650509150_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_374_subset__trans,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C2 )
=> ( ord_less_eq_set_b @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_375_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_376_subset__trans,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ord_less_eq_set_o @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_377_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_378_subset__trans,axiom,
! [A: set_c,B: set_c,C2: set_c] :
( ( ord_less_eq_set_c @ A @ B )
=> ( ( ord_less_eq_set_c @ B @ C2 )
=> ( ord_less_eq_set_c @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_379_set__eq__subset,axiom,
( ( ^ [Y3: set_set_b,Z: set_set_b] : ( Y3 = Z ) )
= ( ^ [A3: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A3 @ B3 )
& ( ord_le3795704787696855135_set_b @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_380_set__eq__subset,axiom,
( ( ^ [Y3: set_set_a,Z: set_set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_381_set__eq__subset,axiom,
( ( ^ [Y3: set_b,Z: set_b] : ( Y3 = Z ) )
= ( ^ [A3: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A3 @ B3 )
& ( ord_less_eq_set_b @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_382_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_383_set__eq__subset,axiom,
( ( ^ [Y3: set_o,Z: set_o] : ( Y3 = Z ) )
= ( ^ [A3: set_o,B3: set_o] :
( ( ord_less_eq_set_o @ A3 @ B3 )
& ( ord_less_eq_set_o @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_384_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_385_set__eq__subset,axiom,
( ( ^ [Y3: set_c,Z: set_c] : ( Y3 = Z ) )
= ( ^ [A3: set_c,B3: set_c] :
( ( ord_less_eq_set_c @ A3 @ B3 )
& ( ord_less_eq_set_c @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_386_Collect__subset,axiom,
! [A: set_b_d,P: ( b > d ) > $o] :
( ord_less_eq_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_387_Collect__subset,axiom,
! [A: set_a_b,P: ( a > b ) > $o] :
( ord_less_eq_set_a_b
@ ( collect_a_b
@ ^ [X3: a > b] :
( ( member_a_b @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_388_Collect__subset,axiom,
! [A: set_set_d,P: set_d > $o] :
( ord_le3937772867789547105_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_389_Collect__subset,axiom,
! [A: set_set_b,P: set_b > $o] :
( ord_le3795704787696855135_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_390_Collect__subset,axiom,
! [A: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_391_Collect__subset,axiom,
! [A: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_392_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_393_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_394_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_395_Collect__subset,axiom,
! [A: set_c,P: c > $o] :
( ord_less_eq_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_396_Collect__mono__iff,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) )
= ( ! [X3: set_b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_397_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_398_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_399_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_400_Collect__mono__iff,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) )
= ( ! [X3: $o] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_401_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_402_Collect__mono__iff,axiom,
! [P: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ ( collect_c @ P ) @ ( collect_c @ Q ) )
= ( ! [X3: c] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_403_measurable__compose,axiom,
! [F2: a > a,M: sigma_measure_a,N: sigma_measure_a,G2: a > b,L: sigma_measure_b] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M @ N ) )
=> ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N @ L ) )
=> ( member_a_b
@ ^ [X3: a] : ( G2 @ ( F2 @ X3 ) )
@ ( sigma_measurable_a_b @ M @ L ) ) ) ) ).
% measurable_compose
thf(fact_404_measurable__compose,axiom,
! [F2: b > b,M: sigma_measure_b,N: sigma_measure_b,G2: b > d,L: sigma_measure_d] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ M @ N ) )
=> ( ( member_b_d @ G2 @ ( sigma_measurable_b_d @ N @ L ) )
=> ( member_b_d
@ ^ [X3: b] : ( G2 @ ( F2 @ X3 ) )
@ ( sigma_measurable_b_d @ M @ L ) ) ) ) ).
% measurable_compose
thf(fact_405_measurable__compose,axiom,
! [F2: a > b,M: sigma_measure_a,N: sigma_measure_b,G2: b > b,L: sigma_measure_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M @ N ) )
=> ( ( member_b_b @ G2 @ ( sigma_measurable_b_b @ N @ L ) )
=> ( member_a_b
@ ^ [X3: a] : ( G2 @ ( F2 @ X3 ) )
@ ( sigma_measurable_a_b @ M @ L ) ) ) ) ).
% measurable_compose
thf(fact_406_measurable__compose,axiom,
! [F2: a > b,M: sigma_measure_a,N: sigma_measure_b,G2: b > d,L: sigma_measure_d] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M @ N ) )
=> ( ( member_b_d @ G2 @ ( sigma_measurable_b_d @ N @ L ) )
=> ( member_a_d
@ ^ [X3: a] : ( G2 @ ( F2 @ X3 ) )
@ ( sigma_measurable_a_d @ M @ L ) ) ) ) ).
% measurable_compose
thf(fact_407_measurable__compose,axiom,
! [F2: b > d,M: sigma_measure_b,N: sigma_measure_d,G2: d > d,L: sigma_measure_d] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ M @ N ) )
=> ( ( member_d_d @ G2 @ ( sigma_measurable_d_d @ N @ L ) )
=> ( member_b_d
@ ^ [X3: b] : ( G2 @ ( F2 @ X3 ) )
@ ( sigma_measurable_b_d @ M @ L ) ) ) ) ).
% measurable_compose
thf(fact_408_measurable__empty__iff,axiom,
! [N: sigma_measure_d,F2: d > d,M: sigma_measure_d] :
( ( ( sigma_space_d @ N )
= bot_bot_set_d )
=> ( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ M @ N ) )
= ( ( sigma_space_d @ M )
= bot_bot_set_d ) ) ) ).
% measurable_empty_iff
thf(fact_409_measurable__empty__iff,axiom,
! [N: sigma_measure_d,F2: a > d,M: sigma_measure_a] :
( ( ( sigma_space_d @ N )
= bot_bot_set_d )
=> ( ( member_a_d @ F2 @ ( sigma_measurable_a_d @ M @ N ) )
= ( ( sigma_space_a @ M )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_410_measurable__empty__iff,axiom,
! [N: sigma_measure_d,F2: $o > d,M: sigma_measure_o] :
( ( ( sigma_space_d @ N )
= bot_bot_set_d )
=> ( ( member_o_d @ F2 @ ( sigma_measurable_o_d @ M @ N ) )
= ( ( sigma_space_o @ M )
= bot_bot_set_o ) ) ) ).
% measurable_empty_iff
thf(fact_411_measurable__empty__iff,axiom,
! [N: sigma_measure_b,F2: d > b,M: sigma_measure_d] :
( ( ( sigma_space_b @ N )
= bot_bot_set_b )
=> ( ( member_d_b @ F2 @ ( sigma_measurable_d_b @ M @ N ) )
= ( ( sigma_space_d @ M )
= bot_bot_set_d ) ) ) ).
% measurable_empty_iff
thf(fact_412_measurable__empty__iff,axiom,
! [N: sigma_measure_b,F2: b > b,M: sigma_measure_b] :
( ( ( sigma_space_b @ N )
= bot_bot_set_b )
=> ( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ M @ N ) )
= ( ( sigma_space_b @ M )
= bot_bot_set_b ) ) ) ).
% measurable_empty_iff
thf(fact_413_measurable__empty__iff,axiom,
! [N: sigma_measure_b,F2: $o > b,M: sigma_measure_o] :
( ( ( sigma_space_b @ N )
= bot_bot_set_b )
=> ( ( member_o_b @ F2 @ ( sigma_measurable_o_b @ M @ N ) )
= ( ( sigma_space_o @ M )
= bot_bot_set_o ) ) ) ).
% measurable_empty_iff
thf(fact_414_measurable__empty__iff,axiom,
! [N: sigma_measure_a,F2: d > a,M: sigma_measure_d] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
=> ( ( member_d_a @ F2 @ ( sigma_measurable_d_a @ M @ N ) )
= ( ( sigma_space_d @ M )
= bot_bot_set_d ) ) ) ).
% measurable_empty_iff
thf(fact_415_measurable__empty__iff,axiom,
! [N: sigma_measure_a,F2: b > a,M: sigma_measure_b] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
=> ( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ M @ N ) )
= ( ( sigma_space_b @ M )
= bot_bot_set_b ) ) ) ).
% measurable_empty_iff
thf(fact_416_measurable__empty__iff,axiom,
! [N: sigma_measure_a,F2: a > a,M: sigma_measure_a] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
=> ( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M @ N ) )
= ( ( sigma_space_a @ M )
= bot_bot_set_a ) ) ) ).
% measurable_empty_iff
thf(fact_417_measurable__empty__iff,axiom,
! [N: sigma_measure_a,F2: $o > a,M: sigma_measure_o] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
=> ( ( member_o_a @ F2 @ ( sigma_measurable_o_a @ M @ N ) )
= ( ( sigma_space_o @ M )
= bot_bot_set_o ) ) ) ).
% measurable_empty_iff
thf(fact_418_measurable__compose__rev,axiom,
! [F2: b > b,L: sigma_measure_b,N: sigma_measure_b,G2: a > b,M: sigma_measure_a] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ L @ N ) )
=> ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ M @ L ) )
=> ( member_a_b
@ ^ [X3: a] : ( F2 @ ( G2 @ X3 ) )
@ ( sigma_measurable_a_b @ M @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_419_measurable__compose__rev,axiom,
! [F2: d > d,L: sigma_measure_d,N: sigma_measure_d,G2: b > d,M: sigma_measure_b] :
( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ L @ N ) )
=> ( ( member_b_d @ G2 @ ( sigma_measurable_b_d @ M @ L ) )
=> ( member_b_d
@ ^ [X3: b] : ( F2 @ ( G2 @ X3 ) )
@ ( sigma_measurable_b_d @ M @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_420_measurable__compose__rev,axiom,
! [F2: a > b,L: sigma_measure_a,N: sigma_measure_b,G2: a > a,M: sigma_measure_a] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ L @ N ) )
=> ( ( member_a_a @ G2 @ ( sigma_measurable_a_a @ M @ L ) )
=> ( member_a_b
@ ^ [X3: a] : ( F2 @ ( G2 @ X3 ) )
@ ( sigma_measurable_a_b @ M @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_421_measurable__compose__rev,axiom,
! [F2: b > d,L: sigma_measure_b,N: sigma_measure_d,G2: b > b,M: sigma_measure_b] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ L @ N ) )
=> ( ( member_b_b @ G2 @ ( sigma_measurable_b_b @ M @ L ) )
=> ( member_b_d
@ ^ [X3: b] : ( F2 @ ( G2 @ X3 ) )
@ ( sigma_measurable_b_d @ M @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_422_measurable__compose__rev,axiom,
! [F2: b > d,L: sigma_measure_b,N: sigma_measure_d,G2: a > b,M: sigma_measure_a] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ L @ N ) )
=> ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ M @ L ) )
=> ( member_a_d
@ ^ [X3: a] : ( F2 @ ( G2 @ X3 ) )
@ ( sigma_measurable_a_d @ M @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_423_measurable__mono,axiom,
! [N2: sigma_measure_d,N: sigma_measure_d,M: sigma_measure_d,M2: sigma_measure_d] :
( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ N2 ) @ ( sigma_sets_d @ N ) )
=> ( ( ( sigma_space_d @ N )
= ( sigma_space_d @ N2 ) )
=> ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ ( sigma_sets_d @ M2 ) )
=> ( ( ( sigma_space_d @ M )
= ( sigma_space_d @ M2 ) )
=> ( ord_less_eq_set_d_d @ ( sigma_measurable_d_d @ M @ N ) @ ( sigma_measurable_d_d @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_424_measurable__mono,axiom,
! [N2: sigma_measure_d,N: sigma_measure_d,M: sigma_measure_b,M2: sigma_measure_b] :
( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ N2 ) @ ( sigma_sets_d @ N ) )
=> ( ( ( sigma_space_d @ N )
= ( sigma_space_d @ N2 ) )
=> ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_space_b @ M )
= ( sigma_space_b @ M2 ) )
=> ( ord_less_eq_set_b_d @ ( sigma_measurable_b_d @ M @ N ) @ ( sigma_measurable_b_d @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_425_measurable__mono,axiom,
! [N2: sigma_measure_d,N: sigma_measure_d,M: sigma_measure_a,M2: sigma_measure_a] :
( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ N2 ) @ ( sigma_sets_d @ N ) )
=> ( ( ( sigma_space_d @ N )
= ( sigma_space_d @ N2 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_space_a @ M )
= ( sigma_space_a @ M2 ) )
=> ( ord_less_eq_set_a_d @ ( sigma_measurable_a_d @ M @ N ) @ ( sigma_measurable_a_d @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_426_measurable__mono,axiom,
! [N2: sigma_measure_b,N: sigma_measure_b,M: sigma_measure_d,M2: sigma_measure_d] :
( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ N2 ) @ ( sigma_sets_b @ N ) )
=> ( ( ( sigma_space_b @ N )
= ( sigma_space_b @ N2 ) )
=> ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ ( sigma_sets_d @ M2 ) )
=> ( ( ( sigma_space_d @ M )
= ( sigma_space_d @ M2 ) )
=> ( ord_less_eq_set_d_b @ ( sigma_measurable_d_b @ M @ N ) @ ( sigma_measurable_d_b @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_427_measurable__mono,axiom,
! [N2: sigma_measure_b,N: sigma_measure_b,M: sigma_measure_b,M2: sigma_measure_b] :
( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ N2 ) @ ( sigma_sets_b @ N ) )
=> ( ( ( sigma_space_b @ N )
= ( sigma_space_b @ N2 ) )
=> ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_space_b @ M )
= ( sigma_space_b @ M2 ) )
=> ( ord_less_eq_set_b_b @ ( sigma_measurable_b_b @ M @ N ) @ ( sigma_measurable_b_b @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_428_measurable__mono,axiom,
! [N2: sigma_measure_b,N: sigma_measure_b,M: sigma_measure_a,M2: sigma_measure_a] :
( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ N2 ) @ ( sigma_sets_b @ N ) )
=> ( ( ( sigma_space_b @ N )
= ( sigma_space_b @ N2 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_space_a @ M )
= ( sigma_space_a @ M2 ) )
=> ( ord_less_eq_set_a_b @ ( sigma_measurable_a_b @ M @ N ) @ ( sigma_measurable_a_b @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_429_measurable__mono,axiom,
! [N2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_d,M2: sigma_measure_d] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N2 ) @ ( sigma_sets_a @ N ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ N2 ) )
=> ( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ M ) @ ( sigma_sets_d @ M2 ) )
=> ( ( ( sigma_space_d @ M )
= ( sigma_space_d @ M2 ) )
=> ( ord_less_eq_set_d_a @ ( sigma_measurable_d_a @ M @ N ) @ ( sigma_measurable_d_a @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_430_measurable__mono,axiom,
! [N2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_b,M2: sigma_measure_b] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N2 ) @ ( sigma_sets_a @ N ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ N2 ) )
=> ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ M ) @ ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_space_b @ M )
= ( sigma_space_b @ M2 ) )
=> ( ord_less_eq_set_b_a @ ( sigma_measurable_b_a @ M @ N ) @ ( sigma_measurable_b_a @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_431_measurable__mono,axiom,
! [N2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_a,M2: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N2 ) @ ( sigma_sets_a @ N ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ N2 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_space_a @ M )
= ( sigma_space_a @ M2 ) )
=> ( ord_less_eq_set_a_a @ ( sigma_measurable_a_a @ M @ N ) @ ( sigma_measurable_a_a @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_432_measurable__cong__sets,axiom,
! [M: sigma_measure_d,M2: sigma_measure_d,N: sigma_measure_d,N2: sigma_measure_d] :
( ( ( sigma_sets_d @ M )
= ( sigma_sets_d @ M2 ) )
=> ( ( ( sigma_sets_d @ N )
= ( sigma_sets_d @ N2 ) )
=> ( ( sigma_measurable_d_d @ M @ N )
= ( sigma_measurable_d_d @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_433_measurable__cong__sets,axiom,
! [M: sigma_measure_d,M2: sigma_measure_d,N: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_sets_d @ M )
= ( sigma_sets_d @ M2 ) )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ N2 ) )
=> ( ( sigma_measurable_d_a @ M @ N )
= ( sigma_measurable_d_a @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_434_measurable__cong__sets,axiom,
! [M: sigma_measure_d,M2: sigma_measure_d,N: sigma_measure_b,N2: sigma_measure_b] :
( ( ( sigma_sets_d @ M )
= ( sigma_sets_d @ M2 ) )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ N2 ) )
=> ( ( sigma_measurable_d_b @ M @ N )
= ( sigma_measurable_d_b @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_435_measurable__cong__sets,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a,N: sigma_measure_d,N2: sigma_measure_d] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_sets_d @ N )
= ( sigma_sets_d @ N2 ) )
=> ( ( sigma_measurable_a_d @ M @ N )
= ( sigma_measurable_a_d @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_436_measurable__cong__sets,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a,N: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ N2 ) )
=> ( ( sigma_measurable_a_a @ M @ N )
= ( sigma_measurable_a_a @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_437_measurable__cong__sets,axiom,
! [M: sigma_measure_b,M2: sigma_measure_b,N: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_sets_b @ M )
= ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ N2 ) )
=> ( ( sigma_measurable_b_a @ M @ N )
= ( sigma_measurable_b_a @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_438_measurable__cong__sets,axiom,
! [M: sigma_measure_b,M2: sigma_measure_b,N: sigma_measure_b,N2: sigma_measure_b] :
( ( ( sigma_sets_b @ M )
= ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ N2 ) )
=> ( ( sigma_measurable_b_b @ M @ N )
= ( sigma_measurable_b_b @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_439_measurable__cong__sets,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a,N: sigma_measure_b,N2: sigma_measure_b] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ N2 ) )
=> ( ( sigma_measurable_a_b @ M @ N )
= ( sigma_measurable_a_b @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_440_measurable__cong__sets,axiom,
! [M: sigma_measure_b,M2: sigma_measure_b,N: sigma_measure_d,N2: sigma_measure_d] :
( ( ( sigma_sets_b @ M )
= ( sigma_sets_b @ M2 ) )
=> ( ( ( sigma_sets_d @ N )
= ( sigma_sets_d @ N2 ) )
=> ( ( sigma_measurable_b_d @ M @ N )
= ( sigma_measurable_b_d @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_441_measurable__cong__simp,axiom,
! [M: sigma_measure_a,N: sigma_measure_a,M2: sigma_measure_b,N2: sigma_measure_b,F2: a > b,G2: a > b] :
( ( M = N )
=> ( ( M2 = N2 )
=> ( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M @ M2 ) )
= ( member_a_b @ G2 @ ( sigma_measurable_a_b @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_442_measurable__cong__simp,axiom,
! [M: sigma_measure_b,N: sigma_measure_b,M2: sigma_measure_d,N2: sigma_measure_d,F2: b > d,G2: b > d] :
( ( M = N )
=> ( ( M2 = N2 )
=> ( ! [W: b] :
( ( member_b @ W @ ( sigma_space_b @ M ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ M @ M2 ) )
= ( member_b_d @ G2 @ ( sigma_measurable_b_d @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_443_measurable__space,axiom,
! [F2: c > c,M: sigma_measure_c,A: sigma_measure_c,X: c] :
( ( member_c_c @ F2 @ ( sigma_measurable_c_c @ M @ A ) )
=> ( ( member_c @ X @ ( sigma_space_c @ M ) )
=> ( member_c @ ( F2 @ X ) @ ( sigma_space_c @ A ) ) ) ) ).
% measurable_space
thf(fact_444_measurable__space,axiom,
! [F2: c > $o,M: sigma_measure_c,A: sigma_measure_o,X: c] :
( ( member_c_o @ F2 @ ( sigma_measurable_c_o @ M @ A ) )
=> ( ( member_c @ X @ ( sigma_space_c @ M ) )
=> ( member_o @ ( F2 @ X ) @ ( sigma_space_o @ A ) ) ) ) ).
% measurable_space
thf(fact_445_measurable__space,axiom,
! [F2: c > nat,M: sigma_measure_c,A: sigma_measure_nat,X: c] :
( ( member_c_nat @ F2 @ ( sigma_2544038740538346112_c_nat @ M @ A ) )
=> ( ( member_c @ X @ ( sigma_space_c @ M ) )
=> ( member_nat @ ( F2 @ X ) @ ( sigma_space_nat @ A ) ) ) ) ).
% measurable_space
thf(fact_446_measurable__space,axiom,
! [F2: $o > c,M: sigma_measure_o,A: sigma_measure_c,X: $o] :
( ( member_o_c @ F2 @ ( sigma_measurable_o_c @ M @ A ) )
=> ( ( member_o @ X @ ( sigma_space_o @ M ) )
=> ( member_c @ ( F2 @ X ) @ ( sigma_space_c @ A ) ) ) ) ).
% measurable_space
thf(fact_447_measurable__space,axiom,
! [F2: $o > $o,M: sigma_measure_o,A: sigma_measure_o,X: $o] :
( ( member_o_o @ F2 @ ( sigma_measurable_o_o @ M @ A ) )
=> ( ( member_o @ X @ ( sigma_space_o @ M ) )
=> ( member_o @ ( F2 @ X ) @ ( sigma_space_o @ A ) ) ) ) ).
% measurable_space
thf(fact_448_measurable__space,axiom,
! [F2: $o > nat,M: sigma_measure_o,A: sigma_measure_nat,X: $o] :
( ( member_o_nat @ F2 @ ( sigma_1999164137574644376_o_nat @ M @ A ) )
=> ( ( member_o @ X @ ( sigma_space_o @ M ) )
=> ( member_nat @ ( F2 @ X ) @ ( sigma_space_nat @ A ) ) ) ) ).
% measurable_space
thf(fact_449_measurable__space,axiom,
! [F2: nat > c,M: sigma_measure_nat,A: sigma_measure_c,X: nat] :
( ( member_nat_c @ F2 @ ( sigma_4105081583803843550_nat_c @ M @ A ) )
=> ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
=> ( member_c @ ( F2 @ X ) @ ( sigma_space_c @ A ) ) ) ) ).
% measurable_space
thf(fact_450_measurable__space,axiom,
! [F2: nat > $o,M: sigma_measure_nat,A: sigma_measure_o,X: nat] :
( ( member_nat_o @ F2 @ ( sigma_5101835498682829686_nat_o @ M @ A ) )
=> ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
=> ( member_o @ ( F2 @ X ) @ ( sigma_space_o @ A ) ) ) ) ).
% measurable_space
thf(fact_451_measurable__space,axiom,
! [F2: nat > nat,M: sigma_measure_nat,A: sigma_measure_nat,X: nat] :
( ( member_nat_nat @ F2 @ ( sigma_4350458207664084850at_nat @ M @ A ) )
=> ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
=> ( member_nat @ ( F2 @ X ) @ ( sigma_space_nat @ A ) ) ) ) ).
% measurable_space
thf(fact_452_measurable__space,axiom,
! [F2: c > b,M: sigma_measure_c,A: sigma_measure_b,X: c] :
( ( member_c_b @ F2 @ ( sigma_measurable_c_b @ M @ A ) )
=> ( ( member_c @ X @ ( sigma_space_c @ M ) )
=> ( member_b @ ( F2 @ X ) @ ( sigma_space_b @ A ) ) ) ) ).
% measurable_space
thf(fact_453_measurable__cong,axiom,
! [M: sigma_measure_a,F2: a > b,G2: a > b,M2: sigma_measure_b] :
( ! [W: a] :
( ( member_a @ W @ ( sigma_space_a @ M ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M @ M2 ) )
= ( member_a_b @ G2 @ ( sigma_measurable_a_b @ M @ M2 ) ) ) ) ).
% measurable_cong
thf(fact_454_measurable__cong,axiom,
! [M: sigma_measure_b,F2: b > d,G2: b > d,M2: sigma_measure_d] :
( ! [W: b] :
( ( member_b @ W @ ( sigma_space_b @ M ) )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ M @ M2 ) )
= ( member_b_d @ G2 @ ( sigma_measurable_b_d @ M @ M2 ) ) ) ) ).
