%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP117^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:41 AM UTC 2026
% Result : Theorem 2.91s 1.70s
% Output : Refutation 2.91s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 2
% Syntax : Number of formulae : 11 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 11 ( 10 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 105 ( 7 ~; 0 |; 0 &; 98 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 2 ( 2 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 122 ( 120 usr; 16 con; 0-2 aty)
% Number of variables : 16 ( 14 ^; 2 !; 0 ?; 16 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_o: $tType ).
thf(type_def_7,type,
set_Ex113815278nnreal: $tType ).
thf(type_def_8,type,
extend1728876344nnreal: $tType ).
thf(type_def_9,type,
set_Fi1058188332real_n: $tType ).
thf(type_def_10,type,
finite1489363574real_n: $tType ).
thf(type_def_11,type,
set_set_o: $tType ).
thf(type_def_12,type,
set_se944069346_int_n: $tType ).
thf(type_def_13,type,
set_Fi160064172_int_n: $tType ).
thf(type_def_14,type,
set_se2111327970real_n: $tType ).
thf(type_def_15,type,
set_set_nat: $tType ).
thf(type_def_16,type,
set_nat: $tType ).
thf(type_def_17,type,
set_se820660888real_n: $tType ).
thf(type_def_18,type,
set_Si1125517487real_n: $tType ).
thf(type_def_19,type,
sigma_1466784463real_n: $tType ).
thf(type_def_20,type,
nat: $tType ).
thf(type_def_21,type,
finite964658038_int_n: $tType ).
thf(type_def_22,type,
sigma_measure_o: $tType ).
thf(type_def_23,type,
sigma_1422848389real_n: $tType ).
thf(func_def_0,type,
sigma_1422848389real_n: $tType ).
thf(func_def_1,type,
set_Si1125517487real_n: $tType ).
thf(func_def_2,type,
set_se820660888real_n: $tType ).
thf(func_def_3,type,
sigma_1466784463real_n: $tType ).
thf(func_def_4,type,
set_se2111327970real_n: $tType ).
thf(func_def_5,type,
set_se944069346_int_n: $tType ).
thf(func_def_6,type,
set_Fi1058188332real_n: $tType ).
thf(func_def_7,type,
set_Fi160064172_int_n: $tType ).
thf(func_def_8,type,
finite1489363574real_n: $tType ).
thf(func_def_9,type,
finite964658038_int_n: $tType ).
thf(func_def_10,type,
set_Ex113815278nnreal: $tType ).
thf(func_def_11,type,
set_set_nat: $tType ).
thf(func_def_12,type,
extend1728876344nnreal: $tType ).
thf(func_def_13,type,
sigma_measure_o: $tType ).
thf(func_def_14,type,
set_set_o: $tType ).
thf(func_def_15,type,
set_nat: $tType ).
thf(func_def_16,type,
set_o: $tType ).
thf(func_def_17,type,
nat: $tType ).
thf(func_def_18,type,
complete_Sup_Sup_o: set_o > $o ).
thf(func_def_19,type,
comple1413366923nnreal: set_Ex113815278nnreal > extend1728876344nnreal ).
thf(func_def_20,type,
comple2042271945real_n: set_Fi1058188332real_n > finite1489363574real_n ).
thf(func_def_21,type,
comple1665300069_set_o: set_set_o > set_o ).
thf(func_def_22,type,
comple970917503_int_n: set_se944069346_int_n > set_Fi160064172_int_n ).
thf(func_def_23,type,
comple825005695real_n: set_se2111327970real_n > set_Fi1058188332real_n ).
thf(func_def_24,type,
comple1682161881et_nat: set_set_nat > set_nat ).
thf(func_def_25,type,
comple1917283637real_n: set_se820660888real_n > set_se2111327970real_n ).
thf(func_def_26,type,
comple488165692real_n: set_Si1125517487real_n > sigma_1466784463real_n ).
thf(func_def_27,type,
comple230862828real_n: sigma_1466784463real_n > sigma_1466784463real_n ).
thf(func_def_28,type,
counta1142393929_int_n: set_Fi160064172_int_n > nat > finite964658038_int_n ).
thf(func_def_29,type,
minus_1196255695_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n ).
thf(func_def_30,type,
minus_1037315151real_n: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ).
thf(func_def_31,type,
minus_1686442501real_n: set_Fi1058188332real_n > set_Fi1058188332real_n > set_Fi1058188332real_n ).
thf(func_def_32,type,
one_on705384445nnreal: extend1728876344nnreal ).
thf(func_def_33,type,
one_on1253059131real_n: finite1489363574real_n ).
thf(func_def_34,type,
plus_p1763960001nnreal: extend1728876344nnreal > extend1728876344nnreal > extend1728876344nnreal ).
thf(func_def_35,type,
plus_p1654784127_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n ).
thf(func_def_36,type,
plus_p585657087real_n: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ).
thf(func_def_37,type,
zero_z1963244097nnreal: extend1728876344nnreal ).
thf(func_def_38,type,
zero_z200130687real_n: finite1489363574real_n ).
thf(func_def_39,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(func_def_40,type,
inf_in1108485182_int_n: set_Fi160064172_int_n > set_Fi160064172_int_n > set_Fi160064172_int_n ).
thf(func_def_41,type,
inf_in1974387902real_n: set_Fi1058188332real_n > set_Fi1058188332real_n > set_Fi1058188332real_n ).
thf(func_def_42,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_43,type,
inf_in632889204real_n: set_se2111327970real_n > set_se2111327970real_n > set_se2111327970real_n ).
thf(func_def_44,type,
lebesg260170249real_n: sigma_1466784463real_n ).
thf(func_def_45,type,
measure_null_sets_o: sigma_measure_o > set_set_o ).
thf(func_def_46,type,
measur1402256771real_n: sigma_1466784463real_n > set_se2111327970real_n ).
thf(func_def_47,type,
measur2126959417real_n: sigma_1422848389real_n > set_se820660888real_n ).
thf(func_def_48,type,
minkow1134813771n_real: finite964658038_int_n > finite1489363574real_n ).
thf(func_def_49,type,
ord_le2133614988nnreal: extend1728876344nnreal > extend1728876344nnreal > $o ).
thf(func_def_50,type,
top_to287930409nt_n_o: finite964658038_int_n > $o ).
thf(func_def_51,type,
top_top_nat_o: nat > $o ).
thf(func_def_53,type,
top_to1845833192nnreal: extend1728876344nnreal ).
thf(func_def_54,type,
top_top_set_o: set_o ).
thf(func_def_55,type,
top_to131672412_int_n: set_Fi160064172_int_n ).
thf(func_def_56,type,
top_to1292442332real_n: set_Fi1058188332real_n ).
thf(func_def_57,type,
top_top_set_nat: set_nat ).
thf(func_def_58,type,
top_to1587634578_int_n: set_se944069346_int_n ).
thf(func_def_59,type,
top_to20708754real_n: set_se2111327970real_n ).
thf(func_def_60,type,
top_top_set_set_nat: set_set_nat ).
thf(func_def_61,type,
sums_E1192373732nnreal: ( nat > extend1728876344nnreal ) > extend1728876344nnreal > $o ).
thf(func_def_62,type,
collect_o: ( $o > $o ) > set_o ).
thf(func_def_63,type,
collec1941932235_int_n: ( finite964658038_int_n > $o ) > set_Fi160064172_int_n ).
thf(func_def_64,type,
collec321817931real_n: ( finite1489363574real_n > $o ) > set_Fi1058188332real_n ).
thf(func_def_65,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_66,type,
collec452821761real_n: ( set_Fi1058188332real_n > $o ) > set_se2111327970real_n ).
thf(func_def_67,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(func_def_68,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(func_def_69,type,
image_1759008383real_n: ( $o > set_Fi1058188332real_n ) > set_o > set_se2111327970real_n ).
thf(func_def_70,type,
image_452144437real_n: ( $o > set_se2111327970real_n ) > set_o > set_se820660888real_n ).
thf(func_def_71,type,
image_1599934780real_n: ( $o > sigma_1466784463real_n ) > set_o > set_Si1125517487real_n ).
thf(func_def_72,type,
image_2066995319nnreal: ( extend1728876344nnreal > extend1728876344nnreal ) > set_Ex113815278nnreal > set_Ex113815278nnreal ).
thf(func_def_73,type,
image_216309723nt_n_o: ( finite964658038_int_n > $o ) > set_Fi160064172_int_n > set_o ).
thf(func_def_74,type,
image_1278151539_int_n: ( finite964658038_int_n > finite964658038_int_n ) > set_Fi160064172_int_n > set_Fi160064172_int_n ).
thf(func_def_75,type,
image_2058828787real_n: ( finite964658038_int_n > finite1489363574real_n ) > set_Fi160064172_int_n > set_Fi1058188332real_n ).
thf(func_def_76,type,
image_497739341_n_nat: ( finite964658038_int_n > nat ) > set_Fi160064172_int_n > set_nat ).
thf(func_def_77,type,
image_1819506345_int_n: ( finite964658038_int_n > set_Fi160064172_int_n ) > set_Fi160064172_int_n > set_se944069346_int_n ).
thf(func_def_78,type,
image_355963305real_n: ( finite964658038_int_n > set_Fi1058188332real_n ) > set_Fi160064172_int_n > set_se2111327970real_n ).
thf(func_def_79,type,
image_1085873667et_nat: ( finite964658038_int_n > set_nat ) > set_Fi160064172_int_n > set_set_nat ).
thf(func_def_80,type,
image_439535603real_n: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n > set_Fi1058188332real_n ).
thf(func_def_81,type,
image_545463721real_n: ( finite1489363574real_n > set_Fi1058188332real_n ) > set_Fi1058188332real_n > set_se2111327970real_n ).
thf(func_def_82,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(func_def_83,type,
image_1961920973_int_n: ( nat > finite964658038_int_n ) > set_nat > set_Fi160064172_int_n ).
thf(func_def_84,type,
image_183184717real_n: ( nat > finite1489363574real_n ) > set_nat > set_Fi1058188332real_n ).
thf(func_def_85,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_86,type,
image_968789251_int_n: ( nat > set_Fi160064172_int_n ) > set_nat > set_se944069346_int_n ).
thf(func_def_87,type,
image_1856576259real_n: ( nat > set_Fi1058188332real_n ) > set_nat > set_se2111327970real_n ).
thf(func_def_88,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(func_def_89,type,
image_set_o_o: ( set_o > $o ) > set_set_o > set_o ).
thf(func_def_90,type,
image_792439519real_n: ( set_Fi160064172_int_n > set_Fi1058188332real_n ) > set_se944069346_int_n > set_se2111327970real_n ).
thf(func_def_91,type,
image_1054146965real_n: ( set_Fi160064172_int_n > set_se2111327970real_n ) > set_se944069346_int_n > set_se820660888real_n ).
thf(func_def_92,type,
image_1648361637al_n_o: ( set_Fi1058188332real_n > $o ) > set_se2111327970real_n > set_o ).
thf(func_def_93,type,
image_1687589765_set_o: ( set_Fi1058188332real_n > set_o ) > set_se2111327970real_n > set_set_o ).
thf(func_def_94,type,
image_1661509983real_n: ( set_Fi1058188332real_n > set_Fi1058188332real_n ) > set_se2111327970real_n > set_se2111327970real_n ).
thf(func_def_95,type,
image_797440021real_n: ( set_Fi1058188332real_n > set_se2111327970real_n ) > set_se2111327970real_n > set_se820660888real_n ).
thf(func_def_96,type,
image_987430492real_n: ( set_Fi1058188332real_n > sigma_1466784463real_n ) > set_se2111327970real_n > set_Si1125517487real_n ).
thf(func_def_97,type,
image_933134521real_n: ( set_nat > set_Fi1058188332real_n ) > set_set_nat > set_se2111327970real_n ).
thf(func_def_98,type,
image_1587769199real_n: ( set_nat > set_se2111327970real_n ) > set_set_nat > set_se820660888real_n ).
thf(func_def_99,type,
image_1681970287al_n_o: ( set_se2111327970real_n > $o ) > set_se820660888real_n > set_o ).
thf(func_def_100,type,
image_1298280374real_n: ( sigma_1466784463real_n > set_Fi1058188332real_n ) > set_Si1125517487real_n > set_se2111327970real_n ).
thf(func_def_101,type,
vimage_o_o: ( $o > $o ) > set_o > set_o ).
thf(func_def_102,type,
vimage961837641real_n: ( $o > set_Fi1058188332real_n ) > set_se2111327970real_n > set_o ).
thf(func_def_103,type,
vimage1122713129_int_n: ( finite964658038_int_n > finite964658038_int_n ) > set_Fi160064172_int_n > set_Fi160064172_int_n ).
thf(func_def_104,type,
vimage1276736425real_n: ( finite964658038_int_n > finite1489363574real_n ) > set_Fi1058188332real_n > set_Fi160064172_int_n ).
thf(func_def_105,type,
vimage1398021123_n_nat: ( finite964658038_int_n > nat ) > set_nat > set_Fi160064172_int_n ).
thf(func_def_106,type,
vimage464515423real_n: ( finite964658038_int_n > set_Fi1058188332real_n ) > set_se2111327970real_n > set_Fi160064172_int_n ).
thf(func_def_107,type,
vimage1233683625real_n: ( finite1489363574real_n > finite1489363574real_n ) > set_Fi1058188332real_n > set_Fi1058188332real_n ).
thf(func_def_108,type,
vimage714719107_int_n: ( nat > finite964658038_int_n ) > set_Fi160064172_int_n > set_nat ).
thf(func_def_109,type,
vimage1860757507real_n: ( nat > finite1489363574real_n ) > set_Fi1058188332real_n > set_nat ).
thf(func_def_110,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_111,type,
vimage3210681real_n: ( nat > set_Fi1058188332real_n ) > set_se2111327970real_n > set_nat ).
thf(func_def_112,type,
vimage851190895al_n_o: ( set_Fi1058188332real_n > $o ) > set_o > set_se2111327970real_n ).
thf(func_def_113,type,
vimage784510485real_n: ( set_Fi1058188332real_n > set_Fi1058188332real_n ) > set_se2111327970real_n > set_se2111327970real_n ).
thf(func_def_114,type,
sigma_1536574303real_n: sigma_1466784463real_n > set_Fi1058188332real_n > extend1728876344nnreal ).
thf(func_def_115,type,
sigma_1235138647real_n: sigma_1466784463real_n > set_se2111327970real_n ).
thf(func_def_116,type,
sigma_space_o: sigma_measure_o > set_o ).
thf(func_def_117,type,
sigma_476185326real_n: sigma_1466784463real_n > set_Fi1058188332real_n ).
thf(func_def_118,type,
sigma_607186084real_n: sigma_1422848389real_n > set_se2111327970real_n ).
thf(func_def_119,type,
member_o: $o > set_o > $o ).
thf(func_def_120,type,
member1217042383nnreal: extend1728876344nnreal > set_Ex113815278nnreal > $o ).
thf(func_def_121,type,
member27055245_int_n: finite964658038_int_n > set_Fi160064172_int_n > $o ).
thf(func_def_122,type,
member1352538125real_n: finite1489363574real_n > set_Fi1058188332real_n > $o ).
thf(func_def_123,type,
member_nat: nat > set_nat > $o ).
thf(func_def_124,type,
member_set_o: set_o > set_set_o > $o ).
thf(func_def_125,type,
member223413699real_n: set_Fi1058188332real_n > set_se2111327970real_n > $o ).
thf(func_def_126,type,
member1475136633real_n: set_se2111327970real_n > set_se820660888real_n > $o ).
thf(func_def_127,type,
member1000184real_n: sigma_1466784463real_n > set_Si1125517487real_n > $o ).
thf(func_def_128,type,
r: finite964658038_int_n > set_Fi1058188332real_n ).
thf(func_def_129,type,
s: set_Fi1058188332real_n ).
thf(func_def_130,type,
t: finite964658038_int_n > set_Fi1058188332real_n ).
thf(func_def_131,type,
t2: finite964658038_int_n > set_Fi1058188332real_n ).
thf(func_def_132,type,
f: nat > finite964658038_int_n ).
thf(func_def_142,type,
sK0: set_Fi1058188332real_n > ( finite964658038_int_n > set_Fi1058188332real_n ) > finite964658038_int_n ).
thf(func_def_143,type,
sK1: ( nat > set_Fi1058188332real_n ) > set_Fi1058188332real_n > nat ).
thf(func_def_144,type,
sK2: ( finite964658038_int_n > set_Fi1058188332real_n ) > sigma_1466784463real_n > finite964658038_int_n ).
thf(func_def_145,type,
sK3: sigma_1466784463real_n > ( nat > set_Fi1058188332real_n ) > nat ).
thf(func_def_147,type,
ph5:
!>[X0: $tType] : X0 ).
thf(func_def_148,type,
sK6: nat ).
thf(f986,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f985,f970]) ).
thf(f970,plain,
! [X0: finite964658038_int_n] :
( ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ X0 ) )
= ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: finite964658038_int_n] :
( ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ X0 ) )
= ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uys0jgiiMJ/DTF2THF_7164.p',fact_0_emeasure__T_H) ).
thf(f985,plain,
( ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ sK6 ) ) )
!= ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ sK6 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f984]) ).
thf(f984,plain,
( ( ^ [Y0: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ Y0 ) ) )
@ sK6 )
!= ( ^ [Y0: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ Y0 ) ) )
@ sK6 ) ),
inference(negative_extensionality,[status(thm)],[f953]) ).
thf(f953,plain,
( ( ^ [Y0: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ Y0 ) ) ) )
!= ( ^ [Y0: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f900]) ).
thf(f900,plain,
( ( ^ [Y0: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ Y0 ) ) ) )
!= ( ^ [Y0: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ Y0 ) ) ) ) ),
inference(flattening,[status(thm)],[f404]) ).
thf(f404,plain,
( ( ^ [Y0: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ Y0 ) ) ) )
!= ( ^ [Y0: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f403]) ).
thf(f403,plain,
( ( ^ [X0: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ X0 ) ) ) )
!= ( ^ [X1: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f355]) ).
thf(f355,negated_conjecture,
( ( ^ [X1: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ X1 ) ) ) )
!= ( ^ [X1: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ X1 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f354]) ).
thf(f354,conjecture,
( ( ^ [X1: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ X1 ) ) ) )
= ( ^ [X1: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ X1 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uys0jgiiMJ/DTF2THF_7164.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP117^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n016.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:27:06 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.47 thf(spec,logic,$$dhol).
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % File : ITP117^1 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.47 % Domain : Interactive Theorem Proving
% 0.30/0.47 % Problem : Sledgehammer Minkowskis_Theorem problem prob_290__6248976_1
% 0.30/0.47 % Version : Especial.
% 0.30/0.47 % English :
% 0.30/0.47
% 0.30/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.47 % Source : [Des21]
% 0.30/0.47 % Names : Minkowskis_Theorem/prob_290__6248976_1 [Des21]
% 0.30/0.47
% 0.30/0.47 % Status : Theorem
% 0.30/0.47 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% 0.30/0.47 % Syntax : Number of formulae : 487 ( 187 unt; 133 typ; 0 def)
% 0.30/0.47 % Number of atoms : 807 ( 345 equ; 0 cnn)
% 0.30/0.47 % Maximal formula atoms : 6 ( 2 avg)
% 0.30/0.47 % Number of connectives : 3013 ( 72 ~; 3 |; 79 &;2559 @)
% 0.30/0.47 % ( 0 <=>; 300 =>; 0 <=; 0 <~>)
% 0.30/0.47 % Maximal formula depth : 14 ( 6 avg)
% 0.30/0.47 % Number of types : 19 ( 18 usr)
% 0.30/0.47 % Number of type conns : 541 ( 541 >; 0 *; 0 +; 0 <<)
% 0.30/0.47 % Number of symbols : 117 ( 115 usr; 16 con; 0-2 aty)
% 0.30/0.47 % Number of variables : 1094 ( 122 ^; 923 !; 49 ?;1094 :)
% 0.30/0.47 % SPC : TH0_THM_EQU_NAR
% 0.30/0.47
% 0.30/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:47:06.896
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % Could-be-implicit typings (18)
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% 0.30/0.47 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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% 0.30/0.47 thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
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% 0.30/0.47 thf(ty_n_t__Sigma____Algebra__Omeasure_I_Eo_J,type,
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% 0.30/0.47 thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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% 0.30/0.47 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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% 0.30/0.47 thf(ty_n_t__Set__Oset_I_Eo_J,type,
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% 0.30/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.30/0.47 nat: $tType ).
% 0.30/0.47
% 0.30/0.47 % Explicit typings (115)
% 0.30/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
% 0.30/0.47 complete_Sup_Sup_o: set_o > $o ).
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% 0.30/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 0.30/0.47 comple2042271945real_n: set_Fi1058188332real_n > finite1489363574real_n ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
% 0.30/0.47 comple1665300069_set_o: set_set_o > set_o ).
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% 0.30/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_J,type,
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% 0.30/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
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% 0.30/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.30/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
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% 0.30/0.47 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Sigma____Algebra__Omeasure_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
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% 0.30/0.47 thf(sy_c_Complete__Measure_Ocompletion_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Countable__Set_Ofrom__nat__into_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
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% 0.30/0.47 thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
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% 0.30/0.47 thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
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% 0.30/0.47 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
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% 0.30/0.47 thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
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% 0.30/0.47 thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
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% 0.30/0.47 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
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% 0.30/0.47 thf(sy_c_Measure__Space_Onull__sets_001_Eo,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_v_S,type,
% 0.30/0.47 s: set_Fi1058188332real_n ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_T_H____,type,
% 0.30/0.47 t: finite964658038_int_n > set_Fi1058188332real_n ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_T____,type,
% 0.30/0.47 t2: finite964658038_int_n > set_Fi1058188332real_n ).
% 0.30/0.47
% 0.30/0.47 thf(sy_v_f____,type,
% 0.30/0.47 f: nat > finite964658038_int_n ).
% 0.30/0.47
% 0.30/0.47 % Relevant facts (353)
% 0.30/0.47 thf(fact_0_emeasure__T_H,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n] :
% 0.30/0.47 ( ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ A ) )
% 0.30/0.47 = ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % emeasure_T'
% 0.30/0.47 thf(fact_1__092_060open_062_092_060And_062a_O_AT_H_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n] : ( member223413699real_n @ ( t @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<And>a. T' a \<in> sets lebesgue\<close>
% 0.30/0.47 thf(fact_2__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n] : ( member223413699real_n @ ( t2 @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<And>a. T a \<in> sets lebesgue\<close>
% 0.30/0.47 thf(fact_3_calculation,axiom,
% 0.30/0.47 ( sums_E1192373732nnreal
% 0.30/0.47 @ ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ N ) ) )
% 0.30/0.47 @ ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ s ) ) ).
% 0.30/0.47
% 0.30/0.47 % calculation
% 0.30/0.47 thf(fact_4_assms_I1_J,axiom,
% 0.30/0.47 member223413699real_n @ s @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % assms(1)
% 0.30/0.47 thf(fact_5_emeasure__T__Int,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 0.30/0.47 ( ( A != B )
% 0.30/0.47 => ( ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( inf_in1974387902real_n @ ( t2 @ A ) @ ( t2 @ B ) ) )
% 0.30/0.47 = zero_z1963244097nnreal ) ) ).
% 0.30/0.47
% 0.30/0.47 % emeasure_T_Int
% 0.30/0.47 thf(fact_6_T__Int,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 0.30/0.47 ( ( A != B )
% 0.30/0.47 => ( member223413699real_n @ ( inf_in1974387902real_n @ ( t2 @ A ) @ ( t2 @ B ) ) @ ( measur1402256771real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % T_Int
% 0.30/0.47 thf(fact_7_assms_I2_J,axiom,
% 0.30/0.47 ord_le2133614988nnreal @ one_on705384445nnreal @ ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ s ) ).
% 0.30/0.47
% 0.30/0.47 % assms(2)
% 0.30/0.47 thf(fact_8__092_060open_062_092_060And_062a_O_AR_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n] : ( member223413699real_n @ ( r @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<And>a. R a \<in> sets lebesgue\<close>
% 0.30/0.47 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,
% 0.30/0.47 ( sums_E1192373732nnreal
% 0.30/0.47 @ ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ N ) ) )
% 0.30/0.47 @ ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n )
% 0.30/0.47 @ ( comple825005695real_n
% 0.30/0.47 @ ( image_1856576259real_n
% 0.30/0.47 @ ^ [N: nat] : ( t2 @ ( f @ N ) )
% 0.30/0.47 @ top_top_set_nat ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>(\<lambda>n. emeasure lebesgue (T (f n))) sums emeasure lebesgue (\<Union>n. T (f n))\<close>
% 0.30/0.47 thf(fact_10_T_H__def,axiom,
% 0.30/0.47 ( t
% 0.30/0.47 = ( ^ [A2: finite964658038_int_n] :
% 0.30/0.47 ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ ( minkow1134813771n_real @ A2 ) )
% 0.30/0.47 @ ( t2 @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % T'_def
% 0.30/0.47 thf(fact_11_f__def,axiom,
% 0.30/0.47 ( f
% 0.30/0.47 = ( counta1142393929_int_n @ top_to131672412_int_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % f_def
% 0.30/0.47 thf(fact_12_T_H__altdef,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n] :
% 0.30/0.47 ( ( t @ A )
% 0.30/0.47 = ( vimage1233683625real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ X @ ( minkow1134813771n_real @ A ) )
% 0.30/0.47 @ ( t2 @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % T'_altdef
% 0.30/0.47 thf(fact_13_of__int__vec__eq__iff,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 0.30/0.47 ( ( ( minkow1134813771n_real @ A )
% 0.30/0.47 = ( minkow1134813771n_real @ B ) )
% 0.30/0.47 = ( A = B ) ) ).
% 0.30/0.47
% 0.30/0.47 % of_int_vec_eq_iff
% 0.30/0.47 thf(fact_14_T__def,axiom,
% 0.30/0.47 ( t2
% 0.30/0.47 = ( ^ [A2: finite964658038_int_n] : ( inf_in1974387902real_n @ s @ ( r @ A2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % T_def
% 0.30/0.47 thf(fact_15_R__Int,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 0.30/0.47 ( ( A != B )
% 0.30/0.47 => ( member223413699real_n @ ( inf_in1974387902real_n @ ( r @ A ) @ ( r @ B ) ) @ ( measur1402256771real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % R_Int
% 0.30/0.47 thf(fact_16_sums__emeasure_H,axiom,
% 0.30/0.47 ! [B2: nat > set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 0.30/0.47 ( ! [X2: nat] : ( member223413699real_n @ ( B2 @ X2 ) @ ( sigma_1235138647real_n @ M ) )
% 0.30/0.47 => ( ! [X2: nat,Y: nat] :
% 0.30/0.47 ( ( X2 != Y )
% 0.30/0.47 => ( ( sigma_1536574303real_n @ M @ ( inf_in1974387902real_n @ ( B2 @ X2 ) @ ( B2 @ Y ) ) )
% 0.30/0.47 = zero_z1963244097nnreal ) )
% 0.30/0.47 => ( sums_E1192373732nnreal
% 0.30/0.47 @ ^ [X: nat] : ( sigma_1536574303real_n @ M @ ( B2 @ X ) )
% 0.30/0.47 @ ( sigma_1536574303real_n @ M @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ top_top_set_nat ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sums_emeasure'
% 0.30/0.47 thf(fact_17_null__sets__UN,axiom,
% 0.30/0.47 ! [N2: nat > set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 0.30/0.47 ( ! [I: nat] : ( member223413699real_n @ ( N2 @ I ) @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( member223413699real_n @ ( comple825005695real_n @ ( image_1856576259real_n @ N2 @ top_top_set_nat ) ) @ ( measur1402256771real_n @ M ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets_UN
% 0.30/0.47 thf(fact_18_null__sets__UN,axiom,
% 0.30/0.47 ! [N2: finite964658038_int_n > set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 0.30/0.47 ( ! [I: finite964658038_int_n] : ( member223413699real_n @ ( N2 @ I ) @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( member223413699real_n @ ( comple825005695real_n @ ( image_355963305real_n @ N2 @ top_to131672412_int_n ) ) @ ( measur1402256771real_n @ M ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets_UN
% 0.30/0.47 thf(fact_19_null__setsI,axiom,
% 0.30/0.47 ! [M: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( ( sigma_1536574303real_n @ M @ A3 )
% 0.30/0.47 = zero_z1963244097nnreal )
% 0.30/0.47 => ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) )
% 0.30/0.47 => ( member223413699real_n @ A3 @ ( measur1402256771real_n @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_setsI
% 0.30/0.47 thf(fact_20_image__vimage__eq,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_439535603real_n @ F @ ( vimage1233683625real_n @ F @ A3 ) )
% 0.30/0.47 = ( inf_in1974387902real_n @ A3 @ ( image_439535603real_n @ F @ top_to1292442332real_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_vimage_eq
% 0.30/0.47 thf(fact_21_image__vimage__eq,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( image_1856576259real_n @ F @ ( vimage3210681real_n @ F @ A3 ) )
% 0.30/0.47 = ( inf_in632889204real_n @ A3 @ ( image_1856576259real_n @ F @ top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_vimage_eq
% 0.30/0.47 thf(fact_22_image__vimage__eq,axiom,
% 0.30/0.47 ! [F: nat > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_183184717real_n @ F @ ( vimage1860757507real_n @ F @ A3 ) )
% 0.30/0.47 = ( inf_in1974387902real_n @ A3 @ ( image_183184717real_n @ F @ top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_vimage_eq
% 0.30/0.47 thf(fact_23_image__vimage__eq,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( image_355963305real_n @ F @ ( vimage464515423real_n @ F @ A3 ) )
% 0.30/0.47 = ( inf_in632889204real_n @ A3 @ ( image_355963305real_n @ F @ top_to131672412_int_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_vimage_eq
% 0.30/0.47 thf(fact_24_image__vimage__eq,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_2058828787real_n @ F @ ( vimage1276736425real_n @ F @ A3 ) )
% 0.30/0.47 = ( inf_in1974387902real_n @ A3 @ ( image_2058828787real_n @ F @ top_to131672412_int_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_vimage_eq
% 0.30/0.47 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,
% 0.30/0.47 ! [A: finite964658038_int_n] :
% 0.30/0.47 ( ( member223413699real_n @ ( t2 @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) )
% 0.30/0.47 => ( member223413699real_n
% 0.30/0.47 @ ( inf_in1974387902real_n
% 0.30/0.47 @ ( vimage1233683625real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ X @ ( minkow1134813771n_real @ A ) )
% 0.30/0.47 @ ( t2 @ A ) )
% 0.30/0.47 @ ( sigma_476185326real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) )
% 0.30/0.47 @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<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>
% 0.30/0.47 thf(fact_26_surj__diff__right,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n] :
% 0.30/0.47 ( ( image_1278151539_int_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : ( minus_1196255695_int_n @ X @ A )
% 0.30/0.47 @ top_to131672412_int_n )
% 0.30/0.47 = top_to131672412_int_n ) ).
% 0.30/0.47
% 0.30/0.47 % surj_diff_right
% 0.30/0.47 thf(fact_27_surj__diff__right,axiom,
% 0.30/0.47 ! [A: finite1489363574real_n] :
% 0.30/0.47 ( ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
% 0.30/0.47 @ top_to1292442332real_n )
% 0.30/0.47 = top_to1292442332real_n ) ).
% 0.30/0.47
% 0.30/0.47 % surj_diff_right
% 0.30/0.47 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,
% 0.30/0.47 ( ( comple825005695real_n @ ( image_355963305real_n @ t2 @ top_to131672412_int_n ) )
% 0.30/0.47 = ( comple825005695real_n
% 0.30/0.47 @ ( image_1856576259real_n
% 0.30/0.47 @ ^ [N: nat] : ( t2 @ ( f @ N ) )
% 0.30/0.47 @ top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % \<open>\<Union> (range T) = (\<Union>n. T (f n))\<close>
% 0.30/0.47 thf(fact_29_Sup__UNIV,axiom,
% 0.30/0.47 ( ( comple1682161881et_nat @ top_top_set_set_nat )
% 0.30/0.47 = top_top_set_nat ) ).
% 0.30/0.47
% 0.30/0.47 % Sup_UNIV
% 0.30/0.47 thf(fact_30_Sup__UNIV,axiom,
% 0.30/0.47 ( ( comple970917503_int_n @ top_to1587634578_int_n )
% 0.30/0.47 = top_to131672412_int_n ) ).
% 0.30/0.47
% 0.30/0.47 % Sup_UNIV
% 0.30/0.47 thf(fact_31_Sup__UNIV,axiom,
% 0.30/0.47 ( ( comple825005695real_n @ top_to20708754real_n )
% 0.30/0.47 = top_to1292442332real_n ) ).
% 0.30/0.47
% 0.30/0.47 % Sup_UNIV
% 0.30/0.47 thf(fact_32_Sup__UNIV,axiom,
% 0.30/0.47 ( ( complete_Sup_Sup_o @ top_top_set_o )
% 0.30/0.47 = top_top_o ) ).
% 0.30/0.47
% 0.30/0.47 % Sup_UNIV
% 0.30/0.47 thf(fact_33_Sup__eq__top__iff,axiom,
% 0.30/0.47 ! [A3: set_Ex113815278nnreal] :
% 0.30/0.47 ( ( ( comple1413366923nnreal @ A3 )
% 0.30/0.47 = top_to1845833192nnreal )
% 0.30/0.47 = ( ! [X: extend1728876344nnreal] :
% 0.30/0.47 ( ( ord_le2133614988nnreal @ X @ top_to1845833192nnreal )
% 0.30/0.47 => ? [Y2: extend1728876344nnreal] :
% 0.30/0.47 ( ( member1217042383nnreal @ Y2 @ A3 )
% 0.30/0.47 & ( ord_le2133614988nnreal @ X @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup_eq_top_iff
% 0.30/0.47 thf(fact_34_surj__plus,axiom,
% 0.30/0.47 ! [A: finite1489363574real_n] :
% 0.30/0.47 ( ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ top_to1292442332real_n )
% 0.30/0.47 = top_to1292442332real_n ) ).
% 0.30/0.47
% 0.30/0.47 % surj_plus
% 0.30/0.47 thf(fact_35_surj__plus,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n] :
% 0.30/0.47 ( ( image_1278151539_int_n @ ( plus_p1654784127_int_n @ A ) @ top_to131672412_int_n )
% 0.30/0.47 = top_to131672412_int_n ) ).
% 0.30/0.47
% 0.30/0.47 % surj_plus
% 0.30/0.47 thf(fact_36_diff__numeral__special_I9_J,axiom,
% 0.30/0.47 ( ( minus_1037315151real_n @ one_on1253059131real_n @ one_on1253059131real_n )
% 0.30/0.47 = zero_z200130687real_n ) ).
