TPTP Problem File: SLH0152^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Median_Method/0000_Median/prob_00075_002599__14641004_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1653 ( 516 unt; 375 typ; 0 def)
% Number of atoms : 3698 (1147 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 12857 ( 280 ~; 12 |; 157 &;10702 @)
% ( 0 <=>;1706 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 52 ( 51 usr)
% Number of type conns : 1049 (1049 >; 0 *; 0 +; 0 <<)
% Number of symbols : 325 ( 324 usr; 28 con; 0-3 aty)
% Number of variables : 3340 ( 109 ^;3173 !; 58 ?;3340 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:43:18.741
%------------------------------------------------------------------------------
% Could-be-implicit typings (51)
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
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thf(ty_n_t__Real__Oreal,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (324)
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thf(sy_c_Complete__Measure_Ocomplete__measure_001tf__a,type,
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thf(sy_c_Complete__Measure_Ocompletion_001tf__a,type,
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thf(sy_c_Complete__Measure_Omain__part_001tf__a,type,
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thf(sy_c_Complete__Measure_Onull__part_001t__Real__Oreal,type,
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thf(sy_c_Complete__Measure_Onull__part_001tf__a,type,
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thf(sy_c_Countable__Set_Ocountable_001tf__a,type,
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thf(sy_c_Equivalence__Measurable__On__Borel_Omeasurable__on_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Extended__Nonnegative__Real_Oennreal,type,
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thf(sy_c_Fun_Oid_001tf__a,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
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sigma_sets_set_a: sigma_measure_set_a > set_set_set_a ).
thf(sy_c_Sigma__Algebra_Osets_001tf__a,type,
sigma_sets_a: sigma_measure_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Extended____Nonnegative____Real__Oennreal,type,
sigma_3147302497200244656nnreal: sigma_7234349610311085201nnreal > set_Ex3793607809372303086nnreal ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Real__Oreal,type,
sigma_space_real: sigma_measure_real > set_real ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
sigma_2539764534872131430nnreal: sigma_523634232904505671nnreal > set_se4580700918925141924nnreal ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_It__Real__Oreal_J,type,
sigma_space_set_real: sigma_3733394171116455995t_real > set_set_real ).
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_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
topolo3503219976281768444nnreal: set_re5328672808648366137nnreal > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Extended____Nonnegative____Real__Oennreal,type,
topolo1037242508615874353nnreal: set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Real__Oreal,type,
topolo4860482606490270245n_real: set_real > $o ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001tf__a,type,
topolo8477419352202985285open_a: set_a > $o ).
thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
member8329810500450651686nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > set_Ex7514979451064110021nnreal > $o ).
thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Real__Oreal_J,type,
member2874014351250825754l_real: ( extend8495563244428889912nnreal > real ) > set_Ex5658717452565810105l_real > $o ).
thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
member4416662722526258908nnreal: ( extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal ) > set_Ex4629243626970651003nnreal > $o ).
thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Set__Oset_It__Real__Oreal_J_J,type,
member6764088077590758224t_real: ( extend8495563244428889912nnreal > set_real ) > set_Ex1976581565604454895t_real > $o ).
thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Set__Oset_Itf__a_J_J,type,
member6799942265337811078_set_a: ( extend8495563244428889912nnreal > set_a ) > set_Ex166883028395225405_set_a > $o ).
thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_Mtf__a_J,type,
member4924430693770431270real_a: ( extend8495563244428889912nnreal > a ) > set_Ex2249781601450085341real_a > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
member2919562650594848410nnreal: ( real > extend8495563244428889912nnreal ) > set_re5328672808648366137nnreal > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
member_real_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
member8689841359643572048nnreal: ( real > set_Ex3793607809372303086nnreal ) > set_re634636480907793903nnreal > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Set__Oset_It__Real__Oreal_J_J,type,
member_real_set_real: ( real > set_real ) > set_real_set_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Set__Oset_Itf__a_J_J,type,
member_real_set_a: ( real > set_a ) > set_real_set_a > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mtf__a_J,type,
member_real_a: ( real > a ) > set_real_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_Mt__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
member9048868947896282770nnreal: ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ) > set_se5993948446613689905nnreal > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
member7536123422392205318t_real: ( set_Ex3793607809372303086nnreal > set_real ) > set_se3106664747148989349t_real > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_Mt__Set__Oset_Itf__a_J_J,type,
member8849812955461179984_set_a: ( set_Ex3793607809372303086nnreal > set_a ) > set_se2341910093884376583_set_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
member1248903934968170374nnreal: ( set_real > set_Ex3793607809372303086nnreal ) > set_se6192103498290354981nnreal > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
member8820419758626733818t_real: ( set_real > set_real ) > set_se3821091506293227161t_real > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_Itf__a_J_J,type,
member9134392423035811420_set_a: ( set_real > set_a ) > set_set_real_set_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
member8552228822107236582nnreal: ( set_a > set_Ex3793607809372303086nnreal ) > set_se2858003755320519069nnreal > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
member6119777607969566810t_real: ( set_a > set_real ) > set_set_a_set_real > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member_set_a_set_a: ( set_a > set_a ) > set_set_a_set_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
member298456594901751504nnreal: ( a > extend8495563244428889912nnreal ) > set_a_7161065143582548615nnreal > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Real__Oreal_J,type,
member_a_real: ( a > real ) > set_a_real > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
member2532357421736347526nnreal: ( a > set_Ex3793607809372303086nnreal ) > set_a_7828589535950383165nnreal > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__Real__Oreal_J_J,type,
member_a_set_real: ( a > set_real ) > set_a_set_real > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
member_a_set_a: ( a > set_a ) > set_a_set_a > $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_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_J,type,
member524040414084610768nnreal: set_re5328672808648366137nnreal > set_se2490721793304844655nnreal > $o ).
thf(sy_c_member_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
member603777416030116741nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
member6568240578637133883nnreal: set_se4580700918925141924nnreal > set_se8256708918794385754nnreal > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
member_set_set_real: set_set_real > set_set_set_real > $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_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Sigma____Algebra__Omeasure_It__Extended____Nonnegative____Real__Oennreal_J,type,
member6261374078160781754nnreal: sigma_7234349610311085201nnreal > set_Si97717610131227249nnreal > $o ).
thf(sy_c_member_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
member4553183543495551918e_real: sigma_measure_real > set_Si6059263944882162789e_real > $o ).
thf(sy_c_member_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
member3534519376729797778sure_a: sigma_measure_a > set_Sigma_measure_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_I,type,
i: set_a ).
% Relevant facts (1277)
thf(fact_0_assms,axiom,
down_ray_a @ i ).
% assms
thf(fact_1__092_060open_062_N_AI_A_092_060in_062_Asets_Aborel_092_060close_062,axiom,
member_set_a @ ( uminus_uminus_set_a @ i ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ).
% \<open>- I \<in> sets borel\<close>
thf(fact_2__092_060open_062up__ray_A_I_N_AI_J_092_060close_062,axiom,
up_ray_a @ ( uminus_uminus_set_a @ i ) ).
% \<open>up_ray (- I)\<close>
thf(fact_3_up__ray__borel,axiom,
! [I: set_a] :
( ( up_ray_a @ I )
=> ( member_set_a @ I @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% up_ray_borel
thf(fact_4_up__ray__borel,axiom,
! [I: set_real] :
( ( up_ray_real @ I )
=> ( member_set_real @ I @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% up_ray_borel
thf(fact_5_up__ray__borel,axiom,
! [I: set_Ex3793607809372303086nnreal] :
( ( up_ray4546996785294415186nnreal @ I )
=> ( member603777416030116741nnreal @ I @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).
% up_ray_borel
thf(fact_6_borel__comp,axiom,
! [A: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
=> ( member_set_a @ ( uminus_uminus_set_a @ A ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% borel_comp
thf(fact_7_borel__comp,axiom,
! [A: set_real] :
( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ ( uminus612125837232591019t_real @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% borel_comp
thf(fact_8_borel__comp,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
=> ( member603777416030116741nnreal @ ( uminus5517552291522096439nnreal @ A ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).
% borel_comp
thf(fact_9_sets__Ball,axiom,
! [I: set_set_a,A: set_a > set_real,M: set_a > sigma_measure_real,I2: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ I )
=> ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_10_sets__Ball,axiom,
! [I: set_set_real,A: set_real > set_real,M: set_real > sigma_measure_real,I2: set_real] :
( ! [X: set_real] :
( ( member_set_real @ X @ I )
=> ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
=> ( ( member_set_real @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_11_sets__Ball,axiom,
! [I: set_set_a,A: set_a > set_a,M: set_a > sigma_measure_a,I2: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ I )
=> ( member_set_a @ ( A @ X ) @ ( sigma_sets_a @ ( M @ X ) ) ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_12_sets__Ball,axiom,
! [I: set_set_real,A: set_real > set_a,M: set_real > sigma_measure_a,I2: set_real] :
( ! [X: set_real] :
( ( member_set_real @ X @ I )
=> ( member_set_a @ ( A @ X ) @ ( sigma_sets_a @ ( M @ X ) ) ) )
=> ( ( member_set_real @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_13_sets__Ball,axiom,
! [I: set_re5328672808648366137nnreal,A: ( real > extend8495563244428889912nnreal ) > set_real,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_real,I2: real > extend8495563244428889912nnreal] :
( ! [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ I )
=> ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
=> ( ( member2919562650594848410nnreal @ I2 @ I )
=> ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_14_sets__Ball,axiom,
! [I: set_re5328672808648366137nnreal,A: ( real > extend8495563244428889912nnreal ) > set_a,M: ( real > extend8495563244428889912nnreal ) > sigma_measure_a,I2: real > extend8495563244428889912nnreal] :
( ! [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ I )
=> ( member_set_a @ ( A @ X ) @ ( sigma_sets_a @ ( M @ X ) ) ) )
=> ( ( member2919562650594848410nnreal @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_15_sets__Ball,axiom,
! [I: set_se4580700918925141924nnreal,A: set_Ex3793607809372303086nnreal > set_a,M: set_Ex3793607809372303086nnreal > sigma_measure_a,I2: set_Ex3793607809372303086nnreal] :
( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ I )
=> ( member_set_a @ ( A @ X ) @ ( sigma_sets_a @ ( M @ X ) ) ) )
=> ( ( member603777416030116741nnreal @ I2 @ I )
=> ( member_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_16_sets__Ball,axiom,
! [I: set_set_a,A: set_a > set_Ex3793607809372303086nnreal,M: set_a > sigma_7234349610311085201nnreal,I2: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ I )
=> ( member603777416030116741nnreal @ ( A @ X ) @ ( sigma_5465916536984168985nnreal @ ( M @ X ) ) ) )
=> ( ( member_set_a @ I2 @ I )
=> ( member603777416030116741nnreal @ ( A @ I2 ) @ ( sigma_5465916536984168985nnreal @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_17_sets__Ball,axiom,
! [I: set_set_real,A: set_real > set_Ex3793607809372303086nnreal,M: set_real > sigma_7234349610311085201nnreal,I2: set_real] :
( ! [X: set_real] :
( ( member_set_real @ X @ I )
=> ( member603777416030116741nnreal @ ( A @ X ) @ ( sigma_5465916536984168985nnreal @ ( M @ X ) ) ) )
=> ( ( member_set_real @ I2 @ I )
=> ( member603777416030116741nnreal @ ( A @ I2 ) @ ( sigma_5465916536984168985nnreal @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_18_sets__Ball,axiom,
! [I: set_se4580700918925141924nnreal,A: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,M: set_Ex3793607809372303086nnreal > sigma_7234349610311085201nnreal,I2: set_Ex3793607809372303086nnreal] :
( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ I )
=> ( member603777416030116741nnreal @ ( A @ X ) @ ( sigma_5465916536984168985nnreal @ ( M @ X ) ) ) )
=> ( ( member603777416030116741nnreal @ I2 @ I )
=> ( member603777416030116741nnreal @ ( A @ I2 ) @ ( sigma_5465916536984168985nnreal @ ( M @ I2 ) ) ) ) ) ).
% sets_Ball
thf(fact_19_sets__diff__measure,axiom,
! [M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal] :
( ( sigma_5465916536984168985nnreal @ ( radon_8175693640339213217nnreal @ M @ N ) )
= ( sigma_5465916536984168985nnreal @ M ) ) ).
% sets_diff_measure
thf(fact_20_sets__diff__measure,axiom,
! [M: sigma_measure_real,N: sigma_measure_real] :
( ( sigma_sets_real @ ( radon_5355578694595008149e_real @ M @ N ) )
= ( sigma_sets_real @ M ) ) ).
% sets_diff_measure
thf(fact_21_sets__diff__measure,axiom,
! [M: sigma_measure_a,N: sigma_measure_a] :
( ( sigma_sets_a @ ( radon_diff_measure_a @ M @ N ) )
= ( sigma_sets_a @ M ) ) ).
% sets_diff_measure
thf(fact_22_sets__lborel,axiom,
( ( sigma_sets_real @ lebesgue_lborel_real )
= ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).
% sets_lborel
thf(fact_23_space__in__borel,axiom,
member_set_a @ top_top_set_a @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ).
% space_in_borel
thf(fact_24_space__in__borel,axiom,
member_set_real @ top_top_set_real @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).
% space_in_borel
thf(fact_25_space__in__borel,axiom,
member603777416030116741nnreal @ top_to7994903218803871134nnreal @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ).
% space_in_borel
thf(fact_26_lborelD,axiom,
! [A: set_real] :
( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ A @ ( sigma_sets_real @ lebesgue_lborel_real ) ) ) ).
% lborelD
thf(fact_27_greaterThanLessThan__borel,axiom,
! [A2: a,B: a] : ( member_set_a @ ( set_or5939364468397584554Than_a @ A2 @ B ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% greaterThanLessThan_borel
thf(fact_28_greaterThanLessThan__borel,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( member603777416030116741nnreal @ ( set_or1838662946377535116nnreal @ A2 @ B ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ).
% greaterThanLessThan_borel
thf(fact_29_greaterThanLessThan__borel,axiom,
! [A2: real,B: real] : ( member_set_real @ ( set_or1633881224788618240n_real @ A2 @ B ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).
% greaterThanLessThan_borel
thf(fact_30_greaterThanAtMost__borel,axiom,
! [A2: a,B: a] : ( member_set_a @ ( set_or4472690218693186638Most_a @ A2 @ B ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ).
% greaterThanAtMost_borel
thf(fact_31_greaterThanAtMost__borel,axiom,
! [A2: real,B: real] : ( member_set_real @ ( set_or2392270231875598684t_real @ A2 @ B ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).
% greaterThanAtMost_borel
thf(fact_32_greaterThanAtMost__borel,axiom,
! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( member603777416030116741nnreal @ ( set_or4532334673728481768nnreal @ A2 @ B ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ).
% greaterThanAtMost_borel
thf(fact_33_borel__singleton,axiom,
! [A: set_a,X2: a] :
( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
=> ( member_set_a @ ( insert_a @ X2 @ A ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% borel_singleton
thf(fact_34_borel__singleton,axiom,
! [A: set_real,X2: real] :
( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ ( insert_real @ X2 @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% borel_singleton
thf(fact_35_borel__singleton,axiom,
! [A: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal] :
( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ A ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).
% borel_singleton
thf(fact_36_borel__open,axiom,
! [A: set_a] :
( ( topolo8477419352202985285open_a @ A )
=> ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ).
% borel_open
thf(fact_37_borel__open,axiom,
! [A: set_real] :
( ( topolo4860482606490270245n_real @ A )
=> ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).
% borel_open
thf(fact_38_borel__open,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( topolo1037242508615874353nnreal @ A )
=> ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).
% borel_open
thf(fact_39_open__UNIV,axiom,
topolo1037242508615874353nnreal @ top_to7994903218803871134nnreal ).
% open_UNIV
thf(fact_40_open__UNIV,axiom,
topolo4860482606490270245n_real @ top_top_set_real ).
% open_UNIV
thf(fact_41_open__UNIV,axiom,
topolo8477419352202985285open_a @ top_top_set_a ).
% open_UNIV
thf(fact_42_ComplI,axiom,
! [C: set_a,A: set_set_a] :
( ~ ( member_set_a @ C @ A )
=> ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A ) ) ) ).
% ComplI
thf(fact_43_ComplI,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ~ ( member2919562650594848410nnreal @ C @ A )
=> ( member2919562650594848410nnreal @ C @ ( uminus2275888197404385410nnreal @ A ) ) ) ).
% ComplI
thf(fact_44_ComplI,axiom,
! [C: set_real,A: set_set_real] :
( ~ ( member_set_real @ C @ A )
=> ( member_set_real @ C @ ( uminus708787163358948833t_real @ A ) ) ) ).
% ComplI
thf(fact_45_ComplI,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ~ ( member603777416030116741nnreal @ C @ A )
=> ( member603777416030116741nnreal @ C @ ( uminus4762152451731718637nnreal @ A ) ) ) ).
% ComplI
thf(fact_46_ComplI,axiom,
! [C: a,A: set_a] :
( ~ ( member_a @ C @ A )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_47_ComplI,axiom,
! [C: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
( ~ ( member7908768830364227535nnreal @ C @ A )
=> ( member7908768830364227535nnreal @ C @ ( uminus5517552291522096439nnreal @ A ) ) ) ).
% ComplI
thf(fact_48_ComplI,axiom,
! [C: real,A: set_real] :
( ~ ( member_real @ C @ A )
=> ( member_real @ C @ ( uminus612125837232591019t_real @ A ) ) ) ).
% ComplI
thf(fact_49_Compl__iff,axiom,
! [C: set_a,A: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A ) )
= ( ~ ( member_set_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_50_Compl__iff,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( uminus2275888197404385410nnreal @ A ) )
= ( ~ ( member2919562650594848410nnreal @ C @ A ) ) ) ).
% Compl_iff
thf(fact_51_Compl__iff,axiom,
! [C: set_real,A: set_set_real] :
( ( member_set_real @ C @ ( uminus708787163358948833t_real @ A ) )
= ( ~ ( member_set_real @ C @ A ) ) ) ).
% Compl_iff
thf(fact_52_Compl__iff,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( uminus4762152451731718637nnreal @ A ) )
= ( ~ ( member603777416030116741nnreal @ C @ A ) ) ) ).
% Compl_iff
thf(fact_53_Compl__iff,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a @ C @ A ) ) ) ).
% Compl_iff
thf(fact_54_Compl__iff,axiom,
! [C: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C @ ( uminus5517552291522096439nnreal @ A ) )
= ( ~ ( member7908768830364227535nnreal @ C @ A ) ) ) ).
% Compl_iff
thf(fact_55_Compl__iff,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
= ( ~ ( member_real @ C @ A ) ) ) ).
% Compl_iff
thf(fact_56_Compl__eq__Compl__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( uminus_uminus_set_a @ A )
= ( uminus_uminus_set_a @ B2 ) )
= ( A = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_57_Compl__eq__Compl__iff,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ( uminus5517552291522096439nnreal @ A )
= ( uminus5517552291522096439nnreal @ B2 ) )
= ( A = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_58_Compl__eq__Compl__iff,axiom,
! [A: set_real,B2: set_real] :
( ( ( uminus612125837232591019t_real @ A )
= ( uminus612125837232591019t_real @ B2 ) )
= ( A = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_59_insertCI,axiom,
! [A2: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_60_insertCI,axiom,
! [A2: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal,B: real > extend8495563244428889912nnreal] :
( ( ~ ( member2919562650594848410nnreal @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member2919562650594848410nnreal @ A2 @ ( insert152533262698245683nnreal @ B @ B2 ) ) ) ).
% insertCI
thf(fact_61_insertCI,axiom,
! [A2: set_real,B2: set_set_real,B: set_real] :
( ( ~ ( member_set_real @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_real @ A2 @ ( insert_set_real @ B @ B2 ) ) ) ).
% insertCI
thf(fact_62_insertCI,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal,B: set_Ex3793607809372303086nnreal] :
( ( ~ ( member603777416030116741nnreal @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ B @ B2 ) ) ) ).
% insertCI
thf(fact_63_insert__iff,axiom,
! [A2: set_a,B: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B @ A ) )
= ( ( A2 = B )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_64_insert__iff,axiom,
! [A2: real > extend8495563244428889912nnreal,B: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ A2 @ ( insert152533262698245683nnreal @ B @ A ) )
= ( ( A2 = B )
| ( member2919562650594848410nnreal @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_65_insert__iff,axiom,
! [A2: set_real,B: set_real,A: set_set_real] :
( ( member_set_real @ A2 @ ( insert_set_real @ B @ A ) )
= ( ( A2 = B )
| ( member_set_real @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_66_insert__iff,axiom,
! [A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ B @ A ) )
= ( ( A2 = B )
| ( member603777416030116741nnreal @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_67_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_68_add_Oinverse__inverse,axiom,
! [A2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_69_neg__equal__iff__equal,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) )
= ( A2 = B ) ) ).
% neg_equal_iff_equal
thf(fact_70_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X2: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ X2 ) )
= X2 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_71_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X2: set_Ex3793607809372303086nnreal] :
( ( uminus5517552291522096439nnreal @ ( uminus5517552291522096439nnreal @ X2 ) )
= X2 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_72_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X2: set_real] :
( ( uminus612125837232591019t_real @ ( uminus612125837232591019t_real @ X2 ) )
= X2 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_73_UNIV__I,axiom,
! [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% UNIV_I
thf(fact_74_UNIV__I,axiom,
! [X2: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X2 @ top_to315565310957491945nnreal ) ).
% UNIV_I
thf(fact_75_UNIV__I,axiom,
! [X2: set_real] : ( member_set_real @ X2 @ top_top_set_set_real ) ).
% UNIV_I
thf(fact_76_UNIV__I,axiom,
! [X2: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X2 @ top_to3356475028079756884nnreal ) ).
% UNIV_I
thf(fact_77_UNIV__I,axiom,
! [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ top_to7994903218803871134nnreal ) ).
% UNIV_I
thf(fact_78_UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_I
thf(fact_79_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
thf(fact_80_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ( uminus_uminus_set_a @ X2 )
= ( uminus_uminus_set_a @ Y ) )
= ( X2 = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_81_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( ( uminus5517552291522096439nnreal @ X2 )
= ( uminus5517552291522096439nnreal @ Y ) )
= ( X2 = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_82_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X2: set_real,Y: set_real] :
( ( ( uminus612125837232591019t_real @ X2 )
= ( uminus612125837232591019t_real @ Y ) )
= ( X2 = Y ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_83_UNIV__witness,axiom,
? [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% UNIV_witness
thf(fact_84_UNIV__witness,axiom,
? [X: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X @ top_to315565310957491945nnreal ) ).
% UNIV_witness
thf(fact_85_UNIV__witness,axiom,
? [X: set_real] : ( member_set_real @ X @ top_top_set_set_real ) ).
% UNIV_witness
thf(fact_86_UNIV__witness,axiom,
? [X: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X @ top_to3356475028079756884nnreal ) ).
% UNIV_witness
thf(fact_87_UNIV__witness,axiom,
? [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ top_to7994903218803871134nnreal ) ).
% UNIV_witness
thf(fact_88_UNIV__witness,axiom,
? [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_witness
thf(fact_89_UNIV__witness,axiom,
? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_witness
thf(fact_90_UNIV__eq__I,axiom,
! [A: set_set_a] :
( ! [X: set_a] : ( member_set_a @ X @ A )
=> ( top_top_set_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_91_UNIV__eq__I,axiom,
! [A: set_re5328672808648366137nnreal] :
( ! [X: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X @ A )
=> ( top_to315565310957491945nnreal = A ) ) ).
% UNIV_eq_I
thf(fact_92_UNIV__eq__I,axiom,
! [A: set_set_real] :
( ! [X: set_real] : ( member_set_real @ X @ A )
=> ( top_top_set_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_93_UNIV__eq__I,axiom,
! [A: set_se4580700918925141924nnreal] :
( ! [X: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X @ A )
=> ( top_to3356475028079756884nnreal = A ) ) ).
% UNIV_eq_I
thf(fact_94_UNIV__eq__I,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ! [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ A )
=> ( top_to7994903218803871134nnreal = A ) ) ).
% UNIV_eq_I
thf(fact_95_UNIV__eq__I,axiom,
! [A: set_real] :
( ! [X: real] : ( member_real @ X @ A )
=> ( top_top_set_real = A ) ) ).
% UNIV_eq_I
thf(fact_96_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X: a] : ( member_a @ X @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_97_minus__equation__iff,axiom,
! [A2: real,B: real] :
( ( ( uminus_uminus_real @ A2 )
= B )
= ( ( uminus_uminus_real @ B )
= A2 ) ) ).
% minus_equation_iff
thf(fact_98_equation__minus__iff,axiom,
! [A2: real,B: real] :
( ( A2
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_99_verit__negate__coefficient_I3_J,axiom,
! [A2: real,B: real] :
( ( A2 = B )
=> ( ( uminus_uminus_real @ A2 )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_100_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B3: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B3 ) )
& ~ ( member_set_a @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_101_mk__disjoint__insert,axiom,
! [A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ A2 @ A )
=> ? [B3: set_re5328672808648366137nnreal] :
( ( A
= ( insert152533262698245683nnreal @ A2 @ B3 ) )
& ~ ( member2919562650594848410nnreal @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_102_mk__disjoint__insert,axiom,
! [A2: set_real,A: set_set_real] :
( ( member_set_real @ A2 @ A )
=> ? [B3: set_set_real] :
( ( A
= ( insert_set_real @ A2 @ B3 ) )
& ~ ( member_set_real @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_103_mk__disjoint__insert,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ A )
=> ? [B3: set_se4580700918925141924nnreal] :
( ( A
= ( insert1343806209672318238nnreal @ A2 @ B3 ) )
& ~ ( member603777416030116741nnreal @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_104_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B: set_a,B2: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B @ B2 )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C2: set_set_a] :
( ( A
= ( insert_set_a @ B @ C2 ) )
& ~ ( member_set_a @ B @ C2 )
& ( B2
= ( insert_set_a @ A2 @ C2 ) )
& ~ ( member_set_a @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_105_insert__eq__iff,axiom,
! [A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal] :
( ~ ( member2919562650594848410nnreal @ A2 @ A )
=> ( ~ ( member2919562650594848410nnreal @ B @ B2 )
=> ( ( ( insert152533262698245683nnreal @ A2 @ A )
= ( insert152533262698245683nnreal @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C2: set_re5328672808648366137nnreal] :
( ( A
= ( insert152533262698245683nnreal @ B @ C2 ) )
& ~ ( member2919562650594848410nnreal @ B @ C2 )
& ( B2
= ( insert152533262698245683nnreal @ A2 @ C2 ) )
& ~ ( member2919562650594848410nnreal @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_106_insert__eq__iff,axiom,
! [A2: set_real,A: set_set_real,B: set_real,B2: set_set_real] :
( ~ ( member_set_real @ A2 @ A )
=> ( ~ ( member_set_real @ B @ B2 )
=> ( ( ( insert_set_real @ A2 @ A )
= ( insert_set_real @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C2: set_set_real] :
( ( A
= ( insert_set_real @ B @ C2 ) )
& ~ ( member_set_real @ B @ C2 )
& ( B2
= ( insert_set_real @ A2 @ C2 ) )
& ~ ( member_set_real @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_107_insert__eq__iff,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ~ ( member603777416030116741nnreal @ A2 @ A )
=> ( ~ ( member603777416030116741nnreal @ B @ B2 )
=> ( ( ( insert1343806209672318238nnreal @ A2 @ A )
= ( insert1343806209672318238nnreal @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C2: set_se4580700918925141924nnreal] :
( ( A
= ( insert1343806209672318238nnreal @ B @ C2 ) )
& ~ ( member603777416030116741nnreal @ B @ C2 )
& ( B2
= ( insert1343806209672318238nnreal @ A2 @ C2 ) )
& ~ ( member603777416030116741nnreal @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_108_insert__absorb,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_109_insert__absorb,axiom,
! [A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ A2 @ A )
=> ( ( insert152533262698245683nnreal @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_110_insert__absorb,axiom,
! [A2: set_real,A: set_set_real] :
( ( member_set_real @ A2 @ A )
=> ( ( insert_set_real @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_111_insert__absorb,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ A )
=> ( ( insert1343806209672318238nnreal @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_112_insert__ident,axiom,
! [X2: set_a,A: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X2 @ A )
=> ( ~ ( member_set_a @ X2 @ B2 )
=> ( ( ( insert_set_a @ X2 @ A )
= ( insert_set_a @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_113_insert__ident,axiom,
! [X2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ~ ( member2919562650594848410nnreal @ X2 @ A )
=> ( ~ ( member2919562650594848410nnreal @ X2 @ B2 )
=> ( ( ( insert152533262698245683nnreal @ X2 @ A )
= ( insert152533262698245683nnreal @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_114_insert__ident,axiom,
! [X2: set_real,A: set_set_real,B2: set_set_real] :
( ~ ( member_set_real @ X2 @ A )
=> ( ~ ( member_set_real @ X2 @ B2 )
=> ( ( ( insert_set_real @ X2 @ A )
= ( insert_set_real @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_115_insert__ident,axiom,
! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ~ ( member603777416030116741nnreal @ X2 @ A )
=> ( ~ ( member603777416030116741nnreal @ X2 @ B2 )
=> ( ( ( insert1343806209672318238nnreal @ X2 @ A )
= ( insert1343806209672318238nnreal @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_116_Set_Oset__insert,axiom,
! [X2: set_a,A: set_set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ! [B3: set_set_a] :
( ( A
= ( insert_set_a @ X2 @ B3 ) )
=> ( member_set_a @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_117_Set_Oset__insert,axiom,
! [X2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ X2 @ A )
=> ~ ! [B3: set_re5328672808648366137nnreal] :
( ( A
= ( insert152533262698245683nnreal @ X2 @ B3 ) )
=> ( member2919562650594848410nnreal @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_118_Set_Oset__insert,axiom,
! [X2: set_real,A: set_set_real] :
( ( member_set_real @ X2 @ A )
=> ~ ! [B3: set_set_real] :
( ( A
= ( insert_set_real @ X2 @ B3 ) )
=> ( member_set_real @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_119_Set_Oset__insert,axiom,
! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ X2 @ A )
=> ~ ! [B3: set_se4580700918925141924nnreal] :
( ( A
= ( insert1343806209672318238nnreal @ X2 @ B3 ) )
=> ( member603777416030116741nnreal @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_120_insertI2,axiom,
! [A2: set_a,B2: set_set_a,B: set_a] :
( ( member_set_a @ A2 @ B2 )
=> ( member_set_a @ A2 @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_121_insertI2,axiom,
! [A2: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal,B: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ A2 @ B2 )
=> ( member2919562650594848410nnreal @ A2 @ ( insert152533262698245683nnreal @ B @ B2 ) ) ) ).
% insertI2
thf(fact_122_insertI2,axiom,
! [A2: set_real,B2: set_set_real,B: set_real] :
( ( member_set_real @ A2 @ B2 )
=> ( member_set_real @ A2 @ ( insert_set_real @ B @ B2 ) ) ) ).
% insertI2
thf(fact_123_insertI2,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal,B: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A2 @ B2 )
=> ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ B @ B2 ) ) ) ).
% insertI2
thf(fact_124_insertI1,axiom,
! [A2: set_a,B2: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_125_insertI1,axiom,
! [A2: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal] : ( member2919562650594848410nnreal @ A2 @ ( insert152533262698245683nnreal @ A2 @ B2 ) ) ).
% insertI1
thf(fact_126_insertI1,axiom,
! [A2: set_real,B2: set_set_real] : ( member_set_real @ A2 @ ( insert_set_real @ A2 @ B2 ) ) ).
% insertI1
thf(fact_127_insertI1,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] : ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ A2 @ B2 ) ) ).
% insertI1
thf(fact_128_insertE,axiom,
! [A2: set_a,B: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_129_insertE,axiom,
! [A2: real > extend8495563244428889912nnreal,B: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ A2 @ ( insert152533262698245683nnreal @ B @ A ) )
=> ( ( A2 != B )
=> ( member2919562650594848410nnreal @ A2 @ A ) ) ) ).
% insertE
thf(fact_130_insertE,axiom,
! [A2: set_real,B: set_real,A: set_set_real] :
( ( member_set_real @ A2 @ ( insert_set_real @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_real @ A2 @ A ) ) ) ).
% insertE
thf(fact_131_insertE,axiom,
! [A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ B @ A ) )
=> ( ( A2 != B )
=> ( member603777416030116741nnreal @ A2 @ A ) ) ) ).
% insertE
thf(fact_132_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_133_mem__Collect__eq,axiom,
! [A2: real > extend8495563244428889912nnreal,P: ( real > extend8495563244428889912nnreal ) > $o] :
( ( member2919562650594848410nnreal @ A2 @ ( collec9130413544115709400nnreal @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A2: set_real,P: set_real > $o] :
( ( member_set_real @ A2 @ ( collect_set_real @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
! [A2: set_Ex3793607809372303086nnreal,P: set_Ex3793607809372303086nnreal > $o] :
( ( member603777416030116741nnreal @ A2 @ ( collec4858231573021281987nnreal @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_136_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_137_Collect__mem__eq,axiom,
! [A: set_re5328672808648366137nnreal] :
( ( collec9130413544115709400nnreal
@ ^ [X3: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A: set_set_real] :
( ( collect_set_real
@ ^ [X3: set_real] : ( member_set_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A: set_se4580700918925141924nnreal] :
( ( collec4858231573021281987nnreal
@ ^ [X3: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_140_double__complement,axiom,
! [A: set_a] :
( ( uminus_uminus_set_a @ ( uminus_uminus_set_a @ A ) )
= A ) ).
% double_complement
thf(fact_141_double__complement,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( uminus5517552291522096439nnreal @ ( uminus5517552291522096439nnreal @ A ) )
= A ) ).
% double_complement
thf(fact_142_double__complement,axiom,
! [A: set_real] :
( ( uminus612125837232591019t_real @ ( uminus612125837232591019t_real @ A ) )
= A ) ).
% double_complement
thf(fact_143_ComplD,axiom,
! [C: set_a,A: set_set_a] :
( ( member_set_a @ C @ ( uminus6103902357914783669_set_a @ A ) )
=> ~ ( member_set_a @ C @ A ) ) ).
% ComplD
thf(fact_144_ComplD,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( uminus2275888197404385410nnreal @ A ) )
=> ~ ( member2919562650594848410nnreal @ C @ A ) ) ).
% ComplD
thf(fact_145_ComplD,axiom,
! [C: set_real,A: set_set_real] :
( ( member_set_real @ C @ ( uminus708787163358948833t_real @ A ) )
=> ~ ( member_set_real @ C @ A ) ) ).
% ComplD
thf(fact_146_ComplD,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( uminus4762152451731718637nnreal @ A ) )
=> ~ ( member603777416030116741nnreal @ C @ A ) ) ).
% ComplD
thf(fact_147_ComplD,axiom,
! [C: a,A: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a @ C @ A ) ) ).
% ComplD
thf(fact_148_ComplD,axiom,
! [C: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C @ ( uminus5517552291522096439nnreal @ A ) )
=> ~ ( member7908768830364227535nnreal @ C @ A ) ) ).
% ComplD
thf(fact_149_ComplD,axiom,
! [C: real,A: set_real] :
( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
=> ~ ( member_real @ C @ A ) ) ).
% ComplD
thf(fact_150_insert__UNIV,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( insert7407984058720857448nnreal @ X2 @ top_to7994903218803871134nnreal )
= top_to7994903218803871134nnreal ) ).
% insert_UNIV
thf(fact_151_insert__UNIV,axiom,
! [X2: real] :
( ( insert_real @ X2 @ top_top_set_real )
= top_top_set_real ) ).
% insert_UNIV
thf(fact_152_insert__UNIV,axiom,
! [X2: a] :
( ( insert_a @ X2 @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_153_open__greaterThanLessThan,axiom,
! [A2: real,B: real] : ( topolo4860482606490270245n_real @ ( set_or1633881224788618240n_real @ A2 @ B ) ) ).
% open_greaterThanLessThan
thf(fact_154_iso__tuple__UNIV__I,axiom,
! [X2: set_a] : ( member_set_a @ X2 @ top_top_set_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_155_iso__tuple__UNIV__I,axiom,
! [X2: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X2 @ top_to315565310957491945nnreal ) ).
% iso_tuple_UNIV_I
thf(fact_156_iso__tuple__UNIV__I,axiom,
! [X2: set_real] : ( member_set_real @ X2 @ top_top_set_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_157_iso__tuple__UNIV__I,axiom,
! [X2: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X2 @ top_to3356475028079756884nnreal ) ).
% iso_tuple_UNIV_I
thf(fact_158_iso__tuple__UNIV__I,axiom,
! [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ top_to7994903218803871134nnreal ) ).
% iso_tuple_UNIV_I
thf(fact_159_iso__tuple__UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_160_iso__tuple__UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_161_exists__diff,axiom,
! [P: set_a > $o] :
( ( ? [S: set_a] : ( P @ ( uminus_uminus_set_a @ S ) ) )
= ( ? [X4: set_a] : ( P @ X4 ) ) ) ).
% exists_diff
thf(fact_162_exists__diff,axiom,
! [P: set_Ex3793607809372303086nnreal > $o] :
( ( ? [S: set_Ex3793607809372303086nnreal] : ( P @ ( uminus5517552291522096439nnreal @ S ) ) )
= ( ? [X4: set_Ex3793607809372303086nnreal] : ( P @ X4 ) ) ) ).
% exists_diff
thf(fact_163_exists__diff,axiom,
! [P: set_real > $o] :
( ( ? [S: set_real] : ( P @ ( uminus612125837232591019t_real @ S ) ) )
= ( ? [X4: set_real] : ( P @ X4 ) ) ) ).
% exists_diff
thf(fact_164_Compl__in__sets__lebesgue,axiom,
! [A: set_real] :
( ( member_set_real @ ( uminus612125837232591019t_real @ A ) @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
= ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) ) ) ).
% Compl_in_sets_lebesgue
thf(fact_165_fsigma__UNIV,axiom,
lebesgue_fsigma_real @ top_top_set_real ).
% fsigma_UNIV
thf(fact_166_space__lborel,axiom,
( ( sigma_space_real @ lebesgue_lborel_real )
= ( sigma_space_real @ borel_5078946678739801102l_real ) ) ).
% space_lborel
thf(fact_167_space__borel,axiom,
( ( sigma_space_a @ borel_5459123734250506524orel_a )
= top_top_set_a ) ).
% space_borel
thf(fact_168_space__borel,axiom,
( ( sigma_space_real @ borel_5078946678739801102l_real )
= top_top_set_real ) ).
% space_borel
thf(fact_169_space__borel,axiom,
( ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal )
= top_to7994903218803871134nnreal ) ).
% space_borel
thf(fact_170_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_5465916536984168985nnreal @ ( nonneg1394255657502361022nnreal @ top_to7994903218803871134nnreal ) )
= top_to3356475028079756884nnreal ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_171_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_sets_real @ ( nonneg387815094551837234e_real @ top_top_set_real ) )
= top_top_set_set_real ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_172_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_sets_a @ ( nonneg7367794086797660664sure_a @ top_top_set_a ) )
= top_top_set_set_a ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_173_space__diff__measure,axiom,
! [M: sigma_measure_a,N: sigma_measure_a] :
( ( sigma_space_a @ ( radon_diff_measure_a @ M @ N ) )
= ( sigma_space_a @ M ) ) ).
% space_diff_measure
thf(fact_174_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_to3356475028079756884nnreal
= ( sigma_5465916536984168985nnreal @ ( nonneg1394255657502361022nnreal @ top_to7994903218803871134nnreal ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_175_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_top_set_set_real
= ( sigma_sets_real @ ( nonneg387815094551837234e_real @ top_top_set_real ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_176_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_top_set_set_a
= ( sigma_sets_a @ ( nonneg7367794086797660664sure_a @ top_top_set_a ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_177_sets__uniform__count__measure__count__space,axiom,
! [A: set_a] :
( ( sigma_sets_a @ ( nonneg7367794086797660664sure_a @ A ) )
= ( sigma_sets_a @ ( sigma_count_space_a @ A ) ) ) ).
% sets_uniform_count_measure_count_space
thf(fact_178_sets__uniform__count__measure__count__space,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( sigma_5465916536984168985nnreal @ ( nonneg1394255657502361022nnreal @ A ) )
= ( sigma_5465916536984168985nnreal @ ( sigma_7204664791115113951nnreal @ A ) ) ) ).
% sets_uniform_count_measure_count_space
thf(fact_179_sets__uniform__count__measure__count__space,axiom,
! [A: set_real] :
( ( sigma_sets_real @ ( nonneg387815094551837234e_real @ A ) )
= ( sigma_sets_real @ ( sigma_8508918144308765139e_real @ A ) ) ) ).
% sets_uniform_count_measure_count_space
thf(fact_180_top__set__def,axiom,
( top_to7994903218803871134nnreal
= ( collec6648975593938027277nnreal @ top_to5118619752887738471real_o ) ) ).
% top_set_def
thf(fact_181_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_182_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_183_lborel__neq__count__space,axiom,
! [A: set_real] :
( lebesgue_lborel_real
!= ( sigma_8508918144308765139e_real @ A ) ) ).
% lborel_neq_count_space
thf(fact_184_sets__UNIV,axiom,
! [A: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( sigma_7204664791115113951nnreal @ top_to7994903218803871134nnreal ) ) ) ).
% sets_UNIV
thf(fact_185_sets__UNIV,axiom,
! [A: set_real] : ( member_set_real @ A @ ( sigma_sets_real @ ( sigma_8508918144308765139e_real @ top_top_set_real ) ) ) ).
% sets_UNIV
thf(fact_186_sets__UNIV,axiom,
! [A: set_a] : ( member_set_a @ A @ ( sigma_sets_a @ ( sigma_count_space_a @ top_top_set_a ) ) ) ).
% sets_UNIV
thf(fact_187_measurable__count__space__insert,axiom,
! [S2: set_a,S3: set_set_a,A: set_set_a] :
( ( member_set_a @ S2 @ S3 )
=> ( ( member_set_set_a @ A @ ( sigma_sets_set_a @ ( sigma_1106005778614564215_set_a @ S3 ) ) )
=> ( member_set_set_a @ ( insert_set_a @ S2 @ A ) @ ( sigma_sets_set_a @ ( sigma_1106005778614564215_set_a @ S3 ) ) ) ) ) ).
