%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP157^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n006.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 08:56:00 AM UTC 2026
% Result : Theorem 3.18s 1.84s
% Output : Refutation 3.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 4
% Syntax : Number of formulae : 30 ( 30 unt; 0 typ; 0 def)
% Number of atoms : 94 ( 30 equ; 0 cnn)
% Maximal formula atoms : 1 ( 3 avg)
% Number of connectives : 177 ( 4 ~; 0 |; 9 &; 158 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 2 avg)
% Number of types : 31 ( 30 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 169 ( 165 usr; 18 con; 0-3 aty)
% ( 6 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 42 ( 30 ^; 12 !; 0 ?; 42 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_Product_prod_a_a: $tType ).
thf(type_def_6,type,
product_prod_a_a: $tType ).
thf(type_def_8,type,
set_Pr1948701895od_a_a: $tType ).
thf(type_def_9,type,
set_a: $tType ).
thf(type_def_10,type,
a: $tType ).
thf(type_def_11,type,
produc1572603623od_a_a: $tType ).
thf(type_def_12,type,
set_Pr1295299783od_a_a: $tType ).
thf(type_def_13,type,
produc2010915495od_a_a: $tType ).
thf(type_def_14,type,
set_Pr231806599od_a_a: $tType ).
thf(type_def_15,type,
set_Pr532616391od_a_a: $tType ).
thf(type_def_16,type,
produc772425383od_a_a: $tType ).
thf(type_def_17,type,
set_Pr1194146439od_a_a: $tType ).
thf(type_def_18,type,
set_Pr2081559751od_a_a: $tType ).
thf(type_def_19,type,
produc1016684094_a_a_a: $tType ).
thf(type_def_20,type,
set_Pr1689873822_a_a_a: $tType ).
thf(type_def_21,type,
set_Pr729215751_a_a_a: $tType ).
thf(type_def_22,type,
produc1555800720od_a_a: $tType ).
thf(type_def_23,type,
set_Pr195866352od_a_a: $tType ).
thf(type_def_24,type,
set_Pr660101255od_a_a: $tType ).
thf(type_def_25,type,
produc1921647824od_a_a: $tType ).
thf(type_def_26,type,
set_Pr681306928od_a_a: $tType ).
thf(type_def_27,type,
produc1056976615od_a_a: $tType ).
thf(type_def_28,type,
produc1423126247od_a_a: $tType ).
thf(type_def_29,type,
produc1990712039od_a_a: $tType ).
thf(type_def_30,type,
produc47586087od_a_a: $tType ).
thf(type_def_31,type,
produc1683075710_a_a_a: $tType ).
thf(type_def_32,type,
produc1120419623_a_a_a: $tType ).
thf(type_def_33,type,
produc1546279079od_a_a: $tType ).
thf(type_def_34,type,
set_Pr173461255od_a_a: $tType ).
thf(type_def_35,type,
set_Pr6137822_a_a_a: $tType ).
thf(func_def_0,type,
set_Pr2081559751od_a_a: $tType ).
thf(func_def_1,type,
set_Pr532616391od_a_a: $tType ).
thf(func_def_2,type,
produc1423126247od_a_a: $tType ).
thf(func_def_3,type,
produc1056976615od_a_a: $tType ).
thf(func_def_4,type,
set_Pr660101255od_a_a: $tType ).
thf(func_def_5,type,
produc1546279079od_a_a: $tType ).
thf(func_def_6,type,
set_Pr1295299783od_a_a: $tType ).
thf(func_def_7,type,
produc1990712039od_a_a: $tType ).
thf(func_def_8,type,
set_Pr1194146439od_a_a: $tType ).
thf(func_def_9,type,
set_Pr231806599od_a_a: $tType ).
thf(func_def_10,type,
set_Pr729215751_a_a_a: $tType ).
thf(func_def_11,type,
set_Pr173461255od_a_a: $tType ).
thf(func_def_12,type,
produc772425383od_a_a: $tType ).
thf(func_def_13,type,
produc2010915495od_a_a: $tType ).
thf(func_def_14,type,
produc1120419623_a_a_a: $tType ).
thf(func_def_15,type,
produc47586087od_a_a: $tType ).
thf(func_def_16,type,
set_Pr195866352od_a_a: $tType ).
thf(func_def_17,type,
set_Pr6137822_a_a_a: $tType ).
thf(func_def_18,type,
produc1555800720od_a_a: $tType ).
thf(func_def_19,type,
produc1683075710_a_a_a: $tType ).
thf(func_def_20,type,
set_Pr1948701895od_a_a: $tType ).
thf(func_def_21,type,
produc1572603623od_a_a: $tType ).
thf(func_def_22,type,
set_Pr681306928od_a_a: $tType ).
thf(func_def_23,type,
set_Pr1689873822_a_a_a: $tType ).
thf(func_def_24,type,
produc1921647824od_a_a: $tType ).
thf(func_def_25,type,
produc1016684094_a_a_a: $tType ).
thf(func_def_26,type,
set_Product_prod_a_a: $tType ).
thf(func_def_27,type,
product_prod_a_a: $tType ).
thf(func_def_28,type,
set_a: $tType ).
thf(func_def_29,type,
a: $tType ).
thf(func_def_30,type,
bNF_Gr1372473186od_a_a: set_Product_prod_a_a > ( product_prod_a_a > product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(func_def_31,type,
bNF_Gr_a_a: set_a > ( a > a ) > set_Product_prod_a_a ).
thf(func_def_32,type,
bNF_Gr533818957od_a_a: set_Pr1948701895od_a_a > ( produc1572603623od_a_a > product_prod_a_a ) > ( produc1572603623od_a_a > product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(func_def_33,type,
bNF_Gr356189709_a_a_a: set_Pr1948701895od_a_a > ( produc1572603623od_a_a > a ) > ( produc1572603623od_a_a > a ) > set_Product_prod_a_a ).
thf(func_def_34,type,
bNF_Gr200450189od_a_a: set_Product_prod_a_a > ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(func_def_35,type,
bNF_Gr1853143117_a_a_a: set_Product_prod_a_a > ( product_prod_a_a > a ) > ( product_prod_a_a > a ) > set_Product_prod_a_a ).
thf(func_def_36,type,
bNF_Gr1715025636od_a_a: set_a > ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(func_def_37,type,
bNF_Gr423046308_a_a_a: set_a > ( a > a ) > ( a > a ) > set_Product_prod_a_a ).
thf(func_def_38,type,
minus_107310383_a_a_o: ( produc1572603623od_a_a > $o ) > ( produc1572603623od_a_a > $o ) > produc1572603623od_a_a > $o ).
thf(func_def_39,type,
minus_1797478127_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > product_prod_a_a > $o ).
thf(func_def_40,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(func_def_41,type,
minus_1034916142od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_42,type,
minus_1507102574od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_43,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(func_def_44,type,
order_491428350od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_45,type,
order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_46,type,
order_preorder_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_47,type,
bot_bo1116205875od_a_a: set_Pr1948701895od_a_a ).