% measurable_cong
thf(fact_455_Int__Collect__mono,axiom,
! [A: set_d,B: set_d,P: d > $o,Q: d > $o] :
( ( ord_less_eq_set_d @ A @ B )
=> ( ! [X2: d] :
( ( member_d @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_d @ ( inf_inf_set_d @ A @ ( collect_d @ P ) ) @ ( inf_inf_set_d @ B @ ( collect_d @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_456_Int__Collect__mono,axiom,
! [A: set_b,B: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_457_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_458_Int__Collect__mono,axiom,
! [A: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_459_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_460_Int__Collect__mono,axiom,
! [A: set_c,B: set_c,P: c > $o,Q: c > $o] :
( ( ord_less_eq_set_c @ A @ B )
=> ( ! [X2: c] :
( ( member_c @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A @ ( collect_c @ P ) ) @ ( inf_inf_set_c @ B @ ( collect_c @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_461_Int__Collect__mono,axiom,
! [A: set_set_d,B: set_set_d,P: set_d > $o,Q: set_d > $o] :
( ( ord_le3937772867789547105_set_d @ A @ B )
=> ( ! [X2: set_d] :
( ( member_set_d @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3937772867789547105_set_d @ ( inf_inf_set_set_d @ A @ ( collect_set_d @ P ) ) @ ( inf_inf_set_set_d @ B @ ( collect_set_d @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_462_Int__Collect__mono,axiom,
! [A: set_set_b,B: set_set_b,P: set_b > $o,Q: set_b > $o] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ! [X2: set_b] :
( ( member_set_b @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ ( collect_set_b @ P ) ) @ ( inf_inf_set_set_b @ B @ ( collect_set_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_463_Int__Collect__mono,axiom,
! [A: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_464_Int__Collect__mono,axiom,
! [A: set_b_d,B: set_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
( ( ord_less_eq_set_b_d @ A @ B )
=> ( ! [X2: b > d] :
( ( member_b_d @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_b_d @ ( inf_inf_set_b_d @ A @ ( collect_b_d @ P ) ) @ ( inf_inf_set_b_d @ B @ ( collect_b_d @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_465_Int__greatest,axiom,
! [C2: set_d,A: set_d,B: set_d] :
( ( ord_less_eq_set_d @ C2 @ A )
=> ( ( ord_less_eq_set_d @ C2 @ B )
=> ( ord_less_eq_set_d @ C2 @ ( inf_inf_set_d @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_466_Int__greatest,axiom,
! [C2: set_set_b,A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ C2 @ A )
=> ( ( ord_le3795704787696855135_set_b @ C2 @ B )
=> ( ord_le3795704787696855135_set_b @ C2 @ ( inf_inf_set_set_b @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_467_Int__greatest,axiom,
! [C2: set_set_a,A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C2 @ B )
=> ( ord_le3724670747650509150_set_a @ C2 @ ( inf_inf_set_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_468_Int__greatest,axiom,
! [C2: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C2 @ A )
=> ( ( ord_less_eq_set_b @ C2 @ B )
=> ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_469_Int__greatest,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_470_Int__greatest,axiom,
! [C2: set_o,A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ C2 @ A )
=> ( ( ord_less_eq_set_o @ C2 @ B )
=> ( ord_less_eq_set_o @ C2 @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_471_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_472_Int__greatest,axiom,
! [C2: set_c,A: set_c,B: set_c] :
( ( ord_less_eq_set_c @ C2 @ A )
=> ( ( ord_less_eq_set_c @ C2 @ B )
=> ( ord_less_eq_set_c @ C2 @ ( inf_inf_set_c @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_473_Int__absorb2,axiom,
! [A: set_d,B: set_d] :
( ( ord_less_eq_set_d @ A @ B )
=> ( ( inf_inf_set_d @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_474_Int__absorb2,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ B )
=> ( ( inf_inf_set_set_b @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_475_Int__absorb2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( inf_inf_set_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_476_Int__absorb2,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( inf_inf_set_b @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_477_Int__absorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_478_Int__absorb2,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( inf_inf_set_o @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_479_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_480_Int__absorb2,axiom,
! [A: set_c,B: set_c] :
( ( ord_less_eq_set_c @ A @ B )
=> ( ( inf_inf_set_c @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_481_Int__absorb1,axiom,
! [B: set_d,A: set_d] :
( ( ord_less_eq_set_d @ B @ A )
=> ( ( inf_inf_set_d @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_482_Int__absorb1,axiom,
! [B: set_set_b,A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ B @ A )
=> ( ( inf_inf_set_set_b @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_483_Int__absorb1,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( inf_inf_set_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_484_Int__absorb1,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( inf_inf_set_b @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_485_Int__absorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_486_Int__absorb1,axiom,
! [B: set_o,A: set_o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( inf_inf_set_o @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_487_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_488_Int__absorb1,axiom,
! [B: set_c,A: set_c] :
( ( ord_less_eq_set_c @ B @ A )
=> ( ( inf_inf_set_c @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_489_Int__lower2,axiom,
! [A: set_d,B: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_490_Int__lower2,axiom,
! [A: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_491_Int__lower2,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_492_Int__lower2,axiom,
! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_493_Int__lower2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_494_Int__lower2,axiom,
! [A: set_o,B: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_495_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_496_Int__lower2,axiom,
! [A: set_c,B: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_497_Int__lower1,axiom,
! [A: set_d,B: set_d] : ( ord_less_eq_set_d @ ( inf_inf_set_d @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_498_Int__lower1,axiom,
! [A: set_set_b,B: set_set_b] : ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_499_Int__lower1,axiom,
! [A: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_500_Int__lower1,axiom,
! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_501_Int__lower1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_502_Int__lower1,axiom,
! [A: set_o,B: set_o] : ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_503_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_504_Int__lower1,axiom,
! [A: set_c,B: set_c] : ( ord_less_eq_set_c @ ( inf_inf_set_c @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_505_Int__mono,axiom,
! [A: set_d,C2: set_d,B: set_d,D: set_d] :
( ( ord_less_eq_set_d @ A @ C2 )
=> ( ( ord_less_eq_set_d @ B @ D )
=> ( ord_less_eq_set_d @ ( inf_inf_set_d @ A @ B ) @ ( inf_inf_set_d @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_506_Int__mono,axiom,
! [A: set_set_b,C2: set_set_b,B: set_set_b,D: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ C2 )
=> ( ( ord_le3795704787696855135_set_b @ B @ D )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ B ) @ ( inf_inf_set_set_b @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_507_Int__mono,axiom,
! [A: set_set_a,C2: set_set_a,B: set_set_a,D: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ B @ D )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ B ) @ ( inf_inf_set_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_508_Int__mono,axiom,
! [A: set_b,C2: set_b,B: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ ( inf_inf_set_b @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_509_Int__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_510_Int__mono,axiom,
! [A: set_o,C2: set_o,B: set_o,D: set_o] :
( ( ord_less_eq_set_o @ A @ C2 )
=> ( ( ord_less_eq_set_o @ B @ D )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ B ) @ ( inf_inf_set_o @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_511_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_512_Int__mono,axiom,
! [A: set_c,C2: set_c,B: set_c,D: set_c] :
( ( ord_less_eq_set_c @ A @ C2 )
=> ( ( ord_less_eq_set_c @ B @ D )
=> ( ord_less_eq_set_c @ ( inf_inf_set_c @ A @ B ) @ ( inf_inf_set_c @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_513_disjoint__iff__not__equal,axiom,
! [A: set_d,B: set_d] :
( ( ( inf_inf_set_d @ A @ B )
= bot_bot_set_d )
= ( ! [X3: d] :
( ( member_d @ X3 @ A )
=> ! [Y: d] :
( ( member_d @ Y @ B )
=> ( X3 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_514_disjoint__iff__not__equal,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
= ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ! [Y: b] :
( ( member_b @ Y @ B )
=> ( X3 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_515_disjoint__iff__not__equal,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y: a] :
( ( member_a @ Y @ B )
=> ( X3 != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_516_disjoint__iff__not__equal,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ! [Y: $o] :
( ( member_o @ Y @ B )
=> ( X3 = (~ Y) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_517_Int__empty__right,axiom,
! [A: set_d] :
( ( inf_inf_set_d @ A @ bot_bot_set_d )
= bot_bot_set_d ) ).
% Int_empty_right
thf(fact_518_Int__empty__right,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_519_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_520_Int__empty__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_521_Int__empty__left,axiom,
! [B: set_d] :
( ( inf_inf_set_d @ bot_bot_set_d @ B )
= bot_bot_set_d ) ).
% Int_empty_left
thf(fact_522_Int__empty__left,axiom,
! [B: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_523_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_524_Int__empty__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_525_disjoint__iff,axiom,
! [A: set_c,B: set_c] :
( ( ( inf_inf_set_c @ A @ B )
= bot_bot_set_c )
= ( ! [X3: c] :
( ( member_c @ X3 @ A )
=> ~ ( member_c @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_526_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_527_disjoint__iff,axiom,
! [A: set_d,B: set_d] :
( ( ( inf_inf_set_d @ A @ B )
= bot_bot_set_d )
= ( ! [X3: d] :
( ( member_d @ X3 @ A )
=> ~ ( member_d @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_528_disjoint__iff,axiom,
! [A: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
= ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ~ ( member_b @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_529_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_530_disjoint__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_531_disjoint__iff,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_532_disjoint__iff,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ( inf_inf_set_set_b @ A @ B )
= bot_bot_set_set_b )
= ( ! [X3: set_b] :
( ( member_set_b @ X3 @ A )
=> ~ ( member_set_b @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_533_disjoint__iff,axiom,
! [A: set_set_d,B: set_set_d] :
( ( ( inf_inf_set_set_d @ A @ B )
= bot_bot_set_set_d )
= ( ! [X3: set_d] :
( ( member_set_d @ X3 @ A )
=> ~ ( member_set_d @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_534_disjoint__iff,axiom,
! [A: set_b_d,B: set_b_d] :
( ( ( inf_inf_set_b_d @ A @ B )
= bot_bot_set_b_d )
= ( ! [X3: b > d] :
( ( member_b_d @ X3 @ A )
=> ~ ( member_b_d @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_535_Int__emptyI,axiom,
! [A: set_c,B: set_c] :
( ! [X2: c] :
( ( member_c @ X2 @ A )
=> ~ ( member_c @ X2 @ B ) )
=> ( ( inf_inf_set_c @ A @ B )
= bot_bot_set_c ) ) ).
% Int_emptyI
thf(fact_536_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_537_Int__emptyI,axiom,
! [A: set_d,B: set_d] :
( ! [X2: d] :
( ( member_d @ X2 @ A )
=> ~ ( member_d @ X2 @ B ) )
=> ( ( inf_inf_set_d @ A @ B )
= bot_bot_set_d ) ) ).
% Int_emptyI
thf(fact_538_Int__emptyI,axiom,
! [A: set_b,B: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ~ ( member_b @ X2 @ B ) )
=> ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_539_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_540_Int__emptyI,axiom,
! [A: set_o,B: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ~ ( member_o @ X2 @ B ) )
=> ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_541_Int__emptyI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ( member_set_a @ X2 @ B ) )
=> ( ( inf_inf_set_set_a @ A @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_542_Int__emptyI,axiom,
! [A: set_set_b,B: set_set_b] :
( ! [X2: set_b] :
( ( member_set_b @ X2 @ A )
=> ~ ( member_set_b @ X2 @ B ) )
=> ( ( inf_inf_set_set_b @ A @ B )
= bot_bot_set_set_b ) ) ).
% Int_emptyI
thf(fact_543_Int__emptyI,axiom,
! [A: set_set_d,B: set_set_d] :
( ! [X2: set_d] :
( ( member_set_d @ X2 @ A )
=> ~ ( member_set_d @ X2 @ B ) )
=> ( ( inf_inf_set_set_d @ A @ B )
= bot_bot_set_set_d ) ) ).
% Int_emptyI
thf(fact_544_Int__emptyI,axiom,
! [A: set_b_d,B: set_b_d] :
( ! [X2: b > d] :
( ( member_b_d @ X2 @ A )
=> ~ ( member_b_d @ X2 @ B ) )
=> ( ( inf_inf_set_b_d @ A @ B )
= bot_bot_set_b_d ) ) ).
% Int_emptyI
thf(fact_545_subset__vimage__iff,axiom,
! [A: set_b,F2: b > c,B: set_c] :
( ( ord_less_eq_set_b @ A @ ( vimage_b_c @ F2 @ B ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_c @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_546_subset__vimage__iff,axiom,
! [A: set_b,F2: b > b,B: set_b] :
( ( ord_less_eq_set_b @ A @ ( vimage_b_b @ F2 @ B ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_b @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_547_subset__vimage__iff,axiom,
! [A: set_b,F2: b > a,B: set_a] :
( ( ord_less_eq_set_b @ A @ ( vimage_b_a @ F2 @ B ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_a @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_548_subset__vimage__iff,axiom,
! [A: set_b,F2: b > $o,B: set_o] :
( ( ord_less_eq_set_b @ A @ ( vimage_b_o @ F2 @ B ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_o @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_549_subset__vimage__iff,axiom,
! [A: set_b,F2: b > nat,B: set_nat] :
( ( ord_less_eq_set_b @ A @ ( vimage_b_nat @ F2 @ B ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_nat @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_550_subset__vimage__iff,axiom,
! [A: set_b,F2: b > d,B: set_d] :
( ( ord_less_eq_set_b @ A @ ( vimage_b_d @ F2 @ B ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_d @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_551_subset__vimage__iff,axiom,
! [A: set_a,F2: a > c,B: set_c] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_c @ F2 @ B ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_c @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_552_subset__vimage__iff,axiom,
! [A: set_a,F2: a > b,B: set_b] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_b @ F2 @ B ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_b @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_553_subset__vimage__iff,axiom,
! [A: set_a,F2: a > a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_a @ F2 @ B ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_554_subset__vimage__iff,axiom,
! [A: set_a,F2: a > $o,B: set_o] :
( ( ord_less_eq_set_a @ A @ ( vimage_a_o @ F2 @ B ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_o @ ( F2 @ X3 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_555_vimage__mono,axiom,
! [A: set_d,B: set_d,F2: b > d] :
( ( ord_less_eq_set_d @ A @ B )
=> ( ord_less_eq_set_b @ ( vimage_b_d @ F2 @ A ) @ ( vimage_b_d @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_556_vimage__mono,axiom,
! [A: set_b,B: set_b,F2: b > b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_b @ ( vimage_b_b @ F2 @ A ) @ ( vimage_b_b @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_557_vimage__mono,axiom,
! [A: set_b,B: set_b,F2: a > b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_b @ F2 @ A ) @ ( vimage_a_b @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_558_vimage__mono,axiom,
! [A: set_b,B: set_b,F2: $o > b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_o @ ( vimage_o_b @ F2 @ A ) @ ( vimage_o_b @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_559_vimage__mono,axiom,
! [A: set_b,B: set_b,F2: nat > b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_nat @ ( vimage_nat_b @ F2 @ A ) @ ( vimage_nat_b @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_560_vimage__mono,axiom,
! [A: set_b,B: set_b,F2: c > b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_c @ ( vimage_c_b @ F2 @ A ) @ ( vimage_c_b @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_561_vimage__mono,axiom,
! [A: set_a,B: set_a,F2: b > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_b @ ( vimage_b_a @ F2 @ A ) @ ( vimage_b_a @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_562_vimage__mono,axiom,
! [A: set_a,B: set_a,F2: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F2 @ A ) @ ( vimage_a_a @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_563_vimage__mono,axiom,
! [A: set_a,B: set_a,F2: $o > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_o @ ( vimage_o_a @ F2 @ A ) @ ( vimage_o_a @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_564_vimage__mono,axiom,
! [A: set_a,B: set_a,F2: nat > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_nat @ ( vimage_nat_a @ F2 @ A ) @ ( vimage_nat_a @ F2 @ B ) ) ) ).
% vimage_mono
thf(fact_565_sets__le__imp__space__le,axiom,
! [A: sigma_measure_d,B: sigma_measure_d] :
( ( ord_le3937772867789547105_set_d @ ( sigma_sets_d @ A ) @ ( sigma_sets_d @ B ) )
=> ( ord_less_eq_set_d @ ( sigma_space_d @ A ) @ ( sigma_space_d @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_566_sets__le__imp__space__le,axiom,
! [A: sigma_measure_set_b,B: sigma_measure_set_b] :
( ( ord_le3201067847557142847_set_b @ ( sigma_sets_set_b @ A ) @ ( sigma_sets_set_b @ B ) )
=> ( ord_le3795704787696855135_set_b @ ( sigma_space_set_b @ A ) @ ( sigma_space_set_b @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_567_sets__le__imp__space__le,axiom,
! [A: sigma_measure_set_a,B: sigma_measure_set_a] :
( ( ord_le5722252365846178494_set_a @ ( sigma_sets_set_a @ A ) @ ( sigma_sets_set_a @ B ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_space_set_a @ A ) @ ( sigma_space_set_a @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_568_sets__le__imp__space__le,axiom,
! [A: sigma_measure_o,B: sigma_measure_o] :
( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ A ) @ ( sigma_sets_o @ B ) )
=> ( ord_less_eq_set_o @ ( sigma_space_o @ A ) @ ( sigma_space_o @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_569_sets__le__imp__space__le,axiom,
! [A: sigma_measure_nat,B: sigma_measure_nat] :
( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ A ) @ ( sigma_sets_nat @ B ) )
=> ( ord_less_eq_set_nat @ ( sigma_space_nat @ A ) @ ( sigma_space_nat @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_570_sets__le__imp__space__le,axiom,
! [A: sigma_measure_c,B: sigma_measure_c] :
( ( ord_le3866738827743201120_set_c @ ( sigma_sets_c @ A ) @ ( sigma_sets_c @ B ) )
=> ( ord_less_eq_set_c @ ( sigma_space_c @ A ) @ ( sigma_space_c @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_571_sets__le__imp__space__le,axiom,
! [A: sigma_measure_b,B: sigma_measure_b] :
( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ A ) @ ( sigma_sets_b @ B ) )
=> ( ord_less_eq_set_b @ ( sigma_space_b @ A ) @ ( sigma_space_b @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_572_sets__le__imp__space__le,axiom,
! [A: sigma_measure_a,B: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A ) @ ( sigma_sets_a @ B ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A ) @ ( sigma_space_a @ B ) ) ) ).
% sets_le_imp_space_le
thf(fact_573_measurable__ident__sets,axiom,
! [M: sigma_measure_d,M2: sigma_measure_d] :
( ( ( sigma_sets_d @ M )
= ( sigma_sets_d @ M2 ) )
=> ( member_d_d
@ ^ [X3: d] : X3
@ ( sigma_measurable_d_d @ M @ M2 ) ) ) ).
% measurable_ident_sets
thf(fact_574_measurable__ident__sets,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( member_a_a
@ ^ [X3: a] : X3
@ ( sigma_measurable_a_a @ M @ M2 ) ) ) ).
% measurable_ident_sets
thf(fact_575_measurable__ident__sets,axiom,
! [M: sigma_measure_b,M2: sigma_measure_b] :
( ( ( sigma_sets_b @ M )
= ( sigma_sets_b @ M2 ) )
=> ( member_b_b
@ ^ [X3: b] : X3
@ ( sigma_measurable_b_b @ M @ M2 ) ) ) ).
% measurable_ident_sets
thf(fact_576_measurable__const,axiom,
! [C: b,M2: sigma_measure_b,M: sigma_measure_a] :
( ( member_b @ C @ ( sigma_space_b @ M2 ) )
=> ( member_a_b
@ ^ [X3: a] : C
@ ( sigma_measurable_a_b @ M @ M2 ) ) ) ).
% measurable_const
thf(fact_577_measurable__const,axiom,
! [C: d,M2: sigma_measure_d,M: sigma_measure_b] :
( ( member_d @ C @ ( sigma_space_d @ M2 ) )
=> ( member_b_d
@ ^ [X3: b] : C
@ ( sigma_measurable_b_d @ M @ M2 ) ) ) ).
% measurable_const
thf(fact_578_sets_Osets__into__space,axiom,
! [X: set_d,M: sigma_measure_d] :
( ( member_set_d @ X @ ( sigma_sets_d @ M ) )
=> ( ord_less_eq_set_d @ X @ ( sigma_space_d @ M ) ) ) ).
% sets.sets_into_space
thf(fact_579_sets_Osets__into__space,axiom,
! [X: set_set_b,M: sigma_measure_set_b] :
( ( member_set_set_b @ X @ ( sigma_sets_set_b @ M ) )
=> ( ord_le3795704787696855135_set_b @ X @ ( sigma_space_set_b @ M ) ) ) ).
% sets.sets_into_space
thf(fact_580_sets_Osets__into__space,axiom,
! [X: set_set_a,M: sigma_measure_set_a] :
( ( member_set_set_a @ X @ ( sigma_sets_set_a @ M ) )
=> ( ord_le3724670747650509150_set_a @ X @ ( sigma_space_set_a @ M ) ) ) ).
% sets.sets_into_space
thf(fact_581_sets_Osets__into__space,axiom,
! [X: set_b,M: sigma_measure_b] :
( ( member_set_b @ X @ ( sigma_sets_b @ M ) )
=> ( ord_less_eq_set_b @ X @ ( sigma_space_b @ M ) ) ) ).
% sets.sets_into_space
thf(fact_582_sets_Osets__into__space,axiom,
! [X: set_a,M: sigma_measure_a] :
( ( member_set_a @ X @ ( sigma_sets_a @ M ) )
=> ( ord_less_eq_set_a @ X @ ( sigma_space_a @ M ) ) ) ).
% sets.sets_into_space
thf(fact_583_sets_Osets__into__space,axiom,
! [X: set_o,M: sigma_measure_o] :
( ( member_set_o @ X @ ( sigma_sets_o @ M ) )
=> ( ord_less_eq_set_o @ X @ ( sigma_space_o @ M ) ) ) ).
% sets.sets_into_space
thf(fact_584_sets_Osets__into__space,axiom,
! [X: set_nat,M: sigma_measure_nat] :
( ( member_set_nat @ X @ ( sigma_sets_nat @ M ) )
=> ( ord_less_eq_set_nat @ X @ ( sigma_space_nat @ M ) ) ) ).
% sets.sets_into_space
thf(fact_585_sets_Osets__into__space,axiom,
! [X: set_c,M: sigma_measure_c] :
( ( member_set_c @ X @ ( sigma_sets_c @ M ) )
=> ( ord_less_eq_set_c @ X @ ( sigma_space_c @ M ) ) ) ).
% sets.sets_into_space
thf(fact_586_measurable__sets__Collect,axiom,
! [F2: c > c,M: sigma_measure_c,N: sigma_measure_c,P: c > $o] :
( ( member_c_c @ F2 @ ( sigma_measurable_c_c @ M @ N ) )
=> ( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_c @ N ) )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_c @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_587_measurable__sets__Collect,axiom,
! [F2: $o > c,M: sigma_measure_o,N: sigma_measure_c,P: c > $o] :
( ( member_o_c @ F2 @ ( sigma_measurable_o_c @ M @ N ) )
=> ( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_c @ N ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_588_measurable__sets__Collect,axiom,
! [F2: c > $o,M: sigma_measure_c,N: sigma_measure_o,P: $o > $o] :
( ( member_c_o @ F2 @ ( sigma_measurable_c_o @ M @ N ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_o @ N ) )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_c @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_589_measurable__sets__Collect,axiom,
! [F2: $o > $o,M: sigma_measure_o,N: sigma_measure_o,P: $o > $o] :
( ( member_o_o @ F2 @ ( sigma_measurable_o_o @ M @ N ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_o @ N ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_590_measurable__sets__Collect,axiom,
! [F2: nat > c,M: sigma_measure_nat,N: sigma_measure_c,P: c > $o] :
( ( member_nat_c @ F2 @ ( sigma_4105081583803843550_nat_c @ M @ N ) )
=> ( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_c @ N ) )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_nat @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_591_measurable__sets__Collect,axiom,
! [F2: nat > $o,M: sigma_measure_nat,N: sigma_measure_o,P: $o > $o] :
( ( member_nat_o @ F2 @ ( sigma_5101835498682829686_nat_o @ M @ N ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_o @ N ) )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_nat @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_592_measurable__sets__Collect,axiom,
! [F2: c > nat,M: sigma_measure_c,N: sigma_measure_nat,P: nat > $o] :
( ( member_c_nat @ F2 @ ( sigma_2544038740538346112_c_nat @ M @ N ) )
=> ( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_nat @ N ) )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_c @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_593_measurable__sets__Collect,axiom,
! [F2: $o > nat,M: sigma_measure_o,N: sigma_measure_nat,P: nat > $o] :
( ( member_o_nat @ F2 @ ( sigma_1999164137574644376_o_nat @ M @ N ) )
=> ( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_nat @ N ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_594_measurable__sets__Collect,axiom,
! [F2: nat > nat,M: sigma_measure_nat,N: sigma_measure_nat,P: nat > $o] :
( ( member_nat_nat @ F2 @ ( sigma_4350458207664084850at_nat @ M @ N ) )
=> ( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_nat @ N ) )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_nat @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_595_measurable__sets__Collect,axiom,
! [F2: b > c,M: sigma_measure_b,N: sigma_measure_c,P: c > $o] :
( ( member_b_c @ F2 @ ( sigma_measurable_b_c @ M @ N ) )
=> ( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ N ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_c @ N ) )
=> ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( P @ ( F2 @ X3 ) ) ) )
@ ( sigma_sets_b @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_596_measurable__If,axiom,
! [F2: a > b,M: sigma_measure_a,M2: sigma_measure_b,G2: a > b,P: a > $o] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M @ M2 ) )
=> ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ M @ M2 ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_a @ M ) )
=> ( member_a_b
@ ^ [X3: a] : ( if_b @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ ( sigma_measurable_a_b @ M @ M2 ) ) ) ) ) ).
% measurable_If
thf(fact_597_measurable__If,axiom,
! [F2: b > d,M: sigma_measure_b,M2: sigma_measure_d,G2: b > d,P: b > $o] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ M @ M2 ) )
=> ( ( member_b_d @ G2 @ ( sigma_measurable_b_d @ M @ M2 ) )
=> ( ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b @ M ) )
=> ( member_b_d
@ ^ [X3: b] : ( if_d @ ( P @ X3 ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ ( sigma_measurable_b_d @ M @ M2 ) ) ) ) ) ).
% measurable_If
thf(fact_598_measurable__If__set,axiom,
! [F2: a > b,M: sigma_measure_a,M2: sigma_measure_b,G2: a > b,A: set_a] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M @ M2 ) )
=> ( ( member_a_b @ G2 @ ( sigma_measurable_a_b @ M @ M2 ) )
=> ( ( member_set_a @ ( inf_inf_set_a @ A @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( member_a_b
@ ^ [X3: a] : ( if_b @ ( member_a @ X3 @ A ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ ( sigma_measurable_a_b @ M @ M2 ) ) ) ) ) ).
% measurable_If_set
thf(fact_599_measurable__If__set,axiom,
! [F2: b > d,M: sigma_measure_b,M2: sigma_measure_d,G2: b > d,A: set_b] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ M @ M2 ) )
=> ( ( member_b_d @ G2 @ ( sigma_measurable_b_d @ M @ M2 ) )
=> ( ( member_set_b @ ( inf_inf_set_b @ A @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) )
=> ( member_b_d
@ ^ [X3: b] : ( if_d @ ( member_b @ X3 @ A ) @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
@ ( sigma_measurable_b_d @ M @ M2 ) ) ) ) ) ).
% measurable_If_set
thf(fact_600_measurable__sets,axiom,
! [F2: b > a,M: sigma_measure_b,A: sigma_measure_a,S: set_a] :
( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ M @ A ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ A ) )
=> ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_a @ F2 @ S ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) ) ) ).
% measurable_sets
thf(fact_601_measurable__sets,axiom,
! [F2: b > b,M: sigma_measure_b,A: sigma_measure_b,S: set_b] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ M @ A ) )
=> ( ( member_set_b @ S @ ( sigma_sets_b @ A ) )
=> ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_b @ F2 @ S ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) ) ) ).