% 0.30/0.47
% 0.30/0.47 % diff_numeral_special(9)
% 0.30/0.47 thf(fact_37_image__eqI,axiom,
% 0.30/0.47 ! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,X3: finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( ( member1352538125real_n @ X3 @ A3 )
% 0.30/0.47 => ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_eqI
% 0.30/0.47 thf(fact_38_image__eqI,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,X3: nat,A3: set_nat] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( ( member_nat @ X3 @ A3 )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_eqI
% 0.30/0.47 thf(fact_39_image__eqI,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,X3: finite964658038_int_n,A3: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( ( member27055245_int_n @ X3 @ A3 )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_355963305real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_eqI
% 0.30/0.47 thf(fact_40_image__eqI,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,X3: set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( ( member223413699real_n @ X3 @ A3 )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_eqI
% 0.30/0.47 thf(fact_41_image__eqI,axiom,
% 0.30/0.47 ! [B: $o,F: set_Fi1058188332real_n > $o,X3: set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( ( member223413699real_n @ X3 @ A3 )
% 0.30/0.47 => ( member_o @ B @ ( image_1648361637al_n_o @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_eqI
% 0.30/0.47 thf(fact_42_image__eqI,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: $o > set_Fi1058188332real_n,X3: $o,A3: set_o] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( ( member_o @ X3 @ A3 )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_1759008383real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_eqI
% 0.30/0.47 thf(fact_43_image__eqI,axiom,
% 0.30/0.47 ! [B: $o,F: $o > $o,X3: $o,A3: set_o] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( ( member_o @ X3 @ A3 )
% 0.30/0.47 => ( member_o @ B @ ( image_o_o @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_eqI
% 0.30/0.47 thf(fact_44_UNIV__I,axiom,
% 0.30/0.47 ! [X3: set_Fi1058188332real_n] : ( member223413699real_n @ X3 @ top_to20708754real_n ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_I
% 0.30/0.47 thf(fact_45_UNIV__I,axiom,
% 0.30/0.47 ! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_I
% 0.30/0.47 thf(fact_46_UNIV__I,axiom,
% 0.30/0.47 ! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_I
% 0.30/0.47 thf(fact_47_UNIV__I,axiom,
% 0.30/0.47 ! [X3: finite964658038_int_n] : ( member27055245_int_n @ X3 @ top_to131672412_int_n ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_I
% 0.30/0.47 thf(fact_48_Int__iff,axiom,
% 0.30/0.47 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
% 0.30/0.47 = ( ( member223413699real_n @ C @ A3 )
% 0.30/0.47 & ( member223413699real_n @ C @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_iff
% 0.30/0.47 thf(fact_49_Int__iff,axiom,
% 0.30/0.47 ! [C: $o,A3: set_o,B2: set_o] :
% 0.30/0.47 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.30/0.47 = ( ( member_o @ C @ A3 )
% 0.30/0.47 & ( member_o @ C @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_iff
% 0.30/0.47 thf(fact_50_Int__iff,axiom,
% 0.30/0.47 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 0.30/0.47 = ( ( member1352538125real_n @ C @ A3 )
% 0.30/0.47 & ( member1352538125real_n @ C @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_iff
% 0.30/0.47 thf(fact_51_IntI,axiom,
% 0.30/0.47 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ C @ A3 )
% 0.30/0.47 => ( ( member223413699real_n @ C @ B2 )
% 0.30/0.47 => ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntI
% 0.30/0.47 thf(fact_52_IntI,axiom,
% 0.30/0.47 ! [C: $o,A3: set_o,B2: set_o] :
% 0.30/0.47 ( ( member_o @ C @ A3 )
% 0.30/0.47 => ( ( member_o @ C @ B2 )
% 0.30/0.47 => ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntI
% 0.30/0.47 thf(fact_53_IntI,axiom,
% 0.30/0.47 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ C @ A3 )
% 0.30/0.47 => ( ( member1352538125real_n @ C @ B2 )
% 0.30/0.47 => ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntI
% 0.30/0.47 thf(fact_54_Union__iff,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,C2: set_se820660888real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A3 @ ( comple1917283637real_n @ C2 ) )
% 0.30/0.47 = ( ? [X: set_se2111327970real_n] :
% 0.30/0.47 ( ( member1475136633real_n @ X @ C2 )
% 0.30/0.47 & ( member223413699real_n @ A3 @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Union_iff
% 0.30/0.47 thf(fact_55_Union__iff,axiom,
% 0.30/0.47 ! [A3: $o,C2: set_set_o] :
% 0.30/0.47 ( ( member_o @ A3 @ ( comple1665300069_set_o @ C2 ) )
% 0.30/0.47 = ( ? [X: set_o] :
% 0.30/0.47 ( ( member_set_o @ X @ C2 )
% 0.30/0.47 & ( member_o @ A3 @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Union_iff
% 0.30/0.47 thf(fact_56_Union__iff,axiom,
% 0.30/0.47 ! [A3: finite1489363574real_n,C2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ A3 @ ( comple825005695real_n @ C2 ) )
% 0.30/0.47 = ( ? [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ C2 )
% 0.30/0.47 & ( member1352538125real_n @ A3 @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Union_iff
% 0.30/0.47 thf(fact_57_UnionI,axiom,
% 0.30/0.47 ! [X4: set_se2111327970real_n,C2: set_se820660888real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1475136633real_n @ X4 @ C2 )
% 0.30/0.47 => ( ( member223413699real_n @ A3 @ X4 )
% 0.30/0.47 => ( member223413699real_n @ A3 @ ( comple1917283637real_n @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnionI
% 0.30/0.47 thf(fact_58_UnionI,axiom,
% 0.30/0.47 ! [X4: set_o,C2: set_set_o,A3: $o] :
% 0.30/0.47 ( ( member_set_o @ X4 @ C2 )
% 0.30/0.47 => ( ( member_o @ A3 @ X4 )
% 0.30/0.47 => ( member_o @ A3 @ ( comple1665300069_set_o @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnionI
% 0.30/0.47 thf(fact_59_UnionI,axiom,
% 0.30/0.47 ! [X4: set_Fi1058188332real_n,C2: set_se2111327970real_n,A3: finite1489363574real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X4 @ C2 )
% 0.30/0.47 => ( ( member1352538125real_n @ A3 @ X4 )
% 0.30/0.47 => ( member1352538125real_n @ A3 @ ( comple825005695real_n @ C2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnionI
% 0.30/0.47 thf(fact_60_UN__ball__bex__simps_I1_J,axiom,
% 0.30/0.47 ! [A3: set_se2111327970real_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ! [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ A3 ) )
% 0.30/0.47 => ( P @ X ) ) )
% 0.30/0.47 = ( ! [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ A3 )
% 0.30/0.47 => ! [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ X )
% 0.30/0.47 => ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_ball_bex_simps(1)
% 0.30/0.47 thf(fact_61_UN__ball__bex__simps_I3_J,axiom,
% 0.30/0.47 ! [A3: set_se2111327970real_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ? [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ A3 ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( ? [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ A3 )
% 0.30/0.47 & ? [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ X )
% 0.30/0.47 & ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_ball_bex_simps(3)
% 0.30/0.47 thf(fact_62_null__sets_ODiff,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,M: sigma_1466784463real_n,B: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( ( member223413699real_n @ B @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( member223413699real_n @ ( minus_1686442501real_n @ A @ B ) @ ( measur1402256771real_n @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.Diff
% 0.30/0.47 thf(fact_63_vimage__eq,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ B2 ) )
% 0.30/0.47 = ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_eq
% 0.30/0.47 thf(fact_64_vimage__eq,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > $o,B2: set_o] :
% 0.30/0.47 ( ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ B2 ) )
% 0.30/0.47 = ( member_o @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_eq
% 0.30/0.47 thf(fact_65_vimage__eq,axiom,
% 0.30/0.47 ! [A: $o,F: $o > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member_o @ A @ ( vimage961837641real_n @ F @ B2 ) )
% 0.30/0.47 = ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_eq
% 0.30/0.47 thf(fact_66_vimage__eq,axiom,
% 0.30/0.47 ! [A: $o,F: $o > $o,B2: set_o] :
% 0.30/0.47 ( ( member_o @ A @ ( vimage_o_o @ F @ B2 ) )
% 0.30/0.47 = ( member_o @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_eq
% 0.30/0.47 thf(fact_67_vimage__eq,axiom,
% 0.30/0.47 ! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ B2 ) )
% 0.30/0.47 = ( member1352538125real_n @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_eq
% 0.30/0.47 thf(fact_68_vimageI,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A: set_Fi1058188332real_n,B: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( ( F @ A )
% 0.30/0.47 = B )
% 0.30/0.47 => ( ( member223413699real_n @ B @ B2 )
% 0.30/0.47 => ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI
% 0.30/0.47 thf(fact_69_vimageI,axiom,
% 0.30/0.47 ! [F: $o > set_Fi1058188332real_n,A: $o,B: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( ( F @ A )
% 0.30/0.47 = B )
% 0.30/0.47 => ( ( member223413699real_n @ B @ B2 )
% 0.30/0.47 => ( member_o @ A @ ( vimage961837641real_n @ F @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI
% 0.30/0.47 thf(fact_70_vimageI,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > $o,A: set_Fi1058188332real_n,B: $o,B2: set_o] :
% 0.30/0.47 ( ( ( F @ A )
% 0.30/0.47 = B )
% 0.30/0.47 => ( ( member_o @ B @ B2 )
% 0.30/0.47 => ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI
% 0.30/0.47 thf(fact_71_vimageI,axiom,
% 0.30/0.47 ! [F: $o > $o,A: $o,B: $o,B2: set_o] :
% 0.30/0.47 ( ( ( F @ A )
% 0.30/0.47 = B )
% 0.30/0.47 => ( ( member_o @ B @ B2 )
% 0.30/0.47 => ( member_o @ A @ ( vimage_o_o @ F @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI
% 0.30/0.47 thf(fact_72_vimageI,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,A: finite1489363574real_n,B: finite1489363574real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( ( F @ A )
% 0.30/0.47 = B )
% 0.30/0.47 => ( ( member1352538125real_n @ B @ B2 )
% 0.30/0.47 => ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ B2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI
% 0.30/0.47 thf(fact_73_image__ident,axiom,
% 0.30/0.47 ! [Y3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : X
% 0.30/0.47 @ Y3 )
% 0.30/0.47 = Y3 ) ).
% 0.30/0.47
% 0.30/0.47 % image_ident
% 0.30/0.47 thf(fact_74_vimage__Collect__eq,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( vimage1233683625real_n @ F @ ( collec321817931real_n @ P ) )
% 0.30/0.47 = ( collec321817931real_n
% 0.30/0.47 @ ^ [Y2: finite1489363574real_n] : ( P @ ( F @ Y2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_Collect_eq
% 0.30/0.47 thf(fact_75_vimage__ident,axiom,
% 0.30/0.47 ! [Y3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( vimage1233683625real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : X
% 0.30/0.47 @ Y3 )
% 0.30/0.47 = Y3 ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_ident
% 0.30/0.47 thf(fact_76_Int__UNIV,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( ( inf_in1974387902real_n @ A3 @ B2 )
% 0.30/0.47 = top_to1292442332real_n )
% 0.30/0.47 = ( ( A3 = top_to1292442332real_n )
% 0.30/0.47 & ( B2 = top_to1292442332real_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_UNIV
% 0.30/0.47 thf(fact_77_Int__UNIV,axiom,
% 0.30/0.47 ! [A3: set_nat,B2: set_nat] :
% 0.30/0.47 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.30/0.47 = top_top_set_nat )
% 0.30/0.47 = ( ( A3 = top_top_set_nat )
% 0.30/0.47 & ( B2 = top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_UNIV
% 0.30/0.47 thf(fact_78_Int__UNIV,axiom,
% 0.30/0.47 ! [A3: set_Fi160064172_int_n,B2: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( ( inf_in1108485182_int_n @ A3 @ B2 )
% 0.30/0.47 = top_to131672412_int_n )
% 0.30/0.47 = ( ( A3 = top_to131672412_int_n )
% 0.30/0.47 & ( B2 = top_to131672412_int_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_UNIV
% 0.30/0.47 thf(fact_79_ball__UN,axiom,
% 0.30/0.47 ! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ! [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 0.30/0.47 => ( P @ X ) ) )
% 0.30/0.47 = ( ! [X: nat] :
% 0.30/0.47 ( ( member_nat @ X @ A3 )
% 0.30/0.47 => ! [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 0.30/0.47 => ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ball_UN
% 0.30/0.47 thf(fact_80_ball__UN,axiom,
% 0.30/0.47 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ! [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 0.30/0.47 => ( P @ X ) ) )
% 0.30/0.47 = ( ! [X: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X @ A3 )
% 0.30/0.47 => ! [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 0.30/0.47 => ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ball_UN
% 0.30/0.47 thf(fact_81_mem__Collect__eq,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( member223413699real_n @ A @ ( collec452821761real_n @ P ) )
% 0.30/0.47 = ( P @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Collect_eq
% 0.30/0.47 thf(fact_82_mem__Collect__eq,axiom,
% 0.30/0.47 ! [A: $o,P: $o > $o] :
% 0.30/0.47 ( ( member_o @ A @ ( collect_o @ P ) )
% 0.30/0.47 = ( P @ A ) ) ).
% 0.30/0.47
% 0.30/0.47 % mem_Collect_eq
% 0.30/0.47 thf(fact_83_Collect__mem__eq,axiom,
% 0.30/0.47 ! [A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ A3 ) )
% 0.30/0.47 = A3 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mem_eq
% 0.30/0.47 thf(fact_84_Collect__mem__eq,axiom,
% 0.30/0.47 ! [A3: set_o] :
% 0.30/0.47 ( ( collect_o
% 0.30/0.47 @ ^ [X: $o] : ( member_o @ X @ A3 ) )
% 0.30/0.47 = A3 ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_mem_eq
% 0.30/0.47 thf(fact_85_bex__UN,axiom,
% 0.30/0.47 ! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ? [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( ? [X: nat] :
% 0.30/0.47 ( ( member_nat @ X @ A3 )
% 0.30/0.47 & ? [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 0.30/0.47 & ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_UN
% 0.30/0.47 thf(fact_86_bex__UN,axiom,
% 0.30/0.47 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ? [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( ? [X: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X @ A3 )
% 0.30/0.47 & ? [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 0.30/0.47 & ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_UN
% 0.30/0.47 thf(fact_87_UN__ball__bex__simps_I2_J,axiom,
% 0.30/0.47 ! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ! [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 0.30/0.47 => ( P @ X ) ) )
% 0.30/0.47 = ( ! [X: nat] :
% 0.30/0.47 ( ( member_nat @ X @ A3 )
% 0.30/0.47 => ! [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 0.30/0.47 => ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_ball_bex_simps(2)
% 0.30/0.47 thf(fact_88_UN__ball__bex__simps_I2_J,axiom,
% 0.30/0.47 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ! [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 0.30/0.47 => ( P @ X ) ) )
% 0.30/0.47 = ( ! [X: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X @ A3 )
% 0.30/0.47 => ! [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 0.30/0.47 => ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_ball_bex_simps(2)
% 0.30/0.47 thf(fact_89_UN__ball__bex__simps_I4_J,axiom,
% 0.30/0.47 ! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ? [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( ? [X: nat] :
% 0.30/0.47 ( ( member_nat @ X @ A3 )
% 0.30/0.47 & ? [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 0.30/0.47 & ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_ball_bex_simps(4)
% 0.30/0.47 thf(fact_90_UN__ball__bex__simps_I4_J,axiom,
% 0.30/0.47 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( ? [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( ? [X: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X @ A3 )
% 0.30/0.47 & ? [Y2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 0.30/0.47 & ( P @ Y2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_ball_bex_simps(4)
% 0.30/0.47 thf(fact_91_vimage__UNIV,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n] :
% 0.30/0.47 ( ( vimage1233683625real_n @ F @ top_to1292442332real_n )
% 0.30/0.47 = top_to1292442332real_n ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_UNIV
% 0.30/0.47 thf(fact_92_vimage__UNIV,axiom,
% 0.30/0.47 ! [F: nat > nat] :
% 0.30/0.47 ( ( vimage_nat_nat @ F @ top_top_set_nat )
% 0.30/0.47 = top_top_set_nat ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_UNIV
% 0.30/0.47 thf(fact_93_vimage__UNIV,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > nat] :
% 0.30/0.47 ( ( vimage1398021123_n_nat @ F @ top_top_set_nat )
% 0.30/0.47 = top_to131672412_int_n ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_UNIV
% 0.30/0.47 thf(fact_94_vimage__UNIV,axiom,
% 0.30/0.47 ! [F: nat > finite964658038_int_n] :
% 0.30/0.47 ( ( vimage714719107_int_n @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_top_set_nat ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_UNIV
% 0.30/0.47 thf(fact_95_vimage__UNIV,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > finite964658038_int_n] :
% 0.30/0.47 ( ( vimage1122713129_int_n @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_to131672412_int_n ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_UNIV
% 0.30/0.47 thf(fact_96_null__sets_OInt,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,M: sigma_1466784463real_n,B: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( ( member223413699real_n @ B @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( member223413699real_n @ ( inf_in1974387902real_n @ A @ B ) @ ( measur1402256771real_n @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.Int
% 0.30/0.47 thf(fact_97_vimage__Int,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( vimage1233683625real_n @ F @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 0.30/0.47 = ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ A3 ) @ ( vimage1233683625real_n @ F @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_Int
% 0.30/0.47 thf(fact_98_SUP__identity__eq,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( comple2042271945real_n
% 0.30/0.47 @ ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : X
% 0.30/0.47 @ A3 ) )
% 0.30/0.47 = ( comple2042271945real_n @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_identity_eq
% 0.30/0.47 thf(fact_99_SUP__identity__eq,axiom,
% 0.30/0.47 ! [A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( comple825005695real_n
% 0.30/0.47 @ ( image_1661509983real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] : X
% 0.30/0.47 @ A3 ) )
% 0.30/0.47 = ( comple825005695real_n @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_identity_eq
% 0.30/0.47 thf(fact_100_SUP__identity__eq,axiom,
% 0.30/0.47 ! [A3: set_o] :
% 0.30/0.47 ( ( complete_Sup_Sup_o
% 0.30/0.47 @ ( image_o_o
% 0.30/0.47 @ ^ [X: $o] : X
% 0.30/0.47 @ A3 ) )
% 0.30/0.47 = ( complete_Sup_Sup_o @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_identity_eq
% 0.30/0.47 thf(fact_101_UN__iff,axiom,
% 0.30/0.47 ! [B: finite1489363574real_n,B2: nat > set_Fi1058188332real_n,A3: set_nat] :
% 0.30/0.47 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 0.30/0.47 = ( ? [X: nat] :
% 0.30/0.47 ( ( member_nat @ X @ A3 )
% 0.30/0.47 & ( member1352538125real_n @ B @ ( B2 @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_iff
% 0.30/0.47 thf(fact_102_UN__iff,axiom,
% 0.30/0.47 ! [B: finite1489363574real_n,B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 0.30/0.47 = ( ? [X: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X @ A3 )
% 0.30/0.47 & ( member1352538125real_n @ B @ ( B2 @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_iff
% 0.30/0.47 thf(fact_103_UN__I,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: set_Fi1058188332real_n,B2: set_Fi1058188332real_n > set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A @ A3 )
% 0.30/0.47 => ( ( member223413699real_n @ B @ ( B2 @ A ) )
% 0.30/0.47 => ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_797440021real_n @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_I
% 0.30/0.47 thf(fact_104_UN__I,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: $o,B2: set_Fi1058188332real_n > set_o] :
% 0.30/0.47 ( ( member223413699real_n @ A @ A3 )
% 0.30/0.47 => ( ( member_o @ B @ ( B2 @ A ) )
% 0.30/0.47 => ( member_o @ B @ ( comple1665300069_set_o @ ( image_1687589765_set_o @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_I
% 0.30/0.47 thf(fact_105_UN__I,axiom,
% 0.30/0.47 ! [A: $o,A3: set_o,B: set_Fi1058188332real_n,B2: $o > set_se2111327970real_n] :
% 0.30/0.47 ( ( member_o @ A @ A3 )
% 0.30/0.47 => ( ( member223413699real_n @ B @ ( B2 @ A ) )
% 0.30/0.47 => ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_452144437real_n @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_I
% 0.30/0.47 thf(fact_106_UN__I,axiom,
% 0.30/0.47 ! [A: $o,A3: set_o,B: $o,B2: $o > set_o] :
% 0.30/0.47 ( ( member_o @ A @ A3 )
% 0.30/0.47 => ( ( member_o @ B @ ( B2 @ A ) )
% 0.30/0.47 => ( member_o @ B @ ( comple1665300069_set_o @ ( image_o_set_o @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_I
% 0.30/0.47 thf(fact_107_UN__I,axiom,
% 0.30/0.47 ! [A: nat,A3: set_nat,B: finite1489363574real_n,B2: nat > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member_nat @ A @ A3 )
% 0.30/0.47 => ( ( member1352538125real_n @ B @ ( B2 @ A ) )
% 0.30/0.47 => ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_I
% 0.30/0.47 thf(fact_108_UN__I,axiom,
% 0.30/0.47 ! [A: finite964658038_int_n,A3: set_Fi160064172_int_n,B: finite1489363574real_n,B2: finite964658038_int_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member27055245_int_n @ A @ A3 )
% 0.30/0.47 => ( ( member1352538125real_n @ B @ ( B2 @ A ) )
% 0.30/0.47 => ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_I
% 0.30/0.47 thf(fact_109_UN__I,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: finite1489363574real_n,B2: set_Fi1058188332real_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A @ A3 )
% 0.30/0.47 => ( ( member1352538125real_n @ B @ ( B2 @ A ) )
% 0.30/0.47 => ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1661509983real_n @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_I
% 0.30/0.47 thf(fact_110_UN__I,axiom,
% 0.30/0.47 ! [A: $o,A3: set_o,B: finite1489363574real_n,B2: $o > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member_o @ A @ A3 )
% 0.30/0.47 => ( ( member1352538125real_n @ B @ ( B2 @ A ) )
% 0.30/0.47 => ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1759008383real_n @ B2 @ A3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_I
% 0.30/0.47 thf(fact_111_S__decompose,axiom,
% 0.30/0.47 ( s
% 0.30/0.47 = ( comple825005695real_n @ ( image_355963305real_n @ t2 @ top_to131672412_int_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % S_decompose
% 0.30/0.47 thf(fact_112_image__add__0,axiom,
% 0.30/0.47 ! [S: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_439535603real_n @ ( plus_p585657087real_n @ zero_z200130687real_n ) @ S )
% 0.30/0.47 = S ) ).
% 0.30/0.47
% 0.30/0.47 % image_add_0
% 0.30/0.47 thf(fact_113_image__add__0,axiom,
% 0.30/0.47 ! [S: set_Ex113815278nnreal] :
% 0.30/0.47 ( ( image_2066995319nnreal @ ( plus_p1763960001nnreal @ zero_z1963244097nnreal ) @ S )
% 0.30/0.47 = S ) ).
% 0.30/0.47
% 0.30/0.47 % image_add_0
% 0.30/0.47 thf(fact_114_null__sets_OInt__space__eq2,axiom,
% 0.30/0.47 ! [X3: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X3 @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( ( inf_in1974387902real_n @ X3 @ ( sigma_476185326real_n @ M ) )
% 0.30/0.47 = X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.Int_space_eq2
% 0.30/0.47 thf(fact_115_null__sets_OInt__space__eq1,axiom,
% 0.30/0.47 ! [X3: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X3 @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( ( inf_in1974387902real_n @ ( sigma_476185326real_n @ M ) @ X3 )
% 0.30/0.47 = X3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.Int_space_eq1
% 0.30/0.47 thf(fact_116_in__sets__SUP,axiom,
% 0.30/0.47 ! [I2: set_Fi1058188332real_n,I3: set_se2111327970real_n,M: set_Fi1058188332real_n > sigma_1466784463real_n,Y3: set_Fi1058188332real_n,X4: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ I2 @ I3 )
% 0.30/0.47 => ( ! [I: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ I @ I3 )
% 0.30/0.47 => ( ( sigma_476185326real_n @ ( M @ I ) )
% 0.30/0.47 = Y3 ) )
% 0.30/0.47 => ( ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( M @ I2 ) ) )
% 0.30/0.47 => ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_987430492real_n @ M @ I3 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_sets_SUP
% 0.30/0.47 thf(fact_117_in__sets__SUP,axiom,
% 0.30/0.47 ! [I2: $o,I3: set_o,M: $o > sigma_1466784463real_n,Y3: set_Fi1058188332real_n,X4: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member_o @ I2 @ I3 )
% 0.30/0.47 => ( ! [I: $o] :
% 0.30/0.47 ( ( member_o @ I @ I3 )
% 0.30/0.47 => ( ( sigma_476185326real_n @ ( M @ I ) )
% 0.30/0.47 = Y3 ) )
% 0.30/0.47 => ( ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( M @ I2 ) ) )
% 0.30/0.47 => ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_1599934780real_n @ M @ I3 ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_sets_SUP
% 0.30/0.47 thf(fact_118_in__sets__Sup,axiom,
% 0.30/0.47 ! [M: set_Si1125517487real_n,X4: set_Fi1058188332real_n,M2: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ! [M3: sigma_1466784463real_n] :
% 0.30/0.47 ( ( member1000184real_n @ M3 @ M )
% 0.30/0.47 => ( ( sigma_476185326real_n @ M3 )
% 0.30/0.47 = X4 ) )
% 0.30/0.47 => ( ( member1000184real_n @ M2 @ M )
% 0.30/0.47 => ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M2 ) )
% 0.30/0.47 => ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ ( comple488165692real_n @ M ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % in_sets_Sup
% 0.30/0.47 thf(fact_119_sets__SUP__cong,axiom,
% 0.30/0.47 ! [I3: set_se2111327970real_n,M: set_Fi1058188332real_n > sigma_1466784463real_n,N2: set_Fi1058188332real_n > sigma_1466784463real_n] :
% 0.30/0.47 ( ! [I: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ I @ I3 )
% 0.30/0.47 => ( ( sigma_1235138647real_n @ ( M @ I ) )
% 0.30/0.47 = ( sigma_1235138647real_n @ ( N2 @ I ) ) ) )
% 0.30/0.47 => ( ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_987430492real_n @ M @ I3 ) ) )
% 0.30/0.47 = ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_987430492real_n @ N2 @ I3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sets_SUP_cong
% 0.30/0.47 thf(fact_120_sets__SUP__cong,axiom,
% 0.30/0.47 ! [I3: set_o,M: $o > sigma_1466784463real_n,N2: $o > sigma_1466784463real_n] :
% 0.30/0.47 ( ! [I: $o] :
% 0.30/0.47 ( ( member_o @ I @ I3 )
% 0.30/0.47 => ( ( sigma_1235138647real_n @ ( M @ I ) )
% 0.30/0.47 = ( sigma_1235138647real_n @ ( N2 @ I ) ) ) )
% 0.30/0.47 => ( ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_1599934780real_n @ M @ I3 ) ) )
% 0.30/0.47 = ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_1599934780real_n @ N2 @ I3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % sets_SUP_cong
% 0.30/0.47 thf(fact_121_Sup__set__def,axiom,
% 0.30/0.47 ( comple1917283637real_n
% 0.30/0.47 = ( ^ [A4: set_se820660888real_n] :
% 0.30/0.47 ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] : ( complete_Sup_Sup_o @ ( image_1681970287al_n_o @ ( member223413699real_n @ X ) @ A4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup_set_def
% 0.30/0.47 thf(fact_122_Sup__set__def,axiom,
% 0.30/0.47 ( comple1665300069_set_o
% 0.30/0.47 = ( ^ [A4: set_set_o] :
% 0.30/0.47 ( collect_o
% 0.30/0.47 @ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup_set_def
% 0.30/0.47 thf(fact_123_Sup__set__def,axiom,
% 0.30/0.47 ( comple825005695real_n
% 0.30/0.47 = ( ^ [A4: set_se2111327970real_n] :
% 0.30/0.47 ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( complete_Sup_Sup_o @ ( image_1648361637al_n_o @ ( member1352538125real_n @ X ) @ A4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup_set_def
% 0.30/0.47 thf(fact_124_space__Sup__eq__UN,axiom,
% 0.30/0.47 ! [M: set_Si1125517487real_n] :
% 0.30/0.47 ( ( sigma_476185326real_n @ ( comple488165692real_n @ M ) )
% 0.30/0.47 = ( comple825005695real_n @ ( image_1298280374real_n @ sigma_476185326real_n @ M ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % space_Sup_eq_UN
% 0.30/0.47 thf(fact_125_top__set__def,axiom,
% 0.30/0.47 ( top_top_set_nat
% 0.30/0.47 = ( collect_nat @ top_top_nat_o ) ) ).
% 0.30/0.47
% 0.30/0.47 % top_set_def
% 0.30/0.47 thf(fact_126_top__set__def,axiom,
% 0.30/0.47 ( top_to131672412_int_n
% 0.30/0.47 = ( collec1941932235_int_n @ top_to287930409nt_n_o ) ) ).
