TPTP Problem File: ITP116^1.p
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%------------------------------------------------------------------------------
% File : ITP116^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Minkowskis_Theorem problem prob_203__6248138_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Minkowskis_Theorem/prob_203__6248138_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.38 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 476 ( 175 unt; 114 typ; 0 def)
% Number of atoms : 964 ( 319 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 3818 ( 35 ~; 4 |; 118 &;3341 @)
% ( 0 <=>; 320 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 646 ( 646 >; 0 *; 0 +; 0 <<)
% Number of symbols : 97 ( 94 usr; 8 con; 0-4 aty)
% Number of variables : 1206 ( 215 ^; 988 !; 3 ?;1206 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:45:44.564
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
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% Explicit typings (94)
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vimage1059850558real_n: ( finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ) > set_Fi1326602817real_n > set_Fi1058188332real_n ).
thf(sy_c_Set_Ovimage_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
vimage1233683625real_n: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n > set_Fi1058188332real_n ).
thf(sy_c_Set_Ovimage_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_001t__Nat__Onat,type,
vimage281029891_n_nat: ( finite1489363574real_n > nat ) > set_nat > set_Fi1058188332real_n ).
thf(sy_c_Set_Ovimage_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
vimage207290975real_n: ( finite1489363574real_n > set_Fi1058188332real_n ) > set_se2111327970real_n > set_Fi1058188332real_n ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
vimage2134951412real_n: ( set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n ) > set_Fi1326602817real_n > set_se2111327970real_n ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
vimage973736031real_n: ( set_Fi1058188332real_n > finite1489363574real_n ) > set_Fi1058188332real_n > set_se2111327970real_n ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Nat__Onat,type,
vimage501526201_n_nat: ( set_Fi1058188332real_n > nat ) > set_nat > set_se2111327970real_n ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
vimage784510485real_n: ( set_Fi1058188332real_n > set_Fi1058188332real_n ) > set_se2111327970real_n > set_se2111327970real_n ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_1185134568real_n: sigma_107786596real_n > sigma_107786596real_n > set_Fi1066397675real_n ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_2028985427real_n: sigma_107786596real_n > sigma_1466784463real_n > set_Fi1491909078real_n ).
thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_364818953real_n: sigma_107786596real_n > sigma_1422848389real_n > set_Fi909698444real_n ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_2016438227real_n: sigma_1466784463real_n > sigma_107786596real_n > set_Fi1260307670real_n ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_439801790real_n: sigma_1466784463real_n > sigma_1466784463real_n > set_Fi1326602817real_n ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_566919540real_n: sigma_1466784463real_n > sigma_1422848389real_n > set_Fi1645173239real_n ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_588796041real_n: sigma_1422848389real_n > sigma_107786596real_n > set_se830533260real_n ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_1333364596real_n: sigma_1422848389real_n > sigma_1466784463real_n > set_se955370231real_n ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_239294762real_n: sigma_1422848389real_n > sigma_1422848389real_n > set_se1738601133real_n ).
thf(sy_c_Sigma__Algebra_Orestrict__space_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_1052429895real_n: sigma_107786596real_n > set_Fi1326602817real_n > sigma_107786596real_n ).
thf(sy_c_Sigma__Algebra_Orestrict__space_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_346513458real_n: sigma_1466784463real_n > set_Fi1058188332real_n > sigma_1466784463real_n ).
thf(sy_c_Sigma__Algebra_Orestrict__space_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_993999336real_n: sigma_1422848389real_n > set_se2111327970real_n > sigma_1422848389real_n ).
thf(sy_c_Sigma__Algebra_Osets_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_522684908real_n: sigma_107786596real_n > set_se221767415real_n ).
thf(sy_c_Sigma__Algebra_Osets_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_1235138647real_n: sigma_1466784463real_n > set_se2111327970real_n ).
thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_433815053real_n: sigma_1422848389real_n > set_se820660888real_n ).
thf(sy_c_Sigma__Algebra_Ospace_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_1483971331real_n: sigma_107786596real_n > set_Fi1326602817real_n ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_476185326real_n: sigma_1466784463real_n > set_Fi1058188332real_n ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
sigma_607186084real_n: sigma_1422848389real_n > set_se2111327970real_n ).
thf(sy_c_Sigma__Algebra_Ovimage__algebra_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_136294295real_n: set_Fi1326602817real_n > ( ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n ) > sigma_1466784463real_n > sigma_107786596real_n ).
thf(sy_c_Sigma__Algebra_Ovimage__algebra_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_821351682real_n: set_Fi1058188332real_n > ( finite1489363574real_n > finite1489363574real_n ) > sigma_1466784463real_n > sigma_1466784463real_n ).
thf(sy_c_Sigma__Algebra_Ovimage__algebra_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_1384150200real_n: set_se2111327970real_n > ( set_Fi1058188332real_n > finite1489363574real_n ) > sigma_1466784463real_n > sigma_1422848389real_n ).
thf(sy_c_member_001_062_I_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_M_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
member117715276real_n: ( ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n ) > set_Fi1066397675real_n > $o ).
thf(sy_c_member_001_062_I_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
member1695588023real_n: ( ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n ) > set_Fi1491909078real_n > $o ).
thf(sy_c_member_001_062_I_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_Mt__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
member640587117real_n: ( ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n ) > set_Fi909698444real_n > $o ).
thf(sy_c_member_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_M_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
member408431031real_n: ( finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ) > set_Fi1260307670real_n > $o ).
thf(sy_c_member_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
member1746150050real_n: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1326602817real_n > $o ).
thf(sy_c_member_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
member966061400real_n: ( finite1489363574real_n > set_Fi1058188332real_n ) > set_Fi1645173239real_n > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_M_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
member1764433517real_n: ( set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n ) > set_se830533260real_n > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
member1759501912real_n: ( set_Fi1058188332real_n > finite1489363574real_n ) > set_se955370231real_n > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_Mt__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
member1734791438real_n: ( set_Fi1058188332real_n > set_Fi1058188332real_n ) > set_se1738601133real_n > $o ).
thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
member1352538125real_n: finite1489363574real_n > set_Fi1058188332real_n > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
member2104752728real_n: set_Fi1326602817real_n > set_se221767415real_n > $o ).
thf(sy_c_member_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
member223413699real_n: set_Fi1058188332real_n > set_se2111327970real_n > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
member1475136633real_n: set_se2111327970real_n > set_se820660888real_n > $o ).
thf(sy_v_R____,type,
r: finite964658038_int_n > set_Fi1058188332real_n ).
thf(sy_v_S,type,
s: set_Fi1058188332real_n ).
thf(sy_v_T_H____,type,
t: finite964658038_int_n > set_Fi1058188332real_n ).
thf(sy_v_T____,type,
t2: finite964658038_int_n > set_Fi1058188332real_n ).
thf(sy_v_a____,type,
a: finite964658038_int_n ).
% Relevant facts (354)
thf(fact_0_assms_I1_J,axiom,
member223413699real_n @ s @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ).
% assms(1)
thf(fact_1_of__int__vec__eq__iff,axiom,
! [A: finite964658038_int_n,B: finite964658038_int_n] :
( ( ( minkow1134813771n_real @ A )
= ( minkow1134813771n_real @ B ) )
= ( A = B ) ) ).
% of_int_vec_eq_iff
thf(fact_2__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_A_092_060Longrightarrow_062_A_I_092_060lambda_062x_O_Ax_A_L_Aof__int__vec_Aa_J_A_N_096_AT_Aa_A_092_060inter_062_Aspace_Alebesgue_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
! [A: finite964658038_int_n] :
( ( member223413699real_n @ ( t2 @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) )
=> ( member223413699real_n
@ ( inf_in1974387902real_n
@ ( vimage1233683625real_n
@ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ X @ ( minkow1134813771n_real @ A ) )
@ ( t2 @ A ) )
@ ( sigma_476185326real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) )
@ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% \<open>\<And>a. T a \<in> sets lebesgue \<Longrightarrow> (\<lambda>x. x + of_int_vec a) -` T a \<inter> space lebesgue \<in> sets lebesgue\<close>
thf(fact_3__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
! [A: finite964658038_int_n] : ( member223413699real_n @ ( t2 @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% \<open>\<And>a. T a \<in> sets lebesgue\<close>
thf(fact_4_T_H__altdef,axiom,
! [A: finite964658038_int_n] :
( ( t @ A )
= ( vimage1233683625real_n
@ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ X @ ( minkow1134813771n_real @ A ) )
@ ( t2 @ A ) ) ) ).
% T'_altdef
thf(fact_5__092_060open_062_092_060And_062a_O_AR_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
! [A: finite964658038_int_n] : ( member223413699real_n @ ( r @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% \<open>\<And>a. R a \<in> sets lebesgue\<close>
thf(fact_6_sets__completionI__sets,axiom,
! [A2: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ A2 @ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n @ A2 @ ( sigma_1235138647real_n @ ( comple230862828real_n @ M ) ) ) ) ).
% sets_completionI_sets
thf(fact_7__092_060open_062_092_060And_062a_O_A_I_092_060lambda_062x_O_Ax_A_L_Aof__int__vec_Aa_J_A_092_060in_062_Alebesgue_A_092_060rightarrow_062_092_060_094sub_062M_Alebesgue_092_060close_062,axiom,
! [A: finite964658038_int_n] :
( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ X @ ( minkow1134813771n_real @ A ) )
@ ( sigma_439801790real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% \<open>\<And>a. (\<lambda>x. x + of_int_vec a) \<in> lebesgue \<rightarrow>\<^sub>M lebesgue\<close>
thf(fact_8_vimage__ident,axiom,
! [Y: set_Fi1058188332real_n] :
( ( vimage1233683625real_n
@ ^ [X: finite1489363574real_n] : X
@ Y )
= Y ) ).
% vimage_ident
thf(fact_9_vimage__Collect__eq,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,P: finite1489363574real_n > $o] :
( ( vimage1233683625real_n @ F @ ( collec321817931real_n @ P ) )
= ( collec321817931real_n
@ ^ [Y2: finite1489363574real_n] : ( P @ ( F @ Y2 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_10_vimageI,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A: set_Fi1058188332real_n,B: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( ( F @ A )
= B )
=> ( ( member223413699real_n @ B @ B2 )
=> ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_11_vimageI,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,A: finite1489363574real_n > finite1489363574real_n,B: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( ( F @ A )
= B )
=> ( ( member223413699real_n @ B @ B2 )
=> ( member1746150050real_n @ A @ ( vimage1910974324real_n @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_12_vimageI,axiom,
! [F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,A: set_Fi1058188332real_n,B: finite1489363574real_n > finite1489363574real_n,B2: set_Fi1326602817real_n] :
( ( ( F @ A )
= B )
=> ( ( member1746150050real_n @ B @ B2 )
=> ( member223413699real_n @ A @ ( vimage2134951412real_n @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_13_vimageI,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,A: finite1489363574real_n > finite1489363574real_n,B: finite1489363574real_n > finite1489363574real_n,B2: set_Fi1326602817real_n] :
( ( ( F @ A )
= B )
=> ( ( member1746150050real_n @ B @ B2 )
=> ( member1746150050real_n @ A @ ( vimage180751827real_n @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_14_vimageI,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A: finite1489363574real_n,B: finite1489363574real_n,B2: set_Fi1058188332real_n] :
( ( ( F @ A )
= B )
=> ( ( member1352538125real_n @ B @ B2 )
=> ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_15_vimage__eq,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ B2 ) )
= ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_16_vimage__eq,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,B2: set_Fi1326602817real_n] :
( ( member223413699real_n @ A @ ( vimage2134951412real_n @ F @ B2 ) )
= ( member1746150050real_n @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_17_vimage__eq,axiom,
! [A: finite1489363574real_n > finite1489363574real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( member1746150050real_n @ A @ ( vimage1910974324real_n @ F @ B2 ) )
= ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_18_vimage__eq,axiom,
! [A: finite1489363574real_n > finite1489363574real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ A @ ( vimage180751827real_n @ F @ B2 ) )
= ( member1746150050real_n @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_19_vimage__eq,axiom,
! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ B2 ) )
= ( member1352538125real_n @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_20_set__plus__intro,axiom,
! [A: set_Fi1058188332real_n,C: set_se2111327970real_n,B: set_Fi1058188332real_n,D: set_se2111327970real_n] :
( ( member223413699real_n @ A @ C )
=> ( ( member223413699real_n @ B @ D )
=> ( member223413699real_n @ ( plus_p1606848693real_n @ A @ B ) @ ( plus_p565022571real_n @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_21_set__plus__intro,axiom,
! [A: finite1489363574real_n,C: set_Fi1058188332real_n,B: finite1489363574real_n,D: set_Fi1058188332real_n] :
( ( member1352538125real_n @ A @ C )
=> ( ( member1352538125real_n @ B @ D )
=> ( member1352538125real_n @ ( plus_p585657087real_n @ A @ B ) @ ( plus_p1606848693real_n @ C @ D ) ) ) ) ).