% measurable_sets
thf(fact_602_measurable__sets,axiom,
! [F2: a > d,M: sigma_measure_a,A: sigma_measure_d,S: set_d] :
( ( member_a_d @ F2 @ ( sigma_measurable_a_d @ M @ A ) )
=> ( ( member_set_d @ S @ ( sigma_sets_d @ A ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_d @ F2 @ S ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ).
% measurable_sets
thf(fact_603_measurable__sets,axiom,
! [F2: a > a,M: sigma_measure_a,A: sigma_measure_a,S: set_a] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M @ A ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ A ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F2 @ S ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ).
% measurable_sets
thf(fact_604_measurable__sets,axiom,
! [F2: d > d,M: sigma_measure_d,A: sigma_measure_d,S: set_d] :
( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ M @ A ) )
=> ( ( member_set_d @ S @ ( sigma_sets_d @ A ) )
=> ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_d @ F2 @ S ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) ) ) ).
% measurable_sets
thf(fact_605_measurable__sets,axiom,
! [F2: d > a,M: sigma_measure_d,A: sigma_measure_a,S: set_a] :
( ( member_d_a @ F2 @ ( sigma_measurable_d_a @ M @ A ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ A ) )
=> ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_a @ F2 @ S ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) ) ) ).
% measurable_sets
thf(fact_606_measurable__sets,axiom,
! [F2: d > b,M: sigma_measure_d,A: sigma_measure_b,S: set_b] :
( ( member_d_b @ F2 @ ( sigma_measurable_d_b @ M @ A ) )
=> ( ( member_set_b @ S @ ( sigma_sets_b @ A ) )
=> ( member_set_d @ ( inf_inf_set_d @ ( vimage_d_b @ F2 @ S ) @ ( sigma_space_d @ M ) ) @ ( sigma_sets_d @ M ) ) ) ) ).
% measurable_sets
thf(fact_607_measurable__sets,axiom,
! [F2: a > b,M: sigma_measure_a,A: sigma_measure_b,S: set_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M @ A ) )
=> ( ( member_set_b @ S @ ( sigma_sets_b @ A ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_b @ F2 @ S ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ).
% measurable_sets
thf(fact_608_measurable__sets,axiom,
! [F2: b > d,M: sigma_measure_b,A: sigma_measure_d,S: set_d] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ M @ A ) )
=> ( ( member_set_d @ S @ ( sigma_sets_d @ A ) )
=> ( member_set_b @ ( inf_inf_set_b @ ( vimage_b_d @ F2 @ S ) @ ( sigma_space_b @ M ) ) @ ( sigma_sets_b @ M ) ) ) ) ).
% measurable_sets
thf(fact_609_measurableI,axiom,
! [M: sigma_measure_c,F2: c > c,N: sigma_measure_c] :
( ! [X2: c] :
( ( member_c @ X2 @ ( sigma_space_c @ M ) )
=> ( member_c @ ( F2 @ X2 ) @ ( sigma_space_c @ N ) ) )
=> ( ! [A4: set_c] :
( ( member_set_c @ A4 @ ( sigma_sets_c @ N ) )
=> ( member_set_c @ ( inf_inf_set_c @ ( vimage_c_c @ F2 @ A4 ) @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) ) )
=> ( member_c_c @ F2 @ ( sigma_measurable_c_c @ M @ N ) ) ) ) ).
% measurableI
thf(fact_610_measurableI,axiom,
! [M: sigma_measure_c,F2: c > $o,N: sigma_measure_o] :
( ! [X2: c] :
( ( member_c @ X2 @ ( sigma_space_c @ M ) )
=> ( member_o @ ( F2 @ X2 ) @ ( sigma_space_o @ N ) ) )
=> ( ! [A4: set_o] :
( ( member_set_o @ A4 @ ( sigma_sets_o @ N ) )
=> ( member_set_c @ ( inf_inf_set_c @ ( vimage_c_o @ F2 @ A4 ) @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) ) )
=> ( member_c_o @ F2 @ ( sigma_measurable_c_o @ M @ N ) ) ) ) ).
% measurableI
thf(fact_611_measurableI,axiom,
! [M: sigma_measure_c,F2: c > nat,N: sigma_measure_nat] :
( ! [X2: c] :
( ( member_c @ X2 @ ( sigma_space_c @ M ) )
=> ( member_nat @ ( F2 @ X2 ) @ ( sigma_space_nat @ N ) ) )
=> ( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ ( sigma_sets_nat @ N ) )
=> ( member_set_c @ ( inf_inf_set_c @ ( vimage_c_nat @ F2 @ A4 ) @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) ) )
=> ( member_c_nat @ F2 @ ( sigma_2544038740538346112_c_nat @ M @ N ) ) ) ) ).
% measurableI
thf(fact_612_measurableI,axiom,
! [M: sigma_measure_o,F2: $o > c,N: sigma_measure_c] :
( ! [X2: $o] :
( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( member_c @ ( F2 @ X2 ) @ ( sigma_space_c @ N ) ) )
=> ( ! [A4: set_c] :
( ( member_set_c @ A4 @ ( sigma_sets_c @ N ) )
=> ( member_set_o @ ( inf_inf_set_o @ ( vimage_o_c @ F2 @ A4 ) @ ( sigma_space_o @ M ) ) @ ( sigma_sets_o @ M ) ) )
=> ( member_o_c @ F2 @ ( sigma_measurable_o_c @ M @ N ) ) ) ) ).
% measurableI
thf(fact_613_measurableI,axiom,
! [M: sigma_measure_o,F2: $o > $o,N: sigma_measure_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( member_o @ ( F2 @ X2 ) @ ( sigma_space_o @ N ) ) )
=> ( ! [A4: set_o] :
( ( member_set_o @ A4 @ ( sigma_sets_o @ N ) )
=> ( member_set_o @ ( inf_inf_set_o @ ( vimage_o_o @ F2 @ A4 ) @ ( sigma_space_o @ M ) ) @ ( sigma_sets_o @ M ) ) )
=> ( member_o_o @ F2 @ ( sigma_measurable_o_o @ M @ N ) ) ) ) ).
% measurableI
thf(fact_614_measurableI,axiom,
! [M: sigma_measure_o,F2: $o > nat,N: sigma_measure_nat] :
( ! [X2: $o] :
( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( member_nat @ ( F2 @ X2 ) @ ( sigma_space_nat @ N ) ) )
=> ( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ ( sigma_sets_nat @ N ) )
=> ( member_set_o @ ( inf_inf_set_o @ ( vimage_o_nat @ F2 @ A4 ) @ ( sigma_space_o @ M ) ) @ ( sigma_sets_o @ M ) ) )
=> ( member_o_nat @ F2 @ ( sigma_1999164137574644376_o_nat @ M @ N ) ) ) ) ).
% measurableI
thf(fact_615_measurableI,axiom,
! [M: sigma_measure_nat,F2: nat > c,N: sigma_measure_c] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( sigma_space_nat @ M ) )
=> ( member_c @ ( F2 @ X2 ) @ ( sigma_space_c @ N ) ) )
=> ( ! [A4: set_c] :
( ( member_set_c @ A4 @ ( sigma_sets_c @ N ) )
=> ( member_set_nat @ ( inf_inf_set_nat @ ( vimage_nat_c @ F2 @ A4 ) @ ( sigma_space_nat @ M ) ) @ ( sigma_sets_nat @ M ) ) )
=> ( member_nat_c @ F2 @ ( sigma_4105081583803843550_nat_c @ M @ N ) ) ) ) ).
% measurableI
thf(fact_616_measurableI,axiom,
! [M: sigma_measure_nat,F2: nat > $o,N: sigma_measure_o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( sigma_space_nat @ M ) )
=> ( member_o @ ( F2 @ X2 ) @ ( sigma_space_o @ N ) ) )
=> ( ! [A4: set_o] :
( ( member_set_o @ A4 @ ( sigma_sets_o @ N ) )
=> ( member_set_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F2 @ A4 ) @ ( sigma_space_nat @ M ) ) @ ( sigma_sets_nat @ M ) ) )
=> ( member_nat_o @ F2 @ ( sigma_5101835498682829686_nat_o @ M @ N ) ) ) ) ).
% measurableI
thf(fact_617_measurableI,axiom,
! [M: sigma_measure_nat,F2: nat > nat,N: sigma_measure_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( sigma_space_nat @ M ) )
=> ( member_nat @ ( F2 @ X2 ) @ ( sigma_space_nat @ N ) ) )
=> ( ! [A4: set_nat] :
( ( member_set_nat @ A4 @ ( sigma_sets_nat @ N ) )
=> ( member_set_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F2 @ A4 ) @ ( sigma_space_nat @ M ) ) @ ( sigma_sets_nat @ M ) ) )
=> ( member_nat_nat @ F2 @ ( sigma_4350458207664084850at_nat @ M @ N ) ) ) ) ).
% measurableI
thf(fact_618_measurableI,axiom,
! [M: sigma_measure_c,F2: c > b,N: sigma_measure_b] :
( ! [X2: c] :
( ( member_c @ X2 @ ( sigma_space_c @ M ) )
=> ( member_b @ ( F2 @ X2 ) @ ( sigma_space_b @ N ) ) )
=> ( ! [A4: set_b] :
( ( member_set_b @ A4 @ ( sigma_sets_b @ N ) )
=> ( member_set_c @ ( inf_inf_set_c @ ( vimage_c_b @ F2 @ A4 ) @ ( sigma_space_c @ M ) ) @ ( sigma_sets_c @ M ) ) )
=> ( member_c_b @ F2 @ ( sigma_measurable_c_b @ M @ N ) ) ) ) ).
% measurableI
thf(fact_619_Int__left__commute,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) )
= ( inf_inf_set_b @ B @ ( inf_inf_set_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_620_Int__left__commute,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_621_Int__left__commute,axiom,
! [A: set_d,B: set_d,C2: set_d] :
( ( inf_inf_set_d @ A @ ( inf_inf_set_d @ B @ C2 ) )
= ( inf_inf_set_d @ B @ ( inf_inf_set_d @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_622_Int__left__absorb,axiom,
! [A: set_b,B: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ A @ B ) )
= ( inf_inf_set_b @ A @ B ) ) ).
% Int_left_absorb
thf(fact_623_Int__left__absorb,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% Int_left_absorb
thf(fact_624_Int__left__absorb,axiom,
! [A: set_d,B: set_d] :
( ( inf_inf_set_d @ A @ ( inf_inf_set_d @ A @ B ) )
= ( inf_inf_set_d @ A @ B ) ) ).
% Int_left_absorb
thf(fact_625_Int__commute,axiom,
( inf_inf_set_b
= ( ^ [A3: set_b,B3: set_b] : ( inf_inf_set_b @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_626_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_627_Int__commute,axiom,
( inf_inf_set_d
= ( ^ [A3: set_d,B3: set_d] : ( inf_inf_set_d @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_628_Int__absorb,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_629_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_630_Int__absorb,axiom,
! [A: set_d] :
( ( inf_inf_set_d @ A @ A )
= A ) ).
% Int_absorb
thf(fact_631_Int__assoc,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B ) @ C2 )
= ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_632_Int__assoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_633_Int__assoc,axiom,
! [A: set_d,B: set_d,C2: set_d] :
( ( inf_inf_set_d @ ( inf_inf_set_d @ A @ B ) @ C2 )
= ( inf_inf_set_d @ A @ ( inf_inf_set_d @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_634_IntD2,axiom,
! [C: c,A: set_c,B: set_c] :
( ( member_c @ C @ ( inf_inf_set_c @ A @ B ) )
=> ( member_c @ C @ B ) ) ).
% IntD2
thf(fact_635_IntD2,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C @ B ) ) ).
% IntD2
thf(fact_636_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_637_IntD2,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ( member_b @ C @ B ) ) ).
% IntD2
thf(fact_638_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_639_IntD2,axiom,
! [C: d,A: set_d,B: set_d] :
( ( member_d @ C @ ( inf_inf_set_d @ A @ B ) )
=> ( member_d @ C @ B ) ) ).
% IntD2
thf(fact_640_IntD2,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ B ) ) ).
% IntD2
thf(fact_641_IntD2,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
=> ( member_set_b @ C @ B ) ) ).
% IntD2
thf(fact_642_IntD2,axiom,
! [C: set_d,A: set_set_d,B: set_set_d] :
( ( member_set_d @ C @ ( inf_inf_set_set_d @ A @ B ) )
=> ( member_set_d @ C @ B ) ) ).
% IntD2
thf(fact_643_IntD2,axiom,
! [C: b > d,A: set_b_d,B: set_b_d] :
( ( member_b_d @ C @ ( inf_inf_set_b_d @ A @ B ) )
=> ( member_b_d @ C @ B ) ) ).
% IntD2
thf(fact_644_IntD1,axiom,
! [C: c,A: set_c,B: set_c] :
( ( member_c @ C @ ( inf_inf_set_c @ A @ B ) )
=> ( member_c @ C @ A ) ) ).
% IntD1
thf(fact_645_IntD1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C @ A ) ) ).
% IntD1
thf(fact_646_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_647_IntD1,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ( member_b @ C @ A ) ) ).
% IntD1
thf(fact_648_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_649_IntD1,axiom,
! [C: d,A: set_d,B: set_d] :
( ( member_d @ C @ ( inf_inf_set_d @ A @ B ) )
=> ( member_d @ C @ A ) ) ).
% IntD1
thf(fact_650_IntD1,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_651_IntD1,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
=> ( member_set_b @ C @ A ) ) ).
% IntD1
thf(fact_652_IntD1,axiom,
! [C: set_d,A: set_set_d,B: set_set_d] :
( ( member_set_d @ C @ ( inf_inf_set_set_d @ A @ B ) )
=> ( member_set_d @ C @ A ) ) ).
% IntD1
thf(fact_653_IntD1,axiom,
! [C: b > d,A: set_b_d,B: set_b_d] :
( ( member_b_d @ C @ ( inf_inf_set_b_d @ A @ B ) )
=> ( member_b_d @ C @ A ) ) ).
% IntD1
thf(fact_654_IntE,axiom,
! [C: c,A: set_c,B: set_c] :
( ( member_c @ C @ ( inf_inf_set_c @ A @ B ) )
=> ~ ( ( member_c @ C @ A )
=> ~ ( member_c @ C @ B ) ) ) ).
% IntE
thf(fact_655_IntE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ~ ( ( member_o @ C @ A )
=> ~ ( member_o @ C @ B ) ) ) ).
% IntE
thf(fact_656_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_657_IntE,axiom,
! [C: b,A: set_b,B: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B ) )
=> ~ ( ( member_b @ C @ A )
=> ~ ( member_b @ C @ B ) ) ) ).
% IntE
thf(fact_658_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_659_IntE,axiom,
! [C: d,A: set_d,B: set_d] :
( ( member_d @ C @ ( inf_inf_set_d @ A @ B ) )
=> ~ ( ( member_d @ C @ A )
=> ~ ( member_d @ C @ B ) ) ) ).
% IntE
thf(fact_660_IntE,axiom,
! [C: set_a,A: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B ) ) ) ).
% IntE
thf(fact_661_IntE,axiom,
! [C: set_b,A: set_set_b,B: set_set_b] :
( ( member_set_b @ C @ ( inf_inf_set_set_b @ A @ B ) )
=> ~ ( ( member_set_b @ C @ A )
=> ~ ( member_set_b @ C @ B ) ) ) ).
% IntE
thf(fact_662_IntE,axiom,
! [C: set_d,A: set_set_d,B: set_set_d] :
( ( member_set_d @ C @ ( inf_inf_set_set_d @ A @ B ) )
=> ~ ( ( member_set_d @ C @ A )
=> ~ ( member_set_d @ C @ B ) ) ) ).
% IntE
thf(fact_663_IntE,axiom,
! [C: b > d,A: set_b_d,B: set_b_d] :
( ( member_b_d @ C @ ( inf_inf_set_b_d @ A @ B ) )
=> ~ ( ( member_b_d @ C @ A )
=> ~ ( member_b_d @ C @ B ) ) ) ).
% IntE
thf(fact_664_vimage__Collect,axiom,
! [P: d > $o,F2: b > d,Q: b > $o] :
( ! [X2: b] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_b_d @ F2 @ ( collect_d @ P ) )
= ( collect_b @ Q ) ) ) ).
% vimage_Collect
thf(fact_665_vimage__Collect,axiom,
! [P: a > $o,F2: a > a,Q: a > $o] :
( ! [X2: a] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_a_a @ F2 @ ( collect_a @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_666_vimage__Collect,axiom,
! [P: a > $o,F2: b > a,Q: b > $o] :
( ! [X2: b] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_b_a @ F2 @ ( collect_a @ P ) )
= ( collect_b @ Q ) ) ) ).
% vimage_Collect
thf(fact_667_vimage__Collect,axiom,
! [P: a > $o,F2: nat > a,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_nat_a @ F2 @ ( collect_a @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_668_vimage__Collect,axiom,
! [P: b > $o,F2: a > b,Q: a > $o] :
( ! [X2: a] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_a_b @ F2 @ ( collect_b @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_669_vimage__Collect,axiom,
! [P: b > $o,F2: b > b,Q: b > $o] :
( ! [X2: b] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_b_b @ F2 @ ( collect_b @ P ) )
= ( collect_b @ Q ) ) ) ).
% vimage_Collect
thf(fact_670_vimage__Collect,axiom,
! [P: b > $o,F2: nat > b,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_nat_b @ F2 @ ( collect_b @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_671_vimage__Collect,axiom,
! [P: nat > $o,F2: a > nat,Q: a > $o] :
( ! [X2: a] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_a_nat @ F2 @ ( collect_nat @ P ) )
= ( collect_a @ Q ) ) ) ).
% vimage_Collect
thf(fact_672_vimage__Collect,axiom,
! [P: nat > $o,F2: b > nat,Q: b > $o] :
( ! [X2: b] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_b_nat @ F2 @ ( collect_nat @ P ) )
= ( collect_b @ Q ) ) ) ).
% vimage_Collect
thf(fact_673_vimage__Collect,axiom,
! [P: nat > $o,F2: nat > nat,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ ( F2 @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage_nat_nat @ F2 @ ( collect_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_674_vimageI2,axiom,
! [F2: c > c,A2: c,A: set_c] :
( ( member_c @ ( F2 @ A2 ) @ A )
=> ( member_c @ A2 @ ( vimage_c_c @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_675_vimageI2,axiom,
! [F2: b > c,A2: b,A: set_c] :
( ( member_c @ ( F2 @ A2 ) @ A )
=> ( member_b @ A2 @ ( vimage_b_c @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_676_vimageI2,axiom,
! [F2: a > c,A2: a,A: set_c] :
( ( member_c @ ( F2 @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_c @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_677_vimageI2,axiom,
! [F2: $o > c,A2: $o,A: set_c] :
( ( member_c @ ( F2 @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage_o_c @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_678_vimageI2,axiom,
! [F2: nat > c,A2: nat,A: set_c] :
( ( member_c @ ( F2 @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_c @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_679_vimageI2,axiom,
! [F2: c > b,A2: c,A: set_b] :
( ( member_b @ ( F2 @ A2 ) @ A )
=> ( member_c @ A2 @ ( vimage_c_b @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_680_vimageI2,axiom,
! [F2: b > b,A2: b,A: set_b] :
( ( member_b @ ( F2 @ A2 ) @ A )
=> ( member_b @ A2 @ ( vimage_b_b @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_681_vimageI2,axiom,
! [F2: a > b,A2: a,A: set_b] :
( ( member_b @ ( F2 @ A2 ) @ A )
=> ( member_a @ A2 @ ( vimage_a_b @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_682_vimageI2,axiom,
! [F2: $o > b,A2: $o,A: set_b] :
( ( member_b @ ( F2 @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage_o_b @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_683_vimageI2,axiom,
! [F2: nat > b,A2: nat,A: set_b] :
( ( member_b @ ( F2 @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_b @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_684_vimageE,axiom,
! [A2: c,F2: c > c,B: set_c] :
( ( member_c @ A2 @ ( vimage_c_c @ F2 @ B ) )
=> ( member_c @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_685_vimageE,axiom,
! [A2: c,F2: c > b,B: set_b] :
( ( member_c @ A2 @ ( vimage_c_b @ F2 @ B ) )
=> ( member_b @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_686_vimageE,axiom,
! [A2: c,F2: c > a,B: set_a] :
( ( member_c @ A2 @ ( vimage_c_a @ F2 @ B ) )
=> ( member_a @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_687_vimageE,axiom,
! [A2: c,F2: c > $o,B: set_o] :
( ( member_c @ A2 @ ( vimage_c_o @ F2 @ B ) )
=> ( member_o @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_688_vimageE,axiom,
! [A2: c,F2: c > nat,B: set_nat] :
( ( member_c @ A2 @ ( vimage_c_nat @ F2 @ B ) )
=> ( member_nat @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_689_vimageE,axiom,
! [A2: b,F2: b > c,B: set_c] :
( ( member_b @ A2 @ ( vimage_b_c @ F2 @ B ) )
=> ( member_c @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_690_vimageE,axiom,
! [A2: b,F2: b > b,B: set_b] :
( ( member_b @ A2 @ ( vimage_b_b @ F2 @ B ) )
=> ( member_b @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_691_vimageE,axiom,
! [A2: b,F2: b > a,B: set_a] :
( ( member_b @ A2 @ ( vimage_b_a @ F2 @ B ) )
=> ( member_a @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_692_vimageE,axiom,
! [A2: b,F2: b > $o,B: set_o] :
( ( member_b @ A2 @ ( vimage_b_o @ F2 @ B ) )
=> ( member_o @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_693_vimageE,axiom,
! [A2: b,F2: b > nat,B: set_nat] :
( ( member_b @ A2 @ ( vimage_b_nat @ F2 @ B ) )
=> ( member_nat @ ( F2 @ A2 ) @ B ) ) ).
% vimageE
thf(fact_694_vimageD,axiom,
! [A2: c,F2: c > c,A: set_c] :
( ( member_c @ A2 @ ( vimage_c_c @ F2 @ A ) )
=> ( member_c @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_695_vimageD,axiom,
! [A2: c,F2: c > b,A: set_b] :
( ( member_c @ A2 @ ( vimage_c_b @ F2 @ A ) )
=> ( member_b @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_696_vimageD,axiom,
! [A2: c,F2: c > a,A: set_a] :
( ( member_c @ A2 @ ( vimage_c_a @ F2 @ A ) )
=> ( member_a @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_697_vimageD,axiom,
! [A2: c,F2: c > $o,A: set_o] :
( ( member_c @ A2 @ ( vimage_c_o @ F2 @ A ) )
=> ( member_o @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_698_vimageD,axiom,
! [A2: c,F2: c > nat,A: set_nat] :
( ( member_c @ A2 @ ( vimage_c_nat @ F2 @ A ) )
=> ( member_nat @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_699_vimageD,axiom,
! [A2: b,F2: b > c,A: set_c] :
( ( member_b @ A2 @ ( vimage_b_c @ F2 @ A ) )
=> ( member_c @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_700_vimageD,axiom,
! [A2: b,F2: b > b,A: set_b] :
( ( member_b @ A2 @ ( vimage_b_b @ F2 @ A ) )
=> ( member_b @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_701_vimageD,axiom,
! [A2: b,F2: b > a,A: set_a] :
( ( member_b @ A2 @ ( vimage_b_a @ F2 @ A ) )
=> ( member_a @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_702_vimageD,axiom,
! [A2: b,F2: b > $o,A: set_o] :
( ( member_b @ A2 @ ( vimage_b_o @ F2 @ A ) )
=> ( member_o @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_703_vimageD,axiom,
! [A2: b,F2: b > nat,A: set_nat] :
( ( member_b @ A2 @ ( vimage_b_nat @ F2 @ A ) )
=> ( member_nat @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_704_Collect__conj__eq,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ( collect_set_b
@ ^ [X3: set_b] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_705_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_706_Collect__conj__eq,axiom,
! [P: b > $o,Q: b > $o] :
( ( collect_b
@ ^ [X3: b] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_707_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_708_Collect__conj__eq,axiom,
! [P: d > $o,Q: d > $o] :
( ( collect_d
@ ^ [X3: d] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_d @ ( collect_d @ P ) @ ( collect_d @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_709_Int__Collect,axiom,
! [X: c,A: set_c,P: c > $o] :
( ( member_c @ X @ ( inf_inf_set_c @ A @ ( collect_c @ P ) ) )
= ( ( member_c @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_710_Int__Collect,axiom,
! [X: $o,A: set_o,P: $o > $o] :
( ( member_o @ X @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) )
= ( ( member_o @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_711_Int__Collect,axiom,
! [X: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_712_Int__Collect,axiom,
! [X: b,A: set_b,P: b > $o] :
( ( member_b @ X @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) )
= ( ( member_b @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_713_Int__Collect,axiom,
! [X: a,A: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_714_Int__Collect,axiom,
! [X: d,A: set_d,P: d > $o] :
( ( member_d @ X @ ( inf_inf_set_d @ A @ ( collect_d @ P ) ) )
= ( ( member_d @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_715_Int__Collect,axiom,
! [X: set_a,A: set_set_a,P: set_a > $o] :
( ( member_set_a @ X @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_716_Int__Collect,axiom,
! [X: set_d,A: set_set_d,P: set_d > $o] :
( ( member_set_d @ X @ ( inf_inf_set_set_d @ A @ ( collect_set_d @ P ) ) )
= ( ( member_set_d @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_717_Int__Collect,axiom,
! [X: set_b,A: set_set_b,P: set_b > $o] :
( ( member_set_b @ X @ ( inf_inf_set_set_b @ A @ ( collect_set_b @ P ) ) )
= ( ( member_set_b @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_718_Int__Collect,axiom,
! [X: b > d,A: set_b_d,P: ( b > d ) > $o] :
( ( member_b_d @ X @ ( inf_inf_set_b_d @ A @ ( collect_b_d @ P ) ) )
= ( ( member_b_d @ X @ A )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_719_Int__def,axiom,
( inf_inf_set_c
= ( ^ [A3: set_c,B3: set_c] :
( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ A3 )
& ( member_c @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_720_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A3: set_o,B3: set_o] :
( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A3 )
& ( member_o @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_721_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( member_nat @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_722_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A3: set_b,B3: set_b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A3 )
& ( member_b @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_723_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A3 )
& ( member_a @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_724_Int__def,axiom,
( inf_inf_set_d
= ( ^ [A3: set_d,B3: set_d] :
( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ A3 )
& ( member_d @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_725_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( member_set_a @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_726_Int__def,axiom,
( inf_inf_set_set_d
= ( ^ [A3: set_set_d,B3: set_set_d] :
( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ A3 )
& ( member_set_d @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_727_Int__def,axiom,
( inf_inf_set_set_b
= ( ^ [A3: set_set_b,B3: set_set_b] :
( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ A3 )
& ( member_set_b @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_728_Int__def,axiom,
( inf_inf_set_b_d
= ( ^ [A3: set_b_d,B3: set_b_d] :
( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ A3 )
& ( member_b_d @ X3 @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_729_vimage__def,axiom,
( vimage_a_c
= ( ^ [F3: a > c,B3: set_c] :
( collect_a
@ ^ [X3: a] : ( member_c @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_730_vimage__def,axiom,
( vimage_a_b
= ( ^ [F3: a > b,B3: set_b] :
( collect_a
@ ^ [X3: a] : ( member_b @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_731_vimage__def,axiom,
( vimage_a_a
= ( ^ [F3: a > a,B3: set_a] :
( collect_a
@ ^ [X3: a] : ( member_a @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_732_vimage__def,axiom,
( vimage_a_o
= ( ^ [F3: a > $o,B3: set_o] :
( collect_a
@ ^ [X3: a] : ( member_o @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_733_vimage__def,axiom,
( vimage_a_nat
= ( ^ [F3: a > nat,B3: set_nat] :
( collect_a
@ ^ [X3: a] : ( member_nat @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_734_vimage__def,axiom,
( vimage_b_c
= ( ^ [F3: b > c,B3: set_c] :
( collect_b
@ ^ [X3: b] : ( member_c @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_735_vimage__def,axiom,
( vimage_b_b
= ( ^ [F3: b > b,B3: set_b] :
( collect_b
@ ^ [X3: b] : ( member_b @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_736_vimage__def,axiom,
( vimage_b_a
= ( ^ [F3: b > a,B3: set_a] :
( collect_b
@ ^ [X3: b] : ( member_a @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_737_vimage__def,axiom,
( vimage_b_o
= ( ^ [F3: b > $o,B3: set_o] :
( collect_b
@ ^ [X3: b] : ( member_o @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_738_vimage__def,axiom,
( vimage_b_nat
= ( ^ [F3: b > nat,B3: set_nat] :
( collect_b
@ ^ [X3: b] : ( member_nat @ ( F3 @ X3 ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_739_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_b,M2: sigma_measure_b] :
( ( ( sigma_sets_b @ M )
= ( sigma_sets_b @ M2 ) )
=> ( ( sigma_space_b @ M )
= ( sigma_space_b @ M2 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_740_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( ( sigma_space_a @ M )
= ( sigma_space_a @ M2 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_741_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_d,M2: sigma_measure_d] :
( ( ( sigma_sets_d @ M )
= ( sigma_sets_d @ M2 ) )
=> ( ( sigma_space_d @ M )
= ( sigma_space_d @ M2 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_742_vimage__inter__cong,axiom,
! [S: set_b,F2: b > d,G2: b > d,Y4: set_d] :
( ! [W: b] :
( ( member_b @ W @ S )
=> ( ( F2 @ W )
= ( G2 @ W ) ) )
=> ( ( inf_inf_set_b @ ( vimage_b_d @ F2 @ Y4 ) @ S )
= ( inf_inf_set_b @ ( vimage_b_d @ G2 @ Y4 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_743_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_c,P: $o] :
( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& P ) )
@ ( sigma_sets_c @ M ) ) ).