% measurable_count_space_insert
thf(fact_188_measurable__count__space__insert,axiom,
! [S2: real > extend8495563244428889912nnreal,S3: set_re5328672808648366137nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ S2 @ S3 )
=> ( ( member524040414084610768nnreal @ A @ ( sigma_3125015092713243876nnreal @ ( sigma_2847285985465031850nnreal @ S3 ) ) )
=> ( member524040414084610768nnreal @ ( insert152533262698245683nnreal @ S2 @ A ) @ ( sigma_3125015092713243876nnreal @ ( sigma_2847285985465031850nnreal @ S3 ) ) ) ) ) ).
% measurable_count_space_insert
thf(fact_189_measurable__count__space__insert,axiom,
! [S2: set_real,S3: set_set_real,A: set_set_real] :
( ( member_set_real @ S2 @ S3 )
=> ( ( member_set_set_real @ A @ ( sigma_sets_set_real @ ( sigma_3507695683712708105t_real @ S3 ) ) )
=> ( member_set_set_real @ ( insert_set_real @ S2 @ A ) @ ( sigma_sets_set_real @ ( sigma_3507695683712708105t_real @ S3 ) ) ) ) ) ).
% measurable_count_space_insert
thf(fact_190_measurable__count__space__insert,axiom,
! [S2: set_Ex3793607809372303086nnreal,S3: set_se4580700918925141924nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ S2 @ S3 )
=> ( ( member6568240578637133883nnreal @ A @ ( sigma_5308793447563920847nnreal @ ( sigma_384769719376554389nnreal @ S3 ) ) )
=> ( member6568240578637133883nnreal @ ( insert1343806209672318238nnreal @ S2 @ A ) @ ( sigma_5308793447563920847nnreal @ ( sigma_384769719376554389nnreal @ S3 ) ) ) ) ) ).
% measurable_count_space_insert
thf(fact_191_measurable__count__space__insert,axiom,
! [S2: a,S3: set_a,A: set_a] :
( ( member_a @ S2 @ S3 )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( sigma_count_space_a @ S3 ) ) )
=> ( member_set_a @ ( insert_a @ S2 @ A ) @ ( sigma_sets_a @ ( sigma_count_space_a @ S3 ) ) ) ) ) ).
% measurable_count_space_insert
thf(fact_192_measurable__count__space__insert,axiom,
! [S2: extend8495563244428889912nnreal,S3: set_Ex3793607809372303086nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ S2 @ S3 )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( sigma_7204664791115113951nnreal @ S3 ) ) )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ S2 @ A ) @ ( sigma_5465916536984168985nnreal @ ( sigma_7204664791115113951nnreal @ S3 ) ) ) ) ) ).
% measurable_count_space_insert
thf(fact_193_measurable__count__space__insert,axiom,
! [S2: real,S3: set_real,A: set_real] :
( ( member_real @ S2 @ S3 )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( sigma_8508918144308765139e_real @ S3 ) ) )
=> ( member_set_real @ ( insert_real @ S2 @ A ) @ ( sigma_sets_real @ ( sigma_8508918144308765139e_real @ S3 ) ) ) ) ) ).
% measurable_count_space_insert
thf(fact_194_space__completion,axiom,
! [M: sigma_measure_real] :
( ( sigma_space_real @ ( comple3506806835435775778n_real @ M ) )
= ( sigma_space_real @ M ) ) ).
% space_completion
thf(fact_195_sets__completionI__sets,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ).
% sets_completionI_sets
thf(fact_196_sets__completionI__sets,axiom,
! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ).
% sets_completionI_sets
thf(fact_197_sets__completionI__sets,axiom,
! [A: set_real,M: sigma_measure_real] :
( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ).
% sets_completionI_sets
thf(fact_198_sets_Otop,axiom,
! [M: sigma_measure_a] : ( member_set_a @ ( sigma_space_a @ M ) @ ( sigma_sets_a @ M ) ) ).
% sets.top
thf(fact_199_sets_Otop,axiom,
! [M: sigma_7234349610311085201nnreal] : ( member603777416030116741nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) ) ).
% sets.top
thf(fact_200_sets_Otop,axiom,
! [M: sigma_measure_real] : ( member_set_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).
% sets.top
thf(fact_201_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_202_sets__eq__imp__space__eq,axiom,
! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
( ( ( sigma_5465916536984168985nnreal @ M )
= ( sigma_5465916536984168985nnreal @ M2 ) )
=> ( ( sigma_3147302497200244656nnreal @ M )
= ( sigma_3147302497200244656nnreal @ M2 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_203_sets__eq__imp__space__eq,axiom,
! [M: sigma_measure_real,M2: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M2 ) )
=> ( ( sigma_space_real @ M )
= ( sigma_space_real @ M2 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_204_fsigma__imp__gdelta,axiom,
! [S3: set_a] :
( ( lebesgue_fsigma_a @ S3 )
=> ( lebesgue_gdelta_a @ ( uminus_uminus_set_a @ S3 ) ) ) ).
% fsigma_imp_gdelta
thf(fact_205_fsigma__imp__gdelta,axiom,
! [S3: set_Ex3793607809372303086nnreal] :
( ( lebesg3344469894415766602nnreal @ S3 )
=> ( lebesg8314085847218018492nnreal @ ( uminus5517552291522096439nnreal @ S3 ) ) ) ).
% fsigma_imp_gdelta
thf(fact_206_fsigma__imp__gdelta,axiom,
! [S3: set_real] :
( ( lebesgue_fsigma_real @ S3 )
=> ( lebesgue_gdelta_real @ ( uminus612125837232591019t_real @ S3 ) ) ) ).
% fsigma_imp_gdelta
thf(fact_207_gdelta__complement,axiom,
! [S3: set_a] :
( ( lebesgue_gdelta_a @ ( uminus_uminus_set_a @ S3 ) )
= ( lebesgue_fsigma_a @ S3 ) ) ).
% gdelta_complement
thf(fact_208_gdelta__complement,axiom,
! [S3: set_Ex3793607809372303086nnreal] :
( ( lebesg8314085847218018492nnreal @ ( uminus5517552291522096439nnreal @ S3 ) )
= ( lebesg3344469894415766602nnreal @ S3 ) ) ).
% gdelta_complement
thf(fact_209_gdelta__complement,axiom,
! [S3: set_real] :
( ( lebesgue_gdelta_real @ ( uminus612125837232591019t_real @ S3 ) )
= ( lebesgue_fsigma_real @ S3 ) ) ).
% gdelta_complement
thf(fact_210_gdelta__imp__fsigma,axiom,
! [S3: set_a] :
( ( lebesgue_gdelta_a @ S3 )
=> ( lebesgue_fsigma_a @ ( uminus_uminus_set_a @ S3 ) ) ) ).
% gdelta_imp_fsigma
thf(fact_211_gdelta__imp__fsigma,axiom,
! [S3: set_Ex3793607809372303086nnreal] :
( ( lebesg8314085847218018492nnreal @ S3 )
=> ( lebesg3344469894415766602nnreal @ ( uminus5517552291522096439nnreal @ S3 ) ) ) ).
% gdelta_imp_fsigma
thf(fact_212_gdelta__imp__fsigma,axiom,
! [S3: set_real] :
( ( lebesgue_gdelta_real @ S3 )
=> ( lebesgue_fsigma_real @ ( uminus612125837232591019t_real @ S3 ) ) ) ).
% gdelta_imp_fsigma
thf(fact_213_insert__null__sets__iff,axiom,
! [A2: real,N: set_real] :
( ( member_set_real @ ( insert_real @ A2 @ N ) @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
= ( member_set_real @ N @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) ) ) ).
% insert_null_sets_iff
thf(fact_214_space__lebesgue__on,axiom,
! [S3: set_real] :
( ( sigma_space_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) )
= S3 ) ).
% space_lebesgue_on
thf(fact_215_sets__restrict__UNIV,axiom,
! [M: sigma_7234349610311085201nnreal] :
( ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ M @ top_to7994903218803871134nnreal ) )
= ( sigma_5465916536984168985nnreal @ M ) ) ).
% sets_restrict_UNIV
thf(fact_216_sets__restrict__UNIV,axiom,
! [M: sigma_measure_real] :
( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ top_top_set_real ) )
= ( sigma_sets_real @ M ) ) ).
% sets_restrict_UNIV
thf(fact_217_sets__restrict__UNIV,axiom,
! [M: sigma_measure_a] :
( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ top_top_set_a ) )
= ( sigma_sets_a @ M ) ) ).
% sets_restrict_UNIV
thf(fact_218_space__restrict__space2,axiom,
! [Omega: set_a,M: sigma_measure_a] :
( ( member_set_a @ Omega @ ( sigma_sets_a @ M ) )
=> ( ( sigma_space_a @ ( sigma_8692839461743104066pace_a @ M @ Omega ) )
= Omega ) ) ).
% space_restrict_space2
thf(fact_219_space__restrict__space2,axiom,
! [Omega: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ Omega @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( sigma_3147302497200244656nnreal @ ( sigma_4884701650823297268nnreal @ M @ Omega ) )
= Omega ) ) ).
% space_restrict_space2
thf(fact_220_space__restrict__space2,axiom,
! [Omega: set_real,M: sigma_measure_real] :
( ( member_set_real @ Omega @ ( sigma_sets_real @ M ) )
=> ( ( sigma_space_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) )
= Omega ) ) ).
% space_restrict_space2
thf(fact_221_sets__lebesgue__on__refl,axiom,
! [S3: set_real] : ( member_set_real @ S3 @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) ) ) ).
% sets_lebesgue_on_refl
thf(fact_222_insert__null__sets__lebesgue__on__iff,axiom,
! [A2: real,S3: set_real,N: set_real] :
( ( member_real @ A2 @ S3 )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( ( member_set_real @ ( insert_real @ A2 @ N ) @ ( measur3710062792471635001s_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) ) )
= ( member_set_real @ N @ ( measur3710062792471635001s_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) ) ) ) ) ) ).
% insert_null_sets_lebesgue_on_iff
thf(fact_223_restrict__space__sets__cong,axiom,
! [A: set_a,B2: set_a,M: sigma_measure_a,N: sigma_measure_a] :
( ( A = B2 )
=> ( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N ) )
=> ( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ A ) )
= ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ N @ B2 ) ) ) ) ) ).
% restrict_space_sets_cong
thf(fact_224_restrict__space__sets__cong,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal] :
( ( A = B2 )
=> ( ( ( sigma_5465916536984168985nnreal @ M )
= ( sigma_5465916536984168985nnreal @ N ) )
=> ( ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ M @ A ) )
= ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ N @ B2 ) ) ) ) ) ).
% restrict_space_sets_cong
thf(fact_225_restrict__space__sets__cong,axiom,
! [A: set_real,B2: set_real,M: sigma_measure_real,N: sigma_measure_real] :
( ( A = B2 )
=> ( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ N ) )
=> ( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ A ) )
= ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ N @ B2 ) ) ) ) ) ).
% restrict_space_sets_cong
thf(fact_226_sets__restrict__space__cong,axiom,
! [M: sigma_measure_a,N: sigma_measure_a,Omega: set_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N ) )
=> ( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ Omega ) )
= ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ N @ Omega ) ) ) ) ).
% sets_restrict_space_cong
thf(fact_227_sets__restrict__space__cong,axiom,
! [M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,Omega: set_Ex3793607809372303086nnreal] :
( ( ( sigma_5465916536984168985nnreal @ M )
= ( sigma_5465916536984168985nnreal @ N ) )
=> ( ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ M @ Omega ) )
= ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ N @ Omega ) ) ) ) ).
% sets_restrict_space_cong
thf(fact_228_sets__restrict__space__cong,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,Omega: set_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ N ) )
=> ( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) )
= ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ N @ Omega ) ) ) ) ).
% sets_restrict_space_cong
thf(fact_229_null__sets__completionI,axiom,
! [N: set_a,M: sigma_measure_a] :
( ( member_set_a @ N @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ N @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ).
% null_sets_completionI
thf(fact_230_null__sets__completionI,axiom,
! [N: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ N @ ( measur1209175464439008069nnreal @ M ) )
=> ( member603777416030116741nnreal @ N @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ).
% null_sets_completionI
thf(fact_231_null__sets__completionI,axiom,
! [N: set_real,M: sigma_measure_real] :
( ( member_set_real @ N @ ( measur3710062792471635001s_real @ M ) )
=> ( member_set_real @ N @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ).
% null_sets_completionI
thf(fact_232_null__sets__completion__iff,axiom,
! [N: set_a,M: sigma_measure_a] :
( ( member_set_a @ N @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ N @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
= ( member_set_a @ N @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets_completion_iff
thf(fact_233_null__sets__completion__iff,axiom,
! [N: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ N @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ N @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
= ( member603777416030116741nnreal @ N @ ( measur1209175464439008069nnreal @ M ) ) ) ) ).
% null_sets_completion_iff
thf(fact_234_null__sets__completion__iff,axiom,
! [N: set_real,M: sigma_measure_real] :
( ( member_set_real @ N @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ N @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) )
= ( member_set_real @ N @ ( measur3710062792471635001s_real @ M ) ) ) ) ).
% null_sets_completion_iff
thf(fact_235_lebesgue__on__UNIV__eq,axiom,
( ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ top_top_set_real )
= ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) ).
% lebesgue_on_UNIV_eq
thf(fact_236_top__empty__eq,axiom,
( top_top_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ top_top_set_set_a ) ) ) ).
% top_empty_eq
thf(fact_237_top__empty__eq,axiom,
( top_to199868804852128988real_o
= ( ^ [X3: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X3 @ top_to315565310957491945nnreal ) ) ) ).
% top_empty_eq
thf(fact_238_top__empty__eq,axiom,
( top_top_set_real_o
= ( ^ [X3: set_real] : ( member_set_real @ X3 @ top_top_set_set_real ) ) ) ).
% top_empty_eq
thf(fact_239_top__empty__eq,axiom,
( top_to5272770551662541617real_o
= ( ^ [X3: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X3 @ top_to3356475028079756884nnreal ) ) ) ).
% top_empty_eq
thf(fact_240_top__empty__eq,axiom,
( top_to5118619752887738471real_o
= ( ^ [X3: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X3 @ top_to7994903218803871134nnreal ) ) ) ).
% top_empty_eq
thf(fact_241_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X3: real] : ( member_real @ X3 @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_242_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_243_null__setsD2,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ A @ ( sigma_sets_a @ M ) ) ) ).
% null_setsD2
thf(fact_244_null__setsD2,axiom,
! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ M ) )
=> ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% null_setsD2
thf(fact_245_null__setsD2,axiom,
! [A: set_real,M: sigma_measure_real] :
( ( member_set_real @ A @ ( measur3710062792471635001s_real @ M ) )
=> ( member_set_real @ A @ ( sigma_sets_real @ M ) ) ) ).
% null_setsD2
thf(fact_246_null__part__null__sets,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ ( complete_null_part_a @ M @ S3 ) @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ).
% null_part_null_sets
thf(fact_247_null__part__null__sets,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
=> ( member603777416030116741nnreal @ ( comple6358047150840085292nnreal @ M @ S3 ) @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ).
% null_part_null_sets
thf(fact_248_null__part__null__sets,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ( member_set_real @ ( comple4917500974405109920t_real @ M @ S3 ) @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ).
% null_part_null_sets
thf(fact_249_main__part__sets,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ ( complete_main_part_a @ M @ S3 ) @ ( sigma_sets_a @ M ) ) ) ).
% main_part_sets
thf(fact_250_main__part__sets,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
=> ( member603777416030116741nnreal @ ( comple2904675884154540190nnreal @ M @ S3 ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% main_part_sets
thf(fact_251_main__part__sets,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ( member_set_real @ ( comple5203310272383980818t_real @ M @ S3 ) @ ( sigma_sets_real @ M ) ) ) ).
% main_part_sets
thf(fact_252_restrict__space__singleton,axiom,
! [X2: a,M: sigma_measure_a] :
( ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
= ( sigma_sets_a @ ( sigma_count_space_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% restrict_space_singleton
thf(fact_253_restrict__space__singleton,axiom,
! [X2: extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ M @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) )
= ( sigma_5465916536984168985nnreal @ ( sigma_7204664791115113951nnreal @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) ) ) ) ).
% restrict_space_singleton
thf(fact_254_restrict__space__singleton,axiom,
! [X2: real,M: sigma_measure_real] :
( ( member_set_real @ ( insert_real @ X2 @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) )
=> ( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
= ( sigma_sets_real @ ( sigma_8508918144308765139e_real @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ).
% restrict_space_singleton
thf(fact_255_lebesgue__measurable__diff__null,axiom,
! [N: set_real,F: real > real] :
( ( member_set_real @ N @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ ( uminus612125837232591019t_real @ N ) ) @ borel_5078946678739801102l_real ) )
= ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ borel_5078946678739801102l_real ) ) ) ) ).
% lebesgue_measurable_diff_null
thf(fact_256_measurable__top,axiom,
! [M: sigma_measure_real] : ( member2919562650594848410nnreal @ top_to4050977978985836211nnreal @ ( sigma_9017504469962657078nnreal @ M @ ( sigma_7204664791115113951nnreal @ top_to7994903218803871134nnreal ) ) ) ).
% measurable_top
thf(fact_257_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_258_all__not__in__conv,axiom,
! [A: set_re5328672808648366137nnreal] :
( ( ! [X3: real > extend8495563244428889912nnreal] :
~ ( member2919562650594848410nnreal @ X3 @ A ) )
= ( A = bot_bo6037503491064675021nnreal ) ) ).
% all_not_in_conv
thf(fact_259_all__not__in__conv,axiom,
! [A: set_set_real] :
( ( ! [X3: set_real] :
~ ( member_set_real @ X3 @ A ) )
= ( A = bot_bot_set_set_real ) ) ).
% all_not_in_conv
thf(fact_260_all__not__in__conv,axiom,
! [A: set_se4580700918925141924nnreal] :
( ( ! [X3: set_Ex3793607809372303086nnreal] :
~ ( member603777416030116741nnreal @ X3 @ A ) )
= ( A = bot_bo2988155216863113784nnreal ) ) ).
% all_not_in_conv
thf(fact_261_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_262_empty__iff,axiom,
! [C: real > extend8495563244428889912nnreal] :
~ ( member2919562650594848410nnreal @ C @ bot_bo6037503491064675021nnreal ) ).
% empty_iff
thf(fact_263_empty__iff,axiom,
! [C: set_real] :
~ ( member_set_real @ C @ bot_bot_set_set_real ) ).
% empty_iff
thf(fact_264_empty__iff,axiom,
! [C: set_Ex3793607809372303086nnreal] :
~ ( member603777416030116741nnreal @ C @ bot_bo2988155216863113784nnreal ) ).
% empty_iff
thf(fact_265_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_266_singletonI,axiom,
! [A2: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ A2 @ ( insert152533262698245683nnreal @ A2 @ bot_bo6037503491064675021nnreal ) ) ).
% singletonI
thf(fact_267_singletonI,axiom,
! [A2: set_real] : ( member_set_real @ A2 @ ( insert_set_real @ A2 @ bot_bot_set_set_real ) ) ).
% singletonI
thf(fact_268_singletonI,axiom,
! [A2: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ A2 @ bot_bo2988155216863113784nnreal ) ) ).
% singletonI
thf(fact_269_sets_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( sigma_sets_a @ M ) ) ).
% sets.empty_sets
thf(fact_270_sets_Oempty__sets,axiom,
! [M: sigma_7234349610311085201nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( sigma_5465916536984168985nnreal @ M ) ) ).
% sets.empty_sets
thf(fact_271_sets_Oempty__sets,axiom,
! [M: sigma_measure_real] : ( member_set_real @ bot_bot_set_real @ ( sigma_sets_real @ M ) ) ).
% sets.empty_sets
thf(fact_272_null__sets_Oempty__sets,axiom,
! [M: sigma_measure_a] : ( member_set_a @ bot_bot_set_a @ ( measure_null_sets_a @ M ) ) ).
% null_sets.empty_sets
thf(fact_273_null__sets_Oempty__sets,axiom,
! [M: sigma_measure_real] : ( member_set_real @ bot_bot_set_real @ ( measur3710062792471635001s_real @ M ) ) ).
% null_sets.empty_sets
thf(fact_274_null__sets_Oempty__sets,axiom,
! [M: sigma_7234349610311085201nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( measur1209175464439008069nnreal @ M ) ) ).
% null_sets.empty_sets
thf(fact_275_main__part,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ M ) )
=> ( ( complete_main_part_a @ M @ S3 )
= S3 ) ) ).
% main_part
thf(fact_276_main__part,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( comple2904675884154540190nnreal @ M @ S3 )
= S3 ) ) ).
% main_part
thf(fact_277_main__part,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ M ) )
=> ( ( comple5203310272383980818t_real @ M @ S3 )
= S3 ) ) ).
% main_part
thf(fact_278_null__part__sets_I1_J,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( complete_null_part_a @ M @ S3 ) @ ( sigma_sets_a @ M ) ) ) ).
% null_part_sets(1)
thf(fact_279_null__part__sets_I1_J,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( comple6358047150840085292nnreal @ M @ S3 ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% null_part_sets(1)
thf(fact_280_null__part__sets_I1_J,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( comple4917500974405109920t_real @ M @ S3 ) @ ( sigma_sets_real @ M ) ) ) ).
% null_part_sets(1)
thf(fact_281_boolean__algebra_Ocompl__one,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.compl_one
thf(fact_282_boolean__algebra_Ocompl__one,axiom,
( ( uminus5517552291522096439nnreal @ top_to7994903218803871134nnreal )
= bot_bo4854962954004695426nnreal ) ).
% boolean_algebra.compl_one
thf(fact_283_boolean__algebra_Ocompl__one,axiom,
( ( uminus612125837232591019t_real @ top_top_set_real )
= bot_bot_set_real ) ).
% boolean_algebra.compl_one
thf(fact_284_boolean__algebra_Ocompl__zero,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% boolean_algebra.compl_zero
thf(fact_285_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5517552291522096439nnreal @ bot_bo4854962954004695426nnreal )
= top_to7994903218803871134nnreal ) ).
% boolean_algebra.compl_zero
thf(fact_286_boolean__algebra_Ocompl__zero,axiom,
( ( uminus612125837232591019t_real @ bot_bot_set_real )
= top_top_set_real ) ).
% boolean_algebra.compl_zero
thf(fact_287_measurable__lborel2,axiom,
! [M: sigma_7234349610311085201nnreal] :
( ( sigma_9017504469962657078nnreal @ lebesgue_lborel_real @ M )
= ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ M ) ) ).
% measurable_lborel2
thf(fact_288_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_289_ex__in__conv,axiom,
! [A: set_re5328672808648366137nnreal] :
( ( ? [X3: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X3 @ A ) )
= ( A != bot_bo6037503491064675021nnreal ) ) ).
% ex_in_conv
thf(fact_290_ex__in__conv,axiom,
! [A: set_set_real] :
( ( ? [X3: set_real] : ( member_set_real @ X3 @ A ) )
= ( A != bot_bot_set_set_real ) ) ).
% ex_in_conv
thf(fact_291_ex__in__conv,axiom,
! [A: set_se4580700918925141924nnreal] :
( ( ? [X3: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X3 @ A ) )
= ( A != bot_bo2988155216863113784nnreal ) ) ).
% ex_in_conv
thf(fact_292_equals0I,axiom,
! [A: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_293_equals0I,axiom,
! [A: set_re5328672808648366137nnreal] :
( ! [Y2: real > extend8495563244428889912nnreal] :
~ ( member2919562650594848410nnreal @ Y2 @ A )
=> ( A = bot_bo6037503491064675021nnreal ) ) ).
% equals0I
thf(fact_294_equals0I,axiom,
! [A: set_set_real] :
( ! [Y2: set_real] :
~ ( member_set_real @ Y2 @ A )
=> ( A = bot_bot_set_set_real ) ) ).
% equals0I
thf(fact_295_equals0I,axiom,
! [A: set_se4580700918925141924nnreal] :
( ! [Y2: set_Ex3793607809372303086nnreal] :
~ ( member603777416030116741nnreal @ Y2 @ A )
=> ( A = bot_bo2988155216863113784nnreal ) ) ).
% equals0I
thf(fact_296_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_297_equals0D,axiom,
! [A: set_re5328672808648366137nnreal,A2: real > extend8495563244428889912nnreal] :
( ( A = bot_bo6037503491064675021nnreal )
=> ~ ( member2919562650594848410nnreal @ A2 @ A ) ) ).
% equals0D
thf(fact_298_equals0D,axiom,
! [A: set_set_real,A2: set_real] :
( ( A = bot_bot_set_set_real )
=> ~ ( member_set_real @ A2 @ A ) ) ).
% equals0D
thf(fact_299_equals0D,axiom,
! [A: set_se4580700918925141924nnreal,A2: set_Ex3793607809372303086nnreal] :
( ( A = bot_bo2988155216863113784nnreal )
=> ~ ( member603777416030116741nnreal @ A2 @ A ) ) ).
% equals0D
thf(fact_300_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_301_emptyE,axiom,
! [A2: real > extend8495563244428889912nnreal] :
~ ( member2919562650594848410nnreal @ A2 @ bot_bo6037503491064675021nnreal ) ).
% emptyE
thf(fact_302_emptyE,axiom,
! [A2: set_real] :
~ ( member_set_real @ A2 @ bot_bot_set_set_real ) ).
% emptyE
thf(fact_303_emptyE,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
~ ( member603777416030116741nnreal @ A2 @ bot_bo2988155216863113784nnreal ) ).
% emptyE
thf(fact_304_measurable__empty__iff,axiom,
! [N: sigma_7234349610311085201nnreal,F: real > extend8495563244428889912nnreal,M: sigma_measure_real] :
( ( ( sigma_3147302497200244656nnreal @ N )
= bot_bo4854962954004695426nnreal )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
= ( ( sigma_space_real @ M )
= bot_bot_set_real ) ) ) ).
% measurable_empty_iff
thf(fact_305_space__empty__iff,axiom,
! [N: sigma_measure_a] :
( ( ( sigma_space_a @ N )
= bot_bot_set_a )
= ( ( sigma_sets_a @ N )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ) ).
% space_empty_iff
thf(fact_306_space__empty__iff,axiom,
! [N: sigma_7234349610311085201nnreal] :
( ( ( sigma_3147302497200244656nnreal @ N )
= bot_bo4854962954004695426nnreal )
= ( ( sigma_5465916536984168985nnreal @ N )
= ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) ) ) ).
% space_empty_iff
thf(fact_307_space__empty__iff,axiom,
! [N: sigma_measure_real] :
( ( ( sigma_space_real @ N )
= bot_bot_set_real )
= ( ( sigma_sets_real @ N )
= ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ) ).
% space_empty_iff
thf(fact_308_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_309_measurable__cong__sets,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a,N: sigma_7234349610311085201nnreal,N2: sigma_7234349610311085201nnreal] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_5465916536984168985nnreal @ N )
= ( sigma_5465916536984168985nnreal @ N2 ) )
=> ( ( sigma_214952329563889126nnreal @ M @ N )
= ( sigma_214952329563889126nnreal @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_310_measurable__cong__sets,axiom,
! [M: sigma_measure_a,M2: sigma_measure_a,N: sigma_measure_real,N2: sigma_measure_real] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ N2 ) )
=> ( ( sigma_9116425665531756122a_real @ M @ N )
= ( sigma_9116425665531756122a_real @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_311_measurable__cong__sets,axiom,
! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal,N: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_5465916536984168985nnreal @ M )
= ( sigma_5465916536984168985nnreal @ M2 ) )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ N2 ) )
=> ( ( sigma_3031480723531659892real_a @ M @ N )
= ( sigma_3031480723531659892real_a @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_312_measurable__cong__sets,axiom,
! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,N2: sigma_7234349610311085201nnreal] :
( ( ( sigma_5465916536984168985nnreal @ M )
= ( sigma_5465916536984168985nnreal @ M2 ) )
=> ( ( ( sigma_5465916536984168985nnreal @ N )
= ( sigma_5465916536984168985nnreal @ N2 ) )
=> ( ( sigma_7926153774531450434nnreal @ M @ N )
= ( sigma_7926153774531450434nnreal @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_313_measurable__cong__sets,axiom,
! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal,N: sigma_measure_real,N2: sigma_measure_real] :
( ( ( sigma_5465916536984168985nnreal @ M )
= ( sigma_5465916536984168985nnreal @ M2 ) )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ N2 ) )
=> ( ( sigma_7049758200512112822l_real @ M @ N )
= ( sigma_7049758200512112822l_real @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_314_measurable__cong__sets,axiom,
! [M: sigma_measure_real,M2: sigma_measure_real,N: sigma_measure_a,N2: sigma_measure_a] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ N2 ) )
=> ( ( sigma_523072396149930112real_a @ M @ N )
= ( sigma_523072396149930112real_a @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_315_measurable__cong__sets,axiom,
! [M: sigma_measure_real,M2: sigma_measure_real,N: sigma_measure_real,N2: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ N2 ) )
=> ( ( sigma_5267869275261027754l_real @ M @ N )
= ( sigma_5267869275261027754l_real @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_316_measurable__cong__sets,axiom,
! [M: sigma_measure_real,M2: sigma_measure_real,N: sigma_7234349610311085201nnreal,N2: sigma_7234349610311085201nnreal] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_5465916536984168985nnreal @ N )
= ( sigma_5465916536984168985nnreal @ N2 ) )
=> ( ( sigma_9017504469962657078nnreal @ M @ N )
= ( sigma_9017504469962657078nnreal @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_317_measurable__cong,axiom,
! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal,M2: sigma_7234349610311085201nnreal] :
( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ M2 ) )
= ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ M2 ) ) ) ) ).
% measurable_cong
thf(fact_318_measurable__space,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,A: sigma_7234349610311085201nnreal,X2: real] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ A ) )
=> ( ( member_real @ X2 @ ( sigma_space_real @ M ) )
=> ( member7908768830364227535nnreal @ ( F @ X2 ) @ ( sigma_3147302497200244656nnreal @ A ) ) ) ) ).
% measurable_space
thf(fact_319_measurable__space,axiom,
! [F: set_a > set_a,M: sigma_measure_set_a,A: sigma_measure_set_a,X2: set_a] :
( ( member_set_a_set_a @ F @ ( sigma_5212894042034225104_set_a @ M @ A ) )
=> ( ( member_set_a @ X2 @ ( sigma_space_set_a @ M ) )
=> ( member_set_a @ ( F @ X2 ) @ ( sigma_space_set_a @ A ) ) ) ) ).
% measurable_space
thf(fact_320_measurable__space,axiom,
! [F: set_a > set_real,M: sigma_measure_set_a,A: sigma_3733394171116455995t_real,X2: set_a] :
( ( member6119777607969566810t_real @ F @ ( sigma_5529004876658666480t_real @ M @ A ) )
=> ( ( member_set_a @ X2 @ ( sigma_space_set_a @ M ) )
=> ( member_set_real @ ( F @ X2 ) @ ( sigma_space_set_real @ A ) ) ) ) ).
% measurable_space
thf(fact_321_measurable__space,axiom,
! [F: set_a > set_Ex3793607809372303086nnreal,M: sigma_measure_set_a,A: sigma_523634232904505671nnreal,X2: set_a] :
( ( member8552228822107236582nnreal @ F @ ( sigma_2316796825407894268nnreal @ M @ A ) )
=> ( ( member_set_a @ X2 @ ( sigma_space_set_a @ M ) )
=> ( member603777416030116741nnreal @ ( F @ X2 ) @ ( sigma_2539764534872131430nnreal @ A ) ) ) ) ).
% measurable_space
thf(fact_322_measurable__space,axiom,
! [F: set_real > set_a,M: sigma_3733394171116455995t_real,A: sigma_measure_set_a,X2: set_real] :
( ( member9134392423035811420_set_a @ F @ ( sigma_8826535904794920746_set_a @ M @ A ) )
=> ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M ) )
=> ( member_set_a @ ( F @ X2 ) @ ( sigma_space_set_a @ A ) ) ) ) ).
% measurable_space
thf(fact_323_measurable__space,axiom,
! [F: set_real > set_real,M: sigma_3733394171116455995t_real,A: sigma_3733394171116455995t_real,X2: set_real] :
( ( member8820419758626733818t_real @ F @ ( sigma_8759315257289043734t_real @ M @ A ) )
=> ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M ) )
=> ( member_set_real @ ( F @ X2 ) @ ( sigma_space_set_real @ A ) ) ) ) ).
% measurable_space
thf(fact_324_measurable__space,axiom,
! [F: set_real > set_Ex3793607809372303086nnreal,M: sigma_3733394171116455995t_real,A: sigma_523634232904505671nnreal,X2: set_real] :
( ( member1248903934968170374nnreal @ F @ ( sigma_4962942689157396770nnreal @ M @ A ) )
=> ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M ) )
=> ( member603777416030116741nnreal @ ( F @ X2 ) @ ( sigma_2539764534872131430nnreal @ A ) ) ) ) ).
% measurable_space
thf(fact_325_measurable__space,axiom,
! [F: set_Ex3793607809372303086nnreal > set_a,M: sigma_523634232904505671nnreal,A: sigma_measure_set_a,X2: set_Ex3793607809372303086nnreal] :
( ( member8849812955461179984_set_a @ F @ ( sigma_7598581795090538910_set_a @ M @ A ) )
=> ( ( member603777416030116741nnreal @ X2 @ ( sigma_2539764534872131430nnreal @ M ) )
=> ( member_set_a @ ( F @ X2 ) @ ( sigma_space_set_a @ A ) ) ) ) ).
% measurable_space
thf(fact_326_measurable__space,axiom,
! [F: set_Ex3793607809372303086nnreal > set_real,M: sigma_523634232904505671nnreal,A: sigma_3733394171116455995t_real,X2: set_Ex3793607809372303086nnreal] :
( ( member7536123422392205318t_real @ F @ ( sigma_6728074762985347490t_real @ M @ A ) )
=> ( ( member603777416030116741nnreal @ X2 @ ( sigma_2539764534872131430nnreal @ M ) )
=> ( member_set_real @ ( F @ X2 ) @ ( sigma_space_set_real @ A ) ) ) ) ).
% measurable_space
thf(fact_327_measurable__space,axiom,
! [F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,M: sigma_523634232904505671nnreal,A: sigma_523634232904505671nnreal,X2: set_Ex3793607809372303086nnreal] :
( ( member9048868947896282770nnreal @ F @ ( sigma_8870595382113208750nnreal @ M @ A ) )
=> ( ( member603777416030116741nnreal @ X2 @ ( sigma_2539764534872131430nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( F @ X2 ) @ ( sigma_2539764534872131430nnreal @ A ) ) ) ) ).
% measurable_space
thf(fact_328_measurable__cong__simp,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_7234349610311085201nnreal,N2: sigma_7234349610311085201nnreal,F: real > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal] :
( ( M = N )
=> ( ( M2 = N2 )
=> ( ! [W: real] :
( ( member_real @ W @ ( sigma_space_real @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ M2 ) )
= ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_329_measurable__completion,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( comple3506806835435775778n_real @ M ) @ N ) ) ) ).
% measurable_completion
thf(fact_330_measurable__restrict__space1,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,Omega: set_real] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( sigma_5414646170262037096e_real @ M @ Omega ) @ N ) ) ) ).
% measurable_restrict_space1
thf(fact_331_empty__not__UNIV,axiom,
bot_bo4854962954004695426nnreal != top_to7994903218803871134nnreal ).
% empty_not_UNIV
thf(fact_332_empty__not__UNIV,axiom,
bot_bot_set_real != top_top_set_real ).
% empty_not_UNIV
thf(fact_333_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_334_singletonD,axiom,
! [B: set_a,A2: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_335_singletonD,axiom,
! [B: real > extend8495563244428889912nnreal,A2: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ B @ ( insert152533262698245683nnreal @ A2 @ bot_bo6037503491064675021nnreal ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_336_singletonD,axiom,
! [B: set_real,A2: set_real] :
( ( member_set_real @ B @ ( insert_set_real @ A2 @ bot_bot_set_set_real ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_337_singletonD,axiom,
! [B: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ B @ ( insert1343806209672318238nnreal @ A2 @ bot_bo2988155216863113784nnreal ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_338_singleton__iff,axiom,
! [B: set_a,A2: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_339_singleton__iff,axiom,
! [B: real > extend8495563244428889912nnreal,A2: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ B @ ( insert152533262698245683nnreal @ A2 @ bot_bo6037503491064675021nnreal ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_340_singleton__iff,axiom,
! [B: set_real,A2: set_real] :
( ( member_set_real @ B @ ( insert_set_real @ A2 @ bot_bot_set_set_real ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_341_singleton__iff,axiom,
! [B: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ B @ ( insert1343806209672318238nnreal @ A2 @ bot_bo2988155216863113784nnreal ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_342_null__sets_Oinsert__in__sets,axiom,
! [X2: a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( insert_a @ X2 @ A ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.insert_in_sets
thf(fact_343_null__sets_Oinsert__in__sets,axiom,
! [X2: real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ ( insert_real @ X2 @ bot_bot_set_real ) @ ( measur3710062792471635001s_real @ M ) )
=> ( ( member_set_real @ A @ ( measur3710062792471635001s_real @ M ) )
=> ( member_set_real @ ( insert_real @ X2 @ A ) @ ( measur3710062792471635001s_real @ M ) ) ) ) ).
% null_sets.insert_in_sets
thf(fact_344_null__sets_Oinsert__in__sets,axiom,
! [X2: extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ A ) @ ( measur1209175464439008069nnreal @ M ) ) ) ) ).
% null_sets.insert_in_sets
thf(fact_345_measurable__count__space,axiom,
! [F: real > extend8495563244428889912nnreal,A: set_real] : ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( sigma_8508918144308765139e_real @ A ) @ ( sigma_7204664791115113951nnreal @ top_to7994903218803871134nnreal ) ) ) ).
% measurable_count_space
thf(fact_346_borel__measurable__count__space,axiom,
! [F: real > extend8495563244428889912nnreal,S3: set_real] : ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( sigma_8508918144308765139e_real @ S3 ) @ borel_6524799422816628122nnreal ) ) ).
% borel_measurable_count_space
thf(fact_347_perfect__space__class_OUNIV__not__singleton,axiom,
! [X2: extend8495563244428889912nnreal] :
( top_to7994903218803871134nnreal
!= ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_348_perfect__space__class_OUNIV__not__singleton,axiom,
! [X2: real] :
( top_top_set_real
!= ( insert_real @ X2 @ bot_bot_set_real ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_349_sets_Oinsert__in__sets,axiom,
! [X2: a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( insert_a @ X2 @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( insert_a @ X2 @ A ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_350_sets_Oinsert__in__sets,axiom,
! [X2: extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ A ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_351_sets_Oinsert__in__sets,axiom,
! [X2: real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ ( insert_real @ X2 @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( insert_real @ X2 @ A ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_352_Compl__UNIV__eq,axiom,
( ( uminus_uminus_set_a @ top_top_set_a )
= bot_bot_set_a ) ).
% Compl_UNIV_eq
thf(fact_353_Compl__UNIV__eq,axiom,
( ( uminus5517552291522096439nnreal @ top_to7994903218803871134nnreal )
= bot_bo4854962954004695426nnreal ) ).
% Compl_UNIV_eq
thf(fact_354_Compl__UNIV__eq,axiom,
( ( uminus612125837232591019t_real @ top_top_set_real )
= bot_bot_set_real ) ).
% Compl_UNIV_eq
thf(fact_355_Compl__empty__eq,axiom,
( ( uminus_uminus_set_a @ bot_bot_set_a )
= top_top_set_a ) ).
% Compl_empty_eq
thf(fact_356_Compl__empty__eq,axiom,
( ( uminus5517552291522096439nnreal @ bot_bo4854962954004695426nnreal )
= top_to7994903218803871134nnreal ) ).
% Compl_empty_eq
thf(fact_357_Compl__empty__eq,axiom,
( ( uminus612125837232591019t_real @ bot_bot_set_real )
= top_top_set_real ) ).
% Compl_empty_eq
thf(fact_358_measurable__lebesgue__cong,axiom,
! [S3: set_real,F: real > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal] :
( ! [X: real] :
( ( member_real @ X @ S3 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) @ M ) )
= ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) @ M ) ) ) ) ).
% measurable_lebesgue_cong
thf(fact_359_is__borel__def,axiom,
( borel_3656262399657348386nnreal
= ( ^ [F2: real > extend8495563244428889912nnreal,M3: sigma_measure_real] : ( member2919562650594848410nnreal @ F2 @ ( sigma_9017504469962657078nnreal @ M3 @ borel_6524799422816628122nnreal ) ) ) ) ).
% is_borel_def
thf(fact_360_borel__measurable__diff__null,axiom,
! [N: set_real,S3: set_real,F: real > real] :
( ( member_set_real @ N @ ( measur3710062792471635001s_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) ) )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ ( minus_minus_set_real @ S3 @ N ) ) @ borel_5078946678739801102l_real ) )
= ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) @ borel_5078946678739801102l_real ) ) ) ) ) ).
% borel_measurable_diff_null
thf(fact_361_main__part__null__part__Int,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( inf_inf_set_a @ ( complete_main_part_a @ M @ S3 ) @ ( complete_null_part_a @ M @ S3 ) )
= bot_bot_set_a ) ) ).
% main_part_null_part_Int
thf(fact_362_main__part__null__part__Int,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
=> ( ( inf_in3368558534146122112nnreal @ ( comple2904675884154540190nnreal @ M @ S3 ) @ ( comple6358047150840085292nnreal @ M @ S3 ) )
= bot_bo4854962954004695426nnreal ) ) ).
% main_part_null_part_Int
thf(fact_363_main__part__null__part__Int,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ( ( inf_inf_set_real @ ( comple5203310272383980818t_real @ M @ S3 ) @ ( comple4917500974405109920t_real @ M @ S3 ) )
= bot_bot_set_real ) ) ).
