TPTP Problem File: ITP117^1.p
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%------------------------------------------------------------------------------
% File : ITP117^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Minkowskis_Theorem problem prob_290__6248976_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_290__6248976_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 487 ( 187 unt; 133 typ; 0 def)
% Number of atoms : 807 ( 345 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 3013 ( 72 ~; 3 |; 79 &;2559 @)
% ( 0 <=>; 300 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 541 ( 541 >; 0 *; 0 +; 0 <<)
% Number of symbols : 117 ( 115 usr; 16 con; 0-2 aty)
% Number of variables : 1094 ( 122 ^; 923 !; 49 ?;1094 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:47:06.896
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
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% Explicit typings (115)
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image_1661509983real_n: ( set_Fi1058188332real_n > set_Fi1058188332real_n ) > set_se2111327970real_n > set_se2111327970real_n ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
image_797440021real_n: ( set_Fi1058188332real_n > set_se2111327970real_n ) > set_se2111327970real_n > set_se820660888real_n ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001t__Sigma____Algebra__Omeasure_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
image_987430492real_n: ( set_Fi1058188332real_n > sigma_1466784463real_n ) > set_se2111327970real_n > set_Si1125517487real_n ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
image_933134521real_n: ( set_nat > set_Fi1058188332real_n ) > set_set_nat > set_se2111327970real_n ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
image_1587769199real_n: ( set_nat > set_se2111327970real_n ) > set_set_nat > set_se820660888real_n ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J_001_Eo,type,
image_1681970287al_n_o: ( set_se2111327970real_n > $o ) > set_se820660888real_n > set_o ).
thf(sy_c_Set_Oimage_001t__Sigma____Algebra__Omeasure_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,
image_1298280374real_n: ( sigma_1466784463real_n > set_Fi1058188332real_n ) > set_Si1125517487real_n > set_se2111327970real_n ).
thf(sy_c_Set_Ovimage_001_Eo_001_Eo,type,
vimage_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
vimage961837641real_n: ( $o > set_Fi1058188332real_n ) > set_se2111327970real_n > set_o ).
thf(sy_c_Set_Ovimage_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
vimage1122713129_int_n: ( finite964658038_int_n > finite964658038_int_n ) > set_Fi160064172_int_n > set_Fi160064172_int_n ).
thf(sy_c_Set_Ovimage_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
vimage1276736425real_n: ( finite964658038_int_n > finite1489363574real_n ) > set_Fi1058188332real_n > set_Fi160064172_int_n ).
thf(sy_c_Set_Ovimage_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_001t__Nat__Onat,type,
vimage1398021123_n_nat: ( finite964658038_int_n > nat ) > set_nat > set_Fi160064172_int_n ).
thf(sy_c_Set_Ovimage_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
vimage464515423real_n: ( finite964658038_int_n > set_Fi1058188332real_n ) > set_se2111327970real_n > set_Fi160064172_int_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__Nat__Onat_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
vimage714719107_int_n: ( nat > finite964658038_int_n ) > set_Fi160064172_int_n > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
vimage1860757507real_n: ( nat > finite1489363574real_n ) > set_Fi1058188332real_n > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
vimage3210681real_n: ( nat > set_Fi1058188332real_n ) > set_se2111327970real_n > set_nat ).
thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001_Eo,type,
vimage851190895al_n_o: ( set_Fi1058188332real_n > $o ) > set_o > 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_Oemeasure_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
sigma_1536574303real_n: sigma_1466784463real_n > set_Fi1058188332real_n > extend1728876344nnreal ).
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_Ospace_001_Eo,type,
sigma_space_o: sigma_measure_o > set_o ).
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_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member1217042383nnreal: extend1728876344nnreal > set_Ex113815278nnreal > $o ).
thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
member27055245_int_n: finite964658038_int_n > set_Fi160064172_int_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__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__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_c_member_001t__Sigma____Algebra__Omeasure_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
member1000184real_n: sigma_1466784463real_n > set_Si1125517487real_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_f____,type,
f: nat > finite964658038_int_n ).
% Relevant facts (353)
thf(fact_0_emeasure__T_H,axiom,
! [A: finite964658038_int_n] :
( ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ A ) )
= ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ A ) ) ) ).
% emeasure_T'
thf(fact_1__092_060open_062_092_060And_062a_O_AT_H_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
! [A: finite964658038_int_n] : ( member223413699real_n @ ( t @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% \<open>\<And>a. T' a \<in> sets lebesgue\<close>
thf(fact_2__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_3_calculation,axiom,
( sums_E1192373732nnreal
@ ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ N ) ) )
@ ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ s ) ) ).
% calculation
thf(fact_4_assms_I1_J,axiom,
member223413699real_n @ s @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ).
% assms(1)
thf(fact_5_emeasure__T__Int,axiom,
! [A: finite964658038_int_n,B: finite964658038_int_n] :
( ( A != B )
=> ( ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( inf_in1974387902real_n @ ( t2 @ A ) @ ( t2 @ B ) ) )
= zero_z1963244097nnreal ) ) ).
% emeasure_T_Int
thf(fact_6_T__Int,axiom,
! [A: finite964658038_int_n,B: finite964658038_int_n] :
( ( A != B )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ ( t2 @ A ) @ ( t2 @ B ) ) @ ( measur1402256771real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% T_Int
thf(fact_7_assms_I2_J,axiom,
ord_le2133614988nnreal @ one_on705384445nnreal @ ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ s ) ).
% assms(2)
thf(fact_8__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_9__092_060open_062_I_092_060lambda_062n_O_Aemeasure_Alebesgue_A_IT_A_If_An_J_J_J_Asums_Aemeasure_Alebesgue_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062,axiom,
( sums_E1192373732nnreal
@ ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ N ) ) )
@ ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n )
@ ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [N: nat] : ( t2 @ ( f @ N ) )
@ top_top_set_nat ) ) ) ) ).
% \<open>(\<lambda>n. emeasure lebesgue (T (f n))) sums emeasure lebesgue (\<Union>n. T (f n))\<close>
thf(fact_10_T_H__def,axiom,
( t
= ( ^ [A2: finite964658038_int_n] :
( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ ( minkow1134813771n_real @ A2 ) )
@ ( t2 @ A2 ) ) ) ) ).
% T'_def
thf(fact_11_f__def,axiom,
( f
= ( counta1142393929_int_n @ top_to131672412_int_n ) ) ).
% f_def
thf(fact_12_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_13_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_14_T__def,axiom,
( t2
= ( ^ [A2: finite964658038_int_n] : ( inf_in1974387902real_n @ s @ ( r @ A2 ) ) ) ) ).
% T_def
thf(fact_15_R__Int,axiom,
! [A: finite964658038_int_n,B: finite964658038_int_n] :
( ( A != B )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ ( r @ A ) @ ( r @ B ) ) @ ( measur1402256771real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% R_Int
thf(fact_16_sums__emeasure_H,axiom,
! [B2: nat > set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ! [X2: nat] : ( member223413699real_n @ ( B2 @ X2 ) @ ( sigma_1235138647real_n @ M ) )
=> ( ! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ( sigma_1536574303real_n @ M @ ( inf_in1974387902real_n @ ( B2 @ X2 ) @ ( B2 @ Y ) ) )
= zero_z1963244097nnreal ) )
=> ( sums_E1192373732nnreal
@ ^ [X: nat] : ( sigma_1536574303real_n @ M @ ( B2 @ X ) )
@ ( sigma_1536574303real_n @ M @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ top_top_set_nat ) ) ) ) ) ) ).
% sums_emeasure'
thf(fact_17_null__sets__UN,axiom,
! [N2: nat > set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ! [I: nat] : ( member223413699real_n @ ( N2 @ I ) @ ( measur1402256771real_n @ M ) )
=> ( member223413699real_n @ ( comple825005695real_n @ ( image_1856576259real_n @ N2 @ top_top_set_nat ) ) @ ( measur1402256771real_n @ M ) ) ) ).
% null_sets_UN
thf(fact_18_null__sets__UN,axiom,
! [N2: finite964658038_int_n > set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ! [I: finite964658038_int_n] : ( member223413699real_n @ ( N2 @ I ) @ ( measur1402256771real_n @ M ) )
=> ( member223413699real_n @ ( comple825005695real_n @ ( image_355963305real_n @ N2 @ top_to131672412_int_n ) ) @ ( measur1402256771real_n @ M ) ) ) ).
% null_sets_UN
thf(fact_19_null__setsI,axiom,
! [M: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
( ( ( sigma_1536574303real_n @ M @ A3 )
= zero_z1963244097nnreal )
=> ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n @ A3 @ ( measur1402256771real_n @ M ) ) ) ) ).
% null_setsI
thf(fact_20_image__vimage__eq,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( image_439535603real_n @ F @ ( vimage1233683625real_n @ F @ A3 ) )
= ( inf_in1974387902real_n @ A3 @ ( image_439535603real_n @ F @ top_to1292442332real_n ) ) ) ).
% image_vimage_eq
thf(fact_21_image__vimage__eq,axiom,
! [F: nat > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( image_1856576259real_n @ F @ ( vimage3210681real_n @ F @ A3 ) )
= ( inf_in632889204real_n @ A3 @ ( image_1856576259real_n @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_22_image__vimage__eq,axiom,
! [F: nat > finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( image_183184717real_n @ F @ ( vimage1860757507real_n @ F @ A3 ) )
= ( inf_in1974387902real_n @ A3 @ ( image_183184717real_n @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_23_image__vimage__eq,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( image_355963305real_n @ F @ ( vimage464515423real_n @ F @ A3 ) )
= ( inf_in632889204real_n @ A3 @ ( image_355963305real_n @ F @ top_to131672412_int_n ) ) ) ).
% image_vimage_eq
thf(fact_24_image__vimage__eq,axiom,
! [F: finite964658038_int_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( image_2058828787real_n @ F @ ( vimage1276736425real_n @ F @ A3 ) )
= ( inf_in1974387902real_n @ A3 @ ( image_2058828787real_n @ F @ top_to131672412_int_n ) ) ) ).
% image_vimage_eq
thf(fact_25__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_26_surj__diff__right,axiom,
! [A: finite964658038_int_n] :
( ( image_1278151539_int_n
@ ^ [X: finite964658038_int_n] : ( minus_1196255695_int_n @ X @ A )
@ top_to131672412_int_n )
= top_to131672412_int_n ) ).
% surj_diff_right
thf(fact_27_surj__diff__right,axiom,
! [A: finite1489363574real_n] :
( ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
@ top_to1292442332real_n )
= top_to1292442332real_n ) ).
% surj_diff_right
thf(fact_28__092_060open_062_092_060Union_062_A_Irange_AT_J_A_061_A_I_092_060Union_062n_O_AT_A_If_An_J_J_092_060close_062,axiom,
( ( comple825005695real_n @ ( image_355963305real_n @ t2 @ top_to131672412_int_n ) )
= ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [N: nat] : ( t2 @ ( f @ N ) )
@ top_top_set_nat ) ) ) ).
% \<open>\<Union> (range T) = (\<Union>n. T (f n))\<close>
thf(fact_29_Sup__UNIV,axiom,
( ( comple1682161881et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_30_Sup__UNIV,axiom,
( ( comple970917503_int_n @ top_to1587634578_int_n )
= top_to131672412_int_n ) ).