thf(func_def_48,type,
bot_bo2131659635od_a_a: set_Product_prod_a_a ).
thf(func_def_49,type,
bot_bot_set_a: set_a ).
thf(func_def_50,type,
ord_le1525442358_a_a_o: ( produc1572603623od_a_a > $o ) > ( produc1572603623od_a_a > $o ) > $o ).
thf(func_def_51,type,
ord_le252190158_a_a_o: ( product_prod_a_a > product_prod_a_a > $o ) > ( product_prod_a_a > product_prod_a_a > $o ) > $o ).
thf(func_def_52,type,
ord_le1347718902_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > $o ).
thf(func_def_53,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $o ).
thf(func_def_54,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(func_def_55,type,
ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_56,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_57,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_58,type,
top_to2011536774_a_a_o: produc1572603623od_a_a > $o ).
thf(func_def_59,type,
top_to993969022_a_a_o: product_prod_a_a > product_prod_a_a > $o ).
thf(func_def_60,type,
top_to1483957574_a_a_o: product_prod_a_a > $o ).
thf(func_def_61,type,
top_top_a_a_o: a > a > $o ).
thf(func_def_62,type,
top_top_a_o: a > $o ).
thf(func_def_63,type,
top_to52711959od_a_a: set_Pr1948701895od_a_a ).
thf(func_def_64,type,
top_to398455383od_a_a: set_Product_prod_a_a ).
thf(func_def_65,type,
top_top_set_a: set_a ).
thf(func_def_66,type,
prefer1867335523od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_67,type,
prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_68,type,
prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_69,type,
prefer468349667od_a_a: produc2010915495od_a_a > set_Pr231806599od_a_a > set_Pr532616391od_a_a > set_Pr231806599od_a_a ).
thf(func_def_70,type,
prefer1377343203od_a_a: produc772425383od_a_a > set_Pr1194146439od_a_a > set_Pr2081559751od_a_a > set_Pr1194146439od_a_a ).
thf(func_def_71,type,
prefer55414563od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_72,type,
prefer1944486522_a_a_a: produc1016684094_a_a_a > set_Pr1689873822_a_a_a > set_Pr729215751_a_a_a > set_Pr1689873822_a_a_a ).
thf(func_def_73,type,
prefer445484748od_a_a: produc1555800720od_a_a > set_Pr195866352od_a_a > set_Pr660101255od_a_a > set_Pr195866352od_a_a ).
thf(func_def_74,type,
prefer701966604od_a_a: produc1921647824od_a_a > set_Pr681306928od_a_a > set_Pr231806599od_a_a > set_Pr681306928od_a_a ).
thf(func_def_75,type,
prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_76,type,
prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_77,type,
prefer926864306od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_78,type,
prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_79,type,
prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_80,type,
prefer1409946857od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_81,type,
prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_82,type,
prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_83,type,
produc539587415od_a_a: produc2010915495od_a_a > produc2010915495od_a_a > produc1056976615od_a_a ).
thf(func_def_84,type,
produc852390743od_a_a: produc772425383od_a_a > produc772425383od_a_a > produc1423126247od_a_a ).
thf(func_def_85,type,
produc1935643479od_a_a: produc1572603623od_a_a > produc1572603623od_a_a > produc1990712039od_a_a ).
thf(func_def_86,type,
produc524982679od_a_a: produc1572603623od_a_a > product_prod_a_a > produc47586087od_a_a ).
thf(func_def_87,type,
produc436797294_a_a_a: produc1572603623od_a_a > a > produc1683075710_a_a_a ).
thf(func_def_88,type,
produc2054708503_a_a_a: produc1016684094_a_a_a > produc1016684094_a_a_a > produc1120419623_a_a_a ).
thf(func_def_89,type,
produc1573596055od_a_a: produc1555800720od_a_a > produc1555800720od_a_a > produc1546279079od_a_a ).
thf(func_def_90,type,
produc563767703od_a_a: produc1921647824od_a_a > produc1921647824od_a_a > produc2010915495od_a_a ).
thf(func_def_91,type,
produc1555606807od_a_a: product_prod_a_a > produc1572603623od_a_a > produc772425383od_a_a ).
thf(func_def_92,type,
produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
thf(func_def_93,type,
produc2061588782_a_a_a: product_prod_a_a > a > produc1016684094_a_a_a ).
thf(func_def_94,type,
produc1095491584od_a_a: a > produc1572603623od_a_a > produc1555800720od_a_a ).
thf(func_def_95,type,
produc1299253312od_a_a: a > product_prod_a_a > produc1921647824od_a_a ).
thf(func_def_96,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_97,type,
produc1460477704od_a_a: set_Pr1948701895od_a_a > ( produc1572603623od_a_a > set_Pr1948701895od_a_a ) > set_Pr1295299783od_a_a ).
thf(func_def_98,type,
produc197318472od_a_a: set_Pr1948701895od_a_a > ( produc1572603623od_a_a > set_Product_prod_a_a ) > set_Pr173461255od_a_a ).
thf(func_def_99,type,
produc4724255_a_a_a: set_Pr1948701895od_a_a > ( produc1572603623od_a_a > set_a ) > set_Pr6137822_a_a_a ).
thf(func_def_100,type,
produc1227942600od_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_Pr1948701895od_a_a ) > set_Pr1194146439od_a_a ).
thf(func_def_101,type,
produc304751368od_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(func_def_102,type,
produc1282482655_a_a_a: set_Product_prod_a_a > ( product_prod_a_a > set_a ) > set_Pr1689873822_a_a_a ).
thf(func_def_103,type,
produc663418545od_a_a: set_a > ( a > set_Pr1948701895od_a_a ) > set_Pr195866352od_a_a ).
thf(func_def_104,type,
produc520147185od_a_a: set_a > ( a > set_Product_prod_a_a ) > set_Pr681306928od_a_a ).
thf(func_def_105,type,
product_Sigma_a_a: set_a > ( a > set_a ) > set_Product_prod_a_a ).
thf(func_def_106,type,
produc94438221_a_a_o: ( produc1572603623od_a_a > $o ) > product_prod_a_a > product_prod_a_a > $o ).
thf(func_def_107,type,
product_curry_a_a_o: ( product_prod_a_a > $o ) > a > a > $o ).
thf(func_def_108,type,
produc1779756150od_a_a: ( product_prod_a_a > product_prod_a_a ) > ( product_prod_a_a > product_prod_a_a ) > produc1572603623od_a_a > produc1572603623od_a_a ).
thf(func_def_109,type,
produc363290532_a_a_a: ( product_prod_a_a > a ) > ( product_prod_a_a > a ) > produc1572603623od_a_a > product_prod_a_a ).