% 0.30/0.47
% 0.30/0.47 % top_set_def
% 0.30/0.47 thf(fact_127_Diff__Int__distrib2,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ ( minus_1686442501real_n @ A3 @ B2 ) @ C2 )
% 0.30/0.47 = ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ C2 ) @ ( inf_in1974387902real_n @ B2 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_Int_distrib2
% 0.30/0.47 thf(fact_128_Diff__Int__distrib,axiom,
% 0.30/0.47 ! [C2: set_Fi1058188332real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ C2 @ ( minus_1686442501real_n @ A3 @ B2 ) )
% 0.30/0.47 = ( minus_1686442501real_n @ ( inf_in1974387902real_n @ C2 @ A3 ) @ ( inf_in1974387902real_n @ C2 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_Int_distrib
% 0.30/0.47 thf(fact_129_Diff__Diff__Int,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( minus_1686442501real_n @ A3 @ ( minus_1686442501real_n @ A3 @ B2 ) )
% 0.30/0.47 = ( inf_in1974387902real_n @ A3 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_Diff_Int
% 0.30/0.47 thf(fact_130_Diff__Int2,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,C2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ C2 ) @ ( inf_in1974387902real_n @ B2 @ C2 ) )
% 0.30/0.47 = ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ C2 ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Diff_Int2
% 0.30/0.47 thf(fact_131_Int__Diff,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ B2 ) @ C2 )
% 0.30/0.47 = ( inf_in1974387902real_n @ A3 @ ( minus_1686442501real_n @ B2 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_Diff
% 0.30/0.47 thf(fact_132_vimage__Diff,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( vimage1233683625real_n @ F @ ( minus_1686442501real_n @ A3 @ B2 ) )
% 0.30/0.47 = ( minus_1686442501real_n @ ( vimage1233683625real_n @ F @ A3 ) @ ( vimage1233683625real_n @ F @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_Diff
% 0.30/0.47 thf(fact_133_null__sets_Osets__Collect__disj,axiom,
% 0.30/0.47 ! [M: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o,Q: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( member1475136633real_n
% 0.30/0.47 @ ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 @ ( measur2126959417real_n @ M ) )
% 0.30/0.47 => ( ( member1475136633real_n
% 0.30/0.47 @ ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 0.30/0.47 & ( Q @ X ) ) )
% 0.30/0.47 @ ( measur2126959417real_n @ M ) )
% 0.30/0.47 => ( member1475136633real_n
% 0.30/0.47 @ ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 0.30/0.47 & ( ( Q @ X )
% 0.30/0.47 | ( P @ X ) ) ) )
% 0.30/0.47 @ ( measur2126959417real_n @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.sets_Collect_disj
% 0.30/0.47 thf(fact_134_null__sets_Osets__Collect__disj,axiom,
% 0.30/0.47 ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
% 0.30/0.47 ( ( member_set_o
% 0.30/0.47 @ ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 @ ( measure_null_sets_o @ M ) )
% 0.30/0.47 => ( ( member_set_o
% 0.30/0.47 @ ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 0.30/0.47 & ( Q @ X ) ) )
% 0.30/0.47 @ ( measure_null_sets_o @ M ) )
% 0.30/0.47 => ( member_set_o
% 0.30/0.47 @ ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 0.30/0.47 & ( ( Q @ X )
% 0.30/0.47 | ( P @ X ) ) ) )
% 0.30/0.47 @ ( measure_null_sets_o @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.sets_Collect_disj
% 0.30/0.47 thf(fact_135_null__sets_Osets__Collect__disj,axiom,
% 0.30/0.47 ! [M: sigma_1466784463real_n,P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( member223413699real_n
% 0.30/0.47 @ ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( ( member223413699real_n
% 0.30/0.47 @ ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 0.30/0.47 & ( Q @ X ) ) )
% 0.30/0.47 @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( member223413699real_n
% 0.30/0.47 @ ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 0.30/0.47 & ( ( Q @ X )
% 0.30/0.47 | ( P @ X ) ) ) )
% 0.30/0.47 @ ( measur1402256771real_n @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.sets_Collect_disj
% 0.30/0.47 thf(fact_136_null__sets_Osets__Collect__conj,axiom,
% 0.30/0.47 ! [M: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o,Q: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( member1475136633real_n
% 0.30/0.47 @ ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 @ ( measur2126959417real_n @ M ) )
% 0.30/0.47 => ( ( member1475136633real_n
% 0.30/0.47 @ ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 0.30/0.47 & ( Q @ X ) ) )
% 0.30/0.47 @ ( measur2126959417real_n @ M ) )
% 0.30/0.47 => ( member1475136633real_n
% 0.30/0.47 @ ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 0.30/0.47 & ( Q @ X )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 @ ( measur2126959417real_n @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.sets_Collect_conj
% 0.30/0.47 thf(fact_137_null__sets_Osets__Collect__conj,axiom,
% 0.30/0.47 ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
% 0.30/0.47 ( ( member_set_o
% 0.30/0.47 @ ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 @ ( measure_null_sets_o @ M ) )
% 0.30/0.47 => ( ( member_set_o
% 0.30/0.47 @ ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 0.30/0.47 & ( Q @ X ) ) )
% 0.30/0.47 @ ( measure_null_sets_o @ M ) )
% 0.30/0.47 => ( member_set_o
% 0.30/0.47 @ ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 0.30/0.47 & ( Q @ X )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 @ ( measure_null_sets_o @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.sets_Collect_conj
% 0.30/0.47 thf(fact_138_null__sets_Osets__Collect__conj,axiom,
% 0.30/0.47 ! [M: sigma_1466784463real_n,P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( member223413699real_n
% 0.30/0.47 @ ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( ( member223413699real_n
% 0.30/0.47 @ ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 0.30/0.47 & ( Q @ X ) ) )
% 0.30/0.47 @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( member223413699real_n
% 0.30/0.47 @ ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 0.30/0.47 & ( Q @ X )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 @ ( measur1402256771real_n @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_sets.sets_Collect_conj
% 0.30/0.47 thf(fact_139_translation__diff,axiom,
% 0.30/0.47 ! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ ( minus_1686442501real_n @ S2 @ T ) )
% 0.30/0.47 = ( minus_1686442501real_n @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ S2 ) @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % translation_diff
% 0.30/0.47 thf(fact_140_null__set__Diff,axiom,
% 0.30/0.47 ! [B2: set_Fi1058188332real_n,M: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ B2 @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) )
% 0.30/0.47 => ( member223413699real_n @ ( minus_1686442501real_n @ B2 @ A3 ) @ ( measur1402256771real_n @ M ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_set_Diff
% 0.30/0.47 thf(fact_141_Int__Union2,axiom,
% 0.30/0.47 ! [B2: set_se2111327970real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ ( comple825005695real_n @ B2 ) @ A3 )
% 0.30/0.47 = ( comple825005695real_n
% 0.30/0.47 @ ( image_1661509983real_n
% 0.30/0.47 @ ^ [C3: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ C3 @ A3 )
% 0.30/0.47 @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_Union2
% 0.30/0.47 thf(fact_142_Int__Union,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ A3 @ ( comple825005695real_n @ B2 ) )
% 0.30/0.47 = ( comple825005695real_n @ ( image_1661509983real_n @ ( inf_in1974387902real_n @ A3 ) @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_Union
% 0.30/0.47 thf(fact_143_vimage__Union,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( vimage1233683625real_n @ F @ ( comple825005695real_n @ A3 ) )
% 0.30/0.47 = ( comple825005695real_n @ ( image_1661509983real_n @ ( vimage1233683625real_n @ F ) @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_Union
% 0.30/0.47 thf(fact_144_UN__extend__simps_I6_J,axiom,
% 0.30/0.47 ! [A3: nat > set_Fi1058188332real_n,C2: set_nat,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( minus_1686442501real_n @ ( comple825005695real_n @ ( image_1856576259real_n @ A3 @ C2 ) ) @ B2 )
% 0.30/0.47 = ( comple825005695real_n
% 0.30/0.47 @ ( image_1856576259real_n
% 0.30/0.47 @ ^ [X: nat] : ( minus_1686442501real_n @ ( A3 @ X ) @ B2 )
% 0.30/0.47 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_extend_simps(6)
% 0.30/0.47 thf(fact_145_UN__extend__simps_I6_J,axiom,
% 0.30/0.47 ! [A3: finite964658038_int_n > set_Fi1058188332real_n,C2: set_Fi160064172_int_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( minus_1686442501real_n @ ( comple825005695real_n @ ( image_355963305real_n @ A3 @ C2 ) ) @ B2 )
% 0.30/0.47 = ( comple825005695real_n
% 0.30/0.47 @ ( image_355963305real_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : ( minus_1686442501real_n @ ( A3 @ X ) @ B2 )
% 0.30/0.47 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UN_extend_simps(6)
% 0.30/0.47 thf(fact_146_emeasure__Diff__null__set,axiom,
% 0.30/0.47 ! [B2: set_Fi1058188332real_n,M: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ B2 @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) )
% 0.30/0.47 => ( ( sigma_1536574303real_n @ M @ ( minus_1686442501real_n @ A3 @ B2 ) )
% 0.30/0.47 = ( sigma_1536574303real_n @ M @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % emeasure_Diff_null_set
% 0.30/0.47 thf(fact_147_Sup_OSUP__cong,axiom,
% 0.30/0.47 ! [A3: set_nat,B2: set_nat,C2: nat > set_Fi1058188332real_n,D: nat > set_Fi1058188332real_n,Sup: set_se2111327970real_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: nat] :
% 0.30/0.47 ( ( member_nat @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( Sup @ ( image_1856576259real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( Sup @ ( image_1856576259real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup.SUP_cong
% 0.30/0.47 thf(fact_148_Sup_OSUP__cong,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: finite1489363574real_n > finite1489363574real_n,D: finite1489363574real_n > finite1489363574real_n,Sup: set_Fi1058188332real_n > finite1489363574real_n] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( Sup @ ( image_439535603real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( Sup @ ( image_439535603real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup.SUP_cong
% 0.30/0.47 thf(fact_149_Sup_OSUP__cong,axiom,
% 0.30/0.47 ! [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] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( Sup @ ( image_355963305real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( Sup @ ( image_355963305real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup.SUP_cong
% 0.30/0.47 thf(fact_150_Inf_OINF__cong,axiom,
% 0.30/0.47 ! [A3: set_nat,B2: set_nat,C2: nat > set_Fi1058188332real_n,D: nat > set_Fi1058188332real_n,Inf: set_se2111327970real_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: nat] :
% 0.30/0.47 ( ( member_nat @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( Inf @ ( image_1856576259real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( Inf @ ( image_1856576259real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inf.INF_cong
% 0.30/0.47 thf(fact_151_Inf_OINF__cong,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: finite1489363574real_n > finite1489363574real_n,D: finite1489363574real_n > finite1489363574real_n,Inf: set_Fi1058188332real_n > finite1489363574real_n] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( Inf @ ( image_439535603real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( Inf @ ( image_439535603real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inf.INF_cong
% 0.30/0.47 thf(fact_152_Inf_OINF__cong,axiom,
% 0.30/0.47 ! [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] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( Inf @ ( image_355963305real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( Inf @ ( image_355963305real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inf.INF_cong
% 0.30/0.47 thf(fact_153_rev__image__eqI,axiom,
% 0.30/0.47 ! [X3: finite1489363574real_n,A3: set_Fi1058188332real_n,B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X3 @ A3 )
% 0.30/0.47 => ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_image_eqI
% 0.30/0.47 thf(fact_154_rev__image__eqI,axiom,
% 0.30/0.47 ! [X3: nat,A3: set_nat,B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member_nat @ X3 @ A3 )
% 0.30/0.47 => ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_image_eqI
% 0.30/0.47 thf(fact_155_rev__image__eqI,axiom,
% 0.30/0.47 ! [X3: finite964658038_int_n,A3: set_Fi160064172_int_n,B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X3 @ A3 )
% 0.30/0.47 => ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_355963305real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_image_eqI
% 0.30/0.47 thf(fact_156_rev__image__eqI,axiom,
% 0.30/0.47 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X3 @ A3 )
% 0.30/0.47 => ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_image_eqI
% 0.30/0.47 thf(fact_157_rev__image__eqI,axiom,
% 0.30/0.47 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: $o,F: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( member223413699real_n @ X3 @ A3 )
% 0.30/0.47 => ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member_o @ B @ ( image_1648361637al_n_o @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_image_eqI
% 0.30/0.47 thf(fact_158_rev__image__eqI,axiom,
% 0.30/0.47 ! [X3: $o,A3: set_o,B: set_Fi1058188332real_n,F: $o > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member_o @ X3 @ A3 )
% 0.30/0.47 => ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_1759008383real_n @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_image_eqI
% 0.30/0.47 thf(fact_159_rev__image__eqI,axiom,
% 0.30/0.47 ! [X3: $o,A3: set_o,B: $o,F: $o > $o] :
% 0.30/0.47 ( ( member_o @ X3 @ A3 )
% 0.30/0.47 => ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member_o @ B @ ( image_o_o @ F @ A3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % rev_image_eqI
% 0.30/0.47 thf(fact_160_ball__imageD,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,A3: set_nat,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ! [X2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X2 @ ( image_1856576259real_n @ F @ A3 ) )
% 0.30/0.47 => ( P @ X2 ) )
% 0.30/0.47 => ! [X5: nat] :
% 0.30/0.47 ( ( member_nat @ X5 @ A3 )
% 0.30/0.47 => ( P @ ( F @ X5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ball_imageD
% 0.30/0.47 thf(fact_161_ball__imageD,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ! [X2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X2 @ ( image_439535603real_n @ F @ A3 ) )
% 0.30/0.47 => ( P @ X2 ) )
% 0.30/0.47 => ! [X5: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X5 @ A3 )
% 0.30/0.47 => ( P @ ( F @ X5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ball_imageD
% 0.30/0.47 thf(fact_162_ball__imageD,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ! [X2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X2 @ ( image_355963305real_n @ F @ A3 ) )
% 0.30/0.47 => ( P @ X2 ) )
% 0.30/0.47 => ! [X5: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X5 @ A3 )
% 0.30/0.47 => ( P @ ( F @ X5 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % ball_imageD
% 0.30/0.47 thf(fact_163_image__cong,axiom,
% 0.30/0.47 ! [M: set_nat,N2: set_nat,F: nat > set_Fi1058188332real_n,G: nat > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( M = N2 )
% 0.30/0.47 => ( ! [X2: nat] :
% 0.30/0.47 ( ( member_nat @ X2 @ N2 )
% 0.30/0.47 => ( ( F @ X2 )
% 0.30/0.47 = ( G @ X2 ) ) )
% 0.30/0.47 => ( ( image_1856576259real_n @ F @ M )
% 0.30/0.47 = ( image_1856576259real_n @ G @ N2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_cong
% 0.30/0.47 thf(fact_164_image__cong,axiom,
% 0.30/0.47 ! [M: set_Fi1058188332real_n,N2: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n] :
% 0.30/0.47 ( ( M = N2 )
% 0.30/0.47 => ( ! [X2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X2 @ N2 )
% 0.30/0.47 => ( ( F @ X2 )
% 0.30/0.47 = ( G @ X2 ) ) )
% 0.30/0.47 => ( ( image_439535603real_n @ F @ M )
% 0.30/0.47 = ( image_439535603real_n @ G @ N2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_cong
% 0.30/0.47 thf(fact_165_image__cong,axiom,
% 0.30/0.47 ! [M: set_Fi160064172_int_n,N2: set_Fi160064172_int_n,F: finite964658038_int_n > set_Fi1058188332real_n,G: finite964658038_int_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( M = N2 )
% 0.30/0.47 => ( ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X2 @ N2 )
% 0.30/0.47 => ( ( F @ X2 )
% 0.30/0.47 = ( G @ X2 ) ) )
% 0.30/0.47 => ( ( image_355963305real_n @ F @ M )
% 0.30/0.47 = ( image_355963305real_n @ G @ N2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_cong
% 0.30/0.47 thf(fact_166_bex__imageD,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,A3: set_nat,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ? [X5: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X5 @ ( image_1856576259real_n @ F @ A3 ) )
% 0.30/0.47 & ( P @ X5 ) )
% 0.30/0.47 => ? [X2: nat] :
% 0.30/0.47 ( ( member_nat @ X2 @ A3 )
% 0.30/0.47 & ( P @ ( F @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_imageD
% 0.30/0.47 thf(fact_167_bex__imageD,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ? [X5: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X5 @ ( image_439535603real_n @ F @ A3 ) )
% 0.30/0.47 & ( P @ X5 ) )
% 0.30/0.47 => ? [X2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X2 @ A3 )
% 0.30/0.47 & ( P @ ( F @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_imageD
% 0.30/0.47 thf(fact_168_bex__imageD,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ? [X5: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X5 @ ( image_355963305real_n @ F @ A3 ) )
% 0.30/0.47 & ( P @ X5 ) )
% 0.30/0.47 => ? [X2: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X2 @ A3 )
% 0.30/0.47 & ( P @ ( F @ X2 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_imageD
% 0.30/0.47 thf(fact_169_image__iff,axiom,
% 0.30/0.47 ! [Z: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ Z @ ( image_439535603real_n @ F @ A3 ) )
% 0.30/0.47 = ( ? [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ A3 )
% 0.30/0.47 & ( Z
% 0.30/0.47 = ( F @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_iff
% 0.30/0.47 thf(fact_170_image__iff,axiom,
% 0.30/0.47 ! [Z: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,A3: set_nat] :
% 0.30/0.47 ( ( member223413699real_n @ Z @ ( image_1856576259real_n @ F @ A3 ) )
% 0.30/0.47 = ( ? [X: nat] :
% 0.30/0.47 ( ( member_nat @ X @ A3 )
% 0.30/0.47 & ( Z
% 0.30/0.47 = ( F @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_iff
% 0.30/0.47 thf(fact_171_image__iff,axiom,
% 0.30/0.47 ! [Z: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( member223413699real_n @ Z @ ( image_355963305real_n @ F @ A3 ) )
% 0.30/0.47 = ( ? [X: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X @ A3 )
% 0.30/0.47 & ( Z
% 0.30/0.47 = ( F @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_iff
% 0.30/0.47 thf(fact_172_imageI,axiom,
% 0.30/0.47 ! [X3: finite1489363574real_n,A3: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X3 @ A3 )
% 0.30/0.47 => ( member1352538125real_n @ ( F @ X3 ) @ ( image_439535603real_n @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_173_imageI,axiom,
% 0.30/0.47 ! [X3: nat,A3: set_nat,F: nat > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member_nat @ X3 @ A3 )
% 0.30/0.47 => ( member223413699real_n @ ( F @ X3 ) @ ( image_1856576259real_n @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_174_imageI,axiom,
% 0.30/0.47 ! [X3: finite964658038_int_n,A3: set_Fi160064172_int_n,F: finite964658038_int_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X3 @ A3 )
% 0.30/0.47 => ( member223413699real_n @ ( F @ X3 ) @ ( image_355963305real_n @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_175_imageI,axiom,
% 0.30/0.47 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X3 @ A3 )
% 0.30/0.47 => ( member223413699real_n @ ( F @ X3 ) @ ( image_1661509983real_n @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_176_imageI,axiom,
% 0.30/0.47 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,F: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( member223413699real_n @ X3 @ A3 )
% 0.30/0.47 => ( member_o @ ( F @ X3 ) @ ( image_1648361637al_n_o @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_177_imageI,axiom,
% 0.30/0.47 ! [X3: $o,A3: set_o,F: $o > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member_o @ X3 @ A3 )
% 0.30/0.47 => ( member223413699real_n @ ( F @ X3 ) @ ( image_1759008383real_n @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_178_imageI,axiom,
% 0.30/0.47 ! [X3: $o,A3: set_o,F: $o > $o] :
% 0.30/0.47 ( ( member_o @ X3 @ A3 )
% 0.30/0.47 => ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_179_is__num__normalize_I1_J,axiom,
% 0.30/0.47 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] :
% 0.30/0.47 ( ( plus_p585657087real_n @ ( plus_p585657087real_n @ A @ B ) @ C )
% 0.30/0.47 = ( plus_p585657087real_n @ A @ ( plus_p585657087real_n @ B @ C ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % is_num_normalize(1)
% 0.30/0.47 thf(fact_180_UNIV__witness,axiom,
% 0.30/0.47 ? [X2: set_Fi1058188332real_n] : ( member223413699real_n @ X2 @ top_to20708754real_n ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_witness
% 0.30/0.47 thf(fact_181_UNIV__witness,axiom,
% 0.30/0.47 ? [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_witness
% 0.30/0.47 thf(fact_182_UNIV__witness,axiom,
% 0.30/0.47 ? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_witness
% 0.30/0.47 thf(fact_183_UNIV__witness,axiom,
% 0.30/0.47 ? [X2: finite964658038_int_n] : ( member27055245_int_n @ X2 @ top_to131672412_int_n ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_witness
% 0.30/0.47 thf(fact_184_UNIV__eq__I,axiom,
% 0.30/0.47 ! [A3: set_se2111327970real_n] :
% 0.30/0.47 ( ! [X2: set_Fi1058188332real_n] : ( member223413699real_n @ X2 @ A3 )
% 0.30/0.47 => ( top_to20708754real_n = A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_eq_I
% 0.30/0.47 thf(fact_185_UNIV__eq__I,axiom,
% 0.30/0.47 ! [A3: set_o] :
% 0.30/0.47 ( ! [X2: $o] : ( member_o @ X2 @ A3 )
% 0.30/0.47 => ( top_top_set_o = A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_eq_I
% 0.30/0.47 thf(fact_186_UNIV__eq__I,axiom,
% 0.30/0.47 ! [A3: set_nat] :
% 0.30/0.47 ( ! [X2: nat] : ( member_nat @ X2 @ A3 )
% 0.30/0.47 => ( top_top_set_nat = A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_eq_I
% 0.30/0.47 thf(fact_187_UNIV__eq__I,axiom,
% 0.30/0.47 ! [A3: set_Fi160064172_int_n] :
% 0.30/0.47 ( ! [X2: finite964658038_int_n] : ( member27055245_int_n @ X2 @ A3 )
% 0.30/0.47 => ( top_to131672412_int_n = A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_eq_I
% 0.30/0.47 thf(fact_188_Int__left__commute,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ A3 @ ( inf_in1974387902real_n @ B2 @ C2 ) )
% 0.30/0.47 = ( inf_in1974387902real_n @ B2 @ ( inf_in1974387902real_n @ A3 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_left_commute
% 0.30/0.47 thf(fact_189_Int__left__absorb,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ A3 @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 0.30/0.47 = ( inf_in1974387902real_n @ A3 @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_left_absorb
% 0.30/0.47 thf(fact_190_Int__commute,axiom,
% 0.30/0.47 ( inf_in1974387902real_n
% 0.30/0.47 = ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ B3 @ A4 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_commute
% 0.30/0.47 thf(fact_191_Int__absorb,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ A3 @ A3 )
% 0.30/0.47 = A3 ) ).
% 0.30/0.47
% 0.30/0.47 % Int_absorb
% 0.30/0.47 thf(fact_192_Int__assoc,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ ( inf_in1974387902real_n @ A3 @ B2 ) @ C2 )
% 0.30/0.47 = ( inf_in1974387902real_n @ A3 @ ( inf_in1974387902real_n @ B2 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_assoc
% 0.30/0.47 thf(fact_193_IntD2,axiom,
% 0.30/0.47 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
% 0.30/0.47 => ( member223413699real_n @ C @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntD2
% 0.30/0.47 thf(fact_194_IntD2,axiom,
% 0.30/0.47 ! [C: $o,A3: set_o,B2: set_o] :
% 0.30/0.47 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.30/0.47 => ( member_o @ C @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntD2
% 0.30/0.47 thf(fact_195_IntD2,axiom,
% 0.30/0.47 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 0.30/0.47 => ( member1352538125real_n @ C @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntD2
% 0.30/0.47 thf(fact_196_IntD1,axiom,
% 0.30/0.47 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
% 0.30/0.47 => ( member223413699real_n @ C @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntD1
% 0.30/0.47 thf(fact_197_IntD1,axiom,
% 0.30/0.47 ! [C: $o,A3: set_o,B2: set_o] :
% 0.30/0.47 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.30/0.47 => ( member_o @ C @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntD1
% 0.30/0.47 thf(fact_198_IntD1,axiom,
% 0.30/0.47 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 0.30/0.47 => ( member1352538125real_n @ C @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntD1
% 0.30/0.47 thf(fact_199_IntE,axiom,
% 0.30/0.47 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
% 0.30/0.47 => ~ ( ( member223413699real_n @ C @ A3 )
% 0.30/0.47 => ~ ( member223413699real_n @ C @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntE
% 0.30/0.47 thf(fact_200_IntE,axiom,
% 0.30/0.47 ! [C: $o,A3: set_o,B2: set_o] :
% 0.30/0.47 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.30/0.47 => ~ ( ( member_o @ C @ A3 )
% 0.30/0.47 => ~ ( member_o @ C @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntE
% 0.30/0.47 thf(fact_201_IntE,axiom,
% 0.30/0.47 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 0.30/0.47 => ~ ( ( member1352538125real_n @ C @ A3 )
% 0.30/0.47 => ~ ( member1352538125real_n @ C @ B2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % IntE
% 0.30/0.47 thf(fact_202_UnionE,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,C2: set_se820660888real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A3 @ ( comple1917283637real_n @ C2 ) )
% 0.30/0.47 => ~ ! [X6: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A3 @ X6 )
% 0.30/0.47 => ~ ( member1475136633real_n @ X6 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnionE
% 0.30/0.47 thf(fact_203_UnionE,axiom,
% 0.30/0.47 ! [A3: $o,C2: set_set_o] :
% 0.30/0.47 ( ( member_o @ A3 @ ( comple1665300069_set_o @ C2 ) )
% 0.30/0.47 => ~ ! [X6: set_o] :
% 0.30/0.47 ( ( member_o @ A3 @ X6 )
% 0.30/0.47 => ~ ( member_set_o @ X6 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnionE
% 0.30/0.47 thf(fact_204_UnionE,axiom,
% 0.30/0.47 ! [A3: finite1489363574real_n,C2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ A3 @ ( comple825005695real_n @ C2 ) )
% 0.30/0.47 => ~ ! [X6: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ A3 @ X6 )
% 0.30/0.47 => ~ ( member223413699real_n @ X6 @ C2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % UnionE
% 0.30/0.47 thf(fact_205_vimage__Collect,axiom,
% 0.30/0.47 ! [P: finite1489363574real_n > $o,F: finite1489363574real_n > finite1489363574real_n,Q: finite1489363574real_n > $o] :
% 0.30/0.47 ( ! [X2: finite1489363574real_n] :
% 0.30/0.47 ( ( P @ ( F @ X2 ) )
% 0.30/0.47 = ( Q @ X2 ) )
% 0.30/0.47 => ( ( vimage1233683625real_n @ F @ ( collec321817931real_n @ P ) )
% 0.30/0.47 = ( collec321817931real_n @ Q ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_Collect
% 0.30/0.47 thf(fact_206_vimageI2,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A: set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ ( F @ A ) @ A3 )
% 0.30/0.47 => ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI2
% 0.30/0.47 thf(fact_207_vimageI2,axiom,
% 0.30/0.47 ! [F: $o > set_Fi1058188332real_n,A: $o,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ ( F @ A ) @ A3 )
% 0.30/0.47 => ( member_o @ A @ ( vimage961837641real_n @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI2
% 0.30/0.47 thf(fact_208_vimageI2,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > $o,A: set_Fi1058188332real_n,A3: set_o] :
% 0.30/0.47 ( ( member_o @ ( F @ A ) @ A3 )
% 0.30/0.47 => ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI2
% 0.30/0.47 thf(fact_209_vimageI2,axiom,
% 0.30/0.47 ! [F: $o > $o,A: $o,A3: set_o] :
% 0.30/0.47 ( ( member_o @ ( F @ A ) @ A3 )
% 0.30/0.47 => ( member_o @ A @ ( vimage_o_o @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI2
% 0.30/0.47 thf(fact_210_vimageI2,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,A: finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ ( F @ A ) @ A3 )
% 0.30/0.47 => ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageI2
% 0.30/0.47 thf(fact_211_vimageE,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ B2 ) )
% 0.30/0.47 => ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageE
% 0.30/0.47 thf(fact_212_vimageE,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > $o,B2: set_o] :
% 0.30/0.47 ( ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ B2 ) )
% 0.30/0.47 => ( member_o @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageE
% 0.30/0.47 thf(fact_213_vimageE,axiom,
% 0.30/0.47 ! [A: $o,F: $o > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 0.30/0.47 ( ( member_o @ A @ ( vimage961837641real_n @ F @ B2 ) )
% 0.30/0.47 => ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageE
% 0.30/0.47 thf(fact_214_vimageE,axiom,
% 0.30/0.47 ! [A: $o,F: $o > $o,B2: set_o] :
% 0.30/0.47 ( ( member_o @ A @ ( vimage_o_o @ F @ B2 ) )
% 0.30/0.47 => ( member_o @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageE
% 0.30/0.47 thf(fact_215_vimageE,axiom,
% 0.30/0.47 ! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ B2 ) )
% 0.30/0.47 => ( member1352538125real_n @ ( F @ A ) @ B2 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageE
% 0.30/0.47 thf(fact_216_vimageD,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ A3 ) )
% 0.30/0.47 => ( member223413699real_n @ ( F @ A ) @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageD
% 0.30/0.47 thf(fact_217_vimageD,axiom,
% 0.30/0.47 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > $o,A3: set_o] :
% 0.30/0.47 ( ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ A3 ) )
% 0.30/0.47 => ( member_o @ ( F @ A ) @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageD
% 0.30/0.47 thf(fact_218_vimageD,axiom,
% 0.30/0.47 ! [A: $o,F: $o > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( member_o @ A @ ( vimage961837641real_n @ F @ A3 ) )
% 0.30/0.47 => ( member223413699real_n @ ( F @ A ) @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageD
% 0.30/0.47 thf(fact_219_vimageD,axiom,
% 0.30/0.47 ! [A: $o,F: $o > $o,A3: set_o] :
% 0.30/0.47 ( ( member_o @ A @ ( vimage_o_o @ F @ A3 ) )
% 0.30/0.47 => ( member_o @ ( F @ A ) @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageD
% 0.30/0.47 thf(fact_220_vimageD,axiom,
% 0.30/0.47 ! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ A3 ) )
% 0.30/0.47 => ( member1352538125real_n @ ( F @ A ) @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimageD
% 0.30/0.47 thf(fact_221_Sup_OSUP__identity__eq,axiom,
% 0.30/0.47 ! [Sup: set_Fi1058188332real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( Sup
% 0.30/0.47 @ ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : X
% 0.30/0.47 @ A3 ) )
% 0.30/0.47 = ( Sup @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Sup.SUP_identity_eq
% 0.30/0.47 thf(fact_222_Inf_OINF__identity__eq,axiom,
% 0.30/0.47 ! [Inf: set_Fi1058188332real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( Inf
% 0.30/0.47 @ ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : X
% 0.30/0.47 @ A3 ) )
% 0.30/0.47 = ( Inf @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % Inf.INF_identity_eq
% 0.30/0.47 thf(fact_223_Compr__image__eq,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ ( image_439535603real_n @ F @ A3 ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( image_439535603real_n @ F
% 0.30/0.47 @ ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ A3 )
% 0.30/0.47 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compr_image_eq
% 0.30/0.47 thf(fact_224_Compr__image__eq,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,A3: set_nat,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( image_1856576259real_n @ F @ A3 ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( image_1856576259real_n @ F
% 0.30/0.47 @ ( collect_nat
% 0.30/0.47 @ ^ [X: nat] :
% 0.30/0.47 ( ( member_nat @ X @ A3 )
% 0.30/0.47 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compr_image_eq
% 0.30/0.47 thf(fact_225_Compr__image__eq,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( image_355963305real_n @ F @ A3 ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( image_355963305real_n @ F
% 0.30/0.47 @ ( collec1941932235_int_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X @ A3 )
% 0.30/0.47 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compr_image_eq
% 0.30/0.47 thf(fact_226_Compr__image__eq,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( image_1661509983real_n @ F @ A3 ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( image_1661509983real_n @ F
% 0.30/0.47 @ ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ A3 )
% 0.30/0.47 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compr_image_eq
% 0.30/0.47 thf(fact_227_Compr__image__eq,axiom,
% 0.30/0.47 ! [F: $o > set_Fi1058188332real_n,A3: set_o,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ ( image_1759008383real_n @ F @ A3 ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( image_1759008383real_n @ F
% 0.30/0.47 @ ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ A3 )
% 0.30/0.47 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compr_image_eq
% 0.30/0.47 thf(fact_228_Compr__image__eq,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > $o,A3: set_se2111327970real_n,P: $o > $o] :
% 0.30/0.47 ( ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ ( image_1648361637al_n_o @ F @ A3 ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( image_1648361637al_n_o @ F
% 0.30/0.47 @ ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ A3 )
% 0.30/0.47 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compr_image_eq
% 0.30/0.47 thf(fact_229_Compr__image__eq,axiom,
% 0.30/0.47 ! [F: $o > $o,A3: set_o,P: $o > $o] :
% 0.30/0.47 ( ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ ( image_o_o @ F @ A3 ) )
% 0.30/0.47 & ( P @ X ) ) )
% 0.30/0.47 = ( image_o_o @ F
% 0.30/0.47 @ ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ A3 )
% 0.30/0.47 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Compr_image_eq
% 0.30/0.47 thf(fact_230_image__image,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: nat > set_Fi1058188332real_n,A3: set_nat] :
% 0.30/0.47 ( ( image_1661509983real_n @ F @ ( image_1856576259real_n @ G @ A3 ) )
% 0.30/0.47 = ( image_1856576259real_n
% 0.30/0.47 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.30/0.47 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_image
% 0.30/0.47 thf(fact_231_image__image,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( image_1661509983real_n @ F @ ( image_355963305real_n @ G @ A3 ) )
% 0.30/0.47 = ( image_355963305real_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 0.30/0.47 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_image
% 0.30/0.47 thf(fact_232_image__image,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,G: nat > nat,A3: set_nat] :
% 0.30/0.47 ( ( image_1856576259real_n @ F @ ( image_nat_nat @ G @ A3 ) )
% 0.30/0.47 = ( image_1856576259real_n
% 0.30/0.47 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.30/0.47 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_image
% 0.30/0.47 thf(fact_233_image__image,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,G: finite964658038_int_n > nat,A3: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( image_1856576259real_n @ F @ ( image_497739341_n_nat @ G @ A3 ) )
% 0.30/0.47 = ( image_355963305real_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 0.30/0.47 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_image
% 0.30/0.47 thf(fact_234_image__image,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_439535603real_n @ F @ ( image_439535603real_n @ G @ A3 ) )
% 0.30/0.47 = ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( F @ ( G @ X ) )
% 0.30/0.47 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_image
% 0.30/0.47 thf(fact_235_image__image,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,G: nat > finite964658038_int_n,A3: set_nat] :
% 0.30/0.47 ( ( image_355963305real_n @ F @ ( image_1961920973_int_n @ G @ A3 ) )
% 0.30/0.47 = ( image_1856576259real_n
% 0.30/0.47 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.30/0.47 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_image
% 0.30/0.47 thf(fact_236_image__image,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,G: finite964658038_int_n > finite964658038_int_n,A3: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( image_355963305real_n @ F @ ( image_1278151539_int_n @ G @ A3 ) )
% 0.30/0.47 = ( image_355963305real_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 0.30/0.47 @ A3 ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_image
% 0.30/0.47 thf(fact_237_imageE,axiom,
% 0.30/0.47 ! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A3 ) )
% 0.30/0.47 => ~ ! [X2: finite1489363574real_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X2 ) )
% 0.30/0.47 => ~ ( member1352538125real_n @ X2 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageE
% 0.30/0.47 thf(fact_238_imageE,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,A3: set_nat] :
% 0.30/0.47 ( ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ A3 ) )
% 0.30/0.47 => ~ ! [X2: nat] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X2 ) )
% 0.30/0.47 => ~ ( member_nat @ X2 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageE
% 0.30/0.47 thf(fact_239_imageE,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( member223413699real_n @ B @ ( image_355963305real_n @ F @ A3 ) )
% 0.30/0.47 => ~ ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X2 ) )
% 0.30/0.47 => ~ ( member27055245_int_n @ X2 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageE
% 0.30/0.47 thf(fact_240_imageE,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A3 ) )
% 0.30/0.47 => ~ ! [X2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X2 ) )
% 0.30/0.47 => ~ ( member223413699real_n @ X2 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageE
% 0.30/0.47 thf(fact_241_imageE,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: $o > set_Fi1058188332real_n,A3: set_o] :
% 0.30/0.47 ( ( member223413699real_n @ B @ ( image_1759008383real_n @ F @ A3 ) )
% 0.30/0.47 => ~ ! [X2: $o] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X2 ) )
% 0.30/0.47 => ~ ( member_o @ X2 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageE
% 0.30/0.47 thf(fact_242_imageE,axiom,
% 0.30/0.47 ! [B: $o,F: set_Fi1058188332real_n > $o,A3: set_se2111327970real_n] :
% 0.30/0.47 ( ( member_o @ B @ ( image_1648361637al_n_o @ F @ A3 ) )
% 0.30/0.47 => ~ ! [X2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X2 ) )
% 0.30/0.47 => ~ ( member223413699real_n @ X2 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageE
% 0.30/0.47 thf(fact_243_imageE,axiom,
% 0.30/0.47 ! [B: $o,F: $o > $o,A3: set_o] :
% 0.30/0.47 ( ( member_o @ B @ ( image_o_o @ F @ A3 ) )
% 0.30/0.47 => ~ ! [X2: $o] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X2 ) )
% 0.30/0.47 => ~ ( member_o @ X2 @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageE
% 0.30/0.47 thf(fact_244_UNIV__def,axiom,
% 0.30/0.47 ( top_top_set_nat
% 0.30/0.47 = ( collect_nat
% 0.30/0.47 @ ^ [X: nat] : $true ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_def
% 0.30/0.47 thf(fact_245_UNIV__def,axiom,
% 0.30/0.47 ( top_to131672412_int_n
% 0.30/0.47 = ( collec1941932235_int_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : $true ) ) ).
% 0.30/0.47
% 0.30/0.47 % UNIV_def
% 0.30/0.47 thf(fact_246_Collect__conj__eq,axiom,
% 0.30/0.47 ! [P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( P @ X )
% 0.30/0.47 & ( Q @ X ) ) )
% 0.30/0.47 = ( inf_in1974387902real_n @ ( collec321817931real_n @ P ) @ ( collec321817931real_n @ Q ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Collect_conj_eq
% 0.30/0.47 thf(fact_247_Int__Collect,axiom,
% 0.30/0.47 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,P: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( member223413699real_n @ X3 @ ( inf_in632889204real_n @ A3 @ ( collec452821761real_n @ P ) ) )
% 0.30/0.47 = ( ( member223413699real_n @ X3 @ A3 )
% 0.30/0.47 & ( P @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_Collect
% 0.30/0.47 thf(fact_248_Int__Collect,axiom,
% 0.30/0.47 ! [X3: $o,A3: set_o,P: $o > $o] :
% 0.30/0.47 ( ( member_o @ X3 @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
% 0.30/0.47 = ( ( member_o @ X3 @ A3 )
% 0.30/0.47 & ( P @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_Collect
% 0.30/0.47 thf(fact_249_Int__Collect,axiom,
% 0.30/0.47 ! [X3: finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
% 0.30/0.47 ( ( member1352538125real_n @ X3 @ ( inf_in1974387902real_n @ A3 @ ( collec321817931real_n @ P ) ) )
% 0.30/0.47 = ( ( member1352538125real_n @ X3 @ A3 )
% 0.30/0.47 & ( P @ X3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_Collect
% 0.30/0.47 thf(fact_250_Int__def,axiom,
% 0.30/0.47 ( inf_in632889204real_n
% 0.30/0.47 = ( ^ [A4: set_se2111327970real_n,B3: set_se2111327970real_n] :
% 0.30/0.47 ( collec452821761real_n
% 0.30/0.47 @ ^ [X: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X @ A4 )
% 0.30/0.47 & ( member223413699real_n @ X @ B3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_def
% 0.30/0.47 thf(fact_251_Int__def,axiom,
% 0.30/0.47 ( inf_inf_set_o
% 0.30/0.47 = ( ^ [A4: set_o,B3: set_o] :
% 0.30/0.47 ( collect_o
% 0.30/0.47 @ ^ [X: $o] :
% 0.30/0.47 ( ( member_o @ X @ A4 )
% 0.30/0.47 & ( member_o @ X @ B3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_def
% 0.30/0.47 thf(fact_252_Int__def,axiom,
% 0.30/0.47 ( inf_in1974387902real_n
% 0.30/0.47 = ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] :
% 0.30/0.47 ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X @ A4 )
% 0.30/0.47 & ( member1352538125real_n @ X @ B3 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % Int_def
% 0.30/0.47 thf(fact_253_vimage__def,axiom,
% 0.30/0.47 ( vimage1233683625real_n
% 0.30/0.47 = ( ^ [F2: finite1489363574real_n > finite1489363574real_n,B3: set_Fi1058188332real_n] :
% 0.30/0.47 ( collec321817931real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ ( F2 @ X ) @ B3 ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_def
% 0.30/0.47 thf(fact_254_less__numeral__extra_I3_J,axiom,
% 0.30/0.47 ~ ( ord_le2133614988nnreal @ zero_z1963244097nnreal @ zero_z1963244097nnreal ) ).
% 0.30/0.47
% 0.30/0.47 % less_numeral_extra(3)
% 0.30/0.47 thf(fact_255_less__numeral__extra_I4_J,axiom,
% 0.30/0.47 ~ ( ord_le2133614988nnreal @ one_on705384445nnreal @ one_on705384445nnreal ) ).