% Sup_UNIV
thf(fact_31_Sup__UNIV,axiom,
( ( comple825005695real_n @ top_to20708754real_n )
= top_to1292442332real_n ) ).
% Sup_UNIV
thf(fact_32_Sup__UNIV,axiom,
( ( complete_Sup_Sup_o @ top_top_set_o )
= top_top_o ) ).
% Sup_UNIV
thf(fact_33_Sup__eq__top__iff,axiom,
! [A3: set_Ex113815278nnreal] :
( ( ( comple1413366923nnreal @ A3 )
= top_to1845833192nnreal )
= ( ! [X: extend1728876344nnreal] :
( ( ord_le2133614988nnreal @ X @ top_to1845833192nnreal )
=> ? [Y2: extend1728876344nnreal] :
( ( member1217042383nnreal @ Y2 @ A3 )
& ( ord_le2133614988nnreal @ X @ Y2 ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_34_surj__plus,axiom,
! [A: finite1489363574real_n] :
( ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ top_to1292442332real_n )
= top_to1292442332real_n ) ).
% surj_plus
thf(fact_35_surj__plus,axiom,
! [A: finite964658038_int_n] :
( ( image_1278151539_int_n @ ( plus_p1654784127_int_n @ A ) @ top_to131672412_int_n )
= top_to131672412_int_n ) ).
% surj_plus
thf(fact_36_diff__numeral__special_I9_J,axiom,
( ( minus_1037315151real_n @ one_on1253059131real_n @ one_on1253059131real_n )
= zero_z200130687real_n ) ).
% diff_numeral_special(9)
thf(fact_37_image__eqI,axiom,
! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,X3: finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( B
= ( F @ X3 ) )
=> ( ( member1352538125real_n @ X3 @ A3 )
=> ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_38_image__eqI,axiom,
! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,X3: nat,A3: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A3 )
=> ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_39_image__eqI,axiom,
! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,X3: finite964658038_int_n,A3: set_Fi160064172_int_n] :
( ( B
= ( F @ X3 ) )
=> ( ( member27055245_int_n @ X3 @ A3 )
=> ( member223413699real_n @ B @ ( image_355963305real_n @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_40_image__eqI,axiom,
! [B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,X3: set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( B
= ( F @ X3 ) )
=> ( ( member223413699real_n @ X3 @ A3 )
=> ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_41_image__eqI,axiom,
! [B: $o,F: set_Fi1058188332real_n > $o,X3: set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( B
= ( F @ X3 ) )
=> ( ( member223413699real_n @ X3 @ A3 )
=> ( member_o @ B @ ( image_1648361637al_n_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_42_image__eqI,axiom,
! [B: set_Fi1058188332real_n,F: $o > set_Fi1058188332real_n,X3: $o,A3: set_o] :
( ( B
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A3 )
=> ( member223413699real_n @ B @ ( image_1759008383real_n @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_43_image__eqI,axiom,
! [B: $o,F: $o > $o,X3: $o,A3: set_o] :
( ( B
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A3 )
=> ( member_o @ B @ ( image_o_o @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_44_UNIV__I,axiom,
! [X3: set_Fi1058188332real_n] : ( member223413699real_n @ X3 @ top_to20708754real_n ) ).
% UNIV_I
thf(fact_45_UNIV__I,axiom,
! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_I
thf(fact_46_UNIV__I,axiom,
! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_47_UNIV__I,axiom,
! [X3: finite964658038_int_n] : ( member27055245_int_n @ X3 @ top_to131672412_int_n ) ).
% UNIV_I
thf(fact_48_Int__iff,axiom,
! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
= ( ( member223413699real_n @ C @ A3 )
& ( member223413699real_n @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_49_Int__iff,axiom,
! [C: $o,A3: set_o,B2: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
= ( ( member_o @ C @ A3 )
& ( member_o @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_50_Int__iff,axiom,
! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
= ( ( member1352538125real_n @ C @ A3 )
& ( member1352538125real_n @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_51_IntI,axiom,
! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C @ A3 )
=> ( ( member223413699real_n @ C @ B2 )
=> ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_52_IntI,axiom,
! [C: $o,A3: set_o,B2: set_o] :
( ( member_o @ C @ A3 )
=> ( ( member_o @ C @ B2 )
=> ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_53_IntI,axiom,
! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C @ A3 )
=> ( ( member1352538125real_n @ C @ B2 )
=> ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_54_Union__iff,axiom,
! [A3: set_Fi1058188332real_n,C2: set_se820660888real_n] :
( ( member223413699real_n @ A3 @ ( comple1917283637real_n @ C2 ) )
= ( ? [X: set_se2111327970real_n] :
( ( member1475136633real_n @ X @ C2 )
& ( member223413699real_n @ A3 @ X ) ) ) ) ).
% Union_iff
thf(fact_55_Union__iff,axiom,
! [A3: $o,C2: set_set_o] :
( ( member_o @ A3 @ ( comple1665300069_set_o @ C2 ) )
= ( ? [X: set_o] :
( ( member_set_o @ X @ C2 )
& ( member_o @ A3 @ X ) ) ) ) ).
% Union_iff
thf(fact_56_Union__iff,axiom,
! [A3: finite1489363574real_n,C2: set_se2111327970real_n] :
( ( member1352538125real_n @ A3 @ ( comple825005695real_n @ C2 ) )
= ( ? [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ C2 )
& ( member1352538125real_n @ A3 @ X ) ) ) ) ).
% Union_iff
thf(fact_57_UnionI,axiom,
! [X4: set_se2111327970real_n,C2: set_se820660888real_n,A3: set_Fi1058188332real_n] :
( ( member1475136633real_n @ X4 @ C2 )
=> ( ( member223413699real_n @ A3 @ X4 )
=> ( member223413699real_n @ A3 @ ( comple1917283637real_n @ C2 ) ) ) ) ).
% UnionI
thf(fact_58_UnionI,axiom,
! [X4: set_o,C2: set_set_o,A3: $o] :
( ( member_set_o @ X4 @ C2 )
=> ( ( member_o @ A3 @ X4 )
=> ( member_o @ A3 @ ( comple1665300069_set_o @ C2 ) ) ) ) ).
% UnionI
thf(fact_59_UnionI,axiom,
! [X4: set_Fi1058188332real_n,C2: set_se2111327970real_n,A3: finite1489363574real_n] :
( ( member223413699real_n @ X4 @ C2 )
=> ( ( member1352538125real_n @ A3 @ X4 )
=> ( member1352538125real_n @ A3 @ ( comple825005695real_n @ C2 ) ) ) ) ).
% UnionI
thf(fact_60_UN__ball__bex__simps_I1_J,axiom,
! [A3: set_se2111327970real_n,P: finite1489363574real_n > $o] :
( ( ! [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ A3 ) )
=> ( P @ X ) ) )
= ( ! [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ A3 )
=> ! [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ X )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_61_UN__ball__bex__simps_I3_J,axiom,
! [A3: set_se2111327970real_n,P: finite1489363574real_n > $o] :
( ( ? [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ A3 ) )
& ( P @ X ) ) )
= ( ? [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ A3 )
& ? [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ X )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_62_null__sets_ODiff,axiom,
! [A: set_Fi1058188332real_n,M: sigma_1466784463real_n,B: set_Fi1058188332real_n] :
( ( member223413699real_n @ A @ ( measur1402256771real_n @ M ) )
=> ( ( member223413699real_n @ B @ ( measur1402256771real_n @ M ) )
=> ( member223413699real_n @ ( minus_1686442501real_n @ A @ B ) @ ( measur1402256771real_n @ M ) ) ) ) ).
% null_sets.Diff
thf(fact_63_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_64_vimage__eq,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > $o,B2: set_o] :
( ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ B2 ) )
= ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_65_vimage__eq,axiom,
! [A: $o,F: $o > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( member_o @ A @ ( vimage961837641real_n @ F @ B2 ) )
= ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_66_vimage__eq,axiom,
! [A: $o,F: $o > $o,B2: set_o] :
( ( member_o @ A @ ( vimage_o_o @ F @ B2 ) )
= ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_67_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_68_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_69_vimageI,axiom,
! [F: $o > set_Fi1058188332real_n,A: $o,B: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( ( F @ A )
= B )
=> ( ( member223413699real_n @ B @ B2 )
=> ( member_o @ A @ ( vimage961837641real_n @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_70_vimageI,axiom,
! [F: set_Fi1058188332real_n > $o,A: set_Fi1058188332real_n,B: $o,B2: set_o] :
( ( ( F @ A )
= B )
=> ( ( member_o @ B @ B2 )
=> ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_71_vimageI,axiom,
! [F: $o > $o,A: $o,B: $o,B2: set_o] :
( ( ( F @ A )
= B )
=> ( ( member_o @ B @ B2 )
=> ( member_o @ A @ ( vimage_o_o @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_72_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_73_image__ident,axiom,
! [Y3: set_Fi1058188332real_n] :
( ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : X
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_74_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_75_vimage__ident,axiom,
! [Y3: set_Fi1058188332real_n] :
( ( vimage1233683625real_n
@ ^ [X: finite1489363574real_n] : X
@ Y3 )
= Y3 ) ).
% vimage_ident
thf(fact_76_Int__UNIV,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( ( inf_in1974387902real_n @ A3 @ B2 )
= top_to1292442332real_n )
= ( ( A3 = top_to1292442332real_n )
& ( B2 = top_to1292442332real_n ) ) ) ).
% Int_UNIV
thf(fact_77_Int__UNIV,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B2 )
= top_top_set_nat )
= ( ( A3 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_78_Int__UNIV,axiom,
! [A3: set_Fi160064172_int_n,B2: set_Fi160064172_int_n] :
( ( ( inf_in1108485182_int_n @ A3 @ B2 )
= top_to131672412_int_n )
= ( ( A3 = top_to131672412_int_n )
& ( B2 = top_to131672412_int_n ) ) ) ).
% Int_UNIV
thf(fact_79_ball__UN,axiom,
! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
( ( ! [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ! [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_80_ball__UN,axiom,
! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
( ( ! [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
=> ( P @ X ) ) )
= ( ! [X: finite964658038_int_n] :
( ( member27055245_int_n @ X @ A3 )
=> ! [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_81_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_82_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A3: set_se2111327970real_n] :
( ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A3: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_85_bex__UN,axiom,
! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
( ( ? [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A3 )
& ? [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_86_bex__UN,axiom,
! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
( ( ? [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
& ( P @ X ) ) )
= ( ? [X: finite964658038_int_n] :
( ( member27055245_int_n @ X @ A3 )
& ? [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_87_UN__ball__bex__simps_I2_J,axiom,
! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
( ( ! [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ! [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_88_UN__ball__bex__simps_I2_J,axiom,
! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
( ( ! [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
=> ( P @ X ) ) )
= ( ! [X: finite964658038_int_n] :
( ( member27055245_int_n @ X @ A3 )
=> ! [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_89_UN__ball__bex__simps_I4_J,axiom,
! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
( ( ? [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A3 )
& ? [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_90_UN__ball__bex__simps_I4_J,axiom,
! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
( ( ? [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
& ( P @ X ) ) )
= ( ? [X: finite964658038_int_n] :
( ( member27055245_int_n @ X @ A3 )
& ? [Y2: finite1489363574real_n] :
( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_91_vimage__UNIV,axiom,
! [F: finite1489363574real_n > finite1489363574real_n] :
( ( vimage1233683625real_n @ F @ top_to1292442332real_n )
= top_to1292442332real_n ) ).