thf(func_def_110,type,
produc2012818120od_a_a: ( a > product_prod_a_a ) > ( a > product_prod_a_a ) > product_prod_a_a > produc1572603623od_a_a ).
thf(func_def_111,type,
produc700022774_a_a_a: ( a > a ) > ( a > a ) > product_prod_a_a > product_prod_a_a ).
thf(func_def_112,type,
produc303632707od_a_a: produc1990712039od_a_a > produc1572603623od_a_a ).
thf(func_def_113,type,
produc745123715od_a_a: produc47586087od_a_a > produc1572603623od_a_a ).
thf(func_def_114,type,
produc2019352922_a_a_a: produc1683075710_a_a_a > produc1572603623od_a_a ).
thf(func_def_115,type,
produc1775747843od_a_a: produc772425383od_a_a > product_prod_a_a ).
thf(func_def_116,type,
produc2115494211od_a_a: produc1572603623od_a_a > product_prod_a_a ).
thf(func_def_117,type,
produc1805832986_a_a_a: produc1016684094_a_a_a > product_prod_a_a ).
thf(func_def_118,type,
produc530563564od_a_a: produc1555800720od_a_a > a ).
thf(func_def_119,type,
produc1043497516od_a_a: produc1921647824od_a_a > a ).
thf(func_def_120,type,
product_fst_a_a: product_prod_a_a > a ).
thf(func_def_121,type,
produc1441227909od_a_a: produc1990712039od_a_a > produc1572603623od_a_a ).
thf(func_def_122,type,
produc24848069od_a_a: produc47586087od_a_a > product_prod_a_a ).
thf(func_def_123,type,
produc262335132_a_a_a: produc1683075710_a_a_a > a ).
thf(func_def_124,type,
produc1055472197od_a_a: produc772425383od_a_a > produc1572603623od_a_a ).
thf(func_def_125,type,
produc368344197od_a_a: produc1572603623od_a_a > product_prod_a_a ).
thf(func_def_126,type,
produc1954021468_a_a_a: produc1016684094_a_a_a > a ).
thf(func_def_127,type,
produc921029422od_a_a: produc1555800720od_a_a > produc1572603623od_a_a ).
thf(func_def_128,type,
produc1191685998od_a_a: produc1921647824od_a_a > product_prod_a_a ).
thf(func_def_129,type,
product_snd_a_a: product_prod_a_a > a ).
thf(func_def_130,type,
produc1641853175od_a_a: produc1572603623od_a_a > produc1572603623od_a_a ).
thf(func_def_131,type,
product_swap_a_a: product_prod_a_a > product_prod_a_a ).
thf(func_def_132,type,
produc905674258od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a ).
thf(func_def_133,type,
product_product_a_a: set_a > set_a > set_Product_prod_a_a ).
thf(func_def_134,type,
collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
thf(func_def_135,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_136,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_137,type,
image_236793797od_a_a: ( produc1572603623od_a_a > produc1572603623od_a_a ) > set_Pr1948701895od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_138,type,
image_734753285od_a_a: ( produc1572603623od_a_a > product_prod_a_a ) > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_139,type,
image_1105229276_a_a_a: ( produc1572603623od_a_a > a ) > set_Pr1948701895od_a_a > set_a ).
thf(func_def_140,type,
image_1765377413od_a_a: ( product_prod_a_a > produc1572603623od_a_a ) > set_Product_prod_a_a > set_Pr1948701895od_a_a ).
thf(func_def_141,type,
image_794773445od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_142,type,
image_340790172_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).
thf(func_def_143,type,
image_1763923566od_a_a: ( a > produc1572603623od_a_a ) > set_a > set_Pr1948701895od_a_a ).
thf(func_def_144,type,
image_1725938350od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).
thf(func_def_145,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(func_def_146,type,
lex_pr942790511od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a ).
thf(func_def_147,type,
lex_pr2084296006_a_a_a: set_Pr1948701895od_a_a > set_Product_prod_a_a > set_Pr729215751_a_a_a ).
thf(func_def_148,type,
lex_pr1321960536od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Pr231806599od_a_a ).
thf(func_def_149,type,
lex_prod_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a ).
thf(func_def_150,type,
same_f1138477100od_a_a: ( product_prod_a_a > $o ) > ( product_prod_a_a > set_Pr1948701895od_a_a ) > set_Pr1295299783od_a_a ).
thf(func_def_151,type,
same_fst_a_a: ( a > $o ) > ( a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(func_def_152,type,
member180906768od_a_a: produc1056976615od_a_a > set_Pr532616391od_a_a > $o ).
thf(func_def_153,type,
member547056400od_a_a: produc1423126247od_a_a > set_Pr2081559751od_a_a > $o ).
thf(func_def_154,type,
member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_155,type,
member1545452880od_a_a: produc47586087od_a_a > set_Pr173461255od_a_a > $o ).
thf(func_def_156,type,
member575366567_a_a_a: produc1683075710_a_a_a > set_Pr6137822_a_a_a > $o ).
thf(func_def_157,type,
member470802768_a_a_a: produc1120419623_a_a_a > set_Pr729215751_a_a_a > $o ).
thf(func_def_158,type,
member1424209616od_a_a: produc1546279079od_a_a > set_Pr660101255od_a_a > $o ).
thf(func_def_159,type,
member1361298640od_a_a: produc2010915495od_a_a > set_Pr231806599od_a_a > $o ).
thf(func_def_160,type,
member122808528od_a_a: produc772425383od_a_a > set_Pr1194146439od_a_a > $o ).
thf(func_def_161,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_162,type,
member1403138919_a_a_a: produc1016684094_a_a_a > set_Pr1689873822_a_a_a > $o ).
thf(func_def_163,type,
member448091577od_a_a: produc1555800720od_a_a > set_Pr195866352od_a_a > $o ).
thf(func_def_164,type,
member160619001od_a_a: produc1921647824od_a_a > set_Pr681306928od_a_a > $o ).
thf(func_def_165,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_166,type,
member_a: a > set_a > $o ).
thf(func_def_167,type,
b: set_a ).
thf(func_def_168,type,
pr: set_Product_prod_a_a ).
thf(func_def_169,type,
y: a ).
thf(func_def_170,type,
z: a ).
thf(func_def_190,type,
ph1:
!>[X0: $tType] : X0 ).
thf(func_def_191,type,
sK2: a ).
thf(func_def_192,type,
sK3: product_prod_a_a ).
thf(func_def_193,type,
sK4: produc1572603623od_a_a ).
thf(func_def_194,type,
sK5: product_prod_a_a > a ).
thf(func_def_195,type,
sK6: product_prod_a_a > a ).
thf(func_def_196,type,
sK7: produc1016684094_a_a_a > a ).
thf(func_def_197,type,
sK8: produc1016684094_a_a_a > product_prod_a_a ).
thf(func_def_198,type,
sK9: produc1921647824od_a_a > product_prod_a_a ).
thf(func_def_199,type,
sK10: produc1921647824od_a_a > a ).