% main_part_null_part_Int
thf(fact_364_measurable__restrict__countable,axiom,
! [X5: set_real,M: sigma_measure_real,F: real > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal] :
( ( counta7319604579010473777e_real @ X5 )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member7908768830364227535nnreal @ ( F @ X ) @ ( sigma_3147302497200244656nnreal @ N ) ) )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( sigma_5414646170262037096e_real @ M @ ( uminus612125837232591019t_real @ X5 ) ) @ N ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_365_measurable__restrict__countable,axiom,
! [X5: set_a,M: sigma_measure_a,F: a > set_a,N: sigma_measure_set_a] :
( ( counta4098120917673242425able_a @ X5 )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_a @ ( F @ X ) @ ( sigma_space_set_a @ N ) ) )
=> ( ( member_a_set_a @ F @ ( sigma_3685133166752798000_set_a @ ( sigma_8692839461743104066pace_a @ M @ ( uminus_uminus_set_a @ X5 ) ) @ N ) )
=> ( member_a_set_a @ F @ ( sigma_3685133166752798000_set_a @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_366_measurable__restrict__countable,axiom,
! [X5: set_a,M: sigma_measure_a,F: a > set_real,N: sigma_3733394171116455995t_real] :
( ( counta4098120917673242425able_a @ X5 )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_real @ ( F @ X ) @ ( sigma_space_set_real @ N ) ) )
=> ( ( member_a_set_real @ F @ ( sigma_739038748264640144t_real @ ( sigma_8692839461743104066pace_a @ M @ ( uminus_uminus_set_a @ X5 ) ) @ N ) )
=> ( member_a_set_real @ F @ ( sigma_739038748264640144t_real @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_367_measurable__restrict__countable,axiom,
! [X5: set_a,M: sigma_measure_a,F: a > set_Ex3793607809372303086nnreal,N: sigma_523634232904505671nnreal] :
( ( counta4098120917673242425able_a @ X5 )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member603777416030116741nnreal @ ( F @ X ) @ ( sigma_2539764534872131430nnreal @ N ) ) )
=> ( ( member2532357421736347526nnreal @ F @ ( sigma_7596264061814621596nnreal @ ( sigma_8692839461743104066pace_a @ M @ ( uminus_uminus_set_a @ X5 ) ) @ N ) )
=> ( member2532357421736347526nnreal @ F @ ( sigma_7596264061814621596nnreal @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_368_measurable__restrict__countable,axiom,
! [X5: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > set_a,N: sigma_measure_set_a] :
( ( counta8439243037236335165nnreal @ X5 )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member_set_a @ ( F @ X ) @ ( sigma_space_set_a @ N ) ) )
=> ( ( member6799942265337811078_set_a @ F @ ( sigma_7624677704890010580_set_a @ ( sigma_4884701650823297268nnreal @ M @ ( uminus5517552291522096439nnreal @ X5 ) ) @ N ) )
=> ( member6799942265337811078_set_a @ F @ ( sigma_7624677704890010580_set_a @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_369_measurable__restrict__countable,axiom,
! [X5: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > set_real,N: sigma_3733394171116455995t_real] :
( ( counta8439243037236335165nnreal @ X5 )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member_set_real @ ( F @ X ) @ ( sigma_space_set_real @ N ) ) )
=> ( ( member6764088077590758224t_real @ F @ ( sigma_5175731160935721196t_real @ ( sigma_4884701650823297268nnreal @ M @ ( uminus5517552291522096439nnreal @ X5 ) ) @ N ) )
=> ( member6764088077590758224t_real @ F @ ( sigma_5175731160935721196t_real @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_370_measurable__restrict__countable,axiom,
! [X5: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal,N: sigma_523634232904505671nnreal] :
( ( counta8439243037236335165nnreal @ X5 )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member603777416030116741nnreal @ ( F @ X ) @ ( sigma_2539764534872131430nnreal @ N ) ) )
=> ( ( member4416662722526258908nnreal @ F @ ( sigma_1941770836459163128nnreal @ ( sigma_4884701650823297268nnreal @ M @ ( uminus5517552291522096439nnreal @ X5 ) ) @ N ) )
=> ( member4416662722526258908nnreal @ F @ ( sigma_1941770836459163128nnreal @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_371_measurable__restrict__countable,axiom,
! [X5: set_real,M: sigma_measure_real,F: real > set_a,N: sigma_measure_set_a] :
( ( counta7319604579010473777e_real @ X5 )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_a @ ( F @ X ) @ ( sigma_space_set_a @ N ) ) )
=> ( ( member_real_set_a @ F @ ( sigma_4283435981211228640_set_a @ ( sigma_5414646170262037096e_real @ M @ ( uminus612125837232591019t_real @ X5 ) ) @ N ) )
=> ( member_real_set_a @ F @ ( sigma_4283435981211228640_set_a @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_372_measurable__restrict__countable,axiom,
! [X5: set_real,M: sigma_measure_real,F: real > set_real,N: sigma_3733394171116455995t_real] :
( ( counta7319604579010473777e_real @ X5 )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( F @ X ) @ ( sigma_space_set_real @ N ) ) )
=> ( ( member_real_set_real @ F @ ( sigma_6606012509476713952t_real @ ( sigma_5414646170262037096e_real @ M @ ( uminus612125837232591019t_real @ X5 ) ) @ N ) )
=> ( member_real_set_real @ F @ ( sigma_6606012509476713952t_real @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_373_measurable__restrict__countable,axiom,
! [X5: set_real,M: sigma_measure_real,F: real > set_Ex3793607809372303086nnreal,N: sigma_523634232904505671nnreal] :
( ( counta7319604579010473777e_real @ X5 )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member603777416030116741nnreal @ ( F @ X ) @ ( sigma_2539764534872131430nnreal @ N ) ) )
=> ( ( member8689841359643572048nnreal @ F @ ( sigma_2400199819729843436nnreal @ ( sigma_5414646170262037096e_real @ M @ ( uminus612125837232591019t_real @ X5 ) ) @ N ) )
=> ( member8689841359643572048nnreal @ F @ ( sigma_2400199819729843436nnreal @ M @ N ) ) ) ) ) ) ).
% measurable_restrict_countable
thf(fact_374_space__sup__measure_H,axiom,
! [B2: sigma_measure_a,A: sigma_measure_a] :
( ( ( sigma_sets_a @ B2 )
= ( sigma_sets_a @ A ) )
=> ( ( sigma_space_a @ ( measur3004909623614618064sure_a @ A @ B2 ) )
= ( sigma_space_a @ A ) ) ) ).
% space_sup_measure'
thf(fact_375_space__sup__measure_H,axiom,
! [B2: sigma_7234349610311085201nnreal,A: sigma_7234349610311085201nnreal] :
( ( ( sigma_5465916536984168985nnreal @ B2 )
= ( sigma_5465916536984168985nnreal @ A ) )
=> ( ( sigma_3147302497200244656nnreal @ ( measur4473656680840910822nnreal @ A @ B2 ) )
= ( sigma_3147302497200244656nnreal @ A ) ) ) ).
% space_sup_measure'
thf(fact_376_space__sup__measure_H,axiom,
! [B2: sigma_measure_real,A: sigma_measure_real] :
( ( ( sigma_sets_real @ B2 )
= ( sigma_sets_real @ A ) )
=> ( ( sigma_space_real @ ( measur2147279183506585690e_real @ A @ B2 ) )
= ( sigma_space_real @ A ) ) ) ).
% space_sup_measure'
thf(fact_377_measurable__on__iff__borel__measurable,axiom,
! [S3: set_real,F: real > real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( ( equiva5980327992511004390l_real @ F @ S3 )
= ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) @ borel_5078946678739801102l_real ) ) ) ) ).
% measurable_on_iff_borel_measurable
thf(fact_378_measurable__on__imp__borel__measurable__lebesgue,axiom,
! [F: real > real,S3: set_real] :
( ( equiva5980327992511004390l_real @ F @ S3 )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) @ borel_5078946678739801102l_real ) ) ) ) ).
% measurable_on_imp_borel_measurable_lebesgue
thf(fact_379_id__borel__measurable__lebesgue__on,axiom,
! [S3: set_real] : ( member_real_real @ id_real @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) @ borel_5078946678739801102l_real ) ) ).
% id_borel_measurable_lebesgue_on
thf(fact_380_IntI,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_381_IntI,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ A )
=> ( ( member2919562650594848410nnreal @ C @ B2 )
=> ( member2919562650594848410nnreal @ C @ ( inf_in8454409011496165067nnreal @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_382_IntI,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ A )
=> ( ( member_set_real @ C @ B2 )
=> ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_383_IntI,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ A )
=> ( ( member603777416030116741nnreal @ C @ B2 )
=> ( member603777416030116741nnreal @ C @ ( inf_in5190865051653673526nnreal @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_384_Int__iff,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B2 ) )
= ( ( member_set_a @ C @ A )
& ( member_set_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_385_Int__iff,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( inf_in8454409011496165067nnreal @ A @ B2 ) )
= ( ( member2919562650594848410nnreal @ C @ A )
& ( member2919562650594848410nnreal @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_386_Int__iff,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) )
= ( ( member_set_real @ C @ A )
& ( member_set_real @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_387_Int__iff,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( inf_in5190865051653673526nnreal @ A @ B2 ) )
= ( ( member603777416030116741nnreal @ C @ A )
& ( member603777416030116741nnreal @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_388_DiffI,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_389_DiffI,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ A )
=> ( ~ ( member2919562650594848410nnreal @ C @ B2 )
=> ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_390_DiffI,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ A )
=> ( ~ ( member_set_real @ C @ B2 )
=> ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_391_DiffI,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ A )
=> ( ~ ( member603777416030116741nnreal @ C @ B2 )
=> ( member603777416030116741nnreal @ C @ ( minus_5908140721592501885nnreal @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_392_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_393_Diff__iff,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A @ B2 ) )
= ( ( member2919562650594848410nnreal @ C @ A )
& ~ ( member2919562650594848410nnreal @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_394_Diff__iff,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A @ B2 ) )
= ( ( member_set_real @ C @ A )
& ~ ( member_set_real @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_395_Diff__iff,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( minus_5908140721592501885nnreal @ A @ B2 ) )
= ( ( member603777416030116741nnreal @ C @ A )
& ~ ( member603777416030116741nnreal @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_396_minus__diff__eq,axiom,
! [A2: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B ) )
= ( minus_minus_real @ B @ A2 ) ) ).
% minus_diff_eq
thf(fact_397_inf__top_Oright__neutral,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ A2 @ top_to7994903218803871134nnreal )
= A2 ) ).
% inf_top.right_neutral
thf(fact_398_inf__top_Oright__neutral,axiom,
! [A2: set_real] :
( ( inf_inf_set_real @ A2 @ top_top_set_real )
= A2 ) ).
% inf_top.right_neutral
thf(fact_399_inf__top_Oright__neutral,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ top_top_set_a )
= A2 ) ).
% inf_top.right_neutral
thf(fact_400_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( top_to7994903218803871134nnreal
= ( inf_in3368558534146122112nnreal @ A2 @ B ) )
= ( ( A2 = top_to7994903218803871134nnreal )
& ( B = top_to7994903218803871134nnreal ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_401_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_real,B: set_real] :
( ( top_top_set_real
= ( inf_inf_set_real @ A2 @ B ) )
= ( ( A2 = top_top_set_real )
& ( B = top_top_set_real ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_402_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_a,B: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A2 @ B ) )
= ( ( A2 = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_403_inf__top_Oleft__neutral,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ top_to7994903218803871134nnreal @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_404_inf__top_Oleft__neutral,axiom,
! [A2: set_real] :
( ( inf_inf_set_real @ top_top_set_real @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_405_inf__top_Oleft__neutral,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_406_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ A2 @ B )
= top_to7994903218803871134nnreal )
= ( ( A2 = top_to7994903218803871134nnreal )
& ( B = top_to7994903218803871134nnreal ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_407_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A2 @ B )
= top_top_set_real )
= ( ( A2 = top_top_set_real )
& ( B = top_top_set_real ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_408_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= top_top_set_a )
= ( ( A2 = top_top_set_a )
& ( B = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_409_top__eq__inf__iff,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( top_to7994903218803871134nnreal
= ( inf_in3368558534146122112nnreal @ X2 @ Y ) )
= ( ( X2 = top_to7994903218803871134nnreal )
& ( Y = top_to7994903218803871134nnreal ) ) ) ).
% top_eq_inf_iff
thf(fact_410_top__eq__inf__iff,axiom,
! [X2: set_real,Y: set_real] :
( ( top_top_set_real
= ( inf_inf_set_real @ X2 @ Y ) )
= ( ( X2 = top_top_set_real )
& ( Y = top_top_set_real ) ) ) ).
% top_eq_inf_iff
thf(fact_411_top__eq__inf__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X2 @ Y ) )
= ( ( X2 = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_412_inf__eq__top__iff,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ X2 @ Y )
= top_to7994903218803871134nnreal )
= ( ( X2 = top_to7994903218803871134nnreal )
& ( Y = top_to7994903218803871134nnreal ) ) ) ).
% inf_eq_top_iff
thf(fact_413_inf__eq__top__iff,axiom,
! [X2: set_real,Y: set_real] :
( ( ( inf_inf_set_real @ X2 @ Y )
= top_top_set_real )
= ( ( X2 = top_top_set_real )
& ( Y = top_top_set_real ) ) ) ).
% inf_eq_top_iff
thf(fact_414_inf__eq__top__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X2 @ Y )
= top_top_set_a )
= ( ( X2 = top_top_set_a )
& ( Y = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_415_inf__top__right,axiom,
! [X2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ X2 @ top_to7994903218803871134nnreal )
= X2 ) ).
% inf_top_right
thf(fact_416_inf__top__right,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ X2 @ top_top_set_real )
= X2 ) ).
% inf_top_right
thf(fact_417_inf__top__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ top_top_set_a )
= X2 ) ).
% inf_top_right
thf(fact_418_inf__top__left,axiom,
! [X2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ top_to7994903218803871134nnreal @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_419_inf__top__left,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ top_top_set_real @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_420_inf__top__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X2 )
= X2 ) ).
% inf_top_left
thf(fact_421_Int__UNIV,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ A @ B2 )
= top_to7994903218803871134nnreal )
= ( ( A = top_to7994903218803871134nnreal )
& ( B2 = top_to7994903218803871134nnreal ) ) ) ).
% Int_UNIV
thf(fact_422_Int__UNIV,axiom,
! [A: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A @ B2 )
= top_top_set_real )
= ( ( A = top_top_set_real )
& ( B2 = top_top_set_real ) ) ) ).
% Int_UNIV
thf(fact_423_Int__UNIV,axiom,
! [A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A @ B2 )
= top_top_set_a )
= ( ( A = top_top_set_a )
& ( B2 = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_424_Int__insert__right__if1,axiom,
! [A2: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B2 ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_425_Int__insert__right__if1,axiom,
! [A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ A2 @ A )
=> ( ( inf_in8454409011496165067nnreal @ A @ ( insert152533262698245683nnreal @ A2 @ B2 ) )
= ( insert152533262698245683nnreal @ A2 @ ( inf_in8454409011496165067nnreal @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_426_Int__insert__right__if1,axiom,
! [A2: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ A2 @ A )
=> ( ( inf_inf_set_set_real @ A @ ( insert_set_real @ A2 @ B2 ) )
= ( insert_set_real @ A2 @ ( inf_inf_set_set_real @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_427_Int__insert__right__if1,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ A )
=> ( ( inf_in5190865051653673526nnreal @ A @ ( insert1343806209672318238nnreal @ A2 @ B2 ) )
= ( insert1343806209672318238nnreal @ A2 @ ( inf_in5190865051653673526nnreal @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_428_Int__insert__right__if0,axiom,
! [A2: set_a,A: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B2 ) )
= ( inf_inf_set_set_a @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_429_Int__insert__right__if0,axiom,
! [A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ~ ( member2919562650594848410nnreal @ A2 @ A )
=> ( ( inf_in8454409011496165067nnreal @ A @ ( insert152533262698245683nnreal @ A2 @ B2 ) )
= ( inf_in8454409011496165067nnreal @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_430_Int__insert__right__if0,axiom,
! [A2: set_real,A: set_set_real,B2: set_set_real] :
( ~ ( member_set_real @ A2 @ A )
=> ( ( inf_inf_set_set_real @ A @ ( insert_set_real @ A2 @ B2 ) )
= ( inf_inf_set_set_real @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_431_Int__insert__right__if0,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ~ ( member603777416030116741nnreal @ A2 @ A )
=> ( ( inf_in5190865051653673526nnreal @ A @ ( insert1343806209672318238nnreal @ A2 @ B2 ) )
= ( inf_in5190865051653673526nnreal @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_432_Int__insert__left__if1,axiom,
! [A2: set_a,C3: set_set_a,B2: set_set_a] :
( ( member_set_a @ A2 @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B2 ) @ C3 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_433_Int__insert__left__if1,axiom,
! [A2: real > extend8495563244428889912nnreal,C3: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ A2 @ C3 )
=> ( ( inf_in8454409011496165067nnreal @ ( insert152533262698245683nnreal @ A2 @ B2 ) @ C3 )
= ( insert152533262698245683nnreal @ A2 @ ( inf_in8454409011496165067nnreal @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_434_Int__insert__left__if1,axiom,
! [A2: set_real,C3: set_set_real,B2: set_set_real] :
( ( member_set_real @ A2 @ C3 )
=> ( ( inf_inf_set_set_real @ ( insert_set_real @ A2 @ B2 ) @ C3 )
= ( insert_set_real @ A2 @ ( inf_inf_set_set_real @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_435_Int__insert__left__if1,axiom,
! [A2: set_Ex3793607809372303086nnreal,C3: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ C3 )
=> ( ( inf_in5190865051653673526nnreal @ ( insert1343806209672318238nnreal @ A2 @ B2 ) @ C3 )
= ( insert1343806209672318238nnreal @ A2 @ ( inf_in5190865051653673526nnreal @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_436_Int__insert__left__if0,axiom,
! [A2: set_a,C3: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A2 @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_set_a @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_437_Int__insert__left__if0,axiom,
! [A2: real > extend8495563244428889912nnreal,C3: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ~ ( member2919562650594848410nnreal @ A2 @ C3 )
=> ( ( inf_in8454409011496165067nnreal @ ( insert152533262698245683nnreal @ A2 @ B2 ) @ C3 )
= ( inf_in8454409011496165067nnreal @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_438_Int__insert__left__if0,axiom,
! [A2: set_real,C3: set_set_real,B2: set_set_real] :
( ~ ( member_set_real @ A2 @ C3 )
=> ( ( inf_inf_set_set_real @ ( insert_set_real @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_set_real @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_439_Int__insert__left__if0,axiom,
! [A2: set_Ex3793607809372303086nnreal,C3: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ~ ( member603777416030116741nnreal @ A2 @ C3 )
=> ( ( inf_in5190865051653673526nnreal @ ( insert1343806209672318238nnreal @ A2 @ B2 ) @ C3 )
= ( inf_in5190865051653673526nnreal @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_440_sets_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Int
thf(fact_441_sets_OInt,axiom,
! [A2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,B: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ B @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ A2 @ B ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).
% sets.Int
thf(fact_442_sets_OInt,axiom,
! [A2: set_real,M: sigma_measure_real,B: set_real] :
( ( member_set_real @ A2 @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ B @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( inf_inf_set_real @ A2 @ B ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets.Int
thf(fact_443_insert__Diff1,axiom,
! [X2: set_a,B2: set_set_a,A: set_set_a] :
( ( member_set_a @ X2 @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A ) @ B2 )
= ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_444_insert__Diff1,axiom,
! [X2: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ X2 @ B2 )
=> ( ( minus_3708639258518406418nnreal @ ( insert152533262698245683nnreal @ X2 @ A ) @ B2 )
= ( minus_3708639258518406418nnreal @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_445_insert__Diff1,axiom,
! [X2: set_real,B2: set_set_real,A: set_set_real] :
( ( member_set_real @ X2 @ B2 )
=> ( ( minus_5467046032205032049t_real @ ( insert_set_real @ X2 @ A ) @ B2 )
= ( minus_5467046032205032049t_real @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_446_insert__Diff1,axiom,
! [X2: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ X2 @ B2 )
=> ( ( minus_5908140721592501885nnreal @ ( insert1343806209672318238nnreal @ X2 @ A ) @ B2 )
= ( minus_5908140721592501885nnreal @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_447_Diff__insert0,axiom,
! [X2: set_a,A: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X2 @ A )
=> ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ B2 ) )
= ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_448_Diff__insert0,axiom,
! [X2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ~ ( member2919562650594848410nnreal @ X2 @ A )
=> ( ( minus_3708639258518406418nnreal @ A @ ( insert152533262698245683nnreal @ X2 @ B2 ) )
= ( minus_3708639258518406418nnreal @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_449_Diff__insert0,axiom,
! [X2: set_real,A: set_set_real,B2: set_set_real] :
( ~ ( member_set_real @ X2 @ A )
=> ( ( minus_5467046032205032049t_real @ A @ ( insert_set_real @ X2 @ B2 ) )
= ( minus_5467046032205032049t_real @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_450_Diff__insert0,axiom,
! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ~ ( member603777416030116741nnreal @ X2 @ A )
=> ( ( minus_5908140721592501885nnreal @ A @ ( insert1343806209672318238nnreal @ X2 @ B2 ) )
= ( minus_5908140721592501885nnreal @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_451_sets_ODiff,axiom,
! [A2: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.Diff
thf(fact_452_sets_ODiff,axiom,
! [A2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,B: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ B @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( minus_104578273773384135nnreal @ A2 @ B ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).
% sets.Diff
thf(fact_453_sets_ODiff,axiom,
! [A2: set_real,M: sigma_measure_real,B: set_real] :
( ( member_set_real @ A2 @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ B @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( minus_minus_set_real @ A2 @ B ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets.Diff
thf(fact_454_null__sets_OInt,axiom,
! [A2: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.Int
thf(fact_455_null__sets_OInt,axiom,
! [A2: set_real,M: sigma_measure_real,B: set_real] :
( ( member_set_real @ A2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( member_set_real @ B @ ( measur3710062792471635001s_real @ M ) )
=> ( member_set_real @ ( inf_inf_set_real @ A2 @ B ) @ ( measur3710062792471635001s_real @ M ) ) ) ) ).
% null_sets.Int
thf(fact_456_null__sets_OInt,axiom,
! [A2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,B: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ B @ ( measur1209175464439008069nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ A2 @ B ) @ ( measur1209175464439008069nnreal @ M ) ) ) ) ).
% null_sets.Int
thf(fact_457_null__sets_ODiff,axiom,
! [A2: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ A2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ B @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_sets.Diff
thf(fact_458_null__sets_ODiff,axiom,
! [A2: set_real,M: sigma_measure_real,B: set_real] :
( ( member_set_real @ A2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( member_set_real @ B @ ( measur3710062792471635001s_real @ M ) )
=> ( member_set_real @ ( minus_minus_set_real @ A2 @ B ) @ ( measur3710062792471635001s_real @ M ) ) ) ) ).
% null_sets.Diff
thf(fact_459_null__sets_ODiff,axiom,
! [A2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,B: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ B @ ( measur1209175464439008069nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( minus_104578273773384135nnreal @ A2 @ B ) @ ( measur1209175464439008069nnreal @ M ) ) ) ) ).
% null_sets.Diff
thf(fact_460_sets__sup__measure_H,axiom,
! [B2: sigma_measure_a,A: sigma_measure_a] :
( ( ( sigma_sets_a @ B2 )
= ( sigma_sets_a @ A ) )
=> ( ( sigma_sets_a @ ( measur3004909623614618064sure_a @ A @ B2 ) )
= ( sigma_sets_a @ A ) ) ) ).
% sets_sup_measure'
thf(fact_461_sets__sup__measure_H,axiom,
! [B2: sigma_7234349610311085201nnreal,A: sigma_7234349610311085201nnreal] :
( ( ( sigma_5465916536984168985nnreal @ B2 )
= ( sigma_5465916536984168985nnreal @ A ) )
=> ( ( sigma_5465916536984168985nnreal @ ( measur4473656680840910822nnreal @ A @ B2 ) )
= ( sigma_5465916536984168985nnreal @ A ) ) ) ).
% sets_sup_measure'
thf(fact_462_sets__sup__measure_H,axiom,
! [B2: sigma_measure_real,A: sigma_measure_real] :
( ( ( sigma_sets_real @ B2 )
= ( sigma_sets_real @ A ) )
=> ( ( sigma_sets_real @ ( measur2147279183506585690e_real @ A @ B2 ) )
= ( sigma_sets_real @ A ) ) ) ).
% sets_sup_measure'
thf(fact_463_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ ( uminus_uminus_set_a @ X2 ) )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_right
thf(fact_464_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ X2 @ ( uminus5517552291522096439nnreal @ X2 ) )
= bot_bo4854962954004695426nnreal ) ).
% boolean_algebra.conj_cancel_right
thf(fact_465_boolean__algebra_Oconj__cancel__right,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ X2 @ ( uminus612125837232591019t_real @ X2 ) )
= bot_bot_set_real ) ).
% boolean_algebra.conj_cancel_right
thf(fact_466_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_cancel_left
thf(fact_467_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ ( uminus5517552291522096439nnreal @ X2 ) @ X2 )
= bot_bo4854962954004695426nnreal ) ).
% boolean_algebra.conj_cancel_left
thf(fact_468_boolean__algebra_Oconj__cancel__left,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ X2 )
= bot_bot_set_real ) ).
% boolean_algebra.conj_cancel_left
thf(fact_469_inf__compl__bot__right,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ ( uminus_uminus_set_a @ X2 ) ) )
= bot_bot_set_a ) ).
% inf_compl_bot_right
thf(fact_470_inf__compl__bot__right,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ X2 @ ( inf_in3368558534146122112nnreal @ Y @ ( uminus5517552291522096439nnreal @ X2 ) ) )
= bot_bo4854962954004695426nnreal ) ).
% inf_compl_bot_right
thf(fact_471_inf__compl__bot__right,axiom,
! [X2: set_real,Y: set_real] :
( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ Y @ ( uminus612125837232591019t_real @ X2 ) ) )
= bot_bot_set_real ) ).
% inf_compl_bot_right
thf(fact_472_inf__compl__bot__left2,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left2
thf(fact_473_inf__compl__bot__left2,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ X2 @ ( inf_in3368558534146122112nnreal @ ( uminus5517552291522096439nnreal @ X2 ) @ Y ) )
= bot_bo4854962954004695426nnreal ) ).
% inf_compl_bot_left2
thf(fact_474_inf__compl__bot__left2,axiom,
! [X2: set_real,Y: set_real] :
( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ Y ) )
= bot_bot_set_real ) ).
% inf_compl_bot_left2
thf(fact_475_inf__compl__bot__left1,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ ( inf_inf_set_a @ X2 @ Y ) )
= bot_bot_set_a ) ).
% inf_compl_bot_left1
thf(fact_476_inf__compl__bot__left1,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ ( uminus5517552291522096439nnreal @ X2 ) @ ( inf_in3368558534146122112nnreal @ X2 @ Y ) )
= bot_bo4854962954004695426nnreal ) ).
% inf_compl_bot_left1
thf(fact_477_inf__compl__bot__left1,axiom,
! [X2: set_real,Y: set_real] :
( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ ( inf_inf_set_real @ X2 @ Y ) )
= bot_bot_set_real ) ).
% inf_compl_bot_left1
thf(fact_478_disjoint__insert_I2_J,axiom,
! [A: set_set_a,B: set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ ( insert_set_a @ B @ B2 ) ) )
= ( ~ ( member_set_a @ B @ A )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_479_disjoint__insert_I2_J,axiom,
! [A: set_re5328672808648366137nnreal,B: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal] :
( ( bot_bo6037503491064675021nnreal
= ( inf_in8454409011496165067nnreal @ A @ ( insert152533262698245683nnreal @ B @ B2 ) ) )
= ( ~ ( member2919562650594848410nnreal @ B @ A )
& ( bot_bo6037503491064675021nnreal
= ( inf_in8454409011496165067nnreal @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_480_disjoint__insert_I2_J,axiom,
! [A: set_set_real,B: set_real,B2: set_set_real] :
( ( bot_bot_set_set_real
= ( inf_inf_set_set_real @ A @ ( insert_set_real @ B @ B2 ) ) )
= ( ~ ( member_set_real @ B @ A )
& ( bot_bot_set_set_real
= ( inf_inf_set_set_real @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_481_disjoint__insert_I2_J,axiom,
! [A: set_se4580700918925141924nnreal,B: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( bot_bo2988155216863113784nnreal
= ( inf_in5190865051653673526nnreal @ A @ ( insert1343806209672318238nnreal @ B @ B2 ) ) )
= ( ~ ( member603777416030116741nnreal @ B @ A )
& ( bot_bo2988155216863113784nnreal
= ( inf_in5190865051653673526nnreal @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_482_disjoint__insert_I1_J,axiom,
! [B2: set_set_a,A2: set_a,A: set_set_a] :
( ( ( inf_inf_set_set_a @ B2 @ ( insert_set_a @ A2 @ A ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B2 )
& ( ( inf_inf_set_set_a @ B2 @ A )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_483_disjoint__insert_I1_J,axiom,
! [B2: set_re5328672808648366137nnreal,A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( ( inf_in8454409011496165067nnreal @ B2 @ ( insert152533262698245683nnreal @ A2 @ A ) )
= bot_bo6037503491064675021nnreal )
= ( ~ ( member2919562650594848410nnreal @ A2 @ B2 )
& ( ( inf_in8454409011496165067nnreal @ B2 @ A )
= bot_bo6037503491064675021nnreal ) ) ) ).
% disjoint_insert(1)
thf(fact_484_disjoint__insert_I1_J,axiom,
! [B2: set_set_real,A2: set_real,A: set_set_real] :
( ( ( inf_inf_set_set_real @ B2 @ ( insert_set_real @ A2 @ A ) )
= bot_bot_set_set_real )
= ( ~ ( member_set_real @ A2 @ B2 )
& ( ( inf_inf_set_set_real @ B2 @ A )
= bot_bot_set_set_real ) ) ) ).
% disjoint_insert(1)
thf(fact_485_disjoint__insert_I1_J,axiom,
! [B2: set_se4580700918925141924nnreal,A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( ( inf_in5190865051653673526nnreal @ B2 @ ( insert1343806209672318238nnreal @ A2 @ A ) )
= bot_bo2988155216863113784nnreal )
= ( ~ ( member603777416030116741nnreal @ A2 @ B2 )
& ( ( inf_in5190865051653673526nnreal @ B2 @ A )
= bot_bo2988155216863113784nnreal ) ) ) ).
% disjoint_insert(1)
thf(fact_486_insert__disjoint_I2_J,axiom,
! [A2: set_a,A: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B2 ) )
= ( ~ ( member_set_a @ A2 @ B2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_487_insert__disjoint_I2_J,axiom,
! [A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( bot_bo6037503491064675021nnreal
= ( inf_in8454409011496165067nnreal @ ( insert152533262698245683nnreal @ A2 @ A ) @ B2 ) )
= ( ~ ( member2919562650594848410nnreal @ A2 @ B2 )
& ( bot_bo6037503491064675021nnreal
= ( inf_in8454409011496165067nnreal @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_488_insert__disjoint_I2_J,axiom,
! [A2: set_real,A: set_set_real,B2: set_set_real] :
( ( bot_bot_set_set_real
= ( inf_inf_set_set_real @ ( insert_set_real @ A2 @ A ) @ B2 ) )
= ( ~ ( member_set_real @ A2 @ B2 )
& ( bot_bot_set_set_real
= ( inf_inf_set_set_real @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_489_insert__disjoint_I2_J,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( bot_bo2988155216863113784nnreal
= ( inf_in5190865051653673526nnreal @ ( insert1343806209672318238nnreal @ A2 @ A ) @ B2 ) )
= ( ~ ( member603777416030116741nnreal @ A2 @ B2 )
& ( bot_bo2988155216863113784nnreal
= ( inf_in5190865051653673526nnreal @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_490_insert__disjoint_I1_J,axiom,
! [A2: set_a,A: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ A ) @ B2 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A2 @ B2 )
& ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_491_insert__disjoint_I1_J,axiom,
! [A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( ( inf_in8454409011496165067nnreal @ ( insert152533262698245683nnreal @ A2 @ A ) @ B2 )
= bot_bo6037503491064675021nnreal )
= ( ~ ( member2919562650594848410nnreal @ A2 @ B2 )
& ( ( inf_in8454409011496165067nnreal @ A @ B2 )
= bot_bo6037503491064675021nnreal ) ) ) ).
% insert_disjoint(1)
thf(fact_492_insert__disjoint_I1_J,axiom,
! [A2: set_real,A: set_set_real,B2: set_set_real] :
( ( ( inf_inf_set_set_real @ ( insert_set_real @ A2 @ A ) @ B2 )
= bot_bot_set_set_real )
= ( ~ ( member_set_real @ A2 @ B2 )
& ( ( inf_inf_set_set_real @ A @ B2 )
= bot_bot_set_set_real ) ) ) ).
% insert_disjoint(1)
thf(fact_493_insert__disjoint_I1_J,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( ( inf_in5190865051653673526nnreal @ ( insert1343806209672318238nnreal @ A2 @ A ) @ B2 )
= bot_bo2988155216863113784nnreal )
= ( ~ ( member603777416030116741nnreal @ A2 @ B2 )
& ( ( inf_in5190865051653673526nnreal @ A @ B2 )
= bot_bo2988155216863113784nnreal ) ) ) ).
% insert_disjoint(1)
thf(fact_494_Diff__UNIV,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( minus_104578273773384135nnreal @ A @ top_to7994903218803871134nnreal )
= bot_bo4854962954004695426nnreal ) ).
% Diff_UNIV
thf(fact_495_Diff__UNIV,axiom,
! [A: set_real] :
( ( minus_minus_set_real @ A @ top_top_set_real )
= bot_bot_set_real ) ).
% Diff_UNIV
thf(fact_496_Diff__UNIV,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ top_top_set_a )
= bot_bot_set_a ) ).
% Diff_UNIV
thf(fact_497_sets_OInt__space__eq1,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X2 )
= X2 ) ) ).
% sets.Int_space_eq1
thf(fact_498_sets_OInt__space__eq1,axiom,
! [X2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ X2 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( inf_in3368558534146122112nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ X2 )
= X2 ) ) ).
% sets.Int_space_eq1
thf(fact_499_sets_OInt__space__eq1,axiom,
! [X2: set_real,M: sigma_measure_real] :
( ( member_set_real @ X2 @ ( sigma_sets_real @ M ) )
=> ( ( inf_inf_set_real @ ( sigma_space_real @ M ) @ X2 )
= X2 ) ) ).
% sets.Int_space_eq1
thf(fact_500_sets_OInt__space__eq2,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
=> ( ( inf_inf_set_a @ X2 @ ( sigma_space_a @ M ) )
= X2 ) ) ).
% sets.Int_space_eq2
thf(fact_501_sets_OInt__space__eq2,axiom,
! [X2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ X2 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( inf_in3368558534146122112nnreal @ X2 @ ( sigma_3147302497200244656nnreal @ M ) )
= X2 ) ) ).
% sets.Int_space_eq2
thf(fact_502_sets_OInt__space__eq2,axiom,
! [X2: set_real,M: sigma_measure_real] :
( ( member_set_real @ X2 @ ( sigma_sets_real @ M ) )
=> ( ( inf_inf_set_real @ X2 @ ( sigma_space_real @ M ) )
= X2 ) ) ).
% sets.Int_space_eq2
thf(fact_503_sets_Ocompl__sets,axiom,
! [A2: set_a,M: sigma_measure_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ ( sigma_space_a @ M ) @ A2 ) @ ( sigma_sets_a @ M ) ) ) ).
% sets.compl_sets
thf(fact_504_sets_Ocompl__sets,axiom,
! [A2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( minus_104578273773384135nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ A2 ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% sets.compl_sets
thf(fact_505_sets_Ocompl__sets,axiom,
! [A2: set_real,M: sigma_measure_real] :
( ( member_set_real @ A2 @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( minus_minus_set_real @ ( sigma_space_real @ M ) @ A2 ) @ ( sigma_sets_real @ M ) ) ) ).
% sets.compl_sets
thf(fact_506_Compl__disjoint,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= bot_bot_set_a ) ).
% Compl_disjoint
thf(fact_507_Compl__disjoint,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ A @ ( uminus5517552291522096439nnreal @ A ) )
= bot_bo4854962954004695426nnreal ) ).
% Compl_disjoint
thf(fact_508_Compl__disjoint,axiom,
! [A: set_real] :
( ( inf_inf_set_real @ A @ ( uminus612125837232591019t_real @ A ) )
= bot_bot_set_real ) ).
% Compl_disjoint
thf(fact_509_Compl__disjoint2,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ ( uminus_uminus_set_a @ A ) @ A )
= bot_bot_set_a ) ).
% Compl_disjoint2
thf(fact_510_Compl__disjoint2,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ ( uminus5517552291522096439nnreal @ A ) @ A )
= bot_bo4854962954004695426nnreal ) ).
% Compl_disjoint2
thf(fact_511_Compl__disjoint2,axiom,
! [A: set_real] :
( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ A ) @ A )
= bot_bot_set_real ) ).
% Compl_disjoint2
thf(fact_512_null__sets_OInt__space__eq1,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( measure_null_sets_a @ M ) )
=> ( ( inf_inf_set_a @ ( sigma_space_a @ M ) @ X2 )
= X2 ) ) ).
% null_sets.Int_space_eq1
thf(fact_513_null__sets_OInt__space__eq1,axiom,
! [X2: set_real,M: sigma_measure_real] :
( ( member_set_real @ X2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( inf_inf_set_real @ ( sigma_space_real @ M ) @ X2 )
= X2 ) ) ).
% null_sets.Int_space_eq1
thf(fact_514_null__sets_OInt__space__eq1,axiom,
! [X2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ X2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( inf_in3368558534146122112nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ X2 )
= X2 ) ) ).
% null_sets.Int_space_eq1
thf(fact_515_null__sets_OInt__space__eq2,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( measure_null_sets_a @ M ) )
=> ( ( inf_inf_set_a @ X2 @ ( sigma_space_a @ M ) )
= X2 ) ) ).
% null_sets.Int_space_eq2
thf(fact_516_null__sets_OInt__space__eq2,axiom,
! [X2: set_real,M: sigma_measure_real] :
( ( member_set_real @ X2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( inf_inf_set_real @ X2 @ ( sigma_space_real @ M ) )
= X2 ) ) ).
% null_sets.Int_space_eq2
thf(fact_517_null__sets_OInt__space__eq2,axiom,
! [X2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ X2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( inf_in3368558534146122112nnreal @ X2 @ ( sigma_3147302497200244656nnreal @ M ) )
= X2 ) ) ).
% null_sets.Int_space_eq2
thf(fact_518_Diff__Compl,axiom,
! [A: set_a,B2: set_a] :
( ( minus_minus_set_a @ A @ ( uminus_uminus_set_a @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% Diff_Compl
thf(fact_519_Diff__Compl,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( minus_104578273773384135nnreal @ A @ ( uminus5517552291522096439nnreal @ B2 ) )
= ( inf_in3368558534146122112nnreal @ A @ B2 ) ) ).
% Diff_Compl
thf(fact_520_Diff__Compl,axiom,
! [A: set_real,B2: set_real] :
( ( minus_minus_set_real @ A @ ( uminus612125837232591019t_real @ B2 ) )
= ( inf_inf_set_real @ A @ B2 ) ) ).
% Diff_Compl
thf(fact_521_sets__bot,axiom,
( ( sigma_sets_a @ bot_bo2108912051383640591sure_a )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% sets_bot
thf(fact_522_sets__bot,axiom,
( ( sigma_5465916536984168985nnreal @ bot_bo1740529460517930749nnreal )
= ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) ) ).
% sets_bot
thf(fact_523_sets__bot,axiom,
( ( sigma_sets_real @ bot_bo5982154664989874033e_real )
= ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ).
% sets_bot
thf(fact_524_id__borel__measurable__lebesgue,axiom,
member_real_real @ id_real @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ borel_5078946678739801102l_real ) ).
% id_borel_measurable_lebesgue
thf(fact_525_Diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A3: set_a,B4: set_a] : ( inf_inf_set_a @ A3 @ ( uminus_uminus_set_a @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_526_Diff__eq,axiom,
( minus_104578273773384135nnreal
= ( ^ [A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] : ( inf_in3368558534146122112nnreal @ A3 @ ( uminus5517552291522096439nnreal @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_527_Diff__eq,axiom,
( minus_minus_set_real
= ( ^ [A3: set_real,B4: set_real] : ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ B4 ) ) ) ) ).
% Diff_eq
thf(fact_528_diff__eq,axiom,
( minus_minus_set_a
= ( ^ [X3: set_a,Y3: set_a] : ( inf_inf_set_a @ X3 @ ( uminus_uminus_set_a @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_529_diff__eq,axiom,
( minus_104578273773384135nnreal
= ( ^ [X3: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] : ( inf_in3368558534146122112nnreal @ X3 @ ( uminus5517552291522096439nnreal @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_530_diff__eq,axiom,
( minus_minus_set_real
= ( ^ [X3: set_real,Y3: set_real] : ( inf_inf_set_real @ X3 @ ( uminus612125837232591019t_real @ Y3 ) ) ) ) ).
% diff_eq
thf(fact_531_diff__eq__diff__eq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A2 = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_532_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A2: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A2 @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_533_IntE,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B2 ) )
=> ~ ( ( member_set_a @ C @ A )
=> ~ ( member_set_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_534_IntE,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( inf_in8454409011496165067nnreal @ A @ B2 ) )
=> ~ ( ( member2919562650594848410nnreal @ C @ A )
=> ~ ( member2919562650594848410nnreal @ C @ B2 ) ) ) ).
% IntE
thf(fact_535_IntE,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) )
=> ~ ( ( member_set_real @ C @ A )
=> ~ ( member_set_real @ C @ B2 ) ) ) ).
% IntE
thf(fact_536_IntE,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( inf_in5190865051653673526nnreal @ A @ B2 ) )
=> ~ ( ( member603777416030116741nnreal @ C @ A )
=> ~ ( member603777416030116741nnreal @ C @ B2 ) ) ) ).
% IntE
thf(fact_537_DiffE,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
=> ~ ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_538_DiffE,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A @ B2 ) )
=> ~ ( ( member2919562650594848410nnreal @ C @ A )
=> ( member2919562650594848410nnreal @ C @ B2 ) ) ) ).
% DiffE
thf(fact_539_DiffE,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A @ B2 ) )
=> ~ ( ( member_set_real @ C @ A )
=> ( member_set_real @ C @ B2 ) ) ) ).
% DiffE
thf(fact_540_DiffE,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( minus_5908140721592501885nnreal @ A @ B2 ) )
=> ~ ( ( member603777416030116741nnreal @ C @ A )
=> ( member603777416030116741nnreal @ C @ B2 ) ) ) ).
% DiffE
thf(fact_541_IntD1,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B2 ) )
=> ( member_set_a @ C @ A ) ) ).
% IntD1
thf(fact_542_IntD1,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( inf_in8454409011496165067nnreal @ A @ B2 ) )
=> ( member2919562650594848410nnreal @ C @ A ) ) ).
% IntD1
thf(fact_543_IntD1,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) )
=> ( member_set_real @ C @ A ) ) ).
% IntD1
thf(fact_544_IntD1,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( inf_in5190865051653673526nnreal @ A @ B2 ) )
=> ( member603777416030116741nnreal @ C @ A ) ) ).