% 0.30/0.47
% 0.30/0.47 % less_numeral_extra(4)
% 0.30/0.47 thf(fact_256_range__eqI,axiom,
% 0.30/0.47 ! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,X3: finite1489363574real_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ top_to1292442332real_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_eqI
% 0.30/0.47 thf(fact_257_range__eqI,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,X3: nat] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_eqI
% 0.30/0.47 thf(fact_258_range__eqI,axiom,
% 0.30/0.47 ! [B: $o,F: nat > $o,X3: nat] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_eqI
% 0.30/0.47 thf(fact_259_range__eqI,axiom,
% 0.30/0.47 ! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,X3: finite964658038_int_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member223413699real_n @ B @ ( image_355963305real_n @ F @ top_to131672412_int_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_eqI
% 0.30/0.47 thf(fact_260_range__eqI,axiom,
% 0.30/0.47 ! [B: $o,F: finite964658038_int_n > $o,X3: finite964658038_int_n] :
% 0.30/0.47 ( ( B
% 0.30/0.47 = ( F @ X3 ) )
% 0.30/0.47 => ( member_o @ B @ ( image_216309723nt_n_o @ F @ top_to131672412_int_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_eqI
% 0.30/0.47 thf(fact_261_surj__def,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n] :
% 0.30/0.47 ( ( ( image_439535603real_n @ F @ top_to1292442332real_n )
% 0.30/0.47 = top_to1292442332real_n )
% 0.30/0.47 = ( ! [Y2: finite1489363574real_n] :
% 0.30/0.47 ? [X: finite1489363574real_n] :
% 0.30/0.47 ( Y2
% 0.30/0.47 = ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_def
% 0.30/0.47 thf(fact_262_surj__def,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( ( image_1856576259real_n @ F @ top_top_set_nat )
% 0.30/0.47 = top_to20708754real_n )
% 0.30/0.47 = ( ! [Y2: set_Fi1058188332real_n] :
% 0.30/0.47 ? [X: nat] :
% 0.30/0.47 ( Y2
% 0.30/0.47 = ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_def
% 0.30/0.47 thf(fact_263_surj__def,axiom,
% 0.30/0.47 ! [F: nat > nat] :
% 0.30/0.47 ( ( ( image_nat_nat @ F @ top_top_set_nat )
% 0.30/0.47 = top_top_set_nat )
% 0.30/0.47 = ( ! [Y2: nat] :
% 0.30/0.47 ? [X: nat] :
% 0.30/0.47 ( Y2
% 0.30/0.47 = ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_def
% 0.30/0.47 thf(fact_264_surj__def,axiom,
% 0.30/0.47 ! [F: nat > finite964658038_int_n] :
% 0.30/0.47 ( ( ( image_1961920973_int_n @ F @ top_top_set_nat )
% 0.30/0.47 = top_to131672412_int_n )
% 0.30/0.47 = ( ! [Y2: finite964658038_int_n] :
% 0.30/0.47 ? [X: nat] :
% 0.30/0.47 ( Y2
% 0.30/0.47 = ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_def
% 0.30/0.47 thf(fact_265_surj__def,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( ( image_355963305real_n @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_to20708754real_n )
% 0.30/0.47 = ( ! [Y2: set_Fi1058188332real_n] :
% 0.30/0.47 ? [X: finite964658038_int_n] :
% 0.30/0.47 ( Y2
% 0.30/0.47 = ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_def
% 0.30/0.47 thf(fact_266_surj__def,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > nat] :
% 0.30/0.47 ( ( ( image_497739341_n_nat @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_top_set_nat )
% 0.30/0.47 = ( ! [Y2: nat] :
% 0.30/0.47 ? [X: finite964658038_int_n] :
% 0.30/0.47 ( Y2
% 0.30/0.47 = ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_def
% 0.30/0.47 thf(fact_267_surj__def,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > finite964658038_int_n] :
% 0.30/0.47 ( ( ( image_1278151539_int_n @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_to131672412_int_n )
% 0.30/0.47 = ( ! [Y2: finite964658038_int_n] :
% 0.30/0.47 ? [X: finite964658038_int_n] :
% 0.30/0.47 ( Y2
% 0.30/0.47 = ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surj_def
% 0.30/0.47 thf(fact_268_rangeI,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,X3: finite1489363574real_n] : ( member1352538125real_n @ ( F @ X3 ) @ ( image_439535603real_n @ F @ top_to1292442332real_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % rangeI
% 0.30/0.47 thf(fact_269_rangeI,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,X3: nat] : ( member223413699real_n @ ( F @ X3 ) @ ( image_1856576259real_n @ F @ top_top_set_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % rangeI
% 0.30/0.47 thf(fact_270_rangeI,axiom,
% 0.30/0.47 ! [F: nat > $o,X3: nat] : ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % rangeI
% 0.30/0.47 thf(fact_271_rangeI,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,X3: finite964658038_int_n] : ( member223413699real_n @ ( F @ X3 ) @ ( image_355963305real_n @ F @ top_to131672412_int_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % rangeI
% 0.30/0.47 thf(fact_272_rangeI,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > $o,X3: finite964658038_int_n] : ( member_o @ ( F @ X3 ) @ ( image_216309723nt_n_o @ F @ top_to131672412_int_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % rangeI
% 0.30/0.47 thf(fact_273_surjI,axiom,
% 0.30/0.47 ! [G: finite1489363574real_n > finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
% 0.30/0.47 ( ! [X2: finite1489363574real_n] :
% 0.30/0.47 ( ( G @ ( F @ X2 ) )
% 0.30/0.47 = X2 )
% 0.30/0.47 => ( ( image_439535603real_n @ G @ top_to1292442332real_n )
% 0.30/0.47 = top_to1292442332real_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjI
% 0.30/0.47 thf(fact_274_surjI,axiom,
% 0.30/0.47 ! [G: nat > set_Fi1058188332real_n,F: set_Fi1058188332real_n > nat] :
% 0.30/0.47 ( ! [X2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( G @ ( F @ X2 ) )
% 0.30/0.47 = X2 )
% 0.30/0.47 => ( ( image_1856576259real_n @ G @ top_top_set_nat )
% 0.30/0.47 = top_to20708754real_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjI
% 0.30/0.47 thf(fact_275_surjI,axiom,
% 0.30/0.47 ! [G: nat > nat,F: nat > nat] :
% 0.30/0.47 ( ! [X2: nat] :
% 0.30/0.47 ( ( G @ ( F @ X2 ) )
% 0.30/0.47 = X2 )
% 0.30/0.47 => ( ( image_nat_nat @ G @ top_top_set_nat )
% 0.30/0.47 = top_top_set_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjI
% 0.30/0.47 thf(fact_276_surjI,axiom,
% 0.30/0.47 ! [G: nat > finite964658038_int_n,F: finite964658038_int_n > nat] :
% 0.30/0.47 ( ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( ( G @ ( F @ X2 ) )
% 0.30/0.47 = X2 )
% 0.30/0.47 => ( ( image_1961920973_int_n @ G @ top_top_set_nat )
% 0.30/0.47 = top_to131672412_int_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjI
% 0.30/0.47 thf(fact_277_surjI,axiom,
% 0.30/0.47 ! [G: finite964658038_int_n > set_Fi1058188332real_n,F: set_Fi1058188332real_n > finite964658038_int_n] :
% 0.30/0.47 ( ! [X2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( G @ ( F @ X2 ) )
% 0.30/0.47 = X2 )
% 0.30/0.47 => ( ( image_355963305real_n @ G @ top_to131672412_int_n )
% 0.30/0.47 = top_to20708754real_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjI
% 0.30/0.47 thf(fact_278_surjI,axiom,
% 0.30/0.47 ! [G: finite964658038_int_n > nat,F: nat > finite964658038_int_n] :
% 0.30/0.47 ( ! [X2: nat] :
% 0.30/0.47 ( ( G @ ( F @ X2 ) )
% 0.30/0.47 = X2 )
% 0.30/0.47 => ( ( image_497739341_n_nat @ G @ top_to131672412_int_n )
% 0.30/0.47 = top_top_set_nat ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjI
% 0.30/0.47 thf(fact_279_surjI,axiom,
% 0.30/0.47 ! [G: finite964658038_int_n > finite964658038_int_n,F: finite964658038_int_n > finite964658038_int_n] :
% 0.30/0.47 ( ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( ( G @ ( F @ X2 ) )
% 0.30/0.47 = X2 )
% 0.30/0.47 => ( ( image_1278151539_int_n @ G @ top_to131672412_int_n )
% 0.30/0.47 = top_to131672412_int_n ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjI
% 0.30/0.47 thf(fact_280_surjE,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,Y4: finite1489363574real_n] :
% 0.30/0.47 ( ( ( image_439535603real_n @ F @ top_to1292442332real_n )
% 0.30/0.47 = top_to1292442332real_n )
% 0.30/0.47 => ~ ! [X2: finite1489363574real_n] :
% 0.30/0.47 ( Y4
% 0.30/0.47 != ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjE
% 0.30/0.47 thf(fact_281_surjE,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( ( image_1856576259real_n @ F @ top_top_set_nat )
% 0.30/0.47 = top_to20708754real_n )
% 0.30/0.47 => ~ ! [X2: nat] :
% 0.30/0.47 ( Y4
% 0.30/0.47 != ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjE
% 0.30/0.47 thf(fact_282_surjE,axiom,
% 0.30/0.47 ! [F: nat > nat,Y4: nat] :
% 0.30/0.47 ( ( ( image_nat_nat @ F @ top_top_set_nat )
% 0.30/0.47 = top_top_set_nat )
% 0.30/0.47 => ~ ! [X2: nat] :
% 0.30/0.47 ( Y4
% 0.30/0.47 != ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjE
% 0.30/0.47 thf(fact_283_surjE,axiom,
% 0.30/0.47 ! [F: nat > finite964658038_int_n,Y4: finite964658038_int_n] :
% 0.30/0.47 ( ( ( image_1961920973_int_n @ F @ top_top_set_nat )
% 0.30/0.47 = top_to131672412_int_n )
% 0.30/0.47 => ~ ! [X2: nat] :
% 0.30/0.47 ( Y4
% 0.30/0.47 != ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjE
% 0.30/0.47 thf(fact_284_surjE,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( ( image_355963305real_n @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_to20708754real_n )
% 0.30/0.47 => ~ ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( Y4
% 0.30/0.47 != ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjE
% 0.30/0.47 thf(fact_285_surjE,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > nat,Y4: nat] :
% 0.30/0.47 ( ( ( image_497739341_n_nat @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_top_set_nat )
% 0.30/0.47 => ~ ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( Y4
% 0.30/0.47 != ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjE
% 0.30/0.47 thf(fact_286_surjE,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > finite964658038_int_n,Y4: finite964658038_int_n] :
% 0.30/0.47 ( ( ( image_1278151539_int_n @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_to131672412_int_n )
% 0.30/0.47 => ~ ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( Y4
% 0.30/0.47 != ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjE
% 0.30/0.47 thf(fact_287_surjD,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,Y4: finite1489363574real_n] :
% 0.30/0.47 ( ( ( image_439535603real_n @ F @ top_to1292442332real_n )
% 0.30/0.47 = top_to1292442332real_n )
% 0.30/0.47 => ? [X2: finite1489363574real_n] :
% 0.30/0.47 ( Y4
% 0.30/0.47 = ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjD
% 0.30/0.47 thf(fact_288_surjD,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( ( image_1856576259real_n @ F @ top_top_set_nat )
% 0.30/0.47 = top_to20708754real_n )
% 0.30/0.47 => ? [X2: nat] :
% 0.30/0.47 ( Y4
% 0.30/0.47 = ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjD
% 0.30/0.47 thf(fact_289_surjD,axiom,
% 0.30/0.47 ! [F: nat > nat,Y4: nat] :
% 0.30/0.47 ( ( ( image_nat_nat @ F @ top_top_set_nat )
% 0.30/0.47 = top_top_set_nat )
% 0.30/0.47 => ? [X2: nat] :
% 0.30/0.47 ( Y4
% 0.30/0.47 = ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjD
% 0.30/0.47 thf(fact_290_surjD,axiom,
% 0.30/0.47 ! [F: nat > finite964658038_int_n,Y4: finite964658038_int_n] :
% 0.30/0.47 ( ( ( image_1961920973_int_n @ F @ top_top_set_nat )
% 0.30/0.47 = top_to131672412_int_n )
% 0.30/0.47 => ? [X2: nat] :
% 0.30/0.47 ( Y4
% 0.30/0.47 = ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjD
% 0.30/0.47 thf(fact_291_surjD,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( ( image_355963305real_n @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_to20708754real_n )
% 0.30/0.47 => ? [X2: finite964658038_int_n] :
% 0.30/0.47 ( Y4
% 0.30/0.47 = ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjD
% 0.30/0.47 thf(fact_292_surjD,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > nat,Y4: nat] :
% 0.30/0.47 ( ( ( image_497739341_n_nat @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_top_set_nat )
% 0.30/0.47 => ? [X2: finite964658038_int_n] :
% 0.30/0.47 ( Y4
% 0.30/0.47 = ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjD
% 0.30/0.47 thf(fact_293_surjD,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > finite964658038_int_n,Y4: finite964658038_int_n] :
% 0.30/0.47 ( ( ( image_1278151539_int_n @ F @ top_to131672412_int_n )
% 0.30/0.47 = top_to131672412_int_n )
% 0.30/0.47 => ? [X2: finite964658038_int_n] :
% 0.30/0.47 ( Y4
% 0.30/0.47 = ( F @ X2 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % surjD
% 0.30/0.47 thf(fact_294_less__Sup__iff,axiom,
% 0.30/0.47 ! [A: extend1728876344nnreal,S: set_Ex113815278nnreal] :
% 0.30/0.47 ( ( ord_le2133614988nnreal @ A @ ( comple1413366923nnreal @ S ) )
% 0.30/0.47 = ( ? [X: extend1728876344nnreal] :
% 0.30/0.47 ( ( member1217042383nnreal @ X @ S )
% 0.30/0.47 & ( ord_le2133614988nnreal @ A @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % less_Sup_iff
% 0.30/0.47 thf(fact_295_SUP__cong,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: finite1489363574real_n > finite1489363574real_n,D: finite1489363574real_n > finite1489363574real_n] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( comple2042271945real_n @ ( image_439535603real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( comple2042271945real_n @ ( image_439535603real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_cong
% 0.30/0.47 thf(fact_296_SUP__cong,axiom,
% 0.30/0.47 ! [A3: set_nat,B2: set_nat,C2: nat > set_Fi1058188332real_n,D: nat > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: nat] :
% 0.30/0.47 ( ( member_nat @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( comple825005695real_n @ ( image_1856576259real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_cong
% 0.30/0.47 thf(fact_297_SUP__cong,axiom,
% 0.30/0.47 ! [A3: set_Fi160064172_int_n,B2: set_Fi160064172_int_n,C2: finite964658038_int_n > set_Fi1058188332real_n,D: finite964658038_int_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: finite964658038_int_n] :
% 0.30/0.47 ( ( member27055245_int_n @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( comple825005695real_n @ ( image_355963305real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( comple825005695real_n @ ( image_355963305real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_cong
% 0.30/0.47 thf(fact_298_SUP__cong,axiom,
% 0.30/0.47 ! [A3: set_se2111327970real_n,B2: set_se2111327970real_n,C2: set_Fi1058188332real_n > set_Fi1058188332real_n,D: set_Fi1058188332real_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( comple825005695real_n @ ( image_1661509983real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( comple825005695real_n @ ( image_1661509983real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_cong
% 0.30/0.47 thf(fact_299_SUP__cong,axiom,
% 0.30/0.47 ! [A3: set_o,B2: set_o,C2: $o > set_Fi1058188332real_n,D: $o > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: $o] :
% 0.30/0.47 ( ( member_o @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( comple825005695real_n @ ( image_1759008383real_n @ C2 @ A3 ) )
% 0.30/0.47 = ( comple825005695real_n @ ( image_1759008383real_n @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_cong
% 0.30/0.47 thf(fact_300_SUP__cong,axiom,
% 0.30/0.47 ! [A3: set_se2111327970real_n,B2: set_se2111327970real_n,C2: set_Fi1058188332real_n > $o,D: set_Fi1058188332real_n > $o] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( member223413699real_n @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( complete_Sup_Sup_o @ ( image_1648361637al_n_o @ C2 @ A3 ) )
% 0.30/0.47 = ( complete_Sup_Sup_o @ ( image_1648361637al_n_o @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_cong
% 0.30/0.47 thf(fact_301_SUP__cong,axiom,
% 0.30/0.47 ! [A3: set_o,B2: set_o,C2: $o > $o,D: $o > $o] :
% 0.30/0.47 ( ( A3 = B2 )
% 0.30/0.47 => ( ! [X2: $o] :
% 0.30/0.47 ( ( member_o @ X2 @ B2 )
% 0.30/0.47 => ( ( C2 @ X2 )
% 0.30/0.47 = ( D @ X2 ) ) )
% 0.30/0.47 => ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A3 ) )
% 0.30/0.47 = ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % SUP_cong
% 0.30/0.47 thf(fact_302_Int__UNIV__right,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ A3 @ top_to1292442332real_n )
% 0.30/0.47 = A3 ) ).
% 0.30/0.47
% 0.30/0.47 % Int_UNIV_right
% 0.30/0.47 thf(fact_303_Int__UNIV__right,axiom,
% 0.30/0.47 ! [A3: set_nat] :
% 0.30/0.47 ( ( inf_inf_set_nat @ A3 @ top_top_set_nat )
% 0.30/0.47 = A3 ) ).
% 0.30/0.47
% 0.30/0.47 % Int_UNIV_right
% 0.30/0.47 thf(fact_304_Int__UNIV__right,axiom,
% 0.30/0.47 ! [A3: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( inf_in1108485182_int_n @ A3 @ top_to131672412_int_n )
% 0.30/0.47 = A3 ) ).
% 0.30/0.47
% 0.30/0.47 % Int_UNIV_right
% 0.30/0.47 thf(fact_305_Int__UNIV__left,axiom,
% 0.30/0.47 ! [B2: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( inf_in1974387902real_n @ top_to1292442332real_n @ B2 )
% 0.30/0.47 = B2 ) ).
% 0.30/0.47
% 0.30/0.47 % Int_UNIV_left
% 0.30/0.47 thf(fact_306_Int__UNIV__left,axiom,
% 0.30/0.47 ! [B2: set_nat] :
% 0.30/0.47 ( ( inf_inf_set_nat @ top_top_set_nat @ B2 )
% 0.30/0.47 = B2 ) ).
% 0.30/0.47
% 0.30/0.47 % Int_UNIV_left
% 0.30/0.47 thf(fact_307_Int__UNIV__left,axiom,
% 0.30/0.47 ! [B2: set_Fi160064172_int_n] :
% 0.30/0.47 ( ( inf_in1108485182_int_n @ top_to131672412_int_n @ B2 )
% 0.30/0.47 = B2 ) ).
% 0.30/0.47
% 0.30/0.47 % Int_UNIV_left
% 0.30/0.47 thf(fact_308_Union__UNIV,axiom,
% 0.30/0.47 ( ( comple1682161881et_nat @ top_top_set_set_nat )
% 0.30/0.47 = top_top_set_nat ) ).
% 0.30/0.47
% 0.30/0.47 % Union_UNIV
% 0.30/0.47 thf(fact_309_Union__UNIV,axiom,
% 0.30/0.47 ( ( comple970917503_int_n @ top_to1587634578_int_n )
% 0.30/0.47 = top_to131672412_int_n ) ).
% 0.30/0.47
% 0.30/0.47 % Union_UNIV
% 0.30/0.47 thf(fact_310_Union__UNIV,axiom,
% 0.30/0.47 ( ( comple825005695real_n @ top_to20708754real_n )
% 0.30/0.47 = top_to1292442332real_n ) ).
% 0.30/0.47
% 0.30/0.47 % Union_UNIV
% 0.30/0.47 thf(fact_311_null__setsD2,axiom,
% 0.30/0.47 ! [A3: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 0.30/0.47 ( ( member223413699real_n @ A3 @ ( measur1402256771real_n @ M ) )
% 0.30/0.47 => ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % null_setsD2
% 0.30/0.47 thf(fact_312_vimage__inter__cong,axiom,
% 0.30/0.47 ! [S: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,Y4: set_Fi1058188332real_n] :
% 0.30/0.47 ( ! [W: finite1489363574real_n] :
% 0.30/0.47 ( ( member1352538125real_n @ W @ S )
% 0.30/0.47 => ( ( F @ W )
% 0.30/0.47 = ( G @ W ) ) )
% 0.30/0.47 => ( ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ Y4 ) @ S )
% 0.30/0.47 = ( inf_in1974387902real_n @ ( vimage1233683625real_n @ G @ Y4 ) @ S ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % vimage_inter_cong
% 0.30/0.47 thf(fact_313_translation__subtract__diff,axiom,
% 0.30/0.47 ! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T: set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
% 0.30/0.47 @ ( minus_1686442501real_n @ S2 @ T ) )
% 0.30/0.47 = ( minus_1686442501real_n
% 0.30/0.47 @ ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
% 0.30/0.47 @ S2 )
% 0.30/0.47 @ ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
% 0.30/0.47 @ T ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % translation_subtract_diff
% 0.30/0.47 thf(fact_314_range__composition,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n] :
% 0.30/0.47 ( ( image_439535603real_n
% 0.30/0.47 @ ^ [X: finite1489363574real_n] : ( F @ ( G @ X ) )
% 0.30/0.47 @ top_to1292442332real_n )
% 0.30/0.47 = ( image_439535603real_n @ F @ ( image_439535603real_n @ G @ top_to1292442332real_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_315_range__composition,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,G: nat > finite1489363574real_n] :
% 0.30/0.47 ( ( image_183184717real_n
% 0.30/0.47 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.30/0.47 @ top_top_set_nat )
% 0.30/0.47 = ( image_439535603real_n @ F @ ( image_183184717real_n @ G @ top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_316_range__composition,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: nat > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_1856576259real_n
% 0.30/0.47 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.30/0.47 @ top_top_set_nat )
% 0.30/0.47 = ( image_1661509983real_n @ F @ ( image_1856576259real_n @ G @ top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_317_range__composition,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,G: nat > nat] :
% 0.30/0.47 ( ( image_1856576259real_n
% 0.30/0.47 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.30/0.47 @ top_top_set_nat )
% 0.30/0.47 = ( image_1856576259real_n @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_318_range__composition,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,G: nat > finite964658038_int_n] :
% 0.30/0.47 ( ( image_1856576259real_n
% 0.30/0.47 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.30/0.47 @ top_top_set_nat )
% 0.30/0.47 = ( image_355963305real_n @ F @ ( image_1961920973_int_n @ G @ top_top_set_nat ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_319_range__composition,axiom,
% 0.30/0.47 ! [F: finite1489363574real_n > finite1489363574real_n,G: finite964658038_int_n > finite1489363574real_n] :
% 0.30/0.47 ( ( image_2058828787real_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 0.30/0.47 @ top_to131672412_int_n )
% 0.30/0.47 = ( image_439535603real_n @ F @ ( image_2058828787real_n @ G @ top_to131672412_int_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_320_range__composition,axiom,
% 0.30/0.47 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: finite964658038_int_n > set_Fi1058188332real_n] :
% 0.30/0.47 ( ( image_355963305real_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 0.30/0.47 @ top_to131672412_int_n )
% 0.30/0.47 = ( image_1661509983real_n @ F @ ( image_355963305real_n @ G @ top_to131672412_int_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_321_range__composition,axiom,
% 0.30/0.47 ! [F: nat > set_Fi1058188332real_n,G: finite964658038_int_n > nat] :
% 0.30/0.47 ( ( image_355963305real_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 0.30/0.47 @ top_to131672412_int_n )
% 0.30/0.47 = ( image_1856576259real_n @ F @ ( image_497739341_n_nat @ G @ top_to131672412_int_n ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % range_composition
% 0.30/0.47 thf(fact_322_range__composition,axiom,
% 0.30/0.47 ! [F: finite964658038_int_n > set_Fi1058188332real_n,G: finite964658038_int_n > finite964658038_int_n] :
% 0.30/0.47 ( ( image_355963305real_n
% 0.30/0.47 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 0.30/0.48 @ top_to131672412_int_n )
% 0.30/0.48 = ( image_355963305real_n @ F @ ( image_1278151539_int_n @ G @ top_to131672412_int_n ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % range_composition
% 0.30/0.48 thf(fact_323_rangeE,axiom,
% 0.30/0.48 ! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
% 0.30/0.48 ( ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ top_to1292442332real_n ) )
% 0.30/0.48 => ~ ! [X2: finite1489363574real_n] :
% 0.30/0.48 ( B
% 0.30/0.48 != ( F @ X2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % rangeE
% 0.30/0.48 thf(fact_324_rangeE,axiom,
% 0.30/0.48 ! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n] :
% 0.30/0.48 ( ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ top_top_set_nat ) )
% 0.30/0.48 => ~ ! [X2: nat] :
% 0.30/0.48 ( B
% 0.30/0.48 != ( F @ X2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % rangeE
% 0.30/0.48 thf(fact_325_rangeE,axiom,
% 0.30/0.48 ! [B: $o,F: nat > $o] :
% 0.30/0.48 ( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
% 0.30/0.48 => ~ ! [X2: nat] :
% 0.30/0.48 ( B
% 0.30/0.48 = ( ~ ( F @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % rangeE
% 0.30/0.48 thf(fact_326_rangeE,axiom,
% 0.30/0.48 ! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n] :
% 0.30/0.48 ( ( member223413699real_n @ B @ ( image_355963305real_n @ F @ top_to131672412_int_n ) )
% 0.30/0.48 => ~ ! [X2: finite964658038_int_n] :
% 0.30/0.48 ( B
% 0.30/0.48 != ( F @ X2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % rangeE
% 0.30/0.48 thf(fact_327_rangeE,axiom,
% 0.30/0.48 ! [B: $o,F: finite964658038_int_n > $o] :
% 0.30/0.48 ( ( member_o @ B @ ( image_216309723nt_n_o @ F @ top_to131672412_int_n ) )
% 0.30/0.48 => ~ ! [X2: finite964658038_int_n] :
% 0.30/0.48 ( B
% 0.30/0.48 = ( ~ ( F @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % rangeE
% 0.30/0.48 thf(fact_328_SUP__commute,axiom,
% 0.30/0.48 ! [F: nat > nat > set_Fi1058188332real_n,B2: set_nat,A3: set_nat] :
% 0.30/0.48 ( ( comple825005695real_n
% 0.30/0.48 @ ( image_1856576259real_n
% 0.30/0.48 @ ^ [I4: nat] : ( comple825005695real_n @ ( image_1856576259real_n @ ( F @ I4 ) @ B2 ) )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_1856576259real_n
% 0.30/0.48 @ ^ [J: nat] :
% 0.30/0.48 ( comple825005695real_n
% 0.30/0.48 @ ( image_1856576259real_n
% 0.30/0.48 @ ^ [I4: nat] : ( F @ I4 @ J )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % SUP_commute
% 0.30/0.48 thf(fact_329_SUP__commute,axiom,
% 0.30/0.48 ! [F: nat > finite964658038_int_n > set_Fi1058188332real_n,B2: set_Fi160064172_int_n,A3: set_nat] :
% 0.30/0.48 ( ( comple825005695real_n
% 0.30/0.48 @ ( image_1856576259real_n
% 0.30/0.48 @ ^ [I4: nat] : ( comple825005695real_n @ ( image_355963305real_n @ ( F @ I4 ) @ B2 ) )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_355963305real_n
% 0.30/0.48 @ ^ [J: finite964658038_int_n] :
% 0.30/0.48 ( comple825005695real_n
% 0.30/0.48 @ ( image_1856576259real_n
% 0.30/0.48 @ ^ [I4: nat] : ( F @ I4 @ J )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % SUP_commute
% 0.30/0.48 thf(fact_330_SUP__commute,axiom,
% 0.30/0.48 ! [F: finite964658038_int_n > nat > set_Fi1058188332real_n,B2: set_nat,A3: set_Fi160064172_int_n] :
% 0.30/0.48 ( ( comple825005695real_n
% 0.30/0.48 @ ( image_355963305real_n
% 0.30/0.48 @ ^ [I4: finite964658038_int_n] : ( comple825005695real_n @ ( image_1856576259real_n @ ( F @ I4 ) @ B2 ) )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_1856576259real_n
% 0.30/0.48 @ ^ [J: nat] :
% 0.30/0.48 ( comple825005695real_n
% 0.30/0.48 @ ( image_355963305real_n
% 0.30/0.48 @ ^ [I4: finite964658038_int_n] : ( F @ I4 @ J )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % SUP_commute
% 0.30/0.48 thf(fact_331_SUP__commute,axiom,
% 0.30/0.48 ! [F: finite964658038_int_n > finite964658038_int_n > set_Fi1058188332real_n,B2: set_Fi160064172_int_n,A3: set_Fi160064172_int_n] :
% 0.30/0.48 ( ( comple825005695real_n
% 0.30/0.48 @ ( image_355963305real_n
% 0.30/0.48 @ ^ [I4: finite964658038_int_n] : ( comple825005695real_n @ ( image_355963305real_n @ ( F @ I4 ) @ B2 ) )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_355963305real_n
% 0.30/0.48 @ ^ [J: finite964658038_int_n] :
% 0.30/0.48 ( comple825005695real_n
% 0.30/0.48 @ ( image_355963305real_n
% 0.30/0.48 @ ^ [I4: finite964658038_int_n] : ( F @ I4 @ J )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 @ B2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % SUP_commute
% 0.30/0.48 thf(fact_332_image__Union,axiom,
% 0.30/0.48 ! [F: nat > set_Fi1058188332real_n,S: set_set_nat] :
% 0.30/0.48 ( ( image_1856576259real_n @ F @ ( comple1682161881et_nat @ S ) )
% 0.30/0.48 = ( comple1917283637real_n @ ( image_1587769199real_n @ ( image_1856576259real_n @ F ) @ S ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % image_Union
% 0.30/0.48 thf(fact_333_image__Union,axiom,
% 0.30/0.48 ! [F: finite964658038_int_n > set_Fi1058188332real_n,S: set_se944069346_int_n] :
% 0.30/0.48 ( ( image_355963305real_n @ F @ ( comple970917503_int_n @ S ) )
% 0.30/0.48 = ( comple1917283637real_n @ ( image_1054146965real_n @ ( image_355963305real_n @ F ) @ S ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % image_Union
% 0.30/0.48 thf(fact_334_image__Union,axiom,
% 0.30/0.48 ! [F: finite1489363574real_n > finite1489363574real_n,S: set_se2111327970real_n] :
% 0.30/0.48 ( ( image_439535603real_n @ F @ ( comple825005695real_n @ S ) )
% 0.30/0.48 = ( comple825005695real_n @ ( image_1661509983real_n @ ( image_439535603real_n @ F ) @ S ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % image_Union
% 0.30/0.48 thf(fact_335_UN__UN__flatten,axiom,
% 0.30/0.48 ! [C2: nat > set_Fi1058188332real_n,B2: nat > set_nat,A3: set_nat] :
% 0.30/0.48 ( ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( comple1682161881et_nat @ ( image_nat_set_nat @ B2 @ A3 ) ) ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_1856576259real_n
% 0.30/0.48 @ ^ [Y2: nat] : ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( B2 @ Y2 ) ) )
% 0.30/0.48 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_UN_flatten
% 0.30/0.48 thf(fact_336_UN__UN__flatten,axiom,
% 0.30/0.48 ! [C2: nat > set_Fi1058188332real_n,B2: finite964658038_int_n > set_nat,A3: set_Fi160064172_int_n] :
% 0.30/0.48 ( ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( comple1682161881et_nat @ ( image_1085873667et_nat @ B2 @ A3 ) ) ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_355963305real_n
% 0.30/0.48 @ ^ [Y2: finite964658038_int_n] : ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( B2 @ Y2 ) ) )
% 0.30/0.48 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_UN_flatten
% 0.30/0.48 thf(fact_337_UN__UN__flatten,axiom,
% 0.30/0.48 ! [C2: finite964658038_int_n > set_Fi1058188332real_n,B2: nat > set_Fi160064172_int_n,A3: set_nat] :
% 0.30/0.48 ( ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( comple970917503_int_n @ ( image_968789251_int_n @ B2 @ A3 ) ) ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_1856576259real_n
% 0.30/0.48 @ ^ [Y2: nat] : ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( B2 @ Y2 ) ) )
% 0.30/0.48 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_UN_flatten
% 0.30/0.48 thf(fact_338_UN__UN__flatten,axiom,
% 0.30/0.48 ! [C2: finite964658038_int_n > set_Fi1058188332real_n,B2: finite964658038_int_n > set_Fi160064172_int_n,A3: set_Fi160064172_int_n] :
% 0.30/0.48 ( ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( comple970917503_int_n @ ( image_1819506345_int_n @ B2 @ A3 ) ) ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_355963305real_n
% 0.30/0.48 @ ^ [Y2: finite964658038_int_n] : ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( B2 @ Y2 ) ) )
% 0.30/0.48 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_UN_flatten
% 0.30/0.48 thf(fact_339_UN__UN__flatten,axiom,
% 0.30/0.48 ! [C2: finite1489363574real_n > set_Fi1058188332real_n,B2: nat > set_Fi1058188332real_n,A3: set_nat] :
% 0.30/0.48 ( ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_1856576259real_n
% 0.30/0.48 @ ^ [Y2: nat] : ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( B2 @ Y2 ) ) )
% 0.30/0.48 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_UN_flatten
% 0.30/0.48 thf(fact_340_UN__UN__flatten,axiom,
% 0.30/0.48 ! [C2: finite1489363574real_n > set_Fi1058188332real_n,B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 0.30/0.48 ( ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) ) )
% 0.30/0.48 = ( comple825005695real_n
% 0.30/0.48 @ ( image_355963305real_n
% 0.30/0.48 @ ^ [Y2: finite964658038_int_n] : ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( B2 @ Y2 ) ) )
% 0.30/0.48 @ A3 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_UN_flatten
% 0.30/0.48 thf(fact_341_UN__E,axiom,
% 0.30/0.48 ! [B: set_Fi1058188332real_n,B2: set_Fi1058188332real_n > set_se2111327970real_n,A3: set_se2111327970real_n] :
% 0.30/0.48 ( ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_797440021real_n @ B2 @ A3 ) ) )
% 0.30/0.48 => ~ ! [X2: set_Fi1058188332real_n] :
% 0.30/0.48 ( ( member223413699real_n @ X2 @ A3 )
% 0.30/0.48 => ~ ( member223413699real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_E
% 0.30/0.48 thf(fact_342_UN__E,axiom,
% 0.30/0.48 ! [B: set_Fi1058188332real_n,B2: $o > set_se2111327970real_n,A3: set_o] :
% 0.30/0.48 ( ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_452144437real_n @ B2 @ A3 ) ) )
% 0.30/0.48 => ~ ! [X2: $o] :
% 0.30/0.48 ( ( member_o @ X2 @ A3 )
% 0.30/0.48 => ~ ( member223413699real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_E
% 0.30/0.48 thf(fact_343_UN__E,axiom,
% 0.30/0.48 ! [B: $o,B2: set_Fi1058188332real_n > set_o,A3: set_se2111327970real_n] :
% 0.30/0.48 ( ( member_o @ B @ ( comple1665300069_set_o @ ( image_1687589765_set_o @ B2 @ A3 ) ) )
% 0.30/0.48 => ~ ! [X2: set_Fi1058188332real_n] :
% 0.30/0.48 ( ( member223413699real_n @ X2 @ A3 )
% 0.30/0.48 => ~ ( member_o @ B @ ( B2 @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_E
% 0.30/0.48 thf(fact_344_UN__E,axiom,
% 0.30/0.48 ! [B: $o,B2: $o > set_o,A3: set_o] :
% 0.30/0.48 ( ( member_o @ B @ ( comple1665300069_set_o @ ( image_o_set_o @ B2 @ A3 ) ) )
% 0.30/0.48 => ~ ! [X2: $o] :
% 0.30/0.48 ( ( member_o @ X2 @ A3 )
% 0.30/0.48 => ~ ( member_o @ B @ ( B2 @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_E
% 0.30/0.48 thf(fact_345_UN__E,axiom,
% 0.30/0.48 ! [B: finite1489363574real_n,B2: nat > set_Fi1058188332real_n,A3: set_nat] :
% 0.30/0.48 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 0.30/0.48 => ~ ! [X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2 @ A3 )
% 0.30/0.48 => ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_E
% 0.30/0.48 thf(fact_346_UN__E,axiom,
% 0.30/0.48 ! [B: finite1489363574real_n,B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 0.30/0.48 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 0.30/0.48 => ~ ! [X2: finite964658038_int_n] :
% 0.30/0.48 ( ( member27055245_int_n @ X2 @ A3 )
% 0.30/0.48 => ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_E
% 0.30/0.48 thf(fact_347_UN__E,axiom,
% 0.30/0.48 ! [B: finite1489363574real_n,B2: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.48 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1661509983real_n @ B2 @ A3 ) ) )
% 0.30/0.48 => ~ ! [X2: set_Fi1058188332real_n] :
% 0.30/0.48 ( ( member223413699real_n @ X2 @ A3 )
% 0.30/0.48 => ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_E
% 0.30/0.48 thf(fact_348_UN__E,axiom,
% 0.30/0.48 ! [B: finite1489363574real_n,B2: $o > set_Fi1058188332real_n,A3: set_o] :
% 0.30/0.48 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1759008383real_n @ B2 @ A3 ) ) )
% 0.30/0.48 => ~ ! [X2: $o] :
% 0.30/0.48 ( ( member_o @ X2 @ A3 )
% 0.30/0.48 => ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_E
% 0.30/0.48 thf(fact_349_UN__extend__simps_I8_J,axiom,
% 0.30/0.48 ! [B2: nat > set_Fi1058188332real_n,A3: set_set_nat] :
% 0.30/0.48 ( ( comple825005695real_n
% 0.30/0.48 @ ( image_933134521real_n
% 0.30/0.48 @ ^ [Y2: set_nat] : ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ Y2 ) )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 = ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ ( comple1682161881et_nat @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_extend_simps(8)
% 0.30/0.48 thf(fact_350_UN__extend__simps_I8_J,axiom,
% 0.30/0.48 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_se944069346_int_n] :
% 0.30/0.48 ( ( comple825005695real_n
% 0.30/0.48 @ ( image_792439519real_n
% 0.30/0.48 @ ^ [Y2: set_Fi160064172_int_n] : ( comple825005695real_n @ ( image_355963305real_n @ B2 @ Y2 ) )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 = ( comple825005695real_n @ ( image_355963305real_n @ B2 @ ( comple970917503_int_n @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_extend_simps(8)
% 0.30/0.48 thf(fact_351_UN__extend__simps_I8_J,axiom,
% 0.30/0.48 ! [B2: finite1489363574real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 0.30/0.48 ( ( comple825005695real_n
% 0.30/0.48 @ ( image_1661509983real_n
% 0.30/0.48 @ ^ [Y2: set_Fi1058188332real_n] : ( comple825005695real_n @ ( image_545463721real_n @ B2 @ Y2 ) )
% 0.30/0.48 @ A3 ) )
% 0.30/0.48 = ( comple825005695real_n @ ( image_545463721real_n @ B2 @ ( comple825005695real_n @ A3 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % UN_extend_simps(8)
% 0.30/0.48 thf(fact_352_Sup__bool__def,axiom,
% 0.30/0.48 ( complete_Sup_Sup_o
% 0.30/0.48 = ( member_o @ $true ) ) ).
% 0.30/0.48
% 0.30/0.48 % Sup_bool_def
% 0.30/0.48
% 0.30/0.48 % Conjectures (1)
% 0.30/0.48 thf(conj_0,conjecture,
% 0.30/0.48 ( ( ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ N ) ) ) )
% 0.30/0.48 = ( ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ N ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 %------------------------------------------------------------------------------
% 0.30/0.48 ------------------------
% 2.76/1.55 ---- Embedded in THF ---
% 2.76/1.55 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.76/1.55 %%% Generated on Tue Feb 24 08:27:07 EST 2026
% 2.76/1.55 %%% using '$$dhol' embedding, version 1.3.0.
% 2.76/1.55 %%% Logic specification used:
% 2.76/1.55 %%% thf(spec, logic, $$dhol).