% sets.sets_Collect_const
thf(fact_744_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_o,P: $o] :
( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& P ) )
@ ( sigma_sets_o @ M ) ) ).
% sets.sets_Collect_const
thf(fact_745_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_nat,P: $o] :
( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& P ) )
@ ( sigma_sets_nat @ M ) ) ).
% sets.sets_Collect_const
thf(fact_746_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_b,P: $o] :
( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& P ) )
@ ( sigma_sets_b @ M ) ) ).
% sets.sets_Collect_const
thf(fact_747_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_a,P: $o] :
( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& P ) )
@ ( sigma_sets_a @ M ) ) ).
% sets.sets_Collect_const
thf(fact_748_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_d,P: $o] :
( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& P ) )
@ ( sigma_sets_d @ M ) ) ).
% sets.sets_Collect_const
thf(fact_749_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_set_a,P: $o] :
( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& P ) )
@ ( sigma_sets_set_a @ M ) ) ).
% sets.sets_Collect_const
thf(fact_750_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_set_d,P: $o] :
( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& P ) )
@ ( sigma_sets_set_d @ M ) ) ).
% sets.sets_Collect_const
thf(fact_751_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_set_b,P: $o] :
( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& P ) )
@ ( sigma_sets_set_b @ M ) ) ).
% sets.sets_Collect_const
thf(fact_752_sets_Osets__Collect__const,axiom,
! [M: sigma_measure_b_d,P: $o] :
( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& P ) )
@ ( sigma_sets_b_d @ M ) ) ).
% sets.sets_Collect_const
thf(fact_753_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_c,P: c > $o,Q: c > $o] :
( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_c @ M ) )
=> ( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_c @ M ) )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_c @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_754_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_755_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_nat @ M ) )
=> ( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_nat @ M ) )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_nat @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_756_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
( ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b @ M ) )
=> ( ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_b @ M ) )
=> ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_b @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_757_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_a,P: a > $o,Q: a > $o] :
( ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_a @ M ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_a @ M ) )
=> ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_a @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_758_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_d,P: d > $o,Q: d > $o] :
( ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_d @ M ) )
=> ( ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_d @ M ) )
=> ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_d @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_759_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_set_a,P: set_a > $o,Q: set_a > $o] :
( ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_set_a @ M ) )
=> ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_set_a @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_760_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_set_d,P: set_d > $o,Q: set_d > $o] :
( ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_d @ M ) )
=> ( ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_set_d @ M ) )
=> ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_set_d @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_761_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_set_b,P: set_b > $o,Q: set_b > $o] :
( ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_b @ M ) )
=> ( ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_set_b @ M ) )
=> ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_set_b @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_762_sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
( ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b_d @ M ) )
=> ( ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_b_d @ M ) )
=> ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( ( Q @ X3 )
| ( P @ X3 ) ) ) )
@ ( sigma_sets_b_d @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_763_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_c,P: c > $o,Q: c > $o] :
( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_c @ M ) )
=> ( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_c @ M ) )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_c @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_764_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_765_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_nat @ M ) )
=> ( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_nat @ M ) )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_nat @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_766_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
( ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b @ M ) )
=> ( ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_b @ M ) )
=> ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_b @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_767_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_a,P: a > $o,Q: a > $o] :
( ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_a @ M ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_a @ M ) )
=> ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_a @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_768_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_d,P: d > $o,Q: d > $o] :
( ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_d @ M ) )
=> ( ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_d @ M ) )
=> ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_d @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_769_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_set_a,P: set_a > $o,Q: set_a > $o] :
( ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_set_a @ M ) )
=> ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_a @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_770_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_set_d,P: set_d > $o,Q: set_d > $o] :
( ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_d @ M ) )
=> ( ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_set_d @ M ) )
=> ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_d @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_771_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_set_b,P: set_b > $o,Q: set_b > $o] :
( ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_b @ M ) )
=> ( ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_set_b @ M ) )
=> ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_b @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_772_sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
( ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b_d @ M ) )
=> ( ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_b_d @ M ) )
=> ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( Q @ X3 )
& ( P @ X3 ) ) )
@ ( sigma_sets_b_d @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_773_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_c,P: c > $o] :
( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_c @ M ) )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_c @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_774_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_o,P: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_o @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_775_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_nat,P: nat > $o] :
( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_nat @ M ) )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_nat @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_776_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_b,P: b > $o] :
( ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b @ M ) )
=> ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_b @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_777_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_a,P: a > $o] :
( ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_a @ M ) )
=> ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_a @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_778_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_d,P: d > $o] :
( ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_d @ M ) )
=> ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_d @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_779_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_set_a,P: set_a > $o] :
( ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_a @ M ) )
=> ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_set_a @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_780_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_set_d,P: set_d > $o] :
( ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_d @ M ) )
=> ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_set_d @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_781_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_set_b,P: set_b > $o] :
( ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_b @ M ) )
=> ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_set_b @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_782_sets_Osets__Collect__neg,axiom,
! [M: sigma_measure_b_d,P: ( b > d ) > $o] :
( ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b_d @ M ) )
=> ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ~ ( P @ X3 ) ) )
@ ( sigma_sets_b_d @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_783_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_c,P: c > $o,Q: c > $o] :
( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_c @ M ) )
=> ( ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_c @ M ) )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_c @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_784_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_o @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_785_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_nat @ M ) )
=> ( ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_nat @ M ) )
=> ( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_nat @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_786_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
( ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b @ M ) )
=> ( ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_b @ M ) )
=> ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_b @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_787_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_a,P: a > $o,Q: a > $o] :
( ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_a @ M ) )
=> ( ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_a @ M ) )
=> ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_a @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_788_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_d,P: d > $o,Q: d > $o] :
( ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_d @ M ) )
=> ( ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_d @ M ) )
=> ( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_d @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_789_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_set_a,P: set_a > $o,Q: set_a > $o] :
( ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_a @ M ) )
=> ( ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_set_a @ M ) )
=> ( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_set_a @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_790_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_set_d,P: set_d > $o,Q: set_d > $o] :
( ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_d @ M ) )
=> ( ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_set_d @ M ) )
=> ( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_set_d @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_791_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_set_b,P: set_b > $o,Q: set_b > $o] :
( ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_set_b @ M ) )
=> ( ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_set_b @ M ) )
=> ( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_set_b @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_792_sets_Osets__Collect__imp,axiom,
! [M: sigma_measure_b_d,P: ( b > d ) > $o,Q: ( b > d ) > $o] :
( ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b_d @ M ) )
=> ( ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( Q @ X3 ) ) )
@ ( sigma_sets_b_d @ M ) )
=> ( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& ( ( Q @ X3 )
=> ( P @ X3 ) ) ) )
@ ( sigma_sets_b_d @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_793_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_c,Pb: $o] :
( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ M ) )
& Pb ) )
@ ( sigma_sets_c @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_794_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_o,Pb: $o] :
( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ M ) )
& Pb ) )
@ ( sigma_sets_o @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_795_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_nat,Pb: $o] :
( member_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( sigma_space_nat @ M ) )
& Pb ) )
@ ( sigma_sets_nat @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_796_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_b,Pb: $o] :
( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ M ) )
& Pb ) )
@ ( sigma_sets_b @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_797_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_a,Pb: $o] :
( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ M ) )
& Pb ) )
@ ( sigma_sets_a @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_798_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_d,Pb: $o] :
( member_set_d
@ ( collect_d
@ ^ [X3: d] :
( ( member_d @ X3 @ ( sigma_space_d @ M ) )
& Pb ) )
@ ( sigma_sets_d @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_799_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_set_a,Pb: $o] :
( member_set_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_space_set_a @ M ) )
& Pb ) )
@ ( sigma_sets_set_a @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_800_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_set_d,Pb: $o] :
( member_set_set_d
@ ( collect_set_d
@ ^ [X3: set_d] :
( ( member_set_d @ X3 @ ( sigma_space_set_d @ M ) )
& Pb ) )
@ ( sigma_sets_set_d @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_801_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_set_b,Pb: $o] :
( member_set_set_b
@ ( collect_set_b
@ ^ [X3: set_b] :
( ( member_set_b @ X3 @ ( sigma_space_set_b @ M ) )
& Pb ) )
@ ( sigma_sets_set_b @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_802_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_measure_b_d,Pb: $o] :
( member_set_b_d
@ ( collect_b_d
@ ^ [X3: b > d] :
( ( member_b_d @ X3 @ ( sigma_space_b_d @ M ) )
& Pb ) )
@ ( sigma_sets_b_d @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_803_distr__id2,axiom,
! [M: sigma_measure_d,N: sigma_measure_d] :
( ( ( sigma_sets_d @ M )
= ( sigma_sets_d @ N ) )
=> ( ( measure_distr_d_d @ N @ M
@ ^ [X3: d] : X3 )
= N ) ) ).
% distr_id2
thf(fact_804_distr__id2,axiom,
! [M: sigma_measure_a,N: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N ) )
=> ( ( measure_distr_a_a @ N @ M
@ ^ [X3: a] : X3 )
= N ) ) ).
% distr_id2
thf(fact_805_distr__id2,axiom,
! [M: sigma_measure_b,N: sigma_measure_b] :
( ( ( sigma_sets_b @ M )
= ( sigma_sets_b @ N ) )
=> ( ( measure_distr_b_b @ N @ M
@ ^ [X3: b] : X3 )
= N ) ) ).
% distr_id2
thf(fact_806_distr__cong,axiom,
! [M: sigma_measure_c,K: sigma_measure_c,N: sigma_measure_d,L: sigma_measure_d,F2: c > d,G2: c > d] :
( ( M = K )
=> ( ( ( sigma_sets_d @ N )
= ( sigma_sets_d @ L ) )
=> ( ! [X2: c] :
( ( member_c @ X2 @ ( sigma_space_c @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_c_d @ M @ N @ F2 )
= ( measure_distr_c_d @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_807_distr__cong,axiom,
! [M: sigma_measure_o,K: sigma_measure_o,N: sigma_measure_d,L: sigma_measure_d,F2: $o > d,G2: $o > d] :
( ( M = K )
=> ( ( ( sigma_sets_d @ N )
= ( sigma_sets_d @ L ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_o_d @ M @ N @ F2 )
= ( measure_distr_o_d @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_808_distr__cong,axiom,
! [M: sigma_measure_nat,K: sigma_measure_nat,N: sigma_measure_d,L: sigma_measure_d,F2: nat > d,G2: nat > d] :
( ( M = K )
=> ( ( ( sigma_sets_d @ N )
= ( sigma_sets_d @ L ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sigma_space_nat @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_nat_d @ M @ N @ F2 )
= ( measure_distr_nat_d @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_809_distr__cong,axiom,
! [M: sigma_measure_c,K: sigma_measure_c,N: sigma_measure_a,L: sigma_measure_a,F2: c > a,G2: c > a] :
( ( M = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X2: c] :
( ( member_c @ X2 @ ( sigma_space_c @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_c_a @ M @ N @ F2 )
= ( measure_distr_c_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_810_distr__cong,axiom,
! [M: sigma_measure_o,K: sigma_measure_o,N: sigma_measure_a,L: sigma_measure_a,F2: $o > a,G2: $o > a] :
( ( M = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_o_a @ M @ N @ F2 )
= ( measure_distr_o_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_811_distr__cong,axiom,
! [M: sigma_measure_nat,K: sigma_measure_nat,N: sigma_measure_a,L: sigma_measure_a,F2: nat > a,G2: nat > a] :
( ( M = K )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sigma_space_nat @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_nat_a @ M @ N @ F2 )
= ( measure_distr_nat_a @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_812_distr__cong,axiom,
! [M: sigma_measure_c,K: sigma_measure_c,N: sigma_measure_b,L: sigma_measure_b,F2: c > b,G2: c > b] :
( ( M = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X2: c] :
( ( member_c @ X2 @ ( sigma_space_c @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_c_b @ M @ N @ F2 )
= ( measure_distr_c_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_813_distr__cong,axiom,
! [M: sigma_measure_o,K: sigma_measure_o,N: sigma_measure_b,L: sigma_measure_b,F2: $o > b,G2: $o > b] :
( ( M = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ ( sigma_space_o @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_o_b @ M @ N @ F2 )
= ( measure_distr_o_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_814_distr__cong,axiom,
! [M: sigma_measure_nat,K: sigma_measure_nat,N: sigma_measure_b,L: sigma_measure_b,F2: nat > b,G2: nat > b] :
( ( M = K )
=> ( ( ( sigma_sets_b @ N )
= ( sigma_sets_b @ L ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sigma_space_nat @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_nat_b @ M @ N @ F2 )
= ( measure_distr_nat_b @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_815_distr__cong,axiom,
! [M: sigma_measure_b,K: sigma_measure_b,N: sigma_measure_d,L: sigma_measure_d,F2: b > d,G2: b > d] :
( ( M = K )
=> ( ( ( sigma_sets_d @ N )
= ( sigma_sets_d @ L ) )
=> ( ! [X2: b] :
( ( member_b @ X2 @ ( sigma_space_b @ M ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( measure_distr_b_d @ M @ N @ F2 )
= ( measure_distr_b_d @ K @ L @ G2 ) ) ) ) ) ).
% distr_cong
thf(fact_816_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: c > sigma_measure_a,X4: c > b > a,I: set_c,Y5: c > a > b,N: c > sigma_measure_b] :
( ( prob_k159476456662262088_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k159476456662262089_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: c,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_817_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: b > sigma_measure_a,X4: b > b > a,I: set_b,Y5: b > a > b,N: b > sigma_measure_b] :
( ( prob_k2946932038634938121_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k2946932038634938122_b_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: b,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_818_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: a > sigma_measure_a,X4: a > b > a,I: set_a,Y5: a > a > b,N: a > sigma_measure_b] :
( ( prob_k5734387620607614154_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k5734387620607614155_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: a,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_819_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: $o > sigma_measure_a,X4: $o > b > a,I: set_o,Y5: $o > a > b,N: $o > sigma_measure_b] :
( ( prob_k64534356759028784_b_o_a @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k64534356759028785_b_o_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: $o,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_820_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: nat > sigma_measure_a,X4: nat > b > a,I: set_nat,Y5: nat > a > b,N: nat > sigma_measure_b] :
( ( prob_k7670433671732998242_nat_a @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k7670433671732998243_nat_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: nat,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_821_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: c > sigma_measure_b,X4: c > b > b,I: set_c,Y5: c > b > d,N: c > sigma_measure_d] :
( ( prob_k159476456662262089_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k159476456662262091_b_c_d @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: c,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_822_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: b > sigma_measure_b,X4: b > b > b,I: set_b,Y5: b > b > d,N: b > sigma_measure_d] :
( ( prob_k2946932038634938122_b_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k2946932038634938124_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: b,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_823_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: a > sigma_measure_b,X4: a > b > b,I: set_a,Y5: a > b > d,N: a > sigma_measure_d] :
( ( prob_k5734387620607614155_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k5734387620607614157_b_a_d @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: a,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_824_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: $o > sigma_measure_b,X4: $o > b > b,I: set_o,Y5: $o > b > d,N: $o > sigma_measure_d] :
( ( prob_k64534356759028785_b_o_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k64534356759028787_b_o_d @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: $o,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_825_D_Ok__wise__indep__vars__compose,axiom,
! [K2: nat,M2: nat > sigma_measure_b,X4: nat > b > b,I: set_nat,Y5: nat > b > d,N: nat > sigma_measure_d] :
( ( prob_k7670433671732998243_nat_b @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k7670433671732998245_nat_d @ ( measure_distr_a_b @ m @ n @ f ) @ K2 @ N
@ ^ [I3: nat,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.k_wise_indep_vars_compose
thf(fact_826_D_Oindep__setD__ev2,axiom,
! [A: set_set_b,B: set_set_b] :
( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
=> ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) ) ).
% D.indep_setD_ev2
thf(fact_827_D_Oindep__setD__ev1,axiom,
! [A: set_set_b,B: set_set_b] :
( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
=> ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) ) ).
% D.indep_setD_ev1
thf(fact_828_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: c > sigma_measure_a,X4: c > a > a,I: set_c,Y5: c > a > b,N: c > sigma_measure_b] :
( ( prob_k3786629878328907207_a_c_a @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k3786629878328907208_a_c_b @ m @ K2 @ N
@ ^ [I3: c,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_829_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: b > sigma_measure_a,X4: b > a > a,I: set_b,Y5: b > a > b,N: b > sigma_measure_b] :
( ( prob_k6574085460301583240_a_b_a @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k6574085460301583241_a_b_b @ m @ K2 @ N
@ ^ [I3: b,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_830_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: a > sigma_measure_a,X4: a > a > a,I: set_a,Y5: a > a > b,N: a > sigma_measure_b] :
( ( prob_k138169005419483465_a_a_a @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k138169005419483466_a_a_b @ m @ K2 @ N
@ ^ [I3: a,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_831_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: $o > sigma_measure_a,X4: $o > a > a,I: set_o,Y5: $o > a > b,N: $o > sigma_measure_b] :
( ( prob_k8052462064657410095_a_o_a @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k8052462064657410096_a_o_b @ m @ K2 @ N
@ ^ [I3: $o,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_832_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: nat > sigma_measure_a,X4: nat > a > a,I: set_nat,Y5: nat > a > b,N: nat > sigma_measure_b] :
( ( prob_k6325968634923510307_nat_a @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k6325968634923510308_nat_b @ m @ K2 @ N
@ ^ [I3: nat,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_833_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: c > sigma_measure_b,X4: c > a > b,I: set_c,Y5: c > b > d,N: c > sigma_measure_d] :
( ( prob_k3786629878328907208_a_c_b @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k3786629878328907210_a_c_d @ m @ K2 @ N
@ ^ [I3: c,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_834_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: b > sigma_measure_b,X4: b > a > b,I: set_b,Y5: b > b > d,N: b > sigma_measure_d] :
( ( prob_k6574085460301583241_a_b_b @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k6574085460301583243_a_b_d @ m @ K2 @ N
@ ^ [I3: b,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_835_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: a > sigma_measure_b,X4: a > a > b,I: set_a,Y5: a > b > d,N: a > sigma_measure_d] :
( ( prob_k138169005419483466_a_a_b @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k138169005419483468_a_a_d @ m @ K2 @ N
@ ^ [I3: a,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_836_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: $o > sigma_measure_b,X4: $o > a > b,I: set_o,Y5: $o > b > d,N: $o > sigma_measure_d] :
( ( prob_k8052462064657410096_a_o_b @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k8052462064657410098_a_o_d @ m @ K2 @ N
@ ^ [I3: $o,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_837_k__wise__indep__vars__compose,axiom,
! [K2: nat,M2: nat > sigma_measure_b,X4: nat > a > b,I: set_nat,Y5: nat > b > d,N: nat > sigma_measure_d] :
( ( prob_k6325968634923510308_nat_b @ m @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k6325968634923510310_nat_d @ m @ K2 @ N
@ ^ [I3: nat,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_838_D_Oindep__vars__def2,axiom,
! [M2: c > sigma_measure_a,X4: c > b > a,I: set_c] :
( ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: c] :
( ( member_c @ X3 @ I )
=> ( member_b_a @ ( X4 @ X3 ) @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_a] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_a @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_a @ A3 @ ( sigma_sets_a @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_839_D_Oindep__vars__def2,axiom,
! [M2: c > sigma_measure_b,X4: c > b > b,I: set_c] :
( ( indepe1224748351465797814_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: c] :
( ( member_c @ X3 @ I )
=> ( member_b_b @ ( X4 @ X3 ) @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_b] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_b @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_b @ A3 @ ( sigma_sets_b @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_840_D_Oindep__vars__def2,axiom,
! [M2: c > sigma_measure_d,X4: c > b > d,I: set_c] :
( ( indepe1224748351465797816_b_c_d @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: c] :
( ( member_c @ X3 @ I )
=> ( member_b_d @ ( X4 @ X3 ) @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_d] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_d @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_d @ A3 @ ( sigma_sets_d @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_841_D_Oindep__vars__def2,axiom,
! [M2: nat > sigma_measure_a,X4: nat > b > a,I: set_nat] :
( ( indepe4589662937738594229_nat_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I )
=> ( member_b_a @ ( X4 @ X3 ) @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: nat] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_a] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_a @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_a @ A3 @ ( sigma_sets_a @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_842_D_Oindep__vars__def2,axiom,
! [M2: nat > sigma_measure_b,X4: nat > b > b,I: set_nat] :
( ( indepe4589662937738594230_nat_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I )
=> ( member_b_b @ ( X4 @ X3 ) @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: nat] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_b] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_b @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_b @ A3 @ ( sigma_sets_b @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_843_D_Oindep__vars__def2,axiom,
! [M2: nat > sigma_measure_d,X4: nat > b > d,I: set_nat] :
( ( indepe4589662937738594232_nat_d @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I )
=> ( member_b_d @ ( X4 @ X3 ) @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: nat] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_d] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_d @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_d @ A3 @ ( sigma_sets_d @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_844_D_Oindep__vars__def2,axiom,
! [M2: $o > sigma_measure_a,X4: $o > b > a,I: set_o] :
( ( indepe4488128152570241565_b_o_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I )
=> ( member_b_a @ ( X4 @ X3 ) @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: $o] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_a] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_a @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_a @ A3 @ ( sigma_sets_a @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_845_D_Oindep__vars__def2,axiom,
! [M2: $o > sigma_measure_b,X4: $o > b > b,I: set_o] :
( ( indepe4488128152570241566_b_o_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I )
=> ( member_b_b @ ( X4 @ X3 ) @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: $o] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_b] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_b @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_b @ A3 @ ( sigma_sets_b @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_846_D_Oindep__vars__def2,axiom,
! [M2: $o > sigma_measure_d,X4: $o > b > d,I: set_o] :
( ( indepe4488128152570241568_b_o_d @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: $o] :
( ( member_o @ X3 @ I )
=> ( member_b_d @ ( X4 @ X3 ) @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: $o] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_d] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_d @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_d @ A3 @ ( sigma_sets_d @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_847_D_Oindep__vars__def2,axiom,
! [M2: b > sigma_measure_a,X4: b > b > a,I: set_b] :
( ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
= ( ! [X3: b] :
( ( member_b @ X3 @ I )
=> ( member_b_a @ ( X4 @ X3 ) @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ ( M2 @ X3 ) ) ) )
& ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: b] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_a] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_a @ ( X4 @ I3 ) @ A3 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
& ( member_set_a @ A3 @ ( sigma_sets_a @ ( M2 @ I3 ) ) ) ) )
@ I ) ) ) ).