% set_plus_intro
thf(fact_22_add__left__cancel,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( ( plus_p585657087real_n @ A @ B )
= ( plus_p585657087real_n @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_23_add__right__cancel,axiom,
! [B: finite1489363574real_n,A: finite1489363574real_n,C2: finite1489363574real_n] :
( ( ( plus_p585657087real_n @ B @ A )
= ( plus_p585657087real_n @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_24_vimage__def,axiom,
( vimage207290975real_n
= ( ^ [F2: finite1489363574real_n > set_Fi1058188332real_n,B3: set_se2111327970real_n] :
( collec321817931real_n
@ ^ [X: finite1489363574real_n] : ( member223413699real_n @ ( F2 @ X ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_25_vimage__def,axiom,
( vimage1059850558real_n
= ( ^ [F2: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n,B3: set_Fi1326602817real_n] :
( collec321817931real_n
@ ^ [X: finite1489363574real_n] : ( member1746150050real_n @ ( F2 @ X ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_26_vimage__def,axiom,
( vimage1233683625real_n
= ( ^ [F2: finite1489363574real_n > finite1489363574real_n,B3: set_Fi1058188332real_n] :
( collec321817931real_n
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ ( F2 @ X ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_27_vimageD,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,A2: set_se2111327970real_n] :
( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ A2 ) )
=> ( member223413699real_n @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_28_vimageD,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n] :
( ( member223413699real_n @ A @ ( vimage2134951412real_n @ F @ A2 ) )
=> ( member1746150050real_n @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_29_vimageD,axiom,
! [A: finite1489363574real_n > finite1489363574real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,A2: set_se2111327970real_n] :
( ( member1746150050real_n @ A @ ( vimage1910974324real_n @ F @ A2 ) )
=> ( member223413699real_n @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_30_vimageD,axiom,
! [A: finite1489363574real_n > finite1489363574real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ A @ ( vimage180751827real_n @ F @ A2 ) )
=> ( member1746150050real_n @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_31_vimageD,axiom,
! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ A2 ) )
=> ( member1352538125real_n @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_32_vimageE,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ B2 ) )
=> ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_33_vimageE,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,B2: set_Fi1326602817real_n] :
( ( member223413699real_n @ A @ ( vimage2134951412real_n @ F @ B2 ) )
=> ( member1746150050real_n @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_34_vimageE,axiom,
! [A: finite1489363574real_n > finite1489363574real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( member1746150050real_n @ A @ ( vimage1910974324real_n @ F @ B2 ) )
=> ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_35_vimageE,axiom,
! [A: finite1489363574real_n > finite1489363574real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ A @ ( vimage180751827real_n @ F @ B2 ) )
=> ( member1746150050real_n @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_36_vimageE,axiom,
! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ B2 ) )
=> ( member1352538125real_n @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_37_IntI,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ A2 )
=> ( ( member223413699real_n @ C2 @ B2 )
=> ( member223413699real_n @ C2 @ ( inf_in632889204real_n @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_38_IntI,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ A2 )
=> ( ( member1746150050real_n @ C2 @ B2 )
=> ( member1746150050real_n @ C2 @ ( inf_in146441683real_n @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_39_IntI,axiom,
! [C2: finite1489363574real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C2 @ A2 )
=> ( ( member1352538125real_n @ C2 @ B2 )
=> ( member1352538125real_n @ C2 @ ( inf_in1974387902real_n @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_40_Int__iff,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ ( inf_in632889204real_n @ A2 @ B2 ) )
= ( ( member223413699real_n @ C2 @ A2 )
& ( member223413699real_n @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_41_Int__iff,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ ( inf_in146441683real_n @ A2 @ B2 ) )
= ( ( member1746150050real_n @ C2 @ A2 )
& ( member1746150050real_n @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_42_Int__iff,axiom,
! [C2: finite1489363574real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C2 @ ( inf_in1974387902real_n @ A2 @ B2 ) )
= ( ( member1352538125real_n @ C2 @ A2 )
& ( member1352538125real_n @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_43_space__completion,axiom,
! [M: sigma_1466784463real_n] :
( ( sigma_476185326real_n @ ( comple230862828real_n @ M ) )
= ( sigma_476185326real_n @ M ) ) ).
% space_completion
thf(fact_44_vimage__Int,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( vimage1233683625real_n @ F @ ( inf_in1974387902real_n @ A2 @ B2 ) )
= ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ A2 ) @ ( vimage1233683625real_n @ F @ B2 ) ) ) ).
% vimage_Int
thf(fact_45_T__def,axiom,
( t2
= ( ^ [A3: finite964658038_int_n] : ( inf_in1974387902real_n @ s @ ( r @ A3 ) ) ) ) ).
% T_def
thf(fact_46_IntE,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ ( inf_in632889204real_n @ A2 @ B2 ) )
=> ~ ( ( member223413699real_n @ C2 @ A2 )
=> ~ ( member223413699real_n @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_47_IntE,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ ( inf_in146441683real_n @ A2 @ B2 ) )
=> ~ ( ( member1746150050real_n @ C2 @ A2 )
=> ~ ( member1746150050real_n @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_48_IntE,axiom,
! [C2: finite1489363574real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C2 @ ( inf_in1974387902real_n @ A2 @ B2 ) )
=> ~ ( ( member1352538125real_n @ C2 @ A2 )
=> ~ ( member1352538125real_n @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_49_IntD1,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ ( inf_in632889204real_n @ A2 @ B2 ) )
=> ( member223413699real_n @ C2 @ A2 ) ) ).
% IntD1
thf(fact_50_IntD1,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ ( inf_in146441683real_n @ A2 @ B2 ) )
=> ( member1746150050real_n @ C2 @ A2 ) ) ).
% IntD1
thf(fact_51_IntD1,axiom,
! [C2: finite1489363574real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C2 @ ( inf_in1974387902real_n @ A2 @ B2 ) )
=> ( member1352538125real_n @ C2 @ A2 ) ) ).
% IntD1
thf(fact_52_IntD2,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ ( inf_in632889204real_n @ A2 @ B2 ) )
=> ( member223413699real_n @ C2 @ B2 ) ) ).
% IntD2
thf(fact_53_IntD2,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ ( inf_in146441683real_n @ A2 @ B2 ) )
=> ( member1746150050real_n @ C2 @ B2 ) ) ).
% IntD2
thf(fact_54_IntD2,axiom,
! [C2: finite1489363574real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C2 @ ( inf_in1974387902real_n @ A2 @ B2 ) )
=> ( member1352538125real_n @ C2 @ B2 ) ) ).
% IntD2
thf(fact_55_Int__assoc,axiom,
! [A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( inf_in1974387902real_n @ A2 @ B2 ) @ C )
= ( inf_in1974387902real_n @ A2 @ ( inf_in1974387902real_n @ B2 @ C ) ) ) ).
% Int_assoc
thf(fact_56_Int__absorb,axiom,
! [A2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_57_Int__commute,axiom,
( inf_in1974387902real_n
= ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_58_Int__left__absorb,axiom,
! [A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ A2 @ ( inf_in1974387902real_n @ A2 @ B2 ) )
= ( inf_in1974387902real_n @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_59_Int__left__commute,axiom,
! [A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ A2 @ ( inf_in1974387902real_n @ B2 @ C ) )
= ( inf_in1974387902real_n @ B2 @ ( inf_in1974387902real_n @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_60_Collect__conj__eq,axiom,
! [P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
( ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_in1974387902real_n @ ( collec321817931real_n @ P ) @ ( collec321817931real_n @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_61_Int__Collect,axiom,
! [X2: set_Fi1058188332real_n,A2: set_se2111327970real_n,P: set_Fi1058188332real_n > $o] :
( ( member223413699real_n @ X2 @ ( inf_in632889204real_n @ A2 @ ( collec452821761real_n @ P ) ) )
= ( ( member223413699real_n @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_62_Int__Collect,axiom,
! [X2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member1746150050real_n @ X2 @ ( inf_in146441683real_n @ A2 @ ( collec1190264032real_n @ P ) ) )
= ( ( member1746150050real_n @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_63_Int__Collect,axiom,
! [X2: finite1489363574real_n,A2: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
( ( member1352538125real_n @ X2 @ ( inf_in1974387902real_n @ A2 @ ( collec321817931real_n @ P ) ) )
= ( ( member1352538125real_n @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_64_Int__def,axiom,
( inf_in632889204real_n
= ( ^ [A4: set_se2111327970real_n,B3: set_se2111327970real_n] :
( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ A4 )
& ( member223413699real_n @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_65_Int__def,axiom,
( inf_in146441683real_n
= ( ^ [A4: set_Fi1326602817real_n,B3: set_Fi1326602817real_n] :
( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ A4 )
& ( member1746150050real_n @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_66_Int__def,axiom,
( inf_in1974387902real_n
= ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] :
( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ A4 )
& ( member1352538125real_n @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_67_measurable__completion,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,N: sigma_1466784463real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ N ) )
=> ( member1746150050real_n @ F @ ( sigma_439801790real_n @ ( comple230862828real_n @ M ) @ N ) ) ) ).
% measurable_completion
thf(fact_68_vimage__inter__cong,axiom,
! [S: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,Y3: set_Fi1058188332real_n] :
( ! [W: finite1489363574real_n] :
( ( member1352538125real_n @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ Y3 ) @ S )
= ( inf_in1974387902real_n @ ( vimage1233683625real_n @ G @ Y3 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_69_add__right__imp__eq,axiom,
! [B: finite1489363574real_n,A: finite1489363574real_n,C2: finite1489363574real_n] :
( ( ( plus_p585657087real_n @ B @ A )
= ( plus_p585657087real_n @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_70_add__left__imp__eq,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( ( plus_p585657087real_n @ A @ B )
= ( plus_p585657087real_n @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_71_add_Oleft__commute,axiom,
! [B: finite1489363574real_n,A: finite1489363574real_n,C2: finite1489363574real_n] :
( ( plus_p585657087real_n @ B @ ( plus_p585657087real_n @ A @ C2 ) )
= ( plus_p585657087real_n @ A @ ( plus_p585657087real_n @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_72_add_Ocommute,axiom,
( plus_p585657087real_n
= ( ^ [A3: finite1489363574real_n,B4: finite1489363574real_n] : ( plus_p585657087real_n @ B4 @ A3 ) ) ) ).
% add.commute
thf(fact_73_add_Oright__cancel,axiom,
! [B: finite1489363574real_n,A: finite1489363574real_n,C2: finite1489363574real_n] :
( ( ( plus_p585657087real_n @ B @ A )
= ( plus_p585657087real_n @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_74_add_Oleft__cancel,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( ( plus_p585657087real_n @ A @ B )
= ( plus_p585657087real_n @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_75_add_Oassoc,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( plus_p585657087real_n @ ( plus_p585657087real_n @ A @ B ) @ C2 )
= ( plus_p585657087real_n @ A @ ( plus_p585657087real_n @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_76_set__plus__elim,axiom,
! [X2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ X2 @ ( plus_p565022571real_n @ A2 @ B2 ) )
=> ~ ! [A5: set_Fi1058188332real_n,B5: set_Fi1058188332real_n] :
( ( X2
= ( plus_p1606848693real_n @ A5 @ B5 ) )
=> ( ( member223413699real_n @ A5 @ A2 )
=> ~ ( member223413699real_n @ B5 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_77_set__plus__elim,axiom,
! [X2: finite1489363574real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ X2 @ ( plus_p1606848693real_n @ A2 @ B2 ) )
=> ~ ! [A5: finite1489363574real_n,B5: finite1489363574real_n] :
( ( X2
= ( plus_p585657087real_n @ A5 @ B5 ) )
=> ( ( member1352538125real_n @ A5 @ A2 )
=> ~ ( member1352538125real_n @ B5 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_78_group__cancel_Oadd2,axiom,
! [B2: finite1489363574real_n,K: finite1489363574real_n,B: finite1489363574real_n,A: finite1489363574real_n] :
( ( B2
= ( plus_p585657087real_n @ K @ B ) )
=> ( ( plus_p585657087real_n @ A @ B2 )
= ( plus_p585657087real_n @ K @ ( plus_p585657087real_n @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_79_mem__Collect__eq,axiom,
! [A: set_Fi1058188332real_n,P: set_Fi1058188332real_n > $o] :
( ( member223413699real_n @ A @ ( collec452821761real_n @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: finite1489363574real_n > finite1489363574real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member1746150050real_n @ A @ ( collec1190264032real_n @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
! [A: finite1489363574real_n,P: finite1489363574real_n > $o] :
( ( member1352538125real_n @ A @ ( collec321817931real_n @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_se2111327970real_n] :
( ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_Fi1326602817real_n] :
( ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( member1746150050real_n @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A2: set_Fi1058188332real_n] :
( ( collec321817931real_n
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_85_Collect__cong,axiom,
! [P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
( ! [X3: finite1489363574real_n] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec321817931real_n @ P )
= ( collec321817931real_n @ Q ) ) ) ).
% Collect_cong
thf(fact_86_group__cancel_Oadd1,axiom,
! [A2: finite1489363574real_n,K: finite1489363574real_n,A: finite1489363574real_n,B: finite1489363574real_n] :
( ( A2
= ( plus_p585657087real_n @ K @ A ) )
=> ( ( plus_p585657087real_n @ A2 @ B )
= ( plus_p585657087real_n @ K @ ( plus_p585657087real_n @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_87_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: finite1489363574real_n,J: finite1489363574real_n,K: finite1489363574real_n,L: finite1489363574real_n] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p585657087real_n @ I @ K )
= ( plus_p585657087real_n @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_88_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( plus_p585657087real_n @ ( plus_p585657087real_n @ A @ B ) @ C2 )
= ( plus_p585657087real_n @ A @ ( plus_p585657087real_n @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_89_vimage__Collect,axiom,
! [P: finite1489363574real_n > $o,F: finite1489363574real_n > finite1489363574real_n,Q: finite1489363574real_n > $o] :
( ! [X3: finite1489363574real_n] :
( ( P @ ( F @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage1233683625real_n @ F @ ( collec321817931real_n @ P ) )
= ( collec321817931real_n @ Q ) ) ) ).