% 2.76/1.55
% 2.76/1.55 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.uys0jgiiMJ/DTF2DTF7164.p
% See solution above
% 2.76/1.55 ------------------------
% 2.86/1.61 ---- Cleaned THF ---
% 2.86/1.61 thf(ty_n_t__Sigma____Algebra__Omeasure_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
% 2.86/1.61 sigma_1422848389real_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Sigma____Algebra__Omeasure_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
% 2.86/1.61 set_Si1125517487real_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J_J,type,
% 2.86/1.61 set_se820660888real_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Sigma____Algebra__Omeasure_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 sigma_1466784463real_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
% 2.86/1.61 set_se2111327970real_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_J_J,type,
% 2.86/1.61 set_se944069346_int_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 set_Fi1058188332real_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_J,type,
% 2.86/1.61 set_Fi160064172_int_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 finite1489363574real_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 2.86/1.61 finite964658038_int_n: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
% 2.86/1.61 set_Ex113815278nnreal: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.86/1.61 set_set_nat: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Extended____Nonnegative____Real__Oennreal,type,
% 2.86/1.61 extend1728876344nnreal: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Sigma____Algebra__Omeasure_I_Eo_J,type,
% 2.86/1.61 sigma_measure_o: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
% 2.86/1.61 set_set_o: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.86/1.61 set_nat: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Set__Oset_I_Eo_J,type,
% 2.86/1.61 set_o: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(ty_n_t__Nat__Onat,type,
% 2.86/1.61 nat: $tType ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
% 2.86/1.61 complete_Sup_Sup_o: set_o > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nonnegative____Real__Oennreal,type,
% 2.86/1.61 comple1413366923nnreal: set_Ex113815278nnreal > extend1728876344nnreal ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 comple2042271945real_n: set_Fi1058188332real_n > finite1489363574real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
% 2.86/1.61 comple1665300069_set_o: set_set_o > set_o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_J,type,
% 2.86/1.61 comple970917503_int_n: set_se944069346_int_n > set_Fi160064172_int_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 comple825005695real_n: set_se2111327970real_n > set_Fi1058188332real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.86/1.61 comple1682161881et_nat: set_set_nat > set_nat ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
% 2.86/1.61 comple1917283637real_n: set_se820660888real_n > set_se2111327970real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Sigma____Algebra__Omeasure_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 comple488165692real_n: set_Si1125517487real_n > sigma_1466784463real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Complete__Measure_Ocompletion_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 comple230862828real_n: sigma_1466784463real_n > sigma_1466784463real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Countable__Set_Ofrom__nat__into_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 2.86/1.61 counta1142393929_int_n: set_Fi160064172_int_n > nat > finite964658038_int_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 2.86/1.61 minus_1196255695_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 minus_1037315151real_n: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 minus_1686442501real_n: set_Fi1058188332real_n > set_Fi1058188332real_n > set_Fi1058188332real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
% 2.86/1.61 one_on705384445nnreal: extend1728876344nnreal ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Oone__class_Oone_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 one_on1253059131real_n: finite1489363574real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nonnegative____Real__Oennreal,type,
% 2.86/1.61 plus_p1763960001nnreal: extend1728876344nnreal > extend1728876344nnreal > extend1728876344nnreal ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Oplus__class_Oplus_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 2.86/1.61 plus_p1654784127_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Oplus__class_Oplus_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 plus_p585657087real_n: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
% 2.86/1.61 zero_z1963244097nnreal: extend1728876344nnreal ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Groups_Ozero__class_Ozero_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 zero_z200130687real_n: finite1489363574real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
% 2.86/1.61 inf_inf_set_o: set_o > set_o > set_o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_J,type,
% 2.86/1.61 inf_in1108485182_int_n: set_Fi160064172_int_n > set_Fi160064172_int_n > set_Fi160064172_int_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 inf_in1974387902real_n: set_Fi1058188332real_n > set_Fi1058188332real_n > set_Fi1058188332real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.86/1.61 inf_inf_set_nat: set_nat > set_nat > set_nat ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
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% 2.86/1.61
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% 2.86/1.61 vimage1860757507real_n: ( nat > finite1489363574real_n ) > set_Fi1058188332real_n > set_nat ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.86/1.61 vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 vimage3210681real_n: ( nat > set_Fi1058188332real_n ) > set_se2111327970real_n > set_nat ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_001_Eo,type,
% 2.86/1.61 vimage851190895al_n_o: ( set_Fi1058188332real_n > $o ) > set_o > set_se2111327970real_n ).
% 2.86/1.61
% 2.86/1.61 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,
% 2.86/1.61 vimage784510485real_n: ( set_Fi1058188332real_n > set_Fi1058188332real_n ) > set_se2111327970real_n > set_se2111327970real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Sigma__Algebra_Oemeasure_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 sigma_1536574303real_n: sigma_1466784463real_n > set_Fi1058188332real_n > extend1728876344nnreal ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Sigma__Algebra_Osets_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 sigma_1235138647real_n: sigma_1466784463real_n > set_se2111327970real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Sigma__Algebra_Ospace_001_Eo,type,
% 2.86/1.61 sigma_space_o: sigma_measure_o > set_o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Sigma__Algebra_Ospace_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 sigma_476185326real_n: sigma_1466784463real_n > set_Fi1058188332real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 sigma_607186084real_n: sigma_1422848389real_n > set_se2111327970real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_member_001_Eo,type,
% 2.86/1.61 member_o: $o > set_o > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
% 2.86/1.61 member1217042383nnreal: extend1728876344nnreal > set_Ex113815278nnreal > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 2.86/1.61 member27055245_int_n: finite964658038_int_n > set_Fi160064172_int_n > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 2.86/1.61 member1352538125real_n: finite1489363574real_n > set_Fi1058188332real_n > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_member_001t__Nat__Onat,type,
% 2.86/1.61 member_nat: nat > set_nat > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
% 2.86/1.61 member_set_o: set_o > set_set_o > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_member_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 member223413699real_n: set_Fi1058188332real_n > set_se2111327970real_n > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J_J,type,
% 2.86/1.61 member1475136633real_n: set_se2111327970real_n > set_se820660888real_n > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_c_member_001t__Sigma____Algebra__Omeasure_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 2.86/1.61 member1000184real_n: sigma_1466784463real_n > set_Si1125517487real_n > $o ).
% 2.86/1.61
% 2.86/1.61 thf(sy_v_R____,type,
% 2.86/1.61 r: finite964658038_int_n > set_Fi1058188332real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_v_S,type,
% 2.86/1.61 s: set_Fi1058188332real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_v_T_H____,type,
% 2.86/1.61 t: finite964658038_int_n > set_Fi1058188332real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_v_T____,type,
% 2.86/1.61 t2: finite964658038_int_n > set_Fi1058188332real_n ).
% 2.86/1.61
% 2.86/1.61 thf(sy_v_f____,type,
% 2.86/1.61 f: nat > finite964658038_int_n ).
% 2.86/1.61
% 2.86/1.61 thf(fact_0_emeasure__T_H,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n] :
% 2.86/1.61 ( ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ A ) )
% 2.86/1.61 = ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ A ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_1__092_060open_062_092_060And_062a_O_AT_H_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n] : ( member223413699real_n @ ( t @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_2__092_060open_062_092_060And_062a_O_AT_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n] : ( member223413699real_n @ ( t2 @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_3_calculation,axiom,
% 2.86/1.61 ( sums_E1192373732nnreal
% 2.86/1.61 @ ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ N ) ) )
% 2.86/1.61 @ ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ s ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_4_assms_I1_J,axiom,
% 2.86/1.61 member223413699real_n @ s @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_5_emeasure__T__Int,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 2.86/1.61 ( ( A != B )
% 2.86/1.61 => ( ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( inf_in1974387902real_n @ ( t2 @ A ) @ ( t2 @ B ) ) )
% 2.86/1.61 = zero_z1963244097nnreal ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_6_T__Int,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 2.86/1.61 ( ( A != B )
% 2.86/1.61 => ( member223413699real_n @ ( inf_in1974387902real_n @ ( t2 @ A ) @ ( t2 @ B ) ) @ ( measur1402256771real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_7_assms_I2_J,axiom,
% 2.86/1.61 ord_le2133614988nnreal @ one_on705384445nnreal @ ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ s ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_8__092_060open_062_092_060And_062a_O_AR_Aa_A_092_060in_062_Asets_Alebesgue_092_060close_062,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n] : ( member223413699real_n @ ( r @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ).
% 2.86/1.61
% 2.86/1.61 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,
% 2.86/1.61 ( sums_E1192373732nnreal
% 2.86/1.61 @ ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ N ) ) )
% 2.86/1.61 @ ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n )
% 2.86/1.61 @ ( comple825005695real_n
% 2.86/1.61 @ ( image_1856576259real_n
% 2.86/1.61 @ ^ [N: nat] : ( t2 @ ( f @ N ) )
% 2.86/1.61 @ top_top_set_nat ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_10_T_H__def,axiom,
% 2.86/1.61 ( t
% 2.86/1.61 = ( ^ [A2: finite964658038_int_n] :
% 2.86/1.61 ( image_439535603real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ ( minkow1134813771n_real @ A2 ) )
% 2.86/1.61 @ ( t2 @ A2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_11_f__def,axiom,
% 2.86/1.61 ( f
% 2.86/1.61 = ( counta1142393929_int_n @ top_to131672412_int_n ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_12_T_H__altdef,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n] :
% 2.86/1.61 ( ( t @ A )
% 2.86/1.61 = ( vimage1233683625real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ X @ ( minkow1134813771n_real @ A ) )
% 2.86/1.61 @ ( t2 @ A ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_13_of__int__vec__eq__iff,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 2.86/1.61 ( ( ( minkow1134813771n_real @ A )
% 2.86/1.61 = ( minkow1134813771n_real @ B ) )
% 2.86/1.61 = ( A = B ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_14_T__def,axiom,
% 2.86/1.61 ( t2
% 2.86/1.61 = ( ^ [A2: finite964658038_int_n] : ( inf_in1974387902real_n @ s @ ( r @ A2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_15_R__Int,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 2.86/1.61 ( ( A != B )
% 2.86/1.61 => ( member223413699real_n @ ( inf_in1974387902real_n @ ( r @ A ) @ ( r @ B ) ) @ ( measur1402256771real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_16_sums__emeasure_H,axiom,
% 2.86/1.61 ! [B2: nat > set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 2.86/1.61 ( ! [X2: nat] : ( member223413699real_n @ ( B2 @ X2 ) @ ( sigma_1235138647real_n @ M ) )
% 2.86/1.61 => ( ! [X2: nat,Y: nat] :
% 2.86/1.61 ( ( X2 != Y )
% 2.86/1.61 => ( ( sigma_1536574303real_n @ M @ ( inf_in1974387902real_n @ ( B2 @ X2 ) @ ( B2 @ Y ) ) )
% 2.86/1.61 = zero_z1963244097nnreal ) )
% 2.86/1.61 => ( sums_E1192373732nnreal
% 2.86/1.61 @ ^ [X: nat] : ( sigma_1536574303real_n @ M @ ( B2 @ X ) )
% 2.86/1.61 @ ( sigma_1536574303real_n @ M @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ top_top_set_nat ) ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_17_null__sets__UN,axiom,
% 2.86/1.61 ! [N2: nat > set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 2.86/1.61 ( ! [I: nat] : ( member223413699real_n @ ( N2 @ I ) @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( member223413699real_n @ ( comple825005695real_n @ ( image_1856576259real_n @ N2 @ top_top_set_nat ) ) @ ( measur1402256771real_n @ M ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_18_null__sets__UN,axiom,
% 2.86/1.61 ! [N2: finite964658038_int_n > set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 2.86/1.61 ( ! [I: finite964658038_int_n] : ( member223413699real_n @ ( N2 @ I ) @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( member223413699real_n @ ( comple825005695real_n @ ( image_355963305real_n @ N2 @ top_to131672412_int_n ) ) @ ( measur1402256771real_n @ M ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_19_null__setsI,axiom,
% 2.86/1.61 ! [M: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( ( sigma_1536574303real_n @ M @ A3 )
% 2.86/1.61 = zero_z1963244097nnreal )
% 2.86/1.61 => ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) )
% 2.86/1.61 => ( member223413699real_n @ A3 @ ( measur1402256771real_n @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_20_image__vimage__eq,axiom,
% 2.86/1.61 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( image_439535603real_n @ F @ ( vimage1233683625real_n @ F @ A3 ) )
% 2.86/1.61 = ( inf_in1974387902real_n @ A3 @ ( image_439535603real_n @ F @ top_to1292442332real_n ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_21_image__vimage__eq,axiom,
% 2.86/1.61 ! [F: nat > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.61 ( ( image_1856576259real_n @ F @ ( vimage3210681real_n @ F @ A3 ) )
% 2.86/1.61 = ( inf_in632889204real_n @ A3 @ ( image_1856576259real_n @ F @ top_top_set_nat ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_22_image__vimage__eq,axiom,
% 2.86/1.61 ! [F: nat > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( image_183184717real_n @ F @ ( vimage1860757507real_n @ F @ A3 ) )
% 2.86/1.61 = ( inf_in1974387902real_n @ A3 @ ( image_183184717real_n @ F @ top_top_set_nat ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_23_image__vimage__eq,axiom,
% 2.86/1.61 ! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.61 ( ( image_355963305real_n @ F @ ( vimage464515423real_n @ F @ A3 ) )
% 2.86/1.61 = ( inf_in632889204real_n @ A3 @ ( image_355963305real_n @ F @ top_to131672412_int_n ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_24_image__vimage__eq,axiom,
% 2.86/1.61 ! [F: finite964658038_int_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( image_2058828787real_n @ F @ ( vimage1276736425real_n @ F @ A3 ) )
% 2.86/1.61 = ( inf_in1974387902real_n @ A3 @ ( image_2058828787real_n @ F @ top_to131672412_int_n ) ) ) ).
% 2.86/1.61
% 2.86/1.61 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,
% 2.86/1.61 ! [A: finite964658038_int_n] :
% 2.86/1.61 ( ( member223413699real_n @ ( t2 @ A ) @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) )
% 2.86/1.61 => ( member223413699real_n
% 2.86/1.61 @ ( inf_in1974387902real_n
% 2.86/1.61 @ ( vimage1233683625real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] : ( plus_p585657087real_n @ X @ ( minkow1134813771n_real @ A ) )
% 2.86/1.61 @ ( t2 @ A ) )
% 2.86/1.61 @ ( sigma_476185326real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) )
% 2.86/1.61 @ ( sigma_1235138647real_n @ ( comple230862828real_n @ lebesg260170249real_n ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_26_surj__diff__right,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n] :
% 2.86/1.61 ( ( image_1278151539_int_n
% 2.86/1.61 @ ^ [X: finite964658038_int_n] : ( minus_1196255695_int_n @ X @ A )
% 2.86/1.61 @ top_to131672412_int_n )
% 2.86/1.61 = top_to131672412_int_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_27_surj__diff__right,axiom,
% 2.86/1.61 ! [A: finite1489363574real_n] :
% 2.86/1.61 ( ( image_439535603real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
% 2.86/1.61 @ top_to1292442332real_n )
% 2.86/1.61 = top_to1292442332real_n ) ).
% 2.86/1.61
% 2.86/1.61 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,
% 2.86/1.61 ( ( comple825005695real_n @ ( image_355963305real_n @ t2 @ top_to131672412_int_n ) )
% 2.86/1.61 = ( comple825005695real_n
% 2.86/1.61 @ ( image_1856576259real_n
% 2.86/1.61 @ ^ [N: nat] : ( t2 @ ( f @ N ) )
% 2.86/1.61 @ top_top_set_nat ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_29_Sup__UNIV,axiom,
% 2.86/1.61 ( ( comple1682161881et_nat @ top_top_set_set_nat )
% 2.86/1.61 = top_top_set_nat ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_30_Sup__UNIV,axiom,
% 2.86/1.61 ( ( comple970917503_int_n @ top_to1587634578_int_n )
% 2.86/1.61 = top_to131672412_int_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_31_Sup__UNIV,axiom,
% 2.86/1.61 ( ( comple825005695real_n @ top_to20708754real_n )
% 2.86/1.61 = top_to1292442332real_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_32_Sup__UNIV,axiom,
% 2.86/1.61 ( ( complete_Sup_Sup_o @ top_top_set_o )
% 2.86/1.61 = top_top_o ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_33_Sup__eq__top__iff,axiom,
% 2.86/1.61 ! [A3: set_Ex113815278nnreal] :
% 2.86/1.61 ( ( ( comple1413366923nnreal @ A3 )
% 2.86/1.61 = top_to1845833192nnreal )
% 2.86/1.61 = ( ! [X: extend1728876344nnreal] :
% 2.86/1.61 ( ( ord_le2133614988nnreal @ X @ top_to1845833192nnreal )
% 2.86/1.61 => ? [Y2: extend1728876344nnreal] :
% 2.86/1.61 ( ( member1217042383nnreal @ Y2 @ A3 )
% 2.86/1.61 & ( ord_le2133614988nnreal @ X @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_34_surj__plus,axiom,
% 2.86/1.61 ! [A: finite1489363574real_n] :
% 2.86/1.61 ( ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ top_to1292442332real_n )
% 2.86/1.61 = top_to1292442332real_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_35_surj__plus,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n] :
% 2.86/1.61 ( ( image_1278151539_int_n @ ( plus_p1654784127_int_n @ A ) @ top_to131672412_int_n )
% 2.86/1.61 = top_to131672412_int_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_36_diff__numeral__special_I9_J,axiom,
% 2.86/1.61 ( ( minus_1037315151real_n @ one_on1253059131real_n @ one_on1253059131real_n )
% 2.86/1.61 = zero_z200130687real_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_37_image__eqI,axiom,
% 2.86/1.61 ! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,X3: finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( B
% 2.86/1.61 = ( F @ X3 ) )
% 2.86/1.61 => ( ( member1352538125real_n @ X3 @ A3 )
% 2.86/1.61 => ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_38_image__eqI,axiom,
% 2.86/1.61 ! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,X3: nat,A3: set_nat] :
% 2.86/1.61 ( ( B
% 2.86/1.61 = ( F @ X3 ) )
% 2.86/1.61 => ( ( member_nat @ X3 @ A3 )
% 2.86/1.61 => ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_39_image__eqI,axiom,
% 2.86/1.61 ! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,X3: finite964658038_int_n,A3: set_Fi160064172_int_n] :
% 2.86/1.61 ( ( B
% 2.86/1.61 = ( F @ X3 ) )
% 2.86/1.61 => ( ( member27055245_int_n @ X3 @ A3 )
% 2.86/1.61 => ( member223413699real_n @ B @ ( image_355963305real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_40_image__eqI,axiom,
% 2.86/1.61 ! [B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,X3: set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.61 ( ( B
% 2.86/1.61 = ( F @ X3 ) )
% 2.86/1.61 => ( ( member223413699real_n @ X3 @ A3 )
% 2.86/1.61 => ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_41_image__eqI,axiom,
% 2.86/1.61 ! [B: $o,F: set_Fi1058188332real_n > $o,X3: set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.61 ( ( B
% 2.86/1.61 = ( F @ X3 ) )
% 2.86/1.61 => ( ( member223413699real_n @ X3 @ A3 )
% 2.86/1.61 => ( member_o @ B @ ( image_1648361637al_n_o @ F @ A3 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_42_image__eqI,axiom,
% 2.86/1.61 ! [B: set_Fi1058188332real_n,F: $o > set_Fi1058188332real_n,X3: $o,A3: set_o] :
% 2.86/1.61 ( ( B
% 2.86/1.61 = ( F @ X3 ) )
% 2.86/1.61 => ( ( member_o @ X3 @ A3 )
% 2.86/1.61 => ( member223413699real_n @ B @ ( image_1759008383real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_43_image__eqI,axiom,
% 2.86/1.61 ! [B: $o,F: $o > $o,X3: $o,A3: set_o] :
% 2.86/1.61 ( ( B
% 2.86/1.61 = ( F @ X3 ) )
% 2.86/1.61 => ( ( member_o @ X3 @ A3 )
% 2.86/1.61 => ( member_o @ B @ ( image_o_o @ F @ A3 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_44_UNIV__I,axiom,
% 2.86/1.61 ! [X3: set_Fi1058188332real_n] : ( member223413699real_n @ X3 @ top_to20708754real_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_45_UNIV__I,axiom,
% 2.86/1.61 ! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_46_UNIV__I,axiom,
% 2.86/1.61 ! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_47_UNIV__I,axiom,
% 2.86/1.61 ! [X3: finite964658038_int_n] : ( member27055245_int_n @ X3 @ top_to131672412_int_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_48_Int__iff,axiom,
% 2.86/1.61 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 2.86/1.61 ( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
% 2.86/1.61 = ( ( member223413699real_n @ C @ A3 )
% 2.86/1.61 & ( member223413699real_n @ C @ B2 ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_49_Int__iff,axiom,
% 2.86/1.61 ! [C: $o,A3: set_o,B2: set_o] :
% 2.86/1.61 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 2.86/1.61 = ( ( member_o @ C @ A3 )
% 2.86/1.61 & ( member_o @ C @ B2 ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_50_Int__iff,axiom,
% 2.86/1.61 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 2.86/1.61 = ( ( member1352538125real_n @ C @ A3 )
% 2.86/1.61 & ( member1352538125real_n @ C @ B2 ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_51_IntI,axiom,
% 2.86/1.61 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 2.86/1.61 ( ( member223413699real_n @ C @ A3 )
% 2.86/1.61 => ( ( member223413699real_n @ C @ B2 )
% 2.86/1.61 => ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_52_IntI,axiom,
% 2.86/1.61 ! [C: $o,A3: set_o,B2: set_o] :
% 2.86/1.61 ( ( member_o @ C @ A3 )
% 2.86/1.61 => ( ( member_o @ C @ B2 )
% 2.86/1.61 => ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_53_IntI,axiom,
% 2.86/1.61 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ C @ A3 )
% 2.86/1.61 => ( ( member1352538125real_n @ C @ B2 )
% 2.86/1.61 => ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_54_Union__iff,axiom,
% 2.86/1.61 ! [A3: set_Fi1058188332real_n,C2: set_se820660888real_n] :
% 2.86/1.61 ( ( member223413699real_n @ A3 @ ( comple1917283637real_n @ C2 ) )
% 2.86/1.61 = ( ? [X: set_se2111327970real_n] :
% 2.86/1.61 ( ( member1475136633real_n @ X @ C2 )
% 2.86/1.61 & ( member223413699real_n @ A3 @ X ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_55_Union__iff,axiom,
% 2.86/1.61 ! [A3: $o,C2: set_set_o] :
% 2.86/1.61 ( ( member_o @ A3 @ ( comple1665300069_set_o @ C2 ) )
% 2.86/1.61 = ( ? [X: set_o] :
% 2.86/1.61 ( ( member_set_o @ X @ C2 )
% 2.86/1.61 & ( member_o @ A3 @ X ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_56_Union__iff,axiom,
% 2.86/1.61 ! [A3: finite1489363574real_n,C2: set_se2111327970real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ A3 @ ( comple825005695real_n @ C2 ) )
% 2.86/1.61 = ( ? [X: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X @ C2 )
% 2.86/1.61 & ( member1352538125real_n @ A3 @ X ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_57_UnionI,axiom,
% 2.86/1.61 ! [X4: set_se2111327970real_n,C2: set_se820660888real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member1475136633real_n @ X4 @ C2 )
% 2.86/1.61 => ( ( member223413699real_n @ A3 @ X4 )
% 2.86/1.61 => ( member223413699real_n @ A3 @ ( comple1917283637real_n @ C2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_58_UnionI,axiom,
% 2.86/1.61 ! [X4: set_o,C2: set_set_o,A3: $o] :
% 2.86/1.61 ( ( member_set_o @ X4 @ C2 )
% 2.86/1.61 => ( ( member_o @ A3 @ X4 )
% 2.86/1.61 => ( member_o @ A3 @ ( comple1665300069_set_o @ C2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_59_UnionI,axiom,
% 2.86/1.61 ! [X4: set_Fi1058188332real_n,C2: set_se2111327970real_n,A3: finite1489363574real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X4 @ C2 )
% 2.86/1.61 => ( ( member1352538125real_n @ A3 @ X4 )
% 2.86/1.61 => ( member1352538125real_n @ A3 @ ( comple825005695real_n @ C2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_60_UN__ball__bex__simps_I1_J,axiom,
% 2.86/1.61 ! [A3: set_se2111327970real_n,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ! [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ A3 ) )
% 2.86/1.61 => ( P @ X ) ) )
% 2.86/1.61 = ( ! [X: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X @ A3 )
% 2.86/1.61 => ! [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ X )
% 2.86/1.61 => ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_61_UN__ball__bex__simps_I3_J,axiom,
% 2.86/1.61 ! [A3: set_se2111327970real_n,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ? [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ A3 ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 = ( ? [X: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X @ A3 )
% 2.86/1.61 & ? [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ X )
% 2.86/1.61 & ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_62_null__sets_ODiff,axiom,
% 2.86/1.61 ! [A: set_Fi1058188332real_n,M: sigma_1466784463real_n,B: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ A @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( ( member223413699real_n @ B @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( member223413699real_n @ ( minus_1686442501real_n @ A @ B ) @ ( measur1402256771real_n @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_63_vimage__eq,axiom,
% 2.86/1.61 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 2.86/1.61 ( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ B2 ) )
% 2.86/1.61 = ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_64_vimage__eq,axiom,
% 2.86/1.61 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > $o,B2: set_o] :
% 2.86/1.61 ( ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ B2 ) )
% 2.86/1.61 = ( member_o @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_65_vimage__eq,axiom,
% 2.86/1.61 ! [A: $o,F: $o > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 2.86/1.61 ( ( member_o @ A @ ( vimage961837641real_n @ F @ B2 ) )
% 2.86/1.61 = ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_66_vimage__eq,axiom,
% 2.86/1.61 ! [A: $o,F: $o > $o,B2: set_o] :
% 2.86/1.61 ( ( member_o @ A @ ( vimage_o_o @ F @ B2 ) )
% 2.86/1.61 = ( member_o @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_67_vimage__eq,axiom,
% 2.86/1.61 ! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ B2 ) )
% 2.86/1.61 = ( member1352538125real_n @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_68_vimageI,axiom,
% 2.86/1.61 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A: set_Fi1058188332real_n,B: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 2.86/1.61 ( ( ( F @ A )
% 2.86/1.61 = B )
% 2.86/1.61 => ( ( member223413699real_n @ B @ B2 )
% 2.86/1.61 => ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ B2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_69_vimageI,axiom,
% 2.86/1.61 ! [F: $o > set_Fi1058188332real_n,A: $o,B: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 2.86/1.61 ( ( ( F @ A )
% 2.86/1.61 = B )
% 2.86/1.61 => ( ( member223413699real_n @ B @ B2 )
% 2.86/1.61 => ( member_o @ A @ ( vimage961837641real_n @ F @ B2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_70_vimageI,axiom,
% 2.86/1.61 ! [F: set_Fi1058188332real_n > $o,A: set_Fi1058188332real_n,B: $o,B2: set_o] :
% 2.86/1.61 ( ( ( F @ A )
% 2.86/1.61 = B )
% 2.86/1.61 => ( ( member_o @ B @ B2 )
% 2.86/1.61 => ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ B2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_71_vimageI,axiom,
% 2.86/1.61 ! [F: $o > $o,A: $o,B: $o,B2: set_o] :
% 2.86/1.61 ( ( ( F @ A )
% 2.86/1.61 = B )
% 2.86/1.61 => ( ( member_o @ B @ B2 )
% 2.86/1.61 => ( member_o @ A @ ( vimage_o_o @ F @ B2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_72_vimageI,axiom,
% 2.86/1.61 ! [F: finite1489363574real_n > finite1489363574real_n,A: finite1489363574real_n,B: finite1489363574real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( ( F @ A )
% 2.86/1.61 = B )
% 2.86/1.61 => ( ( member1352538125real_n @ B @ B2 )
% 2.86/1.61 => ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ B2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_73_image__ident,axiom,
% 2.86/1.61 ! [Y3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( image_439535603real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] : X
% 2.86/1.61 @ Y3 )
% 2.86/1.61 = Y3 ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_74_vimage__Collect__eq,axiom,
% 2.86/1.61 ! [F: finite1489363574real_n > finite1489363574real_n,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( vimage1233683625real_n @ F @ ( collec321817931real_n @ P ) )
% 2.86/1.61 = ( collec321817931real_n
% 2.86/1.61 @ ^ [Y2: finite1489363574real_n] : ( P @ ( F @ Y2 ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_75_vimage__ident,axiom,
% 2.86/1.61 ! [Y3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( vimage1233683625real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] : X
% 2.86/1.61 @ Y3 )
% 2.86/1.61 = Y3 ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_76_Int__UNIV,axiom,
% 2.86/1.61 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( ( inf_in1974387902real_n @ A3 @ B2 )
% 2.86/1.61 = top_to1292442332real_n )
% 2.86/1.61 = ( ( A3 = top_to1292442332real_n )
% 2.86/1.61 & ( B2 = top_to1292442332real_n ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_77_Int__UNIV,axiom,
% 2.86/1.61 ! [A3: set_nat,B2: set_nat] :
% 2.86/1.61 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 2.86/1.61 = top_top_set_nat )
% 2.86/1.61 = ( ( A3 = top_top_set_nat )
% 2.86/1.61 & ( B2 = top_top_set_nat ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_78_Int__UNIV,axiom,
% 2.86/1.61 ! [A3: set_Fi160064172_int_n,B2: set_Fi160064172_int_n] :
% 2.86/1.61 ( ( ( inf_in1108485182_int_n @ A3 @ B2 )
% 2.86/1.61 = top_to131672412_int_n )
% 2.86/1.61 = ( ( A3 = top_to131672412_int_n )
% 2.86/1.61 & ( B2 = top_to131672412_int_n ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_79_ball__UN,axiom,
% 2.86/1.61 ! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ! [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 2.86/1.61 => ( P @ X ) ) )
% 2.86/1.61 = ( ! [X: nat] :
% 2.86/1.61 ( ( member_nat @ X @ A3 )
% 2.86/1.61 => ! [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 2.86/1.61 => ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_80_ball__UN,axiom,
% 2.86/1.61 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ! [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 2.86/1.61 => ( P @ X ) ) )
% 2.86/1.61 = ( ! [X: finite964658038_int_n] :
% 2.86/1.61 ( ( member27055245_int_n @ X @ A3 )
% 2.86/1.61 => ! [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 2.86/1.61 => ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_81_mem__Collect__eq,axiom,
% 2.86/1.61 ! [A: set_Fi1058188332real_n,P: set_Fi1058188332real_n > $o] :
% 2.86/1.61 ( ( member223413699real_n @ A @ ( collec452821761real_n @ P ) )
% 2.86/1.61 = ( P @ A ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_82_mem__Collect__eq,axiom,
% 2.86/1.61 ! [A: $o,P: $o > $o] :
% 2.86/1.61 ( ( member_o @ A @ ( collect_o @ P ) )
% 2.86/1.61 = ( P @ A ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_83_Collect__mem__eq,axiom,
% 2.86/1.61 ! [A3: set_se2111327970real_n] :
% 2.86/1.61 ( ( collec452821761real_n
% 2.86/1.61 @ ^ [X: set_Fi1058188332real_n] : ( member223413699real_n @ X @ A3 ) )
% 2.86/1.61 = A3 ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_84_Collect__mem__eq,axiom,
% 2.86/1.61 ! [A3: set_o] :
% 2.86/1.61 ( ( collect_o
% 2.86/1.61 @ ^ [X: $o] : ( member_o @ X @ A3 ) )
% 2.86/1.61 = A3 ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_85_bex__UN,axiom,
% 2.86/1.61 ! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ? [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 = ( ? [X: nat] :
% 2.86/1.61 ( ( member_nat @ X @ A3 )
% 2.86/1.61 & ? [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 2.86/1.61 & ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_86_bex__UN,axiom,
% 2.86/1.61 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ? [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 = ( ? [X: finite964658038_int_n] :
% 2.86/1.61 ( ( member27055245_int_n @ X @ A3 )
% 2.86/1.61 & ? [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 2.86/1.61 & ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_87_UN__ball__bex__simps_I2_J,axiom,
% 2.86/1.61 ! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ! [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 2.86/1.61 => ( P @ X ) ) )
% 2.86/1.61 = ( ! [X: nat] :
% 2.86/1.61 ( ( member_nat @ X @ A3 )
% 2.86/1.61 => ! [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 2.86/1.61 => ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_88_UN__ball__bex__simps_I2_J,axiom,
% 2.86/1.61 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ! [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 2.86/1.61 => ( P @ X ) ) )
% 2.86/1.61 = ( ! [X: finite964658038_int_n] :
% 2.86/1.61 ( ( member27055245_int_n @ X @ A3 )
% 2.86/1.61 => ! [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 2.86/1.61 => ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_89_UN__ball__bex__simps_I4_J,axiom,
% 2.86/1.61 ! [B2: nat > set_Fi1058188332real_n,A3: set_nat,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ? [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 = ( ? [X: nat] :
% 2.86/1.61 ( ( member_nat @ X @ A3 )
% 2.86/1.61 & ? [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 2.86/1.61 & ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_90_UN__ball__bex__simps_I4_J,axiom,
% 2.86/1.61 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( ? [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 = ( ? [X: finite964658038_int_n] :
% 2.86/1.61 ( ( member27055245_int_n @ X @ A3 )
% 2.86/1.61 & ? [Y2: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ Y2 @ ( B2 @ X ) )
% 2.86/1.61 & ( P @ Y2 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_91_vimage__UNIV,axiom,
% 2.86/1.61 ! [F: finite1489363574real_n > finite1489363574real_n] :
% 2.86/1.61 ( ( vimage1233683625real_n @ F @ top_to1292442332real_n )
% 2.86/1.61 = top_to1292442332real_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_92_vimage__UNIV,axiom,
% 2.86/1.61 ! [F: nat > nat] :
% 2.86/1.61 ( ( vimage_nat_nat @ F @ top_top_set_nat )
% 2.86/1.61 = top_top_set_nat ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_93_vimage__UNIV,axiom,
% 2.86/1.61 ! [F: finite964658038_int_n > nat] :
% 2.86/1.61 ( ( vimage1398021123_n_nat @ F @ top_top_set_nat )
% 2.86/1.61 = top_to131672412_int_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_94_vimage__UNIV,axiom,
% 2.86/1.61 ! [F: nat > finite964658038_int_n] :
% 2.86/1.61 ( ( vimage714719107_int_n @ F @ top_to131672412_int_n )
% 2.86/1.61 = top_top_set_nat ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_95_vimage__UNIV,axiom,
% 2.86/1.61 ! [F: finite964658038_int_n > finite964658038_int_n] :
% 2.86/1.61 ( ( vimage1122713129_int_n @ F @ top_to131672412_int_n )
% 2.86/1.61 = top_to131672412_int_n ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_96_null__sets_OInt,axiom,
% 2.86/1.61 ! [A: set_Fi1058188332real_n,M: sigma_1466784463real_n,B: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ A @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( ( member223413699real_n @ B @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( member223413699real_n @ ( inf_in1974387902real_n @ A @ B ) @ ( measur1402256771real_n @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_97_vimage__Int,axiom,
% 2.86/1.61 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( vimage1233683625real_n @ F @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 2.86/1.61 = ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ A3 ) @ ( vimage1233683625real_n @ F @ B2 ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_98_SUP__identity__eq,axiom,
% 2.86/1.61 ! [A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( comple2042271945real_n
% 2.86/1.61 @ ( image_439535603real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] : X
% 2.86/1.61 @ A3 ) )
% 2.86/1.61 = ( comple2042271945real_n @ A3 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_99_SUP__identity__eq,axiom,
% 2.86/1.61 ! [A3: set_se2111327970real_n] :
% 2.86/1.61 ( ( comple825005695real_n
% 2.86/1.61 @ ( image_1661509983real_n
% 2.86/1.61 @ ^ [X: set_Fi1058188332real_n] : X
% 2.86/1.61 @ A3 ) )
% 2.86/1.61 = ( comple825005695real_n @ A3 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_100_SUP__identity__eq,axiom,
% 2.86/1.61 ! [A3: set_o] :
% 2.86/1.61 ( ( complete_Sup_Sup_o
% 2.86/1.61 @ ( image_o_o
% 2.86/1.61 @ ^ [X: $o] : X
% 2.86/1.61 @ A3 ) )
% 2.86/1.61 = ( complete_Sup_Sup_o @ A3 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_101_UN__iff,axiom,
% 2.86/1.61 ! [B: finite1489363574real_n,B2: nat > set_Fi1058188332real_n,A3: set_nat] :
% 2.86/1.61 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 2.86/1.61 = ( ? [X: nat] :
% 2.86/1.61 ( ( member_nat @ X @ A3 )
% 2.86/1.61 & ( member1352538125real_n @ B @ ( B2 @ X ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_102_UN__iff,axiom,
% 2.86/1.61 ! [B: finite1489363574real_n,B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 2.86/1.61 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 2.86/1.61 = ( ? [X: finite964658038_int_n] :
% 2.86/1.61 ( ( member27055245_int_n @ X @ A3 )
% 2.86/1.61 & ( member1352538125real_n @ B @ ( B2 @ X ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_103_UN__I,axiom,
% 2.86/1.61 ! [A: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: set_Fi1058188332real_n,B2: set_Fi1058188332real_n > set_se2111327970real_n] :
% 2.86/1.61 ( ( member223413699real_n @ A @ A3 )
% 2.86/1.61 => ( ( member223413699real_n @ B @ ( B2 @ A ) )
% 2.86/1.61 => ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_797440021real_n @ B2 @ A3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_104_UN__I,axiom,
% 2.86/1.61 ! [A: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: $o,B2: set_Fi1058188332real_n > set_o] :
% 2.86/1.61 ( ( member223413699real_n @ A @ A3 )
% 2.86/1.61 => ( ( member_o @ B @ ( B2 @ A ) )
% 2.86/1.61 => ( member_o @ B @ ( comple1665300069_set_o @ ( image_1687589765_set_o @ B2 @ A3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_105_UN__I,axiom,
% 2.86/1.61 ! [A: $o,A3: set_o,B: set_Fi1058188332real_n,B2: $o > set_se2111327970real_n] :
% 2.86/1.61 ( ( member_o @ A @ A3 )
% 2.86/1.61 => ( ( member223413699real_n @ B @ ( B2 @ A ) )
% 2.86/1.61 => ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_452144437real_n @ B2 @ A3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_106_UN__I,axiom,
% 2.86/1.61 ! [A: $o,A3: set_o,B: $o,B2: $o > set_o] :
% 2.86/1.61 ( ( member_o @ A @ A3 )
% 2.86/1.61 => ( ( member_o @ B @ ( B2 @ A ) )
% 2.86/1.61 => ( member_o @ B @ ( comple1665300069_set_o @ ( image_o_set_o @ B2 @ A3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_107_UN__I,axiom,
% 2.86/1.61 ! [A: nat,A3: set_nat,B: finite1489363574real_n,B2: nat > set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member_nat @ A @ A3 )
% 2.86/1.61 => ( ( member1352538125real_n @ B @ ( B2 @ A ) )
% 2.86/1.61 => ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_108_UN__I,axiom,