thf(func_def_200,type,
sK11: produc1572603623od_a_a > product_prod_a_a ).
thf(func_def_201,type,
sK12: produc1572603623od_a_a > product_prod_a_a ).
thf(func_def_202,type,
sK13: produc2010915495od_a_a > produc1921647824od_a_a ).
thf(func_def_203,type,
sK14: produc2010915495od_a_a > produc1921647824od_a_a ).
thf(func_def_204,type,
sK15: produc1555800720od_a_a > produc1572603623od_a_a ).
thf(func_def_205,type,
sK16: produc1555800720od_a_a > a ).
thf(func_def_206,type,
sK17: produc772425383od_a_a > produc1572603623od_a_a ).
thf(func_def_207,type,
sK18: produc772425383od_a_a > product_prod_a_a ).
thf(func_def_208,type,
sK19: produc1572603623od_a_a ).
thf(func_def_209,type,
sK20: product_prod_a_a ).
thf(func_def_210,type,
sK21: product_prod_a_a > a ).
thf(func_def_211,type,
sK22: product_prod_a_a > a ).
thf(func_def_212,type,
sK23: produc1572603623od_a_a > product_prod_a_a ).
thf(func_def_213,type,
sK24: produc1572603623od_a_a > product_prod_a_a ).
thf(f2153,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2152]) ).
thf(f2152,plain,
$false = $true,
inference(boolean_simplification,[status(thm)],[f2151]) ).
thf(f2151,plain,
( $true
= ( ( member_a @ z @ b )
& $false ) ),
inference(forward_demodulation,[status(thm)],[f2147,f1608]) ).
thf(f1608,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ pr )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f1212]) ).
thf(f1212,plain,
( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ pr ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f492]) ).
thf(f492,plain,
( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ pr ) )
= $true ),
inference(fool_elimination,[status(thm)],[f491]) ).
thf(f491,plain,
~ ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ pr ),
inference(rectify,[status(thm)],[f357]) ).
thf(f357,negated_conjecture,
~ ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ pr ),
inference(negated_conjecture,[status(cth)],[f356]) ).
thf(f356,conjecture,
member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ pr,
file('/export/starexec/sandbox2/tmp/tmp.d5SnneaLK1/DTF2THF_13046.p',conj_0) ).
thf(f2147,plain,
( ( ( member_a @ z @ b )
& ( member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ pr ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f2143]) ).
thf(f2143,plain,
( ( ^ [Y0: a] :
( ( member_a @ Y0 @ b )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y0 @ y ) @ pr ) )
@ z )
= $true ),
inference(superposition,[status(thm)],[f1509,f2126]) ).
thf(f2126,plain,
! [X2: a > $o,X1: a] :
( ( member_a @ X1 @ ( collect_a @ X2 ) )
= ( X2 @ X1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f2122]) ).
thf(f2122,plain,
! [X2: a > $o,X1: a] :
( $true
= ( ( member_a @ X1 @ ( collect_a @ X2 ) )
= ( X2 @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2121]) ).
thf(f2121,plain,
! [X2: a > $o,X1: a] :
( ( ^ [Y0: a > $o] :
( ( member_a @ X1 @ ( collect_a @ Y0 ) )
= ( Y0 @ X1 ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f2108]) ).
thf(f2108,plain,
! [X1: a] :
( ( !! @ ( a > $o )
@ ^ [Y0: a > $o] :
( ( member_a @ X1 @ ( collect_a @ Y0 ) )
= ( Y0 @ X1 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f2107]) ).
thf(f2107,plain,
! [X1: a] :
( ( ^ [Y0: a] :
( !! @ ( a > $o )
@ ^ [Y1: a > $o] :
( ( member_a @ Y0 @ ( collect_a @ Y1 ) )
= ( Y1 @ Y0 ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1166]) ).
thf(f1166,plain,
( ( !! @ a
@ ^ [Y0: a] :
( !! @ ( a > $o )
@ ^ [Y1: a > $o] :
( ( member_a @ Y0 @ ( collect_a @ Y1 ) )
= ( Y1 @ Y0 ) ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f915]) ).
thf(f915,plain,
( ( !! @ a
@ ^ [Y0: a] :
( !! @ ( a > $o )
@ ^ [Y1: a > $o] :
( ( member_a @ Y0 @ ( collect_a @ Y1 ) )
= ( Y1 @ Y0 ) ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f914]) ).
thf(f914,plain,
! [X0: a > $o,X1: a] :
( ( member_a @ X1 @ ( collect_a @ X0 ) )
= ( X0 @ X1 ) ),
inference(rectify,[status(thm)],[f92]) ).
thf(f92,axiom,
! [X15: a > $o,X8: a] :
( ( member_a @ X8 @ ( collect_a @ X15 ) )
= ( X15 @ X8 ) ),
file('/export/starexec/sandbox2/tmp/tmp.d5SnneaLK1/DTF2THF_13046.p',fact_91_mem__Collect__eq) ).
thf(f1509,plain,
( ( member_a @ z
@ ( collect_a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ b )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y0 @ y ) @ pr ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1502]) ).
thf(f1502,plain,
( ( member_a @ z
@ ( ^ [Y0: a,Y1: set_a,Y2: set_Product_prod_a_a] :
( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ Y1 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Y0 ) @ Y2 ) ) )
@ y
@ b
@ pr ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1402,f1309]) ).
thf(f1309,plain,
( prefer161218362good_a
= ( ^ [Y0: a,Y1: set_a,Y2: set_Product_prod_a_a] :
( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ Y1 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Y0 ) @ Y2 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f532]) ).
thf(f532,plain,
( prefer161218362good_a
= ( ^ [Y0: a,Y1: set_a,Y2: set_Product_prod_a_a] :
( collect_a
@ ^ [Y3: a] :
( ( member_a @ Y3 @ Y1 )
& ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Y0 ) @ Y2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f531]) ).
thf(f531,plain,
( prefer161218362good_a
= ( ^ [X0: a,X1: set_a,X2: set_Product_prod_a_a] :
( collect_a
@ ^ [X3: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ X0 ) @ X2 )
& ( member_a @ X3 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f55]) ).
thf(f55,axiom,
( prefer161218362good_a
= ( ^ [X20: a,X21: set_a,X22: set_Product_prod_a_a] :
( collect_a
@ ^ [X23: a] :
( ( member449909584od_a_a @ ( product_Pair_a_a @ X23 @ X20 ) @ X22 )
& ( member_a @ X23 @ X21 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.d5SnneaLK1/DTF2THF_13046.p',fact_54_at__least__as__good__def) ).
thf(f1402,plain,
( ( member_a @ z @ ( prefer161218362good_a @ y @ b @ pr ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f502]) ).
thf(f502,plain,
( ( member_a @ z @ ( prefer161218362good_a @ y @ b @ pr ) )
= $true ),
inference(fool_elimination,[status(thm)],[f501]) ).
thf(f501,plain,
member_a @ z @ ( prefer161218362good_a @ y @ b @ pr ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
member_a @ z @ ( prefer161218362good_a @ y @ b @ pr ),
file('/export/starexec/sandbox2/tmp/tmp.d5SnneaLK1/DTF2THF_13046.p',fact_0_assms) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP157^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n006.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:28:00 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.47 ---- Original DTF file ---
% 0.28/0.47 thf(spec,logic,$$dhol).