% vimage_UNIV
thf(fact_92_vimage__UNIV,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_93_vimage__UNIV,axiom,
! [F: finite964658038_int_n > nat] :
( ( vimage1398021123_n_nat @ F @ top_top_set_nat )
= top_to131672412_int_n ) ).
% vimage_UNIV
thf(fact_94_vimage__UNIV,axiom,
! [F: nat > finite964658038_int_n] :
( ( vimage714719107_int_n @ F @ top_to131672412_int_n )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_95_vimage__UNIV,axiom,
! [F: finite964658038_int_n > finite964658038_int_n] :
( ( vimage1122713129_int_n @ F @ top_to131672412_int_n )
= top_to131672412_int_n ) ).
% vimage_UNIV
thf(fact_96_null__sets_OInt,axiom,
! [A: set_Fi1058188332real_n,M: sigma_1466784463real_n,B: set_Fi1058188332real_n] :
( ( member223413699real_n @ A @ ( measur1402256771real_n @ M ) )
=> ( ( member223413699real_n @ B @ ( measur1402256771real_n @ M ) )
=> ( member223413699real_n @ ( inf_in1974387902real_n @ A @ B ) @ ( measur1402256771real_n @ M ) ) ) ) ).
% null_sets.Int
thf(fact_97_vimage__Int,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( vimage1233683625real_n @ F @ ( inf_in1974387902real_n @ A3 @ B2 ) )
= ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ A3 ) @ ( vimage1233683625real_n @ F @ B2 ) ) ) ).
% vimage_Int
thf(fact_98_SUP__identity__eq,axiom,
! [A3: set_Fi1058188332real_n] :
( ( comple2042271945real_n
@ ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : X
@ A3 ) )
= ( comple2042271945real_n @ A3 ) ) ).
% SUP_identity_eq
thf(fact_99_SUP__identity__eq,axiom,
! [A3: set_se2111327970real_n] :
( ( comple825005695real_n
@ ( image_1661509983real_n
@ ^ [X: set_Fi1058188332real_n] : X
@ A3 ) )
= ( comple825005695real_n @ A3 ) ) ).
% SUP_identity_eq
thf(fact_100_SUP__identity__eq,axiom,
! [A3: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X: $o] : X
@ A3 ) )
= ( complete_Sup_Sup_o @ A3 ) ) ).
% SUP_identity_eq
thf(fact_101_UN__iff,axiom,
! [B: finite1489363574real_n,B2: nat > set_Fi1058188332real_n,A3: set_nat] :
( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A3 )
& ( member1352538125real_n @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_102_UN__iff,axiom,
! [B: finite1489363574real_n,B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
= ( ? [X: finite964658038_int_n] :
( ( member27055245_int_n @ X @ A3 )
& ( member1352538125real_n @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_103_UN__I,axiom,
! [A: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: set_Fi1058188332real_n,B2: set_Fi1058188332real_n > set_se2111327970real_n] :
( ( member223413699real_n @ A @ A3 )
=> ( ( member223413699real_n @ B @ ( B2 @ A ) )
=> ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_797440021real_n @ B2 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_104_UN__I,axiom,
! [A: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: $o,B2: set_Fi1058188332real_n > set_o] :
( ( member223413699real_n @ A @ A3 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple1665300069_set_o @ ( image_1687589765_set_o @ B2 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_105_UN__I,axiom,
! [A: $o,A3: set_o,B: set_Fi1058188332real_n,B2: $o > set_se2111327970real_n] :
( ( member_o @ A @ A3 )
=> ( ( member223413699real_n @ B @ ( B2 @ A ) )
=> ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_452144437real_n @ B2 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_106_UN__I,axiom,
! [A: $o,A3: set_o,B: $o,B2: $o > set_o] :
( ( member_o @ A @ A3 )
=> ( ( member_o @ B @ ( B2 @ A ) )
=> ( member_o @ B @ ( comple1665300069_set_o @ ( image_o_set_o @ B2 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_107_UN__I,axiom,
! [A: nat,A3: set_nat,B: finite1489363574real_n,B2: nat > set_Fi1058188332real_n] :
( ( member_nat @ A @ A3 )
=> ( ( member1352538125real_n @ B @ ( B2 @ A ) )
=> ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_108_UN__I,axiom,
! [A: finite964658038_int_n,A3: set_Fi160064172_int_n,B: finite1489363574real_n,B2: finite964658038_int_n > set_Fi1058188332real_n] :
( ( member27055245_int_n @ A @ A3 )
=> ( ( member1352538125real_n @ B @ ( B2 @ A ) )
=> ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_109_UN__I,axiom,
! [A: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: finite1489363574real_n,B2: set_Fi1058188332real_n > set_Fi1058188332real_n] :
( ( member223413699real_n @ A @ A3 )
=> ( ( member1352538125real_n @ B @ ( B2 @ A ) )
=> ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1661509983real_n @ B2 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_110_UN__I,axiom,
! [A: $o,A3: set_o,B: finite1489363574real_n,B2: $o > set_Fi1058188332real_n] :
( ( member_o @ A @ A3 )
=> ( ( member1352538125real_n @ B @ ( B2 @ A ) )
=> ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1759008383real_n @ B2 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_111_S__decompose,axiom,
( s
= ( comple825005695real_n @ ( image_355963305real_n @ t2 @ top_to131672412_int_n ) ) ) ).
% S_decompose
thf(fact_112_image__add__0,axiom,
! [S: set_Fi1058188332real_n] :
( ( image_439535603real_n @ ( plus_p585657087real_n @ zero_z200130687real_n ) @ S )
= S ) ).
% image_add_0
thf(fact_113_image__add__0,axiom,
! [S: set_Ex113815278nnreal] :
( ( image_2066995319nnreal @ ( plus_p1763960001nnreal @ zero_z1963244097nnreal ) @ S )
= S ) ).
% image_add_0
thf(fact_114_null__sets_OInt__space__eq2,axiom,
! [X3: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ X3 @ ( measur1402256771real_n @ M ) )
=> ( ( inf_in1974387902real_n @ X3 @ ( sigma_476185326real_n @ M ) )
= X3 ) ) ).
% null_sets.Int_space_eq2
thf(fact_115_null__sets_OInt__space__eq1,axiom,
! [X3: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ X3 @ ( measur1402256771real_n @ M ) )
=> ( ( inf_in1974387902real_n @ ( sigma_476185326real_n @ M ) @ X3 )
= X3 ) ) ).
% null_sets.Int_space_eq1
thf(fact_116_in__sets__SUP,axiom,
! [I2: set_Fi1058188332real_n,I3: set_se2111327970real_n,M: set_Fi1058188332real_n > sigma_1466784463real_n,Y3: set_Fi1058188332real_n,X4: set_Fi1058188332real_n] :
( ( member223413699real_n @ I2 @ I3 )
=> ( ! [I: set_Fi1058188332real_n] :
( ( member223413699real_n @ I @ I3 )
=> ( ( sigma_476185326real_n @ ( M @ I ) )
= Y3 ) )
=> ( ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( M @ I2 ) ) )
=> ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_987430492real_n @ M @ I3 ) ) ) ) ) ) ) ).
% in_sets_SUP
thf(fact_117_in__sets__SUP,axiom,
! [I2: $o,I3: set_o,M: $o > sigma_1466784463real_n,Y3: set_Fi1058188332real_n,X4: set_Fi1058188332real_n] :
( ( member_o @ I2 @ I3 )
=> ( ! [I: $o] :
( ( member_o @ I @ I3 )
=> ( ( sigma_476185326real_n @ ( M @ I ) )
= Y3 ) )
=> ( ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( M @ I2 ) ) )
=> ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_1599934780real_n @ M @ I3 ) ) ) ) ) ) ) ).
% in_sets_SUP
thf(fact_118_in__sets__Sup,axiom,
! [M: set_Si1125517487real_n,X4: set_Fi1058188332real_n,M2: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
( ! [M3: sigma_1466784463real_n] :
( ( member1000184real_n @ M3 @ M )
=> ( ( sigma_476185326real_n @ M3 )
= X4 ) )
=> ( ( member1000184real_n @ M2 @ M )
=> ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M2 ) )
=> ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ ( comple488165692real_n @ M ) ) ) ) ) ) ).
% in_sets_Sup
thf(fact_119_sets__SUP__cong,axiom,
! [I3: set_se2111327970real_n,M: set_Fi1058188332real_n > sigma_1466784463real_n,N2: set_Fi1058188332real_n > sigma_1466784463real_n] :
( ! [I: set_Fi1058188332real_n] :
( ( member223413699real_n @ I @ I3 )
=> ( ( sigma_1235138647real_n @ ( M @ I ) )
= ( sigma_1235138647real_n @ ( N2 @ I ) ) ) )
=> ( ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_987430492real_n @ M @ I3 ) ) )
= ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_987430492real_n @ N2 @ I3 ) ) ) ) ) ).
% sets_SUP_cong
thf(fact_120_sets__SUP__cong,axiom,
! [I3: set_o,M: $o > sigma_1466784463real_n,N2: $o > sigma_1466784463real_n] :
( ! [I: $o] :
( ( member_o @ I @ I3 )
=> ( ( sigma_1235138647real_n @ ( M @ I ) )
= ( sigma_1235138647real_n @ ( N2 @ I ) ) ) )
=> ( ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_1599934780real_n @ M @ I3 ) ) )
= ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_1599934780real_n @ N2 @ I3 ) ) ) ) ) ).
% sets_SUP_cong
thf(fact_121_Sup__set__def,axiom,
( comple1917283637real_n
= ( ^ [A4: set_se820660888real_n] :
( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] : ( complete_Sup_Sup_o @ ( image_1681970287al_n_o @ ( member223413699real_n @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_122_Sup__set__def,axiom,
( comple1665300069_set_o
= ( ^ [A4: set_set_o] :
( collect_o
@ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_123_Sup__set__def,axiom,
( comple825005695real_n
= ( ^ [A4: set_se2111327970real_n] :
( collec321817931real_n
@ ^ [X: finite1489363574real_n] : ( complete_Sup_Sup_o @ ( image_1648361637al_n_o @ ( member1352538125real_n @ X ) @ A4 ) ) ) ) ) ).
% Sup_set_def
thf(fact_124_space__Sup__eq__UN,axiom,
! [M: set_Si1125517487real_n] :
( ( sigma_476185326real_n @ ( comple488165692real_n @ M ) )
= ( comple825005695real_n @ ( image_1298280374real_n @ sigma_476185326real_n @ M ) ) ) ).
% space_Sup_eq_UN
thf(fact_125_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_126_top__set__def,axiom,
( top_to131672412_int_n
= ( collec1941932235_int_n @ top_to287930409nt_n_o ) ) ).