% D.indep_vars_def2
thf(fact_848_D_Oindep__sets__distr,axiom,
! [F2: b > d,N: sigma_measure_d,A: c > set_set_d,I: set_c] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] :
( image_set_d_set_b
@ ^ [A5: set_d] : ( inf_inf_set_b @ ( vimage_b_d @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe565075120755390507ts_d_c @ ( measure_distr_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_849_D_Oindep__sets__distr,axiom,
! [F2: b > a,N: sigma_measure_a,A: c > set_set_a,I: set_c] :
( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] :
( image_set_a_set_b
@ ^ [A5: set_a] : ( inf_inf_set_b @ ( vimage_b_a @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ N ) ) )
=> ( indepe8927441866673418606ts_a_c @ ( measure_distr_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_850_D_Oindep__sets__distr,axiom,
! [F2: b > b,N: sigma_measure_b,A: c > set_set_b,I: set_c] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] :
( image_set_b_set_b
@ ^ [A5: set_b] : ( inf_inf_set_b @ ( vimage_b_b @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( A @ I2 ) @ ( sigma_sets_b @ N ) ) )
=> ( indepe6139986284700742573ts_b_c @ ( measure_distr_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_851_D_Oindep__sets__distr,axiom,
! [F2: b > d,N: sigma_measure_d,A: nat > set_set_d,I: set_nat] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: nat] :
( image_set_d_set_b
@ ^ [A5: set_d] : ( inf_inf_set_b @ ( vimage_b_d @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe750690977103256037_d_nat @ ( measure_distr_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_852_D_Oindep__sets__distr,axiom,
! [F2: b > b,N: sigma_measure_b,A: nat > set_set_b,I: set_nat] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: nat] :
( image_set_b_set_b
@ ^ [A5: set_b] : ( inf_inf_set_b @ ( vimage_b_b @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( A @ I2 ) @ ( sigma_sets_b @ N ) ) )
=> ( indepe7503174356045242851_b_nat @ ( measure_distr_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_853_D_Oindep__sets__distr,axiom,
! [F2: b > a,N: sigma_measure_a,A: nat > set_set_a,I: set_nat] :
( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: nat] :
( image_set_a_set_b
@ ^ [A5: set_a] : ( inf_inf_set_b @ ( vimage_b_a @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ N ) ) )
=> ( indepe6267730027088848354_a_nat @ ( measure_distr_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_854_D_Oindep__sets__distr,axiom,
! [F2: b > d,N: sigma_measure_d,A: $o > set_set_d,I: set_o] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: $o] :
( image_set_d_set_b
@ ^ [A5: set_d] : ( inf_inf_set_b @ ( vimage_b_d @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe8305812671657367107ts_d_o @ ( measure_distr_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_855_D_Oindep__sets__distr,axiom,
! [F2: b > a,N: sigma_measure_a,A: $o > set_set_a,I: set_o] :
( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: $o] :
( image_set_a_set_b
@ ^ [A5: set_a] : ( inf_inf_set_b @ ( vimage_b_a @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ N ) ) )
=> ( indepe7780107833195774214ts_a_o @ ( measure_distr_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_856_D_Oindep__sets__distr,axiom,
! [F2: b > b,N: sigma_measure_b,A: $o > set_set_b,I: set_o] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: $o] :
( image_set_b_set_b
@ ^ [A5: set_b] : ( inf_inf_set_b @ ( vimage_b_b @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3795704787696855135_set_b @ ( A @ I2 ) @ ( sigma_sets_b @ N ) ) )
=> ( indepe4880885433731379909ts_b_o @ ( measure_distr_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_857_D_Oindep__sets__distr,axiom,
! [F2: b > d,N: sigma_measure_d,A: b > set_set_d,I: set_b] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N ) )
=> ( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: b] :
( image_set_d_set_b
@ ^ [A5: set_d] : ( inf_inf_set_b @ ( vimage_b_d @ F2 @ A5 ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe565075120755390506ts_d_b @ ( measure_distr_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N @ F2 ) @ A @ I ) ) ) ) ).
% D.indep_sets_distr
thf(fact_858_D_Oindep__var__rv2,axiom,
! [S: sigma_measure_d,X4: b > d,T2: sigma_measure_d,Y5: b > d] :
( ( indepe8876569649573725966ar_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ S @ X4 @ T2 @ Y5 )
=> ( member_b_d @ Y5 @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ T2 ) ) ) ).
% D.indep_var_rv2
thf(fact_859_D_Oindep__var__rv1,axiom,
! [S: sigma_measure_d,X4: b > d,T2: sigma_measure_d,Y5: b > d] :
( ( indepe8876569649573725966ar_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ S @ X4 @ T2 @ Y5 )
=> ( member_b_d @ X4 @ ( sigma_measurable_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ S ) ) ) ).
% D.indep_var_rv1
thf(fact_860_D_Oindep__sets__finite__index__sets,axiom,
! [F: set_b > set_set_b,I: set_set_b] :
( ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ J2 @ I )
=> ( ( J2 != bot_bot_set_set_b )
=> ( ( finite_finite_set_b @ J2 )
=> ( indepe6311571491924490380_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_sets_finite_index_sets
thf(fact_861_D_Oindep__sets__finite__index__sets,axiom,
! [F: c > set_set_b,I: set_c] :
( ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_c] :
( ( ord_less_eq_set_c @ J2 @ I )
=> ( ( J2 != bot_bot_set_c )
=> ( ( finite_finite_c @ J2 )
=> ( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_sets_finite_index_sets
thf(fact_862_D_Oindep__sets__finite__index__sets,axiom,
! [F: nat > set_set_b,I: set_nat] :
( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_nat] :
( ( ord_less_eq_set_nat @ J2 @ I )
=> ( ( J2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ J2 )
=> ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_sets_finite_index_sets
thf(fact_863_D_Oindep__sets__finite__index__sets,axiom,
! [F: $o > set_set_b,I: set_o] :
( ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_o] :
( ( ord_less_eq_set_o @ J2 @ I )
=> ( ( J2 != bot_bot_set_o )
=> ( ( finite_finite_o @ J2 )
=> ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_sets_finite_index_sets
thf(fact_864_D_Oindep__sets__finite__index__sets,axiom,
! [F: set_a > set_set_b,I: set_set_a] :
( ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ J2 @ I )
=> ( ( J2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ J2 )
=> ( indepe6311571487621261579_set_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_sets_finite_index_sets
thf(fact_865_D_Oindep__sets__finite__index__sets,axiom,
! [F: ( b > d ) > set_set_b,I: set_b_d] :
( ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_b_d] :
( ( ord_less_eq_set_b_d @ J2 @ I )
=> ( ( J2 != bot_bot_set_b_d )
=> ( ( finite_finite_b_d @ J2 )
=> ( indepe4285407111846563642_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_sets_finite_index_sets
thf(fact_866_D_Oindep__sets__finite__index__sets,axiom,
! [F: ( a > b ) > set_set_b,I: set_a_b] :
( ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_a_b] :
( ( ord_less_eq_set_a_b @ J2 @ I )
=> ( ( J2 != bot_bot_set_a_b )
=> ( ( finite_finite_a_b @ J2 )
=> ( indepe7072862685212782073_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_sets_finite_index_sets
thf(fact_867_D_Oindep__sets__finite__index__sets,axiom,
! [F: b > set_set_b,I: set_b] :
( ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_b] :
( ( ord_less_eq_set_b @ J2 @ I )
=> ( ( J2 != bot_bot_set_b )
=> ( ( finite_finite_b @ J2 )
=> ( indepe6139986284700742572ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_sets_finite_index_sets
thf(fact_868_D_Oindep__sets__finite__index__sets,axiom,
! [F: a > set_set_b,I: set_a] :
( ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_a] :
( ( ord_less_eq_set_a @ J2 @ I )
=> ( ( J2 != bot_bot_set_a )
=> ( ( finite_finite_a @ J2 )
=> ( indepe6139986284700742571ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_sets_finite_index_sets
thf(fact_869_D_Omeasure__space__inter,axiom,
! [S2: set_b,T3: set_b] :
( ( member_set_b @ S2 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( member_set_b @ T3 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ T3 )
= ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ S2 @ T3 ) )
= ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ S2 ) ) ) ) ) ).
% D.measure_space_inter
thf(fact_870_D_Oindep__vars__compose2,axiom,
! [M2: c > sigma_measure_a,X4: c > b > a,I: set_c,Y5: c > a > b,N: c > sigma_measure_b] :
( ( indepe1224748351465797813_b_c_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe1224748351465797814_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: c,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_871_D_Oindep__vars__compose2,axiom,
! [M2: b > sigma_measure_a,X4: b > b > a,I: set_b,Y5: b > a > b,N: b > sigma_measure_b] :
( ( indepe4012203933438473846_b_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe4012203933438473847_b_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: b,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_872_D_Oindep__vars__compose2,axiom,
! [M2: a > sigma_measure_a,X4: a > b > a,I: set_a,Y5: a > a > b,N: a > sigma_measure_b] :
( ( indepe6799659515411149879_b_a_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe6799659515411149880_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: a,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_873_D_Oindep__vars__compose2,axiom,
! [M2: $o > sigma_measure_a,X4: $o > b > a,I: set_o,Y5: $o > a > b,N: $o > sigma_measure_b] :
( ( indepe4488128152570241565_b_o_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe4488128152570241566_b_o_b @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: $o,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_874_D_Oindep__vars__compose2,axiom,
! [M2: nat > sigma_measure_a,X4: nat > b > a,I: set_nat,Y5: nat > a > b,N: nat > sigma_measure_b] :
( ( indepe4589662937738594229_nat_a @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe4589662937738594230_nat_b @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: nat,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_875_D_Oindep__vars__compose2,axiom,
! [M2: c > sigma_measure_b,X4: c > b > b,I: set_c,Y5: c > b > d,N: c > sigma_measure_d] :
( ( indepe1224748351465797814_b_c_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe1224748351465797816_b_c_d @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: c,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_876_D_Oindep__vars__compose2,axiom,
! [M2: b > sigma_measure_b,X4: b > b > b,I: set_b,Y5: b > b > d,N: b > sigma_measure_d] :
( ( indepe4012203933438473847_b_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe4012203933438473849_b_b_d @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: b,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_877_D_Oindep__vars__compose2,axiom,
! [M2: a > sigma_measure_b,X4: a > b > b,I: set_a,Y5: a > b > d,N: a > sigma_measure_d] :
( ( indepe6799659515411149880_b_a_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe6799659515411149882_b_a_d @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: a,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_878_D_Oindep__vars__compose2,axiom,
! [M2: $o > sigma_measure_b,X4: $o > b > b,I: set_o,Y5: $o > b > d,N: $o > sigma_measure_d] :
( ( indepe4488128152570241566_b_o_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe4488128152570241568_b_o_d @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: $o,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_879_D_Oindep__vars__compose2,axiom,
! [M2: nat > sigma_measure_b,X4: nat > b > b,I: set_nat,Y5: nat > b > d,N: nat > sigma_measure_d] :
( ( indepe4589662937738594230_nat_b @ ( measure_distr_a_b @ m @ n @ f ) @ M2 @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( indepe4589662937738594232_nat_d @ ( measure_distr_a_b @ m @ n @ f ) @ N
@ ^ [I3: nat,X3: b] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ).
% D.indep_vars_compose2
thf(fact_880_indep__sets__cong,axiom,
! [I: set_b_d,J: set_b_d,F: ( b > d ) > set_set_a,G: ( b > d ) > set_set_a] :
( ( I = J )
=> ( ! [I2: b > d] :
( ( member_b_d @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7654921378804716857_a_b_d @ m @ F @ I )
= ( indepe7654921378804716857_a_b_d @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_881_indep__sets__cong,axiom,
! [I: set_a_b,J: set_a_b,F: ( a > b ) > set_set_a,G: ( a > b ) > set_set_a] :
( ( I = J )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe1219004915316159480_a_a_b @ m @ F @ I )
= ( indepe1219004915316159480_a_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_882_indep__sets__cong,axiom,
! [I: set_c,J: set_c,F: c > set_set_a,G: c > set_set_a] :
( ( I = J )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe8927441866673418606ts_a_c @ m @ F @ I )
= ( indepe8927441866673418606ts_a_c @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_883_indep__sets__cong,axiom,
! [I: set_b,J: set_b,F: b > set_set_a,G: b > set_set_a] :
( ( I = J )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe8927441866673418605ts_a_b @ m @ F @ I )
= ( indepe8927441866673418605ts_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_884_indep__sets__cong,axiom,
! [I: set_a,J: set_a,F: a > set_set_a,G: a > set_set_a] :
( ( I = J )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe8927441866673418604ts_a_a @ m @ F @ I )
= ( indepe8927441866673418604ts_a_a @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_885_indep__sets__cong,axiom,
! [I: set_set_a,J: set_set_a,F: set_a > set_set_a,G: set_a > set_set_a] :
( ( I = J )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe4967106450811773644_set_a @ m @ F @ I )
= ( indepe4967106450811773644_set_a @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_886_indep__sets__cong,axiom,
! [I: set_set_b,J: set_set_b,F: set_b > set_set_a,G: set_b > set_set_a] :
( ( I = J )
=> ( ! [I2: set_b] :
( ( member_set_b @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe4967106455115002445_set_b @ m @ F @ I )
= ( indepe4967106455115002445_set_b @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_887_indep__sets__cong,axiom,
! [I: set_set_d,J: set_set_d,F: set_d > set_set_a,G: set_d > set_set_a] :
( ( I = J )
=> ( ! [I2: set_d] :
( ( member_set_d @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe4967106463721460047_set_d @ m @ F @ I )
= ( indepe4967106463721460047_set_d @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_888_indep__sets__cong,axiom,
! [I: set_nat,J: set_nat,F: nat > set_set_a,G: nat > set_set_a] :
( ( I = J )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6267730027088848354_a_nat @ m @ F @ I )
= ( indepe6267730027088848354_a_nat @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_889_indep__sets__cong,axiom,
! [I: set_o,J: set_o,F: $o > set_set_a,G: $o > set_set_a] :
( ( I = J )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7780107833195774214ts_a_o @ m @ F @ I )
= ( indepe7780107833195774214ts_a_o @ m @ G @ J ) ) ) ) ).
% indep_sets_cong
thf(fact_890_subprob__space__axioms,axiom,
giry_subprob_space_a @ m ).
% subprob_space_axioms
thf(fact_891_subprob__not__empty,axiom,
( ( sigma_space_a @ m )
!= bot_bot_set_a ) ).
% subprob_not_empty
thf(fact_892_indep__sets__Dynkin,axiom,
! [F: nat > set_set_a,I: set_nat] :
( ( indepe6267730027088848354_a_nat @ m @ F @ I )
=> ( indepe6267730027088848354_a_nat @ m
@ ^ [I3: nat] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F @ I3 ) )
@ I ) ) ).
% indep_sets_Dynkin
thf(fact_893_indep__sets__Dynkin,axiom,
! [F: $o > set_set_a,I: set_o] :
( ( indepe7780107833195774214ts_a_o @ m @ F @ I )
=> ( indepe7780107833195774214ts_a_o @ m
@ ^ [I3: $o] : ( sigma_Dynkin_a @ ( sigma_space_a @ m ) @ ( F @ I3 ) )
@ I ) ) ).
% indep_sets_Dynkin
thf(fact_894_indep__sets__mono__index,axiom,
! [J: set_set_b,I: set_set_b,F: set_b > set_set_a] :
( ( ord_le3795704787696855135_set_b @ J @ I )
=> ( ( indepe4967106455115002445_set_b @ m @ F @ I )
=> ( indepe4967106455115002445_set_b @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_895_indep__sets__mono__index,axiom,
! [J: set_set_a,I: set_set_a,F: set_a > set_set_a] :
( ( ord_le3724670747650509150_set_a @ J @ I )
=> ( ( indepe4967106450811773644_set_a @ m @ F @ I )
=> ( indepe4967106450811773644_set_a @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_896_indep__sets__mono__index,axiom,
! [J: set_b,I: set_b,F: b > set_set_a] :
( ( ord_less_eq_set_b @ J @ I )
=> ( ( indepe8927441866673418605ts_a_b @ m @ F @ I )
=> ( indepe8927441866673418605ts_a_b @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_897_indep__sets__mono__index,axiom,
! [J: set_a,I: set_a,F: a > set_set_a] :
( ( ord_less_eq_set_a @ J @ I )
=> ( ( indepe8927441866673418604ts_a_a @ m @ F @ I )
=> ( indepe8927441866673418604ts_a_a @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_898_indep__sets__mono__index,axiom,
! [J: set_c,I: set_c,F: c > set_set_a] :
( ( ord_less_eq_set_c @ J @ I )
=> ( ( indepe8927441866673418606ts_a_c @ m @ F @ I )
=> ( indepe8927441866673418606ts_a_c @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_899_indep__sets__mono__index,axiom,
! [J: set_nat,I: set_nat,F: nat > set_set_a] :
( ( ord_less_eq_set_nat @ J @ I )
=> ( ( indepe6267730027088848354_a_nat @ m @ F @ I )
=> ( indepe6267730027088848354_a_nat @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_900_indep__sets__mono__index,axiom,
! [J: set_o,I: set_o,F: $o > set_set_a] :
( ( ord_less_eq_set_o @ J @ I )
=> ( ( indepe7780107833195774214ts_a_o @ m @ F @ I )
=> ( indepe7780107833195774214ts_a_o @ m @ F @ J ) ) ) ).
% indep_sets_mono_index
thf(fact_901_indep__sets__mono__sets,axiom,
! [F: ( b > d ) > set_set_a,I: set_b_d,G: ( b > d ) > set_set_a] :
( ( indepe7654921378804716857_a_b_d @ m @ F @ I )
=> ( ! [I2: b > d] :
( ( member_b_d @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7654921378804716857_a_b_d @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_902_indep__sets__mono__sets,axiom,
! [F: ( a > b ) > set_set_a,I: set_a_b,G: ( a > b ) > set_set_a] :
( ( indepe1219004915316159480_a_a_b @ m @ F @ I )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe1219004915316159480_a_a_b @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_903_indep__sets__mono__sets,axiom,
! [F: c > set_set_a,I: set_c,G: c > set_set_a] :
( ( indepe8927441866673418606ts_a_c @ m @ F @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe8927441866673418606ts_a_c @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_904_indep__sets__mono__sets,axiom,
! [F: b > set_set_a,I: set_b,G: b > set_set_a] :
( ( indepe8927441866673418605ts_a_b @ m @ F @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe8927441866673418605ts_a_b @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_905_indep__sets__mono__sets,axiom,
! [F: a > set_set_a,I: set_a,G: a > set_set_a] :
( ( indepe8927441866673418604ts_a_a @ m @ F @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe8927441866673418604ts_a_a @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_906_indep__sets__mono__sets,axiom,
! [F: set_a > set_set_a,I: set_set_a,G: set_a > set_set_a] :
( ( indepe4967106450811773644_set_a @ m @ F @ I )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4967106450811773644_set_a @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_907_indep__sets__mono__sets,axiom,
! [F: set_b > set_set_a,I: set_set_b,G: set_b > set_set_a] :
( ( indepe4967106455115002445_set_b @ m @ F @ I )
=> ( ! [I2: set_b] :
( ( member_set_b @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4967106455115002445_set_b @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_908_indep__sets__mono__sets,axiom,
! [F: set_d > set_set_a,I: set_set_d,G: set_d > set_set_a] :
( ( indepe4967106463721460047_set_d @ m @ F @ I )
=> ( ! [I2: set_d] :
( ( member_set_d @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4967106463721460047_set_d @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_909_indep__sets__mono__sets,axiom,
! [F: nat > set_set_a,I: set_nat,G: nat > set_set_a] :
( ( indepe6267730027088848354_a_nat @ m @ F @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6267730027088848354_a_nat @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_910_indep__sets__mono__sets,axiom,
! [F: $o > set_set_a,I: set_o,G: $o > set_set_a] :
( ( indepe7780107833195774214ts_a_o @ m @ F @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7780107833195774214ts_a_o @ m @ G @ I ) ) ) ).
% indep_sets_mono_sets
thf(fact_911_indep__sets__mono,axiom,
! [F: ( b > d ) > set_set_a,I: set_b_d,J: set_b_d,G: ( b > d ) > set_set_a] :
( ( indepe7654921378804716857_a_b_d @ m @ F @ I )
=> ( ( ord_less_eq_set_b_d @ J @ I )
=> ( ! [I2: b > d] :
( ( member_b_d @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7654921378804716857_a_b_d @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_912_indep__sets__mono,axiom,
! [F: ( a > b ) > set_set_a,I: set_a_b,J: set_a_b,G: ( a > b ) > set_set_a] :
( ( indepe1219004915316159480_a_a_b @ m @ F @ I )
=> ( ( ord_less_eq_set_a_b @ J @ I )
=> ( ! [I2: a > b] :
( ( member_a_b @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe1219004915316159480_a_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_913_indep__sets__mono,axiom,
! [F: set_d > set_set_a,I: set_set_d,J: set_set_d,G: set_d > set_set_a] :
( ( indepe4967106463721460047_set_d @ m @ F @ I )
=> ( ( ord_le3937772867789547105_set_d @ J @ I )
=> ( ! [I2: set_d] :
( ( member_set_d @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4967106463721460047_set_d @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_914_indep__sets__mono,axiom,
! [F: set_b > set_set_a,I: set_set_b,J: set_set_b,G: set_b > set_set_a] :
( ( indepe4967106455115002445_set_b @ m @ F @ I )
=> ( ( ord_le3795704787696855135_set_b @ J @ I )
=> ( ! [I2: set_b] :
( ( member_set_b @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4967106455115002445_set_b @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_915_indep__sets__mono,axiom,
! [F: set_a > set_set_a,I: set_set_a,J: set_set_a,G: set_a > set_set_a] :
( ( indepe4967106450811773644_set_a @ m @ F @ I )
=> ( ( ord_le3724670747650509150_set_a @ J @ I )
=> ( ! [I2: set_a] :
( ( member_set_a @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe4967106450811773644_set_a @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_916_indep__sets__mono,axiom,
! [F: b > set_set_a,I: set_b,J: set_b,G: b > set_set_a] :
( ( indepe8927441866673418605ts_a_b @ m @ F @ I )
=> ( ( ord_less_eq_set_b @ J @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe8927441866673418605ts_a_b @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_917_indep__sets__mono,axiom,
! [F: a > set_set_a,I: set_a,J: set_a,G: a > set_set_a] :
( ( indepe8927441866673418604ts_a_a @ m @ F @ I )
=> ( ( ord_less_eq_set_a @ J @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe8927441866673418604ts_a_a @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_918_indep__sets__mono,axiom,
! [F: c > set_set_a,I: set_c,J: set_c,G: c > set_set_a] :
( ( indepe8927441866673418606ts_a_c @ m @ F @ I )
=> ( ( ord_less_eq_set_c @ J @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe8927441866673418606ts_a_c @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_919_indep__sets__mono,axiom,
! [F: nat > set_set_a,I: set_nat,J: set_nat,G: nat > set_set_a] :
( ( indepe6267730027088848354_a_nat @ m @ F @ I )
=> ( ( ord_less_eq_set_nat @ J @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6267730027088848354_a_nat @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_920_indep__sets__mono,axiom,
! [F: $o > set_set_a,I: set_o,J: set_o,G: $o > set_set_a] :
( ( indepe7780107833195774214ts_a_o @ m @ F @ I )
=> ( ( ord_less_eq_set_o @ J @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7780107833195774214ts_a_o @ m @ G @ J ) ) ) ) ).
% indep_sets_mono
thf(fact_921_indep__sets__finite__index__sets,axiom,
! [F: set_b > set_set_a,I: set_set_b] :
( ( indepe4967106455115002445_set_b @ m @ F @ I )
= ( ! [J2: set_set_b] :
( ( ord_le3795704787696855135_set_b @ J2 @ I )
=> ( ( J2 != bot_bot_set_set_b )
=> ( ( finite_finite_set_b @ J2 )
=> ( indepe4967106455115002445_set_b @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_922_indep__sets__finite__index__sets,axiom,
! [F: set_a > set_set_a,I: set_set_a] :
( ( indepe4967106450811773644_set_a @ m @ F @ I )
= ( ! [J2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ J2 @ I )
=> ( ( J2 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ J2 )
=> ( indepe4967106450811773644_set_a @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_923_indep__sets__finite__index__sets,axiom,
! [F: b > set_set_a,I: set_b] :
( ( indepe8927441866673418605ts_a_b @ m @ F @ I )
= ( ! [J2: set_b] :
( ( ord_less_eq_set_b @ J2 @ I )
=> ( ( J2 != bot_bot_set_b )
=> ( ( finite_finite_b @ J2 )
=> ( indepe8927441866673418605ts_a_b @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_924_indep__sets__finite__index__sets,axiom,
! [F: a > set_set_a,I: set_a] :
( ( indepe8927441866673418604ts_a_a @ m @ F @ I )
= ( ! [J2: set_a] :
( ( ord_less_eq_set_a @ J2 @ I )
=> ( ( J2 != bot_bot_set_a )
=> ( ( finite_finite_a @ J2 )
=> ( indepe8927441866673418604ts_a_a @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_925_indep__sets__finite__index__sets,axiom,
! [F: c > set_set_a,I: set_c] :
( ( indepe8927441866673418606ts_a_c @ m @ F @ I )
= ( ! [J2: set_c] :
( ( ord_less_eq_set_c @ J2 @ I )
=> ( ( J2 != bot_bot_set_c )
=> ( ( finite_finite_c @ J2 )
=> ( indepe8927441866673418606ts_a_c @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_926_indep__sets__finite__index__sets,axiom,
! [F: nat > set_set_a,I: set_nat] :
( ( indepe6267730027088848354_a_nat @ m @ F @ I )
= ( ! [J2: set_nat] :
( ( ord_less_eq_set_nat @ J2 @ I )
=> ( ( J2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ J2 )
=> ( indepe6267730027088848354_a_nat @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_927_indep__sets__finite__index__sets,axiom,
! [F: $o > set_set_a,I: set_o] :
( ( indepe7780107833195774214ts_a_o @ m @ F @ I )
= ( ! [J2: set_o] :
( ( ord_less_eq_set_o @ J2 @ I )
=> ( ( J2 != bot_bot_set_o )
=> ( ( finite_finite_o @ J2 )
=> ( indepe7780107833195774214ts_a_o @ m @ F @ J2 ) ) ) ) ) ) ).