% 2.86/1.61 ! [A: finite964658038_int_n,A3: set_Fi160064172_int_n,B: finite1489363574real_n,B2: finite964658038_int_n > set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member27055245_int_n @ A @ A3 )
% 2.86/1.61 => ( ( member1352538125real_n @ B @ ( B2 @ A ) )
% 2.86/1.61 => ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_109_UN__I,axiom,
% 2.86/1.61 ! [A: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: finite1489363574real_n,B2: set_Fi1058188332real_n > set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ A @ A3 )
% 2.86/1.61 => ( ( member1352538125real_n @ B @ ( B2 @ A ) )
% 2.86/1.61 => ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1661509983real_n @ B2 @ A3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_110_UN__I,axiom,
% 2.86/1.61 ! [A: $o,A3: set_o,B: finite1489363574real_n,B2: $o > set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member_o @ A @ A3 )
% 2.86/1.61 => ( ( member1352538125real_n @ B @ ( B2 @ A ) )
% 2.86/1.61 => ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1759008383real_n @ B2 @ A3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_111_S__decompose,axiom,
% 2.86/1.61 ( s
% 2.86/1.61 = ( comple825005695real_n @ ( image_355963305real_n @ t2 @ top_to131672412_int_n ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_112_image__add__0,axiom,
% 2.86/1.61 ! [S: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( image_439535603real_n @ ( plus_p585657087real_n @ zero_z200130687real_n ) @ S )
% 2.86/1.61 = S ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_113_image__add__0,axiom,
% 2.86/1.61 ! [S: set_Ex113815278nnreal] :
% 2.86/1.61 ( ( image_2066995319nnreal @ ( plus_p1763960001nnreal @ zero_z1963244097nnreal ) @ S )
% 2.86/1.61 = S ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_114_null__sets_OInt__space__eq2,axiom,
% 2.86/1.61 ! [X3: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X3 @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( ( inf_in1974387902real_n @ X3 @ ( sigma_476185326real_n @ M ) )
% 2.86/1.61 = X3 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_115_null__sets_OInt__space__eq1,axiom,
% 2.86/1.61 ! [X3: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X3 @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( ( inf_in1974387902real_n @ ( sigma_476185326real_n @ M ) @ X3 )
% 2.86/1.61 = X3 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_116_in__sets__SUP,axiom,
% 2.86/1.61 ! [I2: set_Fi1058188332real_n,I3: set_se2111327970real_n,M: set_Fi1058188332real_n > sigma_1466784463real_n,Y3: set_Fi1058188332real_n,X4: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ I2 @ I3 )
% 2.86/1.61 => ( ! [I: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ I @ I3 )
% 2.86/1.61 => ( ( sigma_476185326real_n @ ( M @ I ) )
% 2.86/1.61 = Y3 ) )
% 2.86/1.61 => ( ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( M @ I2 ) ) )
% 2.86/1.61 => ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_987430492real_n @ M @ I3 ) ) ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_117_in__sets__SUP,axiom,
% 2.86/1.61 ! [I2: $o,I3: set_o,M: $o > sigma_1466784463real_n,Y3: set_Fi1058188332real_n,X4: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member_o @ I2 @ I3 )
% 2.86/1.61 => ( ! [I: $o] :
% 2.86/1.61 ( ( member_o @ I @ I3 )
% 2.86/1.61 => ( ( sigma_476185326real_n @ ( M @ I ) )
% 2.86/1.61 = Y3 ) )
% 2.86/1.61 => ( ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( M @ I2 ) ) )
% 2.86/1.61 => ( member223413699real_n @ X4 @ ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_1599934780real_n @ M @ I3 ) ) ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_118_in__sets__Sup,axiom,
% 2.86/1.61 ! [M: set_Si1125517487real_n,X4: set_Fi1058188332real_n,M2: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ! [M3: sigma_1466784463real_n] :
% 2.86/1.61 ( ( member1000184real_n @ M3 @ M )
% 2.86/1.61 => ( ( sigma_476185326real_n @ M3 )
% 2.86/1.61 = X4 ) )
% 2.86/1.61 => ( ( member1000184real_n @ M2 @ M )
% 2.86/1.61 => ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M2 ) )
% 2.86/1.61 => ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ ( comple488165692real_n @ M ) ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_119_sets__SUP__cong,axiom,
% 2.86/1.61 ! [I3: set_se2111327970real_n,M: set_Fi1058188332real_n > sigma_1466784463real_n,N2: set_Fi1058188332real_n > sigma_1466784463real_n] :
% 2.86/1.61 ( ! [I: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ I @ I3 )
% 2.86/1.61 => ( ( sigma_1235138647real_n @ ( M @ I ) )
% 2.86/1.61 = ( sigma_1235138647real_n @ ( N2 @ I ) ) ) )
% 2.86/1.61 => ( ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_987430492real_n @ M @ I3 ) ) )
% 2.86/1.61 = ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_987430492real_n @ N2 @ I3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_120_sets__SUP__cong,axiom,
% 2.86/1.61 ! [I3: set_o,M: $o > sigma_1466784463real_n,N2: $o > sigma_1466784463real_n] :
% 2.86/1.61 ( ! [I: $o] :
% 2.86/1.61 ( ( member_o @ I @ I3 )
% 2.86/1.61 => ( ( sigma_1235138647real_n @ ( M @ I ) )
% 2.86/1.61 = ( sigma_1235138647real_n @ ( N2 @ I ) ) ) )
% 2.86/1.61 => ( ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_1599934780real_n @ M @ I3 ) ) )
% 2.86/1.61 = ( sigma_1235138647real_n @ ( comple488165692real_n @ ( image_1599934780real_n @ N2 @ I3 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_121_Sup__set__def,axiom,
% 2.86/1.61 ( comple1917283637real_n
% 2.86/1.61 = ( ^ [A4: set_se820660888real_n] :
% 2.86/1.61 ( collec452821761real_n
% 2.86/1.61 @ ^ [X: set_Fi1058188332real_n] : ( complete_Sup_Sup_o @ ( image_1681970287al_n_o @ ( member223413699real_n @ X ) @ A4 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_122_Sup__set__def,axiom,
% 2.86/1.61 ( comple1665300069_set_o
% 2.86/1.61 = ( ^ [A4: set_set_o] :
% 2.86/1.61 ( collect_o
% 2.86/1.61 @ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A4 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_123_Sup__set__def,axiom,
% 2.86/1.61 ( comple825005695real_n
% 2.86/1.61 = ( ^ [A4: set_se2111327970real_n] :
% 2.86/1.61 ( collec321817931real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] : ( complete_Sup_Sup_o @ ( image_1648361637al_n_o @ ( member1352538125real_n @ X ) @ A4 ) ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_124_space__Sup__eq__UN,axiom,
% 2.86/1.61 ! [M: set_Si1125517487real_n] :
% 2.86/1.61 ( ( sigma_476185326real_n @ ( comple488165692real_n @ M ) )
% 2.86/1.61 = ( comple825005695real_n @ ( image_1298280374real_n @ sigma_476185326real_n @ M ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_125_top__set__def,axiom,
% 2.86/1.61 ( top_top_set_nat
% 2.86/1.61 = ( collect_nat @ top_top_nat_o ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_126_top__set__def,axiom,
% 2.86/1.61 ( top_to131672412_int_n
% 2.86/1.61 = ( collec1941932235_int_n @ top_to287930409nt_n_o ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_127_Diff__Int__distrib2,axiom,
% 2.86/1.61 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( inf_in1974387902real_n @ ( minus_1686442501real_n @ A3 @ B2 ) @ C2 )
% 2.86/1.61 = ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ C2 ) @ ( inf_in1974387902real_n @ B2 @ C2 ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_128_Diff__Int__distrib,axiom,
% 2.86/1.61 ! [C2: set_Fi1058188332real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( inf_in1974387902real_n @ C2 @ ( minus_1686442501real_n @ A3 @ B2 ) )
% 2.86/1.61 = ( minus_1686442501real_n @ ( inf_in1974387902real_n @ C2 @ A3 ) @ ( inf_in1974387902real_n @ C2 @ B2 ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_129_Diff__Diff__Int,axiom,
% 2.86/1.61 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( minus_1686442501real_n @ A3 @ ( minus_1686442501real_n @ A3 @ B2 ) )
% 2.86/1.61 = ( inf_in1974387902real_n @ A3 @ B2 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_130_Diff__Int2,axiom,
% 2.86/1.61 ! [A3: set_Fi1058188332real_n,C2: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ C2 ) @ ( inf_in1974387902real_n @ B2 @ C2 ) )
% 2.86/1.61 = ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ C2 ) @ B2 ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_131_Int__Diff,axiom,
% 2.86/1.61 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( minus_1686442501real_n @ ( inf_in1974387902real_n @ A3 @ B2 ) @ C2 )
% 2.86/1.61 = ( inf_in1974387902real_n @ A3 @ ( minus_1686442501real_n @ B2 @ C2 ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_132_vimage__Diff,axiom,
% 2.86/1.61 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( vimage1233683625real_n @ F @ ( minus_1686442501real_n @ A3 @ B2 ) )
% 2.86/1.61 = ( minus_1686442501real_n @ ( vimage1233683625real_n @ F @ A3 ) @ ( vimage1233683625real_n @ F @ B2 ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_133_null__sets_Osets__Collect__disj,axiom,
% 2.86/1.61 ! [M: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o,Q: set_Fi1058188332real_n > $o] :
% 2.86/1.61 ( ( member1475136633real_n
% 2.86/1.61 @ ( collec452821761real_n
% 2.86/1.61 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 @ ( measur2126959417real_n @ M ) )
% 2.86/1.61 => ( ( member1475136633real_n
% 2.86/1.61 @ ( collec452821761real_n
% 2.86/1.61 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 2.86/1.61 & ( Q @ X ) ) )
% 2.86/1.61 @ ( measur2126959417real_n @ M ) )
% 2.86/1.61 => ( member1475136633real_n
% 2.86/1.61 @ ( collec452821761real_n
% 2.86/1.61 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 2.86/1.61 & ( ( Q @ X )
% 2.86/1.61 | ( P @ X ) ) ) )
% 2.86/1.61 @ ( measur2126959417real_n @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_134_null__sets_Osets__Collect__disj,axiom,
% 2.86/1.61 ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
% 2.86/1.61 ( ( member_set_o
% 2.86/1.61 @ ( collect_o
% 2.86/1.61 @ ^ [X: $o] :
% 2.86/1.61 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 @ ( measure_null_sets_o @ M ) )
% 2.86/1.61 => ( ( member_set_o
% 2.86/1.61 @ ( collect_o
% 2.86/1.61 @ ^ [X: $o] :
% 2.86/1.61 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 2.86/1.61 & ( Q @ X ) ) )
% 2.86/1.61 @ ( measure_null_sets_o @ M ) )
% 2.86/1.61 => ( member_set_o
% 2.86/1.61 @ ( collect_o
% 2.86/1.61 @ ^ [X: $o] :
% 2.86/1.61 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 2.86/1.61 & ( ( Q @ X )
% 2.86/1.61 | ( P @ X ) ) ) )
% 2.86/1.61 @ ( measure_null_sets_o @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_135_null__sets_Osets__Collect__disj,axiom,
% 2.86/1.61 ! [M: sigma_1466784463real_n,P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( member223413699real_n
% 2.86/1.61 @ ( collec321817931real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( ( member223413699real_n
% 2.86/1.61 @ ( collec321817931real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 2.86/1.61 & ( Q @ X ) ) )
% 2.86/1.61 @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( member223413699real_n
% 2.86/1.61 @ ( collec321817931real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 2.86/1.61 & ( ( Q @ X )
% 2.86/1.61 | ( P @ X ) ) ) )
% 2.86/1.61 @ ( measur1402256771real_n @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_136_null__sets_Osets__Collect__conj,axiom,
% 2.86/1.61 ! [M: sigma_1422848389real_n,P: set_Fi1058188332real_n > $o,Q: set_Fi1058188332real_n > $o] :
% 2.86/1.61 ( ( member1475136633real_n
% 2.86/1.61 @ ( collec452821761real_n
% 2.86/1.61 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 @ ( measur2126959417real_n @ M ) )
% 2.86/1.61 => ( ( member1475136633real_n
% 2.86/1.61 @ ( collec452821761real_n
% 2.86/1.61 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 2.86/1.61 & ( Q @ X ) ) )
% 2.86/1.61 @ ( measur2126959417real_n @ M ) )
% 2.86/1.61 => ( member1475136633real_n
% 2.86/1.61 @ ( collec452821761real_n
% 2.86/1.61 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ X @ ( sigma_607186084real_n @ M ) )
% 2.86/1.61 & ( Q @ X )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 @ ( measur2126959417real_n @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_137_null__sets_Osets__Collect__conj,axiom,
% 2.86/1.61 ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
% 2.86/1.61 ( ( member_set_o
% 2.86/1.61 @ ( collect_o
% 2.86/1.61 @ ^ [X: $o] :
% 2.86/1.61 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 @ ( measure_null_sets_o @ M ) )
% 2.86/1.61 => ( ( member_set_o
% 2.86/1.61 @ ( collect_o
% 2.86/1.61 @ ^ [X: $o] :
% 2.86/1.61 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 2.86/1.61 & ( Q @ X ) ) )
% 2.86/1.61 @ ( measure_null_sets_o @ M ) )
% 2.86/1.61 => ( member_set_o
% 2.86/1.61 @ ( collect_o
% 2.86/1.61 @ ^ [X: $o] :
% 2.86/1.61 ( ( member_o @ X @ ( sigma_space_o @ M ) )
% 2.86/1.61 & ( Q @ X )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 @ ( measure_null_sets_o @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_138_null__sets_Osets__Collect__conj,axiom,
% 2.86/1.61 ! [M: sigma_1466784463real_n,P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
% 2.86/1.61 ( ( member223413699real_n
% 2.86/1.61 @ ( collec321817931real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( ( member223413699real_n
% 2.86/1.61 @ ( collec321817931real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 2.86/1.61 & ( Q @ X ) ) )
% 2.86/1.61 @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( member223413699real_n
% 2.86/1.61 @ ( collec321817931real_n
% 2.86/1.61 @ ^ [X: finite1489363574real_n] :
% 2.86/1.61 ( ( member1352538125real_n @ X @ ( sigma_476185326real_n @ M ) )
% 2.86/1.61 & ( Q @ X )
% 2.86/1.61 & ( P @ X ) ) )
% 2.86/1.61 @ ( measur1402256771real_n @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_139_translation__diff,axiom,
% 2.86/1.61 ! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ ( minus_1686442501real_n @ S2 @ T ) )
% 2.86/1.61 = ( minus_1686442501real_n @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ S2 ) @ ( image_439535603real_n @ ( plus_p585657087real_n @ A ) @ T ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_140_null__set__Diff,axiom,
% 2.86/1.61 ! [B2: set_Fi1058188332real_n,M: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( member223413699real_n @ B2 @ ( measur1402256771real_n @ M ) )
% 2.86/1.61 => ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) )
% 2.86/1.61 => ( member223413699real_n @ ( minus_1686442501real_n @ B2 @ A3 ) @ ( measur1402256771real_n @ M ) ) ) ) ).
% 2.86/1.61
% 2.86/1.61 thf(fact_141_Int__Union2,axiom,
% 2.86/1.61 ! [B2: set_se2111327970real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.61 ( ( inf_in1974387902real_n @ ( comple825005695real_n @ B2 ) @ A3 )
% 2.86/1.61 = ( comple825005695real_n
% 2.86/1.61 @ ( image_1661509983real_n
% 2.86/1.61 @ ^ [C3: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ C3 @ A3 )
% 2.86/1.61 @ B2 ) ) ) ).
% 2.86/1.61
% 2.86/1.62 thf(fact_142_Int__Union,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 2.86/1.62 ( ( inf_in1974387902real_n @ A3 @ ( comple825005695real_n @ B2 ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_1661509983real_n @ ( inf_in1974387902real_n @ A3 ) @ B2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_143_vimage__Union,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( vimage1233683625real_n @ F @ ( comple825005695real_n @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_1661509983real_n @ ( vimage1233683625real_n @ F ) @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_144_UN__extend__simps_I6_J,axiom,
% 2.86/1.62 ! [A3: nat > set_Fi1058188332real_n,C2: set_nat,B2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( minus_1686442501real_n @ ( comple825005695real_n @ ( image_1856576259real_n @ A3 @ C2 ) ) @ B2 )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [X: nat] : ( minus_1686442501real_n @ ( A3 @ X ) @ B2 )
% 2.86/1.62 @ C2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_145_UN__extend__simps_I6_J,axiom,
% 2.86/1.62 ! [A3: finite964658038_int_n > set_Fi1058188332real_n,C2: set_Fi160064172_int_n,B2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( minus_1686442501real_n @ ( comple825005695real_n @ ( image_355963305real_n @ A3 @ C2 ) ) @ B2 )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] : ( minus_1686442501real_n @ ( A3 @ X ) @ B2 )
% 2.86/1.62 @ C2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_146_emeasure__Diff__null__set,axiom,
% 2.86/1.62 ! [B2: set_Fi1058188332real_n,M: sigma_1466784463real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ B2 @ ( measur1402256771real_n @ M ) )
% 2.86/1.62 => ( ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) )
% 2.86/1.62 => ( ( sigma_1536574303real_n @ M @ ( minus_1686442501real_n @ A3 @ B2 ) )
% 2.86/1.62 = ( sigma_1536574303real_n @ M @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_147_Sup_OSUP__cong,axiom,
% 2.86/1.62 ! [A3: set_nat,B2: set_nat,C2: nat > set_Fi1058188332real_n,D: nat > set_Fi1058188332real_n,Sup: set_se2111327970real_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: nat] :
% 2.86/1.62 ( ( member_nat @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( Sup @ ( image_1856576259real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( Sup @ ( image_1856576259real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_148_Sup_OSUP__cong,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: finite1489363574real_n > finite1489363574real_n,D: finite1489363574real_n > finite1489363574real_n,Sup: set_Fi1058188332real_n > finite1489363574real_n] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( Sup @ ( image_439535603real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( Sup @ ( image_439535603real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_149_Sup_OSUP__cong,axiom,
% 2.86/1.62 ! [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] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( Sup @ ( image_355963305real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( Sup @ ( image_355963305real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_150_Inf_OINF__cong,axiom,
% 2.86/1.62 ! [A3: set_nat,B2: set_nat,C2: nat > set_Fi1058188332real_n,D: nat > set_Fi1058188332real_n,Inf: set_se2111327970real_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: nat] :
% 2.86/1.62 ( ( member_nat @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( Inf @ ( image_1856576259real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( Inf @ ( image_1856576259real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_151_Inf_OINF__cong,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: finite1489363574real_n > finite1489363574real_n,D: finite1489363574real_n > finite1489363574real_n,Inf: set_Fi1058188332real_n > finite1489363574real_n] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( Inf @ ( image_439535603real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( Inf @ ( image_439535603real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_152_Inf_OINF__cong,axiom,
% 2.86/1.62 ! [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] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( Inf @ ( image_355963305real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( Inf @ ( image_355963305real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_153_rev__image__eqI,axiom,
% 2.86/1.62 ! [X3: finite1489363574real_n,A3: set_Fi1058188332real_n,B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X3 @ A3 )
% 2.86/1.62 => ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_154_rev__image__eqI,axiom,
% 2.86/1.62 ! [X3: nat,A3: set_nat,B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member_nat @ X3 @ A3 )
% 2.86/1.62 => ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_155_rev__image__eqI,axiom,
% 2.86/1.62 ! [X3: finite964658038_int_n,A3: set_Fi160064172_int_n,B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X3 @ A3 )
% 2.86/1.62 => ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member223413699real_n @ B @ ( image_355963305real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_156_rev__image__eqI,axiom,
% 2.86/1.62 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X3 @ A3 )
% 2.86/1.62 => ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_157_rev__image__eqI,axiom,
% 2.86/1.62 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,B: $o,F: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ( member223413699real_n @ X3 @ A3 )
% 2.86/1.62 => ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member_o @ B @ ( image_1648361637al_n_o @ F @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_158_rev__image__eqI,axiom,
% 2.86/1.62 ! [X3: $o,A3: set_o,B: set_Fi1058188332real_n,F: $o > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member_o @ X3 @ A3 )
% 2.86/1.62 => ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member223413699real_n @ B @ ( image_1759008383real_n @ F @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_159_rev__image__eqI,axiom,
% 2.86/1.62 ! [X3: $o,A3: set_o,B: $o,F: $o > $o] :
% 2.86/1.62 ( ( member_o @ X3 @ A3 )
% 2.86/1.62 => ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member_o @ B @ ( image_o_o @ F @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_160_ball__imageD,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,A3: set_nat,P: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X2 @ ( image_1856576259real_n @ F @ A3 ) )
% 2.86/1.62 => ( P @ X2 ) )
% 2.86/1.62 => ! [X5: nat] :
% 2.86/1.62 ( ( member_nat @ X5 @ A3 )
% 2.86/1.62 => ( P @ ( F @ X5 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_161_ball__imageD,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
% 2.86/1.62 ( ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X2 @ ( image_439535603real_n @ F @ A3 ) )
% 2.86/1.62 => ( P @ X2 ) )
% 2.86/1.62 => ! [X5: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X5 @ A3 )
% 2.86/1.62 => ( P @ ( F @ X5 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_162_ball__imageD,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X2 @ ( image_355963305real_n @ F @ A3 ) )
% 2.86/1.62 => ( P @ X2 ) )
% 2.86/1.62 => ! [X5: finite964658038_int_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X5 @ A3 )
% 2.86/1.62 => ( P @ ( F @ X5 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_163_image__cong,axiom,
% 2.86/1.62 ! [M: set_nat,N2: set_nat,F: nat > set_Fi1058188332real_n,G: nat > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( M = N2 )
% 2.86/1.62 => ( ! [X2: nat] :
% 2.86/1.62 ( ( member_nat @ X2 @ N2 )
% 2.86/1.62 => ( ( F @ X2 )
% 2.86/1.62 = ( G @ X2 ) ) )
% 2.86/1.62 => ( ( image_1856576259real_n @ F @ M )
% 2.86/1.62 = ( image_1856576259real_n @ G @ N2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_164_image__cong,axiom,
% 2.86/1.62 ! [M: set_Fi1058188332real_n,N2: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n] :
% 2.86/1.62 ( ( M = N2 )
% 2.86/1.62 => ( ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X2 @ N2 )
% 2.86/1.62 => ( ( F @ X2 )
% 2.86/1.62 = ( G @ X2 ) ) )
% 2.86/1.62 => ( ( image_439535603real_n @ F @ M )
% 2.86/1.62 = ( image_439535603real_n @ G @ N2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_165_image__cong,axiom,
% 2.86/1.62 ! [M: set_Fi160064172_int_n,N2: set_Fi160064172_int_n,F: finite964658038_int_n > set_Fi1058188332real_n,G: finite964658038_int_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( M = N2 )
% 2.86/1.62 => ( ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X2 @ N2 )
% 2.86/1.62 => ( ( F @ X2 )
% 2.86/1.62 = ( G @ X2 ) ) )
% 2.86/1.62 => ( ( image_355963305real_n @ F @ M )
% 2.86/1.62 = ( image_355963305real_n @ G @ N2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_166_bex__imageD,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,A3: set_nat,P: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ? [X5: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X5 @ ( image_1856576259real_n @ F @ A3 ) )
% 2.86/1.62 & ( P @ X5 ) )
% 2.86/1.62 => ? [X2: nat] :
% 2.86/1.62 ( ( member_nat @ X2 @ A3 )
% 2.86/1.62 & ( P @ ( F @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_167_bex__imageD,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
% 2.86/1.62 ( ? [X5: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X5 @ ( image_439535603real_n @ F @ A3 ) )
% 2.86/1.62 & ( P @ X5 ) )
% 2.86/1.62 => ? [X2: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X2 @ A3 )
% 2.86/1.62 & ( P @ ( F @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_168_bex__imageD,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ? [X5: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X5 @ ( image_355963305real_n @ F @ A3 ) )
% 2.86/1.62 & ( P @ X5 ) )
% 2.86/1.62 => ? [X2: finite964658038_int_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X2 @ A3 )
% 2.86/1.62 & ( P @ ( F @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_169_image__iff,axiom,
% 2.86/1.62 ! [Z: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ Z @ ( image_439535603real_n @ F @ A3 ) )
% 2.86/1.62 = ( ? [X: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X @ A3 )
% 2.86/1.62 & ( Z
% 2.86/1.62 = ( F @ X ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_170_image__iff,axiom,
% 2.86/1.62 ! [Z: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,A3: set_nat] :
% 2.86/1.62 ( ( member223413699real_n @ Z @ ( image_1856576259real_n @ F @ A3 ) )
% 2.86/1.62 = ( ? [X: nat] :
% 2.86/1.62 ( ( member_nat @ X @ A3 )
% 2.86/1.62 & ( Z
% 2.86/1.62 = ( F @ X ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_171_image__iff,axiom,
% 2.86/1.62 ! [Z: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( member223413699real_n @ Z @ ( image_355963305real_n @ F @ A3 ) )
% 2.86/1.62 = ( ? [X: finite964658038_int_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X @ A3 )
% 2.86/1.62 & ( Z
% 2.86/1.62 = ( F @ X ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_172_imageI,axiom,
% 2.86/1.62 ! [X3: finite1489363574real_n,A3: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X3 @ A3 )
% 2.86/1.62 => ( member1352538125real_n @ ( F @ X3 ) @ ( image_439535603real_n @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_173_imageI,axiom,
% 2.86/1.62 ! [X3: nat,A3: set_nat,F: nat > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member_nat @ X3 @ A3 )
% 2.86/1.62 => ( member223413699real_n @ ( F @ X3 ) @ ( image_1856576259real_n @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_174_imageI,axiom,
% 2.86/1.62 ! [X3: finite964658038_int_n,A3: set_Fi160064172_int_n,F: finite964658038_int_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X3 @ A3 )
% 2.86/1.62 => ( member223413699real_n @ ( F @ X3 ) @ ( image_355963305real_n @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_175_imageI,axiom,
% 2.86/1.62 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X3 @ A3 )
% 2.86/1.62 => ( member223413699real_n @ ( F @ X3 ) @ ( image_1661509983real_n @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_176_imageI,axiom,
% 2.86/1.62 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,F: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ( member223413699real_n @ X3 @ A3 )
% 2.86/1.62 => ( member_o @ ( F @ X3 ) @ ( image_1648361637al_n_o @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_177_imageI,axiom,
% 2.86/1.62 ! [X3: $o,A3: set_o,F: $o > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member_o @ X3 @ A3 )
% 2.86/1.62 => ( member223413699real_n @ ( F @ X3 ) @ ( image_1759008383real_n @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_178_imageI,axiom,
% 2.86/1.62 ! [X3: $o,A3: set_o,F: $o > $o] :
% 2.86/1.62 ( ( member_o @ X3 @ A3 )
% 2.86/1.62 => ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_179_is__num__normalize_I1_J,axiom,
% 2.86/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] :
% 2.86/1.62 ( ( plus_p585657087real_n @ ( plus_p585657087real_n @ A @ B ) @ C )
% 2.86/1.62 = ( plus_p585657087real_n @ A @ ( plus_p585657087real_n @ B @ C ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_180_UNIV__witness,axiom,
% 2.86/1.62 ? [X2: set_Fi1058188332real_n] : ( member223413699real_n @ X2 @ top_to20708754real_n ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_181_UNIV__witness,axiom,
% 2.86/1.62 ? [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_182_UNIV__witness,axiom,
% 2.86/1.62 ? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_183_UNIV__witness,axiom,
% 2.86/1.62 ? [X2: finite964658038_int_n] : ( member27055245_int_n @ X2 @ top_to131672412_int_n ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_184_UNIV__eq__I,axiom,
% 2.86/1.62 ! [A3: set_se2111327970real_n] :
% 2.86/1.62 ( ! [X2: set_Fi1058188332real_n] : ( member223413699real_n @ X2 @ A3 )
% 2.86/1.62 => ( top_to20708754real_n = A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_185_UNIV__eq__I,axiom,
% 2.86/1.62 ! [A3: set_o] :
% 2.86/1.62 ( ! [X2: $o] : ( member_o @ X2 @ A3 )
% 2.86/1.62 => ( top_top_set_o = A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_186_UNIV__eq__I,axiom,
% 2.86/1.62 ! [A3: set_nat] :
% 2.86/1.62 ( ! [X2: nat] : ( member_nat @ X2 @ A3 )
% 2.86/1.62 => ( top_top_set_nat = A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_187_UNIV__eq__I,axiom,
% 2.86/1.62 ! [A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ! [X2: finite964658038_int_n] : ( member27055245_int_n @ X2 @ A3 )
% 2.86/1.62 => ( top_to131672412_int_n = A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_188_Int__left__commute,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( inf_in1974387902real_n @ A3 @ ( inf_in1974387902real_n @ B2 @ C2 ) )
% 2.86/1.62 = ( inf_in1974387902real_n @ B2 @ ( inf_in1974387902real_n @ A3 @ C2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_189_Int__left__absorb,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( inf_in1974387902real_n @ A3 @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 2.86/1.62 = ( inf_in1974387902real_n @ A3 @ B2 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_190_Int__commute,axiom,
% 2.86/1.62 ( inf_in1974387902real_n
% 2.86/1.62 = ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] : ( inf_in1974387902real_n @ B3 @ A4 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_191_Int__absorb,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( inf_in1974387902real_n @ A3 @ A3 )
% 2.86/1.62 = A3 ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_192_Int__assoc,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( inf_in1974387902real_n @ ( inf_in1974387902real_n @ A3 @ B2 ) @ C2 )
% 2.86/1.62 = ( inf_in1974387902real_n @ A3 @ ( inf_in1974387902real_n @ B2 @ C2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_193_IntD2,axiom,
% 2.86/1.62 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
% 2.86/1.62 => ( member223413699real_n @ C @ B2 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_194_IntD2,axiom,
% 2.86/1.62 ! [C: $o,A3: set_o,B2: set_o] :
% 2.86/1.62 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 2.86/1.62 => ( member_o @ C @ B2 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_195_IntD2,axiom,
% 2.86/1.62 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 2.86/1.62 => ( member1352538125real_n @ C @ B2 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_196_IntD1,axiom,
% 2.86/1.62 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
% 2.86/1.62 => ( member223413699real_n @ C @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_197_IntD1,axiom,
% 2.86/1.62 ! [C: $o,A3: set_o,B2: set_o] :
% 2.86/1.62 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 2.86/1.62 => ( member_o @ C @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_198_IntD1,axiom,
% 2.86/1.62 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 2.86/1.62 => ( member1352538125real_n @ C @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_199_IntE,axiom,
% 2.86/1.62 ! [C: set_Fi1058188332real_n,A3: set_se2111327970real_n,B2: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ C @ ( inf_in632889204real_n @ A3 @ B2 ) )
% 2.86/1.62 => ~ ( ( member223413699real_n @ C @ A3 )
% 2.86/1.62 => ~ ( member223413699real_n @ C @ B2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_200_IntE,axiom,
% 2.86/1.62 ! [C: $o,A3: set_o,B2: set_o] :
% 2.86/1.62 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 2.86/1.62 => ~ ( ( member_o @ C @ A3 )
% 2.86/1.62 => ~ ( member_o @ C @ B2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_201_IntE,axiom,
% 2.86/1.62 ! [C: finite1489363574real_n,A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ C @ ( inf_in1974387902real_n @ A3 @ B2 ) )
% 2.86/1.62 => ~ ( ( member1352538125real_n @ C @ A3 )
% 2.86/1.62 => ~ ( member1352538125real_n @ C @ B2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_202_UnionE,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n,C2: set_se820660888real_n] :
% 2.86/1.62 ( ( member223413699real_n @ A3 @ ( comple1917283637real_n @ C2 ) )
% 2.86/1.62 => ~ ! [X6: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ A3 @ X6 )
% 2.86/1.62 => ~ ( member1475136633real_n @ X6 @ C2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_203_UnionE,axiom,
% 2.86/1.62 ! [A3: $o,C2: set_set_o] :
% 2.86/1.62 ( ( member_o @ A3 @ ( comple1665300069_set_o @ C2 ) )
% 2.86/1.62 => ~ ! [X6: set_o] :
% 2.86/1.62 ( ( member_o @ A3 @ X6 )
% 2.86/1.62 => ~ ( member_set_o @ X6 @ C2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_204_UnionE,axiom,
% 2.86/1.62 ! [A3: finite1489363574real_n,C2: set_se2111327970real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ A3 @ ( comple825005695real_n @ C2 ) )
% 2.86/1.62 => ~ ! [X6: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ A3 @ X6 )
% 2.86/1.62 => ~ ( member223413699real_n @ X6 @ C2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_205_vimage__Collect,axiom,
% 2.86/1.62 ! [P: finite1489363574real_n > $o,F: finite1489363574real_n > finite1489363574real_n,Q: finite1489363574real_n > $o] :
% 2.86/1.62 ( ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( ( P @ ( F @ X2 ) )
% 2.86/1.62 = ( Q @ X2 ) )
% 2.86/1.62 => ( ( vimage1233683625real_n @ F @ ( collec321817931real_n @ P ) )
% 2.86/1.62 = ( collec321817931real_n @ Q ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_206_vimageI2,axiom,
% 2.86/1.62 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A: set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ ( F @ A ) @ A3 )
% 2.86/1.62 => ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_207_vimageI2,axiom,
% 2.86/1.62 ! [F: $o > set_Fi1058188332real_n,A: $o,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ ( F @ A ) @ A3 )
% 2.86/1.62 => ( member_o @ A @ ( vimage961837641real_n @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_208_vimageI2,axiom,
% 2.86/1.62 ! [F: set_Fi1058188332real_n > $o,A: set_Fi1058188332real_n,A3: set_o] :
% 2.86/1.62 ( ( member_o @ ( F @ A ) @ A3 )
% 2.86/1.62 => ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_209_vimageI2,axiom,
% 2.86/1.62 ! [F: $o > $o,A: $o,A3: set_o] :
% 2.86/1.62 ( ( member_o @ ( F @ A ) @ A3 )
% 2.86/1.62 => ( member_o @ A @ ( vimage_o_o @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_210_vimageI2,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,A: finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ ( F @ A ) @ A3 )
% 2.86/1.62 => ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_211_vimageE,axiom,
% 2.86/1.62 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ B2 ) )
% 2.86/1.62 => ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_212_vimageE,axiom,
% 2.86/1.62 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > $o,B2: set_o] :
% 2.86/1.62 ( ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ B2 ) )
% 2.86/1.62 => ( member_o @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_213_vimageE,axiom,
% 2.86/1.62 ! [A: $o,F: $o > set_Fi1058188332real_n,B2: set_se2111327970real_n] :
% 2.86/1.62 ( ( member_o @ A @ ( vimage961837641real_n @ F @ B2 ) )
% 2.86/1.62 => ( member223413699real_n @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_214_vimageE,axiom,
% 2.86/1.62 ! [A: $o,F: $o > $o,B2: set_o] :
% 2.86/1.62 ( ( member_o @ A @ ( vimage_o_o @ F @ B2 ) )
% 2.86/1.62 => ( member_o @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_215_vimageE,axiom,
% 2.86/1.62 ! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,B2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ B2 ) )
% 2.86/1.62 => ( member1352538125real_n @ ( F @ A ) @ B2 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_216_vimageD,axiom,
% 2.86/1.62 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ A @ ( vimage784510485real_n @ F @ A3 ) )
% 2.86/1.62 => ( member223413699real_n @ ( F @ A ) @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_217_vimageD,axiom,
% 2.86/1.62 ! [A: set_Fi1058188332real_n,F: set_Fi1058188332real_n > $o,A3: set_o] :
% 2.86/1.62 ( ( member223413699real_n @ A @ ( vimage851190895al_n_o @ F @ A3 ) )
% 2.86/1.62 => ( member_o @ ( F @ A ) @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_218_vimageD,axiom,
% 2.86/1.62 ! [A: $o,F: $o > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( member_o @ A @ ( vimage961837641real_n @ F @ A3 ) )
% 2.86/1.62 => ( member223413699real_n @ ( F @ A ) @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_219_vimageD,axiom,
% 2.86/1.62 ! [A: $o,F: $o > $o,A3: set_o] :
% 2.86/1.62 ( ( member_o @ A @ ( vimage_o_o @ F @ A3 ) )
% 2.86/1.62 => ( member_o @ ( F @ A ) @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_220_vimageD,axiom,
% 2.86/1.62 ! [A: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ A @ ( vimage1233683625real_n @ F @ A3 ) )
% 2.86/1.62 => ( member1352538125real_n @ ( F @ A ) @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_221_Sup_OSUP__identity__eq,axiom,
% 2.86/1.62 ! [Sup: set_Fi1058188332real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( Sup
% 2.86/1.62 @ ( image_439535603real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] : X
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 = ( Sup @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_222_Inf_OINF__identity__eq,axiom,
% 2.86/1.62 ! [Inf: set_Fi1058188332real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( Inf
% 2.86/1.62 @ ( image_439535603real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] : X
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 = ( Inf @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_223_Compr__image__eq,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
% 2.86/1.62 ( ( collec321817931real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X @ ( image_439535603real_n @ F @ A3 ) )
% 2.86/1.62 & ( P @ X ) ) )
% 2.86/1.62 = ( image_439535603real_n @ F
% 2.86/1.62 @ ( collec321817931real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X @ A3 )
% 2.86/1.62 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_224_Compr__image__eq,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,A3: set_nat,P: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ( collec452821761real_n
% 2.86/1.62 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X @ ( image_1856576259real_n @ F @ A3 ) )
% 2.86/1.62 & ( P @ X ) ) )
% 2.86/1.62 = ( image_1856576259real_n @ F
% 2.86/1.62 @ ( collect_nat
% 2.86/1.62 @ ^ [X: nat] :
% 2.86/1.62 ( ( member_nat @ X @ A3 )
% 2.86/1.62 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_225_Compr__image__eq,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n,P: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ( collec452821761real_n
% 2.86/1.62 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X @ ( image_355963305real_n @ F @ A3 ) )
% 2.86/1.62 & ( P @ X ) ) )
% 2.86/1.62 = ( image_355963305real_n @ F
% 2.86/1.62 @ ( collec1941932235_int_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X @ A3 )
% 2.86/1.62 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_226_Compr__image__eq,axiom,
% 2.86/1.62 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n,P: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ( collec452821761real_n
% 2.86/1.62 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X @ ( image_1661509983real_n @ F @ A3 ) )
% 2.86/1.62 & ( P @ X ) ) )
% 2.86/1.62 = ( image_1661509983real_n @ F
% 2.86/1.62 @ ( collec452821761real_n
% 2.86/1.62 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X @ A3 )
% 2.86/1.62 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_227_Compr__image__eq,axiom,
% 2.86/1.62 ! [F: $o > set_Fi1058188332real_n,A3: set_o,P: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ( collec452821761real_n
% 2.86/1.62 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X @ ( image_1759008383real_n @ F @ A3 ) )
% 2.86/1.62 & ( P @ X ) ) )
% 2.86/1.62 = ( image_1759008383real_n @ F
% 2.86/1.62 @ ( collect_o
% 2.86/1.62 @ ^ [X: $o] :
% 2.86/1.62 ( ( member_o @ X @ A3 )
% 2.86/1.62 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_228_Compr__image__eq,axiom,
% 2.86/1.62 ! [F: set_Fi1058188332real_n > $o,A3: set_se2111327970real_n,P: $o > $o] :
% 2.86/1.62 ( ( collect_o
% 2.86/1.62 @ ^ [X: $o] :
% 2.86/1.62 ( ( member_o @ X @ ( image_1648361637al_n_o @ F @ A3 ) )
% 2.86/1.62 & ( P @ X ) ) )
% 2.86/1.62 = ( image_1648361637al_n_o @ F
% 2.86/1.62 @ ( collec452821761real_n
% 2.86/1.62 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X @ A3 )
% 2.86/1.62 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_229_Compr__image__eq,axiom,
% 2.86/1.62 ! [F: $o > $o,A3: set_o,P: $o > $o] :
% 2.86/1.62 ( ( collect_o
% 2.86/1.62 @ ^ [X: $o] :
% 2.86/1.62 ( ( member_o @ X @ ( image_o_o @ F @ A3 ) )
% 2.86/1.62 & ( P @ X ) ) )
% 2.86/1.62 = ( image_o_o @ F
% 2.86/1.62 @ ( collect_o
% 2.86/1.62 @ ^ [X: $o] :