% top_set_def
thf(fact_127_Diff__Int__distrib2,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( minus_1686442501real_n @ A3 @ B2 ) @ C2 )
= ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ C2 ) @ ( inf_in1974387902real_n @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_128_Diff__Int__distrib,axiom,
! [C2: set_Fi1058188332real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ C2 @ ( minus_1686442501real_n @ A3 @ B2 ) )
= ( minus_1686442501real_n @ ( inf_in1974387902real_n @ C2 @ A3 ) @ ( inf_in1974387902real_n @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_129_Diff__Diff__Int,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( minus_1686442501real_n @ A3 @ ( minus_1686442501real_n @ A3 @ B2 ) )
= ( inf_in1974387902real_n @ A3 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_130_Diff__Int2,axiom,
! [A3: set_Fi1058188332real_n,C2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ C2 ) @ ( inf_in1974387902real_n @ B2 @ C2 ) )
= ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_131_Int__Diff,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
( ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ B2 ) @ C2 )
= ( inf_in1974387902real_n @ A3 @ ( minus_1686442501real_n @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_132_vimage__Diff,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( vimage1233683625real_n @ F @ ( minus_1686442501real_n @ A3 @ B2 ) )
= ( minus_1686442501real_n @ ( vimage1233683625real_n @ F @ A3 ) @ ( vimage1233683625real_n @ F @ B2 ) ) ) ).
% vimage_Diff
thf(fact_133_null__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 ) ) )
@ ( measur2126959417real_n @ M ) )
=> ( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( Q @ X ) ) )
@ ( measur2126959417real_n @ M ) )
=> ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( ( Q @ X )
| ( P @ X ) ) ) )
@ ( measur2126959417real_n @ M ) ) ) ) ).
% null_sets.sets_Collect_disj
thf(fact_134_null__sets_Osets__Collect__disj,axiom,
! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M ) )
& ( P @ X ) ) )
@ ( measure_null_sets_o @ M ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M ) )
& ( Q @ X ) ) )
@ ( measure_null_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M ) )
& ( ( Q @ X )
| ( P @ X ) ) ) )
@ ( measure_null_sets_o @ M ) ) ) ) ).
% null_sets.sets_Collect_disj
thf(fact_135_null__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 ) ) )
@ ( measur1402256771real_n @ M ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( Q @ X ) ) )
@ ( measur1402256771real_n @ M ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( ( Q @ X )
| ( P @ X ) ) ) )
@ ( measur1402256771real_n @ M ) ) ) ) ).
% null_sets.sets_Collect_disj
thf(fact_136_null__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 ) ) )
@ ( measur2126959417real_n @ M ) )
=> ( ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( Q @ X ) ) )
@ ( measur2126959417real_n @ M ) )
=> ( member1475136633real_n
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
& ( Q @ X )
& ( P @ X ) ) )
@ ( measur2126959417real_n @ M ) ) ) ) ).
% null_sets.sets_Collect_conj
thf(fact_137_null__sets_Osets__Collect__conj,axiom,
! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
( ( member_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M ) )
& ( P @ X ) ) )
@ ( measure_null_sets_o @ M ) )
=> ( ( member_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M ) )
& ( Q @ X ) ) )
@ ( measure_null_sets_o @ M ) )
=> ( member_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( sigma_space_o @ M ) )
& ( Q @ X )
& ( P @ X ) ) )
@ ( measure_null_sets_o @ M ) ) ) ) ).
% null_sets.sets_Collect_conj
thf(fact_138_null__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 ) ) )
@ ( measur1402256771real_n @ M ) )
=> ( ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( Q @ X ) ) )
@ ( measur1402256771real_n @ M ) )
=> ( member223413699real_n
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
& ( Q @ X )
& ( P @ X ) ) )
@ ( measur1402256771real_n @ M ) ) ) ) ).
% null_sets.sets_Collect_conj
thf(fact_139_translation__diff,axiom,
! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T: set_Fi1058188332real_n] :
( ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ ( minus_1686442501real_n @ S2 @ T ) )
= ( minus_1686442501real_n @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ S2 ) @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ T ) ) ) ).
% translation_diff
thf(fact_140_null__set__Diff,axiom,
! [B2: set_Fi1058188332real_n,M: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
( ( member223413699real_n @ B2 @ ( measur1402256771real_n @ M ) )
=> ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) )
=> ( member223413699real_n @ ( minus_1686442501real_n @ B2 @ A3 ) @ ( measur1402256771real_n @ M ) ) ) ) ).
% null_set_Diff
thf(fact_141_Int__Union2,axiom,
! [B2: set_se2111327970real_n,A3: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( comple825005695real_n @ B2 ) @ A3 )
= ( comple825005695real_n
@ ( image_1661509983real_n
@ ^ [C3: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ C3 @ A3 )
@ B2 ) ) ) ).
% Int_Union2
thf(fact_142_Int__Union,axiom,
! [A3: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( inf_in1974387902real_n @ A3 @ ( comple825005695real_n @ B2 ) )
= ( comple825005695real_n @ ( image_1661509983real_n @ ( inf_in1974387902real_n @ A3 ) @ B2 ) ) ) ).
% Int_Union
thf(fact_143_vimage__Union,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A3: set_se2111327970real_n] :
( ( vimage1233683625real_n @ F @ ( comple825005695real_n @ A3 ) )
= ( comple825005695real_n @ ( image_1661509983real_n @ ( vimage1233683625real_n @ F ) @ A3 ) ) ) ).
% vimage_Union
thf(fact_144_UN__extend__simps_I6_J,axiom,
! [A3: nat > set_Fi1058188332real_n,C2: set_nat,B2: set_Fi1058188332real_n] :
( ( minus_1686442501real_n @ ( comple825005695real_n @ ( image_1856576259real_n @ A3 @ C2 ) ) @ B2 )
= ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [X: nat] : ( minus_1686442501real_n @ ( A3 @ X ) @ B2 )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_145_UN__extend__simps_I6_J,axiom,
! [A3: finite964658038_int_n > set_Fi1058188332real_n,C2: set_Fi160064172_int_n,B2: set_Fi1058188332real_n] :
( ( minus_1686442501real_n @ ( comple825005695real_n @ ( image_355963305real_n @ A3 @ C2 ) ) @ B2 )
= ( comple825005695real_n
@ ( image_355963305real_n
@ ^ [X: finite964658038_int_n] : ( minus_1686442501real_n @ ( A3 @ X ) @ B2 )
@ C2 ) ) ) ).
% UN_extend_simps(6)
thf(fact_146_emeasure__Diff__null__set,axiom,
! [B2: set_Fi1058188332real_n,M: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
( ( member223413699real_n @ B2 @ ( measur1402256771real_n @ M ) )
=> ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) )
=> ( ( sigma_1536574303real_n @ M @ ( minus_1686442501real_n @ A3 @ B2 ) )
= ( sigma_1536574303real_n @ M @ A3 ) ) ) ) ).
% emeasure_Diff_null_set
thf(fact_147_Sup_OSUP__cong,axiom,
! [A3: set_nat,B2: set_nat,C2: nat > set_Fi1058188332real_n,D: nat > set_Fi1058188332real_n,Sup: set_se2111327970real_n > set_Fi1058188332real_n] :
( ( A3 = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_1856576259real_n @ C2 @ A3 ) )
= ( Sup @ ( image_1856576259real_n @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_148_Sup_OSUP__cong,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: finite1489363574real_n > finite1489363574real_n,D: finite1489363574real_n > finite1489363574real_n,Sup: set_Fi1058188332real_n > finite1489363574real_n] :
( ( A3 = B2 )
=> ( ! [X2: finite1489363574real_n] :
( ( member1352538125real_n @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_439535603real_n @ C2 @ A3 ) )
= ( Sup @ ( image_439535603real_n @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_149_Sup_OSUP__cong,axiom,
! [A3: set_Fi160064172_int_n,B2: set_Fi160064172_int_n,C2: finite964658038_int_n > set_Fi1058188332real_n,D: finite964658038_int_n > set_Fi1058188332real_n,Sup: set_se2111327970real_n > set_Fi1058188332real_n] :
( ( A3 = B2 )
=> ( ! [X2: finite964658038_int_n] :
( ( member27055245_int_n @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_355963305real_n @ C2 @ A3 ) )
= ( Sup @ ( image_355963305real_n @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_150_Inf_OINF__cong,axiom,
! [A3: set_nat,B2: set_nat,C2: nat > set_Fi1058188332real_n,D: nat > set_Fi1058188332real_n,Inf: set_se2111327970real_n > set_Fi1058188332real_n] :
( ( A3 = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_1856576259real_n @ C2 @ A3 ) )
= ( Inf @ ( image_1856576259real_n @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_151_Inf_OINF__cong,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: finite1489363574real_n > finite1489363574real_n,D: finite1489363574real_n > finite1489363574real_n,Inf: set_Fi1058188332real_n > finite1489363574real_n] :
( ( A3 = B2 )
=> ( ! [X2: finite1489363574real_n] :
( ( member1352538125real_n @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_439535603real_n @ C2 @ A3 ) )
= ( Inf @ ( image_439535603real_n @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_152_Inf_OINF__cong,axiom,
! [A3: set_Fi160064172_int_n,B2: set_Fi160064172_int_n,C2: finite964658038_int_n > set_Fi1058188332real_n,D: finite964658038_int_n > set_Fi1058188332real_n,Inf: set_se2111327970real_n > set_Fi1058188332real_n] :
( ( A3 = B2 )
=> ( ! [X2: finite964658038_int_n] :
( ( member27055245_int_n @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_355963305real_n @ C2 @ A3 ) )
= ( Inf @ ( image_355963305real_n @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_153_rev__image__eqI,axiom,
! [X3: finite1489363574real_n,A3: set_Fi1058188332real_n,B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ A3 )
=> ( ( B
= ( F @ X3 ) )
=> ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_154_rev__image__eqI,axiom,
! [X3: nat,A3: set_nat,B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n] :
( ( member_nat @ X3 @ A3 )
=> ( ( B
= ( F @ X3 ) )
=> ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_155_rev__image__eqI,axiom,
! [X3: finite964658038_int_n,A3: set_Fi160064172_int_n,B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n] :
( ( member27055245_int_n @ X3 @ A3 )
=> ( ( B
= ( F @ X3 ) )
=> ( member223413699real_n @ B @ ( image_355963305real_n @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_156_rev__image__eqI,axiom,
! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n] :
( ( member223413699real_n @ X3 @ A3 )
=> ( ( B
= ( F @ X3 ) )
=> ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_157_rev__image__eqI,axiom,
! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: $o,F: set_Fi1058188332real_n > $o] :
( ( member223413699real_n @ X3 @ A3 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_1648361637al_n_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_158_rev__image__eqI,axiom,
! [X3: $o,A3: set_o,B: set_Fi1058188332real_n,F: $o > set_Fi1058188332real_n] :
( ( member_o @ X3 @ A3 )
=> ( ( B
= ( F @ X3 ) )
=> ( member223413699real_n @ B @ ( image_1759008383real_n @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_159_rev__image__eqI,axiom,
! [X3: $o,A3: set_o,B: $o,F: $o > $o] :
( ( member_o @ X3 @ A3 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_o_o @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_160_ball__imageD,axiom,
! [F: nat > set_Fi1058188332real_n,A3: set_nat,P: set_Fi1058188332real_n > $o] :
( ! [X2: set_Fi1058188332real_n] :
( ( member223413699real_n @ X2 @ ( image_1856576259real_n @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_161_ball__imageD,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
( ! [X2: finite1489363574real_n] :
( ( member1352538125real_n @ X2 @ ( image_439535603real_n @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X5: finite1489363574real_n] :
( ( member1352538125real_n @ X5 @ A3 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_162_ball__imageD,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: set_Fi1058188332real_n > $o] :
( ! [X2: set_Fi1058188332real_n] :
( ( member223413699real_n @ X2 @ ( image_355963305real_n @ F @ A3 ) )
=> ( P @ X2 ) )
=> ! [X5: finite964658038_int_n] :
( ( member27055245_int_n @ X5 @ A3 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_163_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > set_Fi1058188332real_n,G: nat > set_Fi1058188332real_n] :
( ( M = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_1856576259real_n @ F @ M )
= ( image_1856576259real_n @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_164_image__cong,axiom,
! [M: set_Fi1058188332real_n,N2: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n] :
( ( M = N2 )
=> ( ! [X2: finite1489363574real_n] :
( ( member1352538125real_n @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_439535603real_n @ F @ M )
= ( image_439535603real_n @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_165_image__cong,axiom,
! [M: set_Fi160064172_int_n,N2: set_Fi160064172_int_n,F: finite964658038_int_n > set_Fi1058188332real_n,G: finite964658038_int_n > set_Fi1058188332real_n] :
( ( M = N2 )
=> ( ! [X2: finite964658038_int_n] :
( ( member27055245_int_n @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_355963305real_n @ F @ M )
= ( image_355963305real_n @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_166_bex__imageD,axiom,
! [F: nat > set_Fi1058188332real_n,A3: set_nat,P: set_Fi1058188332real_n > $o] :
( ? [X5: set_Fi1058188332real_n] :
( ( member223413699real_n @ X5 @ ( image_1856576259real_n @ F @ A3 ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_167_bex__imageD,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
( ? [X5: finite1489363574real_n] :
( ( member1352538125real_n @ X5 @ ( image_439535603real_n @ F @ A3 ) )
& ( P @ X5 ) )
=> ? [X2: finite1489363574real_n] :
( ( member1352538125real_n @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_168_bex__imageD,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: set_Fi1058188332real_n > $o] :
( ? [X5: set_Fi1058188332real_n] :
( ( member223413699real_n @ X5 @ ( image_355963305real_n @ F @ A3 ) )
& ( P @ X5 ) )
=> ? [X2: finite964658038_int_n] :
( ( member27055245_int_n @ X2 @ A3 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_169_image__iff,axiom,
! [Z: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( member1352538125real_n @ Z @ ( image_439535603real_n @ F @ A3 ) )
= ( ? [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_170_image__iff,axiom,
! [Z: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,A3: set_nat] :
( ( member223413699real_n @ Z @ ( image_1856576259real_n @ F @ A3 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_171_image__iff,axiom,
! [Z: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
( ( member223413699real_n @ Z @ ( image_355963305real_n @ F @ A3 ) )
= ( ? [X: finite964658038_int_n] :
( ( member27055245_int_n @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_172_imageI,axiom,
! [X3: finite1489363574real_n,A3: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n] :
( ( member1352538125real_n @ X3 @ A3 )
=> ( member1352538125real_n @ ( F @ X3 ) @ ( image_439535603real_n @ F @ A3 ) ) ) ).