% indep_sets_finite_index_sets
thf(fact_928_image__eqI,axiom,
! [B2: c,F2: c > c,X: c,A: set_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_c @ X @ A )
=> ( member_c @ B2 @ ( image_c_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_929_image__eqI,axiom,
! [B2: b,F2: c > b,X: c,A: set_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_c @ X @ A )
=> ( member_b @ B2 @ ( image_c_b @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_930_image__eqI,axiom,
! [B2: a,F2: c > a,X: c,A: set_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_c @ X @ A )
=> ( member_a @ B2 @ ( image_c_a @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_931_image__eqI,axiom,
! [B2: $o,F2: c > $o,X: c,A: set_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_c @ X @ A )
=> ( member_o @ B2 @ ( image_c_o @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_932_image__eqI,axiom,
! [B2: nat,F2: c > nat,X: c,A: set_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_c @ X @ A )
=> ( member_nat @ B2 @ ( image_c_nat @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_933_image__eqI,axiom,
! [B2: c,F2: b > c,X: b,A: set_b] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_b @ X @ A )
=> ( member_c @ B2 @ ( image_b_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_934_image__eqI,axiom,
! [B2: b,F2: b > b,X: b,A: set_b] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_b @ X @ A )
=> ( member_b @ B2 @ ( image_b_b @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_935_image__eqI,axiom,
! [B2: a,F2: b > a,X: b,A: set_b] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_b @ X @ A )
=> ( member_a @ B2 @ ( image_b_a @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_936_image__eqI,axiom,
! [B2: $o,F2: b > $o,X: b,A: set_b] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_b @ X @ A )
=> ( member_o @ B2 @ ( image_b_o @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_937_image__eqI,axiom,
! [B2: nat,F2: b > nat,X: b,A: set_b] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_b @ X @ A )
=> ( member_nat @ B2 @ ( image_b_nat @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_938_D_Oprob__space__axioms,axiom,
probab7247484486040049090pace_b @ ( measure_distr_a_b @ m @ n @ f ) ).
% D.prob_space_axioms
thf(fact_939_image__is__empty,axiom,
! [F2: b > b,A: set_b] :
( ( ( image_b_b @ F2 @ A )
= bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_940_image__is__empty,axiom,
! [F2: a > b,A: set_a] :
( ( ( image_a_b @ F2 @ A )
= bot_bot_set_b )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_941_image__is__empty,axiom,
! [F2: $o > b,A: set_o] :
( ( ( image_o_b @ F2 @ A )
= bot_bot_set_b )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_942_image__is__empty,axiom,
! [F2: b > a,A: set_b] :
( ( ( image_b_a @ F2 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_943_image__is__empty,axiom,
! [F2: a > a,A: set_a] :
( ( ( image_a_a @ F2 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_944_image__is__empty,axiom,
! [F2: $o > a,A: set_o] :
( ( ( image_o_a @ F2 @ A )
= bot_bot_set_a )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_945_image__is__empty,axiom,
! [F2: b > $o,A: set_b] :
( ( ( image_b_o @ F2 @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_946_image__is__empty,axiom,
! [F2: a > $o,A: set_a] :
( ( ( image_a_o @ F2 @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_947_image__is__empty,axiom,
! [F2: $o > $o,A: set_o] :
( ( ( image_o_o @ F2 @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_948_image__is__empty,axiom,
! [F2: nat > set_b,A: set_nat] :
( ( ( image_nat_set_b @ F2 @ A )
= bot_bot_set_set_b )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_949_empty__is__image,axiom,
! [F2: b > b,A: set_b] :
( ( bot_bot_set_b
= ( image_b_b @ F2 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_950_empty__is__image,axiom,
! [F2: a > b,A: set_a] :
( ( bot_bot_set_b
= ( image_a_b @ F2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_951_empty__is__image,axiom,
! [F2: $o > b,A: set_o] :
( ( bot_bot_set_b
= ( image_o_b @ F2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_952_empty__is__image,axiom,
! [F2: b > a,A: set_b] :
( ( bot_bot_set_a
= ( image_b_a @ F2 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_953_empty__is__image,axiom,
! [F2: a > a,A: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_954_empty__is__image,axiom,
! [F2: $o > a,A: set_o] :
( ( bot_bot_set_a
= ( image_o_a @ F2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_955_empty__is__image,axiom,
! [F2: b > $o,A: set_b] :
( ( bot_bot_set_o
= ( image_b_o @ F2 @ A ) )
= ( A = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_956_empty__is__image,axiom,
! [F2: a > $o,A: set_a] :
( ( bot_bot_set_o
= ( image_a_o @ F2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_957_empty__is__image,axiom,
! [F2: $o > $o,A: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_958_empty__is__image,axiom,
! [F2: nat > set_b,A: set_nat] :
( ( bot_bot_set_set_b
= ( image_nat_set_b @ F2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_959_image__empty,axiom,
! [F2: b > b] :
( ( image_b_b @ F2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% image_empty
thf(fact_960_image__empty,axiom,
! [F2: b > a] :
( ( image_b_a @ F2 @ bot_bot_set_b )
= bot_bot_set_a ) ).
% image_empty
thf(fact_961_image__empty,axiom,
! [F2: b > $o] :
( ( image_b_o @ F2 @ bot_bot_set_b )
= bot_bot_set_o ) ).
% image_empty
thf(fact_962_image__empty,axiom,
! [F2: a > b] :
( ( image_a_b @ F2 @ bot_bot_set_a )
= bot_bot_set_b ) ).
% image_empty
thf(fact_963_image__empty,axiom,
! [F2: a > a] :
( ( image_a_a @ F2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_964_image__empty,axiom,
! [F2: a > $o] :
( ( image_a_o @ F2 @ bot_bot_set_a )
= bot_bot_set_o ) ).
% image_empty
thf(fact_965_image__empty,axiom,
! [F2: $o > b] :
( ( image_o_b @ F2 @ bot_bot_set_o )
= bot_bot_set_b ) ).
% image_empty
thf(fact_966_image__empty,axiom,
! [F2: $o > a] :
( ( image_o_a @ F2 @ bot_bot_set_o )
= bot_bot_set_a ) ).
% image_empty
thf(fact_967_image__empty,axiom,
! [F2: $o > $o] :
( ( image_o_o @ F2 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_968_image__empty,axiom,
! [F2: nat > set_b] :
( ( image_nat_set_b @ F2 @ bot_bot_set_nat )
= bot_bot_set_set_b ) ).
% image_empty
thf(fact_969_D_Omeasure__increasing,axiom,
measur7372598777031404856b_real @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) ) ).
% D.measure_increasing
thf(fact_970_imageI,axiom,
! [X: c,A: set_c,F2: c > c] :
( ( member_c @ X @ A )
=> ( member_c @ ( F2 @ X ) @ ( image_c_c @ F2 @ A ) ) ) ).
% imageI
thf(fact_971_imageI,axiom,
! [X: c,A: set_c,F2: c > b] :
( ( member_c @ X @ A )
=> ( member_b @ ( F2 @ X ) @ ( image_c_b @ F2 @ A ) ) ) ).
% imageI
thf(fact_972_imageI,axiom,
! [X: c,A: set_c,F2: c > a] :
( ( member_c @ X @ A )
=> ( member_a @ ( F2 @ X ) @ ( image_c_a @ F2 @ A ) ) ) ).
% imageI
thf(fact_973_imageI,axiom,
! [X: c,A: set_c,F2: c > $o] :
( ( member_c @ X @ A )
=> ( member_o @ ( F2 @ X ) @ ( image_c_o @ F2 @ A ) ) ) ).
% imageI
thf(fact_974_imageI,axiom,
! [X: c,A: set_c,F2: c > nat] :
( ( member_c @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ ( image_c_nat @ F2 @ A ) ) ) ).
% imageI
thf(fact_975_imageI,axiom,
! [X: b,A: set_b,F2: b > c] :
( ( member_b @ X @ A )
=> ( member_c @ ( F2 @ X ) @ ( image_b_c @ F2 @ A ) ) ) ).
% imageI
thf(fact_976_imageI,axiom,
! [X: b,A: set_b,F2: b > b] :
( ( member_b @ X @ A )
=> ( member_b @ ( F2 @ X ) @ ( image_b_b @ F2 @ A ) ) ) ).
% imageI
thf(fact_977_imageI,axiom,
! [X: b,A: set_b,F2: b > a] :
( ( member_b @ X @ A )
=> ( member_a @ ( F2 @ X ) @ ( image_b_a @ F2 @ A ) ) ) ).
% imageI
thf(fact_978_imageI,axiom,
! [X: b,A: set_b,F2: b > $o] :
( ( member_b @ X @ A )
=> ( member_o @ ( F2 @ X ) @ ( image_b_o @ F2 @ A ) ) ) ).
% imageI
thf(fact_979_imageI,axiom,
! [X: b,A: set_b,F2: b > nat] :
( ( member_b @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ ( image_b_nat @ F2 @ A ) ) ) ).
% imageI
thf(fact_980_image__iff,axiom,
! [Z2: set_a,F2: nat > set_a,A: set_nat] :
( ( member_set_a @ Z2 @ ( image_nat_set_a @ F2 @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z2
= ( F2 @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_981_image__iff,axiom,
! [Z2: set_b,F2: nat > set_b,A: set_nat] :
( ( member_set_b @ Z2 @ ( image_nat_set_b @ F2 @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z2
= ( F2 @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_982_bex__imageD,axiom,
! [F2: nat > set_b,A: set_nat,P: set_b > $o] :
( ? [X5: set_b] :
( ( member_set_b @ X5 @ ( image_nat_set_b @ F2 @ A ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_983_bex__imageD,axiom,
! [F2: nat > set_a,A: set_nat,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a @ X5 @ ( image_nat_set_a @ F2 @ A ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_984_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > set_b,G2: nat > set_b] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image_nat_set_b @ F2 @ M )
= ( image_nat_set_b @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_985_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > set_a,G2: nat > set_a] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image_nat_set_a @ F2 @ M )
= ( image_nat_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_986_ball__imageD,axiom,
! [F2: nat > set_b,A: set_nat,P: set_b > $o] :
( ! [X2: set_b] :
( ( member_set_b @ X2 @ ( image_nat_set_b @ F2 @ A ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_987_ball__imageD,axiom,
! [F2: nat > set_a,A: set_nat,P: set_a > $o] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ ( image_nat_set_a @ F2 @ A ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_988_rev__image__eqI,axiom,
! [X: c,A: set_c,B2: c,F2: c > c] :
( ( member_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_c @ B2 @ ( image_c_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_989_rev__image__eqI,axiom,
! [X: c,A: set_c,B2: b,F2: c > b] :
( ( member_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_b @ B2 @ ( image_c_b @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_990_rev__image__eqI,axiom,
! [X: c,A: set_c,B2: a,F2: c > a] :
( ( member_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_a @ B2 @ ( image_c_a @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_991_rev__image__eqI,axiom,
! [X: c,A: set_c,B2: $o,F2: c > $o] :
( ( member_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_o @ B2 @ ( image_c_o @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_992_rev__image__eqI,axiom,
! [X: c,A: set_c,B2: nat,F2: c > nat] :
( ( member_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_nat @ B2 @ ( image_c_nat @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_993_rev__image__eqI,axiom,
! [X: b,A: set_b,B2: c,F2: b > c] :
( ( member_b @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_c @ B2 @ ( image_b_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_994_rev__image__eqI,axiom,
! [X: b,A: set_b,B2: b,F2: b > b] :
( ( member_b @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_b @ B2 @ ( image_b_b @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_995_rev__image__eqI,axiom,
! [X: b,A: set_b,B2: a,F2: b > a] :
( ( member_b @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_a @ B2 @ ( image_b_a @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_996_rev__image__eqI,axiom,
! [X: b,A: set_b,B2: $o,F2: b > $o] :
( ( member_b @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_o @ B2 @ ( image_b_o @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_997_rev__image__eqI,axiom,
! [X: b,A: set_b,B2: nat,F2: b > nat] :
( ( member_b @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_nat @ B2 @ ( image_b_nat @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_998_Compr__image__eq,axiom,
! [F2: c > c,A: set_c,P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( image_c_c @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_c_c @ F2
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_999_Compr__image__eq,axiom,
! [F2: $o > c,A: set_o,P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( image_o_c @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_o_c @ F2
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1000_Compr__image__eq,axiom,
! [F2: c > $o,A: set_c,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_c_o @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_c_o @ F2
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1001_Compr__image__eq,axiom,
! [F2: $o > $o,A: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_o_o @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_o_o @ F2
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1002_Compr__image__eq,axiom,
! [F2: a > c,A: set_a,P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( image_a_c @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_a_c @ F2
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1003_Compr__image__eq,axiom,
! [F2: a > $o,A: set_a,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_a_o @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_a_o @ F2
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1004_Compr__image__eq,axiom,
! [F2: b > c,A: set_b,P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( image_b_c @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_b_c @ F2
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1005_Compr__image__eq,axiom,
! [F2: b > $o,A: set_b,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_b_o @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_b_o @ F2
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1006_Compr__image__eq,axiom,
! [F2: nat > c,A: set_nat,P: c > $o] :
( ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( image_nat_c @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_nat_c @ F2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1007_Compr__image__eq,axiom,
! [F2: nat > $o,A: set_nat,P: $o > $o] :
( ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( image_nat_o @ F2 @ A ) )
& ( P @ X3 ) ) )
= ( image_nat_o @ F2
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1008_image__image,axiom,
! [F2: set_b > set_b,G2: nat > set_b,A: set_nat] :
( ( image_set_b_set_b @ F2 @ ( image_nat_set_b @ G2 @ A ) )
= ( image_nat_set_b
@ ^ [X3: nat] : ( F2 @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1009_image__image,axiom,
! [F2: set_b > set_a,G2: nat > set_b,A: set_nat] :
( ( image_set_b_set_a @ F2 @ ( image_nat_set_b @ G2 @ A ) )
= ( image_nat_set_a
@ ^ [X3: nat] : ( F2 @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1010_image__image,axiom,
! [F2: set_a > set_b,G2: nat > set_a,A: set_nat] :
( ( image_set_a_set_b @ F2 @ ( image_nat_set_a @ G2 @ A ) )
= ( image_nat_set_b
@ ^ [X3: nat] : ( F2 @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1011_image__image,axiom,
! [F2: set_a > set_a,G2: nat > set_a,A: set_nat] :
( ( image_set_a_set_a @ F2 @ ( image_nat_set_a @ G2 @ A ) )
= ( image_nat_set_a
@ ^ [X3: nat] : ( F2 @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1012_image__image,axiom,
! [F2: nat > set_b,G2: nat > nat,A: set_nat] :
( ( image_nat_set_b @ F2 @ ( image_nat_nat @ G2 @ A ) )
= ( image_nat_set_b
@ ^ [X3: nat] : ( F2 @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1013_image__image,axiom,
! [F2: nat > set_a,G2: nat > nat,A: set_nat] :
( ( image_nat_set_a @ F2 @ ( image_nat_nat @ G2 @ A ) )
= ( image_nat_set_a
@ ^ [X3: nat] : ( F2 @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_1014_imageE,axiom,
! [B2: c,F2: c > c,A: set_c] :
( ( member_c @ B2 @ ( image_c_c @ F2 @ A ) )
=> ~ ! [X2: c] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_c @ X2 @ A ) ) ) ).
% imageE
thf(fact_1015_imageE,axiom,
! [B2: c,F2: b > c,A: set_b] :
( ( member_c @ B2 @ ( image_b_c @ F2 @ A ) )
=> ~ ! [X2: b] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_b @ X2 @ A ) ) ) ).
% imageE
thf(fact_1016_imageE,axiom,
! [B2: c,F2: a > c,A: set_a] :
( ( member_c @ B2 @ ( image_a_c @ F2 @ A ) )
=> ~ ! [X2: a] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1017_imageE,axiom,
! [B2: c,F2: $o > c,A: set_o] :
( ( member_c @ B2 @ ( image_o_c @ F2 @ A ) )
=> ~ ! [X2: $o] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_o @ X2 @ A ) ) ) ).
% imageE
thf(fact_1018_imageE,axiom,
! [B2: c,F2: nat > c,A: set_nat] :
( ( member_c @ B2 @ ( image_nat_c @ F2 @ A ) )
=> ~ ! [X2: nat] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A ) ) ) ).
% imageE
thf(fact_1019_imageE,axiom,
! [B2: b,F2: c > b,A: set_c] :
( ( member_b @ B2 @ ( image_c_b @ F2 @ A ) )
=> ~ ! [X2: c] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_c @ X2 @ A ) ) ) ).
% imageE
thf(fact_1020_imageE,axiom,
! [B2: b,F2: b > b,A: set_b] :
( ( member_b @ B2 @ ( image_b_b @ F2 @ A ) )
=> ~ ! [X2: b] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_b @ X2 @ A ) ) ) ).
% imageE
thf(fact_1021_imageE,axiom,
! [B2: b,F2: a > b,A: set_a] :
( ( member_b @ B2 @ ( image_a_b @ F2 @ A ) )
=> ~ ! [X2: a] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_a @ X2 @ A ) ) ) ).
% imageE
thf(fact_1022_imageE,axiom,
! [B2: b,F2: $o > b,A: set_o] :
( ( member_b @ B2 @ ( image_o_b @ F2 @ A ) )
=> ~ ! [X2: $o] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_o @ X2 @ A ) ) ) ).
% imageE
thf(fact_1023_imageE,axiom,
! [B2: b,F2: nat > b,A: set_nat] :
( ( member_b @ B2 @ ( image_nat_b @ F2 @ A ) )
=> ~ ! [X2: nat] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member_nat @ X2 @ A ) ) ) ).
% imageE
thf(fact_1024_subset__image__iff,axiom,
! [B: set_b,F2: b > b,A: set_b] :
( ( ord_less_eq_set_b @ B @ ( image_b_b @ F2 @ A ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A )
& ( B
= ( image_b_b @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1025_subset__image__iff,axiom,
! [B: set_b,F2: a > b,A: set_a] :
( ( ord_less_eq_set_b @ B @ ( image_a_b @ F2 @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_b @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1026_subset__image__iff,axiom,
! [B: set_b,F2: $o > b,A: set_o] :
( ( ord_less_eq_set_b @ B @ ( image_o_b @ F2 @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B
= ( image_o_b @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1027_subset__image__iff,axiom,
! [B: set_b,F2: nat > b,A: set_nat] :
( ( ord_less_eq_set_b @ B @ ( image_nat_b @ F2 @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_b @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1028_subset__image__iff,axiom,
! [B: set_b,F2: c > b,A: set_c] :
( ( ord_less_eq_set_b @ B @ ( image_c_b @ F2 @ A ) )
= ( ? [AA: set_c] :
( ( ord_less_eq_set_c @ AA @ A )
& ( B
= ( image_c_b @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1029_subset__image__iff,axiom,
! [B: set_a,F2: b > a,A: set_b] :
( ( ord_less_eq_set_a @ B @ ( image_b_a @ F2 @ A ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A )
& ( B
= ( image_b_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1030_subset__image__iff,axiom,
! [B: set_a,F2: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F2 @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1031_subset__image__iff,axiom,
! [B: set_a,F2: $o > a,A: set_o] :
( ( ord_less_eq_set_a @ B @ ( image_o_a @ F2 @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B
= ( image_o_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1032_subset__image__iff,axiom,
! [B: set_a,F2: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F2 @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1033_subset__image__iff,axiom,
! [B: set_a,F2: c > a,A: set_c] :
( ( ord_less_eq_set_a @ B @ ( image_c_a @ F2 @ A ) )
= ( ? [AA: set_c] :
( ( ord_less_eq_set_c @ AA @ A )
& ( B
= ( image_c_a @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1034_image__subset__iff,axiom,
! [F2: nat > set_b,A: set_nat,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ ( image_nat_set_b @ F2 @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_set_b @ ( F2 @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1035_image__subset__iff,axiom,
! [F2: nat > set_a,A: set_nat,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ F2 @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_set_a @ ( F2 @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_1036_subset__imageE,axiom,
! [B: set_b,F2: b > b,A: set_b] :
( ( ord_less_eq_set_b @ B @ ( image_b_b @ F2 @ A ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A )
=> ( B
!= ( image_b_b @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1037_subset__imageE,axiom,
! [B: set_b,F2: a > b,A: set_a] :
( ( ord_less_eq_set_b @ B @ ( image_a_b @ F2 @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_b @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1038_subset__imageE,axiom,
! [B: set_b,F2: $o > b,A: set_o] :
( ( ord_less_eq_set_b @ B @ ( image_o_b @ F2 @ A ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
=> ( B
!= ( image_o_b @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1039_subset__imageE,axiom,
! [B: set_b,F2: nat > b,A: set_nat] :
( ( ord_less_eq_set_b @ B @ ( image_nat_b @ F2 @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_b @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1040_subset__imageE,axiom,
! [B: set_b,F2: c > b,A: set_c] :
( ( ord_less_eq_set_b @ B @ ( image_c_b @ F2 @ A ) )
=> ~ ! [C3: set_c] :
( ( ord_less_eq_set_c @ C3 @ A )
=> ( B
!= ( image_c_b @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1041_subset__imageE,axiom,
! [B: set_a,F2: b > a,A: set_b] :
( ( ord_less_eq_set_a @ B @ ( image_b_a @ F2 @ A ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A )
=> ( B
!= ( image_b_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1042_subset__imageE,axiom,
! [B: set_a,F2: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F2 @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1043_subset__imageE,axiom,
! [B: set_a,F2: $o > a,A: set_o] :
( ( ord_less_eq_set_a @ B @ ( image_o_a @ F2 @ A ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
=> ( B
!= ( image_o_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1044_subset__imageE,axiom,
! [B: set_a,F2: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F2 @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1045_subset__imageE,axiom,
! [B: set_a,F2: c > a,A: set_c] :
( ( ord_less_eq_set_a @ B @ ( image_c_a @ F2 @ A ) )
=> ~ ! [C3: set_c] :
( ( ord_less_eq_set_c @ C3 @ A )
=> ( B
!= ( image_c_a @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_1046_image__subsetI,axiom,
! [A: set_c,F2: c > b,B: set_b] :
( ! [X2: c] :
( ( member_c @ X2 @ A )
=> ( member_b @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_c_b @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1047_image__subsetI,axiom,
! [A: set_b,F2: b > b,B: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( member_b @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1048_image__subsetI,axiom,
! [A: set_a,F2: a > b,B: set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_b @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1049_image__subsetI,axiom,
! [A: set_o,F2: $o > b,B: set_b] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member_b @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_o_b @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1050_image__subsetI,axiom,
! [A: set_nat,F2: nat > b,B: set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_b @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_nat_b @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1051_image__subsetI,axiom,
! [A: set_c,F2: c > a,B: set_a] :
( ! [X2: c] :
( ( member_c @ X2 @ A )
=> ( member_a @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_c_a @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1052_image__subsetI,axiom,
! [A: set_b,F2: b > a,B: set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A )
=> ( member_a @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1053_image__subsetI,axiom,
! [A: set_a,F2: a > a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1054_image__subsetI,axiom,
! [A: set_o,F2: $o > a,B: set_a] :
( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( member_a @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_o_a @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1055_image__subsetI,axiom,
! [A: set_nat,F2: nat > a,B: set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F2 @ X2 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_1056_image__mono,axiom,
! [A: set_b,B: set_b,F2: b > b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_b @ ( image_b_b @ F2 @ A ) @ ( image_b_b @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1057_image__mono,axiom,
! [A: set_b,B: set_b,F2: b > a] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_a @ ( image_b_a @ F2 @ A ) @ ( image_b_a @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1058_image__mono,axiom,
! [A: set_b,B: set_b,F2: b > $o] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_o @ ( image_b_o @ F2 @ A ) @ ( image_b_o @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1059_image__mono,axiom,
! [A: set_b,B: set_b,F2: b > nat] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_b_nat @ F2 @ A ) @ ( image_b_nat @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1060_image__mono,axiom,
! [A: set_b,B: set_b,F2: b > c] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ord_less_eq_set_c @ ( image_b_c @ F2 @ A ) @ ( image_b_c @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1061_image__mono,axiom,
! [A: set_a,B: set_a,F2: a > b] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_b @ ( image_a_b @ F2 @ A ) @ ( image_a_b @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1062_image__mono,axiom,
! [A: set_a,B: set_a,F2: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ ( image_a_a @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1063_image__mono,axiom,
! [A: set_a,B: set_a,F2: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_o @ ( image_a_o @ F2 @ A ) @ ( image_a_o @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1064_image__mono,axiom,
! [A: set_a,B: set_a,F2: a > nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ ( image_a_nat @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1065_image__mono,axiom,
! [A: set_a,B: set_a,F2: a > c] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_c @ ( image_a_c @ F2 @ A ) @ ( image_a_c @ F2 @ B ) ) ) ).