% IntD1
thf(fact_545_IntD2,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A @ B2 ) )
=> ( member_set_a @ C @ B2 ) ) ).
% IntD2
thf(fact_546_IntD2,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( inf_in8454409011496165067nnreal @ A @ B2 ) )
=> ( member2919562650594848410nnreal @ C @ B2 ) ) ).
% IntD2
thf(fact_547_IntD2,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) )
=> ( member_set_real @ C @ B2 ) ) ).
% IntD2
thf(fact_548_IntD2,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( inf_in5190865051653673526nnreal @ A @ B2 ) )
=> ( member603777416030116741nnreal @ C @ B2 ) ) ).
% IntD2
thf(fact_549_DiffD1,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
=> ( member_set_a @ C @ A ) ) ).
% DiffD1
thf(fact_550_DiffD1,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A @ B2 ) )
=> ( member2919562650594848410nnreal @ C @ A ) ) ).
% DiffD1
thf(fact_551_DiffD1,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A @ B2 ) )
=> ( member_set_real @ C @ A ) ) ).
% DiffD1
thf(fact_552_DiffD1,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( minus_5908140721592501885nnreal @ A @ B2 ) )
=> ( member603777416030116741nnreal @ C @ A ) ) ).
% DiffD1
thf(fact_553_DiffD2,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
=> ~ ( member_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_554_DiffD2,axiom,
! [C: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A @ B2 ) )
=> ~ ( member2919562650594848410nnreal @ C @ B2 ) ) ).
% DiffD2
thf(fact_555_DiffD2,axiom,
! [C: set_real,A: set_set_real,B2: set_set_real] :
( ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A @ B2 ) )
=> ~ ( member_set_real @ C @ B2 ) ) ).
% DiffD2
thf(fact_556_DiffD2,axiom,
! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ C @ ( minus_5908140721592501885nnreal @ A @ B2 ) )
=> ~ ( member603777416030116741nnreal @ C @ B2 ) ) ).
% DiffD2
thf(fact_557_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_558_bot__empty__eq,axiom,
( bot_bo6758561407716789752real_o
= ( ^ [X3: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X3 @ bot_bo6037503491064675021nnreal ) ) ) ).
% bot_empty_eq
thf(fact_559_bot__empty__eq,axiom,
( bot_bot_set_real_o
= ( ^ [X3: set_real] : ( member_set_real @ X3 @ bot_bot_set_set_real ) ) ) ).
% bot_empty_eq
thf(fact_560_bot__empty__eq,axiom,
( bot_bo5002694753204610125real_o
= ( ^ [X3: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X3 @ bot_bo2988155216863113784nnreal ) ) ) ).
% bot_empty_eq
thf(fact_561_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ X2 @ top_to7994903218803871134nnreal )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_562_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_real] :
( ( inf_inf_set_real @ X2 @ top_top_set_real )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_563_boolean__algebra_Oconj__one__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ top_top_set_a )
= X2 ) ).
% boolean_algebra.conj_one_right
thf(fact_564_minus__diff__commute,axiom,
! [B: real,A2: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A2 )
= ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ B ) ) ).
% minus_diff_commute
thf(fact_565_Int__UNIV__right,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ A @ top_to7994903218803871134nnreal )
= A ) ).
% Int_UNIV_right
thf(fact_566_Int__UNIV__right,axiom,
! [A: set_real] :
( ( inf_inf_set_real @ A @ top_top_set_real )
= A ) ).
% Int_UNIV_right
thf(fact_567_Int__UNIV__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ top_top_set_a )
= A ) ).
% Int_UNIV_right
thf(fact_568_Int__UNIV__left,axiom,
! [B2: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ top_to7994903218803871134nnreal @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_569_Int__UNIV__left,axiom,
! [B2: set_real] :
( ( inf_inf_set_real @ top_top_set_real @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_570_Int__UNIV__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_571_Int__emptyI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ~ ( member_set_a @ X @ B2 ) )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_572_Int__emptyI,axiom,
! [A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ! [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ A )
=> ~ ( member2919562650594848410nnreal @ X @ B2 ) )
=> ( ( inf_in8454409011496165067nnreal @ A @ B2 )
= bot_bo6037503491064675021nnreal ) ) ).
% Int_emptyI
thf(fact_573_Int__emptyI,axiom,
! [A: set_set_real,B2: set_set_real] :
( ! [X: set_real] :
( ( member_set_real @ X @ A )
=> ~ ( member_set_real @ X @ B2 ) )
=> ( ( inf_inf_set_set_real @ A @ B2 )
= bot_bot_set_set_real ) ) ).
% Int_emptyI
thf(fact_574_Int__emptyI,axiom,
! [A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ A )
=> ~ ( member603777416030116741nnreal @ X @ B2 ) )
=> ( ( inf_in5190865051653673526nnreal @ A @ B2 )
= bot_bo2988155216863113784nnreal ) ) ).
% Int_emptyI
thf(fact_575_disjoint__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_576_disjoint__iff,axiom,
! [A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( ( inf_in8454409011496165067nnreal @ A @ B2 )
= bot_bo6037503491064675021nnreal )
= ( ! [X3: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X3 @ A )
=> ~ ( member2919562650594848410nnreal @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_577_disjoint__iff,axiom,
! [A: set_set_real,B2: set_set_real] :
( ( ( inf_inf_set_set_real @ A @ B2 )
= bot_bot_set_set_real )
= ( ! [X3: set_real] :
( ( member_set_real @ X3 @ A )
=> ~ ( member_set_real @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_578_disjoint__iff,axiom,
! [A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( ( inf_in5190865051653673526nnreal @ A @ B2 )
= bot_bo2988155216863113784nnreal )
= ( ! [X3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X3 @ A )
=> ~ ( member603777416030116741nnreal @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_579_Int__insert__right,axiom,
! [A2: set_a,A: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B2 ) )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ A @ B2 ) ) ) )
& ( ~ ( member_set_a @ A2 @ A )
=> ( ( inf_inf_set_set_a @ A @ ( insert_set_a @ A2 @ B2 ) )
= ( inf_inf_set_set_a @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_580_Int__insert__right,axiom,
! [A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( ( member2919562650594848410nnreal @ A2 @ A )
=> ( ( inf_in8454409011496165067nnreal @ A @ ( insert152533262698245683nnreal @ A2 @ B2 ) )
= ( insert152533262698245683nnreal @ A2 @ ( inf_in8454409011496165067nnreal @ A @ B2 ) ) ) )
& ( ~ ( member2919562650594848410nnreal @ A2 @ A )
=> ( ( inf_in8454409011496165067nnreal @ A @ ( insert152533262698245683nnreal @ A2 @ B2 ) )
= ( inf_in8454409011496165067nnreal @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_581_Int__insert__right,axiom,
! [A2: set_real,A: set_set_real,B2: set_set_real] :
( ( ( member_set_real @ A2 @ A )
=> ( ( inf_inf_set_set_real @ A @ ( insert_set_real @ A2 @ B2 ) )
= ( insert_set_real @ A2 @ ( inf_inf_set_set_real @ A @ B2 ) ) ) )
& ( ~ ( member_set_real @ A2 @ A )
=> ( ( inf_inf_set_set_real @ A @ ( insert_set_real @ A2 @ B2 ) )
= ( inf_inf_set_set_real @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_582_Int__insert__right,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( ( member603777416030116741nnreal @ A2 @ A )
=> ( ( inf_in5190865051653673526nnreal @ A @ ( insert1343806209672318238nnreal @ A2 @ B2 ) )
= ( insert1343806209672318238nnreal @ A2 @ ( inf_in5190865051653673526nnreal @ A @ B2 ) ) ) )
& ( ~ ( member603777416030116741nnreal @ A2 @ A )
=> ( ( inf_in5190865051653673526nnreal @ A @ ( insert1343806209672318238nnreal @ A2 @ B2 ) )
= ( inf_in5190865051653673526nnreal @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_583_Int__insert__left,axiom,
! [A2: set_a,C3: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A2 @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B2 ) @ C3 )
= ( insert_set_a @ A2 @ ( inf_inf_set_set_a @ B2 @ C3 ) ) ) )
& ( ~ ( member_set_a @ A2 @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_set_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_584_Int__insert__left,axiom,
! [A2: real > extend8495563244428889912nnreal,C3: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( ( member2919562650594848410nnreal @ A2 @ C3 )
=> ( ( inf_in8454409011496165067nnreal @ ( insert152533262698245683nnreal @ A2 @ B2 ) @ C3 )
= ( insert152533262698245683nnreal @ A2 @ ( inf_in8454409011496165067nnreal @ B2 @ C3 ) ) ) )
& ( ~ ( member2919562650594848410nnreal @ A2 @ C3 )
=> ( ( inf_in8454409011496165067nnreal @ ( insert152533262698245683nnreal @ A2 @ B2 ) @ C3 )
= ( inf_in8454409011496165067nnreal @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_585_Int__insert__left,axiom,
! [A2: set_real,C3: set_set_real,B2: set_set_real] :
( ( ( member_set_real @ A2 @ C3 )
=> ( ( inf_inf_set_set_real @ ( insert_set_real @ A2 @ B2 ) @ C3 )
= ( insert_set_real @ A2 @ ( inf_inf_set_set_real @ B2 @ C3 ) ) ) )
& ( ~ ( member_set_real @ A2 @ C3 )
=> ( ( inf_inf_set_set_real @ ( insert_set_real @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_set_real @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_586_Int__insert__left,axiom,
! [A2: set_Ex3793607809372303086nnreal,C3: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( ( member603777416030116741nnreal @ A2 @ C3 )
=> ( ( inf_in5190865051653673526nnreal @ ( insert1343806209672318238nnreal @ A2 @ B2 ) @ C3 )
= ( insert1343806209672318238nnreal @ A2 @ ( inf_in5190865051653673526nnreal @ B2 @ C3 ) ) ) )
& ( ~ ( member603777416030116741nnreal @ A2 @ C3 )
=> ( ( inf_in5190865051653673526nnreal @ ( insert1343806209672318238nnreal @ A2 @ B2 ) @ C3 )
= ( inf_in5190865051653673526nnreal @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_587_insert__Diff__if,axiom,
! [X2: set_a,B2: set_set_a,A: set_set_a] :
( ( ( member_set_a @ X2 @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A ) @ B2 )
= ( minus_5736297505244876581_set_a @ A @ B2 ) ) )
& ( ~ ( member_set_a @ X2 @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A ) @ B2 )
= ( insert_set_a @ X2 @ ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_588_insert__Diff__if,axiom,
! [X2: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal,A: set_re5328672808648366137nnreal] :
( ( ( member2919562650594848410nnreal @ X2 @ B2 )
=> ( ( minus_3708639258518406418nnreal @ ( insert152533262698245683nnreal @ X2 @ A ) @ B2 )
= ( minus_3708639258518406418nnreal @ A @ B2 ) ) )
& ( ~ ( member2919562650594848410nnreal @ X2 @ B2 )
=> ( ( minus_3708639258518406418nnreal @ ( insert152533262698245683nnreal @ X2 @ A ) @ B2 )
= ( insert152533262698245683nnreal @ X2 @ ( minus_3708639258518406418nnreal @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_589_insert__Diff__if,axiom,
! [X2: set_real,B2: set_set_real,A: set_set_real] :
( ( ( member_set_real @ X2 @ B2 )
=> ( ( minus_5467046032205032049t_real @ ( insert_set_real @ X2 @ A ) @ B2 )
= ( minus_5467046032205032049t_real @ A @ B2 ) ) )
& ( ~ ( member_set_real @ X2 @ B2 )
=> ( ( minus_5467046032205032049t_real @ ( insert_set_real @ X2 @ A ) @ B2 )
= ( insert_set_real @ X2 @ ( minus_5467046032205032049t_real @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_590_insert__Diff__if,axiom,
! [X2: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal,A: set_se4580700918925141924nnreal] :
( ( ( member603777416030116741nnreal @ X2 @ B2 )
=> ( ( minus_5908140721592501885nnreal @ ( insert1343806209672318238nnreal @ X2 @ A ) @ B2 )
= ( minus_5908140721592501885nnreal @ A @ B2 ) ) )
& ( ~ ( member603777416030116741nnreal @ X2 @ B2 )
=> ( ( minus_5908140721592501885nnreal @ ( insert1343806209672318238nnreal @ X2 @ A ) @ B2 )
= ( insert1343806209672318238nnreal @ X2 @ ( minus_5908140721592501885nnreal @ A @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_591_inf__cancel__left2,axiom,
! [X2: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ A2 ) @ ( inf_inf_set_a @ X2 @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left2
thf(fact_592_inf__cancel__left2,axiom,
! [X2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ ( inf_in3368558534146122112nnreal @ ( uminus5517552291522096439nnreal @ X2 ) @ A2 ) @ ( inf_in3368558534146122112nnreal @ X2 @ B ) )
= bot_bo4854962954004695426nnreal ) ).
% inf_cancel_left2
thf(fact_593_inf__cancel__left2,axiom,
! [X2: set_real,A2: set_real,B: set_real] :
( ( inf_inf_set_real @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ A2 ) @ ( inf_inf_set_real @ X2 @ B ) )
= bot_bot_set_real ) ).
% inf_cancel_left2
thf(fact_594_inf__cancel__left1,axiom,
! [X2: set_a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ A2 ) @ ( inf_inf_set_a @ ( uminus_uminus_set_a @ X2 ) @ B ) )
= bot_bot_set_a ) ).
% inf_cancel_left1
thf(fact_595_inf__cancel__left1,axiom,
! [X2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
( ( inf_in3368558534146122112nnreal @ ( inf_in3368558534146122112nnreal @ X2 @ A2 ) @ ( inf_in3368558534146122112nnreal @ ( uminus5517552291522096439nnreal @ X2 ) @ B ) )
= bot_bo4854962954004695426nnreal ) ).
% inf_cancel_left1
thf(fact_596_inf__cancel__left1,axiom,
! [X2: set_real,A2: set_real,B: set_real] :
( ( inf_inf_set_real @ ( inf_inf_set_real @ X2 @ A2 ) @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ B ) )
= bot_bot_set_real ) ).
% inf_cancel_left1
thf(fact_597_sets__eq__bot,axiom,
! [M: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) )
= ( M = bot_bo2108912051383640591sure_a ) ) ).
% sets_eq_bot
thf(fact_598_sets__eq__bot,axiom,
! [M: sigma_7234349610311085201nnreal] :
( ( ( sigma_5465916536984168985nnreal @ M )
= ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) )
= ( M = bot_bo1740529460517930749nnreal ) ) ).
% sets_eq_bot
thf(fact_599_sets__eq__bot,axiom,
! [M: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) )
= ( M = bot_bo5982154664989874033e_real ) ) ).
% sets_eq_bot
thf(fact_600_sets__eq__bot2,axiom,
! [M: sigma_measure_a] :
( ( ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a )
= ( sigma_sets_a @ M ) )
= ( M = bot_bo2108912051383640591sure_a ) ) ).
% sets_eq_bot2
thf(fact_601_sets__eq__bot2,axiom,
! [M: sigma_7234349610311085201nnreal] :
( ( ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal )
= ( sigma_5465916536984168985nnreal @ M ) )
= ( M = bot_bo1740529460517930749nnreal ) ) ).
% sets_eq_bot2
thf(fact_602_sets__eq__bot2,axiom,
! [M: sigma_measure_real] :
( ( ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real )
= ( sigma_sets_real @ M ) )
= ( M = bot_bo5982154664989874033e_real ) ) ).
% sets_eq_bot2
thf(fact_603_countable__imp__null__set__lborel,axiom,
! [A: set_real] :
( ( counta7319604579010473777e_real @ A )
=> ( member_set_real @ A @ ( measur3710062792471635001s_real @ lebesgue_lborel_real ) ) ) ).
% countable_imp_null_set_lborel
thf(fact_604_insert__Diff,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_605_insert__Diff,axiom,
! [A2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ A2 @ A )
=> ( ( insert152533262698245683nnreal @ A2 @ ( minus_3708639258518406418nnreal @ A @ ( insert152533262698245683nnreal @ A2 @ bot_bo6037503491064675021nnreal ) ) )
= A ) ) ).
% insert_Diff
thf(fact_606_insert__Diff,axiom,
! [A2: set_real,A: set_set_real] :
( ( member_set_real @ A2 @ A )
=> ( ( insert_set_real @ A2 @ ( minus_5467046032205032049t_real @ A @ ( insert_set_real @ A2 @ bot_bot_set_set_real ) ) )
= A ) ) ).
% insert_Diff
thf(fact_607_insert__Diff,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ A )
=> ( ( insert1343806209672318238nnreal @ A2 @ ( minus_5908140721592501885nnreal @ A @ ( insert1343806209672318238nnreal @ A2 @ bot_bo2988155216863113784nnreal ) ) )
= A ) ) ).
% insert_Diff
thf(fact_608_Diff__insert__absorb,axiom,
! [X2: set_a,A: set_set_a] :
( ~ ( member_set_a @ X2 @ A )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A ) @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_609_Diff__insert__absorb,axiom,
! [X2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ~ ( member2919562650594848410nnreal @ X2 @ A )
=> ( ( minus_3708639258518406418nnreal @ ( insert152533262698245683nnreal @ X2 @ A ) @ ( insert152533262698245683nnreal @ X2 @ bot_bo6037503491064675021nnreal ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_610_Diff__insert__absorb,axiom,
! [X2: set_real,A: set_set_real] :
( ~ ( member_set_real @ X2 @ A )
=> ( ( minus_5467046032205032049t_real @ ( insert_set_real @ X2 @ A ) @ ( insert_set_real @ X2 @ bot_bot_set_set_real ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_611_Diff__insert__absorb,axiom,
! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ~ ( member603777416030116741nnreal @ X2 @ A )
=> ( ( minus_5908140721592501885nnreal @ ( insert1343806209672318238nnreal @ X2 @ A ) @ ( insert1343806209672318238nnreal @ X2 @ bot_bo2988155216863113784nnreal ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_612_Compl__eq__Diff__UNIV,axiom,
( uminus_uminus_set_a
= ( minus_minus_set_a @ top_top_set_a ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_613_Compl__eq__Diff__UNIV,axiom,
( uminus5517552291522096439nnreal
= ( minus_104578273773384135nnreal @ top_to7994903218803871134nnreal ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_614_Compl__eq__Diff__UNIV,axiom,
( uminus612125837232591019t_real
= ( minus_minus_set_real @ top_top_set_real ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_615_null__set__Int1,axiom,
! [B2: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_set_Int1
thf(fact_616_null__set__Int1,axiom,
! [B2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ B2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ A @ B2 ) @ ( measur1209175464439008069nnreal @ M ) ) ) ) ).
% null_set_Int1
thf(fact_617_null__set__Int1,axiom,
! [B2: set_real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ B2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( inf_inf_set_real @ A @ B2 ) @ ( measur3710062792471635001s_real @ M ) ) ) ) ).
% null_set_Int1
thf(fact_618_null__set__Int2,axiom,
! [B2: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( inf_inf_set_a @ B2 @ A ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_set_Int2
thf(fact_619_null__set__Int2,axiom,
! [B2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ B2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ B2 @ A ) @ ( measur1209175464439008069nnreal @ M ) ) ) ) ).
% null_set_Int2
thf(fact_620_null__set__Int2,axiom,
! [B2: set_real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ B2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( inf_inf_set_real @ B2 @ A ) @ ( measur3710062792471635001s_real @ M ) ) ) ) ).
% null_set_Int2
thf(fact_621_null__set__Diff,axiom,
! [B2: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( minus_minus_set_a @ B2 @ A ) @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_set_Diff
thf(fact_622_null__set__Diff,axiom,
! [B2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ B2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( minus_104578273773384135nnreal @ B2 @ A ) @ ( measur1209175464439008069nnreal @ M ) ) ) ) ).
% null_set_Diff
thf(fact_623_null__set__Diff,axiom,
! [B2: set_real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ B2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( minus_minus_set_real @ B2 @ A ) @ ( measur3710062792471635001s_real @ M ) ) ) ) ).
% null_set_Diff
thf(fact_624_sets__restrict__restrict__space,axiom,
! [M: sigma_measure_a,A: set_a,B2: set_a] :
( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ ( sigma_8692839461743104066pace_a @ M @ A ) @ B2 ) )
= ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% sets_restrict_restrict_space
thf(fact_625_sets__restrict__restrict__space,axiom,
! [M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ ( sigma_4884701650823297268nnreal @ M @ A ) @ B2 ) )
= ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ M @ ( inf_in3368558534146122112nnreal @ A @ B2 ) ) ) ) ).
% sets_restrict_restrict_space
thf(fact_626_sets__restrict__restrict__space,axiom,
! [M: sigma_measure_real,A: set_real,B2: set_real] :
( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ ( sigma_5414646170262037096e_real @ M @ A ) @ B2 ) )
= ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ ( inf_inf_set_real @ A @ B2 ) ) ) ) ).
% sets_restrict_restrict_space
thf(fact_627_is__singletonI_H,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X: set_a,Y2: set_a] :
( ( member_set_a @ X @ A )
=> ( ( member_set_a @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_singleton_set_a @ A ) ) ) ).
% is_singletonI'
thf(fact_628_is__singletonI_H,axiom,
! [A: set_re5328672808648366137nnreal] :
( ( A != bot_bo6037503491064675021nnreal )
=> ( ! [X: real > extend8495563244428889912nnreal,Y2: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ A )
=> ( ( member2919562650594848410nnreal @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_sin8880349622731141135nnreal @ A ) ) ) ).
% is_singletonI'
thf(fact_629_is__singletonI_H,axiom,
! [A: set_set_real] :
( ( A != bot_bot_set_set_real )
=> ( ! [X: set_real,Y2: set_real] :
( ( member_set_real @ X @ A )
=> ( ( member_set_real @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_sin3548895728136638702t_real @ A ) ) ) ).
% is_singletonI'
thf(fact_630_is__singletonI_H,axiom,
! [A: set_se4580700918925141924nnreal] :
( ( A != bot_bo2988155216863113784nnreal )
=> ( ! [X: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ A )
=> ( ( member603777416030116741nnreal @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_sin9058363718368806650nnreal @ A ) ) ) ).
% is_singletonI'
thf(fact_631_measurable__bot,axiom,
! [M: sigma_measure_real] : ( member2919562650594848410nnreal @ bot_bo1396364965732655767nnreal @ ( sigma_9017504469962657078nnreal @ M @ ( sigma_7204664791115113951nnreal @ top_to7994903218803871134nnreal ) ) ) ).
% measurable_bot
thf(fact_632_lebesgue__measurable__imp__measurable__on__real,axiom,
! [U: real > real,S3: set_real] :
( ( member_real_real @ U @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ borel_5078946678739801102l_real ) )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( equiva5980327992511004390l_real @ U @ S3 ) ) ) ).
% lebesgue_measurable_imp_measurable_on_real
thf(fact_633_sets_Ocountable,axiom,
! [A: set_set_a,M: sigma_measure_set_a] :
( ! [A4: set_a] :
( ( member_set_a @ A4 @ A )
=> ( member_set_set_a @ ( insert_set_a @ A4 @ bot_bot_set_set_a ) @ ( sigma_sets_set_a @ M ) ) )
=> ( ( counta6168152590877469849_set_a @ A )
=> ( member_set_set_a @ A @ ( sigma_sets_set_a @ M ) ) ) ) ).
% sets.countable
thf(fact_634_sets_Ocountable,axiom,
! [A: set_re5328672808648366137nnreal,M: sigma_5394977995791401948nnreal] :
( ! [A4: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ A4 @ A )
=> ( member524040414084610768nnreal @ ( insert152533262698245683nnreal @ A4 @ bot_bo6037503491064675021nnreal ) @ ( sigma_3125015092713243876nnreal @ M ) ) )
=> ( ( counta6024778199358870792nnreal @ A )
=> ( member524040414084610768nnreal @ A @ ( sigma_3125015092713243876nnreal @ M ) ) ) ) ).
% sets.countable
thf(fact_635_sets_Ocountable,axiom,
! [A: set_set_real,M: sigma_3733394171116455995t_real] :
( ! [A4: set_real] :
( ( member_set_real @ A4 @ A )
=> ( member_set_set_real @ ( insert_set_real @ A4 @ bot_bot_set_set_real ) @ ( sigma_sets_set_real @ M ) ) )
=> ( ( counta8054315614235329383t_real @ A )
=> ( member_set_set_real @ A @ ( sigma_sets_set_real @ M ) ) ) ) ).
% sets.countable
thf(fact_636_sets_Ocountable,axiom,
! [A: set_se4580700918925141924nnreal,M: sigma_523634232904505671nnreal] :
( ! [A4: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A4 @ A )
=> ( member6568240578637133883nnreal @ ( insert1343806209672318238nnreal @ A4 @ bot_bo2988155216863113784nnreal ) @ ( sigma_5308793447563920847nnreal @ M ) ) )
=> ( ( counta2425349316461633011nnreal @ A )
=> ( member6568240578637133883nnreal @ A @ ( sigma_5308793447563920847nnreal @ M ) ) ) ) ).
% sets.countable
thf(fact_637_sets_Ocountable,axiom,
! [A: set_a,M: sigma_measure_a] :
( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( member_set_a @ ( insert_a @ A4 @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) ) )
=> ( ( counta4098120917673242425able_a @ A )
=> ( member_set_a @ A @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.countable
thf(fact_638_sets_Ocountable,axiom,
! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ A )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ A4 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( ( counta8439243037236335165nnreal @ A )
=> ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).
% sets.countable
thf(fact_639_sets_Ocountable,axiom,
! [A: set_real,M: sigma_measure_real] :
( ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( member_set_real @ ( insert_real @ A4 @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
=> ( ( counta7319604579010473777e_real @ A )
=> ( member_set_real @ A @ ( sigma_sets_real @ M ) ) ) ) ).
% sets.countable
thf(fact_640_Compl__insert,axiom,
! [X2: a,A: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X2 @ A ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A ) @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_641_Compl__insert,axiom,
! [X2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
( ( uminus5517552291522096439nnreal @ ( insert7407984058720857448nnreal @ X2 @ A ) )
= ( minus_104578273773384135nnreal @ ( uminus5517552291522096439nnreal @ A ) @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) ) ).
% Compl_insert
thf(fact_642_Compl__insert,axiom,
! [X2: real,A: set_real] :
( ( uminus612125837232591019t_real @ ( insert_real @ X2 @ A ) )
= ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A ) @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ).
% Compl_insert
thf(fact_643_restrict__restrict__space,axiom,
! [A: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ ( inf_inf_set_a @ A @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ ( inf_inf_set_a @ B2 @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( ( sigma_8692839461743104066pace_a @ ( sigma_8692839461743104066pace_a @ M @ A ) @ B2 )
= ( sigma_8692839461743104066pace_a @ M @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ) ).
% restrict_restrict_space
thf(fact_644_restrict__restrict__space,axiom,
! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ A @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ B2 @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( sigma_4884701650823297268nnreal @ ( sigma_4884701650823297268nnreal @ M @ A ) @ B2 )
= ( sigma_4884701650823297268nnreal @ M @ ( inf_in3368558534146122112nnreal @ A @ B2 ) ) ) ) ) ).
% restrict_restrict_space
thf(fact_645_restrict__restrict__space,axiom,
! [A: set_real,M: sigma_measure_real,B2: set_real] :
( ( member_set_real @ ( inf_inf_set_real @ A @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ ( inf_inf_set_real @ B2 @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
=> ( ( sigma_5414646170262037096e_real @ ( sigma_5414646170262037096e_real @ M @ A ) @ B2 )
= ( sigma_5414646170262037096e_real @ M @ ( inf_inf_set_real @ A @ B2 ) ) ) ) ) ).
% restrict_restrict_space
thf(fact_646_sets__restrict__space__count__space,axiom,
! [A: set_a,B2: set_a] :
( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ ( sigma_count_space_a @ A ) @ B2 ) )
= ( sigma_sets_a @ ( sigma_count_space_a @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% sets_restrict_space_count_space
thf(fact_647_sets__restrict__space__count__space,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ ( sigma_7204664791115113951nnreal @ A ) @ B2 ) )
= ( sigma_5465916536984168985nnreal @ ( sigma_7204664791115113951nnreal @ ( inf_in3368558534146122112nnreal @ A @ B2 ) ) ) ) ).
% sets_restrict_space_count_space
thf(fact_648_sets__restrict__space__count__space,axiom,
! [A: set_real,B2: set_real] :
( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ ( sigma_8508918144308765139e_real @ A ) @ B2 ) )
= ( sigma_sets_real @ ( sigma_8508918144308765139e_real @ ( inf_inf_set_real @ A @ B2 ) ) ) ) ).
% sets_restrict_space_count_space
thf(fact_649_measurable__discrete__difference,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,X5: set_real,G: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( ( counta7319604579010473777e_real @ X5 )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member7908768830364227535nnreal @ ( G @ X ) @ ( sigma_3147302497200244656nnreal @ N ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( ~ ( member_real @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_650_measurable__discrete__difference,axiom,
! [F: a > set_a,M: sigma_measure_a,N: sigma_measure_set_a,X5: set_a,G: a > set_a] :
( ( member_a_set_a @ F @ ( sigma_3685133166752798000_set_a @ M @ N ) )
=> ( ( counta4098120917673242425able_a @ X5 )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_a @ ( G @ X ) @ ( sigma_space_set_a @ N ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( ~ ( member_a @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member_a_set_a @ G @ ( sigma_3685133166752798000_set_a @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_651_measurable__discrete__difference,axiom,
! [F: a > set_real,M: sigma_measure_a,N: sigma_3733394171116455995t_real,X5: set_a,G: a > set_real] :
( ( member_a_set_real @ F @ ( sigma_739038748264640144t_real @ M @ N ) )
=> ( ( counta4098120917673242425able_a @ X5 )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_real @ ( G @ X ) @ ( sigma_space_set_real @ N ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( ~ ( member_a @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member_a_set_real @ G @ ( sigma_739038748264640144t_real @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_652_measurable__discrete__difference,axiom,
! [F: a > set_Ex3793607809372303086nnreal,M: sigma_measure_a,N: sigma_523634232904505671nnreal,X5: set_a,G: a > set_Ex3793607809372303086nnreal] :
( ( member2532357421736347526nnreal @ F @ ( sigma_7596264061814621596nnreal @ M @ N ) )
=> ( ( counta4098120917673242425able_a @ X5 )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ X5 )
=> ( member603777416030116741nnreal @ ( G @ X ) @ ( sigma_2539764534872131430nnreal @ N ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( ~ ( member_a @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member2532357421736347526nnreal @ G @ ( sigma_7596264061814621596nnreal @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_653_measurable__discrete__difference,axiom,
! [F: extend8495563244428889912nnreal > set_a,M: sigma_7234349610311085201nnreal,N: sigma_measure_set_a,X5: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > set_a] :
( ( member6799942265337811078_set_a @ F @ ( sigma_7624677704890010580_set_a @ M @ N ) )
=> ( ( counta8439243037236335165nnreal @ X5 )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member_set_a @ ( G @ X ) @ ( sigma_space_set_a @ N ) ) )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
=> ( ~ ( member7908768830364227535nnreal @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member6799942265337811078_set_a @ G @ ( sigma_7624677704890010580_set_a @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_654_measurable__discrete__difference,axiom,
! [F: extend8495563244428889912nnreal > set_real,M: sigma_7234349610311085201nnreal,N: sigma_3733394171116455995t_real,X5: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > set_real] :
( ( member6764088077590758224t_real @ F @ ( sigma_5175731160935721196t_real @ M @ N ) )
=> ( ( counta8439243037236335165nnreal @ X5 )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member_set_real @ ( G @ X ) @ ( sigma_space_set_real @ N ) ) )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
=> ( ~ ( member7908768830364227535nnreal @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member6764088077590758224t_real @ G @ ( sigma_5175731160935721196t_real @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_655_measurable__discrete__difference,axiom,
! [F: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,N: sigma_523634232904505671nnreal,X5: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal] :
( ( member4416662722526258908nnreal @ F @ ( sigma_1941770836459163128nnreal @ M @ N ) )
=> ( ( counta8439243037236335165nnreal @ X5 )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ X5 )
=> ( member603777416030116741nnreal @ ( G @ X ) @ ( sigma_2539764534872131430nnreal @ N ) ) )
=> ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
=> ( ~ ( member7908768830364227535nnreal @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member4416662722526258908nnreal @ G @ ( sigma_1941770836459163128nnreal @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_656_measurable__discrete__difference,axiom,
! [F: real > set_a,M: sigma_measure_real,N: sigma_measure_set_a,X5: set_real,G: real > set_a] :
( ( member_real_set_a @ F @ ( sigma_4283435981211228640_set_a @ M @ N ) )
=> ( ( counta7319604579010473777e_real @ X5 )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_a @ ( G @ X ) @ ( sigma_space_set_a @ N ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( ~ ( member_real @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member_real_set_a @ G @ ( sigma_4283435981211228640_set_a @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_657_measurable__discrete__difference,axiom,
! [F: real > set_real,M: sigma_measure_real,N: sigma_3733394171116455995t_real,X5: set_real,G: real > set_real] :
( ( member_real_set_real @ F @ ( sigma_6606012509476713952t_real @ M @ N ) )
=> ( ( counta7319604579010473777e_real @ X5 )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( G @ X ) @ ( sigma_space_set_real @ N ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( ~ ( member_real @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member_real_set_real @ G @ ( sigma_6606012509476713952t_real @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_658_measurable__discrete__difference,axiom,
! [F: real > set_Ex3793607809372303086nnreal,M: sigma_measure_real,N: sigma_523634232904505671nnreal,X5: set_real,G: real > set_Ex3793607809372303086nnreal] :
( ( member8689841359643572048nnreal @ F @ ( sigma_2400199819729843436nnreal @ M @ N ) )
=> ( ( counta7319604579010473777e_real @ X5 )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( member603777416030116741nnreal @ ( G @ X ) @ ( sigma_2539764534872131430nnreal @ N ) ) )
=> ( ! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( ~ ( member_real @ X @ X5 )
=> ( ( F @ X )
= ( G @ X ) ) ) )
=> ( member8689841359643572048nnreal @ G @ ( sigma_2400199819729843436nnreal @ M @ N ) ) ) ) ) ) ) ).
% measurable_discrete_difference
thf(fact_659_measurable__on__imp__borel__measurable__lebesgue__UNIV,axiom,
! [F: real > real] :
( ( equiva5980327992511004390l_real @ F @ top_top_set_real )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ borel_5078946678739801102l_real ) ) ) ).
% measurable_on_imp_borel_measurable_lebesgue_UNIV
thf(fact_660_lebesgue__measurable__imp__measurable__on,axiom,
! [F: real > real,S3: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ borel_5078946678739801102l_real ) )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( equiva5980327992511004390l_real @ F @ S3 ) ) ) ).
% lebesgue_measurable_imp_measurable_on
thf(fact_661_ivl__disj__int__two_I2_J,axiom,
! [L: real,M4: real,U: real] :
( ( inf_inf_set_real @ ( set_or2392270231875598684t_real @ L @ M4 ) @ ( set_or1633881224788618240n_real @ M4 @ U ) )
= bot_bot_set_real ) ).
% ivl_disj_int_two(2)
thf(fact_662_ivl__disj__int__two_I6_J,axiom,
! [L: real,M4: real,U: real] :
( ( inf_inf_set_real @ ( set_or2392270231875598684t_real @ L @ M4 ) @ ( set_or2392270231875598684t_real @ M4 @ U ) )
= bot_bot_set_real ) ).
% ivl_disj_int_two(6)
thf(fact_663_DEADID_Oin__rel,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A5: set_a,B5: set_a] :
? [Z2: set_a] :
( ( member_set_a @ Z2 @ top_top_set_set_a )
& ( ( id_set_a @ Z2 )
= A5 )
& ( ( id_set_a @ Z2 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_664_DEADID_Oin__rel,axiom,
( ( ^ [Y4: real > extend8495563244428889912nnreal,Z: real > extend8495563244428889912nnreal] : ( Y4 = Z ) )
= ( ^ [A5: real > extend8495563244428889912nnreal,B5: real > extend8495563244428889912nnreal] :
? [Z2: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ Z2 @ top_to315565310957491945nnreal )
& ( ( id_rea5353623948652148818nnreal @ Z2 )
= A5 )
& ( ( id_rea5353623948652148818nnreal @ Z2 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_665_DEADID_Oin__rel,axiom,
( ( ^ [Y4: set_real,Z: set_real] : ( Y4 = Z ) )
= ( ^ [A5: set_real,B5: set_real] :
? [Z2: set_real] :
( ( member_set_real @ Z2 @ top_top_set_set_real )
& ( ( id_set_real @ Z2 )
= A5 )
& ( ( id_set_real @ Z2 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_666_DEADID_Oin__rel,axiom,
( ( ^ [Y4: set_Ex3793607809372303086nnreal,Z: set_Ex3793607809372303086nnreal] : ( Y4 = Z ) )
= ( ^ [A5: set_Ex3793607809372303086nnreal,B5: set_Ex3793607809372303086nnreal] :
? [Z2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ Z2 @ top_to3356475028079756884nnreal )
& ( ( id_set2823833123132642621nnreal @ Z2 )
= A5 )
& ( ( id_set2823833123132642621nnreal @ Z2 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_667_DEADID_Oin__rel,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y4 = Z ) )
= ( ^ [A5: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
? [Z2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Z2 @ top_to7994903218803871134nnreal )
& ( ( id_Ext8331313139072774535nnreal @ Z2 )
= A5 )
& ( ( id_Ext8331313139072774535nnreal @ Z2 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_668_DEADID_Oin__rel,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A5: real,B5: real] :
? [Z2: real] :
( ( member_real @ Z2 @ top_top_set_real )
& ( ( id_real @ Z2 )
= A5 )
& ( ( id_real @ Z2 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_669_DEADID_Oin__rel,axiom,
( ( ^ [Y4: a,Z: a] : ( Y4 = Z ) )
= ( ^ [A5: a,B5: a] :
? [Z2: a] :
( ( member_a @ Z2 @ top_top_set_a )
& ( ( id_a @ Z2 )
= A5 )
& ( ( id_a @ Z2 )
= B5 ) ) ) ) ).
% DEADID.in_rel
thf(fact_670_minus__diff__minus,axiom,
! [A2: real,B: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B ) ) ) ).
% minus_diff_minus
thf(fact_671_diff__null__sets__lebesgue,axiom,
! [N: set_real,S3: set_real,X5: set_real] :
( ( member_set_real @ N @ ( measur3710062792471635001s_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) ) )
=> ( ( member_set_real @ ( minus_minus_set_real @ X5 @ N ) @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) ) )
=> ( ( ord_less_eq_set_real @ N @ X5 )
=> ( member_set_real @ X5 @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ S3 ) ) ) ) ) ) ).
% diff_null_sets_lebesgue
thf(fact_672_borel__measurable__vimage,axiom,
! [F: a > a,M: sigma_measure_a,X2: a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ borel_5459123734250506524orel_a ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_673_borel__measurable__vimage,axiom,
! [F: extend8495563244428889912nnreal > a,M: sigma_7234349610311085201nnreal,X2: a] :
( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ M @ borel_5459123734250506524orel_a ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_674_borel__measurable__vimage,axiom,
! [F: real > a,M: sigma_measure_real,X2: a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_675_borel__measurable__vimage,axiom,
! [F: a > real,M: sigma_measure_a,X2: real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_676_borel__measurable__vimage,axiom,
! [F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,X2: real] :
( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_677_borel__measurable__vimage,axiom,
! [F: real > real,M: sigma_measure_real,X2: real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_678_borel__measurable__vimage,axiom,
! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,X2: extend8495563244428889912nnreal] :
( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_679_borel__measurable__vimage,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,X2: extend8495563244428889912nnreal] :
( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ borel_6524799422816628122nnreal ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_680_borel__measurable__vimage,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,X2: extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ).
% borel_measurable_vimage
thf(fact_681_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_682_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_683_subsetI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( member_set_a @ X @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_684_subsetI,axiom,
! [A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ! [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ A )
=> ( member2919562650594848410nnreal @ X @ B2 ) )
=> ( ord_le2462468573666744473nnreal @ A @ B2 ) ) ).
% subsetI
thf(fact_685_subsetI,axiom,
! [A: set_set_real,B2: set_set_real] :
( ! [X: set_real] :
( ( member_set_real @ X @ A )
=> ( member_set_real @ X @ B2 ) )
=> ( ord_le3558479182127378552t_real @ A @ B2 ) ) ).
% subsetI
thf(fact_686_subsetI,axiom,
! [A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ A )
=> ( member603777416030116741nnreal @ X @ B2 ) )
=> ( ord_le3366939622266546180nnreal @ A @ B2 ) ) ).