% vimage_Collect
thf(fact_90_vimageI2,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A: set_Fi1058188332real_n,A2: set_se2111327970real_n] :
( ( member223413699real_n @ ( F @ A ) @ A2 )
=> ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_91_vimageI2,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,A: finite1489363574real_n > finite1489363574real_n,A2: set_se2111327970real_n] :
( ( member223413699real_n @ ( F @ A ) @ A2 )
=> ( member1746150050real_n @ A @ ( vimage1910974324real_n @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_92_vimageI2,axiom,
! [F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,A: set_Fi1058188332real_n,A2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ ( F @ A ) @ A2 )
=> ( member223413699real_n @ A @ ( vimage2134951412real_n @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_93_vimageI2,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,A: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ ( F @ A ) @ A2 )
=> ( member1746150050real_n @ A @ ( vimage180751827real_n @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_94_vimageI2,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A: finite1489363574real_n,A2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ ( F @ A ) @ A2 )
=> ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_95_sets_OInt__space__eq2,axiom,
! [X2: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ X2 @ ( sigma_1235138647real_n @ M ) )
=> ( ( inf_in1974387902real_n @ X2 @ ( sigma_476185326real_n @ M ) )
= X2 ) ) ).
% sets.Int_space_eq2
thf(fact_96_sets_OInt__space__eq1,axiom,
! [X2: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ X2 @ ( sigma_1235138647real_n @ M ) )
=> ( ( inf_in1974387902real_n @ ( sigma_476185326real_n @ M ) @ X2 )
= X2 ) ) ).
% sets.Int_space_eq1
thf(fact_97_measurable__sets,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,A2: sigma_1466784463real_n,S: set_Fi1058188332real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ A2 ) )
=> ( ( member223413699real_n @ S @ ( sigma_1235138647real_n @ A2 ) )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ S ) @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) ) ) ) ).
% measurable_sets
thf(fact_98_measurableI,axiom,
! [M: sigma_1422848389real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,N: sigma_1422848389real_n] :
( ! [X3: set_Fi1058188332real_n] :
( ( member223413699real_n @ X3 @ ( sigma_607186084real_n @ M ) )
=> ( member223413699real_n @ ( F @ X3 ) @ ( sigma_607186084real_n @ N ) ) )
=> ( ! [A6: set_se2111327970real_n] :
( ( member1475136633real_n @ A6 @ ( sigma_433815053real_n @ N ) )
=> ( member1475136633real_n @ ( inf_in632889204real_n @ ( vimage784510485real_n @ F @ A6 ) @ ( sigma_607186084real_n @ M ) ) @ ( sigma_433815053real_n @ M ) ) )
=> ( member1734791438real_n @ F @ ( sigma_239294762real_n @ M @ N ) ) ) ) ).
% measurableI
thf(fact_99_measurableI,axiom,
! [M: sigma_1422848389real_n,F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,N: sigma_107786596real_n] :
( ! [X3: set_Fi1058188332real_n] :
( ( member223413699real_n @ X3 @ ( sigma_607186084real_n @ M ) )
=> ( member1746150050real_n @ ( F @ X3 ) @ ( sigma_1483971331real_n @ N ) ) )
=> ( ! [A6: set_Fi1326602817real_n] :
( ( member2104752728real_n @ A6 @ ( sigma_522684908real_n @ N ) )
=> ( member1475136633real_n @ ( inf_in632889204real_n @ ( vimage2134951412real_n @ F @ A6 ) @ ( sigma_607186084real_n @ M ) ) @ ( sigma_433815053real_n @ M ) ) )
=> ( member1764433517real_n @ F @ ( sigma_588796041real_n @ M @ N ) ) ) ) ).
% measurableI
thf(fact_100_measurableI,axiom,
! [M: sigma_107786596real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,N: sigma_1422848389real_n] :
( ! [X3: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X3 @ ( sigma_1483971331real_n @ M ) )
=> ( member223413699real_n @ ( F @ X3 ) @ ( sigma_607186084real_n @ N ) ) )
=> ( ! [A6: set_se2111327970real_n] :
( ( member1475136633real_n @ A6 @ ( sigma_433815053real_n @ N ) )
=> ( member2104752728real_n @ ( inf_in146441683real_n @ ( vimage1910974324real_n @ F @ A6 ) @ ( sigma_1483971331real_n @ M ) ) @ ( sigma_522684908real_n @ M ) ) )
=> ( member640587117real_n @ F @ ( sigma_364818953real_n @ M @ N ) ) ) ) ).
% measurableI
thf(fact_101_measurableI,axiom,
! [M: sigma_107786596real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,N: sigma_107786596real_n] :
( ! [X3: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X3 @ ( sigma_1483971331real_n @ M ) )
=> ( member1746150050real_n @ ( F @ X3 ) @ ( sigma_1483971331real_n @ N ) ) )
=> ( ! [A6: set_Fi1326602817real_n] :
( ( member2104752728real_n @ A6 @ ( sigma_522684908real_n @ N ) )
=> ( member2104752728real_n @ ( inf_in146441683real_n @ ( vimage180751827real_n @ F @ A6 ) @ ( sigma_1483971331real_n @ M ) ) @ ( sigma_522684908real_n @ M ) ) )
=> ( member117715276real_n @ F @ ( sigma_1185134568real_n @ M @ N ) ) ) ) ).
% measurableI
thf(fact_102_measurableI,axiom,
! [M: sigma_1422848389real_n,F: set_Fi1058188332real_n > finite1489363574real_n,N: sigma_1466784463real_n] :
( ! [X3: set_Fi1058188332real_n] :
( ( member223413699real_n @ X3 @ ( sigma_607186084real_n @ M ) )
=> ( member1352538125real_n @ ( F @ X3 ) @ ( sigma_476185326real_n @ N ) ) )
=> ( ! [A6: set_Fi1058188332real_n] :
( ( member223413699real_n @ A6 @ ( sigma_1235138647real_n @ N ) )
=> ( member1475136633real_n @ ( inf_in632889204real_n @ ( vimage973736031real_n @ F @ A6 ) @ ( sigma_607186084real_n @ M ) ) @ ( sigma_433815053real_n @ M ) ) )
=> ( member1759501912real_n @ F @ ( sigma_1333364596real_n @ M @ N ) ) ) ) ).
% measurableI
thf(fact_103_measurableI,axiom,
! [M: sigma_107786596real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n,N: sigma_1466784463real_n] :
( ! [X3: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X3 @ ( sigma_1483971331real_n @ M ) )
=> ( member1352538125real_n @ ( F @ X3 ) @ ( sigma_476185326real_n @ N ) ) )
=> ( ! [A6: set_Fi1058188332real_n] :
( ( member223413699real_n @ A6 @ ( sigma_1235138647real_n @ N ) )
=> ( member2104752728real_n @ ( inf_in146441683real_n @ ( vimage1072397758real_n @ F @ A6 ) @ ( sigma_1483971331real_n @ M ) ) @ ( sigma_522684908real_n @ M ) ) )
=> ( member1695588023real_n @ F @ ( sigma_2028985427real_n @ M @ N ) ) ) ) ).
% measurableI
thf(fact_104_measurableI,axiom,
! [M: sigma_1466784463real_n,F: finite1489363574real_n > set_Fi1058188332real_n,N: sigma_1422848389real_n] :
( ! [X3: finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ ( sigma_476185326real_n @ M ) )
=> ( member223413699real_n @ ( F @ X3 ) @ ( sigma_607186084real_n @ N ) ) )
=> ( ! [A6: set_se2111327970real_n] :
( ( member1475136633real_n @ A6 @ ( sigma_433815053real_n @ N ) )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ ( vimage207290975real_n @ F @ A6 ) @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) ) )
=> ( member966061400real_n @ F @ ( sigma_566919540real_n @ M @ N ) ) ) ) ).
% measurableI
thf(fact_105_measurableI,axiom,
! [M: sigma_1466784463real_n,F: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n,N: sigma_107786596real_n] :
( ! [X3: finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ ( sigma_476185326real_n @ M ) )
=> ( member1746150050real_n @ ( F @ X3 ) @ ( sigma_1483971331real_n @ N ) ) )
=> ( ! [A6: set_Fi1326602817real_n] :
( ( member2104752728real_n @ A6 @ ( sigma_522684908real_n @ N ) )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ ( vimage1059850558real_n @ F @ A6 ) @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) ) )
=> ( member408431031real_n @ F @ ( sigma_2016438227real_n @ M @ N ) ) ) ) ).
% measurableI
thf(fact_106_measurableI,axiom,
! [M: sigma_1466784463real_n,F: finite1489363574real_n > finite1489363574real_n,N: sigma_1466784463real_n] :
( ! [X3: finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ ( sigma_476185326real_n @ M ) )
=> ( member1352538125real_n @ ( F @ X3 ) @ ( sigma_476185326real_n @ N ) ) )
=> ( ! [A6: set_Fi1058188332real_n] :
( ( member223413699real_n @ A6 @ ( sigma_1235138647real_n @ N ) )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ A6 ) @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) ) )
=> ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ N ) ) ) ) ).
% measurableI
thf(fact_107_sets_Otop,axiom,
! [M: sigma_1466784463real_n] : ( member223413699real_n @ ( sigma_476185326real_n @ M ) @ ( sigma_1235138647real_n @ M ) ) ).
% sets.top
thf(fact_108_sets_OInt,axiom,
! [A: set_Fi1058188332real_n,M: sigma_1466784463real_n,B: set_Fi1058188332real_n] :
( ( member223413699real_n @ A @ ( sigma_1235138647real_n @ M ) )
=> ( ( member223413699real_n @ B @ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ A @ B ) @ ( sigma_1235138647real_n @ M ) ) ) ) ).
% sets.Int
thf(fact_109_measurable__If__set,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,M2: sigma_1466784463real_n,G: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ M2 ) )
=> ( ( member1746150050real_n @ G @ ( sigma_439801790real_n @ M @ M2 ) )
=> ( ( member223413699real_n @ ( inf_in1974387902real_n @ A2 @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) )
=> ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( if_Fin127821360real_n @ ( member1352538125real_n @ X @ A2 ) @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_439801790real_n @ M @ M2 ) ) ) ) ) ).
% measurable_If_set
thf(fact_110_measurable__sets__Collect,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,M: sigma_1422848389real_n,N: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o] :
( ( member1734791438real_n @ F @ ( sigma_239294762real_n @ M @ N ) )
=> ( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ N ) )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ N ) )
=> ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_433815053real_n @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_111_measurable__sets__Collect,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,M: sigma_107786596real_n,N: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o] :
( ( member640587117real_n @ F @ ( sigma_364818953real_n @ M @ N ) )
=> ( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ N ) )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ N ) )
=> ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_522684908real_n @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_112_measurable__sets__Collect,axiom,
! [F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,M: sigma_1422848389real_n,N: sigma_107786596real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member1764433517real_n @ F @ ( sigma_588796041real_n @ M @ N ) )
=> ( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ N ) )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ N ) )
=> ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_433815053real_n @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_113_measurable__sets__Collect,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,M: sigma_107786596real_n,N: sigma_107786596real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member117715276real_n @ F @ ( sigma_1185134568real_n @ M @ N ) )
=> ( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ N ) )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ N ) )
=> ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_522684908real_n @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_114_measurable__sets__Collect,axiom,
! [F: finite1489363574real_n > set_Fi1058188332real_n,M: sigma_1466784463real_n,N: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o] :
( ( member966061400real_n @ F @ ( sigma_566919540real_n @ M @ N ) )
=> ( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ N ) )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ N ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_1235138647real_n @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_115_measurable__sets__Collect,axiom,
! [F: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,N: sigma_107786596real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member408431031real_n @ F @ ( sigma_2016438227real_n @ M @ N ) )
=> ( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ N ) )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ N ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_1235138647real_n @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_116_measurable__sets__Collect,axiom,
! [F: set_Fi1058188332real_n > finite1489363574real_n,M: sigma_1422848389real_n,N: sigma_1466784463real_n,P: finite1489363574real_n > $o] :
( ( member1759501912real_n @ F @ ( sigma_1333364596real_n @ M @ N ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ N ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ N ) )
=> ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_433815053real_n @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_117_measurable__sets__Collect,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n,M: sigma_107786596real_n,N: sigma_1466784463real_n,P: finite1489363574real_n > $o] :
( ( member1695588023real_n @ F @ ( sigma_2028985427real_n @ M @ N ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ N ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ N ) )
=> ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_522684908real_n @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_118_measurable__sets__Collect,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,N: sigma_1466784463real_n,P: finite1489363574real_n > $o] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ N ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ N ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ N ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ ( F @ X ) ) ) )
@ ( sigma_1235138647real_n @ M ) ) ) ) ).
% measurable_sets_Collect
thf(fact_119_measurable__If,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,M2: sigma_1466784463real_n,G: finite1489363574real_n > finite1489363574real_n,P: finite1489363574real_n > $o] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ M2 ) )
=> ( ( member1746150050real_n @ G @ ( sigma_439801790real_n @ M @ M2 ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( if_Fin127821360real_n @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_439801790real_n @ M @ M2 ) ) ) ) ) ).
% measurable_If
thf(fact_120_inf__right__idem,axiom,
! [X2: set_Fi1058188332real_n,Y3: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( inf_in1974387902real_n @ X2 @ Y3 ) @ Y3 )
= ( inf_in1974387902real_n @ X2 @ Y3 ) ) ).
% inf_right_idem
thf(fact_121_inf_Oright__idem,axiom,
! [A: set_Fi1058188332real_n,B: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( inf_in1974387902real_n @ A @ B ) @ B )
= ( inf_in1974387902real_n @ A @ B ) ) ).
% inf.right_idem
thf(fact_122_inf__left__idem,axiom,
! [X2: set_Fi1058188332real_n,Y3: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ X2 @ ( inf_in1974387902real_n @ X2 @ Y3 ) )
= ( inf_in1974387902real_n @ X2 @ Y3 ) ) ).