% imageI
thf(fact_173_imageI,axiom,
! [X3: nat,A3: set_nat,F: nat > set_Fi1058188332real_n] :
( ( member_nat @ X3 @ A3 )
=> ( member223413699real_n @ ( F @ X3 ) @ ( image_1856576259real_n @ F @ A3 ) ) ) ).
% imageI
thf(fact_174_imageI,axiom,
! [X3: finite964658038_int_n,A3: set_Fi160064172_int_n,F: finite964658038_int_n > set_Fi1058188332real_n] :
( ( member27055245_int_n @ X3 @ A3 )
=> ( member223413699real_n @ ( F @ X3 ) @ ( image_355963305real_n @ F @ A3 ) ) ) ).
% imageI
thf(fact_175_imageI,axiom,
! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n] :
( ( member223413699real_n @ X3 @ A3 )
=> ( member223413699real_n @ ( F @ X3 ) @ ( image_1661509983real_n @ F @ A3 ) ) ) ).
% imageI
thf(fact_176_imageI,axiom,
! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,F: set_Fi1058188332real_n > $o] :
( ( member223413699real_n @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ ( image_1648361637al_n_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_177_imageI,axiom,
! [X3: $o,A3: set_o,F: $o > set_Fi1058188332real_n] :
( ( member_o @ X3 @ A3 )
=> ( member223413699real_n @ ( F @ X3 ) @ ( image_1759008383real_n @ F @ A3 ) ) ) ).
% imageI
thf(fact_178_imageI,axiom,
! [X3: $o,A3: set_o,F: $o > $o] :
( ( member_o @ X3 @ A3 )
=> ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A3 ) ) ) ).
% imageI
thf(fact_179_is__num__normalize_I1_J,axiom,
! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] :
( ( plus_p585657087real_n @ ( plus_p585657087real_n @ A @ B ) @ C )
= ( plus_p585657087real_n @ A @ ( plus_p585657087real_n @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_180_UNIV__witness,axiom,
? [X2: set_Fi1058188332real_n] : ( member223413699real_n @ X2 @ top_to20708754real_n ) ).
% UNIV_witness
thf(fact_181_UNIV__witness,axiom,
? [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_182_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_183_UNIV__witness,axiom,
? [X2: finite964658038_int_n] : ( member27055245_int_n @ X2 @ top_to131672412_int_n ) ).
% UNIV_witness
thf(fact_184_UNIV__eq__I,axiom,
! [A3: set_se2111327970real_n] :
( ! [X2: set_Fi1058188332real_n] : ( member223413699real_n @ X2 @ A3 )
=> ( top_to20708754real_n = A3 ) ) ).
% UNIV_eq_I
thf(fact_185_UNIV__eq__I,axiom,
! [A3: set_o] :
( ! [X2: $o] : ( member_o @ X2 @ A3 )
=> ( top_top_set_o = A3 ) ) ).
% UNIV_eq_I
thf(fact_186_UNIV__eq__I,axiom,
! [A3: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A3 )
=> ( top_top_set_nat = A3 ) ) ).
% UNIV_eq_I
thf(fact_187_UNIV__eq__I,axiom,
! [A3: set_Fi160064172_int_n] :
( ! [X2: finite964658038_int_n] : ( member27055245_int_n @ X2 @ A3 )
=> ( top_to131672412_int_n = A3 ) ) ).
% UNIV_eq_I
thf(fact_188_Int__left__commute,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ A3 @ ( inf_in1974387902real_n @ B2 @ C2 ) )
= ( inf_in1974387902real_n @ B2 @ ( inf_in1974387902real_n @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_189_Int__left__absorb,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ A3 @ ( inf_in1974387902real_n @ A3 @ B2 ) )
= ( inf_in1974387902real_n @ A3 @ B2 ) ) ).
% Int_left_absorb
thf(fact_190_Int__commute,axiom,
( inf_in1974387902real_n
= ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_191_Int__absorb,axiom,
! [A3: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_192_Int__assoc,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ ( inf_in1974387902real_n @ A3 @ B2 ) @ C2 )
= ( inf_in1974387902real_n @ A3 @ ( inf_in1974387902real_n @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_193_IntD2,axiom,
! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
=> ( member223413699real_n @ C @ B2 ) ) ).
% IntD2
thf(fact_194_IntD2,axiom,
! [C: $o,A3: set_o,B2: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
=> ( member_o @ C @ B2 ) ) ).
% IntD2
thf(fact_195_IntD2,axiom,
! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
=> ( member1352538125real_n @ C @ B2 ) ) ).
% IntD2
thf(fact_196_IntD1,axiom,
! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
=> ( member223413699real_n @ C @ A3 ) ) ).
% IntD1
thf(fact_197_IntD1,axiom,
! [C: $o,A3: set_o,B2: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
=> ( member_o @ C @ A3 ) ) ).
% IntD1
thf(fact_198_IntD1,axiom,
! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
=> ( member1352538125real_n @ C @ A3 ) ) ).
% IntD1
thf(fact_199_IntE,axiom,
! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
=> ~ ( ( member223413699real_n @ C @ A3 )
=> ~ ( member223413699real_n @ C @ B2 ) ) ) ).
% IntE
thf(fact_200_IntE,axiom,
! [C: $o,A3: set_o,B2: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
=> ~ ( ( member_o @ C @ A3 )
=> ~ ( member_o @ C @ B2 ) ) ) ).
% IntE
thf(fact_201_IntE,axiom,
! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
=> ~ ( ( member1352538125real_n @ C @ A3 )
=> ~ ( member1352538125real_n @ C @ B2 ) ) ) ).
% IntE
thf(fact_202_UnionE,axiom,
! [A3: set_Fi1058188332real_n,C2: set_se820660888real_n] :
( ( member223413699real_n @ A3 @ ( comple1917283637real_n @ C2 ) )
=> ~ ! [X6: set_se2111327970real_n] :
( ( member223413699real_n @ A3 @ X6 )
=> ~ ( member1475136633real_n @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_203_UnionE,axiom,
! [A3: $o,C2: set_set_o] :
( ( member_o @ A3 @ ( comple1665300069_set_o @ C2 ) )
=> ~ ! [X6: set_o] :
( ( member_o @ A3 @ X6 )
=> ~ ( member_set_o @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_204_UnionE,axiom,
! [A3: finite1489363574real_n,C2: set_se2111327970real_n] :
( ( member1352538125real_n @ A3 @ ( comple825005695real_n @ C2 ) )
=> ~ ! [X6: set_Fi1058188332real_n] :
( ( member1352538125real_n @ A3 @ X6 )
=> ~ ( member223413699real_n @ X6 @ C2 ) ) ) ).
% UnionE
thf(fact_205_vimage__Collect,axiom,
! [P: finite1489363574real_n > $o,F: finite1489363574real_n > finite1489363574real_n,Q: finite1489363574real_n > $o] :
( ! [X2: finite1489363574real_n] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage1233683625real_n @ F @ ( collec321817931real_n @ P ) )
= ( collec321817931real_n @ Q ) ) ) ).
% vimage_Collect
thf(fact_206_vimageI2,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A: set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( member223413699real_n @ ( F @ A ) @ A3 )
=> ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_207_vimageI2,axiom,
! [F: $o > set_Fi1058188332real_n,A: $o,A3: set_se2111327970real_n] :
( ( member223413699real_n @ ( F @ A ) @ A3 )
=> ( member_o @ A @ ( vimage961837641real_n @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_208_vimageI2,axiom,
! [F: set_Fi1058188332real_n > $o,A: set_Fi1058188332real_n,A3: set_o] :
( ( member_o @ ( F @ A ) @ A3 )
=> ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_209_vimageI2,axiom,
! [F: $o > $o,A: $o,A3: set_o] :
( ( member_o @ ( F @ A ) @ A3 )
=> ( member_o @ A @ ( vimage_o_o @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_210_vimageI2,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A: finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( member1352538125real_n @ ( F @ A ) @ A3 )
=> ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ A3 ) ) ) ).