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 % File : ITP157^1 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.47 % Domain : Interactive Theorem Proving
% 0.28/0.47 % Problem : Sledgehammer Preferences problem prob_86__6247862_1
% 0.28/0.47 % Version : Especial.
% 0.28/0.47 % English :
% 0.28/0.47
% 0.28/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.28/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.28/0.47 % Source : [Des21]
% 0.28/0.47 % Names : Preferences/prob_86__6247862_1 [Des21]
% 0.28/0.47
% 0.28/0.47 % Status : Theorem
% 0.28/0.47 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.28/0.47 % Syntax : Number of formulae : 527 ( 166 unt; 171 typ; 0 def)
% 0.28/0.47 % Number of atoms : 982 ( 312 equ; 0 cnn)
% 0.28/0.47 % Maximal formula atoms : 7 ( 2 avg)
% 0.28/0.47 % Number of connectives : 3250 ( 124 ~; 4 |; 110 &;2654 @)
% 0.28/0.47 % ( 0 <=>; 358 =>; 0 <=; 0 <~>)
% 0.28/0.47 % Maximal formula depth : 15 ( 7 avg)
% 0.28/0.47 % Number of types : 31 ( 30 usr)
% 0.28/0.47 % Number of type conns : 530 ( 530 >; 0 *; 0 +; 0 <<)
% 0.28/0.47 % Number of symbols : 142 ( 141 usr; 10 con; 0-3 aty)
% 0.28/0.47 % Number of variables : 1291 ( 171 ^;1096 !; 24 ?;1291 :)
% 0.28/0.47 % SPC : TH0_THM_EQU_NAR
% 0.28/0.47
% 0.28/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:43:15.680
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 % Could-be-implicit typings (30)
% 0.28/0.47 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J_J,type,
% 0.28/0.47 set_Pr2081559751od_a_a: $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J_J,type,
% 0.28/0.47 set_Pr532616391od_a_a: $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 0.28/0.47 produc1423126247od_a_a: $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 0.28/0.47 produc1056976615od_a_a: $tType ).
% 0.28/0.47
% 0.28/0.47 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J_J,type,
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% 0.28/0.47 % at_lst_asgd_ge
% 0.28/0.47 thf(fact_7_at__lst__asgd__ge,axiom,
% 0.28/0.47 ! [X: product_prod_a_a,Y: product_prod_a_a,B: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B @ Pr ) )
% 0.28/0.47 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% 0.28/0.47
% 0.28/0.47 % at_lst_asgd_ge
% 0.28/0.47 thf(fact_8_at__lst__asgd__ge,axiom,
% 0.28/0.47 ! [X: a,Y: a,B: set_a,Pr: set_Product_prod_a_a] :
% 0.28/0.47 ( ( member_a @ X @ ( prefer161218362good_a @ Y @ B @ Pr ) )
% 0.28/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% 0.28/0.47
% 0.28/0.47 % at_lst_asgd_ge
% 0.28/0.47 thf(fact_9_prod_Oinject,axiom,
% 0.28/0.47 ! [X1: produc1921647824od_a_a,X2: produc1921647824od_a_a,Y1: produc1921647824od_a_a,Y2: produc1921647824od_a_a] :
% 0.28/0.47 ( ( ( produc563767703od_a_a @ X1 @ X2 )
% 0.28/0.47 = ( produc563767703od_a_a @ Y1 @ Y2 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X2 = Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod.inject
% 0.28/0.47 thf(fact_10_prod_Oinject,axiom,
% 0.28/0.47 ! [X1: product_prod_a_a,X2: produc1572603623od_a_a,Y1: product_prod_a_a,Y2: produc1572603623od_a_a] :
% 0.28/0.47 ( ( ( produc1555606807od_a_a @ X1 @ X2 )
% 0.28/0.47 = ( produc1555606807od_a_a @ Y1 @ Y2 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X2 = Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod.inject
% 0.28/0.47 thf(fact_11_prod_Oinject,axiom,
% 0.28/0.47 ! [X1: product_prod_a_a,X2: a,Y1: product_prod_a_a,Y2: a] :
% 0.28/0.47 ( ( ( produc2061588782_a_a_a @ X1 @ X2 )
% 0.28/0.47 = ( produc2061588782_a_a_a @ Y1 @ Y2 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X2 = Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod.inject
% 0.28/0.47 thf(fact_12_prod_Oinject,axiom,
% 0.28/0.47 ! [X1: a,X2: produc1572603623od_a_a,Y1: a,Y2: produc1572603623od_a_a] :
% 0.28/0.47 ( ( ( produc1095491584od_a_a @ X1 @ X2 )
% 0.28/0.47 = ( produc1095491584od_a_a @ Y1 @ Y2 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X2 = Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod.inject
% 0.28/0.47 thf(fact_13_prod_Oinject,axiom,
% 0.28/0.47 ! [X1: a,X2: product_prod_a_a,Y1: a,Y2: product_prod_a_a] :
% 0.28/0.47 ( ( ( produc1299253312od_a_a @ X1 @ X2 )
% 0.28/0.47 = ( produc1299253312od_a_a @ Y1 @ Y2 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X2 = Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod.inject
% 0.28/0.47 thf(fact_14_prod_Oinject,axiom,
% 0.28/0.47 ! [X1: product_prod_a_a,X2: product_prod_a_a,Y1: product_prod_a_a,Y2: product_prod_a_a] :
% 0.28/0.47 ( ( ( produc1474507607od_a_a @ X1 @ X2 )
% 0.28/0.47 = ( produc1474507607od_a_a @ Y1 @ Y2 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X2 = Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod.inject
% 0.28/0.47 thf(fact_15_prod_Oinject,axiom,
% 0.28/0.47 ! [X1: a,X2: a,Y1: a,Y2: a] :
% 0.28/0.47 ( ( ( product_Pair_a_a @ X1 @ X2 )
% 0.28/0.47 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 0.28/0.47 = ( ( X1 = Y1 )
% 0.28/0.47 & ( X2 = Y2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod.inject
% 0.28/0.47 thf(fact_16_old_Oprod_Oinject,axiom,
% 0.28/0.47 ! [A: produc1921647824od_a_a,B2: produc1921647824od_a_a,A2: produc1921647824od_a_a,B3: produc1921647824od_a_a] :
% 0.28/0.47 ( ( ( produc563767703od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc563767703od_a_a @ A2 @ B3 ) )
% 0.28/0.47 = ( ( A = A2 )
% 0.28/0.47 & ( B2 = B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inject
% 0.28/0.47 thf(fact_17_old_Oprod_Oinject,axiom,
% 0.28/0.47 ! [A: product_prod_a_a,B2: produc1572603623od_a_a,A2: product_prod_a_a,B3: produc1572603623od_a_a] :
% 0.28/0.47 ( ( ( produc1555606807od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc1555606807od_a_a @ A2 @ B3 ) )