% inf_left_idem
thf(fact_123_inf_Oleft__idem,axiom,
! [A: set_Fi1058188332real_n,B: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ A @ ( inf_in1974387902real_n @ A @ B ) )
= ( inf_in1974387902real_n @ A @ B ) ) ).
% inf.left_idem
thf(fact_124_inf__idem,axiom,
! [X2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_125_inf_Oidem,axiom,
! [A: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ A @ A )
= A ) ).
% inf.idem
thf(fact_126_inf__set__def,axiom,
( inf_in632889204real_n
= ( ^ [A4: set_se2111327970real_n,B3: set_se2111327970real_n] :
( collec452821761real_n
@ ( inf_in409346577al_n_o
@ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ A4 )
@ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_127_inf__set__def,axiom,
( inf_in146441683real_n
= ( ^ [A4: set_Fi1326602817real_n,B3: set_Fi1326602817real_n] :
( collec1190264032real_n
@ ( inf_in32002162al_n_o
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( member1746150050real_n @ X @ A4 )
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( member1746150050real_n @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_128_inf__set__def,axiom,
( inf_in1974387902real_n
= ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] :
( collec321817931real_n
@ ( inf_in1620715847al_n_o
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ X @ A4 )
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_129_inf__left__commute,axiom,
! [X2: set_Fi1058188332real_n,Y3: set_Fi1058188332real_n,Z: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ X2 @ ( inf_in1974387902real_n @ Y3 @ Z ) )
= ( inf_in1974387902real_n @ Y3 @ ( inf_in1974387902real_n @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_130_inf_Oleft__commute,axiom,
! [B: set_Fi1058188332real_n,A: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ B @ ( inf_in1974387902real_n @ A @ C2 ) )
= ( inf_in1974387902real_n @ A @ ( inf_in1974387902real_n @ B @ C2 ) ) ) ).
% inf.left_commute
thf(fact_131_inf__commute,axiom,
( inf_in1974387902real_n
= ( ^ [X: set_Fi1058188332real_n,Y2: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ Y2 @ X ) ) ) ).
% inf_commute
thf(fact_132_inf_Ocommute,axiom,
( inf_in1974387902real_n
= ( ^ [A3: set_Fi1058188332real_n,B4: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ B4 @ A3 ) ) ) ).
% inf.commute
thf(fact_133_inf__assoc,axiom,
! [X2: set_Fi1058188332real_n,Y3: set_Fi1058188332real_n,Z: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( inf_in1974387902real_n @ X2 @ Y3 ) @ Z )
= ( inf_in1974387902real_n @ X2 @ ( inf_in1974387902real_n @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_134_inf_Oassoc,axiom,
! [A: set_Fi1058188332real_n,B: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( inf_in1974387902real_n @ A @ B ) @ C2 )
= ( inf_in1974387902real_n @ A @ ( inf_in1974387902real_n @ B @ C2 ) ) ) ).
% inf.assoc
thf(fact_135_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_Fi1058188332real_n,K: set_Fi1058188332real_n,B: set_Fi1058188332real_n,A: set_Fi1058188332real_n] :
( ( B2
= ( inf_in1974387902real_n @ K @ B ) )
=> ( ( inf_in1974387902real_n @ A @ B2 )
= ( inf_in1974387902real_n @ K @ ( inf_in1974387902real_n @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_136_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_Fi1058188332real_n,K: set_Fi1058188332real_n,A: set_Fi1058188332real_n,B: set_Fi1058188332real_n] :
( ( A2
= ( inf_in1974387902real_n @ K @ A ) )
=> ( ( inf_in1974387902real_n @ A2 @ B )
= ( inf_in1974387902real_n @ K @ ( inf_in1974387902real_n @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_137_inf__sup__aci_I1_J,axiom,
( inf_in1974387902real_n
= ( ^ [X: set_Fi1058188332real_n,Y2: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ Y2 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_138_inf__sup__aci_I2_J,axiom,
! [X2: set_Fi1058188332real_n,Y3: set_Fi1058188332real_n,Z: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( inf_in1974387902real_n @ X2 @ Y3 ) @ Z )
= ( inf_in1974387902real_n @ X2 @ ( inf_in1974387902real_n @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_139_inf__sup__aci_I3_J,axiom,
! [X2: set_Fi1058188332real_n,Y3: set_Fi1058188332real_n,Z: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ X2 @ ( inf_in1974387902real_n @ Y3 @ Z ) )
= ( inf_in1974387902real_n @ Y3 @ ( inf_in1974387902real_n @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_140_inf__sup__aci_I4_J,axiom,
! [X2: set_Fi1058188332real_n,Y3: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ X2 @ ( inf_in1974387902real_n @ X2 @ Y3 ) )
= ( inf_in1974387902real_n @ X2 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_141_measurable__compose__rev,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,L2: sigma_1466784463real_n,N: sigma_1466784463real_n,G: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ L2 @ N ) )
=> ( ( member1746150050real_n @ G @ ( sigma_439801790real_n @ M @ L2 ) )
=> ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( F @ ( G @ X ) )
@ ( sigma_439801790real_n @ M @ N ) ) ) ) ).
% measurable_compose_rev
thf(fact_142_measurable__compose,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,N: sigma_1466784463real_n,G: finite1489363574real_n > finite1489363574real_n,L2: sigma_1466784463real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ N ) )
=> ( ( member1746150050real_n @ G @ ( sigma_439801790real_n @ N @ L2 ) )
=> ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( G @ ( F @ X ) )
@ ( sigma_439801790real_n @ M @ L2 ) ) ) ) ).
% measurable_compose
thf(fact_143_measurable__id,axiom,
! [M: sigma_1466784463real_n] :
( member1746150050real_n
@ ^ [X: finite1489363574real_n] : X
@ ( sigma_439801790real_n @ M @ M ) ) ).
% measurable_id
thf(fact_144_measurable__cong__sets,axiom,
! [M: sigma_1466784463real_n,M2: sigma_1466784463real_n,N: sigma_1466784463real_n,N2: sigma_1466784463real_n] :
( ( ( sigma_1235138647real_n @ M )
= ( sigma_1235138647real_n @ M2 ) )
=> ( ( ( sigma_1235138647real_n @ N )
= ( sigma_1235138647real_n @ N2 ) )
=> ( ( sigma_439801790real_n @ M @ N )
= ( sigma_439801790real_n @ M2 @ N2 ) ) ) ) ).
% measurable_cong_sets
thf(fact_145_sets__eq__imp__space__eq,axiom,
! [M: sigma_1466784463real_n,M2: sigma_1466784463real_n] :
( ( ( sigma_1235138647real_n @ M )
= ( sigma_1235138647real_n @ M2 ) )
=> ( ( sigma_476185326real_n @ M )
= ( sigma_476185326real_n @ M2 ) ) ) ).
% sets_eq_imp_space_eq
thf(fact_146_measurable__cong__simp,axiom,
! [M: sigma_1466784463real_n,N: sigma_1466784463real_n,M2: sigma_1466784463real_n,N2: sigma_1466784463real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n] :
( ( M = N )
=> ( ( M2 = N2 )
=> ( ! [W: finite1489363574real_n] :
( ( member1352538125real_n @ W @ ( sigma_476185326real_n @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ M2 ) )
= ( member1746150050real_n @ G @ ( sigma_439801790real_n @ N @ N2 ) ) ) ) ) ) ).
% measurable_cong_simp
thf(fact_147_measurable__space,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,M: sigma_1422848389real_n,A2: sigma_1422848389real_n,X2: set_Fi1058188332real_n] :
( ( member1734791438real_n @ F @ ( sigma_239294762real_n @ M @ A2 ) )
=> ( ( member223413699real_n @ X2 @ ( sigma_607186084real_n @ M ) )
=> ( member223413699real_n @ ( F @ X2 ) @ ( sigma_607186084real_n @ A2 ) ) ) ) ).
% measurable_space
thf(fact_148_measurable__space,axiom,
! [F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,M: sigma_1422848389real_n,A2: sigma_107786596real_n,X2: set_Fi1058188332real_n] :
( ( member1764433517real_n @ F @ ( sigma_588796041real_n @ M @ A2 ) )
=> ( ( member223413699real_n @ X2 @ ( sigma_607186084real_n @ M ) )
=> ( member1746150050real_n @ ( F @ X2 ) @ ( sigma_1483971331real_n @ A2 ) ) ) ) ).
% measurable_space
thf(fact_149_measurable__space,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,M: sigma_107786596real_n,A2: sigma_1422848389real_n,X2: finite1489363574real_n > finite1489363574real_n] :
( ( member640587117real_n @ F @ ( sigma_364818953real_n @ M @ A2 ) )
=> ( ( member1746150050real_n @ X2 @ ( sigma_1483971331real_n @ M ) )
=> ( member223413699real_n @ ( F @ X2 ) @ ( sigma_607186084real_n @ A2 ) ) ) ) ).
% measurable_space
thf(fact_150_measurable__space,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,M: sigma_107786596real_n,A2: sigma_107786596real_n,X2: finite1489363574real_n > finite1489363574real_n] :
( ( member117715276real_n @ F @ ( sigma_1185134568real_n @ M @ A2 ) )
=> ( ( member1746150050real_n @ X2 @ ( sigma_1483971331real_n @ M ) )
=> ( member1746150050real_n @ ( F @ X2 ) @ ( sigma_1483971331real_n @ A2 ) ) ) ) ).
% measurable_space
thf(fact_151_measurable__space,axiom,
! [F: set_Fi1058188332real_n > finite1489363574real_n,M: sigma_1422848389real_n,A2: sigma_1466784463real_n,X2: set_Fi1058188332real_n] :
( ( member1759501912real_n @ F @ ( sigma_1333364596real_n @ M @ A2 ) )
=> ( ( member223413699real_n @ X2 @ ( sigma_607186084real_n @ M ) )
=> ( member1352538125real_n @ ( F @ X2 ) @ ( sigma_476185326real_n @ A2 ) ) ) ) ).
% measurable_space
thf(fact_152_measurable__space,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n,M: sigma_107786596real_n,A2: sigma_1466784463real_n,X2: finite1489363574real_n > finite1489363574real_n] :
( ( member1695588023real_n @ F @ ( sigma_2028985427real_n @ M @ A2 ) )
=> ( ( member1746150050real_n @ X2 @ ( sigma_1483971331real_n @ M ) )
=> ( member1352538125real_n @ ( F @ X2 ) @ ( sigma_476185326real_n @ A2 ) ) ) ) ).
% measurable_space
thf(fact_153_measurable__space,axiom,
! [F: finite1489363574real_n > set_Fi1058188332real_n,M: sigma_1466784463real_n,A2: sigma_1422848389real_n,X2: finite1489363574real_n] :
( ( member966061400real_n @ F @ ( sigma_566919540real_n @ M @ A2 ) )
=> ( ( member1352538125real_n @ X2 @ ( sigma_476185326real_n @ M ) )
=> ( member223413699real_n @ ( F @ X2 ) @ ( sigma_607186084real_n @ A2 ) ) ) ) ).
% measurable_space
thf(fact_154_measurable__space,axiom,
! [F: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,A2: sigma_107786596real_n,X2: finite1489363574real_n] :
( ( member408431031real_n @ F @ ( sigma_2016438227real_n @ M @ A2 ) )
=> ( ( member1352538125real_n @ X2 @ ( sigma_476185326real_n @ M ) )
=> ( member1746150050real_n @ ( F @ X2 ) @ ( sigma_1483971331real_n @ A2 ) ) ) ) ).
% measurable_space
thf(fact_155_measurable__space,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,A2: sigma_1466784463real_n,X2: finite1489363574real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ A2 ) )
=> ( ( member1352538125real_n @ X2 @ ( sigma_476185326real_n @ M ) )
=> ( member1352538125real_n @ ( F @ X2 ) @ ( sigma_476185326real_n @ A2 ) ) ) ) ).
% measurable_space
thf(fact_156_measurable__cong,axiom,
! [M: sigma_1466784463real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,M2: sigma_1466784463real_n] :
( ! [W: finite1489363574real_n] :
( ( member1352538125real_n @ W @ ( sigma_476185326real_n @ M ) )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ M2 ) )
= ( member1746150050real_n @ G @ ( sigma_439801790real_n @ M @ M2 ) ) ) ) ).
% measurable_cong
thf(fact_157_measurable__ident__sets,axiom,
! [M: sigma_1466784463real_n,M2: sigma_1466784463real_n] :
( ( ( sigma_1235138647real_n @ M )
= ( sigma_1235138647real_n @ M2 ) )
=> ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : X
@ ( sigma_439801790real_n @ M @ M2 ) ) ) ).
% measurable_ident_sets
thf(fact_158_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_1422848389real_n,Pb: $o] :
( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& Pb ) )
@ ( sigma_433815053real_n @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_159_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_107786596real_n,Pb: $o] :
( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& Pb ) )
@ ( sigma_522684908real_n @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_160_sets_Osets__Collect_I5_J,axiom,
! [M: sigma_1466784463real_n,Pb: $o] :
( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& Pb ) )
@ ( sigma_1235138647real_n @ M ) ) ).