% image_mono
thf(fact_1066_Setcompr__eq__image,axiom,
! [F2: c > a,A: set_c] :
( ( collect_a
@ ^ [Uu: a] :
? [X3: c] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_c @ X3 @ A ) ) )
= ( image_c_a @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1067_Setcompr__eq__image,axiom,
! [F2: b > a,A: set_b] :
( ( collect_a
@ ^ [Uu: a] :
? [X3: b] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_b @ X3 @ A ) ) )
= ( image_b_a @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1068_Setcompr__eq__image,axiom,
! [F2: a > a,A: set_a] :
( ( collect_a
@ ^ [Uu: a] :
? [X3: a] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_a @ X3 @ A ) ) )
= ( image_a_a @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1069_Setcompr__eq__image,axiom,
! [F2: $o > a,A: set_o] :
( ( collect_a
@ ^ [Uu: a] :
? [X3: $o] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_o @ X3 @ A ) ) )
= ( image_o_a @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1070_Setcompr__eq__image,axiom,
! [F2: nat > a,A: set_nat] :
( ( collect_a
@ ^ [Uu: a] :
? [X3: nat] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_nat @ X3 @ A ) ) )
= ( image_nat_a @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1071_Setcompr__eq__image,axiom,
! [F2: c > b,A: set_c] :
( ( collect_b
@ ^ [Uu: b] :
? [X3: c] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_c @ X3 @ A ) ) )
= ( image_c_b @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1072_Setcompr__eq__image,axiom,
! [F2: b > b,A: set_b] :
( ( collect_b
@ ^ [Uu: b] :
? [X3: b] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_b @ X3 @ A ) ) )
= ( image_b_b @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1073_Setcompr__eq__image,axiom,
! [F2: a > b,A: set_a] :
( ( collect_b
@ ^ [Uu: b] :
? [X3: a] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_a @ X3 @ A ) ) )
= ( image_a_b @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1074_Setcompr__eq__image,axiom,
! [F2: $o > b,A: set_o] :
( ( collect_b
@ ^ [Uu: b] :
? [X3: $o] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_o @ X3 @ A ) ) )
= ( image_o_b @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1075_Setcompr__eq__image,axiom,
! [F2: nat > b,A: set_nat] :
( ( collect_b
@ ^ [Uu: b] :
? [X3: nat] :
( ( Uu
= ( F2 @ X3 ) )
& ( member_nat @ X3 @ A ) ) )
= ( image_nat_b @ F2 @ A ) ) ).
% Setcompr_eq_image
thf(fact_1076_setcompr__eq__image,axiom,
! [F2: a > a,P: a > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X3: a] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_a_a @ F2 @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1077_setcompr__eq__image,axiom,
! [F2: b > a,P: b > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X3: b] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_b_a @ F2 @ ( collect_b @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1078_setcompr__eq__image,axiom,
! [F2: nat > a,P: nat > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X3: nat] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_nat_a @ F2 @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1079_setcompr__eq__image,axiom,
! [F2: a > b,P: a > $o] :
( ( collect_b
@ ^ [Uu: b] :
? [X3: a] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_a_b @ F2 @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1080_setcompr__eq__image,axiom,
! [F2: b > b,P: b > $o] :
( ( collect_b
@ ^ [Uu: b] :
? [X3: b] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_b_b @ F2 @ ( collect_b @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1081_setcompr__eq__image,axiom,
! [F2: nat > b,P: nat > $o] :
( ( collect_b
@ ^ [Uu: b] :
? [X3: nat] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_nat_b @ F2 @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1082_setcompr__eq__image,axiom,
! [F2: a > nat,P: a > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: a] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_a_nat @ F2 @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1083_setcompr__eq__image,axiom,
! [F2: b > nat,P: b > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: b] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_b_nat @ F2 @ ( collect_b @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1084_setcompr__eq__image,axiom,
! [F2: nat > nat,P: nat > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X3: nat] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F2 @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1085_setcompr__eq__image,axiom,
! [F2: nat > set_a,P: nat > $o] :
( ( collect_set_a
@ ^ [Uu: set_a] :
? [X3: nat] :
( ( Uu
= ( F2 @ X3 ) )
& ( P @ X3 ) ) )
= ( image_nat_set_a @ F2 @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_1086_image__Int__subset,axiom,
! [F2: b > d,A: set_b,B: set_b] : ( ord_less_eq_set_d @ ( image_b_d @ F2 @ ( inf_inf_set_b @ A @ B ) ) @ ( inf_inf_set_d @ ( image_b_d @ F2 @ A ) @ ( image_b_d @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1087_image__Int__subset,axiom,
! [F2: a > d,A: set_a,B: set_a] : ( ord_less_eq_set_d @ ( image_a_d @ F2 @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_d @ ( image_a_d @ F2 @ A ) @ ( image_a_d @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1088_image__Int__subset,axiom,
! [F2: d > d,A: set_d,B: set_d] : ( ord_less_eq_set_d @ ( image_d_d @ F2 @ ( inf_inf_set_d @ A @ B ) ) @ ( inf_inf_set_d @ ( image_d_d @ F2 @ A ) @ ( image_d_d @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1089_image__Int__subset,axiom,
! [F2: b > b,A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( image_b_b @ F2 @ ( inf_inf_set_b @ A @ B ) ) @ ( inf_inf_set_b @ ( image_b_b @ F2 @ A ) @ ( image_b_b @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1090_image__Int__subset,axiom,
! [F2: a > b,A: set_a,B: set_a] : ( ord_less_eq_set_b @ ( image_a_b @ F2 @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_b @ ( image_a_b @ F2 @ A ) @ ( image_a_b @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1091_image__Int__subset,axiom,
! [F2: d > b,A: set_d,B: set_d] : ( ord_less_eq_set_b @ ( image_d_b @ F2 @ ( inf_inf_set_d @ A @ B ) ) @ ( inf_inf_set_b @ ( image_d_b @ F2 @ A ) @ ( image_d_b @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1092_image__Int__subset,axiom,
! [F2: b > a,A: set_b,B: set_b] : ( ord_less_eq_set_a @ ( image_b_a @ F2 @ ( inf_inf_set_b @ A @ B ) ) @ ( inf_inf_set_a @ ( image_b_a @ F2 @ A ) @ ( image_b_a @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1093_image__Int__subset,axiom,
! [F2: a > a,A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F2 @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F2 @ A ) @ ( image_a_a @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1094_image__Int__subset,axiom,
! [F2: d > a,A: set_d,B: set_d] : ( ord_less_eq_set_a @ ( image_d_a @ F2 @ ( inf_inf_set_d @ A @ B ) ) @ ( inf_inf_set_a @ ( image_d_a @ F2 @ A ) @ ( image_d_a @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1095_image__Int__subset,axiom,
! [F2: b > $o,A: set_b,B: set_b] : ( ord_less_eq_set_o @ ( image_b_o @ F2 @ ( inf_inf_set_b @ A @ B ) ) @ ( inf_inf_set_o @ ( image_b_o @ F2 @ A ) @ ( image_b_o @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_1096_image__subset__iff__subset__vimage,axiom,
! [F2: b > d,A: set_b,B: set_d] :
( ( ord_less_eq_set_d @ ( image_b_d @ F2 @ A ) @ B )
= ( ord_less_eq_set_b @ A @ ( vimage_b_d @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1097_image__subset__iff__subset__vimage,axiom,
! [F2: b > b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ ( image_b_b @ F2 @ A ) @ B )
= ( ord_less_eq_set_b @ A @ ( vimage_b_b @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1098_image__subset__iff__subset__vimage,axiom,
! [F2: a > b,A: set_a,B: set_b] :
( ( ord_less_eq_set_b @ ( image_a_b @ F2 @ A ) @ B )
= ( ord_less_eq_set_a @ A @ ( vimage_a_b @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1099_image__subset__iff__subset__vimage,axiom,
! [F2: $o > b,A: set_o,B: set_b] :
( ( ord_less_eq_set_b @ ( image_o_b @ F2 @ A ) @ B )
= ( ord_less_eq_set_o @ A @ ( vimage_o_b @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1100_image__subset__iff__subset__vimage,axiom,
! [F2: nat > b,A: set_nat,B: set_b] :
( ( ord_less_eq_set_b @ ( image_nat_b @ F2 @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_b @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1101_image__subset__iff__subset__vimage,axiom,
! [F2: c > b,A: set_c,B: set_b] :
( ( ord_less_eq_set_b @ ( image_c_b @ F2 @ A ) @ B )
= ( ord_less_eq_set_c @ A @ ( vimage_c_b @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1102_image__subset__iff__subset__vimage,axiom,
! [F2: b > a,A: set_b,B: set_a] :
( ( ord_less_eq_set_a @ ( image_b_a @ F2 @ A ) @ B )
= ( ord_less_eq_set_b @ A @ ( vimage_b_a @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1103_image__subset__iff__subset__vimage,axiom,
! [F2: a > a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A ) @ B )
= ( ord_less_eq_set_a @ A @ ( vimage_a_a @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1104_image__subset__iff__subset__vimage,axiom,
! [F2: $o > a,A: set_o,B: set_a] :
( ( ord_less_eq_set_a @ ( image_o_a @ F2 @ A ) @ B )
= ( ord_less_eq_set_o @ A @ ( vimage_o_a @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1105_image__subset__iff__subset__vimage,axiom,
! [F2: nat > a,A: set_nat,B: set_a] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_a @ F2 @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1106_image__vimage__subset,axiom,
! [F2: b > d,A: set_d] : ( ord_less_eq_set_d @ ( image_b_d @ F2 @ ( vimage_b_d @ F2 @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_1107_image__vimage__subset,axiom,
! [F2: nat > set_b,A: set_set_b] : ( ord_le3795704787696855135_set_b @ ( image_nat_set_b @ F2 @ ( vimage_nat_set_b @ F2 @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_1108_image__vimage__subset,axiom,
! [F2: nat > set_a,A: set_set_a] : ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ F2 @ ( vimage_nat_set_a @ F2 @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_1109_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: c > sigma_measure_a,X4: c > a > a,I: set_c,Y5: c > a > b,N: c > sigma_measure_b] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k3786629878328907207_a_c_a @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k3786629878328907208_a_c_b @ M @ K2 @ N
@ ^ [I3: c,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1110_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: b > sigma_measure_a,X4: b > a > a,I: set_b,Y5: b > a > b,N: b > sigma_measure_b] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k6574085460301583240_a_b_a @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k6574085460301583241_a_b_b @ M @ K2 @ N
@ ^ [I3: b,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1111_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: a > sigma_measure_a,X4: a > a > a,I: set_a,Y5: a > a > b,N: a > sigma_measure_b] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k138169005419483465_a_a_a @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k138169005419483466_a_a_b @ M @ K2 @ N
@ ^ [I3: a,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1112_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: $o > sigma_measure_a,X4: $o > a > a,I: set_o,Y5: $o > a > b,N: $o > sigma_measure_b] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k8052462064657410095_a_o_a @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k8052462064657410096_a_o_b @ M @ K2 @ N
@ ^ [I3: $o,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1113_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: nat > sigma_measure_a,X4: nat > a > a,I: set_nat,Y5: nat > a > b,N: nat > sigma_measure_b] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k6325968634923510307_nat_a @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_a_b @ ( Y5 @ I2 ) @ ( sigma_measurable_a_b @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k6325968634923510308_nat_b @ M @ K2 @ N
@ ^ [I3: nat,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1114_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: c > sigma_measure_b,X4: c > a > b,I: set_c,Y5: c > b > d,N: c > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k3786629878328907208_a_c_b @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k3786629878328907210_a_c_d @ M @ K2 @ N
@ ^ [I3: c,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1115_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: b > sigma_measure_b,X4: b > a > b,I: set_b,Y5: b > b > d,N: b > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k6574085460301583241_a_b_b @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k6574085460301583243_a_b_d @ M @ K2 @ N
@ ^ [I3: b,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1116_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: a > sigma_measure_b,X4: a > a > b,I: set_a,Y5: a > b > d,N: a > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k138169005419483466_a_a_b @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k138169005419483468_a_a_d @ M @ K2 @ N
@ ^ [I3: a,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1117_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: $o > sigma_measure_b,X4: $o > a > b,I: set_o,Y5: $o > b > d,N: $o > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k8052462064657410096_a_o_b @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k8052462064657410098_a_o_d @ M @ K2 @ N
@ ^ [I3: $o,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1118_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K2: nat,M2: nat > sigma_measure_b,X4: nat > a > b,I: set_nat,Y5: nat > b > d,N: nat > sigma_measure_d] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k6325968634923510308_nat_b @ M @ K2 @ M2 @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_b_d @ ( Y5 @ I2 ) @ ( sigma_measurable_b_d @ ( M2 @ I2 ) @ ( N @ I2 ) ) ) )
=> ( prob_k6325968634923510310_nat_d @ M @ K2 @ N
@ ^ [I3: nat,X3: a] : ( Y5 @ I3 @ ( X4 @ I3 @ X3 ) )
@ I ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_1119_measure__distr,axiom,
! [F2: b > b,M: sigma_measure_b,N: sigma_measure_b,S: set_b] :
( ( member_b_b @ F2 @ ( sigma_measurable_b_b @ M @ N ) )
=> ( ( member_set_b @ S @ ( sigma_sets_b @ N ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_b_b @ M @ N @ F2 ) @ S )
= ( sigma_measure_b2 @ M @ ( inf_inf_set_b @ ( vimage_b_b @ F2 @ S ) @ ( sigma_space_b @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1120_measure__distr,axiom,
! [F2: b > a,M: sigma_measure_b,N: sigma_measure_a,S: set_a] :
( ( member_b_a @ F2 @ ( sigma_measurable_b_a @ M @ N ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ N ) )
=> ( ( sigma_measure_a2 @ ( measure_distr_b_a @ M @ N @ F2 ) @ S )
= ( sigma_measure_b2 @ M @ ( inf_inf_set_b @ ( vimage_b_a @ F2 @ S ) @ ( sigma_space_b @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1121_measure__distr,axiom,
! [F2: a > d,M: sigma_measure_a,N: sigma_measure_d,S: set_d] :
( ( member_a_d @ F2 @ ( sigma_measurable_a_d @ M @ N ) )
=> ( ( member_set_d @ S @ ( sigma_sets_d @ N ) )
=> ( ( sigma_measure_d2 @ ( measure_distr_a_d @ M @ N @ F2 ) @ S )
= ( sigma_measure_a2 @ M @ ( inf_inf_set_a @ ( vimage_a_d @ F2 @ S ) @ ( sigma_space_a @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1122_measure__distr,axiom,
! [F2: a > a,M: sigma_measure_a,N: sigma_measure_a,S: set_a] :
( ( member_a_a @ F2 @ ( sigma_measurable_a_a @ M @ N ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ N ) )
=> ( ( sigma_measure_a2 @ ( measure_distr_a_a @ M @ N @ F2 ) @ S )
= ( sigma_measure_a2 @ M @ ( inf_inf_set_a @ ( vimage_a_a @ F2 @ S ) @ ( sigma_space_a @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1123_measure__distr,axiom,
! [F2: d > d,M: sigma_measure_d,N: sigma_measure_d,S: set_d] :
( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ M @ N ) )
=> ( ( member_set_d @ S @ ( sigma_sets_d @ N ) )
=> ( ( sigma_measure_d2 @ ( measure_distr_d_d @ M @ N @ F2 ) @ S )
= ( sigma_measure_d2 @ M @ ( inf_inf_set_d @ ( vimage_d_d @ F2 @ S ) @ ( sigma_space_d @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1124_measure__distr,axiom,
! [F2: d > b,M: sigma_measure_d,N: sigma_measure_b,S: set_b] :
( ( member_d_b @ F2 @ ( sigma_measurable_d_b @ M @ N ) )
=> ( ( member_set_b @ S @ ( sigma_sets_b @ N ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_d_b @ M @ N @ F2 ) @ S )
= ( sigma_measure_d2 @ M @ ( inf_inf_set_d @ ( vimage_d_b @ F2 @ S ) @ ( sigma_space_d @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1125_measure__distr,axiom,
! [F2: d > a,M: sigma_measure_d,N: sigma_measure_a,S: set_a] :
( ( member_d_a @ F2 @ ( sigma_measurable_d_a @ M @ N ) )
=> ( ( member_set_a @ S @ ( sigma_sets_a @ N ) )
=> ( ( sigma_measure_a2 @ ( measure_distr_d_a @ M @ N @ F2 ) @ S )
= ( sigma_measure_d2 @ M @ ( inf_inf_set_d @ ( vimage_d_a @ F2 @ S ) @ ( sigma_space_d @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1126_measure__distr,axiom,
! [F2: a > b,M: sigma_measure_a,N: sigma_measure_b,S: set_b] :
( ( member_a_b @ F2 @ ( sigma_measurable_a_b @ M @ N ) )
=> ( ( member_set_b @ S @ ( sigma_sets_b @ N ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ M @ N @ F2 ) @ S )
= ( sigma_measure_a2 @ M @ ( inf_inf_set_a @ ( vimage_a_b @ F2 @ S ) @ ( sigma_space_a @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1127_measure__distr,axiom,
! [F2: b > d,M: sigma_measure_b,N: sigma_measure_d,S: set_d] :
( ( member_b_d @ F2 @ ( sigma_measurable_b_d @ M @ N ) )
=> ( ( member_set_d @ S @ ( sigma_sets_d @ N ) )
=> ( ( sigma_measure_d2 @ ( measure_distr_b_d @ M @ N @ F2 ) @ S )
= ( sigma_measure_b2 @ M @ ( inf_inf_set_b @ ( vimage_b_d @ F2 @ S ) @ ( sigma_space_b @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1128_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_d,F2: d > d,N: sigma_measure_d,A: c > set_set_d,I: set_c] :
( ( probab7247484486040049092pace_d @ M )
=> ( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ M @ N ) )
=> ( ( indepe565075120755390507ts_d_c @ M
@ ^ [I3: c] :
( image_set_d_set_d
@ ^ [A5: set_d] : ( inf_inf_set_d @ ( vimage_d_d @ F2 @ A5 ) @ ( sigma_space_d @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe565075120755390507ts_d_c @ ( measure_distr_d_d @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1129_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_d,F2: d > d,N: sigma_measure_d,A: b > set_set_d,I: set_b] :
( ( probab7247484486040049092pace_d @ M )
=> ( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ M @ N ) )
=> ( ( indepe565075120755390506ts_d_b @ M
@ ^ [I3: b] :
( image_set_d_set_d
@ ^ [A5: set_d] : ( inf_inf_set_d @ ( vimage_d_d @ F2 @ A5 ) @ ( sigma_space_d @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe565075120755390506ts_d_b @ ( measure_distr_d_d @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1130_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_d,F2: d > d,N: sigma_measure_d,A: a > set_set_d,I: set_a] :
( ( probab7247484486040049092pace_d @ M )
=> ( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ M @ N ) )
=> ( ( indepe565075120755390505ts_d_a @ M
@ ^ [I3: a] :
( image_set_d_set_d
@ ^ [A5: set_d] : ( inf_inf_set_d @ ( vimage_d_d @ F2 @ A5 ) @ ( sigma_space_d @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe565075120755390505ts_d_a @ ( measure_distr_d_d @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1131_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_d,F2: d > d,N: sigma_measure_d,A: $o > set_set_d,I: set_o] :
( ( probab7247484486040049092pace_d @ M )
=> ( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ M @ N ) )
=> ( ( indepe8305812671657367107ts_d_o @ M
@ ^ [I3: $o] :
( image_set_d_set_d
@ ^ [A5: set_d] : ( inf_inf_set_d @ ( vimage_d_d @ F2 @ A5 ) @ ( sigma_space_d @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe8305812671657367107ts_d_o @ ( measure_distr_d_d @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1132_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_d,F2: d > d,N: sigma_measure_d,A: nat > set_set_d,I: set_nat] :
( ( probab7247484486040049092pace_d @ M )
=> ( ( member_d_d @ F2 @ ( sigma_measurable_d_d @ M @ N ) )
=> ( ( indepe750690977103256037_d_nat @ M
@ ^ [I3: nat] :
( image_set_d_set_d
@ ^ [A5: set_d] : ( inf_inf_set_d @ ( vimage_d_d @ F2 @ A5 ) @ ( sigma_space_d @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe750690977103256037_d_nat @ ( measure_distr_d_d @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1133_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_d,F2: d > a,N: sigma_measure_a,A: c > set_set_a,I: set_c] :
( ( probab7247484486040049092pace_d @ M )
=> ( ( member_d_a @ F2 @ ( sigma_measurable_d_a @ M @ N ) )
=> ( ( indepe565075120755390507ts_d_c @ M
@ ^ [I3: c] :
( image_set_a_set_d
@ ^ [A5: set_a] : ( inf_inf_set_d @ ( vimage_d_a @ F2 @ A5 ) @ ( sigma_space_d @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ N ) ) )
=> ( indepe8927441866673418606ts_a_c @ ( measure_distr_d_a @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1134_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_d,F2: d > a,N: sigma_measure_a,A: b > set_set_a,I: set_b] :
( ( probab7247484486040049092pace_d @ M )
=> ( ( member_d_a @ F2 @ ( sigma_measurable_d_a @ M @ N ) )
=> ( ( indepe565075120755390506ts_d_b @ M
@ ^ [I3: b] :
( image_set_a_set_d
@ ^ [A5: set_a] : ( inf_inf_set_d @ ( vimage_d_a @ F2 @ A5 ) @ ( sigma_space_d @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ N ) ) )
=> ( indepe8927441866673418605ts_a_b @ ( measure_distr_d_a @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1135_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_d,F2: d > a,N: sigma_measure_a,A: a > set_set_a,I: set_a] :
( ( probab7247484486040049092pace_d @ M )
=> ( ( member_d_a @ F2 @ ( sigma_measurable_d_a @ M @ N ) )
=> ( ( indepe565075120755390505ts_d_a @ M
@ ^ [I3: a] :
( image_set_a_set_d
@ ^ [A5: set_a] : ( inf_inf_set_d @ ( vimage_d_a @ F2 @ A5 ) @ ( sigma_space_d @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ N ) ) )
=> ( indepe8927441866673418604ts_a_a @ ( measure_distr_d_a @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1136_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_a,F2: a > d,N: sigma_measure_d,A: c > set_set_d,I: set_c] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_d @ F2 @ ( sigma_measurable_a_d @ M @ N ) )
=> ( ( indepe8927441866673418606ts_a_c @ M
@ ^ [I3: c] :
( image_set_d_set_a
@ ^ [A5: set_d] : ( inf_inf_set_a @ ( vimage_a_d @ F2 @ A5 ) @ ( sigma_space_a @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe565075120755390507ts_d_c @ ( measure_distr_a_d @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1137_prob__space_Oindep__sets__distr,axiom,
! [M: sigma_measure_a,F2: a > d,N: sigma_measure_d,A: b > set_set_d,I: set_b] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_d @ F2 @ ( sigma_measurable_a_d @ M @ N ) )
=> ( ( indepe8927441866673418605ts_a_b @ M
@ ^ [I3: b] :
( image_set_d_set_a
@ ^ [A5: set_d] : ( inf_inf_set_a @ ( vimage_a_d @ F2 @ A5 ) @ ( sigma_space_a @ M ) )
@ ( A @ I3 ) )
@ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( ord_le3937772867789547105_set_d @ ( A @ I2 ) @ ( sigma_sets_d @ N ) ) )
=> ( indepe565075120755390506ts_d_b @ ( measure_distr_a_d @ M @ N @ F2 ) @ A @ I ) ) ) ) ) ).
% prob_space.indep_sets_distr
thf(fact_1138_D_Oindep__eventsI__indep__vars,axiom,
! [N: c > sigma_measure_c,X4: c > b > c,I: set_c,P: c > c > $o] :
( ( indepe1224748351465797815_b_c_c @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_c @ ( N @ I2 ) ) ) )
=> ( indepe8384762880973921173ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1139_D_Oindep__eventsI__indep__vars,axiom,
! [N: c > sigma_measure_o,X4: c > b > $o,I: set_c,P: c > $o > $o] :
( ( indepe7992535546961976527_b_c_o @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: c] :
( ( member_c @ I2 @ I )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_o @ ( N @ I2 ) ) ) )
=> ( indepe8384762880973921173ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1140_D_Oindep__eventsI__indep__vars,axiom,
! [N: b > sigma_measure_c,X4: b > b > c,I: set_b,P: b > c > $o] :
( ( indepe4012203933438473848_b_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_c @ ( N @ I2 ) ) ) )
=> ( indepe8384762880973921172ts_b_b @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1141_D_Oindep__eventsI__indep__vars,axiom,
! [N: b > sigma_measure_o,X4: b > b > $o,I: set_b,P: b > $o > $o] :
( ( indepe1668385909571595024_b_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: b] :
( ( member_b @ I2 @ I )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_o @ ( N @ I2 ) ) ) )
=> ( indepe8384762880973921172ts_b_b @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1142_D_Oindep__eventsI__indep__vars,axiom,
! [N: a > sigma_measure_c,X4: a > b > c,I: set_a,P: a > c > $o] :
( ( indepe6799659515411149881_b_a_c @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_c @ ( N @ I2 ) ) ) )
=> ( indepe8384762880973921171ts_b_a @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: a] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1143_D_Oindep__eventsI__indep__vars,axiom,
! [N: a > sigma_measure_o,X4: a > b > $o,I: set_a,P: a > $o > $o] :
( ( indepe4567608309035989329_b_a_o @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_o @ ( N @ I2 ) ) ) )
=> ( indepe8384762880973921171ts_b_a @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: a] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1144_D_Oindep__eventsI__indep__vars,axiom,
! [N: $o > sigma_measure_c,X4: $o > b > c,I: set_o,P: $o > c > $o] :
( ( indepe4488128152570241567_b_o_c @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_c @ ( N @ I2 ) ) ) )
=> ( indepe796274259248320173ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: $o] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1145_D_Oindep__eventsI__indep__vars,axiom,
! [N: $o > sigma_measure_o,X4: $o > b > $o,I: set_o,P: $o > $o > $o] :
( ( indepe2258988419405707831_b_o_o @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_o @ ( N @ I2 ) ) ) )
=> ( indepe796274259248320173ts_b_o @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: $o] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1146_D_Oindep__eventsI__indep__vars,axiom,
! [N: nat > sigma_measure_c,X4: nat > b > c,I: set_nat,P: nat > c > $o] :
( ( indepe4589662937738594231_nat_c @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_set_c
@ ( collect_c
@ ^ [X3: c] :
( ( member_c @ X3 @ ( sigma_space_c @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_c @ ( N @ I2 ) ) ) )
=> ( indepe2786641642957426683_b_nat @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: nat] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1147_D_Oindep__eventsI__indep__vars,axiom,
! [N: nat > sigma_measure_o,X4: nat > b > $o,I: set_nat,P: nat > $o > $o] :
( ( indepe3736482776648739791_nat_o @ ( measure_distr_a_b @ m @ n @ f ) @ N @ X4 @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( member_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ ( sigma_space_o @ ( N @ I2 ) ) )
& ( P @ I2 @ X3 ) ) )
@ ( sigma_sets_o @ ( N @ I2 ) ) ) )
=> ( indepe2786641642957426683_b_nat @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: nat] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ I3 @ ( X4 @ I3 @ X3 ) ) ) )
@ I ) ) ) ).