% subsetI
thf(fact_687_vimageI,axiom,
! [F: set_a > set_a,A2: set_a,B: set_a,B2: set_set_a] :
( ( ( F @ A2 )
= B )
=> ( ( member_set_a @ B @ B2 )
=> ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_688_vimageI,axiom,
! [F: set_real > set_a,A2: set_real,B: set_a,B2: set_set_a] :
( ( ( F @ A2 )
= B )
=> ( ( member_set_a @ B @ B2 )
=> ( member_set_real @ A2 @ ( vimage4921045269514575487_set_a @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_689_vimageI,axiom,
! [F: set_Ex3793607809372303086nnreal > set_a,A2: set_Ex3793607809372303086nnreal,B: set_a,B2: set_set_a] :
( ( ( F @ A2 )
= B )
=> ( ( member_set_a @ B @ B2 )
=> ( member603777416030116741nnreal @ A2 @ ( vimage9012950911561050995_set_a @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_690_vimageI,axiom,
! [F: set_a > set_real,A2: set_a,B: set_real,B2: set_set_real] :
( ( ( F @ A2 )
= B )
=> ( ( member_set_real @ B @ B2 )
=> ( member_set_a @ A2 @ ( vimage1623514241378321221t_real @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_691_vimageI,axiom,
! [F: set_real > set_real,A2: set_real,B: set_real,B2: set_set_real] :
( ( ( F @ A2 )
= B )
=> ( ( member_set_real @ B @ B2 )
=> ( member_set_real @ A2 @ ( vimage2667142749230307073t_real @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_692_vimageI,axiom,
! [F: set_Ex3793607809372303086nnreal > set_real,A2: set_Ex3793607809372303086nnreal,B: set_real,B2: set_set_real] :
( ( ( F @ A2 )
= B )
=> ( ( member_set_real @ B @ B2 )
=> ( member603777416030116741nnreal @ A2 @ ( vimage2976515561842077453t_real @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_693_vimageI,axiom,
! [F: set_a > set_Ex3793607809372303086nnreal,A2: set_a,B: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( ( F @ A2 )
= B )
=> ( ( member603777416030116741nnreal @ B @ B2 )
=> ( member_set_a @ A2 @ ( vimage3731165941878406353nnreal @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_694_vimageI,axiom,
! [F: set_real > set_Ex3793607809372303086nnreal,A2: set_real,B: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( ( F @ A2 )
= B )
=> ( ( member603777416030116741nnreal @ B @ B2 )
=> ( member_set_real @ A2 @ ( vimage1211383488014126733nnreal @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_695_vimageI,axiom,
! [F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( ( F @ A2 )
= B )
=> ( ( member603777416030116741nnreal @ B @ B2 )
=> ( member603777416030116741nnreal @ A2 @ ( vimage7483462650094439577nnreal @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_696_vimageI,axiom,
! [F: ( real > extend8495563244428889912nnreal ) > set_a,A2: real > extend8495563244428889912nnreal,B: set_a,B2: set_set_a] :
( ( ( F @ A2 )
= B )
=> ( ( member_set_a @ B @ B2 )
=> ( member2919562650594848410nnreal @ A2 @ ( vimage1816238162123171230_set_a @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_697_vimage__eq,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_set_a] :
( ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ B2 ) )
= ( member_set_a @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_698_vimage__eq,axiom,
! [A2: set_a,F: set_a > set_real,B2: set_set_real] :
( ( member_set_a @ A2 @ ( vimage1623514241378321221t_real @ F @ B2 ) )
= ( member_set_real @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_699_vimage__eq,axiom,
! [A2: set_a,F: set_a > set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( member_set_a @ A2 @ ( vimage3731165941878406353nnreal @ F @ B2 ) )
= ( member603777416030116741nnreal @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_700_vimage__eq,axiom,
! [A2: set_real,F: set_real > set_a,B2: set_set_a] :
( ( member_set_real @ A2 @ ( vimage4921045269514575487_set_a @ F @ B2 ) )
= ( member_set_a @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_701_vimage__eq,axiom,
! [A2: set_real,F: set_real > set_real,B2: set_set_real] :
( ( member_set_real @ A2 @ ( vimage2667142749230307073t_real @ F @ B2 ) )
= ( member_set_real @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_702_vimage__eq,axiom,
! [A2: set_real,F: set_real > set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( member_set_real @ A2 @ ( vimage1211383488014126733nnreal @ F @ B2 ) )
= ( member603777416030116741nnreal @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_703_vimage__eq,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_a,B2: set_set_a] :
( ( member603777416030116741nnreal @ A2 @ ( vimage9012950911561050995_set_a @ F @ B2 ) )
= ( member_set_a @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_704_vimage__eq,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_real,B2: set_set_real] :
( ( member603777416030116741nnreal @ A2 @ ( vimage2976515561842077453t_real @ F @ B2 ) )
= ( member_set_real @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_705_vimage__eq,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( vimage7483462650094439577nnreal @ F @ B2 ) )
= ( member603777416030116741nnreal @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_706_vimage__eq,axiom,
! [A2: set_a,F: set_a > real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal] :
( ( member_set_a @ A2 @ ( vimage6157622615598462950nnreal @ F @ B2 ) )
= ( member2919562650594848410nnreal @ ( F @ A2 ) @ B2 ) ) ).
% vimage_eq
thf(fact_707_member__remove,axiom,
! [X2: set_a,Y: set_a,A: set_set_a] :
( ( member_set_a @ X2 @ ( remove_set_a @ Y @ A ) )
= ( ( member_set_a @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_708_member__remove,axiom,
! [X2: real > extend8495563244428889912nnreal,Y: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ X2 @ ( remove389474404184257502nnreal @ Y @ A ) )
= ( ( member2919562650594848410nnreal @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_709_member__remove,axiom,
! [X2: set_real,Y: set_real,A: set_set_real] :
( ( member_set_real @ X2 @ ( remove_set_real @ Y @ A ) )
= ( ( member_set_real @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_710_member__remove,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ X2 @ ( remove6680540689449789641nnreal @ Y @ A ) )
= ( ( member603777416030116741nnreal @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_711_compl__le__compl__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ ( uminus_uminus_set_a @ Y ) )
= ( ord_less_eq_set_a @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_712_compl__le__compl__iff,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ ( uminus5517552291522096439nnreal @ X2 ) @ ( uminus5517552291522096439nnreal @ Y ) )
= ( ord_le6787938422905777998nnreal @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_713_compl__le__compl__iff,axiom,
! [X2: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ ( uminus612125837232591019t_real @ X2 ) @ ( uminus612125837232591019t_real @ Y ) )
= ( ord_less_eq_set_real @ Y @ X2 ) ) ).
% compl_le_compl_iff
thf(fact_714_neg__le__iff__le,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ B ) ) ).
% neg_le_iff_le
thf(fact_715_le__inf__iff,axiom,
! [X2: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ ( inf_inf_real @ Y @ Z3 ) )
= ( ( ord_less_eq_real @ X2 @ Y )
& ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% le_inf_iff
thf(fact_716_inf_Obounded__iff,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B @ C ) )
= ( ( ord_less_eq_real @ A2 @ B )
& ( ord_less_eq_real @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_717_insert__subset,axiom,
! [X2: set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ A ) @ B2 )
= ( ( member_set_a @ X2 @ B2 )
& ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_718_insert__subset,axiom,
! [X2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( ord_le2462468573666744473nnreal @ ( insert152533262698245683nnreal @ X2 @ A ) @ B2 )
= ( ( member2919562650594848410nnreal @ X2 @ B2 )
& ( ord_le2462468573666744473nnreal @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_719_insert__subset,axiom,
! [X2: set_real,A: set_set_real,B2: set_set_real] :
( ( ord_le3558479182127378552t_real @ ( insert_set_real @ X2 @ A ) @ B2 )
= ( ( member_set_real @ X2 @ B2 )
& ( ord_le3558479182127378552t_real @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_720_insert__subset,axiom,
! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( ord_le3366939622266546180nnreal @ ( insert1343806209672318238nnreal @ X2 @ A ) @ B2 )
= ( ( member603777416030116741nnreal @ X2 @ B2 )
& ( ord_le3366939622266546180nnreal @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_721_vimage__UNIV,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( vimage3650734033530794285nnreal @ F @ top_to7994903218803871134nnreal )
= top_to7994903218803871134nnreal ) ).
% vimage_UNIV
thf(fact_722_vimage__UNIV,axiom,
! [F: real > extend8495563244428889912nnreal] :
( ( vimage6366802093293386401nnreal @ F @ top_to7994903218803871134nnreal )
= top_top_set_real ) ).
% vimage_UNIV
thf(fact_723_vimage__UNIV,axiom,
! [F: a > extend8495563244428889912nnreal] :
( ( vimage1258658873539170235nnreal @ F @ top_to7994903218803871134nnreal )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_724_vimage__UNIV,axiom,
! [F: extend8495563244428889912nnreal > real] :
( ( vimage4399055823842842145l_real @ F @ top_top_set_real )
= top_to7994903218803871134nnreal ) ).
% vimage_UNIV
thf(fact_725_vimage__UNIV,axiom,
! [F: real > real] :
( ( vimage_real_real @ F @ top_top_set_real )
= top_top_set_real ) ).
% vimage_UNIV
thf(fact_726_vimage__UNIV,axiom,
! [F: a > real] :
( ( vimage_a_real @ F @ top_top_set_real )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_727_vimage__UNIV,axiom,
! [F: extend8495563244428889912nnreal > a] :
( ( vimage4075187267506941001real_a @ F @ top_top_set_a )
= top_to7994903218803871134nnreal ) ).
% vimage_UNIV
thf(fact_728_vimage__UNIV,axiom,
! [F: real > a] :
( ( vimage_real_a @ F @ top_top_set_a )
= top_top_set_real ) ).
% vimage_UNIV
thf(fact_729_vimage__UNIV,axiom,
! [F: a > a] :
( ( vimage_a_a @ F @ top_top_set_a )
= top_top_set_a ) ).
% vimage_UNIV
thf(fact_730_Compl__subset__Compl__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A ) @ ( uminus_uminus_set_a @ B2 ) )
= ( ord_less_eq_set_a @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_731_Compl__subset__Compl__iff,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ ( uminus5517552291522096439nnreal @ A ) @ ( uminus5517552291522096439nnreal @ B2 ) )
= ( ord_le6787938422905777998nnreal @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_732_Compl__subset__Compl__iff,axiom,
! [A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( uminus612125837232591019t_real @ A ) @ ( uminus612125837232591019t_real @ B2 ) )
= ( ord_less_eq_set_real @ B2 @ A ) ) ).
% Compl_subset_Compl_iff
thf(fact_733_Compl__anti__mono,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B2 ) @ ( uminus_uminus_set_a @ A ) ) ) ).
% Compl_anti_mono
thf(fact_734_Compl__anti__mono,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ B2 )
=> ( ord_le6787938422905777998nnreal @ ( uminus5517552291522096439nnreal @ B2 ) @ ( uminus5517552291522096439nnreal @ A ) ) ) ).
% Compl_anti_mono
thf(fact_735_Compl__anti__mono,axiom,
! [A: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ord_less_eq_set_real @ ( uminus612125837232591019t_real @ B2 ) @ ( uminus612125837232591019t_real @ A ) ) ) ).
% Compl_anti_mono
thf(fact_736_greaterThanLessThan__empty,axiom,
! [L: real,K: real] :
( ( ord_less_eq_real @ L @ K )
=> ( ( set_or1633881224788618240n_real @ K @ L )
= bot_bot_set_real ) ) ).
% greaterThanLessThan_empty
thf(fact_737_greaterThanLessThan__empty__iff,axiom,
! [A2: real,B: real] :
( ( ( set_or1633881224788618240n_real @ A2 @ B )
= bot_bot_set_real )
= ( ord_less_eq_real @ B @ A2 ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_738_greaterThanLessThan__empty__iff2,axiom,
! [A2: real,B: real] :
( ( bot_bot_set_real
= ( set_or1633881224788618240n_real @ A2 @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_739_greaterThanAtMost__empty,axiom,
! [L: real,K: real] :
( ( ord_less_eq_real @ L @ K )
=> ( ( set_or2392270231875598684t_real @ K @ L )
= bot_bot_set_real ) ) ).
% greaterThanAtMost_empty
thf(fact_740_subset__Compl__singleton,axiom,
! [A: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( uminus6103902357914783669_set_a @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) )
= ( ~ ( member_set_a @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_741_subset__Compl__singleton,axiom,
! [A: set_re5328672808648366137nnreal,B: real > extend8495563244428889912nnreal] :
( ( ord_le2462468573666744473nnreal @ A @ ( uminus2275888197404385410nnreal @ ( insert152533262698245683nnreal @ B @ bot_bo6037503491064675021nnreal ) ) )
= ( ~ ( member2919562650594848410nnreal @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_742_subset__Compl__singleton,axiom,
! [A: set_set_real,B: set_real] :
( ( ord_le3558479182127378552t_real @ A @ ( uminus708787163358948833t_real @ ( insert_set_real @ B @ bot_bot_set_set_real ) ) )
= ( ~ ( member_set_real @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_743_subset__Compl__singleton,axiom,
! [A: set_se4580700918925141924nnreal,B: set_Ex3793607809372303086nnreal] :
( ( ord_le3366939622266546180nnreal @ A @ ( uminus4762152451731718637nnreal @ ( insert1343806209672318238nnreal @ B @ bot_bo2988155216863113784nnreal ) ) )
= ( ~ ( member603777416030116741nnreal @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_744_subset__Compl__singleton,axiom,
! [A: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_745_subset__Compl__singleton,axiom,
! [A: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ ( uminus5517552291522096439nnreal @ ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) ) )
= ( ~ ( member7908768830364227535nnreal @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_746_subset__Compl__singleton,axiom,
! [A: set_real,B: real] :
( ( ord_less_eq_set_real @ A @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
= ( ~ ( member_real @ B @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_747_Ioc__subset__iff,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or2392270231875598684t_real @ A2 @ B ) @ ( set_or2392270231875598684t_real @ C @ D ) )
= ( ( ord_less_eq_real @ B @ A2 )
| ( ( ord_less_eq_real @ C @ A2 )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_748_nle__le,axiom,
! [A2: real,B: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B ) )
= ( ( ord_less_eq_real @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_749_le__cases3,axiom,
! [X2: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_750_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [X3: real,Y3: real] :
( ( ord_less_eq_real @ X3 @ Y3 )
& ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_751_ord__eq__le__trans,axiom,
! [A2: real,B: real,C: real] :
( ( A2 = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_752_ord__le__eq__trans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_753_order__antisym,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_754_order_Otrans,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_755_order__trans,axiom,
! [X2: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_756_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B: real] :
( ! [A4: real,B6: real] :
( ( ord_less_eq_real @ A4 @ B6 )
=> ( P @ A4 @ B6 ) )
=> ( ! [A4: real,B6: real] :
( ( P @ B6 @ A4 )
=> ( P @ A4 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_757_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A5 )
& ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_758_dual__order_Oantisym,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_759_dual__order_Otrans,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_760_antisym,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_761_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
& ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_762_order__subst1,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_763_order__subst2,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_764_order__eq__refl,axiom,
! [X2: real,Y: real] :
( ( X2 = Y )
=> ( ord_less_eq_real @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_765_linorder__linear,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
| ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_766_ord__eq__le__subst,axiom,
! [A2: real,F: real > real,B: real,C: real] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_767_ord__le__eq__subst,axiom,
! [A2: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_768_linorder__le__cases,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_769_order__antisym__conv,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_770_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_771_in__mono,axiom,
! [A: set_set_a,B2: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_772_in__mono,axiom,
! [A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal,X2: real > extend8495563244428889912nnreal] :
( ( ord_le2462468573666744473nnreal @ A @ B2 )
=> ( ( member2919562650594848410nnreal @ X2 @ A )
=> ( member2919562650594848410nnreal @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_773_in__mono,axiom,
! [A: set_set_real,B2: set_set_real,X2: set_real] :
( ( ord_le3558479182127378552t_real @ A @ B2 )
=> ( ( member_set_real @ X2 @ A )
=> ( member_set_real @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_774_in__mono,axiom,
! [A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal,X2: set_Ex3793607809372303086nnreal] :
( ( ord_le3366939622266546180nnreal @ A @ B2 )
=> ( ( member603777416030116741nnreal @ X2 @ A )
=> ( member603777416030116741nnreal @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_775_subsetD,axiom,
! [A: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_776_subsetD,axiom,
! [A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal,C: real > extend8495563244428889912nnreal] :
( ( ord_le2462468573666744473nnreal @ A @ B2 )
=> ( ( member2919562650594848410nnreal @ C @ A )
=> ( member2919562650594848410nnreal @ C @ B2 ) ) ) ).
% subsetD
thf(fact_777_subsetD,axiom,
! [A: set_set_real,B2: set_set_real,C: set_real] :
( ( ord_le3558479182127378552t_real @ A @ B2 )
=> ( ( member_set_real @ C @ A )
=> ( member_set_real @ C @ B2 ) ) ) ).
% subsetD
thf(fact_778_subsetD,axiom,
! [A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal,C: set_Ex3793607809372303086nnreal] :
( ( ord_le3366939622266546180nnreal @ A @ B2 )
=> ( ( member603777416030116741nnreal @ C @ A )
=> ( member603777416030116741nnreal @ C @ B2 ) ) ) ).
% subsetD
thf(fact_779_vimageD,axiom,
! [A2: set_a,F: set_a > set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ A ) )
=> ( member_set_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_780_vimageD,axiom,
! [A2: set_a,F: set_a > set_real,A: set_set_real] :
( ( member_set_a @ A2 @ ( vimage1623514241378321221t_real @ F @ A ) )
=> ( member_set_real @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_781_vimageD,axiom,
! [A2: set_a,F: set_a > set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member_set_a @ A2 @ ( vimage3731165941878406353nnreal @ F @ A ) )
=> ( member603777416030116741nnreal @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_782_vimageD,axiom,
! [A2: set_real,F: set_real > set_a,A: set_set_a] :
( ( member_set_real @ A2 @ ( vimage4921045269514575487_set_a @ F @ A ) )
=> ( member_set_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_783_vimageD,axiom,
! [A2: set_real,F: set_real > set_real,A: set_set_real] :
( ( member_set_real @ A2 @ ( vimage2667142749230307073t_real @ F @ A ) )
=> ( member_set_real @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_784_vimageD,axiom,
! [A2: set_real,F: set_real > set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member_set_real @ A2 @ ( vimage1211383488014126733nnreal @ F @ A ) )
=> ( member603777416030116741nnreal @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_785_vimageD,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_a,A: set_set_a] :
( ( member603777416030116741nnreal @ A2 @ ( vimage9012950911561050995_set_a @ F @ A ) )
=> ( member_set_a @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_786_vimageD,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_real,A: set_set_real] :
( ( member603777416030116741nnreal @ A2 @ ( vimage2976515561842077453t_real @ F @ A ) )
=> ( member_set_real @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_787_vimageD,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( vimage7483462650094439577nnreal @ F @ A ) )
=> ( member603777416030116741nnreal @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_788_vimageD,axiom,
! [A2: set_a,F: set_a > real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal] :
( ( member_set_a @ A2 @ ( vimage6157622615598462950nnreal @ F @ A ) )
=> ( member2919562650594848410nnreal @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_789_vimageE,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_set_a] :
( ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ B2 ) )
=> ( member_set_a @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_790_vimageE,axiom,
! [A2: set_a,F: set_a > set_real,B2: set_set_real] :
( ( member_set_a @ A2 @ ( vimage1623514241378321221t_real @ F @ B2 ) )
=> ( member_set_real @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_791_vimageE,axiom,
! [A2: set_a,F: set_a > set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( member_set_a @ A2 @ ( vimage3731165941878406353nnreal @ F @ B2 ) )
=> ( member603777416030116741nnreal @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_792_vimageE,axiom,
! [A2: set_real,F: set_real > set_a,B2: set_set_a] :
( ( member_set_real @ A2 @ ( vimage4921045269514575487_set_a @ F @ B2 ) )
=> ( member_set_a @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_793_vimageE,axiom,
! [A2: set_real,F: set_real > set_real,B2: set_set_real] :
( ( member_set_real @ A2 @ ( vimage2667142749230307073t_real @ F @ B2 ) )
=> ( member_set_real @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_794_vimageE,axiom,
! [A2: set_real,F: set_real > set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( member_set_real @ A2 @ ( vimage1211383488014126733nnreal @ F @ B2 ) )
=> ( member603777416030116741nnreal @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_795_vimageE,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_a,B2: set_set_a] :
( ( member603777416030116741nnreal @ A2 @ ( vimage9012950911561050995_set_a @ F @ B2 ) )
=> ( member_set_a @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_796_vimageE,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_real,B2: set_set_real] :
( ( member603777416030116741nnreal @ A2 @ ( vimage2976515561842077453t_real @ F @ B2 ) )
=> ( member_set_real @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_797_vimageE,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A2 @ ( vimage7483462650094439577nnreal @ F @ B2 ) )
=> ( member603777416030116741nnreal @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_798_vimageE,axiom,
! [A2: set_a,F: set_a > real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal] :
( ( member_set_a @ A2 @ ( vimage6157622615598462950nnreal @ F @ B2 ) )
=> ( member2919562650594848410nnreal @ ( F @ A2 ) @ B2 ) ) ).
% vimageE
thf(fact_799_vimageI2,axiom,
! [F: set_a > set_a,A2: set_a,A: set_set_a] :
( ( member_set_a @ ( F @ A2 ) @ A )
=> ( member_set_a @ A2 @ ( vimage_set_a_set_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_800_vimageI2,axiom,
! [F: set_real > set_a,A2: set_real,A: set_set_a] :
( ( member_set_a @ ( F @ A2 ) @ A )
=> ( member_set_real @ A2 @ ( vimage4921045269514575487_set_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_801_vimageI2,axiom,
! [F: set_Ex3793607809372303086nnreal > set_a,A2: set_Ex3793607809372303086nnreal,A: set_set_a] :
( ( member_set_a @ ( F @ A2 ) @ A )
=> ( member603777416030116741nnreal @ A2 @ ( vimage9012950911561050995_set_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_802_vimageI2,axiom,
! [F: set_a > set_real,A2: set_a,A: set_set_real] :
( ( member_set_real @ ( F @ A2 ) @ A )
=> ( member_set_a @ A2 @ ( vimage1623514241378321221t_real @ F @ A ) ) ) ).
% vimageI2
thf(fact_803_vimageI2,axiom,
! [F: set_real > set_real,A2: set_real,A: set_set_real] :
( ( member_set_real @ ( F @ A2 ) @ A )
=> ( member_set_real @ A2 @ ( vimage2667142749230307073t_real @ F @ A ) ) ) ).
% vimageI2
thf(fact_804_vimageI2,axiom,
! [F: set_Ex3793607809372303086nnreal > set_real,A2: set_Ex3793607809372303086nnreal,A: set_set_real] :
( ( member_set_real @ ( F @ A2 ) @ A )
=> ( member603777416030116741nnreal @ A2 @ ( vimage2976515561842077453t_real @ F @ A ) ) ) ).
% vimageI2
thf(fact_805_vimageI2,axiom,
! [F: set_a > set_Ex3793607809372303086nnreal,A2: set_a,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ ( F @ A2 ) @ A )
=> ( member_set_a @ A2 @ ( vimage3731165941878406353nnreal @ F @ A ) ) ) ).
% vimageI2
thf(fact_806_vimageI2,axiom,
! [F: set_real > set_Ex3793607809372303086nnreal,A2: set_real,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ ( F @ A2 ) @ A )
=> ( member_set_real @ A2 @ ( vimage1211383488014126733nnreal @ F @ A ) ) ) ).
% vimageI2
thf(fact_807_vimageI2,axiom,
! [F: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ ( F @ A2 ) @ A )
=> ( member603777416030116741nnreal @ A2 @ ( vimage7483462650094439577nnreal @ F @ A ) ) ) ).
% vimageI2
thf(fact_808_vimageI2,axiom,
! [F: ( real > extend8495563244428889912nnreal ) > set_a,A2: real > extend8495563244428889912nnreal,A: set_set_a] :
( ( member_set_a @ ( F @ A2 ) @ A )
=> ( member2919562650594848410nnreal @ A2 @ ( vimage1816238162123171230_set_a @ F @ A ) ) ) ).
% vimageI2
thf(fact_809_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B4: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
=> ( member_set_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_810_subset__eq,axiom,
( ord_le2462468573666744473nnreal
= ( ^ [A3: set_re5328672808648366137nnreal,B4: set_re5328672808648366137nnreal] :
! [X3: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X3 @ A3 )
=> ( member2919562650594848410nnreal @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_811_subset__eq,axiom,
( ord_le3558479182127378552t_real
= ( ^ [A3: set_set_real,B4: set_set_real] :
! [X3: set_real] :
( ( member_set_real @ X3 @ A3 )
=> ( member_set_real @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_812_subset__eq,axiom,
( ord_le3366939622266546180nnreal
= ( ^ [A3: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal] :
! [X3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X3 @ A3 )
=> ( member603777416030116741nnreal @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_813_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A3 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_814_subset__iff,axiom,
( ord_le2462468573666744473nnreal
= ( ^ [A3: set_re5328672808648366137nnreal,B4: set_re5328672808648366137nnreal] :
! [T: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ T @ A3 )
=> ( member2919562650594848410nnreal @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_815_subset__iff,axiom,
( ord_le3558479182127378552t_real
= ( ^ [A3: set_set_real,B4: set_set_real] :
! [T: set_real] :
( ( member_set_real @ T @ A3 )
=> ( member_set_real @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_816_subset__iff,axiom,
( ord_le3366939622266546180nnreal
= ( ^ [A3: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal] :
! [T: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ T @ A3 )
=> ( member603777416030116741nnreal @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_817_verit__la__disequality,axiom,
! [A2: real,B: real] :
( ( A2 = B )
| ~ ( ord_less_eq_real @ A2 @ B )
| ~ ( ord_less_eq_real @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_818_sets__le__imp__space__le,axiom,
! [A: sigma_measure_a,B2: sigma_measure_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A ) @ ( sigma_sets_a @ B2 ) )
=> ( ord_less_eq_set_a @ ( sigma_space_a @ A ) @ ( sigma_space_a @ B2 ) ) ) ).
% sets_le_imp_space_le
thf(fact_819_sets__le__imp__space__le,axiom,
! [A: sigma_7234349610311085201nnreal,B2: sigma_7234349610311085201nnreal] :
( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ A ) @ ( sigma_5465916536984168985nnreal @ B2 ) )
=> ( ord_le6787938422905777998nnreal @ ( sigma_3147302497200244656nnreal @ A ) @ ( sigma_3147302497200244656nnreal @ B2 ) ) ) ).
% sets_le_imp_space_le
thf(fact_820_sets__le__imp__space__le,axiom,
! [A: sigma_measure_real,B2: sigma_measure_real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B2 ) )
=> ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B2 ) ) ) ).
% sets_le_imp_space_le
thf(fact_821_vimage__Compl,axiom,
! [F: a > a,A: set_a] :
( ( vimage_a_a @ F @ ( uminus_uminus_set_a @ A ) )
= ( uminus_uminus_set_a @ ( vimage_a_a @ F @ A ) ) ) ).
% vimage_Compl
thf(fact_822_vimage__Compl,axiom,
! [F: extend8495563244428889912nnreal > a,A: set_a] :
( ( vimage4075187267506941001real_a @ F @ ( uminus_uminus_set_a @ A ) )
= ( uminus5517552291522096439nnreal @ ( vimage4075187267506941001real_a @ F @ A ) ) ) ).
% vimage_Compl
thf(fact_823_vimage__Compl,axiom,
! [F: real > a,A: set_a] :
( ( vimage_real_a @ F @ ( uminus_uminus_set_a @ A ) )
= ( uminus612125837232591019t_real @ ( vimage_real_a @ F @ A ) ) ) ).
% vimage_Compl
thf(fact_824_vimage__Compl,axiom,
! [F: a > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
( ( vimage1258658873539170235nnreal @ F @ ( uminus5517552291522096439nnreal @ A ) )
= ( uminus_uminus_set_a @ ( vimage1258658873539170235nnreal @ F @ A ) ) ) ).
% vimage_Compl
thf(fact_825_vimage__Compl,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
( ( vimage3650734033530794285nnreal @ F @ ( uminus5517552291522096439nnreal @ A ) )
= ( uminus5517552291522096439nnreal @ ( vimage3650734033530794285nnreal @ F @ A ) ) ) ).
% vimage_Compl
thf(fact_826_vimage__Compl,axiom,
! [F: real > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
( ( vimage6366802093293386401nnreal @ F @ ( uminus5517552291522096439nnreal @ A ) )
= ( uminus612125837232591019t_real @ ( vimage6366802093293386401nnreal @ F @ A ) ) ) ).
% vimage_Compl
thf(fact_827_vimage__Compl,axiom,
! [F: a > real,A: set_real] :
( ( vimage_a_real @ F @ ( uminus612125837232591019t_real @ A ) )
= ( uminus_uminus_set_a @ ( vimage_a_real @ F @ A ) ) ) ).
% vimage_Compl
thf(fact_828_vimage__Compl,axiom,
! [F: extend8495563244428889912nnreal > real,A: set_real] :
( ( vimage4399055823842842145l_real @ F @ ( uminus612125837232591019t_real @ A ) )
= ( uminus5517552291522096439nnreal @ ( vimage4399055823842842145l_real @ F @ A ) ) ) ).
% vimage_Compl
thf(fact_829_vimage__Compl,axiom,
! [F: real > real,A: set_real] :
( ( vimage_real_real @ F @ ( uminus612125837232591019t_real @ A ) )
= ( uminus612125837232591019t_real @ ( vimage_real_real @ F @ A ) ) ) ).
% vimage_Compl
thf(fact_830_top_Oextremum__uniqueI,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ top_to7994903218803871134nnreal @ A2 )
=> ( A2 = top_to7994903218803871134nnreal ) ) ).
% top.extremum_uniqueI
thf(fact_831_top_Oextremum__uniqueI,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
=> ( A2 = top_top_set_real ) ) ).
% top.extremum_uniqueI
thf(fact_832_top_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
=> ( A2 = top_top_set_a ) ) ).
% top.extremum_uniqueI
thf(fact_833_top_Oextremum__unique,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ top_to7994903218803871134nnreal @ A2 )
= ( A2 = top_to7994903218803871134nnreal ) ) ).
% top.extremum_unique
thf(fact_834_top_Oextremum__unique,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
= ( A2 = top_top_set_real ) ) ).
% top.extremum_unique
thf(fact_835_top_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ top_top_set_a @ A2 )
= ( A2 = top_top_set_a ) ) ).
% top.extremum_unique
thf(fact_836_top__greatest,axiom,
! [A2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A2 @ top_to7994903218803871134nnreal ) ).
% top_greatest
thf(fact_837_top__greatest,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).
% top_greatest
thf(fact_838_top__greatest,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ top_top_set_a ) ).
% top_greatest
thf(fact_839_diff__mono,axiom,
! [A2: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_840_diff__left__mono,axiom,
! [B: real,A2: real,C: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_841_diff__right__mono,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_842_diff__eq__diff__less__eq,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A2 @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A2 @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_843_compl__le__swap2,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ X2 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_844_compl__le__swap2,axiom,
! [Y: set_Ex3793607809372303086nnreal,X2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ ( uminus5517552291522096439nnreal @ Y ) @ X2 )
=> ( ord_le6787938422905777998nnreal @ ( uminus5517552291522096439nnreal @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_845_compl__le__swap2,axiom,
! [Y: set_real,X2: set_real] :
( ( ord_less_eq_set_real @ ( uminus612125837232591019t_real @ Y ) @ X2 )
=> ( ord_less_eq_set_real @ ( uminus612125837232591019t_real @ X2 ) @ Y ) ) ).
% compl_le_swap2
thf(fact_846_compl__le__swap1,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ ( uminus_uminus_set_a @ X2 ) )
=> ( ord_less_eq_set_a @ X2 @ ( uminus_uminus_set_a @ Y ) ) ) ).
% compl_le_swap1
thf(fact_847_compl__le__swap1,axiom,
! [Y: set_Ex3793607809372303086nnreal,X2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ Y @ ( uminus5517552291522096439nnreal @ X2 ) )
=> ( ord_le6787938422905777998nnreal @ X2 @ ( uminus5517552291522096439nnreal @ Y ) ) ) ).
% compl_le_swap1
thf(fact_848_compl__le__swap1,axiom,
! [Y: set_real,X2: set_real] :
( ( ord_less_eq_set_real @ Y @ ( uminus612125837232591019t_real @ X2 ) )
=> ( ord_less_eq_set_real @ X2 @ ( uminus612125837232591019t_real @ Y ) ) ) ).
% compl_le_swap1
thf(fact_849_compl__mono,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y ) @ ( uminus_uminus_set_a @ X2 ) ) ) ).
% compl_mono
thf(fact_850_compl__mono,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ X2 @ Y )
=> ( ord_le6787938422905777998nnreal @ ( uminus5517552291522096439nnreal @ Y ) @ ( uminus5517552291522096439nnreal @ X2 ) ) ) ).
% compl_mono
thf(fact_851_compl__mono,axiom,
! [X2: set_real,Y: set_real] :
( ( ord_less_eq_set_real @ X2 @ Y )
=> ( ord_less_eq_set_real @ ( uminus612125837232591019t_real @ Y ) @ ( uminus612125837232591019t_real @ X2 ) ) ) ).
% compl_mono
thf(fact_852_le__imp__neg__le,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_853_minus__le__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A2 ) ) ).
% minus_le_iff
thf(fact_854_le__minus__iff,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A2 ) ) ) ).
% le_minus_iff
thf(fact_855_inf__sup__ord_I2_J,axiom,
! [X2: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_856_inf__sup__ord_I1_J,axiom,
! [X2: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_857_inf__le1,axiom,
! [X2: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_858_inf__le2,axiom,
! [X2: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_859_le__infE,axiom,
! [X2: real,A2: real,B: real] :
( ( ord_less_eq_real @ X2 @ ( inf_inf_real @ A2 @ B ) )
=> ~ ( ( ord_less_eq_real @ X2 @ A2 )
=> ~ ( ord_less_eq_real @ X2 @ B ) ) ) ).
% le_infE
thf(fact_860_le__infI,axiom,
! [X2: real,A2: real,B: real] :
( ( ord_less_eq_real @ X2 @ A2 )
=> ( ( ord_less_eq_real @ X2 @ B )
=> ( ord_less_eq_real @ X2 @ ( inf_inf_real @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_861_inf__mono,axiom,
! [A2: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ A2 @ C )
=> ( ( ord_less_eq_real @ B @ D )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ ( inf_inf_real @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_862_le__infI1,axiom,
! [A2: real,X2: real,B: real] :
( ( ord_less_eq_real @ A2 @ X2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_863_le__infI2,axiom,
! [B: real,X2: real,A2: real] :
( ( ord_less_eq_real @ B @ X2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_864_inf_OorderE,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( A2
= ( inf_inf_real @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_865_inf_OorderI,axiom,
! [A2: real,B: real] :
( ( A2
= ( inf_inf_real @ A2 @ B ) )
=> ( ord_less_eq_real @ A2 @ B ) ) ).
% inf.orderI
thf(fact_866_inf__unique,axiom,
! [F: real > real > real,X2: real,Y: real] :
( ! [X: real,Y2: real] : ( ord_less_eq_real @ ( F @ X @ Y2 ) @ X )
=> ( ! [X: real,Y2: real] : ( ord_less_eq_real @ ( F @ X @ Y2 ) @ Y2 )
=> ( ! [X: real,Y2: real,Z4: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ X @ Z4 )
=> ( ord_less_eq_real @ X @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_real @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_867_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y3: real] :
( ( inf_inf_real @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_868_inf_Oabsorb1,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( inf_inf_real @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_869_inf_Oabsorb2,axiom,
! [B: real,A2: real] :
( ( ord_less_eq_real @ B @ A2 )
=> ( ( inf_inf_real @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_870_inf__absorb1,axiom,
! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( inf_inf_real @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_871_inf__absorb2,axiom,
! [Y: real,X2: real] :
( ( ord_less_eq_real @ Y @ X2 )
=> ( ( inf_inf_real @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_872_inf_OboundedE,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B @ C ) )
=> ~ ( ( ord_less_eq_real @ A2 @ B )
=> ~ ( ord_less_eq_real @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_873_inf_OboundedI,axiom,
! [A2: real,B: real,C: real] :
( ( ord_less_eq_real @ A2 @ B )
=> ( ( ord_less_eq_real @ A2 @ C )
=> ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_874_inf__greatest,axiom,
! [X2: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ( ord_less_eq_real @ X2 @ Z3 )
=> ( ord_less_eq_real @ X2 @ ( inf_inf_real @ Y @ Z3 ) ) ) ) ).
% inf_greatest
thf(fact_875_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] :
( A5
= ( inf_inf_real @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_876_inf_Ocobounded1,axiom,
! [A2: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_877_inf_Ocobounded2,axiom,
! [A2: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_878_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] :
( ( inf_inf_real @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_879_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A5: real] :
( ( inf_inf_real @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_880_inf_OcoboundedI1,axiom,
! [A2: real,C: real,B: real] :
( ( ord_less_eq_real @ A2 @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_881_inf_OcoboundedI2,axiom,
! [B: real,C: real,A2: real] :
( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_882_subset__UNIV,axiom,
! [A: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A @ top_to7994903218803871134nnreal ) ).
% subset_UNIV
thf(fact_883_subset__UNIV,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% subset_UNIV
thf(fact_884_subset__UNIV,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% subset_UNIV
thf(fact_885_subset__insert,axiom,
! [X2: set_a,A: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X2 @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_886_subset__insert,axiom,
! [X2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ~ ( member2919562650594848410nnreal @ X2 @ A )
=> ( ( ord_le2462468573666744473nnreal @ A @ ( insert152533262698245683nnreal @ X2 @ B2 ) )
= ( ord_le2462468573666744473nnreal @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_887_subset__insert,axiom,
! [X2: set_real,A: set_set_real,B2: set_set_real] :
( ~ ( member_set_real @ X2 @ A )
=> ( ( ord_le3558479182127378552t_real @ A @ ( insert_set_real @ X2 @ B2 ) )
= ( ord_le3558479182127378552t_real @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_888_subset__insert,axiom,
! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ~ ( member603777416030116741nnreal @ X2 @ A )
=> ( ( ord_le3366939622266546180nnreal @ A @ ( insert1343806209672318238nnreal @ X2 @ B2 ) )
= ( ord_le3366939622266546180nnreal @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_889_Int__Collect__mono,axiom,
! [A: set_set_a,B2: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B2 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_890_Int__Collect__mono,axiom,
! [A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal,P: ( real > extend8495563244428889912nnreal ) > $o,Q: ( real > extend8495563244428889912nnreal ) > $o] :
( ( ord_le2462468573666744473nnreal @ A @ B2 )
=> ( ! [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le2462468573666744473nnreal @ ( inf_in8454409011496165067nnreal @ A @ ( collec9130413544115709400nnreal @ P ) ) @ ( inf_in8454409011496165067nnreal @ B2 @ ( collec9130413544115709400nnreal @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_891_Int__Collect__mono,axiom,
! [A: set_set_real,B2: set_set_real,P: set_real > $o,Q: set_real > $o] :
( ( ord_le3558479182127378552t_real @ A @ B2 )
=> ( ! [X: set_real] :
( ( member_set_real @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le3558479182127378552t_real @ ( inf_inf_set_set_real @ A @ ( collect_set_real @ P ) ) @ ( inf_inf_set_set_real @ B2 @ ( collect_set_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_892_Int__Collect__mono,axiom,
! [A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal,P: set_Ex3793607809372303086nnreal > $o,Q: set_Ex3793607809372303086nnreal > $o] :
( ( ord_le3366939622266546180nnreal @ A @ B2 )
=> ( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le3366939622266546180nnreal @ ( inf_in5190865051653673526nnreal @ A @ ( collec4858231573021281987nnreal @ P ) ) @ ( inf_in5190865051653673526nnreal @ B2 @ ( collec4858231573021281987nnreal @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_893_open__subopen,axiom,
( topolo3503219976281768444nnreal
= ( ^ [S: set_re5328672808648366137nnreal] :
! [X3: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X3 @ S )
=> ? [T2: set_re5328672808648366137nnreal] :
( ( topolo3503219976281768444nnreal @ T2 )
& ( member2919562650594848410nnreal @ X3 @ T2 )
& ( ord_le2462468573666744473nnreal @ T2 @ S ) ) ) ) ) ).
% open_subopen
thf(fact_894_topological__space__class_OopenI,axiom,
! [S3: set_re5328672808648366137nnreal] :
( ! [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ S3 )
=> ? [T3: set_re5328672808648366137nnreal] :
( ( topolo3503219976281768444nnreal @ T3 )
& ( member2919562650594848410nnreal @ X @ T3 )
& ( ord_le2462468573666744473nnreal @ T3 @ S3 ) ) )
=> ( topolo3503219976281768444nnreal @ S3 ) ) ).
% topological_space_class.openI
thf(fact_895_first__countable__basisE,axiom,
! [X2: real] :
~ ! [A6: set_set_real] :
( ( counta8054315614235329383t_real @ A6 )
=> ( ! [A7: set_real] :
( ( member_set_real @ A7 @ A6 )
=> ( member_real @ X2 @ A7 ) )
=> ( ! [A7: set_real] :
( ( member_set_real @ A7 @ A6 )
=> ( topolo4860482606490270245n_real @ A7 ) )
=> ~ ! [S4: set_real] :
( ( topolo4860482606490270245n_real @ S4 )
=> ( ( member_real @ X2 @ S4 )
=> ? [X: set_real] :
( ( member_set_real @ X @ A6 )
& ( ord_less_eq_set_real @ X @ S4 ) ) ) ) ) ) ) ).
% first_countable_basisE
thf(fact_896_first__countable__basisE,axiom,
! [X2: extend8495563244428889912nnreal] :
~ ! [A6: set_se4580700918925141924nnreal] :
( ( counta2425349316461633011nnreal @ A6 )
=> ( ! [A7: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A7 @ A6 )
=> ( member7908768830364227535nnreal @ X2 @ A7 ) )
=> ( ! [A7: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A7 @ A6 )
=> ( topolo1037242508615874353nnreal @ A7 ) )
=> ~ ! [S4: set_Ex3793607809372303086nnreal] :
( ( topolo1037242508615874353nnreal @ S4 )
=> ( ( member7908768830364227535nnreal @ X2 @ S4 )
=> ? [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ A6 )
& ( ord_le6787938422905777998nnreal @ X @ S4 ) ) ) ) ) ) ) ).
% first_countable_basisE
thf(fact_897_first__countableI,axiom,
! [A8: set_se2490721793304844655nnreal,X2: real > extend8495563244428889912nnreal] :
( ( counta1475001235576645950nnreal @ A8 )
=> ( ! [A9: set_re5328672808648366137nnreal] :
( ( member524040414084610768nnreal @ A9 @ A8 )
=> ( member2919562650594848410nnreal @ X2 @ A9 ) )
=> ( ! [A9: set_re5328672808648366137nnreal] :
( ( member524040414084610768nnreal @ A9 @ A8 )
=> ( topolo3503219976281768444nnreal @ A9 ) )
=> ( ! [S5: set_re5328672808648366137nnreal] :
( ( topolo3503219976281768444nnreal @ S5 )
=> ( ( member2919562650594848410nnreal @ X2 @ S5 )
=> ? [X6: set_re5328672808648366137nnreal] :
( ( member524040414084610768nnreal @ X6 @ A8 )
& ( ord_le2462468573666744473nnreal @ X6 @ S5 ) ) ) )
=> ? [A6: nat > set_re5328672808648366137nnreal] :
( ! [I3: nat] :
( ( member2919562650594848410nnreal @ X2 @ ( A6 @ I3 ) )
& ( topolo3503219976281768444nnreal @ ( A6 @ I3 ) ) )
& ! [S4: set_re5328672808648366137nnreal] :
( ( ( topolo3503219976281768444nnreal @ S4 )
& ( member2919562650594848410nnreal @ X2 @ S4 ) )
=> ? [I4: nat] : ( ord_le2462468573666744473nnreal @ ( A6 @ I4 ) @ S4 ) ) ) ) ) ) ) ).