% vimageI2
thf(fact_211_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_212_vimageE,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > $o,B2: set_o] :
( ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ B2 ) )
=> ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_213_vimageE,axiom,
! [A: $o,F: $o > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
( ( member_o @ A @ ( vimage961837641real_n @ F @ B2 ) )
=> ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_214_vimageE,axiom,
! [A: $o,F: $o > $o,B2: set_o] :
( ( member_o @ A @ ( vimage_o_o @ F @ B2 ) )
=> ( member_o @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_215_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_216_vimageD,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ A3 ) )
=> ( member223413699real_n @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_217_vimageD,axiom,
! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > $o,A3: set_o] :
( ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ A3 ) )
=> ( member_o @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_218_vimageD,axiom,
! [A: $o,F: $o > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( member_o @ A @ ( vimage961837641real_n @ F @ A3 ) )
=> ( member223413699real_n @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_219_vimageD,axiom,
! [A: $o,F: $o > $o,A3: set_o] :
( ( member_o @ A @ ( vimage_o_o @ F @ A3 ) )
=> ( member_o @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_220_vimageD,axiom,
! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ A3 ) )
=> ( member1352538125real_n @ ( F @ A ) @ A3 ) ) ).
% vimageD
thf(fact_221_Sup_OSUP__identity__eq,axiom,
! [Sup: set_Fi1058188332real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( Sup
@ ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : X
@ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_identity_eq
thf(fact_222_Inf_OINF__identity__eq,axiom,
! [Inf: set_Fi1058188332real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( Inf
@ ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : X
@ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_identity_eq
thf(fact_223_Compr__image__eq,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
( ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ ( image_439535603real_n @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_439535603real_n @ F
@ ( collec321817931real_n
@ ^ [X: finite1489363574real_n] :
( ( member1352538125real_n @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_224_Compr__image__eq,axiom,
! [F: nat > set_Fi1058188332real_n,A3: set_nat,P: set_Fi1058188332real_n > $o] :
( ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( image_1856576259real_n @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_1856576259real_n @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_225_Compr__image__eq,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: set_Fi1058188332real_n > $o] :
( ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( image_355963305real_n @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_355963305real_n @ F
@ ( collec1941932235_int_n
@ ^ [X: finite964658038_int_n] :
( ( member27055245_int_n @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_226_Compr__image__eq,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n,P: set_Fi1058188332real_n > $o] :
( ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( image_1661509983real_n @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_1661509983real_n @ F
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_227_Compr__image__eq,axiom,
! [F: $o > set_Fi1058188332real_n,A3: set_o,P: set_Fi1058188332real_n > $o] :
( ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ ( image_1759008383real_n @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_1759008383real_n @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_228_Compr__image__eq,axiom,
! [F: set_Fi1058188332real_n > $o,A3: set_se2111327970real_n,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_1648361637al_n_o @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_1648361637al_n_o @ F
@ ( collec452821761real_n
@ ^ [X: set_Fi1058188332real_n] :
( ( member223413699real_n @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_229_Compr__image__eq,axiom,
! [F: $o > $o,A3: set_o,P: $o > $o] :
( ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ ( image_o_o @ F @ A3 ) )
& ( P @ X ) ) )
= ( image_o_o @ F
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_230_image__image,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: nat > set_Fi1058188332real_n,A3: set_nat] :
( ( image_1661509983real_n @ F @ ( image_1856576259real_n @ G @ A3 ) )
= ( image_1856576259real_n
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_231_image__image,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
( ( image_1661509983real_n @ F @ ( image_355963305real_n @ G @ A3 ) )
= ( image_355963305real_n
@ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_232_image__image,axiom,
! [F: nat > set_Fi1058188332real_n,G: nat > nat,A3: set_nat] :
( ( image_1856576259real_n @ F @ ( image_nat_nat @ G @ A3 ) )
= ( image_1856576259real_n
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_233_image__image,axiom,
! [F: nat > set_Fi1058188332real_n,G: finite964658038_int_n > nat,A3: set_Fi160064172_int_n] :
( ( image_1856576259real_n @ F @ ( image_497739341_n_nat @ G @ A3 ) )
= ( image_355963305real_n
@ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_234_image__image,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( image_439535603real_n @ F @ ( image_439535603real_n @ G @ A3 ) )
= ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_235_image__image,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,G: nat > finite964658038_int_n,A3: set_nat] :
( ( image_355963305real_n @ F @ ( image_1961920973_int_n @ G @ A3 ) )
= ( image_1856576259real_n
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_236_image__image,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,G: finite964658038_int_n > finite964658038_int_n,A3: set_Fi160064172_int_n] :
( ( image_355963305real_n @ F @ ( image_1278151539_int_n @ G @ A3 ) )
= ( image_355963305real_n
@ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_237_imageE,axiom,
! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
( ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A3 ) )
=> ~ ! [X2: finite1489363574real_n] :
( ( B
= ( F @ X2 ) )
=> ~ ( member1352538125real_n @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_238_imageE,axiom,
! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,A3: set_nat] :
( ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ A3 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_239_imageE,axiom,
! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
( ( member223413699real_n @ B @ ( image_355963305real_n @ F @ A3 ) )
=> ~ ! [X2: finite964658038_int_n] :
( ( B
= ( F @ X2 ) )
=> ~ ( member27055245_int_n @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_240_imageE,axiom,
! [B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A3 ) )
=> ~ ! [X2: set_Fi1058188332real_n] :
( ( B
= ( F @ X2 ) )
=> ~ ( member223413699real_n @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_241_imageE,axiom,
! [B: set_Fi1058188332real_n,F: $o > set_Fi1058188332real_n,A3: set_o] :
( ( member223413699real_n @ B @ ( image_1759008383real_n @ F @ A3 ) )
=> ~ ! [X2: $o] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_o @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_242_imageE,axiom,
! [B: $o,F: set_Fi1058188332real_n > $o,A3: set_se2111327970real_n] :
( ( member_o @ B @ ( image_1648361637al_n_o @ F @ A3 ) )
=> ~ ! [X2: set_Fi1058188332real_n] :
( ( B
= ( F @ X2 ) )
=> ~ ( member223413699real_n @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_243_imageE,axiom,
! [B: $o,F: $o > $o,A3: set_o] :
( ( member_o @ B @ ( image_o_o @ F @ A3 ) )
=> ~ ! [X2: $o] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_o @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_244_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_245_UNIV__def,axiom,
( top_to131672412_int_n
= ( collec1941932235_int_n
@ ^ [X: finite964658038_int_n] : $true ) ) ).
% UNIV_def
thf(fact_246_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_247_Int__Collect,axiom,
! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,P: set_Fi1058188332real_n > $o] :
( ( member223413699real_n @ X3 @ ( inf_in632889204real_n @ A3 @ ( collec452821761real_n @ P ) ) )
= ( ( member223413699real_n @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_248_Int__Collect,axiom,
! [X3: $o,A3: set_o,P: $o > $o] :
( ( member_o @ X3 @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
= ( ( member_o @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_249_Int__Collect,axiom,
! [X3: finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
( ( member1352538125real_n @ X3 @ ( inf_in1974387902real_n @ A3 @ ( collec321817931real_n @ P ) ) )
= ( ( member1352538125real_n @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_250_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_251_Int__def,axiom,
( inf_inf_set_o
= ( ^ [A4: set_o,B3: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A4 )
& ( member_o @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_252_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_253_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_254_less__numeral__extra_I3_J,axiom,
~ ( ord_le2133614988nnreal @ zero_z1963244097nnreal @ zero_z1963244097nnreal ) ).
% less_numeral_extra(3)
thf(fact_255_less__numeral__extra_I4_J,axiom,
~ ( ord_le2133614988nnreal @ one_on705384445nnreal @ one_on705384445nnreal ) ).
% less_numeral_extra(4)
thf(fact_256_range__eqI,axiom,
! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,X3: finite1489363574real_n] :
( ( B
= ( F @ X3 ) )
=> ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ top_to1292442332real_n ) ) ) ).
% range_eqI
thf(fact_257_range__eqI,axiom,
! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,X3: nat] :
( ( B
= ( F @ X3 ) )
=> ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_258_range__eqI,axiom,
! [B: $o,F: nat > $o,X3: nat] :
( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_259_range__eqI,axiom,
! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,X3: finite964658038_int_n] :
( ( B
= ( F @ X3 ) )
=> ( member223413699real_n @ B @ ( image_355963305real_n @ F @ top_to131672412_int_n ) ) ) ).
% range_eqI
thf(fact_260_range__eqI,axiom,
! [B: $o,F: finite964658038_int_n > $o,X3: finite964658038_int_n] :
( ( B
= ( F @ X3 ) )
=> ( member_o @ B @ ( image_216309723nt_n_o @ F @ top_to131672412_int_n ) ) ) ).
% range_eqI
thf(fact_261_surj__def,axiom,
! [F: finite1489363574real_n > finite1489363574real_n] :
( ( ( image_439535603real_n @ F @ top_to1292442332real_n )
= top_to1292442332real_n )
= ( ! [Y2: finite1489363574real_n] :
? [X: finite1489363574real_n] :
( Y2
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_262_surj__def,axiom,
! [F: nat > set_Fi1058188332real_n] :
( ( ( image_1856576259real_n @ F @ top_top_set_nat )
= top_to20708754real_n )
= ( ! [Y2: set_Fi1058188332real_n] :
? [X: nat] :
( Y2
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_263_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X: nat] :
( Y2
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_264_surj__def,axiom,
! [F: nat > finite964658038_int_n] :
( ( ( image_1961920973_int_n @ F @ top_top_set_nat )
= top_to131672412_int_n )
= ( ! [Y2: finite964658038_int_n] :
? [X: nat] :
( Y2
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_265_surj__def,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n] :
( ( ( image_355963305real_n @ F @ top_to131672412_int_n )
= top_to20708754real_n )
= ( ! [Y2: set_Fi1058188332real_n] :
? [X: finite964658038_int_n] :
( Y2
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_266_surj__def,axiom,
! [F: finite964658038_int_n > nat] :
( ( ( image_497739341_n_nat @ F @ top_to131672412_int_n )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X: finite964658038_int_n] :
( Y2
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_267_surj__def,axiom,
! [F: finite964658038_int_n > finite964658038_int_n] :
( ( ( image_1278151539_int_n @ F @ top_to131672412_int_n )
= top_to131672412_int_n )
= ( ! [Y2: finite964658038_int_n] :
? [X: finite964658038_int_n] :
( Y2
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_268_rangeI,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,X3: finite1489363574real_n] : ( member1352538125real_n @ ( F @ X3 ) @ ( image_439535603real_n @ F @ top_to1292442332real_n ) ) ).
% rangeI
thf(fact_269_rangeI,axiom,
! [F: nat > set_Fi1058188332real_n,X3: nat] : ( member223413699real_n @ ( F @ X3 ) @ ( image_1856576259real_n @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_270_rangeI,axiom,
! [F: nat > $o,X3: nat] : ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_271_rangeI,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,X3: finite964658038_int_n] : ( member223413699real_n @ ( F @ X3 ) @ ( image_355963305real_n @ F @ top_to131672412_int_n ) ) ).
% rangeI
thf(fact_272_rangeI,axiom,
! [F: finite964658038_int_n > $o,X3: finite964658038_int_n] : ( member_o @ ( F @ X3 ) @ ( image_216309723nt_n_o @ F @ top_to131672412_int_n ) ) ).