% 2.86/1.62 ( ( member_o @ X @ A3 )
% 2.86/1.62 & ( P @ ( F @ X ) ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_230_image__image,axiom,
% 2.86/1.62 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: nat > set_Fi1058188332real_n,A3: set_nat] :
% 2.86/1.62 ( ( image_1661509983real_n @ F @ ( image_1856576259real_n @ G @ A3 ) )
% 2.86/1.62 = ( image_1856576259real_n
% 2.86/1.62 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 2.86/1.62 @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_231_image__image,axiom,
% 2.86/1.62 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( image_1661509983real_n @ F @ ( image_355963305real_n @ G @ A3 ) )
% 2.86/1.62 = ( image_355963305real_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 2.86/1.62 @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_232_image__image,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,G: nat > nat,A3: set_nat] :
% 2.86/1.62 ( ( image_1856576259real_n @ F @ ( image_nat_nat @ G @ A3 ) )
% 2.86/1.62 = ( image_1856576259real_n
% 2.86/1.62 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 2.86/1.62 @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_233_image__image,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,G: finite964658038_int_n > nat,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( image_1856576259real_n @ F @ ( image_497739341_n_nat @ G @ A3 ) )
% 2.86/1.62 = ( image_355963305real_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 2.86/1.62 @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_234_image__image,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( image_439535603real_n @ F @ ( image_439535603real_n @ G @ A3 ) )
% 2.86/1.62 = ( image_439535603real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] : ( F @ ( G @ X ) )
% 2.86/1.62 @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_235_image__image,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,G: nat > finite964658038_int_n,A3: set_nat] :
% 2.86/1.62 ( ( image_355963305real_n @ F @ ( image_1961920973_int_n @ G @ A3 ) )
% 2.86/1.62 = ( image_1856576259real_n
% 2.86/1.62 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 2.86/1.62 @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_236_image__image,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,G: finite964658038_int_n > finite964658038_int_n,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( image_355963305real_n @ F @ ( image_1278151539_int_n @ G @ A3 ) )
% 2.86/1.62 = ( image_355963305real_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 2.86/1.62 @ A3 ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_237_imageE,axiom,
% 2.86/1.62 ! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ A3 ) )
% 2.86/1.62 => ~ ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X2 ) )
% 2.86/1.62 => ~ ( member1352538125real_n @ X2 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_238_imageE,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,A3: set_nat] :
% 2.86/1.62 ( ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ A3 ) )
% 2.86/1.62 => ~ ! [X2: nat] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X2 ) )
% 2.86/1.62 => ~ ( member_nat @ X2 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_239_imageE,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( member223413699real_n @ B @ ( image_355963305real_n @ F @ A3 ) )
% 2.86/1.62 => ~ ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X2 ) )
% 2.86/1.62 => ~ ( member27055245_int_n @ X2 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_240_imageE,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,F: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ B @ ( image_1661509983real_n @ F @ A3 ) )
% 2.86/1.62 => ~ ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X2 ) )
% 2.86/1.62 => ~ ( member223413699real_n @ X2 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_241_imageE,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,F: $o > set_Fi1058188332real_n,A3: set_o] :
% 2.86/1.62 ( ( member223413699real_n @ B @ ( image_1759008383real_n @ F @ A3 ) )
% 2.86/1.62 => ~ ! [X2: $o] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X2 ) )
% 2.86/1.62 => ~ ( member_o @ X2 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_242_imageE,axiom,
% 2.86/1.62 ! [B: $o,F: set_Fi1058188332real_n > $o,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( member_o @ B @ ( image_1648361637al_n_o @ F @ A3 ) )
% 2.86/1.62 => ~ ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X2 ) )
% 2.86/1.62 => ~ ( member223413699real_n @ X2 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_243_imageE,axiom,
% 2.86/1.62 ! [B: $o,F: $o > $o,A3: set_o] :
% 2.86/1.62 ( ( member_o @ B @ ( image_o_o @ F @ A3 ) )
% 2.86/1.62 => ~ ! [X2: $o] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X2 ) )
% 2.86/1.62 => ~ ( member_o @ X2 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_244_UNIV__def,axiom,
% 2.86/1.62 ( top_top_set_nat
% 2.86/1.62 = ( collect_nat
% 2.86/1.62 @ ^ [X: nat] : $true ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_245_UNIV__def,axiom,
% 2.86/1.62 ( top_to131672412_int_n
% 2.86/1.62 = ( collec1941932235_int_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] : $true ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_246_Collect__conj__eq,axiom,
% 2.86/1.62 ! [P: finite1489363574real_n > $o,Q: finite1489363574real_n > $o] :
% 2.86/1.62 ( ( collec321817931real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] :
% 2.86/1.62 ( ( P @ X )
% 2.86/1.62 & ( Q @ X ) ) )
% 2.86/1.62 = ( inf_in1974387902real_n @ ( collec321817931real_n @ P ) @ ( collec321817931real_n @ Q ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_247_Int__Collect,axiom,
% 2.86/1.62 ! [X3: set_Fi1058188332real_n,A3: set_se2111327970real_n,P: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ( member223413699real_n @ X3 @ ( inf_in632889204real_n @ A3 @ ( collec452821761real_n @ P ) ) )
% 2.86/1.62 = ( ( member223413699real_n @ X3 @ A3 )
% 2.86/1.62 & ( P @ X3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_248_Int__Collect,axiom,
% 2.86/1.62 ! [X3: $o,A3: set_o,P: $o > $o] :
% 2.86/1.62 ( ( member_o @ X3 @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
% 2.86/1.62 = ( ( member_o @ X3 @ A3 )
% 2.86/1.62 & ( P @ X3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_249_Int__Collect,axiom,
% 2.86/1.62 ! [X3: finite1489363574real_n,A3: set_Fi1058188332real_n,P: finite1489363574real_n > $o] :
% 2.86/1.62 ( ( member1352538125real_n @ X3 @ ( inf_in1974387902real_n @ A3 @ ( collec321817931real_n @ P ) ) )
% 2.86/1.62 = ( ( member1352538125real_n @ X3 @ A3 )
% 2.86/1.62 & ( P @ X3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_250_Int__def,axiom,
% 2.86/1.62 ( inf_in632889204real_n
% 2.86/1.62 = ( ^ [A4: set_se2111327970real_n,B3: set_se2111327970real_n] :
% 2.86/1.62 ( collec452821761real_n
% 2.86/1.62 @ ^ [X: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X @ A4 )
% 2.86/1.62 & ( member223413699real_n @ X @ B3 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_251_Int__def,axiom,
% 2.86/1.62 ( inf_inf_set_o
% 2.86/1.62 = ( ^ [A4: set_o,B3: set_o] :
% 2.86/1.62 ( collect_o
% 2.86/1.62 @ ^ [X: $o] :
% 2.86/1.62 ( ( member_o @ X @ A4 )
% 2.86/1.62 & ( member_o @ X @ B3 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_252_Int__def,axiom,
% 2.86/1.62 ( inf_in1974387902real_n
% 2.86/1.62 = ( ^ [A4: set_Fi1058188332real_n,B3: set_Fi1058188332real_n] :
% 2.86/1.62 ( collec321817931real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X @ A4 )
% 2.86/1.62 & ( member1352538125real_n @ X @ B3 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_253_vimage__def,axiom,
% 2.86/1.62 ( vimage1233683625real_n
% 2.86/1.62 = ( ^ [F2: finite1489363574real_n > finite1489363574real_n,B3: set_Fi1058188332real_n] :
% 2.86/1.62 ( collec321817931real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] : ( member1352538125real_n @ ( F2 @ X ) @ B3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_254_less__numeral__extra_I3_J,axiom,
% 2.86/1.62 ~ ( ord_le2133614988nnreal @ zero_z1963244097nnreal @ zero_z1963244097nnreal ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_255_less__numeral__extra_I4_J,axiom,
% 2.86/1.62 ~ ( ord_le2133614988nnreal @ one_on705384445nnreal @ one_on705384445nnreal ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_256_range__eqI,axiom,
% 2.86/1.62 ! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n,X3: finite1489363574real_n] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ top_to1292442332real_n ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_257_range__eqI,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n,X3: nat] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ top_top_set_nat ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_258_range__eqI,axiom,
% 2.86/1.62 ! [B: $o,F: nat > $o,X3: nat] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_259_range__eqI,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n,X3: finite964658038_int_n] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member223413699real_n @ B @ ( image_355963305real_n @ F @ top_to131672412_int_n ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_260_range__eqI,axiom,
% 2.86/1.62 ! [B: $o,F: finite964658038_int_n > $o,X3: finite964658038_int_n] :
% 2.86/1.62 ( ( B
% 2.86/1.62 = ( F @ X3 ) )
% 2.86/1.62 => ( member_o @ B @ ( image_216309723nt_n_o @ F @ top_to131672412_int_n ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_261_surj__def,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n] :
% 2.86/1.62 ( ( ( image_439535603real_n @ F @ top_to1292442332real_n )
% 2.86/1.62 = top_to1292442332real_n )
% 2.86/1.62 = ( ! [Y2: finite1489363574real_n] :
% 2.86/1.62 ? [X: finite1489363574real_n] :
% 2.86/1.62 ( Y2
% 2.86/1.62 = ( F @ X ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_262_surj__def,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( ( image_1856576259real_n @ F @ top_top_set_nat )
% 2.86/1.62 = top_to20708754real_n )
% 2.86/1.62 = ( ! [Y2: set_Fi1058188332real_n] :
% 2.86/1.62 ? [X: nat] :
% 2.86/1.62 ( Y2
% 2.86/1.62 = ( F @ X ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_263_surj__def,axiom,
% 2.86/1.62 ! [F: nat > nat] :
% 2.86/1.62 ( ( ( image_nat_nat @ F @ top_top_set_nat )
% 2.86/1.62 = top_top_set_nat )
% 2.86/1.62 = ( ! [Y2: nat] :
% 2.86/1.62 ? [X: nat] :
% 2.86/1.62 ( Y2
% 2.86/1.62 = ( F @ X ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_264_surj__def,axiom,
% 2.86/1.62 ! [F: nat > finite964658038_int_n] :
% 2.86/1.62 ( ( ( image_1961920973_int_n @ F @ top_top_set_nat )
% 2.86/1.62 = top_to131672412_int_n )
% 2.86/1.62 = ( ! [Y2: finite964658038_int_n] :
% 2.86/1.62 ? [X: nat] :
% 2.86/1.62 ( Y2
% 2.86/1.62 = ( F @ X ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_265_surj__def,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( ( image_355963305real_n @ F @ top_to131672412_int_n )
% 2.86/1.62 = top_to20708754real_n )
% 2.86/1.62 = ( ! [Y2: set_Fi1058188332real_n] :
% 2.86/1.62 ? [X: finite964658038_int_n] :
% 2.86/1.62 ( Y2
% 2.86/1.62 = ( F @ X ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_266_surj__def,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > nat] :
% 2.86/1.62 ( ( ( image_497739341_n_nat @ F @ top_to131672412_int_n )
% 2.86/1.62 = top_top_set_nat )
% 2.86/1.62 = ( ! [Y2: nat] :
% 2.86/1.62 ? [X: finite964658038_int_n] :
% 2.86/1.62 ( Y2
% 2.86/1.62 = ( F @ X ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_267_surj__def,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > finite964658038_int_n] :
% 2.86/1.62 ( ( ( image_1278151539_int_n @ F @ top_to131672412_int_n )
% 2.86/1.62 = top_to131672412_int_n )
% 2.86/1.62 = ( ! [Y2: finite964658038_int_n] :
% 2.86/1.62 ? [X: finite964658038_int_n] :
% 2.86/1.62 ( Y2
% 2.86/1.62 = ( F @ X ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_268_rangeI,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,X3: finite1489363574real_n] : ( member1352538125real_n @ ( F @ X3 ) @ ( image_439535603real_n @ F @ top_to1292442332real_n ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_269_rangeI,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,X3: nat] : ( member223413699real_n @ ( F @ X3 ) @ ( image_1856576259real_n @ F @ top_top_set_nat ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_270_rangeI,axiom,
% 2.86/1.62 ! [F: nat > $o,X3: nat] : ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_271_rangeI,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,X3: finite964658038_int_n] : ( member223413699real_n @ ( F @ X3 ) @ ( image_355963305real_n @ F @ top_to131672412_int_n ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_272_rangeI,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > $o,X3: finite964658038_int_n] : ( member_o @ ( F @ X3 ) @ ( image_216309723nt_n_o @ F @ top_to131672412_int_n ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_273_surjI,axiom,
% 2.86/1.62 ! [G: finite1489363574real_n > finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
% 2.86/1.62 ( ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( ( G @ ( F @ X2 ) )
% 2.86/1.62 = X2 )
% 2.86/1.62 => ( ( image_439535603real_n @ G @ top_to1292442332real_n )
% 2.86/1.62 = top_to1292442332real_n ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_274_surjI,axiom,
% 2.86/1.62 ! [G: nat > set_Fi1058188332real_n,F: set_Fi1058188332real_n > nat] :
% 2.86/1.62 ( ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( G @ ( F @ X2 ) )
% 2.86/1.62 = X2 )
% 2.86/1.62 => ( ( image_1856576259real_n @ G @ top_top_set_nat )
% 2.86/1.62 = top_to20708754real_n ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_275_surjI,axiom,
% 2.86/1.62 ! [G: nat > nat,F: nat > nat] :
% 2.86/1.62 ( ! [X2: nat] :
% 2.86/1.62 ( ( G @ ( F @ X2 ) )
% 2.86/1.62 = X2 )
% 2.86/1.62 => ( ( image_nat_nat @ G @ top_top_set_nat )
% 2.86/1.62 = top_top_set_nat ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_276_surjI,axiom,
% 2.86/1.62 ! [G: nat > finite964658038_int_n,F: finite964658038_int_n > nat] :
% 2.86/1.62 ( ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( ( G @ ( F @ X2 ) )
% 2.86/1.62 = X2 )
% 2.86/1.62 => ( ( image_1961920973_int_n @ G @ top_top_set_nat )
% 2.86/1.62 = top_to131672412_int_n ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_277_surjI,axiom,
% 2.86/1.62 ! [G: finite964658038_int_n > set_Fi1058188332real_n,F: set_Fi1058188332real_n > finite964658038_int_n] :
% 2.86/1.62 ( ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( G @ ( F @ X2 ) )
% 2.86/1.62 = X2 )
% 2.86/1.62 => ( ( image_355963305real_n @ G @ top_to131672412_int_n )
% 2.86/1.62 = top_to20708754real_n ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_278_surjI,axiom,
% 2.86/1.62 ! [G: finite964658038_int_n > nat,F: nat > finite964658038_int_n] :
% 2.86/1.62 ( ! [X2: nat] :
% 2.86/1.62 ( ( G @ ( F @ X2 ) )
% 2.86/1.62 = X2 )
% 2.86/1.62 => ( ( image_497739341_n_nat @ G @ top_to131672412_int_n )
% 2.86/1.62 = top_top_set_nat ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_279_surjI,axiom,
% 2.86/1.62 ! [G: finite964658038_int_n > finite964658038_int_n,F: finite964658038_int_n > finite964658038_int_n] :
% 2.86/1.62 ( ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( ( G @ ( F @ X2 ) )
% 2.86/1.62 = X2 )
% 2.86/1.62 => ( ( image_1278151539_int_n @ G @ top_to131672412_int_n )
% 2.86/1.62 = top_to131672412_int_n ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_280_surjE,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,Y4: finite1489363574real_n] :
% 2.86/1.62 ( ( ( image_439535603real_n @ F @ top_to1292442332real_n )
% 2.86/1.62 = top_to1292442332real_n )
% 2.86/1.62 => ~ ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( Y4
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_281_surjE,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( ( image_1856576259real_n @ F @ top_top_set_nat )
% 2.86/1.62 = top_to20708754real_n )
% 2.86/1.62 => ~ ! [X2: nat] :
% 2.86/1.62 ( Y4
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_282_surjE,axiom,
% 2.86/1.62 ! [F: nat > nat,Y4: nat] :
% 2.86/1.62 ( ( ( image_nat_nat @ F @ top_top_set_nat )
% 2.86/1.62 = top_top_set_nat )
% 2.86/1.62 => ~ ! [X2: nat] :
% 2.86/1.62 ( Y4
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_283_surjE,axiom,
% 2.86/1.62 ! [F: nat > finite964658038_int_n,Y4: finite964658038_int_n] :
% 2.86/1.62 ( ( ( image_1961920973_int_n @ F @ top_top_set_nat )
% 2.86/1.62 = top_to131672412_int_n )
% 2.86/1.62 => ~ ! [X2: nat] :
% 2.86/1.62 ( Y4
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_284_surjE,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( ( image_355963305real_n @ F @ top_to131672412_int_n )
% 2.86/1.62 = top_to20708754real_n )
% 2.86/1.62 => ~ ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( Y4
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_285_surjE,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > nat,Y4: nat] :
% 2.86/1.62 ( ( ( image_497739341_n_nat @ F @ top_to131672412_int_n )
% 2.86/1.62 = top_top_set_nat )
% 2.86/1.62 => ~ ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( Y4
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_286_surjE,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > finite964658038_int_n,Y4: finite964658038_int_n] :
% 2.86/1.62 ( ( ( image_1278151539_int_n @ F @ top_to131672412_int_n )
% 2.86/1.62 = top_to131672412_int_n )
% 2.86/1.62 => ~ ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( Y4
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_287_surjD,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,Y4: finite1489363574real_n] :
% 2.86/1.62 ( ( ( image_439535603real_n @ F @ top_to1292442332real_n )
% 2.86/1.62 = top_to1292442332real_n )
% 2.86/1.62 => ? [X2: finite1489363574real_n] :
% 2.86/1.62 ( Y4
% 2.86/1.62 = ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_288_surjD,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( ( image_1856576259real_n @ F @ top_top_set_nat )
% 2.86/1.62 = top_to20708754real_n )
% 2.86/1.62 => ? [X2: nat] :
% 2.86/1.62 ( Y4
% 2.86/1.62 = ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_289_surjD,axiom,
% 2.86/1.62 ! [F: nat > nat,Y4: nat] :
% 2.86/1.62 ( ( ( image_nat_nat @ F @ top_top_set_nat )
% 2.86/1.62 = top_top_set_nat )
% 2.86/1.62 => ? [X2: nat] :
% 2.86/1.62 ( Y4
% 2.86/1.62 = ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_290_surjD,axiom,
% 2.86/1.62 ! [F: nat > finite964658038_int_n,Y4: finite964658038_int_n] :
% 2.86/1.62 ( ( ( image_1961920973_int_n @ F @ top_top_set_nat )
% 2.86/1.62 = top_to131672412_int_n )
% 2.86/1.62 => ? [X2: nat] :
% 2.86/1.62 ( Y4
% 2.86/1.62 = ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_291_surjD,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,Y4: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( ( image_355963305real_n @ F @ top_to131672412_int_n )
% 2.86/1.62 = top_to20708754real_n )
% 2.86/1.62 => ? [X2: finite964658038_int_n] :
% 2.86/1.62 ( Y4
% 2.86/1.62 = ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_292_surjD,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > nat,Y4: nat] :
% 2.86/1.62 ( ( ( image_497739341_n_nat @ F @ top_to131672412_int_n )
% 2.86/1.62 = top_top_set_nat )
% 2.86/1.62 => ? [X2: finite964658038_int_n] :
% 2.86/1.62 ( Y4
% 2.86/1.62 = ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_293_surjD,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > finite964658038_int_n,Y4: finite964658038_int_n] :
% 2.86/1.62 ( ( ( image_1278151539_int_n @ F @ top_to131672412_int_n )
% 2.86/1.62 = top_to131672412_int_n )
% 2.86/1.62 => ? [X2: finite964658038_int_n] :
% 2.86/1.62 ( Y4
% 2.86/1.62 = ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_294_less__Sup__iff,axiom,
% 2.86/1.62 ! [A: extend1728876344nnreal,S: set_Ex113815278nnreal] :
% 2.86/1.62 ( ( ord_le2133614988nnreal @ A @ ( comple1413366923nnreal @ S ) )
% 2.86/1.62 = ( ? [X: extend1728876344nnreal] :
% 2.86/1.62 ( ( member1217042383nnreal @ X @ S )
% 2.86/1.62 & ( ord_le2133614988nnreal @ A @ X ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_295_SUP__cong,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n,B2: set_Fi1058188332real_n,C2: finite1489363574real_n > finite1489363574real_n,D: finite1489363574real_n > finite1489363574real_n] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( comple2042271945real_n @ ( image_439535603real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( comple2042271945real_n @ ( image_439535603real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_296_SUP__cong,axiom,
% 2.86/1.62 ! [A3: set_nat,B2: set_nat,C2: nat > set_Fi1058188332real_n,D: nat > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: nat] :
% 2.86/1.62 ( ( member_nat @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_1856576259real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_297_SUP__cong,axiom,
% 2.86/1.62 ! [A3: set_Fi160064172_int_n,B2: set_Fi160064172_int_n,C2: finite964658038_int_n > set_Fi1058188332real_n,D: finite964658038_int_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( comple825005695real_n @ ( image_355963305real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_355963305real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_298_SUP__cong,axiom,
% 2.86/1.62 ! [A3: set_se2111327970real_n,B2: set_se2111327970real_n,C2: set_Fi1058188332real_n > set_Fi1058188332real_n,D: set_Fi1058188332real_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( comple825005695real_n @ ( image_1661509983real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_1661509983real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_299_SUP__cong,axiom,
% 2.86/1.62 ! [A3: set_o,B2: set_o,C2: $o > set_Fi1058188332real_n,D: $o > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: $o] :
% 2.86/1.62 ( ( member_o @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( comple825005695real_n @ ( image_1759008383real_n @ C2 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_1759008383real_n @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_300_SUP__cong,axiom,
% 2.86/1.62 ! [A3: set_se2111327970real_n,B2: set_se2111327970real_n,C2: set_Fi1058188332real_n > $o,D: set_Fi1058188332real_n > $o] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( complete_Sup_Sup_o @ ( image_1648361637al_n_o @ C2 @ A3 ) )
% 2.86/1.62 = ( complete_Sup_Sup_o @ ( image_1648361637al_n_o @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_301_SUP__cong,axiom,
% 2.86/1.62 ! [A3: set_o,B2: set_o,C2: $o > $o,D: $o > $o] :
% 2.86/1.62 ( ( A3 = B2 )
% 2.86/1.62 => ( ! [X2: $o] :
% 2.86/1.62 ( ( member_o @ X2 @ B2 )
% 2.86/1.62 => ( ( C2 @ X2 )
% 2.86/1.62 = ( D @ X2 ) ) )
% 2.86/1.62 => ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A3 ) )
% 2.86/1.62 = ( complete_Sup_Sup_o @ ( image_o_o @ D @ B2 ) ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_302_Int__UNIV__right,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( inf_in1974387902real_n @ A3 @ top_to1292442332real_n )
% 2.86/1.62 = A3 ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_303_Int__UNIV__right,axiom,
% 2.86/1.62 ! [A3: set_nat] :
% 2.86/1.62 ( ( inf_inf_set_nat @ A3 @ top_top_set_nat )
% 2.86/1.62 = A3 ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_304_Int__UNIV__right,axiom,
% 2.86/1.62 ! [A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( inf_in1108485182_int_n @ A3 @ top_to131672412_int_n )
% 2.86/1.62 = A3 ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_305_Int__UNIV__left,axiom,
% 2.86/1.62 ! [B2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( inf_in1974387902real_n @ top_to1292442332real_n @ B2 )
% 2.86/1.62 = B2 ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_306_Int__UNIV__left,axiom,
% 2.86/1.62 ! [B2: set_nat] :
% 2.86/1.62 ( ( inf_inf_set_nat @ top_top_set_nat @ B2 )
% 2.86/1.62 = B2 ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_307_Int__UNIV__left,axiom,
% 2.86/1.62 ! [B2: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( inf_in1108485182_int_n @ top_to131672412_int_n @ B2 )
% 2.86/1.62 = B2 ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_308_Union__UNIV,axiom,
% 2.86/1.62 ( ( comple1682161881et_nat @ top_top_set_set_nat )
% 2.86/1.62 = top_top_set_nat ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_309_Union__UNIV,axiom,
% 2.86/1.62 ( ( comple970917503_int_n @ top_to1587634578_int_n )
% 2.86/1.62 = top_to131672412_int_n ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_310_Union__UNIV,axiom,
% 2.86/1.62 ( ( comple825005695real_n @ top_to20708754real_n )
% 2.86/1.62 = top_to1292442332real_n ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_311_null__setsD2,axiom,
% 2.86/1.62 ! [A3: set_Fi1058188332real_n,M: sigma_1466784463real_n] :
% 2.86/1.62 ( ( member223413699real_n @ A3 @ ( measur1402256771real_n @ M ) )
% 2.86/1.62 => ( member223413699real_n @ A3 @ ( sigma_1235138647real_n @ M ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_312_vimage__inter__cong,axiom,
% 2.86/1.62 ! [S: set_Fi1058188332real_n,F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n,Y4: set_Fi1058188332real_n] :
% 2.86/1.62 ( ! [W: finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ W @ S )
% 2.86/1.62 => ( ( F @ W )
% 2.86/1.62 = ( G @ W ) ) )
% 2.86/1.62 => ( ( inf_in1974387902real_n @ ( vimage1233683625real_n @ F @ Y4 ) @ S )
% 2.86/1.62 = ( inf_in1974387902real_n @ ( vimage1233683625real_n @ G @ Y4 ) @ S ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_313_translation__subtract__diff,axiom,
% 2.86/1.62 ! [A: finite1489363574real_n,S2: set_Fi1058188332real_n,T: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( image_439535603real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
% 2.86/1.62 @ ( minus_1686442501real_n @ S2 @ T ) )
% 2.86/1.62 = ( minus_1686442501real_n
% 2.86/1.62 @ ( image_439535603real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
% 2.86/1.62 @ S2 )
% 2.86/1.62 @ ( image_439535603real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] : ( minus_1037315151real_n @ X @ A )
% 2.86/1.62 @ T ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_314_range__composition,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,G: finite1489363574real_n > finite1489363574real_n] :
% 2.86/1.62 ( ( image_439535603real_n
% 2.86/1.62 @ ^ [X: finite1489363574real_n] : ( F @ ( G @ X ) )
% 2.86/1.62 @ top_to1292442332real_n )
% 2.86/1.62 = ( image_439535603real_n @ F @ ( image_439535603real_n @ G @ top_to1292442332real_n ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_315_range__composition,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,G: nat > finite1489363574real_n] :
% 2.86/1.62 ( ( image_183184717real_n
% 2.86/1.62 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 2.86/1.62 @ top_top_set_nat )
% 2.86/1.62 = ( image_439535603real_n @ F @ ( image_183184717real_n @ G @ top_top_set_nat ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_316_range__composition,axiom,
% 2.86/1.62 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: nat > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( image_1856576259real_n
% 2.86/1.62 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 2.86/1.62 @ top_top_set_nat )
% 2.86/1.62 = ( image_1661509983real_n @ F @ ( image_1856576259real_n @ G @ top_top_set_nat ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_317_range__composition,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,G: nat > nat] :
% 2.86/1.62 ( ( image_1856576259real_n
% 2.86/1.62 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 2.86/1.62 @ top_top_set_nat )
% 2.86/1.62 = ( image_1856576259real_n @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_318_range__composition,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,G: nat > finite964658038_int_n] :
% 2.86/1.62 ( ( image_1856576259real_n
% 2.86/1.62 @ ^ [X: nat] : ( F @ ( G @ X ) )
% 2.86/1.62 @ top_top_set_nat )
% 2.86/1.62 = ( image_355963305real_n @ F @ ( image_1961920973_int_n @ G @ top_top_set_nat ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_319_range__composition,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,G: finite964658038_int_n > finite1489363574real_n] :
% 2.86/1.62 ( ( image_2058828787real_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 2.86/1.62 @ top_to131672412_int_n )
% 2.86/1.62 = ( image_439535603real_n @ F @ ( image_2058828787real_n @ G @ top_to131672412_int_n ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_320_range__composition,axiom,
% 2.86/1.62 ! [F: set_Fi1058188332real_n > set_Fi1058188332real_n,G: finite964658038_int_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( image_355963305real_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 2.86/1.62 @ top_to131672412_int_n )
% 2.86/1.62 = ( image_1661509983real_n @ F @ ( image_355963305real_n @ G @ top_to131672412_int_n ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_321_range__composition,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,G: finite964658038_int_n > nat] :
% 2.86/1.62 ( ( image_355963305real_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 2.86/1.62 @ top_to131672412_int_n )
% 2.86/1.62 = ( image_1856576259real_n @ F @ ( image_497739341_n_nat @ G @ top_to131672412_int_n ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_322_range__composition,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,G: finite964658038_int_n > finite964658038_int_n] :
% 2.86/1.62 ( ( image_355963305real_n
% 2.86/1.62 @ ^ [X: finite964658038_int_n] : ( F @ ( G @ X ) )
% 2.86/1.62 @ top_to131672412_int_n )
% 2.86/1.62 = ( image_355963305real_n @ F @ ( image_1278151539_int_n @ G @ top_to131672412_int_n ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_323_rangeE,axiom,
% 2.86/1.62 ! [B: finite1489363574real_n,F: finite1489363574real_n > finite1489363574real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ B @ ( image_439535603real_n @ F @ top_to1292442332real_n ) )
% 2.86/1.62 => ~ ! [X2: finite1489363574real_n] :
% 2.86/1.62 ( B
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_324_rangeE,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,F: nat > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ B @ ( image_1856576259real_n @ F @ top_top_set_nat ) )
% 2.86/1.62 => ~ ! [X2: nat] :
% 2.86/1.62 ( B
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_325_rangeE,axiom,
% 2.86/1.62 ! [B: $o,F: nat > $o] :
% 2.86/1.62 ( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
% 2.86/1.62 => ~ ! [X2: nat] :
% 2.86/1.62 ( B
% 2.86/1.62 = ( ~ ( F @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_326_rangeE,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,F: finite964658038_int_n > set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ B @ ( image_355963305real_n @ F @ top_to131672412_int_n ) )
% 2.86/1.62 => ~ ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( B
% 2.86/1.62 != ( F @ X2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_327_rangeE,axiom,
% 2.86/1.62 ! [B: $o,F: finite964658038_int_n > $o] :
% 2.86/1.62 ( ( member_o @ B @ ( image_216309723nt_n_o @ F @ top_to131672412_int_n ) )
% 2.86/1.62 => ~ ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( B
% 2.86/1.62 = ( ~ ( F @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_328_SUP__commute,axiom,
% 2.86/1.62 ! [F: nat > nat > set_Fi1058188332real_n,B2: set_nat,A3: set_nat] :
% 2.86/1.62 ( ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [I4: nat] : ( comple825005695real_n @ ( image_1856576259real_n @ ( F @ I4 ) @ B2 ) )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [J: nat] :
% 2.86/1.62 ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [I4: nat] : ( F @ I4 @ J )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 @ B2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_329_SUP__commute,axiom,
% 2.86/1.62 ! [F: nat > finite964658038_int_n > set_Fi1058188332real_n,B2: set_Fi160064172_int_n,A3: set_nat] :
% 2.86/1.62 ( ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [I4: nat] : ( comple825005695real_n @ ( image_355963305real_n @ ( F @ I4 ) @ B2 ) )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [J: finite964658038_int_n] :
% 2.86/1.62 ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [I4: nat] : ( F @ I4 @ J )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 @ B2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_330_SUP__commute,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > nat > set_Fi1058188332real_n,B2: set_nat,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [I4: finite964658038_int_n] : ( comple825005695real_n @ ( image_1856576259real_n @ ( F @ I4 ) @ B2 ) )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [J: nat] :
% 2.86/1.62 ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [I4: finite964658038_int_n] : ( F @ I4 @ J )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 @ B2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_331_SUP__commute,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > finite964658038_int_n > set_Fi1058188332real_n,B2: set_Fi160064172_int_n,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [I4: finite964658038_int_n] : ( comple825005695real_n @ ( image_355963305real_n @ ( F @ I4 ) @ B2 ) )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [J: finite964658038_int_n] :
% 2.86/1.62 ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [I4: finite964658038_int_n] : ( F @ I4 @ J )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 @ B2 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_332_image__Union,axiom,
% 2.86/1.62 ! [F: nat > set_Fi1058188332real_n,S: set_set_nat] :
% 2.86/1.62 ( ( image_1856576259real_n @ F @ ( comple1682161881et_nat @ S ) )
% 2.86/1.62 = ( comple1917283637real_n @ ( image_1587769199real_n @ ( image_1856576259real_n @ F ) @ S ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_333_image__Union,axiom,
% 2.86/1.62 ! [F: finite964658038_int_n > set_Fi1058188332real_n,S: set_se944069346_int_n] :
% 2.86/1.62 ( ( image_355963305real_n @ F @ ( comple970917503_int_n @ S ) )
% 2.86/1.62 = ( comple1917283637real_n @ ( image_1054146965real_n @ ( image_355963305real_n @ F ) @ S ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_334_image__Union,axiom,
% 2.86/1.62 ! [F: finite1489363574real_n > finite1489363574real_n,S: set_se2111327970real_n] :
% 2.86/1.62 ( ( image_439535603real_n @ F @ ( comple825005695real_n @ S ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_1661509983real_n @ ( image_439535603real_n @ F ) @ S ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_335_UN__UN__flatten,axiom,
% 2.86/1.62 ! [C2: nat > set_Fi1058188332real_n,B2: nat > set_nat,A3: set_nat] :
% 2.86/1.62 ( ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( comple1682161881et_nat @ ( image_nat_set_nat @ B2 @ A3 ) ) ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [Y2: nat] : ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( B2 @ Y2 ) ) )
% 2.86/1.62 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_336_UN__UN__flatten,axiom,
% 2.86/1.62 ! [C2: nat > set_Fi1058188332real_n,B2: finite964658038_int_n > set_nat,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( comple1682161881et_nat @ ( image_1085873667et_nat @ B2 @ A3 ) ) ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [Y2: finite964658038_int_n] : ( comple825005695real_n @ ( image_1856576259real_n @ C2 @ ( B2 @ Y2 ) ) )
% 2.86/1.62 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_337_UN__UN__flatten,axiom,
% 2.86/1.62 ! [C2: finite964658038_int_n > set_Fi1058188332real_n,B2: nat > set_Fi160064172_int_n,A3: set_nat] :
% 2.86/1.62 ( ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( comple970917503_int_n @ ( image_968789251_int_n @ B2 @ A3 ) ) ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [Y2: nat] : ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( B2 @ Y2 ) ) )
% 2.86/1.62 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_338_UN__UN__flatten,axiom,
% 2.86/1.62 ! [C2: finite964658038_int_n > set_Fi1058188332real_n,B2: finite964658038_int_n > set_Fi160064172_int_n,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( comple970917503_int_n @ ( image_1819506345_int_n @ B2 @ A3 ) ) ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [Y2: finite964658038_int_n] : ( comple825005695real_n @ ( image_355963305real_n @ C2 @ ( B2 @ Y2 ) ) )
% 2.86/1.62 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_339_UN__UN__flatten,axiom,
% 2.86/1.62 ! [C2: finite1489363574real_n > set_Fi1058188332real_n,B2: nat > set_Fi1058188332real_n,A3: set_nat] :
% 2.86/1.62 ( ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_1856576259real_n
% 2.86/1.62 @ ^ [Y2: nat] : ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( B2 @ Y2 ) ) )
% 2.86/1.62 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_340_UN__UN__flatten,axiom,
% 2.86/1.62 ! [C2: finite1489363574real_n > set_Fi1058188332real_n,B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) ) )
% 2.86/1.62 = ( comple825005695real_n
% 2.86/1.62 @ ( image_355963305real_n
% 2.86/1.62 @ ^ [Y2: finite964658038_int_n] : ( comple825005695real_n @ ( image_545463721real_n @ C2 @ ( B2 @ Y2 ) ) )
% 2.86/1.62 @ A3 ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_341_UN__E,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,B2: set_Fi1058188332real_n > set_se2111327970real_n,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_797440021real_n @ B2 @ A3 ) ) )
% 2.86/1.62 => ~ ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X2 @ A3 )
% 2.86/1.62 => ~ ( member223413699real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_342_UN__E,axiom,
% 2.86/1.62 ! [B: set_Fi1058188332real_n,B2: $o > set_se2111327970real_n,A3: set_o] :
% 2.86/1.62 ( ( member223413699real_n @ B @ ( comple1917283637real_n @ ( image_452144437real_n @ B2 @ A3 ) ) )
% 2.86/1.62 => ~ ! [X2: $o] :
% 2.86/1.62 ( ( member_o @ X2 @ A3 )
% 2.86/1.62 => ~ ( member223413699real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_343_UN__E,axiom,
% 2.86/1.62 ! [B: $o,B2: set_Fi1058188332real_n > set_o,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( member_o @ B @ ( comple1665300069_set_o @ ( image_1687589765_set_o @ B2 @ A3 ) ) )
% 2.86/1.62 => ~ ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X2 @ A3 )
% 2.86/1.62 => ~ ( member_o @ B @ ( B2 @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_344_UN__E,axiom,
% 2.86/1.62 ! [B: $o,B2: $o > set_o,A3: set_o] :
% 2.86/1.62 ( ( member_o @ B @ ( comple1665300069_set_o @ ( image_o_set_o @ B2 @ A3 ) ) )
% 2.86/1.62 => ~ ! [X2: $o] :
% 2.86/1.62 ( ( member_o @ X2 @ A3 )
% 2.86/1.62 => ~ ( member_o @ B @ ( B2 @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_345_UN__E,axiom,
% 2.86/1.62 ! [B: finite1489363574real_n,B2: nat > set_Fi1058188332real_n,A3: set_nat] :
% 2.86/1.62 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ A3 ) ) )
% 2.86/1.62 => ~ ! [X2: nat] :
% 2.86/1.62 ( ( member_nat @ X2 @ A3 )
% 2.86/1.62 => ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_346_UN__E,axiom,
% 2.86/1.62 ! [B: finite1489363574real_n,B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_Fi160064172_int_n] :
% 2.86/1.62 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_355963305real_n @ B2 @ A3 ) ) )
% 2.86/1.62 => ~ ! [X2: finite964658038_int_n] :
% 2.86/1.62 ( ( member27055245_int_n @ X2 @ A3 )
% 2.86/1.62 => ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_347_UN__E,axiom,
% 2.86/1.62 ! [B: finite1489363574real_n,B2: set_Fi1058188332real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1661509983real_n @ B2 @ A3 ) ) )
% 2.86/1.62 => ~ ! [X2: set_Fi1058188332real_n] :
% 2.86/1.62 ( ( member223413699real_n @ X2 @ A3 )
% 2.86/1.62 => ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_348_UN__E,axiom,
% 2.86/1.62 ! [B: finite1489363574real_n,B2: $o > set_Fi1058188332real_n,A3: set_o] :
% 2.86/1.62 ( ( member1352538125real_n @ B @ ( comple825005695real_n @ ( image_1759008383real_n @ B2 @ A3 ) ) )
% 2.86/1.62 => ~ ! [X2: $o] :
% 2.86/1.62 ( ( member_o @ X2 @ A3 )
% 2.86/1.62 => ~ ( member1352538125real_n @ B @ ( B2 @ X2 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_349_UN__extend__simps_I8_J,axiom,
% 2.86/1.62 ! [B2: nat > set_Fi1058188332real_n,A3: set_set_nat] :
% 2.86/1.62 ( ( comple825005695real_n
% 2.86/1.62 @ ( image_933134521real_n
% 2.86/1.62 @ ^ [Y2: set_nat] : ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ Y2 ) )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_1856576259real_n @ B2 @ ( comple1682161881et_nat @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_350_UN__extend__simps_I8_J,axiom,
% 2.86/1.62 ! [B2: finite964658038_int_n > set_Fi1058188332real_n,A3: set_se944069346_int_n] :
% 2.86/1.62 ( ( comple825005695real_n
% 2.86/1.62 @ ( image_792439519real_n
% 2.86/1.62 @ ^ [Y2: set_Fi160064172_int_n] : ( comple825005695real_n @ ( image_355963305real_n @ B2 @ Y2 ) )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_355963305real_n @ B2 @ ( comple970917503_int_n @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_351_UN__extend__simps_I8_J,axiom,
% 2.86/1.62 ! [B2: finite1489363574real_n > set_Fi1058188332real_n,A3: set_se2111327970real_n] :
% 2.86/1.62 ( ( comple825005695real_n
% 2.86/1.62 @ ( image_1661509983real_n
% 2.86/1.62 @ ^ [Y2: set_Fi1058188332real_n] : ( comple825005695real_n @ ( image_545463721real_n @ B2 @ Y2 ) )
% 2.86/1.62 @ A3 ) )
% 2.86/1.62 = ( comple825005695real_n @ ( image_545463721real_n @ B2 @ ( comple825005695real_n @ A3 ) ) ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(fact_352_Sup__bool__def,axiom,
% 2.86/1.62 ( complete_Sup_Sup_o
% 2.86/1.62 = ( member_o @ $true ) ) ).