% sets.sets_Collect(5)
thf(fact_161_sets_Osets__Collect__imp,axiom,
! [M: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o,Q: set_Fi1058188332real_n > $o] :
( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ M ) )
=> ( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( Q @ X ) ) )
@ ( sigma_433815053real_n @ M ) )
=> ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( ( Q @ X )
=> ( P @ X ) ) ) )
@ ( sigma_433815053real_n @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_162_sets_Osets__Collect__imp,axiom,
! [M: sigma_107786596real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o,Q: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ M ) )
=> ( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( Q @ X ) ) )
@ ( sigma_522684908real_n @ M ) )
=> ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( ( Q @ X )
=> ( P @ X ) ) ) )
@ ( sigma_522684908real_n @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_163_sets_Osets__Collect__imp,axiom,
! [M: sigma_1466784463real_n,P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( Q @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( ( Q @ X )
=> ( P @ X ) ) ) )
@ ( sigma_1235138647real_n @ M ) ) ) ) ).
% sets.sets_Collect_imp
thf(fact_164_sets_Osets__Collect__neg,axiom,
! [M: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o] :
( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ M ) )
=> ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ~ ( P @ X ) ) )
@ ( sigma_433815053real_n @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_165_sets_Osets__Collect__neg,axiom,
! [M: sigma_107786596real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ M ) )
=> ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ~ ( P @ X ) ) )
@ ( sigma_522684908real_n @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_166_sets_Osets__Collect__neg,axiom,
! [M: sigma_1466784463real_n,P: finite1489363574real_n > $o] :
( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ~ ( P @ X ) ) )
@ ( sigma_1235138647real_n @ M ) ) ) ).
% sets.sets_Collect_neg
thf(fact_167_sets_Osets__Collect__conj,axiom,
! [M: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o,Q: set_Fi1058188332real_n > $o] :
( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ M ) )
=> ( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( Q @ X ) ) )
@ ( sigma_433815053real_n @ M ) )
=> ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( Q @ X )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_168_sets_Osets__Collect__conj,axiom,
! [M: sigma_107786596real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o,Q: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ M ) )
=> ( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( Q @ X ) ) )
@ ( sigma_522684908real_n @ M ) )
=> ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( Q @ X )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_169_sets_Osets__Collect__conj,axiom,
! [M: sigma_1466784463real_n,P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( Q @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( Q @ X )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ M ) ) ) ) ).
% sets.sets_Collect_conj
thf(fact_170_sets_Osets__Collect__disj,axiom,
! [M: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o,Q: set_Fi1058188332real_n > $o] :
( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ M ) )
=> ( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( Q @ X ) ) )
@ ( sigma_433815053real_n @ M ) )
=> ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( ( Q @ X )
| ( P @ X ) ) ) )
@ ( sigma_433815053real_n @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_171_sets_Osets__Collect__disj,axiom,
! [M: sigma_107786596real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o,Q: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ M ) )
=> ( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( Q @ X ) ) )
@ ( sigma_522684908real_n @ M ) )
=> ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( ( Q @ X )
| ( P @ X ) ) ) )
@ ( sigma_522684908real_n @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_172_sets_Osets__Collect__disj,axiom,
! [M: sigma_1466784463real_n,P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( Q @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( ( Q @ X )
| ( P @ X ) ) ) )
@ ( sigma_1235138647real_n @ M ) ) ) ) ).
% sets.sets_Collect_disj
thf(fact_173_sets_Osets__Collect__const,axiom,
! [M: sigma_1422848389real_n,P: $o] :
( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& P ) )
@ ( sigma_433815053real_n @ M ) ) ).
% sets.sets_Collect_const
thf(fact_174_sets_Osets__Collect__const,axiom,
! [M: sigma_107786596real_n,P: $o] :
( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& P ) )
@ ( sigma_522684908real_n @ M ) ) ).
% sets.sets_Collect_const
thf(fact_175_sets_Osets__Collect__const,axiom,
! [M: sigma_1466784463real_n,P: $o] :
( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& P ) )
@ ( sigma_1235138647real_n @ M ) ) ).
% sets.sets_Collect_const
thf(fact_176_measurable__const,axiom,
! [C2: finite1489363574real_n,M2: sigma_1466784463real_n,M: sigma_1466784463real_n] :
( ( member1352538125real_n @ C2 @ ( sigma_476185326real_n @ M2 ) )
=> ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : C2
@ ( sigma_439801790real_n @ M @ M2 ) ) ) ).
% measurable_const
thf(fact_177_T_H__def,axiom,
( t
= ( ^ [A3: finite964658038_int_n] :
( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ ( minkow1134813771n_real @ A3 ) )
@ ( t2 @ A3 ) ) ) ) ).
% T'_def
thf(fact_178_inf__Int__eq,axiom,
! [R: set_se2111327970real_n,S: set_se2111327970real_n] :
( ( inf_in409346577al_n_o
@ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ R )
@ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ S ) )
= ( ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ ( inf_in632889204real_n @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_179_inf__Int__eq,axiom,
! [R: set_Fi1326602817real_n,S: set_Fi1326602817real_n] :
( ( inf_in32002162al_n_o
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( member1746150050real_n @ X @ R )
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( member1746150050real_n @ X @ S ) )
= ( ^ [X: finite1489363574real_n > finite1489363574real_n] : ( member1746150050real_n @ X @ ( inf_in146441683real_n @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_180_inf__Int__eq,axiom,
! [R: set_Fi1058188332real_n,S: set_Fi1058188332real_n] :
( ( inf_in1620715847al_n_o
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ X @ R )
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ X @ S ) )
= ( ^ [X: finite1489363574real_n] : ( member1352538125real_n @ X @ ( inf_in1974387902real_n @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_181_main__part__sets,axiom,
! [S: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ S @ ( sigma_1235138647real_n @ ( comple230862828real_n @ M ) ) )
=> ( member223413699real_n @ ( comple1390568924real_n @ M @ S ) @ ( sigma_1235138647real_n @ M ) ) ) ).
% main_part_sets
thf(fact_182_measurable__restrict__space__iff,axiom,
! [Omega: set_se2111327970real_n,M: sigma_1422848389real_n,C2: set_Fi1058188332real_n,N: sigma_1422848389real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n] :
( ( member1475136633real_n @ ( inf_in632889204real_n @ Omega @ ( sigma_607186084real_n @ M ) ) @ ( sigma_433815053real_n @ M ) )
=> ( ( member223413699real_n @ C2 @ ( sigma_607186084real_n @ N ) )
=> ( ( member1734791438real_n @ F @ ( sigma_239294762real_n @ ( sigma_993999336real_n @ M @ Omega ) @ N ) )
= ( member1734791438real_n
@ ^ [X: set_Fi1058188332real_n] : ( if_set11487206real_n @ ( member223413699real_n @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_239294762real_n @ M @ N ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_183_measurable__restrict__space__iff,axiom,
! [Omega: set_Fi1326602817real_n,M: sigma_107786596real_n,C2: set_Fi1058188332real_n,N: sigma_1422848389real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n] :
( ( member2104752728real_n @ ( inf_in146441683real_n @ Omega @ ( sigma_1483971331real_n @ M ) ) @ ( sigma_522684908real_n @ M ) )
=> ( ( member223413699real_n @ C2 @ ( sigma_607186084real_n @ N ) )
=> ( ( member640587117real_n @ F @ ( sigma_364818953real_n @ ( sigma_1052429895real_n @ M @ Omega ) @ N ) )
= ( member640587117real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( if_set11487206real_n @ ( member1746150050real_n @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_364818953real_n @ M @ N ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_184_measurable__restrict__space__iff,axiom,
! [Omega: set_se2111327970real_n,M: sigma_1422848389real_n,C2: finite1489363574real_n > finite1489363574real_n,N: sigma_107786596real_n,F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n] :
( ( member1475136633real_n @ ( inf_in632889204real_n @ Omega @ ( sigma_607186084real_n @ M ) ) @ ( sigma_433815053real_n @ M ) )
=> ( ( member1746150050real_n @ C2 @ ( sigma_1483971331real_n @ N ) )
=> ( ( member1764433517real_n @ F @ ( sigma_588796041real_n @ ( sigma_993999336real_n @ M @ Omega ) @ N ) )
= ( member1764433517real_n
@ ^ [X: set_Fi1058188332real_n] : ( if_Fin413489477real_n @ ( member223413699real_n @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_588796041real_n @ M @ N ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_185_measurable__restrict__space__iff,axiom,
! [Omega: set_Fi1326602817real_n,M: sigma_107786596real_n,C2: finite1489363574real_n > finite1489363574real_n,N: sigma_107786596real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n] :
( ( member2104752728real_n @ ( inf_in146441683real_n @ Omega @ ( sigma_1483971331real_n @ M ) ) @ ( sigma_522684908real_n @ M ) )
=> ( ( member1746150050real_n @ C2 @ ( sigma_1483971331real_n @ N ) )
=> ( ( member117715276real_n @ F @ ( sigma_1185134568real_n @ ( sigma_1052429895real_n @ M @ Omega ) @ N ) )
= ( member117715276real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( if_Fin413489477real_n @ ( member1746150050real_n @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_1185134568real_n @ M @ N ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_186_measurable__restrict__space__iff,axiom,
! [Omega: set_se2111327970real_n,M: sigma_1422848389real_n,C2: finite1489363574real_n,N: sigma_1466784463real_n,F: set_Fi1058188332real_n > finite1489363574real_n] :
( ( member1475136633real_n @ ( inf_in632889204real_n @ Omega @ ( sigma_607186084real_n @ M ) ) @ ( sigma_433815053real_n @ M ) )
=> ( ( member1352538125real_n @ C2 @ ( sigma_476185326real_n @ N ) )
=> ( ( member1759501912real_n @ F @ ( sigma_1333364596real_n @ ( sigma_993999336real_n @ M @ Omega ) @ N ) )
= ( member1759501912real_n
@ ^ [X: set_Fi1058188332real_n] : ( if_Fin127821360real_n @ ( member223413699real_n @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_1333364596real_n @ M @ N ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_187_measurable__restrict__space__iff,axiom,
! [Omega: set_Fi1326602817real_n,M: sigma_107786596real_n,C2: finite1489363574real_n,N: sigma_1466784463real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n] :
( ( member2104752728real_n @ ( inf_in146441683real_n @ Omega @ ( sigma_1483971331real_n @ M ) ) @ ( sigma_522684908real_n @ M ) )
=> ( ( member1352538125real_n @ C2 @ ( sigma_476185326real_n @ N ) )
=> ( ( member1695588023real_n @ F @ ( sigma_2028985427real_n @ ( sigma_1052429895real_n @ M @ Omega ) @ N ) )
= ( member1695588023real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( if_Fin127821360real_n @ ( member1746150050real_n @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_2028985427real_n @ M @ N ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_188_measurable__restrict__space__iff,axiom,
! [Omega: set_Fi1058188332real_n,M: sigma_1466784463real_n,C2: set_Fi1058188332real_n,N: sigma_1422848389real_n,F: finite1489363574real_n > set_Fi1058188332real_n] :
( ( member223413699real_n @ ( inf_in1974387902real_n @ Omega @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) )
=> ( ( member223413699real_n @ C2 @ ( sigma_607186084real_n @ N ) )
=> ( ( member966061400real_n @ F @ ( sigma_566919540real_n @ ( sigma_346513458real_n @ M @ Omega ) @ N ) )
= ( member966061400real_n
@ ^ [X: finite1489363574real_n] : ( if_set11487206real_n @ ( member1352538125real_n @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_566919540real_n @ M @ N ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_189_measurable__restrict__space__iff,axiom,
! [Omega: set_Fi1058188332real_n,M: sigma_1466784463real_n,C2: finite1489363574real_n > finite1489363574real_n,N: sigma_107786596real_n,F: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n] :
( ( member223413699real_n @ ( inf_in1974387902real_n @ Omega @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) )
=> ( ( member1746150050real_n @ C2 @ ( sigma_1483971331real_n @ N ) )
=> ( ( member408431031real_n @ F @ ( sigma_2016438227real_n @ ( sigma_346513458real_n @ M @ Omega ) @ N ) )
= ( member408431031real_n
@ ^ [X: finite1489363574real_n] : ( if_Fin413489477real_n @ ( member1352538125real_n @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_2016438227real_n @ M @ N ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_190_measurable__restrict__space__iff,axiom,
! [Omega: set_Fi1058188332real_n,M: sigma_1466784463real_n,C2: finite1489363574real_n,N: sigma_1466784463real_n,F: finite1489363574real_n > finite1489363574real_n] :
( ( member223413699real_n @ ( inf_in1974387902real_n @ Omega @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) )
=> ( ( member1352538125real_n @ C2 @ ( sigma_476185326real_n @ N ) )
=> ( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ ( sigma_346513458real_n @ M @ Omega ) @ N ) )
= ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( if_Fin127821360real_n @ ( member1352538125real_n @ X @ Omega ) @ ( F @ X ) @ C2 )
@ ( sigma_439801790real_n @ M @ N ) ) ) ) ) ).
% measurable_restrict_space_iff
thf(fact_191_sets__Least,axiom,
! [M: sigma_1422848389real_n,P: nat > set_Fi1058188332real_n > $o,A2: set_nat] :
( ! [I2: nat] :
( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( P @ I2 @ X ) ) )
@ ( sigma_433815053real_n @ M ) )
=> ( member1475136633real_n
@ ( inf_in632889204real_n
@ ( vimage501526201_n_nat
@ ^ [X: set_Fi1058188332real_n] :
( ord_Least_nat
@ ^ [J2: nat] : ( P @ J2 @ X ) )
@ A2 )
@ ( sigma_607186084real_n @ M ) )
@ ( sigma_433815053real_n @ M ) ) ) ).
% sets_Least
thf(fact_192_sets__Least,axiom,
! [M: sigma_107786596real_n,P: nat > ( finite1489363574real_n > finite1489363574real_n ) > $o,A2: set_nat] :
( ! [I2: nat] :
( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( P @ I2 @ X ) ) )
@ ( sigma_522684908real_n @ M ) )
=> ( member2104752728real_n
@ ( inf_in146441683real_n
@ ( vimage2098059032_n_nat
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ord_Least_nat
@ ^ [J2: nat] : ( P @ J2 @ X ) )
@ A2 )
@ ( sigma_1483971331real_n @ M ) )
@ ( sigma_522684908real_n @ M ) ) ) ).