% rangeI
thf(fact_273_surjI,axiom,
! [G: finite1489363574real_n > finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
( ! [X2: finite1489363574real_n] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_439535603real_n @ G @ top_to1292442332real_n )
= top_to1292442332real_n ) ) ).
% surjI
thf(fact_274_surjI,axiom,
! [G: nat > set_Fi1058188332real_n,F: set_Fi1058188332real_n > nat] :
( ! [X2: set_Fi1058188332real_n] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_1856576259real_n @ G @ top_top_set_nat )
= top_to20708754real_n ) ) ).
% surjI
thf(fact_275_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_276_surjI,axiom,
! [G: nat > finite964658038_int_n,F: finite964658038_int_n > nat] :
( ! [X2: finite964658038_int_n] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_1961920973_int_n @ G @ top_top_set_nat )
= top_to131672412_int_n ) ) ).
% surjI
thf(fact_277_surjI,axiom,
! [G: finite964658038_int_n > set_Fi1058188332real_n,F: set_Fi1058188332real_n > finite964658038_int_n] :
( ! [X2: set_Fi1058188332real_n] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_355963305real_n @ G @ top_to131672412_int_n )
= top_to20708754real_n ) ) ).
% surjI
thf(fact_278_surjI,axiom,
! [G: finite964658038_int_n > nat,F: nat > finite964658038_int_n] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_497739341_n_nat @ G @ top_to131672412_int_n )
= top_top_set_nat ) ) ).
% surjI
thf(fact_279_surjI,axiom,
! [G: finite964658038_int_n > finite964658038_int_n,F: finite964658038_int_n > finite964658038_int_n] :
( ! [X2: finite964658038_int_n] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_1278151539_int_n @ G @ top_to131672412_int_n )
= top_to131672412_int_n ) ) ).
% surjI
thf(fact_280_surjE,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,Y4: finite1489363574real_n] :
( ( ( image_439535603real_n @ F @ top_to1292442332real_n )
= top_to1292442332real_n )
=> ~ ! [X2: finite1489363574real_n] :
( Y4
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_281_surjE,axiom,
! [F: nat > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
( ( ( image_1856576259real_n @ F @ top_top_set_nat )
= top_to20708754real_n )
=> ~ ! [X2: nat] :
( Y4
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_282_surjE,axiom,
! [F: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y4
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_283_surjE,axiom,
! [F: nat > finite964658038_int_n,Y4: finite964658038_int_n] :
( ( ( image_1961920973_int_n @ F @ top_top_set_nat )
= top_to131672412_int_n )
=> ~ ! [X2: nat] :
( Y4
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_284_surjE,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
( ( ( image_355963305real_n @ F @ top_to131672412_int_n )
= top_to20708754real_n )
=> ~ ! [X2: finite964658038_int_n] :
( Y4
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_285_surjE,axiom,
! [F: finite964658038_int_n > nat,Y4: nat] :
( ( ( image_497739341_n_nat @ F @ top_to131672412_int_n )
= top_top_set_nat )
=> ~ ! [X2: finite964658038_int_n] :
( Y4
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_286_surjE,axiom,
! [F: finite964658038_int_n > finite964658038_int_n,Y4: finite964658038_int_n] :
( ( ( image_1278151539_int_n @ F @ top_to131672412_int_n )
= top_to131672412_int_n )
=> ~ ! [X2: finite964658038_int_n] :
( Y4
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_287_surjD,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,Y4: finite1489363574real_n] :
( ( ( image_439535603real_n @ F @ top_to1292442332real_n )
= top_to1292442332real_n )
=> ? [X2: finite1489363574real_n] :
( Y4
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_288_surjD,axiom,
! [F: nat > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
( ( ( image_1856576259real_n @ F @ top_top_set_nat )
= top_to20708754real_n )
=> ? [X2: nat] :
( Y4
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_289_surjD,axiom,
! [F: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y4
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_290_surjD,axiom,
! [F: nat > finite964658038_int_n,Y4: finite964658038_int_n] :
( ( ( image_1961920973_int_n @ F @ top_top_set_nat )
= top_to131672412_int_n )
=> ? [X2: nat] :
( Y4
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_291_surjD,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
( ( ( image_355963305real_n @ F @ top_to131672412_int_n )
= top_to20708754real_n )
=> ? [X2: finite964658038_int_n] :
( Y4
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_292_surjD,axiom,
! [F: finite964658038_int_n > nat,Y4: nat] :
( ( ( image_497739341_n_nat @ F @ top_to131672412_int_n )
= top_top_set_nat )
=> ? [X2: finite964658038_int_n] :
( Y4
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_293_surjD,axiom,
! [F: finite964658038_int_n > finite964658038_int_n,Y4: finite964658038_int_n] :
( ( ( image_1278151539_int_n @ F @ top_to131672412_int_n )
= top_to131672412_int_n )
=> ? [X2: finite964658038_int_n] :
( Y4
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_294_less__Sup__iff,axiom,
! [A: extend1728876344nnreal,S: set_Ex113815278nnreal] :
( ( ord_le2133614988nnreal @ A @ ( comple1413366923nnreal @ S ) )
= ( ? [X: extend1728876344nnreal] :
( ( member1217042383nnreal @ X @ S )
& ( ord_le2133614988nnreal @ A @ X ) ) ) ) ).
% less_Sup_iff
thf(fact_295_SUP__cong,axiom,
! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: finite1489363574real_n > finite1489363574real_n,D: finite1489363574real_n > finite1489363574real_n] :
( ( A3 = B2 )
=> ( ! [X2: finite1489363574real_n] :
( ( member1352538125real_n @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple2042271945real_n @ ( image_439535603real_n @ C2 @ A3 ) )
= ( comple2042271945real_n @ ( image_439535603real_n @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_296_SUP__cong,axiom,
! [A3: set_nat,B2: set_nat,C2: nat > set_Fi1058188332real_n,D: nat > set_Fi1058188332real_n] :
( ( A3 = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ A3 ) )
= ( comple825005695real_n @ ( image_1856576259real_n @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_297_SUP__cong,axiom,
! [A3: set_Fi160064172_int_n,B2: set_Fi160064172_int_n,C2: finite964658038_int_n > set_Fi1058188332real_n,D: finite964658038_int_n > set_Fi1058188332real_n] :
( ( A3 = B2 )
=> ( ! [X2: finite964658038_int_n] :
( ( member27055245_int_n @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple825005695real_n @ ( image_355963305real_n @ C2 @ A3 ) )
= ( comple825005695real_n @ ( image_355963305real_n @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_298_SUP__cong,axiom,
! [A3: set_se2111327970real_n,B2: set_se2111327970real_n,C2: set_Fi1058188332real_n > set_Fi1058188332real_n,D: set_Fi1058188332real_n > set_Fi1058188332real_n] :
( ( A3 = B2 )
=> ( ! [X2: set_Fi1058188332real_n] :
( ( member223413699real_n @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple825005695real_n @ ( image_1661509983real_n @ C2 @ A3 ) )
= ( comple825005695real_n @ ( image_1661509983real_n @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_299_SUP__cong,axiom,
! [A3: set_o,B2: set_o,C2: $o > set_Fi1058188332real_n,D: $o > set_Fi1058188332real_n] :
( ( A3 = B2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( comple825005695real_n @ ( image_1759008383real_n @ C2 @ A3 ) )
= ( comple825005695real_n @ ( image_1759008383real_n @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_300_SUP__cong,axiom,
! [A3: set_se2111327970real_n,B2: set_se2111327970real_n,C2: set_Fi1058188332real_n > $o,D: set_Fi1058188332real_n > $o] :
( ( A3 = B2 )
=> ( ! [X2: set_Fi1058188332real_n] :
( ( member223413699real_n @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_1648361637al_n_o @ C2 @ A3 ) )
= ( complete_Sup_Sup_o @ ( image_1648361637al_n_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_301_SUP__cong,axiom,
! [A3: set_o,B2: set_o,C2: $o > $o,D: $o > $o] :
( ( A3 = B2 )
=> ( ! [X2: $o] :
( ( member_o @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A3 ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_302_Int__UNIV__right,axiom,
! [A3: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ A3 @ top_to1292442332real_n )
= A3 ) ).
% Int_UNIV_right
thf(fact_303_Int__UNIV__right,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ top_top_set_nat )
= A3 ) ).
% Int_UNIV_right
thf(fact_304_Int__UNIV__right,axiom,
! [A3: set_Fi160064172_int_n] :
( ( inf_in1108485182_int_n @ A3 @ top_to131672412_int_n )
= A3 ) ).
% Int_UNIV_right
thf(fact_305_Int__UNIV__left,axiom,
! [B2: set_Fi1058188332real_n] :
( ( inf_in1974387902real_n @ top_to1292442332real_n @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_306_Int__UNIV__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_307_Int__UNIV__left,axiom,
! [B2: set_Fi160064172_int_n] :
( ( inf_in1108485182_int_n @ top_to131672412_int_n @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_308_Union__UNIV,axiom,
( ( comple1682161881et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_309_Union__UNIV,axiom,
( ( comple970917503_int_n @ top_to1587634578_int_n )
= top_to131672412_int_n ) ).
% Union_UNIV
thf(fact_310_Union__UNIV,axiom,
( ( comple825005695real_n @ top_to20708754real_n )
= top_to1292442332real_n ) ).
% Union_UNIV
thf(fact_311_null__setsD2,axiom,
! [A3: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
( ( member223413699real_n @ A3 @ ( measur1402256771real_n @ M ) )
=> ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) ) ) ).
% null_setsD2
thf(fact_312_vimage__inter__cong,axiom,
! [S: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,Y4: set_Fi1058188332real_n] :
( ! [W: finite1489363574real_n] :
( ( member1352538125real_n @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ Y4 ) @ S )
= ( inf_in1974387902real_n @ ( vimage1233683625real_n @ G @ Y4 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_313_translation__subtract__diff,axiom,
! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T: set_Fi1058188332real_n] :
( ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
@ ( minus_1686442501real_n @ S2 @ T ) )
= ( 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 )
@ T ) ) ) ).
% translation_subtract_diff
thf(fact_314_range__composition,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n] :
( ( image_439535603real_n
@ ^ [X: finite1489363574real_n] : ( F @ ( G @ X ) )
@ top_to1292442332real_n )
= ( image_439535603real_n @ F @ ( image_439535603real_n @ G @ top_to1292442332real_n ) ) ) ).
% range_composition
thf(fact_315_range__composition,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,G: nat > finite1489363574real_n] :
( ( image_183184717real_n
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_439535603real_n @ F @ ( image_183184717real_n @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_316_range__composition,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: nat > set_Fi1058188332real_n] :
( ( image_1856576259real_n
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_1661509983real_n @ F @ ( image_1856576259real_n @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_317_range__composition,axiom,
! [F: nat > set_Fi1058188332real_n,G: nat > nat] :
( ( image_1856576259real_n
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_1856576259real_n @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_318_range__composition,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,G: nat > finite964658038_int_n] :
( ( image_1856576259real_n
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ top_top_set_nat )
= ( image_355963305real_n @ F @ ( image_1961920973_int_n @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_319_range__composition,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,G: finite964658038_int_n > finite1489363574real_n] :
( ( image_2058828787real_n
@ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
@ top_to131672412_int_n )
= ( image_439535603real_n @ F @ ( image_2058828787real_n @ G @ top_to131672412_int_n ) ) ) ).