% D.indep_eventsI_indep_vars
thf(fact_1148_D_Oindep__events__finite__index__events,axiom,
! [F: b > set_b,I: set_b] :
( ( indepe8384762880973921172ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_b] :
( ( ord_less_eq_set_b @ J2 @ I )
=> ( ( J2 != bot_bot_set_b )
=> ( ( finite_finite_b @ J2 )
=> ( indepe8384762880973921172ts_b_b @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_events_finite_index_events
thf(fact_1149_D_Oindep__events__finite__index__events,axiom,
! [F: a > set_b,I: set_a] :
( ( indepe8384762880973921171ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_a] :
( ( ord_less_eq_set_a @ J2 @ I )
=> ( ( J2 != bot_bot_set_a )
=> ( ( finite_finite_a @ J2 )
=> ( indepe8384762880973921171ts_b_a @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_events_finite_index_events
thf(fact_1150_D_Oindep__events__finite__index__events,axiom,
! [F: $o > set_b,I: set_o] :
( ( indepe796274259248320173ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_o] :
( ( ord_less_eq_set_o @ J2 @ I )
=> ( ( J2 != bot_bot_set_o )
=> ( ( finite_finite_o @ J2 )
=> ( indepe796274259248320173ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_events_finite_index_events
thf(fact_1151_D_Oindep__events__finite__index__events,axiom,
! [F: nat > set_b,I: set_nat] :
( ( indepe2786641642957426683_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_nat] :
( ( ord_less_eq_set_nat @ J2 @ I )
=> ( ( J2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ J2 )
=> ( indepe2786641642957426683_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_events_finite_index_events
thf(fact_1152_D_Oindep__events__finite__index__events,axiom,
! [F: c > set_b,I: set_c] :
( ( indepe8384762880973921173ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ I )
= ( ! [J2: set_c] :
( ( ord_less_eq_set_c @ J2 @ I )
=> ( ( J2 != bot_bot_set_c )
=> ( ( finite_finite_c @ J2 )
=> ( indepe8384762880973921173ts_b_c @ ( measure_distr_a_b @ m @ n @ f ) @ F @ J2 ) ) ) ) ) ) ).
% D.indep_events_finite_index_events
thf(fact_1153_D_Omeasure__exclude,axiom,
! [A: set_b,B: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A )
= ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) )
=> ( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B )
= zero_zero_real ) ) ) ) ) ).
% D.measure_exclude
thf(fact_1154_D_Oindep__setI,axiom,
! [A: set_set_b,B: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ! [A6: set_b,B4: set_b] :
( ( member_set_b @ A6 @ A )
=> ( ( member_set_b @ B4 @ B )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ A6 @ B4 ) )
= ( times_times_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A6 ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B4 ) ) ) ) )
=> ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B ) ) ) ) ).
% D.indep_setI
thf(fact_1155_D_Oindep__sets2__eq,axiom,
! [A: set_set_b,B: set_set_b] :
( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
= ( ( ord_le3795704787696855135_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( ord_le3795704787696855135_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ! [X3: set_b] :
( ( member_set_b @ X3 @ A )
=> ! [Y: set_b] :
( ( member_set_b @ Y @ B )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ X3 @ Y ) )
= ( times_times_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ X3 ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ Y ) ) ) ) ) ) ) ).
% D.indep_sets2_eq
thf(fact_1156_indep__setD__ev1,axiom,
! [A: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev1
thf(fact_1157_indep__setD__ev2,axiom,
! [A: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
=> ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev2
thf(fact_1158_measure__space__inter,axiom,
! [S2: set_a,T3: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ T3 )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ S2 @ T3 ) )
= ( sigma_measure_a2 @ m @ S2 ) ) ) ) ) ).
% measure_space_inter
thf(fact_1159_indep__setD,axiom,
! [A: set_set_a,B: set_set_a,A2: set_a,B2: set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
=> ( ( member_set_a @ A2 @ A )
=> ( ( member_set_a @ B2 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B2 ) ) ) ) ) ) ).
% indep_setD
thf(fact_1160_measure__exclude,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ A )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ m @ B )
= zero_zero_real ) ) ) ) ) ).
% measure_exclude
thf(fact_1161_indep__setI,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ! [A6: set_a,B4: set_a] :
( ( member_set_a @ A6 @ A )
=> ( ( member_set_a @ B4 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A6 @ B4 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A6 ) @ ( sigma_measure_a2 @ m @ B4 ) ) ) ) )
=> ( indepe2041756565122539606_set_a @ m @ A @ B ) ) ) ) ).
% indep_setI
thf(fact_1162_indep__sets2__eq,axiom,
! [A: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
= ( ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ m ) )
& ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
& ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ X3 @ Y ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ X3 ) @ ( sigma_measure_a2 @ m @ Y ) ) ) ) ) ) ) ).
% indep_sets2_eq
thf(fact_1163_D_Oindep__setD,axiom,
! [A: set_set_b,B: set_set_b,A2: set_b,B2: set_b] :
( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
=> ( ( member_set_b @ A2 @ A )
=> ( ( member_set_b @ B2 @ B )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ A2 @ B2 ) )
= ( times_times_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A2 ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B2 ) ) ) ) ) ) ).
% D.indep_setD
thf(fact_1164_measure__increasing,axiom,
measur1776380161843274167a_real @ ( sigma_sets_a @ m ) @ ( sigma_measure_a2 @ m ) ).
% measure_increasing
thf(fact_1165_assms_I3_J,axiom,
( indepe4851901773132442935_a_c_d @ m @ m2
@ ^ [I3: c] : ( comp_b_d_a @ ( x @ I3 ) @ f )
@ i ) ).
% assms(3)
thf(fact_1166_D_Oprob__space,axiom,
( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
= one_one_real ) ).
% D.prob_space
thf(fact_1167_D_Omeasure__eq__compl,axiom,
! [S2: set_b,T3: set_b] :
( ( member_set_b @ S2 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( member_set_b @ T3 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ S2 ) )
= ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ T3 ) ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ S2 )
= ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ T3 ) ) ) ) ) ).
% D.measure_eq_compl
thf(fact_1168_D_Ofinite__measure__mono,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B ) ) ) ) ).
% D.finite_measure_mono
thf(fact_1169_D_Obounded__measure,axiom,
! [A: set_b] : ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) ) ).
% D.bounded_measure
thf(fact_1170_measure__ge__1__iff,axiom,
! [A: set_a] :
( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ m @ A ) )
= ( ( sigma_measure_a2 @ m @ A )
= one_one_real ) ) ).
% measure_ge_1_iff
thf(fact_1171_bounded__measure,axiom,
! [A: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) ) ).
% bounded_measure
thf(fact_1172_finite__measure__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ).
% finite_measure_mono
thf(fact_1173_prob__space,axiom,
( ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) )
= one_one_real ) ).
% prob_space
thf(fact_1174_D_Omeasure__ge__1__iff,axiom,
! [A: set_b] :
( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) )
= ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A )
= one_one_real ) ) ).
% D.measure_ge_1_iff
thf(fact_1175_subprob__measure__le__1,axiom,
! [X4: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ X4 ) @ one_one_real ) ).
% subprob_measure_le_1
thf(fact_1176_prob__le__1,axiom,
! [A: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A ) @ one_one_real ) ).
% prob_le_1
thf(fact_1177_D_Oprob__le__1,axiom,
! [A: set_b] : ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ one_one_real ) ).
% D.prob_le_1
thf(fact_1178_D_Osubprob__measure__le__1,axiom,
! [X4: set_b] : ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ X4 ) @ one_one_real ) ).
% D.subprob_measure_le_1
thf(fact_1179_D_Oindep__set__sigma__sets,axiom,
! [A: set_set_b,B: set_set_b] :
( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
=> ( ( sigma_Int_stable_b @ A )
=> ( ( sigma_Int_stable_b @ B )
=> ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_sigma_sets_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ A ) @ ( sigma_sigma_sets_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ B ) ) ) ) ) ).
% D.indep_set_sigma_sets
thf(fact_1180_D_Oprob__compl,axiom,
! [A: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ A ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) ) ) ) ).
% D.prob_compl
thf(fact_1181_D_Ofinite__measure__Diff,axiom,
! [A: set_b,B: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B ) ) ) ) ) ) ).
% D.finite_measure_Diff
thf(fact_1182_D_Ofinite__measure__Diff_H,axiom,
! [A: set_b,B: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( inf_inf_set_b @ A @ B ) ) ) ) ) ) ).
% D.finite_measure_Diff'
thf(fact_1183_D_Ofinite__measure__compl,axiom,
! [S: set_b] :
( ( member_set_b @ S @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ S ) )
= ( minus_minus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ S ) ) ) ) ).
% D.finite_measure_compl
thf(fact_1184_measure__eq__compl,axiom,
! [S2: set_a,T3: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S2 ) )
= ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ T3 ) ) )
=> ( ( sigma_measure_a2 @ m @ S2 )
= ( sigma_measure_a2 @ m @ T3 ) ) ) ) ) ).
% measure_eq_compl
thf(fact_1185_finite__measure__compl,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) @ ( sigma_measure_a2 @ m @ S ) ) ) ) ).
% finite_measure_compl
thf(fact_1186_finite__measure__Diff_H,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ) ).
% finite_measure_Diff'
thf(fact_1187_finite__measure__Diff,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).
% finite_measure_Diff
thf(fact_1188_prob__compl,axiom,
! [A: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ A ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_a2 @ m @ A ) ) ) ) ).
% prob_compl
thf(fact_1189_prob__neg,axiom,
! [P: a > $o] :
( ( member_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ m ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ m ) )
& ~ ( P @ X3 ) ) ) )
= ( minus_minus_real @ one_one_real
@ ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( sigma_space_a @ m ) )
& ( P @ X3 ) ) ) ) ) ) ) ).
% prob_neg
thf(fact_1190_D_Oprob__neg,axiom,
! [P: b > $o] :
( ( member_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ X3 ) ) )
@ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f )
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ~ ( P @ X3 ) ) ) )
= ( minus_minus_real @ one_one_real
@ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f )
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
& ( P @ X3 ) ) ) ) ) ) ) ).
% D.prob_neg
thf(fact_1191_indep__set__sigma__sets,axiom,
! [A: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
=> ( ( sigma_Int_stable_a @ A )
=> ( ( sigma_Int_stable_a @ B )
=> ( indepe2041756565122539606_set_a @ m @ ( sigma_sigma_sets_a @ ( sigma_space_a @ m ) @ A ) @ ( sigma_sigma_sets_a @ ( sigma_space_a @ m ) @ B ) ) ) ) ) ).
% indep_set_sigma_sets
thf(fact_1192_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1193_D_Omeasure__zero__union,axiom,
! [S2: set_b,T3: set_b] :
( ( member_set_b @ S2 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( member_set_b @ T3 @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ T3 )
= zero_zero_real )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sup_sup_set_b @ S2 @ T3 ) )
= ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ S2 ) ) ) ) ) ).
% D.measure_zero_union
thf(fact_1194_D_Ofinite__measure__Union,axiom,
! [A: set_b,B: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( ( inf_inf_set_b @ A @ B )
= bot_bot_set_b )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sup_sup_set_b @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B ) ) ) ) ) ) ).
% D.finite_measure_Union
thf(fact_1195_D_Ofinite__measure__Union_H,axiom,
! [A: set_b,B: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sup_sup_set_b @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( minus_minus_set_b @ B @ A ) ) ) ) ) ) ).
% D.finite_measure_Union'
thf(fact_1196_D_Ofinite__measure__subadditive,axiom,
! [A: set_b,B: set_b] :
( ( member_set_b @ A @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ( member_set_b @ B @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ord_less_eq_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( sup_sup_set_b @ A @ B ) ) @ ( plus_plus_real @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ A ) @ ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ B ) ) ) ) ) ).
% D.finite_measure_subadditive
thf(fact_1197_diff__is__0__eq_H,axiom,
! [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ( minus_minus_nat @ M3 @ N4 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1198_measure__zero__union,axiom,
! [S2: set_a,T3: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ T3 )
= zero_zero_real )
=> ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ S2 @ T3 ) )
= ( sigma_measure_a2 @ m @ S2 ) ) ) ) ) ).
% measure_zero_union
thf(fact_1199_finite__measure__subadditive,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A @ B ) ) @ ( plus_plus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ).
% finite_measure_subadditive
thf(fact_1200_finite__measure__Union_H,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% finite_measure_Union'
thf(fact_1201_finite__measure__Union,axiom,
! [A: set_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).
% finite_measure_Union
thf(fact_1202_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1203_le0,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% le0
thf(fact_1204_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1205_diff__0__eq__0,axiom,
! [N4: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N4 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1206_diff__diff__cancel,axiom,
! [I4: nat,N4: nat] :
( ( ord_less_eq_nat @ I4 @ N4 )
=> ( ( minus_minus_nat @ N4 @ ( minus_minus_nat @ N4 @ I4 ) )
= I4 ) ) ).
% diff_diff_cancel
thf(fact_1207_diff__is__0__eq,axiom,
! [M3: nat,N4: nat] :
( ( ( minus_minus_nat @ M3 @ N4 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N4 ) ) ).
% diff_is_0_eq
thf(fact_1208_le__cube,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).
% le_cube
thf(fact_1209_le__square,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ).
% le_square
thf(fact_1210_mult__le__mono,axiom,
! [I4: nat,J3: nat,K2: nat,L2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ K2 @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K2 ) @ ( times_times_nat @ J3 @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1211_mult__le__mono1,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I4 @ K2 ) @ ( times_times_nat @ J3 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1212_mult__le__mono2,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I4 ) @ ( times_times_nat @ K2 @ J3 ) ) ) ).
% mult_le_mono2
thf(fact_1213_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1214_nat__le__linear,axiom,
! [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
| ( ord_less_eq_nat @ N4 @ M3 ) ) ).
% nat_le_linear
thf(fact_1215_le__antisym,axiom,
! [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ( ord_less_eq_nat @ N4 @ M3 )
=> ( M3 = N4 ) ) ) ).
% le_antisym
thf(fact_1216_eq__imp__le,axiom,
! [M3: nat,N4: nat] :
( ( M3 = N4 )
=> ( ord_less_eq_nat @ M3 @ N4 ) ) ).
% eq_imp_le
thf(fact_1217_le__trans,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K2 )
=> ( ord_less_eq_nat @ I4 @ K2 ) ) ) ).
% le_trans
thf(fact_1218_le__refl,axiom,
! [N4: nat] : ( ord_less_eq_nat @ N4 @ N4 ) ).
% le_refl
thf(fact_1219_diff__commute,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K2 ) @ J3 ) ) ).
% diff_commute
thf(fact_1220_diff__mult__distrib2,axiom,
! [K2: nat,M3: nat,N4: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M3 @ N4 ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M3 ) @ ( times_times_nat @ K2 @ N4 ) ) ) ).
% diff_mult_distrib2
thf(fact_1221_diff__mult__distrib,axiom,
! [M3: nat,N4: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M3 @ N4 ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M3 @ K2 ) @ ( times_times_nat @ N4 @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1222_less__eq__nat_Osimps_I1_J,axiom,
! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N4 ) ).
% less_eq_nat.simps(1)
thf(fact_1223_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1224_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1225_le__0__eq,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ N4 @ zero_zero_nat )
= ( N4 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1226_diffs0__imp__equal,axiom,
! [M3: nat,N4: nat] :
( ( ( minus_minus_nat @ M3 @ N4 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N4 @ M3 )
= zero_zero_nat )
=> ( M3 = N4 ) ) ) ).
% diffs0_imp_equal
thf(fact_1227_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1228_diff__le__mono2,axiom,
! [M3: nat,N4: nat,L2: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N4 ) @ ( minus_minus_nat @ L2 @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1229_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1230_diff__le__self,axiom,
! [M3: nat,N4: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N4 ) @ M3 ) ).
% diff_le_self
thf(fact_1231_diff__le__mono,axiom,
! [M3: nat,N4: nat,L2: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L2 ) @ ( minus_minus_nat @ N4 @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1232_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N4: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N4 @ K2 ) )
= ( minus_minus_nat @ M3 @ N4 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1233_le__diff__iff,axiom,
! [K2: nat,M3: nat,N4: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N4 @ K2 ) )
= ( ord_less_eq_nat @ M3 @ N4 ) ) ) ) ).
% le_diff_iff
thf(fact_1234_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N4: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N4 )
=> ( ( ( minus_minus_nat @ M3 @ K2 )
= ( minus_minus_nat @ N4 @ K2 ) )
= ( M3 = N4 ) ) ) ) ).
% eq_diff_iff
thf(fact_1235_D_Osigma__algebra__tail__events,axiom,
! [A: nat > set_set_b] :
( ! [I2: nat] : ( sigma_4968961713055010668ebra_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( A @ I2 ) )
=> ( sigma_4968961713055010668ebra_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( indepe8773861029005768663_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ A ) ) ) ).
% D.sigma_algebra_tail_events
thf(fact_1236_nat__add__left__cancel__le,axiom,
! [K2: nat,M3: nat,N4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N4 ) )
= ( ord_less_eq_nat @ M3 @ N4 ) ) ).
% nat_add_left_cancel_le
thf(fact_1237_diff__diff__left,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ K2 )
= ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1238_Nat_Odiff__diff__right,axiom,
! [K2: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J3 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K2 ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1239_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I4 )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I4 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1240_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J3 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1241_diff__add__inverse2,axiom,
! [M3: nat,N4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N4 ) @ N4 )
= M3 ) ).
% diff_add_inverse2
thf(fact_1242_diff__add__inverse,axiom,
! [N4: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N4 @ M3 ) @ N4 )
= M3 ) ).
% diff_add_inverse
thf(fact_1243_diff__cancel2,axiom,
! [M3: nat,K2: nat,N4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K2 ) @ ( plus_plus_nat @ N4 @ K2 ) )
= ( minus_minus_nat @ M3 @ N4 ) ) ).
% diff_cancel2
thf(fact_1244_Nat_Odiff__cancel,axiom,
! [K2: nat,M3: nat,N4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N4 ) )
= ( minus_minus_nat @ M3 @ N4 ) ) ).
% Nat.diff_cancel
thf(fact_1245_add__leE,axiom,
! [M3: nat,K2: nat,N4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N4 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N4 )
=> ~ ( ord_less_eq_nat @ K2 @ N4 ) ) ) ).
% add_leE
thf(fact_1246_le__add1,axiom,
! [N4: nat,M3: nat] : ( ord_less_eq_nat @ N4 @ ( plus_plus_nat @ N4 @ M3 ) ) ).
% le_add1
thf(fact_1247_le__add2,axiom,
! [N4: nat,M3: nat] : ( ord_less_eq_nat @ N4 @ ( plus_plus_nat @ M3 @ N4 ) ) ).
% le_add2
thf(fact_1248_add__leD1,axiom,
! [M3: nat,K2: nat,N4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N4 )
=> ( ord_less_eq_nat @ M3 @ N4 ) ) ).
% add_leD1
thf(fact_1249_add__leD2,axiom,
! [M3: nat,K2: nat,N4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N4 )
=> ( ord_less_eq_nat @ K2 @ N4 ) ) ).
% add_leD2
thf(fact_1250_le__Suc__ex,axiom,
! [K2: nat,L2: nat] :
( ( ord_less_eq_nat @ K2 @ L2 )
=> ? [N5: nat] :
( L2
= ( plus_plus_nat @ K2 @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_1251_add__le__mono,axiom,
! [I4: nat,J3: nat,K2: nat,L2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_eq_nat @ K2 @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J3 @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_1252_add__le__mono1,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1253_trans__le__add1,axiom,
! [I4: nat,J3: nat,M3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J3 @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1254_trans__le__add2,axiom,
! [I4: nat,J3: nat,M3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M3 @ J3 ) ) ) ).
% trans_le_add2
thf(fact_1255_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1256_diff__add__0,axiom,
! [N4: nat,M3: nat] :
( ( minus_minus_nat @ N4 @ ( plus_plus_nat @ N4 @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1257_le__diff__conv,axiom,
! [J3: nat,K2: nat,I4: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I4 )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I4 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1258_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J3 @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K2 ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1259_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K2 )
= ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J3 @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1260_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J3: nat,I4: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I4 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I4 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1261_Nat_Ole__imp__diff__is__add,axiom,
! [I4: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I4 )
= K2 )
= ( J3
= ( plus_plus_nat @ K2 @ I4 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1262_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1263_D_Okolmogorov__0__1__law,axiom,
! [A: nat > set_set_b,X4: set_b] :
( ! [I2: nat] : ( sigma_4968961713055010668ebra_b @ ( sigma_space_b @ ( measure_distr_a_b @ m @ n @ f ) ) @ ( A @ I2 ) )
=> ( ( indepe7503174356045242851_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ A @ top_top_set_nat )
=> ( ( member_set_b @ X4 @ ( indepe8773861029005768663_b_nat @ ( measure_distr_a_b @ m @ n @ f ) @ A ) )
=> ( ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ X4 )
= zero_zero_real )
| ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ X4 )
= one_one_real ) ) ) ) ) ).
% D.kolmogorov_0_1_law
thf(fact_1264_sigma__algebra__tail__events,axiom,
! [A: nat > set_set_a] :
( ! [I2: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A @ I2 ) )
=> ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( indepe7538416700049374166_a_nat @ m @ A ) ) ) ).
% sigma_algebra_tail_events
thf(fact_1265_kolmogorov__0__1__law,axiom,
! [A: nat > set_set_a,X4: set_a] :
( ! [I2: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A @ I2 ) )
=> ( ( indepe6267730027088848354_a_nat @ m @ A @ top_top_set_nat )
=> ( ( member_set_a @ X4 @ ( indepe7538416700049374166_a_nat @ m @ A ) )
=> ( ( ( sigma_measure_a2 @ m @ X4 )
= zero_zero_real )
| ( ( sigma_measure_a2 @ m @ X4 )
= one_one_real ) ) ) ) ) ).
% kolmogorov_0_1_law
thf(fact_1266_D_Omeasure__countably__zero,axiom,
! [C: nat > set_b] :
( ( ord_le3795704787696855135_set_b @ ( image_nat_set_b @ C @ top_top_set_nat ) @ ( sigma_sets_b @ ( measure_distr_a_b @ m @ n @ f ) ) )
=> ( ! [I2: nat] :
( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( C @ I2 ) )
= zero_zero_real )
=> ( ( sigma_measure_b2 @ ( measure_distr_a_b @ m @ n @ f ) @ ( comple2307003614231284044_set_b @ ( image_nat_set_b @ C @ top_top_set_nat ) ) )
= zero_zero_real ) ) ) ).
% D.measure_countably_zero
thf(fact_1267_indep__set__def,axiom,
! [A: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A @ B )
= ( indepe7780107833195774214ts_a_o @ m @ ( produc6113963288868236716_set_a @ A @ B ) @ top_top_set_o ) ) ).
% indep_set_def
thf(fact_1268_D_Oindep__set__def,axiom,
! [A: set_set_b,B: set_set_b] :
( ( indepe2041756565122539607_set_b @ ( measure_distr_a_b @ m @ n @ f ) @ A @ B )
= ( indepe4880885433731379909ts_b_o @ ( measure_distr_a_b @ m @ n @ f ) @ ( produc6184997328914582701_set_b @ A @ B ) @ top_top_set_o ) ) ).
% D.indep_set_def
thf(fact_1269_measure__countably__zero,axiom,
! [C: nat > set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ C @ top_top_set_nat ) @ ( sigma_sets_a @ m ) )
=> ( ! [I2: nat] :
( ( sigma_measure_a2 @ m @ ( C @ I2 ) )
= zero_zero_real )
=> ( ( sigma_measure_a2 @ m @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ C @ top_top_set_nat ) ) )
= zero_zero_real ) ) ) ).
% measure_countably_zero
thf(fact_1270_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
% Helper facts (7)
thf(help_If_2_1_If_001tf__b_T,axiom,
! [X: b,Y4: b] :
( ( if_b @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001tf__b_T,axiom,
! [X: b,Y4: b] :
( ( if_b @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001tf__d_T,axiom,
! [X: d,Y4: d] :
( ( if_d @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001tf__d_T,axiom,
! [X: d,Y4: d] :
( ( if_d @ $true @ X @ Y4 )
= X ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y4: nat] :
( ( if_nat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y4: nat] :
( ( if_nat @ $true @ X @ Y4 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( indepe6139986284700742573ts_b_c @ ( measure_distr_a_b @ m @ n @ f )
@ ^ [I3: c] :
( collect_set_b
@ ^ [Uu: set_b] :
? [A3: set_d] :
( ( Uu
= ( inf_inf_set_b @ ( vimage_b_d @ ( x @ I3 ) @ A3 ) @ ( sigma_space_b @ n ) ) )
& ( member_set_d @ A3 @ ( sigma_sets_d @ ( m2 @ I3 ) ) ) ) )
@ i ) ).
%------------------------------------------------------------------------------