% sets_Least
thf(fact_193_sets__Least,axiom,
! [M: sigma_1466784463real_n,P: nat > finite1489363574real_n > $o,A2: set_nat] :
( ! [I2: nat] :
( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ I2 @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n
@ ( inf_in1974387902real_n
@ ( vimage281029891_n_nat
@ ^ [X: finite1489363574real_n] :
( ord_Least_nat
@ ^ [J2: nat] : ( P @ J2 @ X ) )
@ A2 )
@ ( sigma_476185326real_n @ M ) )
@ ( sigma_1235138647real_n @ M ) ) ) ).
% sets_Least
thf(fact_194_in__vimage__algebra,axiom,
! [A2: set_Fi1058188332real_n,M: sigma_1466784463real_n,F: finite1489363574real_n > finite1489363574real_n,X4: set_Fi1058188332real_n] :
( ( member223413699real_n @ A2 @ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ A2 ) @ X4 ) @ ( sigma_1235138647real_n @ ( sigma_821351682real_n @ X4 @ F @ M ) ) ) ) ).
% in_vimage_algebra
thf(fact_195_in__borel__measurable__borel,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ borel_676189912real_n ) )
= ( ! [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_1235138647real_n @ borel_676189912real_n ) )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ X ) @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) ) ) ) ) ).
% in_borel_measurable_borel
thf(fact_196_image__eqI,axiom,
! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,X2: finite1489363574real_n,A2: set_Fi1058188332real_n] :
( ( B
= ( F @ X2 ) )
=> ( ( member1352538125real_n @ X2 @ A2 )
=> ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_197_image__eqI,axiom,
! [B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,X2: set_Fi1058188332real_n,A2: set_se2111327970real_n] :
( ( B
= ( F @ X2 ) )
=> ( ( member223413699real_n @ X2 @ A2 )
=> ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_198_image__eqI,axiom,
! [B: finite1489363574real_n > finite1489363574real_n,F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,X2: set_Fi1058188332real_n,A2: set_se2111327970real_n] :
( ( B
= ( F @ X2 ) )
=> ( ( member223413699real_n @ X2 @ A2 )
=> ( member1746150050real_n @ B @ ( image_352856126real_n @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_199_image__eqI,axiom,
! [B: set_Fi1058188332real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,X2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n] :
( ( B
= ( F @ X2 ) )
=> ( ( member1746150050real_n @ X2 @ A2 )
=> ( member223413699real_n @ B @ ( image_128879038real_n @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_200_image__eqI,axiom,
! [B: finite1489363574real_n > finite1489363574real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,X2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n] :
( ( B
= ( F @ X2 ) )
=> ( ( member1746150050real_n @ X2 @ A2 )
=> ( member1746150050real_n @ B @ ( image_1123376925real_n @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_201_image__ident,axiom,
! [Y: set_Fi1058188332real_n] :
( ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_202_add__diff__cancel__right_H,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( minus_1037315151real_n @ ( plus_p585657087real_n @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_203_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_204_add__diff__cancel__right,axiom,
! [A: finite1489363574real_n,C2: finite1489363574real_n,B: finite1489363574real_n] :
( ( minus_1037315151real_n @ ( plus_p585657087real_n @ A @ C2 ) @ ( plus_p585657087real_n @ B @ C2 ) )
= ( minus_1037315151real_n @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_205_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_206_add__diff__cancel__left_H,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( minus_1037315151real_n @ ( plus_p585657087real_n @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_207_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_208_add__diff__cancel__left,axiom,
! [C2: finite1489363574real_n,A: finite1489363574real_n,B: finite1489363574real_n] :
( ( minus_1037315151real_n @ ( plus_p585657087real_n @ C2 @ A ) @ ( plus_p585657087real_n @ C2 @ B ) )
= ( minus_1037315151real_n @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_209_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_210_diff__add__cancel,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( plus_p585657087real_n @ ( minus_1037315151real_n @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_211_add__diff__cancel,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( minus_1037315151real_n @ ( plus_p585657087real_n @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_212_main__part,axiom,
! [S: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ S @ ( sigma_1235138647real_n @ M ) )
=> ( ( comple1390568924real_n @ M @ S )
= S ) ) ).
% main_part
thf(fact_213_space__restrict__space2,axiom,
! [Omega: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ Omega @ ( sigma_1235138647real_n @ M ) )
=> ( ( sigma_476185326real_n @ ( sigma_346513458real_n @ M @ Omega ) )
= Omega ) ) ).
% space_restrict_space2
thf(fact_214_borel__measurable__diff,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,G: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ borel_676189912real_n ) )
=> ( ( member1746150050real_n @ G @ ( sigma_439801790real_n @ M @ borel_676189912real_n ) )
=> ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_439801790real_n @ M @ borel_676189912real_n ) ) ) ) ).
% borel_measurable_diff
thf(fact_215_Compr__image__eq,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A2: set_se2111327970real_n,P: set_Fi1058188332real_n > $o] :
( ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( image_1661509983real_n @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_1661509983real_n @ F
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_216_Compr__image__eq,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,A2: set_Fi1326602817real_n,P: set_Fi1058188332real_n > $o] :
( ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( image_128879038real_n @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_128879038real_n @ F
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_217_Compr__image__eq,axiom,
! [F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,A2: set_se2111327970real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( image_352856126real_n @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_352856126real_n @ F
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_218_Compr__image__eq,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( image_1123376925real_n @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_1123376925real_n @ F
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_219_Compr__image__eq,axiom,
! [F: finite1489363574real_n > set_Fi1058188332real_n,A2: set_Fi1058188332real_n,P: set_Fi1058188332real_n > $o] :
( ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( image_545463721real_n @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_545463721real_n @ F
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_220_Compr__image__eq,axiom,
! [F: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( image_437359496real_n @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_437359496real_n @ F
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_221_Compr__image__eq,axiom,
! [F: set_Fi1058188332real_n > finite1489363574real_n,A2: set_se2111327970real_n,P: finite1489363574real_n > $o] :
( ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( image_1311908777real_n @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_1311908777real_n @ F
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_222_Compr__image__eq,axiom,
! [F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n,A2: set_Fi1326602817real_n,P: finite1489363574real_n > $o] :
( ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( image_449906696real_n @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_449906696real_n @ F
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_223_Compr__image__eq,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
( ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( image_439535603real_n @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_439535603real_n @ F
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_224_image__image,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n] :
( ( image_439535603real_n @ F @ ( image_439535603real_n @ G @ A2 ) )
= ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_225_imageE,axiom,
! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A2 ) )
=> ~ ! [X3: finite1489363574real_n] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1352538125real_n @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_226_imageE,axiom,
! [B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,A2: set_se2111327970real_n] :
( ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A2 ) )
=> ~ ! [X3: set_Fi1058188332real_n] :
( ( B
= ( F @ X3 ) )
=> ~ ( member223413699real_n @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_227_imageE,axiom,
! [B: set_Fi1058188332real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n,A2: set_Fi1326602817real_n] :
( ( member223413699real_n @ B @ ( image_128879038real_n @ F @ A2 ) )
=> ~ ! [X3: finite1489363574real_n > finite1489363574real_n] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1746150050real_n @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_228_imageE,axiom,
! [B: finite1489363574real_n > finite1489363574real_n,F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n,A2: set_se2111327970real_n] :
( ( member1746150050real_n @ B @ ( image_352856126real_n @ F @ A2 ) )
=> ~ ! [X3: set_Fi1058188332real_n] :
( ( B
= ( F @ X3 ) )
=> ~ ( member223413699real_n @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_229_imageE,axiom,
! [B: finite1489363574real_n > finite1489363574real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ B @ ( image_1123376925real_n @ F @ A2 ) )
=> ~ ! [X3: finite1489363574real_n > finite1489363574real_n] :
( ( B
= ( F @ X3 ) )
=> ~ ( member1746150050real_n @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_230_diff__right__commute,axiom,
! [A: finite1489363574real_n,C2: finite1489363574real_n,B: finite1489363574real_n] :
( ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ C2 ) @ B )
= ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_231_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_232_diff__eq__diff__eq,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n,D2: finite1489363574real_n] :
( ( ( minus_1037315151real_n @ A @ B )
= ( minus_1037315151real_n @ C2 @ D2 ) )
=> ( ( A = B )
= ( C2 = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_233_rev__image__eqI,axiom,
! [X2: finite1489363574real_n,A2: set_Fi1058188332real_n,B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
( ( member1352538125real_n @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_234_rev__image__eqI,axiom,
! [X2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n] :
( ( member223413699real_n @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_235_rev__image__eqI,axiom,
! [X2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B: finite1489363574real_n > finite1489363574real_n,F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n] :
( ( member223413699real_n @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member1746150050real_n @ B @ ( image_352856126real_n @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_236_rev__image__eqI,axiom,
! [X2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B: set_Fi1058188332real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n] :
( ( member1746150050real_n @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member223413699real_n @ B @ ( image_128879038real_n @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_237_rev__image__eqI,axiom,
! [X2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B: finite1489363574real_n > finite1489363574real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member1746150050real_n @ B @ ( image_1123376925real_n @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_238_ball__imageD,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
( ! [X3: finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ ( image_439535603real_n @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: finite1489363574real_n] :
( ( member1352538125real_n @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_239_image__cong,axiom,
! [M: set_Fi1058188332real_n,N: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n] :
( ( M = N )
=> ( ! [X3: finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_439535603real_n @ F @ M )
= ( image_439535603real_n @ G @ N ) ) ) ) ).
% image_cong
thf(fact_240_bex__imageD,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
( ? [X5: finite1489363574real_n] :
( ( member1352538125real_n @ X5 @ ( image_439535603real_n @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_241_image__iff,axiom,
! [Z: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ Z @ ( image_439535603real_n @ F @ A2 ) )
= ( ? [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_242_imageI,axiom,
! [X2: finite1489363574real_n,A2: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n] :
( ( member1352538125real_n @ X2 @ A2 )
=> ( member1352538125real_n @ ( F @ X2 ) @ ( image_439535603real_n @ F @ A2 ) ) ) ).
% imageI
thf(fact_243_imageI,axiom,
! [X2: set_Fi1058188332real_n,A2: set_se2111327970real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n] :
( ( member223413699real_n @ X2 @ A2 )
=> ( member223413699real_n @ ( F @ X2 ) @ ( image_1661509983real_n @ F @ A2 ) ) ) ).
% imageI
thf(fact_244_imageI,axiom,
! [X2: set_Fi1058188332real_n,A2: set_se2111327970real_n,F: set_Fi1058188332real_n > finite1489363574real_n > finite1489363574real_n] :
( ( member223413699real_n @ X2 @ A2 )
=> ( member1746150050real_n @ ( F @ X2 ) @ ( image_352856126real_n @ F @ A2 ) ) ) ).
% imageI
thf(fact_245_imageI,axiom,
! [X2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n] :
( ( member1746150050real_n @ X2 @ A2 )
=> ( member223413699real_n @ ( F @ X2 ) @ ( image_128879038real_n @ F @ A2 ) ) ) ).
% imageI
thf(fact_246_imageI,axiom,
! [X2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X2 @ A2 )
=> ( member1746150050real_n @ ( F @ X2 ) @ ( image_1123376925real_n @ F @ A2 ) ) ) ).
% imageI
thf(fact_247_borel__measurable__const,axiom,
! [C2: finite1489363574real_n,M: sigma_1466784463real_n] :
( member1746150050real_n
@ ^ [X: finite1489363574real_n] : C2
@ ( sigma_439801790real_n @ M @ borel_676189912real_n ) ) ).
% borel_measurable_const
thf(fact_248_sets__restrict__space__cong,axiom,
! [M: sigma_1466784463real_n,N: sigma_1466784463real_n,Omega: set_Fi1058188332real_n] :
( ( ( sigma_1235138647real_n @ M )
= ( sigma_1235138647real_n @ N ) )
=> ( ( sigma_1235138647real_n @ ( sigma_346513458real_n @ M @ Omega ) )
= ( sigma_1235138647real_n @ ( sigma_346513458real_n @ N @ Omega ) ) ) ) ).
% sets_restrict_space_cong
thf(fact_249_restrict__space__sets__cong,axiom,
! [A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,M: sigma_1466784463real_n,N: sigma_1466784463real_n] :
( ( A2 = B2 )
=> ( ( ( sigma_1235138647real_n @ M )
= ( sigma_1235138647real_n @ N ) )
=> ( ( sigma_1235138647real_n @ ( sigma_346513458real_n @ M @ A2 ) )
= ( sigma_1235138647real_n @ ( sigma_346513458real_n @ N @ B2 ) ) ) ) ) ).
% restrict_space_sets_cong
thf(fact_250_measurable__restrict__space1,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,N: sigma_1466784463real_n,Omega: set_Fi1058188332real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ N ) )
=> ( member1746150050real_n @ F @ ( sigma_439801790real_n @ ( sigma_346513458real_n @ M @ Omega ) @ N ) ) ) ).
% measurable_restrict_space1
thf(fact_251_add__implies__diff,axiom,
! [C2: finite1489363574real_n,B: finite1489363574real_n,A: finite1489363574real_n] :
( ( ( plus_p585657087real_n @ C2 @ B )
= A )
=> ( C2
= ( minus_1037315151real_n @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_252_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_253_diff__diff__add,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ B ) @ C2 )
= ( minus_1037315151real_n @ A @ ( plus_p585657087real_n @ B @ C2 ) ) ) ).