% range_composition
thf(fact_320_range__composition,axiom,
! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: finite964658038_int_n > set_Fi1058188332real_n] :
( ( image_355963305real_n
@ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
@ top_to131672412_int_n )
= ( image_1661509983real_n @ F @ ( image_355963305real_n @ G @ top_to131672412_int_n ) ) ) ).
% range_composition
thf(fact_321_range__composition,axiom,
! [F: nat > set_Fi1058188332real_n,G: finite964658038_int_n > nat] :
( ( image_355963305real_n
@ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
@ top_to131672412_int_n )
= ( image_1856576259real_n @ F @ ( image_497739341_n_nat @ G @ top_to131672412_int_n ) ) ) ).
% range_composition
thf(fact_322_range__composition,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,G: finite964658038_int_n > finite964658038_int_n] :
( ( image_355963305real_n
@ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
@ top_to131672412_int_n )
= ( image_355963305real_n @ F @ ( image_1278151539_int_n @ G @ top_to131672412_int_n ) ) ) ).
% range_composition
thf(fact_323_rangeE,axiom,
! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
( ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ top_to1292442332real_n ) )
=> ~ ! [X2: finite1489363574real_n] :
( B
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_324_rangeE,axiom,
! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n] :
( ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_325_rangeE,axiom,
! [B: $o,F: nat > $o] :
( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
=> ~ ! [X2: nat] :
( B
= ( ~ ( F @ X2 ) ) ) ) ).
% rangeE
thf(fact_326_rangeE,axiom,
! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n] :
( ( member223413699real_n @ B @ ( image_355963305real_n @ F @ top_to131672412_int_n ) )
=> ~ ! [X2: finite964658038_int_n] :
( B
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_327_rangeE,axiom,
! [B: $o,F: finite964658038_int_n > $o] :
( ( member_o @ B @ ( image_216309723nt_n_o @ F @ top_to131672412_int_n ) )
=> ~ ! [X2: finite964658038_int_n] :
( B
= ( ~ ( F @ X2 ) ) ) ) ).
% rangeE
thf(fact_328_SUP__commute,axiom,
! [F: nat > nat > set_Fi1058188332real_n,B2: set_nat,A3: set_nat] :
( ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [I4: nat] : ( comple825005695real_n @ ( image_1856576259real_n @ ( F @ I4 ) @ B2 ) )
@ A3 ) )
= ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [J: nat] :
( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [I4: nat] : ( F @ I4 @ J )
@ A3 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_329_SUP__commute,axiom,
! [F: nat > finite964658038_int_n > set_Fi1058188332real_n,B2: set_Fi160064172_int_n,A3: set_nat] :
( ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [I4: nat] : ( comple825005695real_n @ ( image_355963305real_n @ ( F @ I4 ) @ B2 ) )
@ A3 ) )
= ( comple825005695real_n
@ ( image_355963305real_n
@ ^ [J: finite964658038_int_n] :
( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [I4: nat] : ( F @ I4 @ J )
@ A3 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_330_SUP__commute,axiom,
! [F: finite964658038_int_n > nat > set_Fi1058188332real_n,B2: set_nat,A3: set_Fi160064172_int_n] :
( ( comple825005695real_n
@ ( image_355963305real_n
@ ^ [I4: finite964658038_int_n] : ( comple825005695real_n @ ( image_1856576259real_n @ ( F @ I4 ) @ B2 ) )
@ A3 ) )
= ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [J: nat] :
( comple825005695real_n
@ ( image_355963305real_n
@ ^ [I4: finite964658038_int_n] : ( F @ I4 @ J )
@ A3 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_331_SUP__commute,axiom,
! [F: finite964658038_int_n > finite964658038_int_n > set_Fi1058188332real_n,B2: set_Fi160064172_int_n,A3: set_Fi160064172_int_n] :
( ( comple825005695real_n
@ ( image_355963305real_n
@ ^ [I4: finite964658038_int_n] : ( comple825005695real_n @ ( image_355963305real_n @ ( F @ I4 ) @ B2 ) )
@ A3 ) )
= ( comple825005695real_n
@ ( image_355963305real_n
@ ^ [J: finite964658038_int_n] :
( comple825005695real_n
@ ( image_355963305real_n
@ ^ [I4: finite964658038_int_n] : ( F @ I4 @ J )
@ A3 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_332_image__Union,axiom,
! [F: nat > set_Fi1058188332real_n,S: set_set_nat] :
( ( image_1856576259real_n @ F @ ( comple1682161881et_nat @ S ) )
= ( comple1917283637real_n @ ( image_1587769199real_n @ ( image_1856576259real_n @ F ) @ S ) ) ) ).
% image_Union
thf(fact_333_image__Union,axiom,
! [F: finite964658038_int_n > set_Fi1058188332real_n,S: set_se944069346_int_n] :
( ( image_355963305real_n @ F @ ( comple970917503_int_n @ S ) )
= ( comple1917283637real_n @ ( image_1054146965real_n @ ( image_355963305real_n @ F ) @ S ) ) ) ).
% image_Union
thf(fact_334_image__Union,axiom,
! [F: finite1489363574real_n > finite1489363574real_n,S: set_se2111327970real_n] :
( ( image_439535603real_n @ F @ ( comple825005695real_n @ S ) )
= ( comple825005695real_n @ ( image_1661509983real_n @ ( image_439535603real_n @ F ) @ S ) ) ) ).
% image_Union
thf(fact_335_UN__UN__flatten,axiom,
! [C2: nat > set_Fi1058188332real_n,B2: nat > set_nat,A3: set_nat] :
( ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( comple1682161881et_nat @ ( image_nat_set_nat @ B2 @ A3 ) ) ) )
= ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [Y2: nat] : ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( B2 @ Y2 ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_336_UN__UN__flatten,axiom,
! [C2: nat > set_Fi1058188332real_n,B2: finite964658038_int_n > set_nat,A3: set_Fi160064172_int_n] :
( ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( comple1682161881et_nat @ ( image_1085873667et_nat @ B2 @ A3 ) ) ) )
= ( comple825005695real_n
@ ( image_355963305real_n
@ ^ [Y2: finite964658038_int_n] : ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( B2 @ Y2 ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_337_UN__UN__flatten,axiom,
! [C2: finite964658038_int_n > set_Fi1058188332real_n,B2: nat > set_Fi160064172_int_n,A3: set_nat] :
( ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( comple970917503_int_n @ ( image_968789251_int_n @ B2 @ A3 ) ) ) )
= ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [Y2: nat] : ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( B2 @ Y2 ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_338_UN__UN__flatten,axiom,
! [C2: finite964658038_int_n > set_Fi1058188332real_n,B2: finite964658038_int_n > set_Fi160064172_int_n,A3: set_Fi160064172_int_n] :
( ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( comple970917503_int_n @ ( image_1819506345_int_n @ B2 @ A3 ) ) ) )
= ( comple825005695real_n
@ ( image_355963305real_n
@ ^ [Y2: finite964658038_int_n] : ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( B2 @ Y2 ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_339_UN__UN__flatten,axiom,
! [C2: finite1489363574real_n > set_Fi1058188332real_n,B2: nat > set_Fi1058188332real_n,A3: set_nat] :
( ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) ) )
= ( comple825005695real_n
@ ( image_1856576259real_n
@ ^ [Y2: nat] : ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( B2 @ Y2 ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_340_UN__UN__flatten,axiom,
! [C2: finite1489363574real_n > set_Fi1058188332real_n,B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
( ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) ) )
= ( comple825005695real_n
@ ( image_355963305real_n
@ ^ [Y2: finite964658038_int_n] : ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( B2 @ Y2 ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_341_UN__E,axiom,
! [B: set_Fi1058188332real_n,B2: set_Fi1058188332real_n > set_se2111327970real_n,A3: set_se2111327970real_n] :
( ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_797440021real_n @ B2 @ A3 ) ) )
=> ~ ! [X2: set_Fi1058188332real_n] :
( ( member223413699real_n @ X2 @ A3 )
=> ~ ( member223413699real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_342_UN__E,axiom,
! [B: set_Fi1058188332real_n,B2: $o > set_se2111327970real_n,A3: set_o] :
( ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_452144437real_n @ B2 @ A3 ) ) )
=> ~ ! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ~ ( member223413699real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_343_UN__E,axiom,
! [B: $o,B2: set_Fi1058188332real_n > set_o,A3: set_se2111327970real_n] :
( ( member_o @ B @ ( comple1665300069_set_o @ ( image_1687589765_set_o @ B2 @ A3 ) ) )
=> ~ ! [X2: set_Fi1058188332real_n] :
( ( member223413699real_n @ X2 @ A3 )
=> ~ ( member_o @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_344_UN__E,axiom,
! [B: $o,B2: $o > set_o,A3: set_o] :
( ( member_o @ B @ ( comple1665300069_set_o @ ( image_o_set_o @ B2 @ A3 ) ) )
=> ~ ! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ~ ( member_o @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_345_UN__E,axiom,
! [B: finite1489363574real_n,B2: nat > set_Fi1058188332real_n,A3: set_nat] :
( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_346_UN__E,axiom,
! [B: finite1489363574real_n,B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
=> ~ ! [X2: finite964658038_int_n] :
( ( member27055245_int_n @ X2 @ A3 )
=> ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_347_UN__E,axiom,
! [B: finite1489363574real_n,B2: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1661509983real_n @ B2 @ A3 ) ) )
=> ~ ! [X2: set_Fi1058188332real_n] :
( ( member223413699real_n @ X2 @ A3 )
=> ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_348_UN__E,axiom,
! [B: finite1489363574real_n,B2: $o > set_Fi1058188332real_n,A3: set_o] :
( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1759008383real_n @ B2 @ A3 ) ) )
=> ~ ! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_349_UN__extend__simps_I8_J,axiom,
! [B2: nat > set_Fi1058188332real_n,A3: set_set_nat] :
( ( comple825005695real_n
@ ( image_933134521real_n
@ ^ [Y2: set_nat] : ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ Y2 ) )
@ A3 ) )
= ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ ( comple1682161881et_nat @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_350_UN__extend__simps_I8_J,axiom,
! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_se944069346_int_n] :
( ( comple825005695real_n
@ ( image_792439519real_n
@ ^ [Y2: set_Fi160064172_int_n] : ( comple825005695real_n @ ( image_355963305real_n @ B2 @ Y2 ) )
@ A3 ) )
= ( comple825005695real_n @ ( image_355963305real_n @ B2 @ ( comple970917503_int_n @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_351_UN__extend__simps_I8_J,axiom,
! [B2: finite1489363574real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
( ( comple825005695real_n
@ ( image_1661509983real_n
@ ^ [Y2: set_Fi1058188332real_n] : ( comple825005695real_n @ ( image_545463721real_n @ B2 @ Y2 ) )
@ A3 ) )
= ( comple825005695real_n @ ( image_545463721real_n @ B2 @ ( comple825005695real_n @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_352_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
% Conjectures (1)
thf(conj_0,conjecture,
( ( ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ N ) ) ) )
= ( ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ N ) ) ) ) ) ).
%------------------------------------------------------------------------------