% first_countableI
thf(fact_898_first__countableI,axiom,
! [A8: set_set_a,X2: a] :
( ( counta6168152590877469849_set_a @ A8 )
=> ( ! [A9: set_a] :
( ( member_set_a @ A9 @ A8 )
=> ( member_a @ X2 @ A9 ) )
=> ( ! [A9: set_a] :
( ( member_set_a @ A9 @ A8 )
=> ( topolo8477419352202985285open_a @ A9 ) )
=> ( ! [S5: set_a] :
( ( topolo8477419352202985285open_a @ S5 )
=> ( ( member_a @ X2 @ S5 )
=> ? [X6: set_a] :
( ( member_set_a @ X6 @ A8 )
& ( ord_less_eq_set_a @ X6 @ S5 ) ) ) )
=> ? [A6: nat > set_a] :
( ! [I3: nat] :
( ( member_a @ X2 @ ( A6 @ I3 ) )
& ( topolo8477419352202985285open_a @ ( A6 @ I3 ) ) )
& ! [S4: set_a] :
( ( ( topolo8477419352202985285open_a @ S4 )
& ( member_a @ X2 @ S4 ) )
=> ? [I4: nat] : ( ord_less_eq_set_a @ ( A6 @ I4 ) @ S4 ) ) ) ) ) ) ) ).
% first_countableI
thf(fact_899_first__countableI,axiom,
! [A8: set_set_real,X2: real] :
( ( counta8054315614235329383t_real @ A8 )
=> ( ! [A9: set_real] :
( ( member_set_real @ A9 @ A8 )
=> ( member_real @ X2 @ A9 ) )
=> ( ! [A9: set_real] :
( ( member_set_real @ A9 @ A8 )
=> ( topolo4860482606490270245n_real @ A9 ) )
=> ( ! [S5: set_real] :
( ( topolo4860482606490270245n_real @ S5 )
=> ( ( member_real @ X2 @ S5 )
=> ? [X6: set_real] :
( ( member_set_real @ X6 @ A8 )
& ( ord_less_eq_set_real @ X6 @ S5 ) ) ) )
=> ? [A6: nat > set_real] :
( ! [I3: nat] :
( ( member_real @ X2 @ ( A6 @ I3 ) )
& ( topolo4860482606490270245n_real @ ( A6 @ I3 ) ) )
& ! [S4: set_real] :
( ( ( topolo4860482606490270245n_real @ S4 )
& ( member_real @ X2 @ S4 ) )
=> ? [I4: nat] : ( ord_less_eq_set_real @ ( A6 @ I4 ) @ S4 ) ) ) ) ) ) ) ).
% first_countableI
thf(fact_900_first__countableI,axiom,
! [A8: set_se4580700918925141924nnreal,X2: extend8495563244428889912nnreal] :
( ( counta2425349316461633011nnreal @ A8 )
=> ( ! [A9: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A9 @ A8 )
=> ( member7908768830364227535nnreal @ X2 @ A9 ) )
=> ( ! [A9: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A9 @ A8 )
=> ( topolo1037242508615874353nnreal @ A9 ) )
=> ( ! [S5: set_Ex3793607809372303086nnreal] :
( ( topolo1037242508615874353nnreal @ S5 )
=> ( ( member7908768830364227535nnreal @ X2 @ S5 )
=> ? [X6: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X6 @ A8 )
& ( ord_le6787938422905777998nnreal @ X6 @ S5 ) ) ) )
=> ? [A6: nat > set_Ex3793607809372303086nnreal] :
( ! [I3: nat] :
( ( member7908768830364227535nnreal @ X2 @ ( A6 @ I3 ) )
& ( topolo1037242508615874353nnreal @ ( A6 @ I3 ) ) )
& ! [S4: set_Ex3793607809372303086nnreal] :
( ( ( topolo1037242508615874353nnreal @ S4 )
& ( member7908768830364227535nnreal @ X2 @ S4 ) )
=> ? [I4: nat] : ( ord_le6787938422905777998nnreal @ ( A6 @ I4 ) @ S4 ) ) ) ) ) ) ) ).
% first_countableI
thf(fact_901_Ioc__inj,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( set_or2392270231875598684t_real @ A2 @ B )
= ( set_or2392270231875598684t_real @ C @ D ) )
= ( ( ( ord_less_eq_real @ B @ A2 )
& ( ord_less_eq_real @ D @ C ) )
| ( ( A2 = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_902_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_903_measurable__mono,axiom,
! [N2: sigma_measure_a,N: sigma_measure_a,M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N2 ) @ ( sigma_sets_a @ N ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ N2 ) )
=> ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M2 ) )
=> ( ( ( sigma_3147302497200244656nnreal @ M )
= ( sigma_3147302497200244656nnreal @ M2 ) )
=> ( ord_le5319533700100273021real_a @ ( sigma_3031480723531659892real_a @ M @ N ) @ ( sigma_3031480723531659892real_a @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_904_measurable__mono,axiom,
! [N2: sigma_measure_a,N: sigma_measure_a,M: sigma_measure_real,M2: sigma_measure_real] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N2 ) @ ( sigma_sets_a @ N ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ N2 ) )
=> ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_space_real @ M )
= ( sigma_space_real @ M2 ) )
=> ( ord_le5743406823621094409real_a @ ( sigma_523072396149930112real_a @ M @ N ) @ ( sigma_523072396149930112real_a @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_905_measurable__mono,axiom,
! [N2: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,M: sigma_measure_a,M2: sigma_measure_a] :
( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N2 ) @ ( sigma_5465916536984168985nnreal @ N ) )
=> ( ( ( sigma_3147302497200244656nnreal @ N )
= ( sigma_3147302497200244656nnreal @ N2 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_space_a @ M )
= ( sigma_space_a @ M2 ) )
=> ( ord_le1007445205377960487nnreal @ ( sigma_214952329563889126nnreal @ M @ N ) @ ( sigma_214952329563889126nnreal @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_906_measurable__mono,axiom,
! [N2: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N2 ) @ ( sigma_5465916536984168985nnreal @ N ) )
=> ( ( ( sigma_3147302497200244656nnreal @ N )
= ( sigma_3147302497200244656nnreal @ N2 ) )
=> ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M2 ) )
=> ( ( ( sigma_3147302497200244656nnreal @ M )
= ( sigma_3147302497200244656nnreal @ M2 ) )
=> ( ord_le2847260637007690789nnreal @ ( sigma_7926153774531450434nnreal @ M @ N ) @ ( sigma_7926153774531450434nnreal @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_907_measurable__mono,axiom,
! [N2: sigma_measure_real,N: sigma_measure_real,M: sigma_measure_a,M2: sigma_measure_a] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N2 ) @ ( sigma_sets_real @ N ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ N2 ) )
=> ( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ M2 ) )
=> ( ( ( sigma_space_a @ M )
= ( sigma_space_a @ M2 ) )
=> ( ord_le3334967407727675675a_real @ ( sigma_9116425665531756122a_real @ M @ N ) @ ( sigma_9116425665531756122a_real @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_908_measurable__mono,axiom,
! [N2: sigma_measure_real,N: sigma_measure_real,M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N2 ) @ ( sigma_sets_real @ N ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ N2 ) )
=> ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M2 ) )
=> ( ( ( sigma_3147302497200244656nnreal @ M )
= ( sigma_3147302497200244656nnreal @ M2 ) )
=> ( ord_le2792513217584188441l_real @ ( sigma_7049758200512112822l_real @ M @ N ) @ ( sigma_7049758200512112822l_real @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_909_measurable__mono,axiom,
! [N2: sigma_measure_real,N: sigma_measure_real,M: sigma_measure_real,M2: sigma_measure_real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N2 ) @ ( sigma_sets_real @ N ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ N2 ) )
=> ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_space_real @ M )
= ( sigma_space_real @ M2 ) )
=> ( ord_le4198349162570665613l_real @ ( sigma_5267869275261027754l_real @ M @ N ) @ ( sigma_5267869275261027754l_real @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_910_measurable__mono,axiom,
! [N2: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,M: sigma_measure_real,M2: sigma_measure_real] :
( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N2 ) @ ( sigma_5465916536984168985nnreal @ N ) )
=> ( ( ( sigma_3147302497200244656nnreal @ N )
= ( sigma_3147302497200244656nnreal @ N2 ) )
=> ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ M2 ) )
=> ( ( ( sigma_space_real @ M )
= ( sigma_space_real @ M2 ) )
=> ( ord_le2462468573666744473nnreal @ ( sigma_9017504469962657078nnreal @ M @ N ) @ ( sigma_9017504469962657078nnreal @ M2 @ N2 ) ) ) ) ) ) ).
% measurable_mono
thf(fact_911_up__ray__def,axiom,
( up_ray_a
= ( ^ [I5: set_a] :
! [X3: a,Y3: a] :
( ( member_a @ X3 @ I5 )
=> ( ( ord_less_eq_a @ X3 @ Y3 )
=> ( member_a @ Y3 @ I5 ) ) ) ) ) ).
% up_ray_def
thf(fact_912_up__ray__def,axiom,
( up_ray4546996785294415186nnreal
= ( ^ [I5: set_Ex3793607809372303086nnreal] :
! [X3: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ I5 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Y3 )
=> ( member7908768830364227535nnreal @ Y3 @ I5 ) ) ) ) ) ).
% up_ray_def
thf(fact_913_up__ray__def,axiom,
( up_ray_real
= ( ^ [I5: set_real] :
! [X3: real,Y3: real] :
( ( member_real @ X3 @ I5 )
=> ( ( ord_less_eq_real @ X3 @ Y3 )
=> ( member_real @ Y3 @ I5 ) ) ) ) ) ).
% up_ray_def
thf(fact_914_first__countable__basis__Int__stableE,axiom,
! [X2: real] :
~ ! [A6: set_set_real] :
( ( counta8054315614235329383t_real @ A6 )
=> ( ! [A7: set_real] :
( ( member_set_real @ A7 @ A6 )
=> ( member_real @ X2 @ A7 ) )
=> ( ! [A7: set_real] :
( ( member_set_real @ A7 @ A6 )
=> ( topolo4860482606490270245n_real @ A7 ) )
=> ( ! [S4: set_real] :
( ( topolo4860482606490270245n_real @ S4 )
=> ( ( member_real @ X2 @ S4 )
=> ? [X: set_real] :
( ( member_set_real @ X @ A6 )
& ( ord_less_eq_set_real @ X @ S4 ) ) ) )
=> ~ ! [A7: set_real] :
( ( member_set_real @ A7 @ A6 )
=> ! [B7: set_real] :
( ( member_set_real @ B7 @ A6 )
=> ( member_set_real @ ( inf_inf_set_real @ A7 @ B7 ) @ A6 ) ) ) ) ) ) ) ).
% first_countable_basis_Int_stableE
thf(fact_915_first__countable__basis__Int__stableE,axiom,
! [X2: extend8495563244428889912nnreal] :
~ ! [A6: set_se4580700918925141924nnreal] :
( ( counta2425349316461633011nnreal @ A6 )
=> ( ! [A7: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A7 @ A6 )
=> ( member7908768830364227535nnreal @ X2 @ A7 ) )
=> ( ! [A7: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A7 @ A6 )
=> ( topolo1037242508615874353nnreal @ A7 ) )
=> ( ! [S4: set_Ex3793607809372303086nnreal] :
( ( topolo1037242508615874353nnreal @ S4 )
=> ( ( member7908768830364227535nnreal @ X2 @ S4 )
=> ? [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ A6 )
& ( ord_le6787938422905777998nnreal @ X @ S4 ) ) ) )
=> ~ ! [A7: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A7 @ A6 )
=> ! [B7: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ B7 @ A6 )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ A7 @ B7 ) @ A6 ) ) ) ) ) ) ) ).
% first_countable_basis_Int_stableE
thf(fact_916_sets_Osets__into__space,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( sigma_sets_a @ M ) )
=> ( ord_less_eq_set_a @ X2 @ ( sigma_space_a @ M ) ) ) ).
% sets.sets_into_space
thf(fact_917_sets_Osets__into__space,axiom,
! [X2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ X2 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ord_le6787938422905777998nnreal @ X2 @ ( sigma_3147302497200244656nnreal @ M ) ) ) ).
% sets.sets_into_space
thf(fact_918_sets_Osets__into__space,axiom,
! [X2: set_real,M: sigma_measure_real] :
( ( member_set_real @ X2 @ ( sigma_sets_real @ M ) )
=> ( ord_less_eq_set_real @ X2 @ ( sigma_space_real @ M ) ) ) ).
% sets.sets_into_space
thf(fact_919_subset__Diff__insert,axiom,
! [A: set_set_a,B2: set_set_a,X2: set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B2 @ ( insert_set_a @ X2 @ C3 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_920_subset__Diff__insert,axiom,
! [A: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal,X2: real > extend8495563244428889912nnreal,C3: set_re5328672808648366137nnreal] :
( ( ord_le2462468573666744473nnreal @ A @ ( minus_3708639258518406418nnreal @ B2 @ ( insert152533262698245683nnreal @ X2 @ C3 ) ) )
= ( ( ord_le2462468573666744473nnreal @ A @ ( minus_3708639258518406418nnreal @ B2 @ C3 ) )
& ~ ( member2919562650594848410nnreal @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_921_subset__Diff__insert,axiom,
! [A: set_set_real,B2: set_set_real,X2: set_real,C3: set_set_real] :
( ( ord_le3558479182127378552t_real @ A @ ( minus_5467046032205032049t_real @ B2 @ ( insert_set_real @ X2 @ C3 ) ) )
= ( ( ord_le3558479182127378552t_real @ A @ ( minus_5467046032205032049t_real @ B2 @ C3 ) )
& ~ ( member_set_real @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_922_subset__Diff__insert,axiom,
! [A: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal,X2: set_Ex3793607809372303086nnreal,C3: set_se4580700918925141924nnreal] :
( ( ord_le3366939622266546180nnreal @ A @ ( minus_5908140721592501885nnreal @ B2 @ ( insert1343806209672318238nnreal @ X2 @ C3 ) ) )
= ( ( ord_le3366939622266546180nnreal @ A @ ( minus_5908140721592501885nnreal @ B2 @ C3 ) )
& ~ ( member603777416030116741nnreal @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_923_subset__Compl__self__eq,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ A ) )
= ( A = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_924_subset__Compl__self__eq,axiom,
! [A: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ ( uminus5517552291522096439nnreal @ A ) )
= ( A = bot_bo4854962954004695426nnreal ) ) ).
% subset_Compl_self_eq
thf(fact_925_subset__Compl__self__eq,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ ( uminus612125837232591019t_real @ A ) )
= ( A = bot_bot_set_real ) ) ).
% subset_Compl_self_eq
thf(fact_926_null__sets__subset,axiom,
! [B2: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ B2 @ ( measure_null_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( member_set_a @ A @ ( measure_null_sets_a @ M ) ) ) ) ) ).
% null_sets_subset
thf(fact_927_null__sets__subset,axiom,
! [B2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ B2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( ord_le6787938422905777998nnreal @ A @ B2 )
=> ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ M ) ) ) ) ) ).
% null_sets_subset
thf(fact_928_null__sets__subset,axiom,
! [B2: set_real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ B2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( ( ord_less_eq_set_real @ A @ B2 )
=> ( member_set_real @ A @ ( measur3710062792471635001s_real @ M ) ) ) ) ) ).
% null_sets_subset
thf(fact_929_null__sets_Osets__into__space,axiom,
! [X2: set_a,M: sigma_measure_a] :
( ( member_set_a @ X2 @ ( measure_null_sets_a @ M ) )
=> ( ord_less_eq_set_a @ X2 @ ( sigma_space_a @ M ) ) ) ).
% null_sets.sets_into_space
thf(fact_930_null__sets_Osets__into__space,axiom,
! [X2: set_real,M: sigma_measure_real] :
( ( member_set_real @ X2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ord_less_eq_set_real @ X2 @ ( sigma_space_real @ M ) ) ) ).
% null_sets.sets_into_space
thf(fact_931_null__sets_Osets__into__space,axiom,
! [X2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ X2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ord_le6787938422905777998nnreal @ X2 @ ( sigma_3147302497200244656nnreal @ M ) ) ) ).
% null_sets.sets_into_space
thf(fact_932_mono__restrict__space,axiom,
! [M: sigma_measure_a,N: sigma_measure_a,X5: set_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ N ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ X5 ) ) @ ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ N @ X5 ) ) ) ) ).
% mono_restrict_space
thf(fact_933_mono__restrict__space,axiom,
! [M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,X5: set_Ex3793607809372303086nnreal] :
( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ N ) )
=> ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ M @ X5 ) ) @ ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ N @ X5 ) ) ) ) ).
% mono_restrict_space
thf(fact_934_mono__restrict__space,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,X5: set_real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ N ) )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ X5 ) ) @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ N @ X5 ) ) ) ) ).
% mono_restrict_space
thf(fact_935_measurable__restrict__mono,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,A: set_real,N: sigma_7234349610311085201nnreal,B2: set_real] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( sigma_5414646170262037096e_real @ M @ A ) @ N ) )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( sigma_5414646170262037096e_real @ M @ B2 ) @ N ) ) ) ) ).
% measurable_restrict_mono
thf(fact_936_completion_Ocomplete2,axiom,
! [A: set_a,B2: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_set_a @ B2 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ A @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ).
% completion.complete2
thf(fact_937_completion_Ocomplete2,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( ord_le6787938422905777998nnreal @ A @ B2 )
=> ( ( member603777416030116741nnreal @ B2 @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
=> ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ).
% completion.complete2
thf(fact_938_completion_Ocomplete2,axiom,
! [A: set_real,B2: set_real,M: sigma_measure_real] :
( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( member_set_real @ B2 @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ( member_set_real @ A @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ).
% completion.complete2
thf(fact_939_null__sets__completion__iff2,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ ( measure_null_sets_a @ M ) )
& ( ord_less_eq_set_a @ A @ X3 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_940_null__sets__completion__iff2,axiom,
! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
= ( ? [X3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X3 @ ( measur1209175464439008069nnreal @ M ) )
& ( ord_le6787938422905777998nnreal @ A @ X3 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_941_null__sets__completion__iff2,axiom,
! [A: set_real,M: sigma_measure_real] :
( ( member_set_real @ A @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) )
= ( ? [X3: set_real] :
( ( member_set_real @ X3 @ ( measur3710062792471635001s_real @ M ) )
& ( ord_less_eq_set_real @ A @ X3 ) ) ) ) ).
% null_sets_completion_iff2
thf(fact_942_null__sets__completion__subset,axiom,
! [B2: set_a,A: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ B2 @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ).
% null_sets_completion_subset
thf(fact_943_null__sets__completion__subset,axiom,
! [B2: set_Ex3793607809372303086nnreal,A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( ord_le6787938422905777998nnreal @ B2 @ A )
=> ( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
=> ( member603777416030116741nnreal @ B2 @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ).
% null_sets_completion_subset
thf(fact_944_null__sets__completion__subset,axiom,
! [B2: set_real,A: set_real,M: sigma_measure_real] :
( ( ord_less_eq_set_real @ B2 @ A )
=> ( ( member_set_real @ A @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ( member_set_real @ B2 @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ).
% null_sets_completion_subset
thf(fact_945_down__ray__def,axiom,
( down_ray_a
= ( ^ [I5: set_a] :
! [X3: a,Y3: a] :
( ( member_a @ Y3 @ I5 )
=> ( ( ord_less_eq_a @ X3 @ Y3 )
=> ( member_a @ X3 @ I5 ) ) ) ) ) ).
% down_ray_def
thf(fact_946_down__ray__def,axiom,
( down_ray_real
= ( ^ [I5: set_real] :
! [X3: real,Y3: real] :
( ( member_real @ Y3 @ I5 )
=> ( ( ord_less_eq_real @ X3 @ Y3 )
=> ( member_real @ X3 @ I5 ) ) ) ) ) ).
% down_ray_def
thf(fact_947_measurable__sets__borel,axiom,
! [F: a > a,M: sigma_measure_a,A: set_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ borel_5459123734250506524orel_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( vimage_a_a @ F @ A ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ) ).
% measurable_sets_borel
thf(fact_948_measurable__sets__borel,axiom,
! [F: a > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ borel_5459123734250506524orel_a @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member_set_a @ ( vimage1258658873539170235nnreal @ F @ A ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ) ).
% measurable_sets_borel
thf(fact_949_measurable__sets__borel,axiom,
! [F: a > real,M: sigma_measure_real,A: set_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ borel_5459123734250506524orel_a @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_a @ ( vimage_a_real @ F @ A ) @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ) ) ) ).
% measurable_sets_borel
thf(fact_950_measurable__sets__borel,axiom,
! [F: real > a,M: sigma_measure_a,A: set_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_real @ ( vimage_real_a @ F @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).
% measurable_sets_borel
thf(fact_951_measurable__sets__borel,axiom,
! [F: real > real,M: sigma_measure_real,A: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( vimage_real_real @ F @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).
% measurable_sets_borel
thf(fact_952_measurable__sets__borel,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member_set_real @ ( vimage6366802093293386401nnreal @ F @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).
% measurable_sets_borel
thf(fact_953_measurable__sets__borel,axiom,
! [F: extend8495563244428889912nnreal > a,M: sigma_measure_a,A: set_a] :
( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ borel_6524799422816628122nnreal @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member603777416030116741nnreal @ ( vimage4075187267506941001real_a @ F @ A ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ) ).
% measurable_sets_borel
thf(fact_954_measurable__sets__borel,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ borel_6524799422816628122nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( vimage3650734033530794285nnreal @ F @ A ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ) ).
% measurable_sets_borel
thf(fact_955_measurable__sets__borel,axiom,
! [F: extend8495563244428889912nnreal > real,M: sigma_measure_real,A: set_real] :
( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member603777416030116741nnreal @ ( vimage4399055823842842145l_real @ F @ A ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ) ).
% measurable_sets_borel
thf(fact_956_inf__shunt,axiom,
! [X2: set_a,Y: set_a] :
( ( ( inf_inf_set_a @ X2 @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ ( uminus_uminus_set_a @ Y ) ) ) ).
% inf_shunt
thf(fact_957_inf__shunt,axiom,
! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ X2 @ Y )
= bot_bo4854962954004695426nnreal )
= ( ord_le6787938422905777998nnreal @ X2 @ ( uminus5517552291522096439nnreal @ Y ) ) ) ).
% inf_shunt
thf(fact_958_inf__shunt,axiom,
! [X2: set_real,Y: set_real] :
( ( ( inf_inf_set_real @ X2 @ Y )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ X2 @ ( uminus612125837232591019t_real @ Y ) ) ) ).
% inf_shunt
thf(fact_959_subset__insert__iff,axiom,
! [A: set_set_a,X2: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X2 @ B2 ) )
= ( ( ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X2 @ A )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_960_subset__insert__iff,axiom,
! [A: set_re5328672808648366137nnreal,X2: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal] :
( ( ord_le2462468573666744473nnreal @ A @ ( insert152533262698245683nnreal @ X2 @ B2 ) )
= ( ( ( member2919562650594848410nnreal @ X2 @ A )
=> ( ord_le2462468573666744473nnreal @ ( minus_3708639258518406418nnreal @ A @ ( insert152533262698245683nnreal @ X2 @ bot_bo6037503491064675021nnreal ) ) @ B2 ) )
& ( ~ ( member2919562650594848410nnreal @ X2 @ A )
=> ( ord_le2462468573666744473nnreal @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_961_subset__insert__iff,axiom,
! [A: set_set_real,X2: set_real,B2: set_set_real] :
( ( ord_le3558479182127378552t_real @ A @ ( insert_set_real @ X2 @ B2 ) )
= ( ( ( member_set_real @ X2 @ A )
=> ( ord_le3558479182127378552t_real @ ( minus_5467046032205032049t_real @ A @ ( insert_set_real @ X2 @ bot_bot_set_set_real ) ) @ B2 ) )
& ( ~ ( member_set_real @ X2 @ A )
=> ( ord_le3558479182127378552t_real @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_962_subset__insert__iff,axiom,
! [A: set_se4580700918925141924nnreal,X2: set_Ex3793607809372303086nnreal,B2: set_se4580700918925141924nnreal] :
( ( ord_le3366939622266546180nnreal @ A @ ( insert1343806209672318238nnreal @ X2 @ B2 ) )
= ( ( ( member603777416030116741nnreal @ X2 @ A )
=> ( ord_le3366939622266546180nnreal @ ( minus_5908140721592501885nnreal @ A @ ( insert1343806209672318238nnreal @ X2 @ bot_bo2988155216863113784nnreal ) ) @ B2 ) )
& ( ~ ( member603777416030116741nnreal @ X2 @ A )
=> ( ord_le3366939622266546180nnreal @ A @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_963_Ioc__disjoint,axiom,
! [A2: real,B: real,C: real,D: real] :
( ( ( inf_inf_set_real @ ( set_or2392270231875598684t_real @ A2 @ B ) @ ( set_or2392270231875598684t_real @ C @ D ) )
= bot_bot_set_real )
= ( ( ord_less_eq_real @ B @ A2 )
| ( ord_less_eq_real @ D @ C )
| ( ord_less_eq_real @ B @ C )
| ( ord_less_eq_real @ D @ A2 ) ) ) ).
% Ioc_disjoint
thf(fact_964_disjoint__eq__subset__Compl,axiom,
! [A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ ( uminus_uminus_set_a @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_965_disjoint__eq__subset__Compl,axiom,
! [A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ( inf_in3368558534146122112nnreal @ A @ B2 )
= bot_bo4854962954004695426nnreal )
= ( ord_le6787938422905777998nnreal @ A @ ( uminus5517552291522096439nnreal @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_966_disjoint__eq__subset__Compl,axiom,
! [A: set_real,B2: set_real] :
( ( ( inf_inf_set_real @ A @ B2 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A @ ( uminus612125837232591019t_real @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_967_completion_Ocomplete,axiom,
! [B2: set_a,A: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( member_set_a @ B2 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ).
% completion.complete
thf(fact_968_completion_Ocomplete,axiom,
! [B2: set_Ex3793607809372303086nnreal,A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( ord_le6787938422905777998nnreal @ B2 @ A )
=> ( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
=> ( member603777416030116741nnreal @ B2 @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ).
% completion.complete
thf(fact_969_completion_Ocomplete,axiom,
! [B2: set_real,A: set_real,M: sigma_measure_real] :
( ( ord_less_eq_set_real @ B2 @ A )
=> ( ( member_set_real @ A @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ( member_set_real @ B2 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ).
% completion.complete
thf(fact_970_sets__completionI__sub,axiom,
! [N2: set_a,M: sigma_measure_a,N: set_a] :
( ( member_set_a @ N2 @ ( measure_null_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ N @ N2 )
=> ( member_set_a @ N @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ).
% sets_completionI_sub
thf(fact_971_sets__completionI__sub,axiom,
! [N2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,N: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ N2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( ord_le6787938422905777998nnreal @ N @ N2 )
=> ( member603777416030116741nnreal @ N @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ).
% sets_completionI_sub
thf(fact_972_sets__completionI__sub,axiom,
! [N2: set_real,M: sigma_measure_real,N: set_real] :
( ( member_set_real @ N2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( ord_less_eq_set_real @ N @ N2 )
=> ( member_set_real @ N @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ).
% sets_completionI_sub
thf(fact_973_sets__restrict__space__subset,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ ( comple3428971583294703880tion_a @ M ) @ S3 ) ) @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ).
% sets_restrict_space_subset
thf(fact_974_sets__restrict__space__subset,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
=> ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ ( comple6668017395272084142nnreal @ M ) @ S3 ) ) @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ).
% sets_restrict_space_subset
thf(fact_975_sets__restrict__space__subset,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ ( comple3506806835435775778n_real @ M ) @ S3 ) ) @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ).
% sets_restrict_space_subset
thf(fact_976_null__sets__restrict__space,axiom,
! [Omega: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ Omega @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ ( sigma_8692839461743104066pace_a @ M @ Omega ) ) )
= ( ( ord_less_eq_set_a @ A @ Omega )
& ( member_set_a @ A @ ( measure_null_sets_a @ M ) ) ) ) ) ).
% null_sets_restrict_space
thf(fact_977_null__sets__restrict__space,axiom,
! [Omega: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ Omega @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ ( sigma_4884701650823297268nnreal @ M @ Omega ) ) )
= ( ( ord_le6787938422905777998nnreal @ A @ Omega )
& ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ M ) ) ) ) ) ).
% null_sets_restrict_space
thf(fact_978_null__sets__restrict__space,axiom,
! [Omega: set_real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ Omega @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ A @ ( measur3710062792471635001s_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) ) )
= ( ( ord_less_eq_set_real @ A @ Omega )
& ( member_set_real @ A @ ( measur3710062792471635001s_real @ M ) ) ) ) ) ).
% null_sets_restrict_space
thf(fact_979_measurable__sets,axiom,
! [F: a > a,M: sigma_measure_a,A: sigma_measure_a,S3: set_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ A ) )
=> ( ( member_set_a @ S3 @ ( sigma_sets_a @ A ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ S3 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ).
% measurable_sets
thf(fact_980_measurable__sets,axiom,
! [F: extend8495563244428889912nnreal > a,M: sigma_7234349610311085201nnreal,A: sigma_measure_a,S3: set_a] :
( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ M @ A ) )
=> ( ( member_set_a @ S3 @ ( sigma_sets_a @ A ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ S3 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).
% measurable_sets
thf(fact_981_measurable__sets,axiom,
! [F: real > a,M: sigma_measure_real,A: sigma_measure_a,S3: set_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ A ) )
=> ( ( member_set_a @ S3 @ ( sigma_sets_a @ A ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ S3 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ).
% measurable_sets
thf(fact_982_measurable__sets,axiom,
! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,A: sigma_7234349610311085201nnreal,S3: set_Ex3793607809372303086nnreal] :
( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ A ) )
=> ( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ A ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ S3 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ).
% measurable_sets
thf(fact_983_measurable__sets,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A: sigma_7234349610311085201nnreal,S3: set_Ex3793607809372303086nnreal] :
( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ A ) )
=> ( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ A ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ S3 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).
% measurable_sets
thf(fact_984_measurable__sets,axiom,
! [F: a > real,M: sigma_measure_a,A: sigma_measure_real,S3: set_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ A ) )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ A ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ S3 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ).
% measurable_sets
thf(fact_985_measurable__sets,axiom,
! [F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,A: sigma_measure_real,S3: set_real] :
( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ A ) )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ A ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ S3 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).
% measurable_sets
thf(fact_986_measurable__sets,axiom,
! [F: real > real,M: sigma_measure_real,A: sigma_measure_real,S3: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ A ) )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ A ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ S3 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ).
% measurable_sets
thf(fact_987_measurable__sets,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,A: sigma_7234349610311085201nnreal,S3: set_Ex3793607809372303086nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ A ) )
=> ( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ A ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ S3 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ).
% measurable_sets
thf(fact_988_measurableI,axiom,
! [M: sigma_measure_a,F: a > a,N: sigma_measure_a] :
( ! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( member_a @ ( F @ X ) @ ( sigma_space_a @ N ) ) )
=> ( ! [A9: set_a] :
( ( member_set_a @ A9 @ ( sigma_sets_a @ N ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ A9 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_989_measurableI,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > a,N: sigma_measure_a] :
( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
=> ( member_a @ ( F @ X ) @ ( sigma_space_a @ N ) ) )
=> ( ! [A9: set_a] :
( ( member_set_a @ A9 @ ( sigma_sets_a @ N ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ A9 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_990_measurableI,axiom,
! [M: sigma_measure_real,F: real > a,N: sigma_measure_a] :
( ! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( member_a @ ( F @ X ) @ ( sigma_space_a @ N ) ) )
=> ( ! [A9: set_a] :
( ( member_set_a @ A9 @ ( sigma_sets_a @ N ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ A9 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_991_measurableI,axiom,
! [M: sigma_measure_a,F: a > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal] :
( ! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( member7908768830364227535nnreal @ ( F @ X ) @ ( sigma_3147302497200244656nnreal @ N ) ) )
=> ( ! [A9: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A9 @ ( sigma_5465916536984168985nnreal @ N ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ A9 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) ) ) ) ).
% measurableI
thf(fact_992_measurableI,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal] :
( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
=> ( member7908768830364227535nnreal @ ( F @ X ) @ ( sigma_3147302497200244656nnreal @ N ) ) )
=> ( ! [A9: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A9 @ ( sigma_5465916536984168985nnreal @ N ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ A9 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ N ) ) ) ) ).
% measurableI
thf(fact_993_measurableI,axiom,
! [M: sigma_measure_a,F: a > real,N: sigma_measure_real] :
( ! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( member_real @ ( F @ X ) @ ( sigma_space_real @ N ) ) )
=> ( ! [A9: set_real] :
( ( member_set_real @ A9 @ ( sigma_sets_real @ N ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ A9 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) ) ) ) ).
% measurableI
thf(fact_994_measurableI,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real,N: sigma_measure_real] :
( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
=> ( member_real @ ( F @ X ) @ ( sigma_space_real @ N ) ) )
=> ( ! [A9: set_real] :
( ( member_set_real @ A9 @ ( sigma_sets_real @ N ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ A9 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ N ) ) ) ) ).
% measurableI
thf(fact_995_measurableI,axiom,
! [M: sigma_measure_real,F: real > real,N: sigma_measure_real] :
( ! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( member_real @ ( F @ X ) @ ( sigma_space_real @ N ) ) )
=> ( ! [A9: set_real] :
( ( member_set_real @ A9 @ ( sigma_sets_real @ N ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ A9 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) ) ) ) ).
% measurableI
thf(fact_996_measurableI,axiom,
! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal] :
( ! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( member7908768830364227535nnreal @ ( F @ X ) @ ( sigma_3147302497200244656nnreal @ N ) ) )
=> ( ! [A9: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A9 @ ( sigma_5465916536984168985nnreal @ N ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ A9 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) ) ) ) ).
% measurableI
thf(fact_997_measurableI,axiom,
! [M: sigma_measure_a,F: a > set_a,N: sigma_measure_set_a] :
( ! [X: a] :
( ( member_a @ X @ ( sigma_space_a @ M ) )
=> ( member_set_a @ ( F @ X ) @ ( sigma_space_set_a @ N ) ) )
=> ( ! [A9: set_set_a] :
( ( member_set_set_a @ A9 @ ( sigma_sets_set_a @ N ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_set_a @ F @ A9 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member_a_set_a @ F @ ( sigma_3685133166752798000_set_a @ M @ N ) ) ) ) ).
% measurableI
thf(fact_998_vimage__sets__compl__iff,axiom,
! [F: a > a,A: set_a,M: sigma_measure_a] :
( ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ A ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
= ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( uminus_uminus_set_a @ A ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ).
% vimage_sets_compl_iff
thf(fact_999_vimage__sets__compl__iff,axiom,
! [F: extend8495563244428889912nnreal > a,A: set_a,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ A ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) )
= ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ ( uminus_uminus_set_a @ A ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% vimage_sets_compl_iff
thf(fact_1000_vimage__sets__compl__iff,axiom,
! [F: real > a,A: set_a,M: sigma_measure_real] :
( ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ A ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
= ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ ( uminus_uminus_set_a @ A ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ).
% vimage_sets_compl_iff
thf(fact_1001_vimage__sets__compl__iff,axiom,
! [F: a > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,M: sigma_measure_a] :
( ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ A ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
= ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ ( uminus5517552291522096439nnreal @ A ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ).
% vimage_sets_compl_iff
thf(fact_1002_vimage__sets__compl__iff,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ A ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) )
= ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ ( uminus5517552291522096439nnreal @ A ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% vimage_sets_compl_iff
thf(fact_1003_vimage__sets__compl__iff,axiom,
! [F: real > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,M: sigma_measure_real] :
( ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ A ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
= ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ ( uminus5517552291522096439nnreal @ A ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ).
% vimage_sets_compl_iff
thf(fact_1004_vimage__sets__compl__iff,axiom,
! [F: a > real,A: set_real,M: sigma_measure_a] :
( ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ A ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
= ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ ( uminus612125837232591019t_real @ A ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ).
% vimage_sets_compl_iff
thf(fact_1005_vimage__sets__compl__iff,axiom,
! [F: extend8495563244428889912nnreal > real,A: set_real,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ A ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) )
= ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ ( uminus612125837232591019t_real @ A ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% vimage_sets_compl_iff
thf(fact_1006_vimage__sets__compl__iff,axiom,
! [F: real > real,A: set_real,M: sigma_measure_real] :
( ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ A ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
= ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ ( uminus612125837232591019t_real @ A ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ).
% vimage_sets_compl_iff
thf(fact_1007_borel__measurable__subalgebra,axiom,
! [N: sigma_measure_a,M: sigma_measure_a,F: a > a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N ) @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ M ) )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ N @ borel_5459123734250506524orel_a ) )
=> ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ borel_5459123734250506524orel_a ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_1008_borel__measurable__subalgebra,axiom,
! [N: sigma_7234349610311085201nnreal,M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > a] :
( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N ) @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( ( sigma_3147302497200244656nnreal @ N )
= ( sigma_3147302497200244656nnreal @ M ) )
=> ( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ N @ borel_5459123734250506524orel_a ) )
=> ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ M @ borel_5459123734250506524orel_a ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_1009_borel__measurable__subalgebra,axiom,
! [N: sigma_measure_real,M: sigma_measure_real,F: real > a] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ M ) )
=> ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ N @ borel_5459123734250506524orel_a ) )
=> ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_1010_borel__measurable__subalgebra,axiom,
! [N: sigma_measure_a,M: sigma_measure_a,F: a > real] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N ) @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ M ) )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ N @ borel_5078946678739801102l_real ) )
=> ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_1011_borel__measurable__subalgebra,axiom,
! [N: sigma_7234349610311085201nnreal,M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N ) @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( ( sigma_3147302497200244656nnreal @ N )
= ( sigma_3147302497200244656nnreal @ M ) )
=> ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ N @ borel_5078946678739801102l_real ) )
=> ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_1012_borel__measurable__subalgebra,axiom,
! [N: sigma_measure_real,M: sigma_measure_real,F: real > real] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ M ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_1013_borel__measurable__subalgebra,axiom,
! [N: sigma_measure_a,M: sigma_measure_a,F: a > extend8495563244428889912nnreal] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ N ) @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_space_a @ N )
= ( sigma_space_a @ M ) )
=> ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ N @ borel_6524799422816628122nnreal ) )
=> ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_1014_borel__measurable__subalgebra,axiom,
! [N: sigma_7234349610311085201nnreal,M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N ) @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( ( sigma_3147302497200244656nnreal @ N )
= ( sigma_3147302497200244656nnreal @ M ) )
=> ( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ N @ borel_6524799422816628122nnreal ) )
=> ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_1015_borel__measurable__subalgebra,axiom,
! [N: sigma_measure_real,M: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
=> ( ( ( sigma_space_real @ N )
= ( sigma_space_real @ M ) )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ N @ borel_6524799422816628122nnreal ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ) ).
% borel_measurable_subalgebra
thf(fact_1016_sets__restrict__space__iff,axiom,
! [Omega: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( inf_inf_set_a @ Omega @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ Omega ) ) )
= ( ( ord_less_eq_set_a @ A @ Omega )
& ( member_set_a @ A @ ( sigma_sets_a @ M ) ) ) ) ) ).
% sets_restrict_space_iff
thf(fact_1017_sets__restrict__space__iff,axiom,
! [Omega: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ Omega @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( sigma_4884701650823297268nnreal @ M @ Omega ) ) )
= ( ( ord_le6787938422905777998nnreal @ A @ Omega )
& ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ) ).
% sets_restrict_space_iff
thf(fact_1018_sets__restrict__space__iff,axiom,
! [Omega: set_real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ ( inf_inf_set_real @ Omega @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) ) )
= ( ( ord_less_eq_set_real @ A @ Omega )
& ( member_set_real @ A @ ( sigma_sets_real @ M ) ) ) ) ) ).
% sets_restrict_space_iff
thf(fact_1019_in__borel__measurable__borel,axiom,
! [F: a > a,M: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ borel_5459123734250506524orel_a ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ X3 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_1020_in__borel__measurable__borel,axiom,
! [F: extend8495563244428889912nnreal > a,M: sigma_7234349610311085201nnreal] :
( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ M @ borel_5459123734250506524orel_a ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ X3 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_1021_in__borel__measurable__borel,axiom,
! [F: real > a,M: sigma_measure_real] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ X3 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_1022_in__borel__measurable__borel,axiom,
! [F: a > real,M: sigma_measure_a] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
= ( ! [X3: set_real] :
( ( member_set_real @ X3 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ X3 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_1023_in__borel__measurable__borel,axiom,
! [F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal] :
( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ borel_5078946678739801102l_real ) )
= ( ! [X3: set_real] :
( ( member_set_real @ X3 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ X3 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_1024_in__borel__measurable__borel,axiom,
! [F: real > real,M: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
= ( ! [X3: set_real] :
( ( member_set_real @ X3 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ X3 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_1025_in__borel__measurable__borel,axiom,
! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a] :
( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) )
= ( ! [X3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X3 @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ X3 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_1026_in__borel__measurable__borel,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal] :
( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ borel_6524799422816628122nnreal ) )
= ( ! [X3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X3 @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ X3 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_1027_in__borel__measurable__borel,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
= ( ! [X3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X3 @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ X3 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_1028_null__part,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ? [N3: set_a] :
( ( member_set_a @ N3 @ ( measure_null_sets_a @ M ) )
& ( ord_less_eq_set_a @ ( complete_null_part_a @ M @ S3 ) @ N3 ) ) ) ).
% null_part
thf(fact_1029_null__part,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
=> ? [N3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ N3 @ ( measur1209175464439008069nnreal @ M ) )
& ( ord_le6787938422905777998nnreal @ ( comple6358047150840085292nnreal @ M @ S3 ) @ N3 ) ) ) ).
% null_part
thf(fact_1030_null__part,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ? [N3: set_real] :
( ( member_set_real @ N3 @ ( measur3710062792471635001s_real @ M ) )
& ( ord_less_eq_set_real @ ( comple4917500974405109920t_real @ M @ S3 ) @ N3 ) ) ) ).
% null_part
thf(fact_1031_borel__measurableI,axiom,
! [F: a > a,M: sigma_measure_a] :
( ! [S5: set_a] :
( ( topolo8477419352202985285open_a @ S5 )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ S5 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ borel_5459123734250506524orel_a ) ) ) ).
% borel_measurableI
thf(fact_1032_borel__measurableI,axiom,
! [F: extend8495563244428889912nnreal > a,M: sigma_7234349610311085201nnreal] :
( ! [S5: set_a] :
( ( topolo8477419352202985285open_a @ S5 )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ S5 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ M @ borel_5459123734250506524orel_a ) ) ) ).
% borel_measurableI
thf(fact_1033_borel__measurableI,axiom,
! [F: real > a,M: sigma_measure_real] :
( ! [S5: set_a] :
( ( topolo8477419352202985285open_a @ S5 )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ S5 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) ) ) ).