% 2.86/1.62
% 2.86/1.62 thf(conj_0,conjecture,
% 2.86/1.62 ( ( ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t2 @ ( f @ N ) ) ) )
% 2.86/1.62 = ( ^ [N: nat] : ( sigma_1536574303real_n @ ( comple230862828real_n @ lebesg260170249real_n ) @ ( t @ ( f @ N ) ) ) ) ) ).
% 2.86/1.62 ------------------------
% 2.86/1.66 % (7264)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/4Mi)
% 2.86/1.66 % (7269)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/18Mi)
% 2.86/1.66 % (7264)Instruction limit reached!
% 2.86/1.66 % (7264)------------------------------
% 2.86/1.66 % (7264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.66 % (7264)Termination reason: Unknown
% 2.86/1.66 % (7264)Termination phase: shuffling
% 2.86/1.66
% 2.86/1.66 % (7264)Memory used [KB]: 1535
% 2.86/1.66 % (7264)Time elapsed: 0.003 s
% 2.86/1.66 % (7264)Instructions burned: 6 (million)
% 2.86/1.66 % (7264)------------------------------
% 2.86/1.66 % (7264)------------------------------
% 2.86/1.66 % (7265)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/27Mi)
% 2.86/1.66 % (7263)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/183Mi)
% 2.86/1.66 % (7266)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/2Mi)
% 2.86/1.67 % (7267)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/2Mi)
% 2.86/1.67 % (7268)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/275Mi)
% 2.86/1.67 % (7266)Instruction limit reached!
% 2.86/1.67 % (7266)------------------------------
% 2.86/1.67 % (7266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.67 % (7266)Termination reason: Unknown
% 2.86/1.67 % (7266)Termination phase: shuffling
% 2.86/1.67
% 2.86/1.67 % (7267)Instruction limit reached!
% 2.86/1.67 % (7267)------------------------------
% 2.86/1.67 % (7267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.67 % (7266)Memory used [KB]: 1407
% 2.86/1.67 % (7266)Time elapsed: 0.004 s
% 2.86/1.67 % (7266)Instructions burned: 2 (million)
% 2.86/1.67 % (7266)------------------------------
% 2.86/1.67 % (7266)------------------------------
% 2.86/1.67 % (7267)Termination reason: Unknown
% 2.86/1.67 % (7267)Termination phase: shuffling
% 2.86/1.67
% 2.86/1.67 % (7267)Memory used [KB]: 1407
% 2.86/1.67 % (7267)Time elapsed: 0.004 s
% 2.86/1.67 % (7267)Instructions burned: 2 (million)
% 2.86/1.67 % (7267)------------------------------
% 2.86/1.67 % (7267)------------------------------
% 2.86/1.67 % (7269)Instruction limit reached!
% 2.86/1.67 % (7269)------------------------------
% 2.86/1.67 % (7269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.67 % (7269)Termination reason: Unknown
% 2.86/1.67 % (7269)Termination phase: shuffling
% 2.86/1.67
% 2.86/1.67 % (7269)Memory used [KB]: 1791
% 2.86/1.67 % (7269)Time elapsed: 0.010 s
% 2.86/1.67 % (7269)Instructions burned: 19 (million)
% 2.86/1.67 % (7269)------------------------------
% 2.86/1.67 % (7269)------------------------------
% 2.86/1.67 % (7270)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/3Mi)
% 2.86/1.67 % (7270)Instruction limit reached!
% 2.86/1.67 % (7270)------------------------------
% 2.86/1.67 % (7270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.67 % (7270)Termination reason: Unknown
% 2.86/1.67 % (7270)Termination phase: shuffling
% 2.86/1.67
% 2.86/1.67 % (7270)Memory used [KB]: 1407
% 2.86/1.67 % (7270)Time elapsed: 0.002 s
% 2.86/1.67 % (7270)Instructions burned: 3 (million)
% 2.86/1.67 % (7270)------------------------------
% 2.86/1.67 % (7270)------------------------------
% 2.86/1.68 % (7265)Instruction limit reached!
% 2.86/1.68 % (7265)------------------------------
% 2.86/1.68 % (7265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.68 % (7265)Termination reason: Unknown
% 2.86/1.68 % (7265)Termination phase: shuffling
% 2.86/1.68
% 2.86/1.68 % (7265)Memory used [KB]: 1918
% 2.86/1.68 % (7265)Time elapsed: 0.016 s
% 2.86/1.68 % (7265)Instructions burned: 27 (million)
% 2.86/1.68 % (7265)------------------------------
% 2.86/1.68 % (7265)------------------------------
% 2.86/1.68 % (7274)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/1041Mi)
% 2.86/1.68 % (7271)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/37Mi)
% 2.86/1.68 % (7272)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/15Mi)
% 2.86/1.69 % (7273)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/3Mi)
% 2.86/1.69 % (7273)Instruction limit reached!
% 2.86/1.69 % (7273)------------------------------
% 2.86/1.69 % (7273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.69 % (7273)Termination reason: Unknown
% 2.86/1.69 % (7273)Termination phase: shuffling
% 2.86/1.69
% 2.86/1.69 % (7273)Memory used [KB]: 1535
% 2.86/1.69 % (7273)Time elapsed: 0.003 s
% 2.86/1.69 % (7273)Instructions burned: 3 (million)
% 2.86/1.69 % (7273)------------------------------
% 2.86/1.69 % (7273)------------------------------
% 2.86/1.69 % (7275)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7164 for (2999ds/7Mi)
% 2.86/1.69 % (7275)Instruction limit reached!
% 2.86/1.69 % (7275)------------------------------
% 2.86/1.69 % (7275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.69 % (7275)Termination reason: Unknown
% 2.86/1.69 % (7275)Termination phase: shuffling
% 2.86/1.69
% 2.86/1.69 % (7275)Memory used [KB]: 1663
% 2.86/1.69 % (7275)Time elapsed: 0.004 s
% 2.86/1.69 % (7275)Instructions burned: 8 (million)
% 2.86/1.69 % (7275)------------------------------
% 2.86/1.69 % (7275)------------------------------
% 2.86/1.69 % (7272)Instruction limit reached!
% 2.86/1.69 % (7272)------------------------------
% 2.86/1.69 % (7272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.69 % (7272)Termination reason: Unknown
% 2.86/1.69 % (7272)Termination phase: shuffling
% 2.86/1.69
% 2.86/1.69 % (7272)Memory used [KB]: 1791
% 2.86/1.69 % (7272)Time elapsed: 0.012 s
% 2.86/1.69 % (7272)Instructions burned: 15 (million)
% 2.86/1.69 % (7272)------------------------------
% 2.86/1.69 % (7272)------------------------------
% 2.91/1.70 % (7268)First to succeed.
% 2.91/1.70 % (7268)Refutation found. Thanks to Tanya!
% 2.91/1.70 % SZS status Theorem for DTF2THF_7164
% 2.91/1.70 % SZS output start Proof for DTF2THF_7164
% 2.91/1.70 thf(type_def_5, type, set_o: $tType).
% 2.91/1.70 thf(type_def_7, type, set_Ex113815278nnreal: $tType).
% 2.91/1.70 thf(type_def_8, type, extend1728876344nnreal: $tType).
% 2.91/1.70 thf(type_def_9, type, set_Fi1058188332real_n: $tType).
% 2.91/1.70 thf(type_def_10, type, finite1489363574real_n: $tType).
% 2.91/1.70 thf(type_def_11, type, set_set_o: $tType).
% 2.91/1.70 thf(type_def_12, type, set_se944069346_int_n: $tType).
% 2.91/1.70 thf(type_def_13, type, set_Fi160064172_int_n: $tType).
% 2.91/1.70 thf(type_def_14, type, set_se2111327970real_n: $tType).
% 2.91/1.70 thf(type_def_15, type, set_set_nat: $tType).
% 2.91/1.70 thf(type_def_16, type, set_nat: $tType).
% 2.91/1.70 thf(type_def_17, type, set_se820660888real_n: $tType).
% 2.91/1.70 thf(type_def_18, type, set_Si1125517487real_n: $tType).
% 2.91/1.70 thf(type_def_19, type, sigma_1466784463real_n: $tType).
% 2.91/1.70 thf(type_def_20, type, nat: $tType).
% 2.91/1.70 thf(type_def_21, type, finite964658038_int_n: $tType).
% 2.91/1.70 thf(type_def_22, type, sigma_measure_o: $tType).
% 2.91/1.70 thf(type_def_23, type, sigma_1422848389real_n: $tType).
% 2.91/1.70 thf(func_def_0, type, sigma_1422848389real_n: $tType).
% 2.91/1.70 thf(func_def_1, type, set_Si1125517487real_n: $tType).
% 2.91/1.70 thf(func_def_2, type, set_se820660888real_n: $tType).
% 2.91/1.70 thf(func_def_3, type, sigma_1466784463real_n: $tType).
% 2.91/1.70 thf(func_def_4, type, set_se2111327970real_n: $tType).
% 2.91/1.70 thf(func_def_5, type, set_se944069346_int_n: $tType).
% 2.91/1.70 thf(func_def_6, type, set_Fi1058188332real_n: $tType).
% 2.91/1.70 thf(func_def_7, type, set_Fi160064172_int_n: $tType).
% 2.91/1.70 thf(func_def_8, type, finite1489363574real_n: $tType).
% 2.91/1.70 thf(func_def_9, type, finite964658038_int_n: $tType).
% 2.91/1.70 thf(func_def_10, type, set_Ex113815278nnreal: $tType).
% 2.91/1.70 thf(func_def_11, type, set_set_nat: $tType).
% 2.91/1.70 thf(func_def_12, type, extend1728876344nnreal: $tType).
% 2.91/1.70 thf(func_def_13, type, sigma_measure_o: $tType).
% 2.91/1.70 thf(func_def_14, type, set_set_o: $tType).
% 2.91/1.70 thf(func_def_15, type, set_nat: $tType).
% 2.91/1.70 thf(func_def_16, type, set_o: $tType).
% 2.91/1.70 thf(func_def_17, type, nat: $tType).
% 2.91/1.70 thf(func_def_18, type, complete_Sup_Sup_o: set_o > $o).
% 2.91/1.70 thf(func_def_19, type, comple1413366923nnreal: set_Ex113815278nnreal > extend1728876344nnreal).
% 2.91/1.70 thf(func_def_20, type, comple2042271945real_n: set_Fi1058188332real_n > finite1489363574real_n).
% 2.91/1.70 thf(func_def_21, type, comple1665300069_set_o: set_set_o > set_o).
% 2.91/1.70 thf(func_def_22, type, comple970917503_int_n: set_se944069346_int_n > set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_23, type, comple825005695real_n: set_se2111327970real_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_24, type, comple1682161881et_nat: set_set_nat > set_nat).
% 2.91/1.70 thf(func_def_25, type, comple1917283637real_n: set_se820660888real_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_26, type, comple488165692real_n: set_Si1125517487real_n > sigma_1466784463real_n).
% 2.91/1.70 thf(func_def_27, type, comple230862828real_n: sigma_1466784463real_n > sigma_1466784463real_n).
% 2.91/1.70 thf(func_def_28, type, counta1142393929_int_n: set_Fi160064172_int_n > nat > finite964658038_int_n).
% 2.91/1.70 thf(func_def_29, type, minus_1196255695_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n).
% 2.91/1.70 thf(func_def_30, type, minus_1037315151real_n: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n).
% 2.91/1.70 thf(func_def_31, type, minus_1686442501real_n: set_Fi1058188332real_n > set_Fi1058188332real_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_32, type, one_on705384445nnreal: extend1728876344nnreal).
% 2.91/1.70 thf(func_def_33, type, one_on1253059131real_n: finite1489363574real_n).
% 2.91/1.70 thf(func_def_34, type, plus_p1763960001nnreal: extend1728876344nnreal > extend1728876344nnreal > extend1728876344nnreal).
% 2.91/1.70 thf(func_def_35, type, plus_p1654784127_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n).
% 2.91/1.70 thf(func_def_36, type, plus_p585657087real_n: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n).
% 2.91/1.70 thf(func_def_37, type, zero_z1963244097nnreal: extend1728876344nnreal).
% 2.91/1.70 thf(func_def_38, type, zero_z200130687real_n: finite1489363574real_n).
% 2.91/1.70 thf(func_def_39, type, inf_inf_set_o: set_o > set_o > set_o).
% 2.91/1.70 thf(func_def_40, type, inf_in1108485182_int_n: set_Fi160064172_int_n > set_Fi160064172_int_n > set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_41, type, inf_in1974387902real_n: set_Fi1058188332real_n > set_Fi1058188332real_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_42, type, inf_inf_set_nat: set_nat > set_nat > set_nat).
% 2.91/1.70 thf(func_def_43, type, inf_in632889204real_n: set_se2111327970real_n > set_se2111327970real_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_44, type, lebesg260170249real_n: sigma_1466784463real_n).
% 2.91/1.70 thf(func_def_45, type, measure_null_sets_o: sigma_measure_o > set_set_o).
% 2.91/1.70 thf(func_def_46, type, measur1402256771real_n: sigma_1466784463real_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_47, type, measur2126959417real_n: sigma_1422848389real_n > set_se820660888real_n).
% 2.91/1.70 thf(func_def_48, type, minkow1134813771n_real: finite964658038_int_n > finite1489363574real_n).
% 2.91/1.70 thf(func_def_49, type, ord_le2133614988nnreal: extend1728876344nnreal > extend1728876344nnreal > $o).
% 2.91/1.70 thf(func_def_50, type, top_to287930409nt_n_o: finite964658038_int_n > $o).
% 2.91/1.70 thf(func_def_51, type, top_top_nat_o: nat > $o).
% 2.91/1.70 thf(func_def_53, type, top_to1845833192nnreal: extend1728876344nnreal).
% 2.91/1.70 thf(func_def_54, type, top_top_set_o: set_o).
% 2.91/1.70 thf(func_def_55, type, top_to131672412_int_n: set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_56, type, top_to1292442332real_n: set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_57, type, top_top_set_nat: set_nat).
% 2.91/1.70 thf(func_def_58, type, top_to1587634578_int_n: set_se944069346_int_n).
% 2.91/1.70 thf(func_def_59, type, top_to20708754real_n: set_se2111327970real_n).
% 2.91/1.70 thf(func_def_60, type, top_top_set_set_nat: set_set_nat).
% 2.91/1.70 thf(func_def_61, type, sums_E1192373732nnreal: (nat > extend1728876344nnreal) > extend1728876344nnreal > $o).
% 2.91/1.70 thf(func_def_62, type, collect_o: ($o > $o) > set_o).
% 2.91/1.70 thf(func_def_63, type, collec1941932235_int_n: (finite964658038_int_n > $o) > set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_64, type, collec321817931real_n: (finite1489363574real_n > $o) > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_65, type, collect_nat: (nat > $o) > set_nat).
% 2.91/1.70 thf(func_def_66, type, collec452821761real_n: (set_Fi1058188332real_n > $o) > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_67, type, image_o_o: ($o > $o) > set_o > set_o).
% 2.91/1.70 thf(func_def_68, type, image_o_set_o: ($o > set_o) > set_o > set_set_o).
% 2.91/1.70 thf(func_def_69, type, image_1759008383real_n: ($o > set_Fi1058188332real_n) > set_o > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_70, type, image_452144437real_n: ($o > set_se2111327970real_n) > set_o > set_se820660888real_n).
% 2.91/1.70 thf(func_def_71, type, image_1599934780real_n: ($o > sigma_1466784463real_n) > set_o > set_Si1125517487real_n).
% 2.91/1.70 thf(func_def_72, type, image_2066995319nnreal: (extend1728876344nnreal > extend1728876344nnreal) > set_Ex113815278nnreal > set_Ex113815278nnreal).
% 2.91/1.70 thf(func_def_73, type, image_216309723nt_n_o: (finite964658038_int_n > $o) > set_Fi160064172_int_n > set_o).
% 2.91/1.70 thf(func_def_74, type, image_1278151539_int_n: (finite964658038_int_n > finite964658038_int_n) > set_Fi160064172_int_n > set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_75, type, image_2058828787real_n: (finite964658038_int_n > finite1489363574real_n) > set_Fi160064172_int_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_76, type, image_497739341_n_nat: (finite964658038_int_n > nat) > set_Fi160064172_int_n > set_nat).
% 2.91/1.70 thf(func_def_77, type, image_1819506345_int_n: (finite964658038_int_n > set_Fi160064172_int_n) > set_Fi160064172_int_n > set_se944069346_int_n).
% 2.91/1.70 thf(func_def_78, type, image_355963305real_n: (finite964658038_int_n > set_Fi1058188332real_n) > set_Fi160064172_int_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_79, type, image_1085873667et_nat: (finite964658038_int_n > set_nat) > set_Fi160064172_int_n > set_set_nat).
% 2.91/1.70 thf(func_def_80, type, image_439535603real_n: (finite1489363574real_n > finite1489363574real_n) > set_Fi1058188332real_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_81, type, image_545463721real_n: (finite1489363574real_n > set_Fi1058188332real_n) > set_Fi1058188332real_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_82, type, image_nat_o: (nat > $o) > set_nat > set_o).
% 2.91/1.70 thf(func_def_83, type, image_1961920973_int_n: (nat > finite964658038_int_n) > set_nat > set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_84, type, image_183184717real_n: (nat > finite1489363574real_n) > set_nat > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_85, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 2.91/1.70 thf(func_def_86, type, image_968789251_int_n: (nat > set_Fi160064172_int_n) > set_nat > set_se944069346_int_n).
% 2.91/1.70 thf(func_def_87, type, image_1856576259real_n: (nat > set_Fi1058188332real_n) > set_nat > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_88, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 2.91/1.70 thf(func_def_89, type, image_set_o_o: (set_o > $o) > set_set_o > set_o).
% 2.91/1.70 thf(func_def_90, type, image_792439519real_n: (set_Fi160064172_int_n > set_Fi1058188332real_n) > set_se944069346_int_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_91, type, image_1054146965real_n: (set_Fi160064172_int_n > set_se2111327970real_n) > set_se944069346_int_n > set_se820660888real_n).
% 2.91/1.70 thf(func_def_92, type, image_1648361637al_n_o: (set_Fi1058188332real_n > $o) > set_se2111327970real_n > set_o).
% 2.91/1.70 thf(func_def_93, type, image_1687589765_set_o: (set_Fi1058188332real_n > set_o) > set_se2111327970real_n > set_set_o).
% 2.91/1.70 thf(func_def_94, type, image_1661509983real_n: (set_Fi1058188332real_n > set_Fi1058188332real_n) > set_se2111327970real_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_95, type, image_797440021real_n: (set_Fi1058188332real_n > set_se2111327970real_n) > set_se2111327970real_n > set_se820660888real_n).
% 2.91/1.70 thf(func_def_96, type, image_987430492real_n: (set_Fi1058188332real_n > sigma_1466784463real_n) > set_se2111327970real_n > set_Si1125517487real_n).
% 2.91/1.70 thf(func_def_97, type, image_933134521real_n: (set_nat > set_Fi1058188332real_n) > set_set_nat > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_98, type, image_1587769199real_n: (set_nat > set_se2111327970real_n) > set_set_nat > set_se820660888real_n).
% 2.91/1.70 thf(func_def_99, type, image_1681970287al_n_o: (set_se2111327970real_n > $o) > set_se820660888real_n > set_o).
% 2.91/1.70 thf(func_def_100, type, image_1298280374real_n: (sigma_1466784463real_n > set_Fi1058188332real_n) > set_Si1125517487real_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_101, type, vimage_o_o: ($o > $o) > set_o > set_o).
% 2.91/1.70 thf(func_def_102, type, vimage961837641real_n: ($o > set_Fi1058188332real_n) > set_se2111327970real_n > set_o).
% 2.91/1.70 thf(func_def_103, type, vimage1122713129_int_n: (finite964658038_int_n > finite964658038_int_n) > set_Fi160064172_int_n > set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_104, type, vimage1276736425real_n: (finite964658038_int_n > finite1489363574real_n) > set_Fi1058188332real_n > set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_105, type, vimage1398021123_n_nat: (finite964658038_int_n > nat) > set_nat > set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_106, type, vimage464515423real_n: (finite964658038_int_n > set_Fi1058188332real_n) > set_se2111327970real_n > set_Fi160064172_int_n).
% 2.91/1.70 thf(func_def_107, type, vimage1233683625real_n: (finite1489363574real_n > finite1489363574real_n) > set_Fi1058188332real_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_108, type, vimage714719107_int_n: (nat > finite964658038_int_n) > set_Fi160064172_int_n > set_nat).
% 2.91/1.70 thf(func_def_109, type, vimage1860757507real_n: (nat > finite1489363574real_n) > set_Fi1058188332real_n > set_nat).
% 2.91/1.70 thf(func_def_110, type, vimage_nat_nat: (nat > nat) > set_nat > set_nat).
% 2.91/1.70 thf(func_def_111, type, vimage3210681real_n: (nat > set_Fi1058188332real_n) > set_se2111327970real_n > set_nat).
% 2.91/1.70 thf(func_def_112, type, vimage851190895al_n_o: (set_Fi1058188332real_n > $o) > set_o > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_113, type, vimage784510485real_n: (set_Fi1058188332real_n > set_Fi1058188332real_n) > set_se2111327970real_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_114, type, sigma_1536574303real_n: sigma_1466784463real_n > set_Fi1058188332real_n > extend1728876344nnreal).
% 2.91/1.70 thf(func_def_115, type, sigma_1235138647real_n: sigma_1466784463real_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_116, type, sigma_space_o: sigma_measure_o > set_o).
% 2.91/1.70 thf(func_def_117, type, sigma_476185326real_n: sigma_1466784463real_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_118, type, sigma_607186084real_n: sigma_1422848389real_n > set_se2111327970real_n).
% 2.91/1.70 thf(func_def_119, type, member_o: $o > set_o > $o).
% 2.91/1.70 thf(func_def_120, type, member1217042383nnreal: extend1728876344nnreal > set_Ex113815278nnreal > $o).
% 2.91/1.70 thf(func_def_121, type, member27055245_int_n: finite964658038_int_n > set_Fi160064172_int_n > $o).
% 2.91/1.70 thf(func_def_122, type, member1352538125real_n: finite1489363574real_n > set_Fi1058188332real_n > $o).
% 2.91/1.70 thf(func_def_123, type, member_nat: nat > set_nat > $o).
% 2.91/1.70 thf(func_def_124, type, member_set_o: set_o > set_set_o > $o).
% 2.91/1.70 thf(func_def_125, type, member223413699real_n: set_Fi1058188332real_n > set_se2111327970real_n > $o).
% 2.91/1.70 thf(func_def_126, type, member1475136633real_n: set_se2111327970real_n > set_se820660888real_n > $o).
% 2.91/1.70 thf(func_def_127, type, member1000184real_n: sigma_1466784463real_n > set_Si1125517487real_n > $o).
% 2.91/1.70 thf(func_def_128, type, r: finite964658038_int_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_129, type, s: set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_130, type, t: finite964658038_int_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_131, type, t2: finite964658038_int_n > set_Fi1058188332real_n).
% 2.91/1.70 thf(func_def_132, type, f: nat > finite964658038_int_n).
% 2.91/1.70 thf(func_def_142, type, sK0: set_Fi1058188332real_n > (finite964658038_int_n > set_Fi1058188332real_n) > finite964658038_int_n).
% 2.91/1.70 thf(func_def_143, type, sK1: (nat > set_Fi1058188332real_n) > set_Fi1058188332real_n > nat).
% 2.91/1.70 thf(func_def_144, type, sK2: (finite964658038_int_n > set_Fi1058188332real_n) > sigma_1466784463real_n > finite964658038_int_n).
% 2.91/1.70 thf(func_def_145, type, sK3: sigma_1466784463real_n > (nat > set_Fi1058188332real_n) > nat).
% 2.91/1.70 thf(func_def_147, type, ph5: !>[X0: $tType]:(X0)).
% 2.91/1.70 thf(func_def_148, type, sK6: nat).
% 2.91/1.70 thf(f986,plain,(
% 2.91/1.70 $false),
% 2.91/1.70 inference(subsumption_resolution,[status(thm)],[f985,f970])).
% 2.91/1.70 thf(f970,plain,(
% 2.91/1.70 ( ! [X0 : finite964658038_int_n] : (((sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ X0)) = (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ X0)))) )),
% 2.91/1.70 inference(cnf_transformation,[status(thm)],[f1])).
% 2.91/1.70 thf(f1,axiom,(
% 2.91/1.70 ! [X0 : finite964658038_int_n] : ((sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ X0)) = (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ X0)))),
% 2.91/1.70 file('/export/starexec/sandbox/tmp/tmp.uys0jgiiMJ/DTF2THF_7164.p',fact_0_emeasure__T_H)).
% 2.91/1.70 thf(f985,plain,(
% 2.91/1.70 ((sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ (f @ sK6))) != (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ (f @ sK6))))),
% 2.91/1.70 inference(beta_eta_normalization,[status(thm)],[f984])).
% 2.91/1.70 thf(f984,plain,(
% 2.91/1.70 (((^[Y0 : nat]: (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ (f @ Y0)))) @ sK6) != ((^[Y0 : nat]: (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ (f @ Y0)))) @ sK6))),
% 2.91/1.70 inference(negative_extensionality,[status(thm)],[f953])).
% 2.91/1.70 thf(f953,plain,(
% 2.91/1.70 ((^[Y0 : nat]: (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ (f @ Y0)))) != (^[Y0 : nat]: (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ (f @ Y0)))))),
% 2.91/1.70 inference(cnf_transformation,[status(thm)],[f900])).
% 2.91/1.70 thf(f900,plain,(
% 2.91/1.70 ((^[Y0 : nat]: (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ (f @ Y0)))) != (^[Y0 : nat]: (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ (f @ Y0)))))),
% 2.91/1.70 inference(flattening,[status(thm)],[f404])).
% 2.91/1.70 thf(f404,plain,(
% 2.91/1.70 ~((^[Y0 : nat]: (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ (f @ Y0)))) = (^[Y0 : nat]: (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ (f @ Y0)))))),
% 2.91/1.70 inference(fool_elimination,[status(thm)],[f403])).
% 2.91/1.70 thf(f403,plain,(
% 2.91/1.70 ~((^[X0 : nat] : (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ (f @ X0)))) = (^[X1 : nat] : (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ (f @ X1)))))),
% 2.91/1.70 inference(rectify,[status(thm)],[f355])).
% 2.91/1.70 thf(f355,negated_conjecture,(
% 2.91/1.70 ~((^[X1 : nat] : (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ (f @ X1)))) = (^[X1 : nat] : (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ (f @ X1)))))),
% 2.91/1.70 inference(negated_conjecture,[status(cth)],[f354])).
% 2.91/1.70 thf(f354,conjecture,(
% 2.91/1.70 ((^[X1 : nat] : (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t @ (f @ X1)))) = (^[X1 : nat] : (sigma_1536574303real_n @ (comple230862828real_n @ lebesg260170249real_n) @ (t2 @ (f @ X1)))))),
% 2.91/1.70 file('/export/starexec/sandbox/tmp/tmp.uys0jgiiMJ/DTF2THF_7164.p',conj_0)).
% 2.91/1.70 % SZS output end Proof for DTF2THF_7164
% 2.91/1.70 % (7268)------------------------------
% 2.91/1.70 % (7268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.70 % (7268)Termination reason: Refutation
% 2.91/1.70
% 2.91/1.70 % (7268)Memory used [KB]: 6652
% 2.91/1.70 % (7268)Time elapsed: 0.037 s
% 2.91/1.70 % (7268)Instructions burned: 56 (million)
% 2.91/1.70 % (7268)------------------------------
% 2.91/1.70 % (7268)------------------------------
% 2.91/1.70 % (7262)Success in time 0.066 s
%------------------------------------------------------------------------------