% 0.28/0.47 = ( ( A = A2 )
% 0.28/0.47 & ( B2 = B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inject
% 0.28/0.47 thf(fact_18_old_Oprod_Oinject,axiom,
% 0.28/0.47 ! [A: product_prod_a_a,B2: a,A2: product_prod_a_a,B3: a] :
% 0.28/0.47 ( ( ( produc2061588782_a_a_a @ A @ B2 )
% 0.28/0.47 = ( produc2061588782_a_a_a @ A2 @ B3 ) )
% 0.28/0.47 = ( ( A = A2 )
% 0.28/0.47 & ( B2 = B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inject
% 0.28/0.47 thf(fact_19_old_Oprod_Oinject,axiom,
% 0.28/0.47 ! [A: a,B2: produc1572603623od_a_a,A2: a,B3: produc1572603623od_a_a] :
% 0.28/0.47 ( ( ( produc1095491584od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc1095491584od_a_a @ A2 @ B3 ) )
% 0.28/0.47 = ( ( A = A2 )
% 0.28/0.47 & ( B2 = B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inject
% 0.28/0.47 thf(fact_20_old_Oprod_Oinject,axiom,
% 0.28/0.47 ! [A: a,B2: product_prod_a_a,A2: a,B3: product_prod_a_a] :
% 0.28/0.47 ( ( ( produc1299253312od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc1299253312od_a_a @ A2 @ B3 ) )
% 0.28/0.47 = ( ( A = A2 )
% 0.28/0.47 & ( B2 = B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inject
% 0.28/0.47 thf(fact_21_old_Oprod_Oinject,axiom,
% 0.28/0.47 ! [A: product_prod_a_a,B2: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
% 0.28/0.47 ( ( ( produc1474507607od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc1474507607od_a_a @ A2 @ B3 ) )
% 0.28/0.47 = ( ( A = A2 )
% 0.28/0.47 & ( B2 = B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inject
% 0.28/0.47 thf(fact_22_old_Oprod_Oinject,axiom,
% 0.28/0.47 ! [A: a,B2: a,A2: a,B3: a] :
% 0.28/0.47 ( ( ( product_Pair_a_a @ A @ B2 )
% 0.28/0.47 = ( product_Pair_a_a @ A2 @ B3 ) )
% 0.28/0.47 = ( ( A = A2 )
% 0.28/0.47 & ( B2 = B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inject
% 0.28/0.47 thf(fact_23_surj__pair,axiom,
% 0.28/0.47 ! [P: produc2010915495od_a_a] :
% 0.28/0.47 ? [X3: produc1921647824od_a_a,Y3: produc1921647824od_a_a] :
% 0.28/0.47 ( P
% 0.28/0.47 = ( produc563767703od_a_a @ X3 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % surj_pair
% 0.28/0.47 thf(fact_24_surj__pair,axiom,
% 0.28/0.47 ! [P: produc772425383od_a_a] :
% 0.28/0.47 ? [X3: product_prod_a_a,Y3: produc1572603623od_a_a] :
% 0.28/0.47 ( P
% 0.28/0.47 = ( produc1555606807od_a_a @ X3 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % surj_pair
% 0.28/0.47 thf(fact_25_surj__pair,axiom,
% 0.28/0.47 ! [P: produc1016684094_a_a_a] :
% 0.28/0.47 ? [X3: product_prod_a_a,Y3: a] :
% 0.28/0.47 ( P
% 0.28/0.47 = ( produc2061588782_a_a_a @ X3 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % surj_pair
% 0.28/0.47 thf(fact_26_surj__pair,axiom,
% 0.28/0.47 ! [P: produc1555800720od_a_a] :
% 0.28/0.47 ? [X3: a,Y3: produc1572603623od_a_a] :
% 0.28/0.47 ( P
% 0.28/0.47 = ( produc1095491584od_a_a @ X3 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % surj_pair
% 0.28/0.47 thf(fact_27_surj__pair,axiom,
% 0.28/0.47 ! [P: produc1921647824od_a_a] :
% 0.28/0.47 ? [X3: a,Y3: product_prod_a_a] :
% 0.28/0.47 ( P
% 0.28/0.47 = ( produc1299253312od_a_a @ X3 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % surj_pair
% 0.28/0.47 thf(fact_28_surj__pair,axiom,
% 0.28/0.47 ! [P: produc1572603623od_a_a] :
% 0.28/0.47 ? [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.28/0.47 ( P
% 0.28/0.47 = ( produc1474507607od_a_a @ X3 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % surj_pair
% 0.28/0.47 thf(fact_29_surj__pair,axiom,
% 0.28/0.47 ! [P: product_prod_a_a] :
% 0.28/0.47 ? [X3: a,Y3: a] :
% 0.28/0.47 ( P
% 0.28/0.47 = ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % surj_pair
% 0.28/0.47 thf(fact_30_prod__cases,axiom,
% 0.28/0.47 ! [P2: produc2010915495od_a_a > $o,P: produc2010915495od_a_a] :
% 0.28/0.47 ( ! [A3: produc1921647824od_a_a,B4: produc1921647824od_a_a] : ( P2 @ ( produc563767703od_a_a @ A3 @ B4 ) )
% 0.28/0.47 => ( P2 @ P ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases
% 0.28/0.47 thf(fact_31_prod__cases,axiom,
% 0.28/0.47 ! [P2: produc772425383od_a_a > $o,P: produc772425383od_a_a] :
% 0.28/0.47 ( ! [A3: product_prod_a_a,B4: produc1572603623od_a_a] : ( P2 @ ( produc1555606807od_a_a @ A3 @ B4 ) )
% 0.28/0.47 => ( P2 @ P ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases
% 0.28/0.47 thf(fact_32_prod__cases,axiom,
% 0.28/0.47 ! [P2: produc1016684094_a_a_a > $o,P: produc1016684094_a_a_a] :
% 0.28/0.47 ( ! [A3: product_prod_a_a,B4: a] : ( P2 @ ( produc2061588782_a_a_a @ A3 @ B4 ) )
% 0.28/0.47 => ( P2 @ P ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases
% 0.28/0.47 thf(fact_33_prod__cases,axiom,
% 0.28/0.47 ! [P2: produc1555800720od_a_a > $o,P: produc1555800720od_a_a] :
% 0.28/0.47 ( ! [A3: a,B4: produc1572603623od_a_a] : ( P2 @ ( produc1095491584od_a_a @ A3 @ B4 ) )
% 0.28/0.47 => ( P2 @ P ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases
% 0.28/0.47 thf(fact_34_prod__cases,axiom,
% 0.28/0.47 ! [P2: produc1921647824od_a_a > $o,P: produc1921647824od_a_a] :
% 0.28/0.47 ( ! [A3: a,B4: product_prod_a_a] : ( P2 @ ( produc1299253312od_a_a @ A3 @ B4 ) )
% 0.28/0.47 => ( P2 @ P ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases
% 0.28/0.47 thf(fact_35_prod__cases,axiom,
% 0.28/0.47 ! [P2: produc1572603623od_a_a > $o,P: produc1572603623od_a_a] :
% 0.28/0.47 ( ! [A3: product_prod_a_a,B4: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B4 ) )
% 0.28/0.47 => ( P2 @ P ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases
% 0.28/0.47 thf(fact_36_prod__cases,axiom,
% 0.28/0.47 ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
% 0.28/0.47 ( ! [A3: a,B4: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B4 ) )
% 0.28/0.47 => ( P2 @ P ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases
% 0.28/0.47 thf(fact_37_Pair__inject,axiom,
% 0.28/0.47 ! [A: produc1921647824od_a_a,B2: produc1921647824od_a_a,A2: produc1921647824od_a_a,B3: produc1921647824od_a_a] :
% 0.28/0.47 ( ( ( produc563767703od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc563767703od_a_a @ A2 @ B3 ) )
% 0.28/0.47 => ~ ( ( A = A2 )
% 0.28/0.47 => ( B2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Pair_inject
% 0.28/0.47 thf(fact_38_Pair__inject,axiom,
% 0.28/0.47 ! [A: product_prod_a_a,B2: produc1572603623od_a_a,A2: product_prod_a_a,B3: produc1572603623od_a_a] :
% 0.28/0.47 ( ( ( produc1555606807od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc1555606807od_a_a @ A2 @ B3 ) )
% 0.28/0.47 => ~ ( ( A = A2 )
% 0.28/0.47 => ( B2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Pair_inject
% 0.28/0.47 thf(fact_39_Pair__inject,axiom,
% 0.28/0.47 ! [A: product_prod_a_a,B2: a,A2: product_prod_a_a,B3: a] :
% 0.28/0.47 ( ( ( produc2061588782_a_a_a @ A @ B2 )
% 0.28/0.47 = ( produc2061588782_a_a_a @ A2 @ B3 ) )
% 0.28/0.47 => ~ ( ( A = A2 )
% 0.28/0.47 => ( B2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Pair_inject
% 0.28/0.47 thf(fact_40_Pair__inject,axiom,
% 0.28/0.47 ! [A: a,B2: produc1572603623od_a_a,A2: a,B3: produc1572603623od_a_a] :
% 0.28/0.47 ( ( ( produc1095491584od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc1095491584od_a_a @ A2 @ B3 ) )
% 0.28/0.47 => ~ ( ( A = A2 )
% 0.28/0.47 => ( B2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Pair_inject
% 0.28/0.47 thf(fact_41_Pair__inject,axiom,
% 0.28/0.47 ! [A: a,B2: product_prod_a_a,A2: a,B3: product_prod_a_a] :
% 0.28/0.47 ( ( ( produc1299253312od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc1299253312od_a_a @ A2 @ B3 ) )
% 0.28/0.47 => ~ ( ( A = A2 )
% 0.28/0.47 => ( B2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Pair_inject
% 0.28/0.47 thf(fact_42_Pair__inject,axiom,
% 0.28/0.47 ! [A: product_prod_a_a,B2: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
% 0.28/0.47 ( ( ( produc1474507607od_a_a @ A @ B2 )
% 0.28/0.47 = ( produc1474507607od_a_a @ A2 @ B3 ) )
% 0.28/0.47 => ~ ( ( A = A2 )
% 0.28/0.47 => ( B2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Pair_inject
% 0.28/0.47 thf(fact_43_Pair__inject,axiom,
% 0.28/0.47 ! [A: a,B2: a,A2: a,B3: a] :
% 0.28/0.47 ( ( ( product_Pair_a_a @ A @ B2 )
% 0.28/0.47 = ( product_Pair_a_a @ A2 @ B3 ) )
% 0.28/0.47 => ~ ( ( A = A2 )
% 0.28/0.47 => ( B2 != B3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Pair_inject
% 0.28/0.47 thf(fact_44_prod__cases3,axiom,
% 0.28/0.47 ! [Y: produc772425383od_a_a] :
% 0.28/0.47 ~ ! [A3: product_prod_a_a,B4: product_prod_a_a,C: product_prod_a_a] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( produc1555606807od_a_a @ A3 @ ( produc1474507607od_a_a @ B4 @ C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases3
% 0.28/0.47 thf(fact_45_prod__cases3,axiom,
% 0.28/0.47 ! [Y: produc1555800720od_a_a] :
% 0.28/0.47 ~ ! [A3: a,B4: product_prod_a_a,C: product_prod_a_a] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( produc1095491584od_a_a @ A3 @ ( produc1474507607od_a_a @ B4 @ C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases3
% 0.28/0.47 thf(fact_46_prod__cases3,axiom,
% 0.28/0.47 ! [Y: produc1921647824od_a_a] :
% 0.28/0.47 ~ ! [A3: a,B4: a,C: a] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( produc1299253312od_a_a @ A3 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases3
% 0.28/0.47 thf(fact_47_prod__cases3,axiom,
% 0.28/0.47 ! [Y: produc1572603623od_a_a] :
% 0.28/0.47 ~ ! [A3: product_prod_a_a,B4: a,C: a] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_cases3
% 0.28/0.47 thf(fact_48_prod__induct3,axiom,
% 0.28/0.47 ! [P2: produc1572603623od_a_a > $o,X: produc1572603623od_a_a] :
% 0.28/0.47 ( ! [A3: product_prod_a_a,B4: a,C: a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B4 @ C ) ) )
% 0.28/0.47 => ( P2 @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % prod_induct3
% 0.28/0.47 thf(fact_49_old_Oprod_Oinducts,axiom,
% 0.28/0.47 ! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.28/0.47 ( ! [A3: a,B4: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B4 ) )
% 0.28/0.47 => ( P2 @ Prod ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inducts
% 0.28/0.47 thf(fact_50_old_Oprod_Oinducts,axiom,
% 0.28/0.47 ! [P2: produc1572603623od_a_a > $o,Prod: produc1572603623od_a_a] :
% 0.28/0.47 ( ! [A3: product_prod_a_a,B4: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B4 ) )
% 0.28/0.47 => ( P2 @ Prod ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.inducts
% 0.28/0.47 thf(fact_51_old_Oprod_Oexhaust,axiom,
% 0.28/0.47 ! [Y: product_prod_a_a] :
% 0.28/0.47 ~ ! [A3: a,B4: a] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( product_Pair_a_a @ A3 @ B4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.exhaust
% 0.28/0.47 thf(fact_52_old_Oprod_Oexhaust,axiom,
% 0.28/0.47 ! [Y: produc1572603623od_a_a] :
% 0.28/0.47 ~ ! [A3: product_prod_a_a,B4: product_prod_a_a] :
% 0.28/0.47 ( Y
% 0.28/0.47 != ( produc1474507607od_a_a @ A3 @ B4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % old.prod.exhaust
% 0.28/0.47 thf(fact_53_at__least__as__good__def,axiom,
% 0.28/0.47 ( prefer55414563od_a_a
% 0.28/0.47 = ( ^ [X4: produc1572603623od_a_a,B5: set_Pr1948701895od_a_a,P3: set_Pr1295299783od_a_a] :
% 0.28/0.47 ( collec1635618130od_a_a
% 0.28/0.47 @ ^ [Y4: produc1572603623od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ Y4 @ B5 )
% 0.28/0.47 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % at_least_as_good_def
% 0.28/0.47 thf(fact_54_at__least__as__good__def,axiom,
% 0.28/0.47 ( prefer161218362good_a
% 0.28/0.47 = ( ^ [X4: a,B5: set_a,P3: set_Product_prod_a_a] :