% diff_diff_add
thf(fact_254_diff__diff__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_add
thf(fact_255_diff__add__eq__diff__diff__swap,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( minus_1037315151real_n @ A @ ( plus_p585657087real_n @ B @ C2 ) )
= ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_256_diff__add__eq,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( plus_p585657087real_n @ ( minus_1037315151real_n @ A @ B ) @ C2 )
= ( minus_1037315151real_n @ ( plus_p585657087real_n @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_257_diff__diff__eq2,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( minus_1037315151real_n @ A @ ( minus_1037315151real_n @ B @ C2 ) )
= ( minus_1037315151real_n @ ( plus_p585657087real_n @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_258_add__diff__eq,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( plus_p585657087real_n @ A @ ( minus_1037315151real_n @ B @ C2 ) )
= ( minus_1037315151real_n @ ( plus_p585657087real_n @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_259_eq__diff__eq,axiom,
! [A: finite1489363574real_n,C2: finite1489363574real_n,B: finite1489363574real_n] :
( ( A
= ( minus_1037315151real_n @ C2 @ B ) )
= ( ( plus_p585657087real_n @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_260_diff__eq__eq,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C2: finite1489363574real_n] :
( ( ( minus_1037315151real_n @ A @ B )
= C2 )
= ( A
= ( plus_p585657087real_n @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_261_group__cancel_Osub1,axiom,
! [A2: finite1489363574real_n,K: finite1489363574real_n,A: finite1489363574real_n,B: finite1489363574real_n] :
( ( A2
= ( plus_p585657087real_n @ K @ A ) )
=> ( ( minus_1037315151real_n @ A2 @ B )
= ( plus_p585657087real_n @ K @ ( minus_1037315151real_n @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_262_sets__vimage__algebra__cong,axiom,
! [M: sigma_1466784463real_n,N: sigma_1466784463real_n,X4: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n] :
( ( ( sigma_1235138647real_n @ M )
= ( sigma_1235138647real_n @ N ) )
=> ( ( sigma_1235138647real_n @ ( sigma_821351682real_n @ X4 @ F @ M ) )
= ( sigma_1235138647real_n @ ( sigma_821351682real_n @ X4 @ F @ N ) ) ) ) ).
% sets_vimage_algebra_cong
thf(fact_263_vimage__algebra__cong,axiom,
! [X4: set_se2111327970real_n,Y: set_se2111327970real_n,F: set_Fi1058188332real_n > finite1489363574real_n,G: set_Fi1058188332real_n > finite1489363574real_n,M: sigma_1466784463real_n,N: sigma_1466784463real_n] :
( ( X4 = Y )
=> ( ! [X3: set_Fi1058188332real_n] :
( ( member223413699real_n @ X3 @ Y )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( ( sigma_1235138647real_n @ M )
= ( sigma_1235138647real_n @ N ) )
=> ( ( sigma_1384150200real_n @ X4 @ F @ M )
= ( sigma_1384150200real_n @ Y @ G @ N ) ) ) ) ) ).
% vimage_algebra_cong
thf(fact_264_vimage__algebra__cong,axiom,
! [X4: set_Fi1326602817real_n,Y: set_Fi1326602817real_n,F: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n,G: ( finite1489363574real_n > finite1489363574real_n ) > finite1489363574real_n,M: sigma_1466784463real_n,N: sigma_1466784463real_n] :
( ( X4 = Y )
=> ( ! [X3: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X3 @ Y )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( ( sigma_1235138647real_n @ M )
= ( sigma_1235138647real_n @ N ) )
=> ( ( sigma_136294295real_n @ X4 @ F @ M )
= ( sigma_136294295real_n @ Y @ G @ N ) ) ) ) ) ).
% vimage_algebra_cong
thf(fact_265_borel__measurable__const__add,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,A: finite1489363574real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ borel_676189912real_n ) )
=> ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ A @ ( F @ X ) )
@ ( sigma_439801790real_n @ M @ borel_676189912real_n ) ) ) ).
% borel_measurable_const_add
thf(fact_266_borel__measurable__add,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,G: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ borel_676189912real_n ) )
=> ( ( member1746150050real_n @ G @ ( sigma_439801790real_n @ M @ borel_676189912real_n ) )
=> ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_439801790real_n @ M @ borel_676189912real_n ) ) ) ) ).
% borel_measurable_add
thf(fact_267_sets__restrict__restrict__space,axiom,
! [M: sigma_1466784463real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( sigma_1235138647real_n @ ( sigma_346513458real_n @ ( sigma_346513458real_n @ M @ A2 ) @ B2 ) )
= ( sigma_1235138647real_n @ ( sigma_346513458real_n @ M @ ( inf_in1974387902real_n @ A2 @ B2 ) ) ) ) ).
% sets_restrict_restrict_space
thf(fact_268_space__restrict__space,axiom,
! [M: sigma_1466784463real_n,Omega: set_Fi1058188332real_n] :
( ( sigma_476185326real_n @ ( sigma_346513458real_n @ M @ Omega ) )
= ( inf_in1974387902real_n @ Omega @ ( sigma_476185326real_n @ M ) ) ) ).
% space_restrict_space
thf(fact_269_measurable__sets__borel,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,A2: set_Fi1058188332real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ borel_676189912real_n @ M ) )
=> ( ( member223413699real_n @ A2 @ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n @ ( vimage1233683625real_n @ F @ A2 ) @ ( sigma_1235138647real_n @ borel_676189912real_n ) ) ) ) ).
% measurable_sets_borel
thf(fact_270_sets__Collect__restrict__space__iff,axiom,
! [S: set_se2111327970real_n,M: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o] :
( ( member1475136633real_n @ S @ ( sigma_433815053real_n @ M ) )
=> ( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ ( sigma_993999336real_n @ M @ S ) ) )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ ( sigma_993999336real_n @ M @ S ) ) )
= ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( member223413699real_n @ X @ S )
& ( P @ X ) ) )
@ ( sigma_433815053real_n @ M ) ) ) ) ).
% sets_Collect_restrict_space_iff
thf(fact_271_sets__Collect__restrict__space__iff,axiom,
! [S: set_Fi1326602817real_n,M: sigma_107786596real_n,P: ( finite1489363574real_n > finite1489363574real_n ) > $o] :
( ( member2104752728real_n @ S @ ( sigma_522684908real_n @ M ) )
=> ( ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ ( sigma_1052429895real_n @ M @ S ) ) )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ ( sigma_1052429895real_n @ M @ S ) ) )
= ( member2104752728real_n
@ ( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ ( sigma_1483971331real_n @ M ) )
& ( member1746150050real_n @ X @ S )
& ( P @ X ) ) )
@ ( sigma_522684908real_n @ M ) ) ) ) ).
% sets_Collect_restrict_space_iff
thf(fact_272_sets__Collect__restrict__space__iff,axiom,
! [S: set_Fi1058188332real_n,M: sigma_1466784463real_n,P: finite1489363574real_n > $o] :
( ( member223413699real_n @ S @ ( sigma_1235138647real_n @ M ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ ( sigma_346513458real_n @ M @ S ) ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ ( sigma_346513458real_n @ M @ S ) ) )
= ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( member1352538125real_n @ X @ S )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ M ) ) ) ) ).
% sets_Collect_restrict_space_iff
thf(fact_273_measurable__equality__set,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n,G: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ M @ borel_676189912real_n ) )
=> ( ( member1746150050real_n @ G @ ( sigma_439801790real_n @ M @ borel_676189912real_n ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( ( F @ X )
= ( G @ X ) ) ) )
@ ( sigma_1235138647real_n @ M ) ) ) ) ).
% measurable_equality_set
thf(fact_274_measurable__If__restrict__space__iff,axiom,
! [M: sigma_1466784463real_n,P: finite1489363574real_n > $o,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,N: sigma_1466784463real_n] :
( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ M ) )
=> ( ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( if_Fin127821360real_n @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_439801790real_n @ M @ N ) )
= ( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ ( sigma_346513458real_n @ M @ ( collec321817931real_n @ P ) ) @ N ) )
& ( member1746150050real_n @ G
@ ( sigma_439801790real_n
@ ( sigma_346513458real_n @ M
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
~ ( P @ X ) ) )
@ N ) ) ) ) ) ).
% measurable_If_restrict_space_iff
thf(fact_275_space__lebesgue__on,axiom,
! [S: set_Fi1058188332real_n] :
( ( sigma_476185326real_n @ ( sigma_346513458real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ S ) )
= S ) ).
% space_lebesgue_on
thf(fact_276_sets__lebesgue__on__refl,axiom,
! [S: set_Fi1058188332real_n] : ( member223413699real_n @ S @ ( sigma_1235138647real_n @ ( sigma_346513458real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ S ) ) ) ).
% sets_lebesgue_on_refl
thf(fact_277_space__lborel,axiom,
( ( sigma_476185326real_n @ lebesg260170249real_n )
= ( sigma_476185326real_n @ borel_676189912real_n ) ) ).
% space_lborel
thf(fact_278_measurable__lborel1,axiom,
! [M: sigma_1466784463real_n] :
( ( sigma_439801790real_n @ M @ lebesg260170249real_n )
= ( sigma_439801790real_n @ M @ borel_676189912real_n ) ) ).
% measurable_lborel1
thf(fact_279_measurable__lborel2,axiom,
! [M: sigma_1466784463real_n] :
( ( sigma_439801790real_n @ lebesg260170249real_n @ M )
= ( sigma_439801790real_n @ borel_676189912real_n @ M ) ) ).
% measurable_lborel2
thf(fact_280_sets_ODiff,axiom,
! [A: set_Fi1058188332real_n,M: sigma_1466784463real_n,B: set_Fi1058188332real_n] :
( ( member223413699real_n @ A @ ( sigma_1235138647real_n @ M ) )
=> ( ( member223413699real_n @ B @ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n @ ( minus_1686442501real_n @ A @ B ) @ ( sigma_1235138647real_n @ M ) ) ) ) ).
% sets.Diff
thf(fact_281_sets_Ocompl__sets,axiom,
! [A: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ A @ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n @ ( minus_1686442501real_n @ ( sigma_476185326real_n @ M ) @ A ) @ ( sigma_1235138647real_n @ M ) ) ) ).
% sets.compl_sets
thf(fact_282_sets__lborel,axiom,
( ( sigma_1235138647real_n @ lebesg260170249real_n )
= ( sigma_1235138647real_n @ borel_676189912real_n ) ) ).
% sets_lborel
thf(fact_283_Int__Diff,axiom,
! [A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C: set_Fi1058188332real_n] :
( ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A2 @ B2 ) @ C )
= ( inf_in1974387902real_n @ A2 @ ( minus_1686442501real_n @ B2 @ C ) ) ) ).
% Int_Diff
thf(fact_284_Diff__Int2,axiom,
! [A2: set_Fi1058188332real_n,C: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A2 @ C ) @ ( inf_in1974387902real_n @ B2 @ C ) )
= ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A2 @ C ) @ B2 ) ) ).
% Diff_Int2
thf(fact_285_Diff__Diff__Int,axiom,
! [A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( minus_1686442501real_n @ A2 @ ( minus_1686442501real_n @ A2 @ B2 ) )
= ( inf_in1974387902real_n @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_286_Diff__Int__distrib,axiom,
! [C: set_Fi1058188332real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ C @ ( minus_1686442501real_n @ A2 @ B2 ) )
= ( minus_1686442501real_n @ ( inf_in1974387902real_n @ C @ A2 ) @ ( inf_in1974387902real_n @ C @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_287_Diff__Int__distrib2,axiom,
! [A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( minus_1686442501real_n @ A2 @ B2 ) @ C )
= ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A2 @ C ) @ ( inf_in1974387902real_n @ B2 @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_288_vimage__Diff,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( vimage1233683625real_n @ F @ ( minus_1686442501real_n @ A2 @ B2 ) )
= ( minus_1686442501real_n @ ( vimage1233683625real_n @ F @ A2 ) @ ( vimage1233683625real_n @ F @ B2 ) ) ) ).
% vimage_Diff
thf(fact_289_sets__restrict__space,axiom,
! [M: sigma_1466784463real_n,Omega: set_Fi1058188332real_n] :
( ( sigma_1235138647real_n @ ( sigma_346513458real_n @ M @ Omega ) )
= ( image_1661509983real_n @ ( inf_in1974387902real_n @ Omega ) @ ( sigma_1235138647real_n @ M ) ) ) ).
% sets_restrict_space
thf(fact_290_lborelD,axiom,
! [A2: set_Fi1058188332real_n] :
( ( member223413699real_n @ A2 @ ( sigma_1235138647real_n @ borel_676189912real_n ) )
=> ( member223413699real_n @ A2 @ ( sigma_1235138647real_n @ lebesg260170249real_n ) ) ) ).
% lborelD
thf(fact_291_measurable__lebesgue__cong,axiom,
! [S: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,M: sigma_1466784463real_n] :
( ! [X3: finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ S )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ ( sigma_346513458real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ S ) @ M ) )
= ( member1746150050real_n @ G @ ( sigma_439801790real_n @ ( sigma_346513458real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ S ) @ M ) ) ) ) ).
% measurable_lebesgue_cong
thf(fact_292_lborelD__Collect,axiom,
! [P: finite1489363574real_n > $o] :
( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ borel_676189912real_n ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ borel_676189912real_n ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ lebesg260170249real_n ) )
& ( P @ X ) ) )
@ ( sigma_1235138647real_n @ lebesg260170249real_n ) ) ) ).