% borel_measurableI
thf(fact_1034_borel__measurableI,axiom,
! [F: a > real,M: sigma_measure_a] :
( ! [S5: set_real] :
( ( topolo4860482606490270245n_real @ S5 )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ S5 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurableI
thf(fact_1035_borel__measurableI,axiom,
! [F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal] :
( ! [S5: set_real] :
( ( topolo4860482606490270245n_real @ S5 )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ S5 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurableI
thf(fact_1036_borel__measurableI,axiom,
! [F: real > real,M: sigma_measure_real] :
( ! [S5: set_real] :
( ( topolo4860482606490270245n_real @ S5 )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ S5 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurableI
thf(fact_1037_borel__measurableI,axiom,
! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a] :
( ! [S5: set_Ex3793607809372303086nnreal] :
( ( topolo1037242508615874353nnreal @ S5 )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ S5 ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) )
=> ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).
% borel_measurableI
thf(fact_1038_borel__measurableI,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal] :
( ! [S5: set_Ex3793607809372303086nnreal] :
( ( topolo1037242508615874353nnreal @ S5 )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ S5 ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
=> ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).
% borel_measurableI
thf(fact_1039_borel__measurableI,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real] :
( ! [S5: set_Ex3793607809372303086nnreal] :
( ( topolo1037242508615874353nnreal @ S5 )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ S5 ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).
% borel_measurableI
thf(fact_1040_increasingD,axiom,
! [M: set_set_a,F: set_a > real,X2: set_a,Y: set_a] :
( ( measur1776380161843274167a_real @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y @ M )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_1041_increasingD,axiom,
! [M: set_set_real,F: set_real > real,X2: set_real,Y: set_real] :
( ( measur4480787322886042509l_real @ M @ F )
=> ( ( ord_less_eq_set_real @ X2 @ Y )
=> ( ( member_set_real @ X2 @ M )
=> ( ( member_set_real @ Y @ M )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_1042_increasingD,axiom,
! [M: set_se4580700918925141924nnreal,F: set_Ex3793607809372303086nnreal > real,X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
( ( measur2890506508110839833l_real @ M @ F )
=> ( ( ord_le6787938422905777998nnreal @ X2 @ Y )
=> ( ( member603777416030116741nnreal @ X2 @ M )
=> ( ( member603777416030116741nnreal @ Y @ M )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).
% increasingD
thf(fact_1043_interval__def,axiom,
( interval_real
= ( ^ [I5: set_real] :
! [X3: real,Y3: real,Z2: real] :
( ( member_real @ X3 @ I5 )
=> ( ( member_real @ Z2 @ I5 )
=> ( ( ord_less_eq_real @ X3 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ( member_real @ Y3 @ I5 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_1044_completion_Ovimage__null__part__null__sets,axiom,
! [F: a > a,M: sigma_measure_a,N: sigma_measure_a,A: set_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ ( comple3428971583294703880tion_a @ M ) @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_a_a @ ( comple3428971583294703880tion_a @ M ) @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_space_a @ ( comple3428971583294703880tion_a @ M ) ) ) @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_null_sets
thf(fact_1045_completion_Ovimage__null__part__null__sets,axiom,
! [F: extend8495563244428889912nnreal > a,M: sigma_7234349610311085201nnreal,N: sigma_measure_a,A: set_a] :
( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ ( comple6668017395272084142nnreal @ M ) @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measur7655964997769656268real_a @ ( comple6668017395272084142nnreal @ M ) @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_null_sets
thf(fact_1046_completion_Ovimage__null__part__null__sets,axiom,
! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ ( comple3428971583294703880tion_a @ M ) @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur4839436603801885502nnreal @ ( comple3428971583294703880tion_a @ M ) @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_space_a @ ( comple3428971583294703880tion_a @ M ) ) ) @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_null_sets
thf(fact_1047_completion_Ovimage__null__part__null__sets,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ ( comple6668017395272084142nnreal @ M ) @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur2549461466288632554nnreal @ ( comple6668017395272084142nnreal @ M ) @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_null_sets
thf(fact_1048_completion_Ovimage__null__part__null__sets,axiom,
! [F: a > real,M: sigma_measure_a,N: sigma_measure_real,A: set_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( comple3428971583294703880tion_a @ M ) @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measure_distr_a_real @ ( comple3428971583294703880tion_a @ M ) @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_space_a @ ( comple3428971583294703880tion_a @ M ) ) ) @ ( measure_null_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_null_sets
thf(fact_1049_completion_Ovimage__null__part__null__sets,axiom,
! [F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,N: sigma_measure_real,A: set_real] :
( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ ( comple6668017395272084142nnreal @ M ) @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur6862244029252366686l_real @ ( comple6668017395272084142nnreal @ M ) @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) @ ( measur1209175464439008069nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_null_sets
thf(fact_1050_completion_Ovimage__null__part__null__sets,axiom,
! [F: real > a,M: sigma_measure_real,N: sigma_measure_a,A: set_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_real_a @ ( comple3506806835435775778n_real @ M ) @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_space_real @ ( comple3506806835435775778n_real @ M ) ) ) @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_null_sets
thf(fact_1051_completion_Ovimage__null__part__null__sets,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur8829990298702910942nnreal @ ( comple3506806835435775778n_real @ M ) @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_space_real @ ( comple3506806835435775778n_real @ M ) ) ) @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_null_sets
thf(fact_1052_completion_Ovimage__null__part__null__sets,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,A: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur2993149975067245138l_real @ ( comple3506806835435775778n_real @ M ) @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_space_real @ ( comple3506806835435775778n_real @ M ) ) ) @ ( measur3710062792471635001s_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_null_sets
thf(fact_1053_completion_Ovimage__null__part__sets,axiom,
! [F: a > a,M: sigma_measure_a,N: sigma_measure_a,A: set_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ ( comple3428971583294703880tion_a @ M ) @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_a_a @ ( comple3428971583294703880tion_a @ M ) @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_space_a @ ( comple3428971583294703880tion_a @ M ) ) ) @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_sets
thf(fact_1054_completion_Ovimage__null__part__sets,axiom,
! [F: extend8495563244428889912nnreal > a,M: sigma_7234349610311085201nnreal,N: sigma_measure_a,A: set_a] :
( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ ( comple6668017395272084142nnreal @ M ) @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measur7655964997769656268real_a @ ( comple6668017395272084142nnreal @ M ) @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_sets
thf(fact_1055_completion_Ovimage__null__part__sets,axiom,
! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ ( comple3428971583294703880tion_a @ M ) @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur4839436603801885502nnreal @ ( comple3428971583294703880tion_a @ M ) @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_space_a @ ( comple3428971583294703880tion_a @ M ) ) ) @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_sets
thf(fact_1056_completion_Ovimage__null__part__sets,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ ( comple6668017395272084142nnreal @ M ) @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur2549461466288632554nnreal @ ( comple6668017395272084142nnreal @ M ) @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_sets
thf(fact_1057_completion_Ovimage__null__part__sets,axiom,
! [F: a > real,M: sigma_measure_a,N: sigma_measure_real,A: set_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( comple3428971583294703880tion_a @ M ) @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measure_distr_a_real @ ( comple3428971583294703880tion_a @ M ) @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_space_a @ ( comple3428971583294703880tion_a @ M ) ) ) @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_sets
thf(fact_1058_completion_Ovimage__null__part__sets,axiom,
! [F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,N: sigma_measure_real,A: set_real] :
( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ ( comple6668017395272084142nnreal @ M ) @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur6862244029252366686l_real @ ( comple6668017395272084142nnreal @ M ) @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_sets
thf(fact_1059_completion_Ovimage__null__part__sets,axiom,
! [F: real > a,M: sigma_measure_real,N: sigma_measure_a,A: set_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_real_a @ ( comple3506806835435775778n_real @ M ) @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_space_real @ ( comple3506806835435775778n_real @ M ) ) ) @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_sets
thf(fact_1060_completion_Ovimage__null__part__sets,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur8829990298702910942nnreal @ ( comple3506806835435775778n_real @ M ) @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_space_real @ ( comple3506806835435775778n_real @ M ) ) ) @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_sets
thf(fact_1061_completion_Ovimage__null__part__sets,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,A: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur2993149975067245138l_real @ ( comple3506806835435775778n_real @ M ) @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_space_real @ ( comple3506806835435775778n_real @ M ) ) ) @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) ) ) ) ) ).
% completion.vimage_null_part_sets
thf(fact_1062_sets__distr,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,F: real > real] :
( ( sigma_sets_real @ ( measur2993149975067245138l_real @ M @ N @ F ) )
= ( sigma_sets_real @ N ) ) ).
% sets_distr
thf(fact_1063_measurable__distr__eq1,axiom,
! [Mf: sigma_measure_real,Nf: sigma_measure_real,F: real > real,Mf2: sigma_7234349610311085201nnreal] :
( ( sigma_9017504469962657078nnreal @ ( measur2993149975067245138l_real @ Mf @ Nf @ F ) @ Mf2 )
= ( sigma_9017504469962657078nnreal @ Nf @ Mf2 ) ) ).
% measurable_distr_eq1
thf(fact_1064_space__distr,axiom,
! [M: sigma_measure_real,N: sigma_measure_real,F: real > real] :
( ( sigma_space_real @ ( measur2993149975067245138l_real @ M @ N @ F ) )
= ( sigma_space_real @ N ) ) ).
% space_distr
thf(fact_1065_lborel__distr__uminus,axiom,
( ( measur2993149975067245138l_real @ lebesgue_lborel_real @ borel_5078946678739801102l_real @ uminus_uminus_real )
= lebesgue_lborel_real ) ).
% lborel_distr_uminus
thf(fact_1066_sets__eq__iff__bounded,axiom,
! [A: sigma_measure_a,B2: sigma_measure_a,C3: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B2 )
=> ( ( ord_le254669795585780187sure_a @ B2 @ C3 )
=> ( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ C3 ) )
=> ( ( sigma_sets_a @ B2 )
= ( sigma_sets_a @ A ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_1067_sets__eq__iff__bounded,axiom,
! [A: sigma_7234349610311085201nnreal,B2: sigma_7234349610311085201nnreal,C3: sigma_7234349610311085201nnreal] :
( ( ord_le1854472233513649201nnreal @ A @ B2 )
=> ( ( ord_le1854472233513649201nnreal @ B2 @ C3 )
=> ( ( ( sigma_5465916536984168985nnreal @ A )
= ( sigma_5465916536984168985nnreal @ C3 ) )
=> ( ( sigma_5465916536984168985nnreal @ B2 )
= ( sigma_5465916536984168985nnreal @ A ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_1068_sets__eq__iff__bounded,axiom,
! [A: sigma_measure_real,B2: sigma_measure_real,C3: sigma_measure_real] :
( ( ord_le487379304121309861e_real @ A @ B2 )
=> ( ( ord_le487379304121309861e_real @ B2 @ C3 )
=> ( ( ( sigma_sets_real @ A )
= ( sigma_sets_real @ C3 ) )
=> ( ( sigma_sets_real @ B2 )
= ( sigma_sets_real @ A ) ) ) ) ) ).
% sets_eq_iff_bounded
thf(fact_1069_distr__cong,axiom,
! [M: sigma_measure_real,K2: sigma_measure_real,N: sigma_measure_real,L2: sigma_measure_real,F: real > real,G: real > real] :
( ( M = K2 )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ L2 ) )
=> ( ! [X: real] :
( ( member_real @ X @ ( sigma_space_real @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur2993149975067245138l_real @ M @ N @ F )
= ( measur2993149975067245138l_real @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1070_distr__cong,axiom,
! [M: sigma_measure_set_a,K2: sigma_measure_set_a,N: sigma_measure_a,L2: sigma_measure_a,F: set_a > a,G: set_a > a] :
( ( M = K2 )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L2 ) )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ ( sigma_space_set_a @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur7064479691503150872et_a_a @ M @ N @ F )
= ( measur7064479691503150872et_a_a @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1071_distr__cong,axiom,
! [M: sigma_3733394171116455995t_real,K2: sigma_3733394171116455995t_real,N: sigma_measure_a,L2: sigma_measure_a,F: set_real > a,G: set_real > a] :
( ( M = K2 )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L2 ) )
=> ( ! [X: set_real] :
( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur8448229113228689186real_a @ M @ N @ F )
= ( measur8448229113228689186real_a @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1072_distr__cong,axiom,
! [M: sigma_523634232904505671nnreal,K2: sigma_523634232904505671nnreal,N: sigma_measure_a,L2: sigma_measure_a,F: set_Ex3793607809372303086nnreal > a,G: set_Ex3793607809372303086nnreal > a] :
( ( M = K2 )
=> ( ( ( sigma_sets_a @ N )
= ( sigma_sets_a @ L2 ) )
=> ( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ ( sigma_2539764534872131430nnreal @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur4356243891041408406real_a @ M @ N @ F )
= ( measur4356243891041408406real_a @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1073_distr__cong,axiom,
! [M: sigma_measure_set_a,K2: sigma_measure_set_a,N: sigma_7234349610311085201nnreal,L2: sigma_7234349610311085201nnreal,F: set_a > extend8495563244428889912nnreal,G: set_a > extend8495563244428889912nnreal] :
( ( M = K2 )
=> ( ( ( sigma_5465916536984168985nnreal @ N )
= ( sigma_5465916536984168985nnreal @ L2 ) )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ ( sigma_space_set_a @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur2970674797653026974nnreal @ M @ N @ F )
= ( measur2970674797653026974nnreal @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1074_distr__cong,axiom,
! [M: sigma_3733394171116455995t_real,K2: sigma_3733394171116455995t_real,N: sigma_7234349610311085201nnreal,L2: sigma_7234349610311085201nnreal,F: set_real > extend8495563244428889912nnreal,G: set_real > extend8495563244428889912nnreal] :
( ( M = K2 )
=> ( ( ( sigma_5465916536984168985nnreal @ N )
= ( sigma_5465916536984168985nnreal @ L2 ) )
=> ( ! [X: set_real] :
( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur174208488455492116nnreal @ M @ N @ F )
= ( measur174208488455492116nnreal @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1075_distr__cong,axiom,
! [M: sigma_523634232904505671nnreal,K2: sigma_523634232904505671nnreal,N: sigma_7234349610311085201nnreal,L2: sigma_7234349610311085201nnreal,F: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal,G: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
( ( M = K2 )
=> ( ( ( sigma_5465916536984168985nnreal @ N )
= ( sigma_5465916536984168985nnreal @ L2 ) )
=> ( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ ( sigma_2539764534872131430nnreal @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur5959735445204559520nnreal @ M @ N @ F )
= ( measur5959735445204559520nnreal @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1076_distr__cong,axiom,
! [M: sigma_measure_set_a,K2: sigma_measure_set_a,N: sigma_measure_real,L2: sigma_measure_real,F: set_a > real,G: set_a > real] :
( ( M = K2 )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ L2 ) )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ ( sigma_space_set_a @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur3755146993059376402a_real @ M @ N @ F )
= ( measur3755146993059376402a_real @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1077_distr__cong,axiom,
! [M: sigma_3733394171116455995t_real,K2: sigma_3733394171116455995t_real,N: sigma_measure_real,L2: sigma_measure_real,F: set_real > real,G: set_real > real] :
( ( M = K2 )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ L2 ) )
=> ( ! [X: set_real] :
( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur874465376107064200l_real @ M @ N @ F )
= ( measur874465376107064200l_real @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1078_distr__cong,axiom,
! [M: sigma_523634232904505671nnreal,K2: sigma_523634232904505671nnreal,N: sigma_measure_real,L2: sigma_measure_real,F: set_Ex3793607809372303086nnreal > real,G: set_Ex3793607809372303086nnreal > real] :
( ( M = K2 )
=> ( ( ( sigma_sets_real @ N )
= ( sigma_sets_real @ L2 ) )
=> ( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ ( sigma_2539764534872131430nnreal @ M ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( measur4516463063617886740l_real @ M @ N @ F )
= ( measur4516463063617886740l_real @ K2 @ L2 @ G ) ) ) ) ) ).
% distr_cong
thf(fact_1079_distr__completion,axiom,
! [X5: real > real,M: sigma_measure_real,N: sigma_measure_real] :
( ( member_real_real @ X5 @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( measur2993149975067245138l_real @ ( comple3506806835435775778n_real @ M ) @ N @ X5 )
= ( measur2993149975067245138l_real @ M @ N @ X5 ) ) ) ).
% distr_completion
thf(fact_1080_distr__completion,axiom,
! [X5: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ X5 @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( ( measur8829990298702910942nnreal @ ( comple3506806835435775778n_real @ M ) @ N @ X5 )
= ( measur8829990298702910942nnreal @ M @ N @ X5 ) ) ) ).
% distr_completion
thf(fact_1081_le__measureD2,axiom,
! [A: sigma_measure_a,B2: sigma_measure_a] :
( ( ord_le254669795585780187sure_a @ A @ B2 )
=> ( ( ( sigma_space_a @ A )
= ( sigma_space_a @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ A ) @ ( sigma_sets_a @ B2 ) ) ) ) ).
% le_measureD2
thf(fact_1082_le__measureD2,axiom,
! [A: sigma_7234349610311085201nnreal,B2: sigma_7234349610311085201nnreal] :
( ( ord_le1854472233513649201nnreal @ A @ B2 )
=> ( ( ( sigma_3147302497200244656nnreal @ A )
= ( sigma_3147302497200244656nnreal @ B2 ) )
=> ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ A ) @ ( sigma_5465916536984168985nnreal @ B2 ) ) ) ) ).
% le_measureD2
thf(fact_1083_le__measureD2,axiom,
! [A: sigma_measure_real,B2: sigma_measure_real] :
( ( ord_le487379304121309861e_real @ A @ B2 )
=> ( ( ( sigma_space_real @ A )
= ( sigma_space_real @ B2 ) )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B2 ) ) ) ) ).
% le_measureD2
thf(fact_1084_completion_Ocompletion__distr__eq,axiom,
! [X5: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ X5 @ ( sigma_9017504469962657078nnreal @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ( measur1209175464439008069nnreal @ ( measur8829990298702910942nnreal @ ( comple3506806835435775778n_real @ M ) @ N @ X5 ) )
= ( measur1209175464439008069nnreal @ N ) )
=> ( ( comple6668017395272084142nnreal @ ( measur8829990298702910942nnreal @ ( comple3506806835435775778n_real @ M ) @ N @ X5 ) )
= ( measur8829990298702910942nnreal @ ( comple3506806835435775778n_real @ M ) @ ( comple6668017395272084142nnreal @ N ) @ X5 ) ) ) ) ).
% completion.completion_distr_eq
thf(fact_1085_completion_Ocompletion__distr__eq,axiom,
! [X5: real > real,M: sigma_measure_real,N: sigma_measure_real] :
( ( member_real_real @ X5 @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ( measur3710062792471635001s_real @ ( measur2993149975067245138l_real @ ( comple3506806835435775778n_real @ M ) @ N @ X5 ) )
= ( measur3710062792471635001s_real @ N ) )
=> ( ( comple3506806835435775778n_real @ ( measur2993149975067245138l_real @ ( comple3506806835435775778n_real @ M ) @ N @ X5 ) )
= ( measur2993149975067245138l_real @ ( comple3506806835435775778n_real @ M ) @ ( comple3506806835435775778n_real @ N ) @ X5 ) ) ) ) ).
% completion.completion_distr_eq
thf(fact_1086_completion_Omeasurable__completion2,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur8829990298702910942nnreal @ ( comple3506806835435775778n_real @ M ) @ N @ F ) ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( comple3506806835435775778n_real @ M ) @ ( comple6668017395272084142nnreal @ N ) ) ) ) ) ).
% completion.measurable_completion2
thf(fact_1087_completion_Omeasurable__completion2,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ M ) @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur2993149975067245138l_real @ ( comple3506806835435775778n_real @ M ) @ N @ F ) ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ M ) @ ( comple3506806835435775778n_real @ N ) ) ) ) ) ).
% completion.measurable_completion2
thf(fact_1088_null__sets__distr__iff,axiom,
! [F: a > a,M: sigma_measure_a,N: sigma_measure_a,A: set_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ ( measure_distr_a_a @ M @ N @ F ) ) )
= ( ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ A ) @ ( sigma_space_a @ M ) ) @ ( measure_null_sets_a @ M ) )
& ( member_set_a @ A @ ( sigma_sets_a @ N ) ) ) ) ) ).
% null_sets_distr_iff
thf(fact_1089_null__sets__distr__iff,axiom,
! [F: real > a,M: sigma_measure_real,N: sigma_measure_a,A: set_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ ( measure_distr_real_a @ M @ N @ F ) ) )
= ( ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ A ) @ ( sigma_space_real @ M ) ) @ ( measur3710062792471635001s_real @ M ) )
& ( member_set_a @ A @ ( sigma_sets_a @ N ) ) ) ) ) ).
% null_sets_distr_iff
thf(fact_1090_null__sets__distr__iff,axiom,
! [F: extend8495563244428889912nnreal > a,M: sigma_7234349610311085201nnreal,N: sigma_measure_a,A: set_a] :
( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ M @ N ) )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ ( measur7655964997769656268real_a @ M @ N @ F ) ) )
= ( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ A ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( measur1209175464439008069nnreal @ M ) )
& ( member_set_a @ A @ ( sigma_sets_a @ N ) ) ) ) ) ).
% null_sets_distr_iff
thf(fact_1091_null__sets__distr__iff,axiom,
! [F: a > extend8495563244428889912nnreal,M: sigma_measure_a,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) )
=> ( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ ( measur4839436603801885502nnreal @ M @ N @ F ) ) )
= ( ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ A ) @ ( sigma_space_a @ M ) ) @ ( measure_null_sets_a @ M ) )
& ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ N ) ) ) ) ) ).
% null_sets_distr_iff
thf(fact_1092_null__sets__distr__iff,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ N ) )
=> ( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ ( measur2549461466288632554nnreal @ M @ N @ F ) ) )
= ( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ A ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( measur1209175464439008069nnreal @ M ) )
& ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ N ) ) ) ) ) ).
% null_sets_distr_iff
thf(fact_1093_null__sets__distr__iff,axiom,
! [F: a > real,M: sigma_measure_a,N: sigma_measure_real,A: set_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) )
=> ( ( member_set_real @ A @ ( measur3710062792471635001s_real @ ( measure_distr_a_real @ M @ N @ F ) ) )
= ( ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ A ) @ ( sigma_space_a @ M ) ) @ ( measure_null_sets_a @ M ) )
& ( member_set_real @ A @ ( sigma_sets_real @ N ) ) ) ) ) ).
% null_sets_distr_iff
thf(fact_1094_null__sets__distr__iff,axiom,
! [F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,N: sigma_measure_real,A: set_real] :
( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ N ) )
=> ( ( member_set_real @ A @ ( measur3710062792471635001s_real @ ( measur6862244029252366686l_real @ M @ N @ F ) ) )
= ( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ A ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( measur1209175464439008069nnreal @ M ) )
& ( member_set_real @ A @ ( sigma_sets_real @ N ) ) ) ) ) ).
% null_sets_distr_iff
thf(fact_1095_null__sets__distr__iff,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,A: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( member_set_real @ A @ ( measur3710062792471635001s_real @ ( measur2993149975067245138l_real @ M @ N @ F ) ) )
= ( ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ A ) @ ( sigma_space_real @ M ) ) @ ( measur3710062792471635001s_real @ M ) )
& ( member_set_real @ A @ ( sigma_sets_real @ N ) ) ) ) ) ).
% null_sets_distr_iff
thf(fact_1096_null__sets__distr__iff,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ ( measur8829990298702910942nnreal @ M @ N @ F ) ) )
= ( ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ A ) @ ( sigma_space_real @ M ) ) @ ( measur3710062792471635001s_real @ M ) )
& ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ N ) ) ) ) ) ).
% null_sets_distr_iff
thf(fact_1097_complete__measure_Ovimage__null__part__null__sets,axiom,
! [M: sigma_measure_a,F: a > a,N: sigma_measure_a,A: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_a_a @ M @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_space_a @ M ) ) @ ( measure_null_sets_a @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_null_sets
thf(fact_1098_complete__measure_Ovimage__null__part__null__sets,axiom,
! [M: sigma_measure_real,F: real > a,N: sigma_measure_a,A: set_a] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_real_a @ M @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_space_real @ M ) ) @ ( measur3710062792471635001s_real @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_null_sets
thf(fact_1099_complete__measure_Ovimage__null__part__null__sets,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > a,N: sigma_measure_a,A: set_a] :
( ( comple9105848400330859749nnreal @ M )
=> ( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ M @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measur7655964997769656268real_a @ M @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( measur1209175464439008069nnreal @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_null_sets
thf(fact_1100_complete__measure_Ovimage__null__part__null__sets,axiom,
! [M: sigma_measure_a,F: a > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur4839436603801885502nnreal @ M @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_space_a @ M ) ) @ ( measure_null_sets_a @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_null_sets
thf(fact_1101_complete__measure_Ovimage__null__part__null__sets,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( comple9105848400330859749nnreal @ M )
=> ( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur2549461466288632554nnreal @ M @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( measur1209175464439008069nnreal @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_null_sets
thf(fact_1102_complete__measure_Ovimage__null__part__null__sets,axiom,
! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur8829990298702910942nnreal @ M @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_space_real @ M ) ) @ ( measur3710062792471635001s_real @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_null_sets
thf(fact_1103_complete__measure_Ovimage__null__part__null__sets,axiom,
! [M: sigma_measure_a,F: a > real,N: sigma_measure_real,A: set_real] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measure_distr_a_real @ M @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_space_a @ M ) ) @ ( measure_null_sets_a @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_null_sets
thf(fact_1104_complete__measure_Ovimage__null__part__null__sets,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real,N: sigma_measure_real,A: set_real] :
( ( comple9105848400330859749nnreal @ M )
=> ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur6862244029252366686l_real @ M @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( measur1209175464439008069nnreal @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_null_sets
thf(fact_1105_complete__measure_Ovimage__null__part__null__sets,axiom,
! [M: sigma_measure_real,F: real > real,N: sigma_measure_real,A: set_real] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur2993149975067245138l_real @ M @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_space_real @ M ) ) @ ( measur3710062792471635001s_real @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_null_sets
thf(fact_1106_complete__measure_Ovimage__null__part__sets,axiom,
! [M: sigma_measure_a,F: a > a,N: sigma_measure_a,A: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_a_a @ M @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_sets
thf(fact_1107_complete__measure_Ovimage__null__part__sets,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > a,N: sigma_measure_a,A: set_a] :
( ( comple9105848400330859749nnreal @ M )
=> ( ( member4924430693770431270real_a @ F @ ( sigma_3031480723531659892real_a @ M @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measur7655964997769656268real_a @ M @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4075187267506941001real_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_sets
thf(fact_1108_complete__measure_Ovimage__null__part__sets,axiom,
! [M: sigma_measure_real,F: real > a,N: sigma_measure_a,A: set_a] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
=> ( ( ord_le3724670747650509150_set_a @ ( measure_null_sets_a @ N ) @ ( measure_null_sets_a @ ( measure_distr_real_a @ M @ N @ F ) ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_a @ F @ ( complete_null_part_a @ N @ A ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_sets
thf(fact_1109_complete__measure_Ovimage__null__part__sets,axiom,
! [M: sigma_measure_a,F: a > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( member298456594901751504nnreal @ F @ ( sigma_214952329563889126nnreal @ M @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur4839436603801885502nnreal @ M @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage1258658873539170235nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_sets
thf(fact_1110_complete__measure_Ovimage__null__part__sets,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( comple9105848400330859749nnreal @ M )
=> ( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur2549461466288632554nnreal @ M @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage3650734033530794285nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_sets
thf(fact_1111_complete__measure_Ovimage__null__part__sets,axiom,
! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur8829990298702910942nnreal @ M @ N @ F ) ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ ( comple6358047150840085292nnreal @ N @ A ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_sets
thf(fact_1112_complete__measure_Ovimage__null__part__sets,axiom,
! [M: sigma_measure_a,F: a > real,N: sigma_measure_real,A: set_real] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measure_distr_a_real @ M @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member_set_a @ ( inf_inf_set_a @ ( vimage_a_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_sets
thf(fact_1113_complete__measure_Ovimage__null__part__sets,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real,N: sigma_measure_real,A: set_real] :
( ( comple9105848400330859749nnreal @ M )
=> ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur6862244029252366686l_real @ M @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ ( vimage4399055823842842145l_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_sets
thf(fact_1114_complete__measure_Ovimage__null__part__sets,axiom,
! [M: sigma_measure_real,F: real > real,N: sigma_measure_real,A: set_real] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur2993149975067245138l_real @ M @ N @ F ) ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ N ) ) )
=> ( member_set_real @ ( inf_inf_set_real @ ( vimage_real_real @ F @ ( comple4917500974405109920t_real @ N @ A ) ) @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ) ) ) ).
% complete_measure.vimage_null_part_sets
thf(fact_1115_lebesgue__measurable__imp__measurable__on__nnreal,axiom,
! [U: real > real,S3: set_real] :
( ( member_real_real @ U @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ borel_5078946678739801102l_real ) )
=> ( ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( U @ X ) )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) )
=> ( equiva5980327992511004390l_real @ U @ S3 ) ) ) ) ).
% lebesgue_measurable_imp_measurable_on_nnreal
thf(fact_1116_lebesgue__measurable__imp__measurable__on__nnreal__UNIV,axiom,
! [U: real > real] :
( ( member_real_real @ U @ ( sigma_5267869275261027754l_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) @ borel_5078946678739801102l_real ) )
=> ( ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( U @ X ) )
=> ( equiva5980327992511004390l_real @ U @ top_top_set_real ) ) ) ).
% lebesgue_measurable_imp_measurable_on_nnreal_UNIV
thf(fact_1117_diff__self,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% diff_self
thf(fact_1118_diff__0__right,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_0_right
thf(fact_1119_diff__zero,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ zero_zero_real )
= A2 ) ).
% diff_zero
thf(fact_1120_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: real] :
( ( minus_minus_real @ A2 @ A2 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1121_neg__equal__zero,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= A2 )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_1122_equal__neg__zero,axiom,
! [A2: real] :
( ( A2
= ( uminus_uminus_real @ A2 ) )
= ( A2 = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_1123_neg__equal__0__iff__equal,axiom,
! [A2: real] :
( ( ( uminus_uminus_real @ A2 )
= zero_zero_real )
= ( A2 = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_1124_neg__0__equal__iff__equal,axiom,
! [A2: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A2 ) )
= ( zero_zero_real = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_1125_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_1126_diff__ge__0__iff__ge,axiom,
! [A2: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B ) )
= ( ord_less_eq_real @ B @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_1127_neg__less__eq__nonneg,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_1128_less__eq__neg__nonpos,axiom,
! [A2: real] :
( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_1129_neg__le__0__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_1130_neg__0__le__iff__le,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_1131_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1132_diff__0,axiom,
! [A2: real] :
( ( minus_minus_real @ zero_zero_real @ A2 )
= ( uminus_uminus_real @ A2 ) ) ).
% diff_0
thf(fact_1133_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A5: real,B5: real] :
( ( minus_minus_real @ A5 @ B5 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1134_completion_Ocomplete__measure__axioms,axiom,
! [M: sigma_measure_real] : ( comple9032484589293727193e_real @ ( comple3506806835435775778n_real @ M ) ) ).
% completion.complete_measure_axioms
thf(fact_1135_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_1136_complete__measure_Ocomplete2,axiom,
! [M: sigma_measure_a,A: set_a,B2: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_set_a @ B2 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ A @ ( measure_null_sets_a @ M ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_1137_complete__measure_Ocomplete2,axiom,
! [M: sigma_measure_real,A: set_real,B2: set_real] :
( ( comple9032484589293727193e_real @ M )
=> ( ( ord_less_eq_set_real @ A @ B2 )
=> ( ( member_set_real @ B2 @ ( measur3710062792471635001s_real @ M ) )
=> ( member_set_real @ A @ ( measur3710062792471635001s_real @ M ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_1138_complete__measure_Ocomplete2,axiom,
! [M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( comple9105848400330859749nnreal @ M )
=> ( ( ord_le6787938422905777998nnreal @ A @ B2 )
=> ( ( member603777416030116741nnreal @ B2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ M ) ) ) ) ) ).
% complete_measure.complete2
thf(fact_1139_complete__measure_Ointro,axiom,
! [M: sigma_measure_a] :
( ! [A9: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A9 )
=> ( ( member_set_a @ A9 @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ B3 @ ( sigma_sets_a @ M ) ) ) )
=> ( comple8155536527497655953sure_a @ M ) ) ).
% complete_measure.intro
thf(fact_1140_complete__measure_Ointro,axiom,
! [M: sigma_7234349610311085201nnreal] :
( ! [A9: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B3 @ A9 )
=> ( ( member603777416030116741nnreal @ A9 @ ( measur1209175464439008069nnreal @ M ) )
=> ( member603777416030116741nnreal @ B3 @ ( sigma_5465916536984168985nnreal @ M ) ) ) )
=> ( comple9105848400330859749nnreal @ M ) ) ).
% complete_measure.intro
thf(fact_1141_complete__measure_Ointro,axiom,
! [M: sigma_measure_real] :
( ! [A9: set_real,B3: set_real] :
( ( ord_less_eq_set_real @ B3 @ A9 )
=> ( ( member_set_real @ A9 @ ( measur3710062792471635001s_real @ M ) )
=> ( member_set_real @ B3 @ ( sigma_sets_real @ M ) ) ) )
=> ( comple9032484589293727193e_real @ M ) ) ).
% complete_measure.intro
thf(fact_1142_complete__measure_Ocomplete,axiom,
! [M: sigma_measure_a,B2: set_a,A: set_a] :
( ( comple8155536527497655953sure_a @ M )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( member_set_a @ A @ ( measure_null_sets_a @ M ) )
=> ( member_set_a @ B2 @ ( sigma_sets_a @ M ) ) ) ) ) ).
% complete_measure.complete
thf(fact_1143_complete__measure_Ocomplete,axiom,
! [M: sigma_7234349610311085201nnreal,B2: set_Ex3793607809372303086nnreal,A: set_Ex3793607809372303086nnreal] :
( ( comple9105848400330859749nnreal @ M )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ A )
=> ( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ M ) )
=> ( member603777416030116741nnreal @ B2 @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ) ).
% complete_measure.complete
thf(fact_1144_complete__measure_Ocomplete,axiom,
! [M: sigma_measure_real,B2: set_real,A: set_real] :
( ( comple9032484589293727193e_real @ M )
=> ( ( ord_less_eq_set_real @ B2 @ A )
=> ( ( member_set_real @ A @ ( measur3710062792471635001s_real @ M ) )
=> ( member_set_real @ B2 @ ( sigma_sets_real @ M ) ) ) ) ) ).
% complete_measure.complete
thf(fact_1145_complete__measure__def,axiom,
( comple8155536527497655953sure_a
= ( ^ [M3: sigma_measure_a] :
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ( member_set_a @ A3 @ ( measure_null_sets_a @ M3 ) )
=> ( member_set_a @ B4 @ ( sigma_sets_a @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_1146_complete__measure__def,axiom,
( comple9105848400330859749nnreal
= ( ^ [M3: sigma_7234349610311085201nnreal] :
! [A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ B4 @ A3 )
=> ( ( member603777416030116741nnreal @ A3 @ ( measur1209175464439008069nnreal @ M3 ) )
=> ( member603777416030116741nnreal @ B4 @ ( sigma_5465916536984168985nnreal @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_1147_complete__measure__def,axiom,
( comple9032484589293727193e_real
= ( ^ [M3: sigma_measure_real] :
! [A3: set_real,B4: set_real] :
( ( ord_less_eq_set_real @ B4 @ A3 )
=> ( ( member_set_real @ A3 @ ( measur3710062792471635001s_real @ M3 ) )
=> ( member_set_real @ B4 @ ( sigma_sets_real @ M3 ) ) ) ) ) ) ).
% complete_measure_def
thf(fact_1148_complete__measure_Ocompletion__distr__eq,axiom,
! [M: sigma_measure_real,X5: real > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member2919562650594848410nnreal @ X5 @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( ( ( measur1209175464439008069nnreal @ ( measur8829990298702910942nnreal @ M @ N @ X5 ) )
= ( measur1209175464439008069nnreal @ N ) )
=> ( ( comple6668017395272084142nnreal @ ( measur8829990298702910942nnreal @ M @ N @ X5 ) )
= ( measur8829990298702910942nnreal @ M @ ( comple6668017395272084142nnreal @ N ) @ X5 ) ) ) ) ) ).
% complete_measure.completion_distr_eq
thf(fact_1149_complete__measure_Ocompletion__distr__eq,axiom,
! [M: sigma_measure_real,X5: real > real,N: sigma_measure_real] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member_real_real @ X5 @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( ( measur3710062792471635001s_real @ ( measur2993149975067245138l_real @ M @ N @ X5 ) )
= ( measur3710062792471635001s_real @ N ) )
=> ( ( comple3506806835435775778n_real @ ( measur2993149975067245138l_real @ M @ N @ X5 ) )
= ( measur2993149975067245138l_real @ M @ ( comple3506806835435775778n_real @ N ) @ X5 ) ) ) ) ) ).
% complete_measure.completion_distr_eq
thf(fact_1150_complete__measure_Omeasurable__completion2,axiom,
! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( ( ord_le3366939622266546180nnreal @ ( measur1209175464439008069nnreal @ N ) @ ( measur1209175464439008069nnreal @ ( measur8829990298702910942nnreal @ M @ N @ F ) ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ ( comple6668017395272084142nnreal @ N ) ) ) ) ) ) ).
% complete_measure.measurable_completion2
thf(fact_1151_complete__measure_Omeasurable__completion2,axiom,
! [M: sigma_measure_real,F: real > real,N: sigma_measure_real] :
( ( comple9032484589293727193e_real @ M )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( ord_le3558479182127378552t_real @ ( measur3710062792471635001s_real @ N ) @ ( measur3710062792471635001s_real @ ( measur2993149975067245138l_real @ M @ N @ F ) ) )
=> ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ ( comple3506806835435775778n_real @ N ) ) ) ) ) ) ).
% complete_measure.measurable_completion2
thf(fact_1152_measurable__piecewise__restrict,axiom,
! [C3: set_set_real,M: sigma_measure_real,F: real > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal] :
( ( counta8054315614235329383t_real @ C3 )
=> ( ! [Omega2: set_real] :
( ( member_set_real @ Omega2 @ C3 )
=> ( member_set_real @ ( inf_inf_set_real @ Omega2 @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( ( ord_less_eq_set_real @ ( sigma_space_real @ M ) @ ( comple3096694443085538997t_real @ C3 ) )
=> ( ! [Omega2: set_real] :
( ( member_set_real @ Omega2 @ C3 )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( sigma_5414646170262037096e_real @ M @ Omega2 ) @ N ) ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) ) ) ) ) ) ).
% measurable_piecewise_restrict
thf(fact_1153_measurable__piecewise__restrict__iff,axiom,
! [C3: set_set_real,M: sigma_measure_real,F: real > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal] :
( ( counta8054315614235329383t_real @ C3 )
=> ( ! [Omega2: set_real] :
( ( member_set_real @ Omega2 @ C3 )
=> ( member_set_real @ ( inf_inf_set_real @ Omega2 @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) )
=> ( ( ord_less_eq_set_real @ ( sigma_space_real @ M ) @ ( comple3096694443085538997t_real @ C3 ) )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
= ( ! [X3: set_real] :
( ( member_set_real @ X3 @ C3 )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( sigma_5414646170262037096e_real @ M @ X3 ) @ N ) ) ) ) ) ) ) ) ).
% measurable_piecewise_restrict_iff
thf(fact_1154_insert__subsetI,axiom,
! [X2: set_a,A: set_set_a,X5: set_set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ A )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1155_insert__subsetI,axiom,
! [X2: real > extend8495563244428889912nnreal,A: set_re5328672808648366137nnreal,X5: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ X2 @ A )
=> ( ( ord_le2462468573666744473nnreal @ X5 @ A )
=> ( ord_le2462468573666744473nnreal @ ( insert152533262698245683nnreal @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1156_insert__subsetI,axiom,
! [X2: set_real,A: set_set_real,X5: set_set_real] :
( ( member_set_real @ X2 @ A )
=> ( ( ord_le3558479182127378552t_real @ X5 @ A )
=> ( ord_le3558479182127378552t_real @ ( insert_set_real @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1157_insert__subsetI,axiom,
! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,X5: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ X2 @ A )
=> ( ( ord_le3366939622266546180nnreal @ X5 @ A )
=> ( ord_le3366939622266546180nnreal @ ( insert1343806209672318238nnreal @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1158_sets_Ocountable__Union,axiom,
! [X5: set_set_a,M: sigma_measure_a] :
( ( counta6168152590877469849_set_a @ X5 )
=> ( ( ord_le3724670747650509150_set_a @ X5 @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ ( comple2307003609928055243_set_a @ X5 ) @ ( sigma_sets_a @ M ) ) ) ) ).
% sets.countable_Union
thf(fact_1159_sets_Ocountable__Union,axiom,
! [X5: set_se4580700918925141924nnreal,M: sigma_7234349610311085201nnreal] :
( ( counta2425349316461633011nnreal @ X5 )
=> ( ( ord_le3366939622266546180nnreal @ X5 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ ( comple4226387801268262977nnreal @ X5 ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).
% sets.countable_Union
thf(fact_1160_sets_Ocountable__Union,axiom,
! [X5: set_set_real,M: sigma_measure_real] :
( ( counta8054315614235329383t_real @ X5 )
=> ( ( ord_le3558479182127378552t_real @ X5 @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ ( comple3096694443085538997t_real @ X5 ) @ ( sigma_sets_real @ M ) ) ) ) ).
% sets.countable_Union
thf(fact_1161_univ__second__countable,axiom,
~ ! [B8: set_set_real] :
( ( counta8054315614235329383t_real @ B8 )
=> ( ! [C4: set_real] :
( ( member_set_real @ C4 @ B8 )
=> ( topolo4860482606490270245n_real @ C4 ) )
=> ~ ! [S4: set_real] :
( ( topolo4860482606490270245n_real @ S4 )
=> ? [U2: set_set_real] :
( ( ord_le3558479182127378552t_real @ U2 @ B8 )
& ( S4
= ( comple3096694443085538997t_real @ U2 ) ) ) ) ) ) ).
% univ_second_countable
thf(fact_1162_univ__second__countable,axiom,
~ ! [B8: set_se4580700918925141924nnreal] :
( ( counta2425349316461633011nnreal @ B8 )
=> ( ! [C4: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ C4 @ B8 )
=> ( topolo1037242508615874353nnreal @ C4 ) )
=> ~ ! [S4: set_Ex3793607809372303086nnreal] :
( ( topolo1037242508615874353nnreal @ S4 )
=> ? [U2: set_se4580700918925141924nnreal] :
( ( ord_le3366939622266546180nnreal @ U2 @ B8 )
& ( S4
= ( comple4226387801268262977nnreal @ U2 ) ) ) ) ) ) ).
% univ_second_countable
thf(fact_1163_Lindelof,axiom,
! [F3: set_set_real] :
( ! [S5: set_real] :
( ( member_set_real @ S5 @ F3 )
=> ( topolo4860482606490270245n_real @ S5 ) )
=> ~ ! [F4: set_set_real] :
( ( ord_le3558479182127378552t_real @ F4 @ F3 )
=> ( ( counta8054315614235329383t_real @ F4 )
=> ( ( comple3096694443085538997t_real @ F4 )
!= ( comple3096694443085538997t_real @ F3 ) ) ) ) ) ).
% Lindelof
thf(fact_1164_Lindelof,axiom,
! [F3: set_se4580700918925141924nnreal] :
( ! [S5: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ S5 @ F3 )
=> ( topolo1037242508615874353nnreal @ S5 ) )
=> ~ ! [F4: set_se4580700918925141924nnreal] :
( ( ord_le3366939622266546180nnreal @ F4 @ F3 )
=> ( ( counta2425349316461633011nnreal @ F4 )
=> ( ( comple4226387801268262977nnreal @ F4 )
!= ( comple4226387801268262977nnreal @ F3 ) ) ) ) ) ).
% Lindelof
thf(fact_1165_subset__emptyI,axiom,
! [A: set_set_a] :
( ! [X: set_a] :
~ ( member_set_a @ X @ A )
=> ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a ) ) ).
% subset_emptyI
thf(fact_1166_subset__emptyI,axiom,
! [A: set_re5328672808648366137nnreal] :
( ! [X: real > extend8495563244428889912nnreal] :
~ ( member2919562650594848410nnreal @ X @ A )
=> ( ord_le2462468573666744473nnreal @ A @ bot_bo6037503491064675021nnreal ) ) ).
% subset_emptyI
thf(fact_1167_subset__emptyI,axiom,
! [A: set_set_real] :
( ! [X: set_real] :
~ ( member_set_real @ X @ A )
=> ( ord_le3558479182127378552t_real @ A @ bot_bot_set_set_real ) ) ).
% subset_emptyI
thf(fact_1168_subset__emptyI,axiom,
! [A: set_se4580700918925141924nnreal] :
( ! [X: set_Ex3793607809372303086nnreal] :
~ ( member603777416030116741nnreal @ X @ A )
=> ( ord_le3366939622266546180nnreal @ A @ bot_bo2988155216863113784nnreal ) ) ).
% subset_emptyI
thf(fact_1169_Sup__UNIV,axiom,
( ( comple4226387801268262977nnreal @ top_to3356475028079756884nnreal )
= top_to7994903218803871134nnreal ) ).
% Sup_UNIV
thf(fact_1170_Sup__UNIV,axiom,
( ( comple3096694443085538997t_real @ top_top_set_set_real )
= top_top_set_real ) ).
% Sup_UNIV
thf(fact_1171_Sup__UNIV,axiom,
( ( comple2307003609928055243_set_a @ top_top_set_set_a )
= top_top_set_a ) ).
% Sup_UNIV
thf(fact_1172_Sup__UNIV,axiom,
( ( comple6814414086264997003nnreal @ top_to7994903218803871134nnreal )
= top_to1496364449551166952nnreal ) ).
% Sup_UNIV
thf(fact_1173_in__sets__Sup,axiom,
! [M: set_Sigma_measure_a,X5: set_a,M4: sigma_measure_a,A: set_a] :
( ! [M5: sigma_measure_a] :
( ( member3534519376729797778sure_a @ M5 @ M )
=> ( ( sigma_space_a @ M5 )
= X5 ) )
=> ( ( member3534519376729797778sure_a @ M4 @ M )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M4 ) )
=> ( member_set_a @ A @ ( sigma_sets_a @ ( comple2239804592135895886sure_a @ M ) ) ) ) ) ) ).
% in_sets_Sup
thf(fact_1174_in__sets__Sup,axiom,
! [M: set_Si97717610131227249nnreal,X5: set_Ex3793607809372303086nnreal,M4: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ! [M5: sigma_7234349610311085201nnreal] :
( ( member6261374078160781754nnreal @ M5 @ M )
=> ( ( sigma_3147302497200244656nnreal @ M5 )
= X5 ) )
=> ( ( member6261374078160781754nnreal @ M4 @ M )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M4 ) )
=> ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ ( comple2394123286901040126nnreal @ M ) ) ) ) ) ) ).
% in_sets_Sup
thf(fact_1175_in__sets__Sup,axiom,
! [M: set_Si6059263944882162789e_real,X5: set_real,M4: sigma_measure_real,A: set_real] :
( ! [M5: sigma_measure_real] :
( ( member4553183543495551918e_real @ M5 @ M )
=> ( ( sigma_space_real @ M5 )
= X5 ) )
=> ( ( member4553183543495551918e_real @ M4 @ M )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M4 ) )
=> ( member_set_real @ A @ ( sigma_sets_real @ ( comple1433435454551854066e_real @ M ) ) ) ) ) ) ).
% in_sets_Sup
thf(fact_1176_measurable__Sup1,axiom,
! [M4: sigma_measure_real,M: set_Si6059263944882162789e_real,F: real > extend8495563244428889912nnreal,N: sigma_7234349610311085201nnreal] :
( ( member4553183543495551918e_real @ M4 @ M )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M4 @ N ) )
=> ( ! [M5: sigma_measure_real,N4: sigma_measure_real] :
( ( member4553183543495551918e_real @ M5 @ M )
=> ( ( member4553183543495551918e_real @ N4 @ M )
=> ( ( sigma_space_real @ M5 )
= ( sigma_space_real @ N4 ) ) ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( comple1433435454551854066e_real @ M ) @ N ) ) ) ) ) ).
% measurable_Sup1
thf(fact_1177_measurable__Sup2,axiom,
! [M: set_Si97717610131227249nnreal,F: real > extend8495563244428889912nnreal,N: sigma_measure_real] :
( ( M != bot_bo8227844048696536285nnreal )
=> ( ! [M5: sigma_7234349610311085201nnreal] :
( ( member6261374078160781754nnreal @ M5 @ M )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ N @ M5 ) ) )
=> ( ! [M5: sigma_7234349610311085201nnreal,N4: sigma_7234349610311085201nnreal] :
( ( member6261374078160781754nnreal @ M5 @ M )
=> ( ( member6261374078160781754nnreal @ N4 @ M )
=> ( ( sigma_3147302497200244656nnreal @ M5 )
= ( sigma_3147302497200244656nnreal @ N4 ) ) ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ N @ ( comple2394123286901040126nnreal @ M ) ) ) ) ) ) ).
% measurable_Sup2
thf(fact_1178_sets__Sup__in__sets,axiom,
! [M: set_Sigma_measure_a,N: sigma_measure_a] :
( ( M != bot_bo5171090233048072389sure_a )
=> ( ! [M5: sigma_measure_a] :
( ( member3534519376729797778sure_a @ M5 @ M )
=> ( ( sigma_space_a @ M5 )
= ( sigma_space_a @ N ) ) )
=> ( ! [M5: sigma_measure_a] :
( ( member3534519376729797778sure_a @ M5 @ M )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M5 ) @ ( sigma_sets_a @ N ) ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ ( comple2239804592135895886sure_a @ M ) ) @ ( sigma_sets_a @ N ) ) ) ) ) ).
% sets_Sup_in_sets
thf(fact_1179_sets__Sup__in__sets,axiom,
! [M: set_Si97717610131227249nnreal,N: sigma_7234349610311085201nnreal] :
( ( M != bot_bo8227844048696536285nnreal )
=> ( ! [M5: sigma_7234349610311085201nnreal] :
( ( member6261374078160781754nnreal @ M5 @ M )
=> ( ( sigma_3147302497200244656nnreal @ M5 )
= ( sigma_3147302497200244656nnreal @ N ) ) )
=> ( ! [M5: sigma_7234349610311085201nnreal] :
( ( member6261374078160781754nnreal @ M5 @ M )
=> ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M5 ) @ ( sigma_5465916536984168985nnreal @ N ) ) )
=> ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ ( comple2394123286901040126nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ N ) ) ) ) ) ).
% sets_Sup_in_sets
thf(fact_1180_sets__Sup__in__sets,axiom,
! [M: set_Si6059263944882162789e_real,N: sigma_measure_real] :
( ( M != bot_bo5686449298802467025e_real )
=> ( ! [M5: sigma_measure_real] :
( ( member4553183543495551918e_real @ M5 @ M )
=> ( ( sigma_space_real @ M5 )
= ( sigma_space_real @ N ) ) )
=> ( ! [M5: sigma_measure_real] :
( ( member4553183543495551918e_real @ M5 @ M )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M5 ) @ ( sigma_sets_real @ N ) ) )
=> ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ ( comple1433435454551854066e_real @ M ) ) @ ( sigma_sets_real @ N ) ) ) ) ) ).
% sets_Sup_in_sets
thf(fact_1181_less__eq__Sup,axiom,
! [A: set_set_a,U: set_a] :
( ! [V: set_a] :
( ( member_set_a @ V @ A )
=> ( ord_less_eq_set_a @ U @ V ) )
=> ( ( A != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1182_less__eq__Sup,axiom,
! [A: set_re5328672808648366137nnreal,U: real > extend8495563244428889912nnreal] :
( ! [V: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ V @ A )
=> ( ord_le1618294441215897699nnreal @ U @ V ) )
=> ( ( A != bot_bo6037503491064675021nnreal )
=> ( ord_le1618294441215897699nnreal @ U @ ( comple2814930536884499286nnreal @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1183_less__eq__Sup,axiom,
! [A: set_set_real,U: set_real] :
( ! [V: set_real] :
( ( member_set_real @ V @ A )
=> ( ord_less_eq_set_real @ U @ V ) )
=> ( ( A != bot_bot_set_set_real )
=> ( ord_less_eq_set_real @ U @ ( comple3096694443085538997t_real @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1184_less__eq__Sup,axiom,
! [A: set_se4580700918925141924nnreal,U: set_Ex3793607809372303086nnreal] :
( ! [V: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ V @ A )
=> ( ord_le6787938422905777998nnreal @ U @ V ) )
=> ( ( A != bot_bo2988155216863113784nnreal )
=> ( ord_le6787938422905777998nnreal @ U @ ( comple4226387801268262977nnreal @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_1185_cSup__least,axiom,
! [X5: set_set_a,Z3: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ X5 )
=> ( ord_less_eq_set_a @ X @ Z3 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ X5 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_1186_cSup__least,axiom,
! [X5: set_re5328672808648366137nnreal,Z3: real > extend8495563244428889912nnreal] :
( ( X5 != bot_bo6037503491064675021nnreal )
=> ( ! [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ X5 )
=> ( ord_le1618294441215897699nnreal @ X @ Z3 ) )
=> ( ord_le1618294441215897699nnreal @ ( comple2814930536884499286nnreal @ X5 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_1187_cSup__least,axiom,
! [X5: set_set_real,Z3: set_real] :
( ( X5 != bot_bot_set_set_real )
=> ( ! [X: set_real] :
( ( member_set_real @ X @ X5 )
=> ( ord_less_eq_set_real @ X @ Z3 ) )
=> ( ord_less_eq_set_real @ ( comple3096694443085538997t_real @ X5 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_1188_cSup__least,axiom,
! [X5: set_se4580700918925141924nnreal,Z3: set_Ex3793607809372303086nnreal] :
( ( X5 != bot_bo2988155216863113784nnreal )
=> ( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ X5 )
=> ( ord_le6787938422905777998nnreal @ X @ Z3 ) )
=> ( ord_le6787938422905777998nnreal @ ( comple4226387801268262977nnreal @ X5 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_1189_cSup__least,axiom,
! [X5: set_real,Z3: real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( ord_less_eq_real @ X @ Z3 ) )
=> ( ord_less_eq_real @ ( comple1385675409528146559p_real @ X5 ) @ Z3 ) ) ) ).
% cSup_least
thf(fact_1190_cSup__eq__non__empty,axiom,
! [X5: set_set_a,A2: set_a] :
( ( X5 != bot_bot_set_set_a )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ X5 )
=> ( ord_less_eq_set_a @ X @ A2 ) )
=> ( ! [Y2: set_a] :
( ! [X6: set_a] :
( ( member_set_a @ X6 @ X5 )
=> ( ord_less_eq_set_a @ X6 @ Y2 ) )
=> ( ord_less_eq_set_a @ A2 @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1191_cSup__eq__non__empty,axiom,
! [X5: set_re5328672808648366137nnreal,A2: real > extend8495563244428889912nnreal] :
( ( X5 != bot_bo6037503491064675021nnreal )
=> ( ! [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ X5 )
=> ( ord_le1618294441215897699nnreal @ X @ A2 ) )
=> ( ! [Y2: real > extend8495563244428889912nnreal] :
( ! [X6: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X6 @ X5 )
=> ( ord_le1618294441215897699nnreal @ X6 @ Y2 ) )
=> ( ord_le1618294441215897699nnreal @ A2 @ Y2 ) )
=> ( ( comple2814930536884499286nnreal @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1192_cSup__eq__non__empty,axiom,
! [X5: set_set_real,A2: set_real] :
( ( X5 != bot_bot_set_set_real )
=> ( ! [X: set_real] :
( ( member_set_real @ X @ X5 )
=> ( ord_less_eq_set_real @ X @ A2 ) )
=> ( ! [Y2: set_real] :
( ! [X6: set_real] :
( ( member_set_real @ X6 @ X5 )
=> ( ord_less_eq_set_real @ X6 @ Y2 ) )
=> ( ord_less_eq_set_real @ A2 @ Y2 ) )
=> ( ( comple3096694443085538997t_real @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1193_cSup__eq__non__empty,axiom,
! [X5: set_se4580700918925141924nnreal,A2: set_Ex3793607809372303086nnreal] :
( ( X5 != bot_bo2988155216863113784nnreal )
=> ( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ X5 )
=> ( ord_le6787938422905777998nnreal @ X @ A2 ) )
=> ( ! [Y2: set_Ex3793607809372303086nnreal] :
( ! [X6: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X6 @ X5 )
=> ( ord_le6787938422905777998nnreal @ X6 @ Y2 ) )
=> ( ord_le6787938422905777998nnreal @ A2 @ Y2 ) )
=> ( ( comple4226387801268262977nnreal @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1194_cSup__eq__non__empty,axiom,
! [X5: set_real,A2: real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X: real] :
( ( member_real @ X @ X5 )
=> ( ord_less_eq_real @ X @ A2 ) )
=> ( ! [Y2: real] :
( ! [X6: real] :
( ( member_real @ X6 @ X5 )
=> ( ord_less_eq_real @ X6 @ Y2 ) )
=> ( ord_less_eq_real @ A2 @ Y2 ) )
=> ( ( comple1385675409528146559p_real @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1195_Union__UNIV,axiom,
( ( comple4226387801268262977nnreal @ top_to3356475028079756884nnreal )
= top_to7994903218803871134nnreal ) ).
% Union_UNIV
thf(fact_1196_Union__UNIV,axiom,
( ( comple3096694443085538997t_real @ top_top_set_set_real )
= top_top_set_real ) ).
% Union_UNIV
thf(fact_1197_Union__UNIV,axiom,
( ( comple2307003609928055243_set_a @ top_top_set_set_a )
= top_top_set_a ) ).
% Union_UNIV
thf(fact_1198_insert__partition,axiom,
! [X2: set_a,F5: set_set_a] :
( ~ ( member_set_a @ X2 @ F5 )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ ( insert_set_a @ X2 @ F5 ) )
=> ! [Xa: set_a] :
( ( member_set_a @ Xa @ ( insert_set_a @ X2 @ F5 ) )
=> ( ( X != Xa )
=> ( ( inf_inf_set_a @ X @ Xa )
= bot_bot_set_a ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ ( comple2307003609928055243_set_a @ F5 ) )
= bot_bot_set_a ) ) ) ).
% insert_partition
thf(fact_1199_insert__partition,axiom,
! [X2: set_real,F5: set_set_real] :
( ~ ( member_set_real @ X2 @ F5 )
=> ( ! [X: set_real] :
( ( member_set_real @ X @ ( insert_set_real @ X2 @ F5 ) )
=> ! [Xa: set_real] :
( ( member_set_real @ Xa @ ( insert_set_real @ X2 @ F5 ) )
=> ( ( X != Xa )
=> ( ( inf_inf_set_real @ X @ Xa )
= bot_bot_set_real ) ) ) )
=> ( ( inf_inf_set_real @ X2 @ ( comple3096694443085538997t_real @ F5 ) )
= bot_bot_set_real ) ) ) ).
% insert_partition
thf(fact_1200_insert__partition,axiom,
! [X2: set_Ex3793607809372303086nnreal,F5: set_se4580700918925141924nnreal] :
( ~ ( member603777416030116741nnreal @ X2 @ F5 )
=> ( ! [X: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X @ ( insert1343806209672318238nnreal @ X2 @ F5 ) )
=> ! [Xa: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ Xa @ ( insert1343806209672318238nnreal @ X2 @ F5 ) )
=> ( ( X != Xa )
=> ( ( inf_in3368558534146122112nnreal @ X @ Xa )
= bot_bo4854962954004695426nnreal ) ) ) )
=> ( ( inf_in3368558534146122112nnreal @ X2 @ ( comple4226387801268262977nnreal @ F5 ) )
= bot_bo4854962954004695426nnreal ) ) ) ).
% insert_partition
thf(fact_1201_measure__distr,axiom,
! [F: real > a,M: sigma_measure_real,N: sigma_measure_a,S3: set_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
=> ( ( member_set_a @ S3 @ ( sigma_sets_a @ N ) )
=> ( ( sigma_measure_a2 @ ( measure_distr_real_a @ M @ N @ F ) @ S3 )
= ( sigma_measure_real2 @ M @ ( inf_inf_set_real @ ( vimage_real_a @ F @ S3 ) @ ( sigma_space_real @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1202_measure__distr,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,S3: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( member_set_real @ S3 @ ( sigma_sets_real @ N ) )
=> ( ( sigma_measure_real2 @ ( measur2993149975067245138l_real @ M @ N @ F ) @ S3 )
= ( sigma_measure_real2 @ M @ ( inf_inf_set_real @ ( vimage_real_real @ F @ S3 ) @ ( sigma_space_real @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1203_measure__distr,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,S3: set_Ex3793607809372303086nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ N ) )
=> ( ( sigma_5736856438657861608nnreal @ ( measur8829990298702910942nnreal @ M @ N @ F ) @ S3 )
= ( sigma_measure_real2 @ M @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ S3 ) @ ( sigma_space_real @ M ) ) ) ) ) ) ).
% measure_distr
thf(fact_1204_measure__lborel__Ioo,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or1633881224788618240n_real @ L @ U ) )
= ( minus_minus_real @ U @ L ) ) ) ).
% measure_lborel_Ioo
thf(fact_1205_measure__lborel__Ioc,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( set_or2392270231875598684t_real @ L @ U ) )
= ( minus_minus_real @ U @ L ) ) ) ).
% measure_lborel_Ioc
thf(fact_1206_measure__empty,axiom,
! [M: sigma_measure_real] :
( ( sigma_measure_real2 @ M @ bot_bot_set_real )
= zero_zero_real ) ).
% measure_empty
thf(fact_1207_measure__completion,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_measure_a2 @ ( comple3428971583294703880tion_a @ M ) @ S3 )
= ( sigma_measure_a2 @ M @ S3 ) ) ) ).
% measure_completion
thf(fact_1208_measure__completion,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( sigma_5736856438657861608nnreal @ ( comple6668017395272084142nnreal @ M ) @ S3 )
= ( sigma_5736856438657861608nnreal @ M @ S3 ) ) ) ).
% measure_completion
thf(fact_1209_measure__completion,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ M ) )
=> ( ( sigma_measure_real2 @ ( comple3506806835435775778n_real @ M ) @ S3 )
= ( sigma_measure_real2 @ M @ S3 ) ) ) ).
% measure_completion
thf(fact_1210_measure__lborel__singleton,axiom,
! [X2: real] :
( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( insert_real @ X2 @ bot_bot_set_real ) )
= zero_zero_real ) ).
% measure_lborel_singleton
thf(fact_1211_measure__le__0__iff,axiom,
! [M: sigma_measure_real,X5: set_real] :
( ( ord_less_eq_real @ ( sigma_measure_real2 @ M @ X5 ) @ zero_zero_real )
= ( ( sigma_measure_real2 @ M @ X5 )
= zero_zero_real ) ) ).
% measure_le_0_iff
thf(fact_1212_measure__nonneg,axiom,
! [M: sigma_measure_real,A: set_real] : ( ord_less_eq_real @ zero_zero_real @ ( sigma_measure_real2 @ M @ A ) ) ).
% measure_nonneg
thf(fact_1213_measure__notin__sets,axiom,
! [A: set_a,M: sigma_measure_a] :
( ~ ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( sigma_measure_a2 @ M @ A )
= zero_zero_real ) ) ).
% measure_notin_sets
thf(fact_1214_measure__notin__sets,axiom,
! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ~ ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( sigma_5736856438657861608nnreal @ M @ A )
= zero_zero_real ) ) ).
% measure_notin_sets
thf(fact_1215_measure__notin__sets,axiom,
! [A: set_real,M: sigma_measure_real] :
( ~ ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( ( sigma_measure_real2 @ M @ A )
= zero_zero_real ) ) ).
% measure_notin_sets
thf(fact_1216_measure__Diff__null__set,axiom,
! [A: set_a,M: sigma_measure_a,B2: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ B2 @ ( measure_null_sets_a @ M ) )
=> ( ( sigma_measure_a2 @ M @ ( minus_minus_set_a @ A @ B2 ) )
= ( sigma_measure_a2 @ M @ A ) ) ) ) ).
% measure_Diff_null_set
thf(fact_1217_measure__Diff__null__set,axiom,
! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( member603777416030116741nnreal @ B2 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( sigma_5736856438657861608nnreal @ M @ ( minus_104578273773384135nnreal @ A @ B2 ) )
= ( sigma_5736856438657861608nnreal @ M @ A ) ) ) ) ).
% measure_Diff_null_set
thf(fact_1218_measure__Diff__null__set,axiom,
! [A: set_real,M: sigma_measure_real,B2: set_real] :
( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( ( member_set_real @ B2 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( sigma_measure_real2 @ M @ ( minus_minus_set_real @ A @ B2 ) )
= ( sigma_measure_real2 @ M @ A ) ) ) ) ).
% measure_Diff_null_set
thf(fact_1219_measure__restrict__space,axiom,
! [Omega: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( inf_inf_set_a @ Omega @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ A @ Omega )
=> ( ( sigma_measure_a2 @ ( sigma_8692839461743104066pace_a @ M @ Omega ) @ A )
= ( sigma_measure_a2 @ M @ A ) ) ) ) ).
% measure_restrict_space
thf(fact_1220_measure__restrict__space,axiom,
! [Omega: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ ( inf_in3368558534146122112nnreal @ Omega @ ( sigma_3147302497200244656nnreal @ M ) ) @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( ord_le6787938422905777998nnreal @ A @ Omega )
=> ( ( sigma_5736856438657861608nnreal @ ( sigma_4884701650823297268nnreal @ M @ Omega ) @ A )
= ( sigma_5736856438657861608nnreal @ M @ A ) ) ) ) ).
% measure_restrict_space
thf(fact_1221_measure__restrict__space,axiom,
! [Omega: set_real,M: sigma_measure_real,A: set_real] :
( ( member_set_real @ ( inf_inf_set_real @ Omega @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
=> ( ( ord_less_eq_set_real @ A @ Omega )
=> ( ( sigma_measure_real2 @ ( sigma_5414646170262037096e_real @ M @ Omega ) @ A )
= ( sigma_measure_real2 @ M @ A ) ) ) ) ).
% measure_restrict_space
thf(fact_1222_content__singleton,axiom,
! [A2: real] :
( ( sigma_measure_real2 @ lebesgue_lborel_real @ ( insert_real @ A2 @ bot_bot_set_real ) )
= zero_zero_real ) ).
% content_singleton
thf(fact_1223_content__pos__le,axiom,
! [X5: set_real] : ( ord_less_eq_real @ zero_zero_real @ ( sigma_measure_real2 @ lebesgue_lborel_real @ X5 ) ) ).
% content_pos_le
thf(fact_1224_content__empty,axiom,
( ( sigma_measure_real2 @ lebesgue_lborel_real @ bot_bot_set_real )
= zero_zero_real ) ).
% content_empty
thf(fact_1225_measure__eq__0__null__sets,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( measure_null_sets_a @ M ) )
=> ( ( sigma_measure_a2 @ M @ S3 )
= zero_zero_real ) ) ).
% measure_eq_0_null_sets
thf(fact_1226_measure__eq__0__null__sets,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( sigma_5736856438657861608nnreal @ M @ S3 )
= zero_zero_real ) ) ).
% measure_eq_0_null_sets
thf(fact_1227_measure__eq__0__null__sets,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( measur3710062792471635001s_real @ M ) )
=> ( ( sigma_measure_real2 @ M @ S3 )
= zero_zero_real ) ) ).
% measure_eq_0_null_sets
thf(fact_1228_space__Sup__measure_H,axiom,
! [M: set_Sigma_measure_a,A: sigma_measure_a] :
( ! [M5: sigma_measure_a] :
( ( member3534519376729797778sure_a @ M5 @ M )
=> ( ( sigma_sets_a @ M5 )
= ( sigma_sets_a @ A ) ) )
=> ( ( M != bot_bo5171090233048072389sure_a )
=> ( ( sigma_space_a @ ( measur7752348538237249968sure_a @ M ) )
= ( sigma_space_a @ A ) ) ) ) ).
% space_Sup_measure'
thf(fact_1229_space__Sup__measure_H,axiom,
! [M: set_Si97717610131227249nnreal,A: sigma_7234349610311085201nnreal] :
( ! [M5: sigma_7234349610311085201nnreal] :
( ( member6261374078160781754nnreal @ M5 @ M )
=> ( ( sigma_5465916536984168985nnreal @ M5 )
= ( sigma_5465916536984168985nnreal @ A ) ) )
=> ( ( M != bot_bo8227844048696536285nnreal )
=> ( ( sigma_3147302497200244656nnreal @ ( measur1651139276328235014nnreal @ M ) )
= ( sigma_3147302497200244656nnreal @ A ) ) ) ) ).
% space_Sup_measure'
thf(fact_1230_space__Sup__measure_H,axiom,
! [M: set_Si6059263944882162789e_real,A: sigma_measure_real] :
( ! [M5: sigma_measure_real] :
( ( member4553183543495551918e_real @ M5 @ M )
=> ( ( sigma_sets_real @ M5 )
= ( sigma_sets_real @ A ) ) )
=> ( ( M != bot_bo5686449298802467025e_real )
=> ( ( sigma_space_real @ ( measur8657758558638653562e_real @ M ) )
= ( sigma_space_real @ A ) ) ) ) ).
% space_Sup_measure'
thf(fact_1231_emeasure__distr,axiom,
! [F: real > a,M: sigma_measure_real,N: sigma_measure_a,A: set_a] :
( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ N ) )
=> ( ( sigma_emeasure_a @ ( measure_distr_real_a @ M @ N @ F ) @ A )
= ( sigma_emeasure_real @ M @ ( inf_inf_set_real @ ( vimage_real_a @ F @ A ) @ ( sigma_space_real @ M ) ) ) ) ) ) ).
% emeasure_distr
thf(fact_1232_emeasure__distr,axiom,
! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,A: set_real] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ N ) )
=> ( ( sigma_emeasure_real @ ( measur2993149975067245138l_real @ M @ N @ F ) @ A )
= ( sigma_emeasure_real @ M @ ( inf_inf_set_real @ ( vimage_real_real @ F @ A ) @ ( sigma_space_real @ M ) ) ) ) ) ) ).
% emeasure_distr
thf(fact_1233_emeasure__distr,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ N ) )
=> ( ( sigma_6589832970846575905nnreal @ ( measur8829990298702910942nnreal @ M @ N @ F ) @ A )
= ( sigma_emeasure_real @ M @ ( inf_inf_set_real @ ( vimage6366802093293386401nnreal @ F @ A ) @ ( sigma_space_real @ M ) ) ) ) ) ) ).
% emeasure_distr
thf(fact_1234_emeasure__empty,axiom,
! [M: sigma_measure_real] :
( ( sigma_emeasure_real @ M @ bot_bot_set_real )
= zero_z7100319975126383169nnreal ) ).
% emeasure_empty
thf(fact_1235_emeasure__bot,axiom,
! [X5: set_real] :
( ( sigma_emeasure_real @ bot_bo5982154664989874033e_real @ X5 )
= zero_z7100319975126383169nnreal ) ).
% emeasure_bot
thf(fact_1236_null__setsI,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( ( sigma_emeasure_a @ M @ A )
= zero_z7100319975126383169nnreal )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( member_set_a @ A @ ( measure_null_sets_a @ M ) ) ) ) ).
% null_setsI
thf(fact_1237_null__setsI,axiom,
! [M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( ( sigma_6589832970846575905nnreal @ M @ A )
= zero_z7100319975126383169nnreal )
=> ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ M ) ) ) ) ).
% null_setsI
thf(fact_1238_null__setsI,axiom,
! [M: sigma_measure_real,A: set_real] :
( ( ( sigma_emeasure_real @ M @ A )
= zero_z7100319975126383169nnreal )
=> ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( member_set_real @ A @ ( measur3710062792471635001s_real @ M ) ) ) ) ).
% null_setsI
thf(fact_1239_null__part__sets_I2_J,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ ( complete_null_part_a @ M @ S3 ) )
= zero_z7100319975126383169nnreal ) ) ).
% null_part_sets(2)
thf(fact_1240_null__part__sets_I2_J,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( sigma_6589832970846575905nnreal @ M @ ( comple6358047150840085292nnreal @ M @ S3 ) )
= zero_z7100319975126383169nnreal ) ) ).
% null_part_sets(2)
thf(fact_1241_null__part__sets_I2_J,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ M ) )
=> ( ( sigma_emeasure_real @ M @ ( comple4917500974405109920t_real @ M @ S3 ) )
= zero_z7100319975126383169nnreal ) ) ).
% null_part_sets(2)
thf(fact_1242_emeasure__lborel__singleton,axiom,
! [X2: real] :
( ( sigma_emeasure_real @ lebesgue_lborel_real @ ( insert_real @ X2 @ bot_bot_set_real ) )
= zero_z7100319975126383169nnreal ) ).
% emeasure_lborel_singleton
thf(fact_1243_emeasure__completion,axiom,
! [S3: set_a,M: sigma_measure_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ( sigma_emeasure_a @ ( comple3428971583294703880tion_a @ M ) @ S3 )
= ( sigma_emeasure_a @ M @ ( complete_main_part_a @ M @ S3 ) ) ) ) ).
% emeasure_completion
thf(fact_1244_emeasure__completion,axiom,
! [S3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ S3 @ ( sigma_5465916536984168985nnreal @ ( comple6668017395272084142nnreal @ M ) ) )
=> ( ( sigma_6589832970846575905nnreal @ ( comple6668017395272084142nnreal @ M ) @ S3 )
= ( sigma_6589832970846575905nnreal @ M @ ( comple2904675884154540190nnreal @ M @ S3 ) ) ) ) ).
% emeasure_completion
thf(fact_1245_emeasure__completion,axiom,
! [S3: set_real,M: sigma_measure_real] :
( ( member_set_real @ S3 @ ( sigma_sets_real @ ( comple3506806835435775778n_real @ M ) ) )
=> ( ( sigma_emeasure_real @ ( comple3506806835435775778n_real @ M ) @ S3 )
= ( sigma_emeasure_real @ M @ ( comple5203310272383980818t_real @ M @ S3 ) ) ) ) ).
% emeasure_completion
thf(fact_1246_emeasure__space,axiom,
! [M: sigma_measure_real,A: set_real] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ M @ A ) @ ( sigma_emeasure_real @ M @ ( sigma_space_real @ M ) ) ) ).
% emeasure_space
thf(fact_1247_measure__eqI,axiom,
! [M: sigma_measure_a,N: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N ) )
=> ( ! [A9: set_a] :
( ( member_set_a @ A9 @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A9 )
= ( sigma_emeasure_a @ N @ A9 ) ) )
=> ( M = N ) ) ) ).
% measure_eqI
thf(fact_1248_measure__eqI,axiom,
! [M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal] :
( ( ( sigma_5465916536984168985nnreal @ M )
= ( sigma_5465916536984168985nnreal @ N ) )
=> ( ! [A9: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ A9 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( sigma_6589832970846575905nnreal @ M @ A9 )
= ( sigma_6589832970846575905nnreal @ N @ A9 ) ) )
=> ( M = N ) ) ) ).
% measure_eqI
thf(fact_1249_measure__eqI,axiom,
! [M: sigma_measure_real,N: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ N ) )
=> ( ! [A9: set_real] :
( ( member_set_real @ A9 @ ( sigma_sets_real @ M ) )
=> ( ( sigma_emeasure_real @ M @ A9 )
= ( sigma_emeasure_real @ N @ A9 ) ) )
=> ( M = N ) ) ) ).
% measure_eqI
thf(fact_1250_null__setsD1,axiom,
! [A: set_a,M: sigma_measure_a] :
( ( member_set_a @ A @ ( measure_null_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A )
= zero_z7100319975126383169nnreal ) ) ).
% null_setsD1
thf(fact_1251_null__setsD1,axiom,
! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( member603777416030116741nnreal @ A @ ( measur1209175464439008069nnreal @ M ) )
=> ( ( sigma_6589832970846575905nnreal @ M @ A )
= zero_z7100319975126383169nnreal ) ) ).
% null_setsD1
thf(fact_1252_null__setsD1,axiom,
! [A: set_real,M: sigma_measure_real] :
( ( member_set_real @ A @ ( measur3710062792471635001s_real @ M ) )
=> ( ( sigma_emeasure_real @ M @ A )
= zero_z7100319975126383169nnreal ) ) ).
% null_setsD1
thf(fact_1253_emeasure__neq__0__sets,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( ( sigma_emeasure_a @ M @ A )
!= zero_z7100319975126383169nnreal )
=> ( member_set_a @ A @ ( sigma_sets_a @ M ) ) ) ).
% emeasure_neq_0_sets
thf(fact_1254_emeasure__neq__0__sets,axiom,
! [M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
( ( ( sigma_6589832970846575905nnreal @ M @ A )
!= zero_z7100319975126383169nnreal )
=> ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).
% emeasure_neq_0_sets
thf(fact_1255_emeasure__neq__0__sets,axiom,
! [M: sigma_measure_real,A: set_real] :
( ( ( sigma_emeasure_real @ M @ A )
!= zero_z7100319975126383169nnreal )
=> ( member_set_real @ A @ ( sigma_sets_real @ M ) ) ) ).
% emeasure_neq_0_sets
thf(fact_1256_emeasure__notin__sets,axiom,
! [A: set_a,M: sigma_measure_a] :
( ~ ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( sigma_emeasure_a @ M @ A )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_notin_sets
thf(fact_1257_emeasure__notin__sets,axiom,
! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ~ ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( sigma_6589832970846575905nnreal @ M @ A )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_notin_sets
thf(fact_1258_emeasure__notin__sets,axiom,
! [A: set_real,M: sigma_measure_real] :
( ~ ( member_set_real @ A @ ( sigma_sets_real @ M ) )
=> ( ( sigma_emeasure_real @ M @ A )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_notin_sets
thf(fact_1259_emeasure__eq__0,axiom,
! [N: set_a,M: sigma_measure_a,K2: set_a] :
( ( member_set_a @ N @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ N )
= zero_z7100319975126383169nnreal )
=> ( ( ord_less_eq_set_a @ K2 @ N )
=> ( ( sigma_emeasure_a @ M @ K2 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% emeasure_eq_0
thf(fact_1260_emeasure__eq__0,axiom,
! [N: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,K2: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ N @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ( ( sigma_6589832970846575905nnreal @ M @ N )
= zero_z7100319975126383169nnreal )
=> ( ( ord_le6787938422905777998nnreal @ K2 @ N )
=> ( ( sigma_6589832970846575905nnreal @ M @ K2 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% emeasure_eq_0
thf(fact_1261_emeasure__eq__0,axiom,
! [N: set_real,M: sigma_measure_real,K2: set_real] :
( ( member_set_real @ N @ ( sigma_sets_real @ M ) )
=> ( ( ( sigma_emeasure_real @ M @ N )
= zero_z7100319975126383169nnreal )
=> ( ( ord_less_eq_set_real @ K2 @ N )
=> ( ( sigma_emeasure_real @ M @ K2 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% emeasure_eq_0
thf(fact_1262_emeasure__mono,axiom,
! [A2: set_a,B: set_a,M: sigma_measure_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ A2 ) @ ( sigma_emeasure_a @ M @ B ) ) ) ) ).
% emeasure_mono
thf(fact_1263_emeasure__mono,axiom,
! [A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B )
=> ( ( member603777416030116741nnreal @ B @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ M @ A2 ) @ ( sigma_6589832970846575905nnreal @ M @ B ) ) ) ) ).
% emeasure_mono
thf(fact_1264_emeasure__mono,axiom,
! [A2: set_real,B: set_real,M: sigma_measure_real] :
( ( ord_less_eq_set_real @ A2 @ B )
=> ( ( member_set_real @ B @ ( sigma_sets_real @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ M @ A2 ) @ ( sigma_emeasure_real @ M @ B ) ) ) ) ).
% emeasure_mono
thf(fact_1265_measure__zero__top,axiom,
! [M: sigma_measure_real,A: set_real] :
( ( ( sigma_emeasure_real @ M @ A )
= top_to1496364449551166952nnreal )
=> ( ( sigma_measure_real2 @ M @ A )
= zero_zero_real ) ) ).
% measure_zero_top
thf(fact_1266_emeasure__lborel__countable,axiom,
! [A: set_real] :
( ( counta7319604579010473777e_real @ A )
=> ( ( sigma_emeasure_real @ lebesgue_lborel_real @ A )
= zero_z7100319975126383169nnreal ) ) ).
% emeasure_lborel_countable
thf(fact_1267_le__measure,axiom,
! [M: sigma_measure_a,N: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N ) )
=> ( ( ord_le254669795585780187sure_a @ M @ N )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( sigma_sets_a @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ X3 ) @ ( sigma_emeasure_a @ N @ X3 ) ) ) ) ) ) ).
% le_measure
thf(fact_1268_le__measure,axiom,
! [M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal] :
( ( ( sigma_5465916536984168985nnreal @ M )
= ( sigma_5465916536984168985nnreal @ N ) )
=> ( ( ord_le1854472233513649201nnreal @ M @ N )
= ( ! [X3: set_Ex3793607809372303086nnreal] :
( ( member603777416030116741nnreal @ X3 @ ( sigma_5465916536984168985nnreal @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ M @ X3 ) @ ( sigma_6589832970846575905nnreal @ N @ X3 ) ) ) ) ) ) ).
% le_measure
thf(fact_1269_le__measure,axiom,
! [M: sigma_measure_real,N: sigma_measure_real] :
( ( ( sigma_sets_real @ M )
= ( sigma_sets_real @ N ) )
=> ( ( ord_le487379304121309861e_real @ M @ N )
= ( ! [X3: set_real] :
( ( member_set_real @ X3 @ ( sigma_sets_real @ M ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ M @ X3 ) @ ( sigma_emeasure_real @ N @ X3 ) ) ) ) ) ) ).
% le_measure
thf(fact_1270_le__measureD3,axiom,
! [A: sigma_measure_a,B2: sigma_measure_a,X5: set_a] :
( ( ord_le254669795585780187sure_a @ A @ B2 )
=> ( ( ( sigma_sets_a @ A )
= ( sigma_sets_a @ B2 ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ A @ X5 ) @ ( sigma_emeasure_a @ B2 @ X5 ) ) ) ) ).
% le_measureD3
thf(fact_1271_le__measureD3,axiom,
! [A: sigma_7234349610311085201nnreal,B2: sigma_7234349610311085201nnreal,X5: set_Ex3793607809372303086nnreal] :
( ( ord_le1854472233513649201nnreal @ A @ B2 )
=> ( ( ( sigma_5465916536984168985nnreal @ A )
= ( sigma_5465916536984168985nnreal @ B2 ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ A @ X5 ) @ ( sigma_6589832970846575905nnreal @ B2 @ X5 ) ) ) ) ).
% le_measureD3
thf(fact_1272_le__measureD3,axiom,
! [A: sigma_measure_real,B2: sigma_measure_real,X5: set_real] :
( ( ord_le487379304121309861e_real @ A @ B2 )
=> ( ( ( sigma_sets_real @ A )
= ( sigma_sets_real @ B2 ) )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ A @ X5 ) @ ( sigma_emeasure_real @ B2 @ X5 ) ) ) ) ).
% le_measureD3
thf(fact_1273_emeasure__lborel__Ioo,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sigma_emeasure_real @ lebesgue_lborel_real @ ( set_or1633881224788618240n_real @ L @ U ) )
= ( extend7643940197134561352nnreal @ ( minus_minus_real @ U @ L ) ) ) ) ).
% emeasure_lborel_Ioo
thf(fact_1274_emeasure__lborel__Ioc,axiom,
! [L: real,U: real] :
( ( ord_less_eq_real @ L @ U )
=> ( ( sigma_emeasure_real @ lebesgue_lborel_real @ ( set_or2392270231875598684t_real @ L @ U ) )
= ( extend7643940197134561352nnreal @ ( minus_minus_real @ U @ L ) ) ) ) ).
% emeasure_lborel_Ioc
thf(fact_1275_measurable__ennreal,axiom,
member2919562650594848410nnreal @ extend7643940197134561352nnreal @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).
% measurable_ennreal
thf(fact_1276_lmeasurable__interval_I2_J,axiom,
! [A2: real,B: real] : ( member_set_real @ ( set_or1633881224788618240n_real @ A2 @ B ) @ ( measur3487404108341735616e_real @ ( comple3506806835435775778n_real @ lebesgue_lborel_real ) ) ) ).
% lmeasurable_interval(2)
% Conjectures (1)
thf(conj_0,conjecture,
member_set_a @ i @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ).
%------------------------------------------------------------------------------