% lborelD_Collect
thf(fact_293_lebesgue__sets__translation,axiom,
! [S: set_Fi1058188332real_n,A: finite1489363574real_n] :
( ( member223413699real_n @ S @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) )
=> ( member223413699real_n @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ S ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% lebesgue_sets_translation
thf(fact_294_lebesgue__measurable__vimage__borel,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,T: set_Fi1058188332real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ borel_676189912real_n ) )
=> ( ( member223413699real_n @ T @ ( sigma_1235138647real_n @ borel_676189912real_n ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ ( F @ X ) @ T ) )
@ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ) ).
% lebesgue_measurable_vimage_borel
thf(fact_295_borel__measurable__lebesgue__preimage__borel,axiom,
! [F: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ borel_676189912real_n ) )
= ( ! [T2: set_Fi1058188332real_n] :
( ( member223413699real_n @ T2 @ ( sigma_1235138647real_n @ borel_676189912real_n ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ ( F @ X ) @ T2 ) )
@ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ) ) ).
% borel_measurable_lebesgue_preimage_borel
thf(fact_296_borel__measurable__vimage__borel,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,S: set_Fi1058188332real_n] :
( ( member1746150050real_n @ F @ ( sigma_439801790real_n @ ( sigma_346513458real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ S ) @ borel_676189912real_n ) )
= ( ! [T2: set_Fi1058188332real_n] :
( ( member223413699real_n @ T2 @ ( sigma_1235138647real_n @ borel_676189912real_n ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ S )
& ( member1352538125real_n @ ( F @ X ) @ T2 ) ) )
@ ( sigma_1235138647real_n @ ( sigma_346513458real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ S ) ) ) ) ) ) ).
% borel_measurable_vimage_borel
thf(fact_297_restrict__space__eq__vimage__algebra_H,axiom,
! [M: sigma_1466784463real_n,Omega: set_Fi1058188332real_n] :
( ( sigma_1235138647real_n @ ( sigma_346513458real_n @ M @ Omega ) )
= ( sigma_1235138647real_n
@ ( sigma_821351682real_n @ ( inf_in1974387902real_n @ Omega @ ( sigma_476185326real_n @ M ) )
@ ^ [X: finite1489363574real_n] : X
@ M ) ) ) ).
% restrict_space_eq_vimage_algebra'
thf(fact_298_restrict__restrict__space,axiom,
! [A2: set_Fi1058188332real_n,M: sigma_1466784463real_n,B2: set_Fi1058188332real_n] :
( ( member223413699real_n @ ( inf_in1974387902real_n @ A2 @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) )
=> ( ( member223413699real_n @ ( inf_in1974387902real_n @ B2 @ ( sigma_476185326real_n @ M ) ) @ ( sigma_1235138647real_n @ M ) )
=> ( ( sigma_346513458real_n @ ( sigma_346513458real_n @ M @ A2 ) @ B2 )
= ( sigma_346513458real_n @ M @ ( inf_in1974387902real_n @ A2 @ B2 ) ) ) ) ) ).
% restrict_restrict_space
thf(fact_299_is__borel__def,axiom,
( borel_1962407338real_n
= ( ^ [F2: finite1489363574real_n > finite1489363574real_n,M3: sigma_1466784463real_n] : ( member1746150050real_n @ F2 @ ( sigma_439801790real_n @ M3 @ borel_676189912real_n ) ) ) ) ).
% is_borel_def
thf(fact_300_Diff__iff,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ ( minus_1698615483real_n @ A2 @ B2 ) )
= ( ( member223413699real_n @ C2 @ A2 )
& ~ ( member223413699real_n @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_301_Diff__iff,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ ( minus_725016986real_n @ A2 @ B2 ) )
= ( ( member1746150050real_n @ C2 @ A2 )
& ~ ( member1746150050real_n @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_302_DiffI,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ A2 )
=> ( ~ ( member223413699real_n @ C2 @ B2 )
=> ( member223413699real_n @ C2 @ ( minus_1698615483real_n @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_303_DiffI,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ A2 )
=> ( ~ ( member1746150050real_n @ C2 @ B2 )
=> ( member1746150050real_n @ C2 @ ( minus_725016986real_n @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_304_set__diff__eq,axiom,
( minus_1698615483real_n
= ( ^ [A4: set_se2111327970real_n,B3: set_se2111327970real_n] :
( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ A4 )
& ~ ( member223413699real_n @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_305_set__diff__eq,axiom,
( minus_725016986real_n
= ( ^ [A4: set_Fi1326602817real_n,B3: set_Fi1326602817real_n] :
( collec1190264032real_n
@ ^ [X: finite1489363574real_n > finite1489363574real_n] :
( ( member1746150050real_n @ X @ A4 )
& ~ ( member1746150050real_n @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_306_set__diff__eq,axiom,
( minus_1686442501real_n
= ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] :
( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ A4 )
& ~ ( member1352538125real_n @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_307_minus__set__def,axiom,
( minus_1698615483real_n
= ( ^ [A4: set_se2111327970real_n,B3: set_se2111327970real_n] :
( collec452821761real_n
@ ( minus_1832115082al_n_o
@ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ A4 )
@ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_308_minus__set__def,axiom,
( minus_725016986real_n
= ( ^ [A4: set_Fi1326602817real_n,B3: set_Fi1326602817real_n] :
( collec1190264032real_n
@ ( minus_391085931al_n_o
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( member1746150050real_n @ X @ A4 )
@ ^ [X: finite1489363574real_n > finite1489363574real_n] : ( member1746150050real_n @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_309_minus__set__def,axiom,
( minus_1686442501real_n
= ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] :
( collec321817931real_n
@ ( minus_455231168al_n_o
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ X @ A4 )
@ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_310_DiffD2,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ ( minus_1698615483real_n @ A2 @ B2 ) )
=> ~ ( member223413699real_n @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_311_DiffD2,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ ( minus_725016986real_n @ A2 @ B2 ) )
=> ~ ( member1746150050real_n @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_312_DiffD1,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ ( minus_1698615483real_n @ A2 @ B2 ) )
=> ( member223413699real_n @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_313_DiffD1,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ ( minus_725016986real_n @ A2 @ B2 ) )
=> ( member1746150050real_n @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_314_DiffE,axiom,
! [C2: set_Fi1058188332real_n,A2: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C2 @ ( minus_1698615483real_n @ A2 @ B2 ) )
=> ~ ( ( member223413699real_n @ C2 @ A2 )
=> ( member223413699real_n @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_315_DiffE,axiom,
! [C2: finite1489363574real_n > finite1489363574real_n,A2: set_Fi1326602817real_n,B2: set_Fi1326602817real_n] :
( ( member1746150050real_n @ C2 @ ( minus_725016986real_n @ A2 @ B2 ) )
=> ~ ( ( member1746150050real_n @ C2 @ A2 )
=> ( member1746150050real_n @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_316_translation__subtract__diff,axiom,
! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T3: set_Fi1058188332real_n] :
( ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
@ ( minus_1686442501real_n @ S2 @ T3 ) )
= ( minus_1686442501real_n
@ ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
@ S2 )
@ ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
@ T3 ) ) ) ).
% translation_subtract_diff
thf(fact_317_translation__subtract__Int,axiom,
! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T3: set_Fi1058188332real_n] :
( ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
@ ( inf_in1974387902real_n @ S2 @ T3 ) )
= ( inf_in1974387902real_n
@ ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
@ S2 )
@ ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
@ T3 ) ) ) ).
% translation_subtract_Int
thf(fact_318_translation__diff,axiom,
! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T3: set_Fi1058188332real_n] :
( ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ ( minus_1686442501real_n @ S2 @ T3 ) )
= ( minus_1686442501real_n @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ S2 ) @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ T3 ) ) ) ).
% translation_diff
thf(fact_319_translation__Int,axiom,
! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T3: set_Fi1058188332real_n] :
( ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ ( inf_in1974387902real_n @ S2 @ T3 ) )
= ( inf_in1974387902real_n @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ S2 ) @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ T3 ) ) ) ).
% translation_Int
thf(fact_320_translation__invert,axiom,
! [A: finite1489363574real_n,A2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ A2 )
= ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ B2 ) )
=> ( A2 = B2 ) ) ).
% translation_invert
thf(fact_321_translation__assoc,axiom,
! [B: finite1489363574real_n,A: finite1489363574real_n,S: set_Fi1058188332real_n] :
( ( image_439535603real_n @ ( plus_p585657087real_n @ B ) @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ S ) )
= ( image_439535603real_n @ ( plus_p585657087real_n @ ( plus_p585657087real_n @ A @ B ) ) @ S ) ) ).
% translation_assoc
thf(fact_322_affine__parallel__expl__aux,axiom,
! [S: set_Fi1058188332real_n,A: finite1489363574real_n,T: set_Fi1058188332real_n] :
( ! [X3: finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ S )
= ( member1352538125real_n @ ( plus_p585657087real_n @ A @ X3 ) @ T ) )
=> ( T
= ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ S ) ) ) ).
% affine_parallel_expl_aux
thf(fact_323_borel__measurable__if,axiom,
! [S: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n] :
( ( member223413699real_n @ S @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) )
=> ( ( member1746150050real_n
@ ^ [X: finite1489363574real_n] : ( if_Fin127821360real_n @ ( member1352538125real_n @ X @ S ) @ ( F @ X ) @ zero_z200130687real_n )
@ ( sigma_439801790real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ borel_676189912real_n ) )
= ( member1746150050real_n @ F @ ( sigma_439801790real_n @ ( sigma_346513458real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ S ) @ borel_676189912real_n ) ) ) ) ).
% borel_measurable_if
thf(fact_324_add_Oleft__neutral,axiom,
! [A: finite1489363574real_n] :
( ( plus_p585657087real_n @ zero_z200130687real_n @ A )
= A ) ).
% add.left_neutral
thf(fact_325_add_Oleft__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add.left_neutral
thf(fact_326_add_Oright__neutral,axiom,
! [A: finite1489363574real_n] :
( ( plus_p585657087real_n @ A @ zero_z200130687real_n )
= A ) ).
% add.right_neutral
thf(fact_327_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_328_add__cancel__left__left,axiom,
! [B: finite1489363574real_n,A: finite1489363574real_n] :
( ( ( plus_p585657087real_n @ B @ A )
= A )
= ( B = zero_z200130687real_n ) ) ).
% add_cancel_left_left
thf(fact_329_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_330_add__cancel__left__right,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( ( plus_p585657087real_n @ A @ B )
= A )
= ( B = zero_z200130687real_n ) ) ).
% add_cancel_left_right
thf(fact_331_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_332_add__cancel__right__left,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( A
= ( plus_p585657087real_n @ B @ A ) )
= ( B = zero_z200130687real_n ) ) ).
% add_cancel_right_left
thf(fact_333_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_334_add__cancel__right__right,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n] :
( ( A
= ( plus_p585657087real_n @ A @ B ) )
= ( B = zero_z200130687real_n ) ) ).
% add_cancel_right_right
thf(fact_335_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_336_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_337_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_338_diff__self,axiom,
! [A: finite1489363574real_n] :
( ( minus_1037315151real_n @ A @ A )
= zero_z200130687real_n ) ).
% diff_self
thf(fact_339_diff__0__right,axiom,
! [A: finite1489363574real_n] :
( ( minus_1037315151real_n @ A @ zero_z200130687real_n )
= A ) ).
% diff_0_right
thf(fact_340_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_341_diff__zero,axiom,
! [A: finite1489363574real_n] :
( ( minus_1037315151real_n @ A @ zero_z200130687real_n )
= A ) ).
% diff_zero
thf(fact_342_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_343_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: finite1489363574real_n] :
( ( minus_1037315151real_n @ A @ A )
= zero_z200130687real_n ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_344_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_345_image__add__0,axiom,
! [S: set_Fi1058188332real_n] :
( ( image_439535603real_n @ ( plus_p585657087real_n @ zero_z200130687real_n ) @ S )
= S ) ).
% image_add_0
thf(fact_346_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_347_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_348_eq__add__iff,axiom,
! [X2: finite1489363574real_n,Y3: finite1489363574real_n] :
( ( X2
= ( plus_p585657087real_n @ X2 @ Y3 ) )
= ( Y3 = zero_z200130687real_n ) ) ).
% eq_add_iff
thf(fact_349_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_350_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_351_diff__self__eq__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ M4 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_352_diffs0__imp__equal,axiom,
! [M4: nat,N3: nat] :
( ( ( minus_minus_nat @ M4 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M4 )
= zero_zero_nat )
=> ( M4 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_353_minus__nat_Odiff__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% minus_nat.diff_0
% Helper facts (7)
thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_T,axiom,
! [X2: finite1489363574real_n,Y3: finite1489363574real_n] :
( ( if_Fin127821360real_n @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_T,axiom,
! [X2: finite1489363574real_n,Y3: finite1489363574real_n] :
( ( if_Fin127821360real_n @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_T,axiom,
! [X2: set_Fi1058188332real_n,Y3: set_Fi1058188332real_n] :
( ( if_set11487206real_n @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_T,axiom,
! [X2: set_Fi1058188332real_n,Y3: set_Fi1058188332real_n] :
( ( if_set11487206real_n @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_T,axiom,
! [X2: finite1489363574real_n > finite1489363574real_n,Y3: finite1489363574real_n > finite1489363574real_n] :
( ( if_Fin413489477real_n @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_T,axiom,
! [X2: finite1489363574real_n > finite1489363574real_n,Y3: finite1489363574real_n > finite1489363574real_n] :
( ( if_Fin413489477real_n @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( member223413699real_n
@ ( vimage1233683625real_n
@ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ X @ ( minkow1134813771n_real @ a ) )
@ ( t2 @ a ) )
@ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
%------------------------------------------------------------------------------