% 0.28/0.47 ( collect_a
% 0.28/0.47 @ ^ [Y4: a] :
% 0.28/0.47 ( ( member_a @ Y4 @ B5 )
% 0.28/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % at_least_as_good_def
% 0.28/0.47 thf(fact_55_at__least__as__good__def,axiom,
% 0.28/0.47 ( prefer477097315od_a_a
% 0.28/0.47 = ( ^ [X4: product_prod_a_a,B5: set_Product_prod_a_a,P3: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( collec645855634od_a_a
% 0.28/0.47 @ ^ [Y4: product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ Y4 @ B5 )
% 0.28/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % at_least_as_good_def
% 0.28/0.47 thf(fact_56_ssubst__Pair__rhs,axiom,
% 0.28/0.47 ! [R: a,S: a,R2: set_Product_prod_a_a,S2: a] :
% 0.28/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ R2 )
% 0.28/0.47 => ( ( S2 = S )
% 0.28/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ssubst_Pair_rhs
% 0.28/0.47 thf(fact_57_ssubst__Pair__rhs,axiom,
% 0.28/0.47 ! [R: product_prod_a_a,S: product_prod_a_a,R2: set_Pr1948701895od_a_a,S2: product_prod_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R @ S ) @ R2 )
% 0.28/0.47 => ( ( S2 = S )
% 0.28/0.47 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R @ S2 ) @ R2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ssubst_Pair_rhs
% 0.28/0.47 thf(fact_58_curryI,axiom,
% 0.28/0.47 ! [F: product_prod_a_a > $o,A: a,B2: a] :
% 0.28/0.47 ( ( F @ ( product_Pair_a_a @ A @ B2 ) )
% 0.28/0.47 => ( product_curry_a_a_o @ F @ A @ B2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % curryI
% 0.28/0.47 thf(fact_59_curryI,axiom,
% 0.28/0.47 ! [F: produc1572603623od_a_a > $o,A: product_prod_a_a,B2: product_prod_a_a] :
% 0.28/0.47 ( ( F @ ( produc1474507607od_a_a @ A @ B2 ) )
% 0.28/0.47 => ( produc94438221_a_a_o @ F @ A @ B2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % curryI
% 0.28/0.47 thf(fact_60_preference_Oindiff__trans,axiom,
% 0.28/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.28/0.47 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.28/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.28/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.28/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.28/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 0.28/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 0.28/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference.indiff_trans
% 0.28/0.47 thf(fact_61_preference_Oindiff__trans,axiom,
% 0.28/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.28/0.47 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.28/0.47 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.28/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.28/0.47 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.28/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 0.28/0.47 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 0.28/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference.indiff_trans
% 0.28/0.47 thf(fact_62_preference_Onot__outside,axiom,
% 0.28/0.47 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.28/0.47 ( ( prefer1409946857od_a_a @ Carrier @ Relation )
% 0.28/0.47 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.28/0.47 => ( member2057358096od_a_a @ X @ Carrier ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference.not_outside
% 0.28/0.47 thf(fact_63_preference_Onot__outside,axiom,
% 0.28/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.28/0.47 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.28/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.28/0.47 => ( member_a @ X @ Carrier ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference.not_outside
% 0.28/0.47 thf(fact_64_preference_Onot__outside,axiom,
% 0.28/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.28/0.47 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.28/0.47 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.28/0.47 => ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference.not_outside
% 0.28/0.47 thf(fact_65_swap__simp,axiom,
% 0.28/0.47 ! [X: a,Y: a] :
% 0.28/0.47 ( ( product_swap_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 0.28/0.47 = ( product_Pair_a_a @ Y @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % swap_simp
% 0.28/0.47 thf(fact_66_swap__simp,axiom,
% 0.28/0.47 ! [X: product_prod_a_a,Y: product_prod_a_a] :
% 0.28/0.47 ( ( produc1641853175od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
% 0.28/0.47 = ( produc1474507607od_a_a @ Y @ X ) ) ).
% 0.28/0.47
% 0.28/0.47 % swap_simp
% 0.28/0.47 thf(fact_67_curryD,axiom,
% 0.28/0.47 ! [F: product_prod_a_a > $o,A: a,B2: a] :
% 0.28/0.47 ( ( product_curry_a_a_o @ F @ A @ B2 )
% 0.28/0.47 => ( F @ ( product_Pair_a_a @ A @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % curryD
% 0.28/0.47 thf(fact_68_curryD,axiom,
% 0.28/0.47 ! [F: produc1572603623od_a_a > $o,A: product_prod_a_a,B2: product_prod_a_a] :
% 0.28/0.47 ( ( produc94438221_a_a_o @ F @ A @ B2 )
% 0.28/0.47 => ( F @ ( produc1474507607od_a_a @ A @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % curryD
% 0.28/0.47 thf(fact_69_curryE,axiom,
% 0.28/0.47 ! [F: product_prod_a_a > $o,A: a,B2: a] :
% 0.28/0.47 ( ( product_curry_a_a_o @ F @ A @ B2 )
% 0.28/0.47 => ( F @ ( product_Pair_a_a @ A @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % curryE
% 0.28/0.47 thf(fact_70_curryE,axiom,
% 0.28/0.47 ! [F: produc1572603623od_a_a > $o,A: product_prod_a_a,B2: product_prod_a_a] :
% 0.28/0.47 ( ( produc94438221_a_a_o @ F @ A @ B2 )
% 0.28/0.47 => ( F @ ( produc1474507607od_a_a @ A @ B2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % curryE
% 0.28/0.47 thf(fact_71_no__better__than__def,axiom,
% 0.28/0.47 ( prefer926864306od_a_a
% 0.28/0.47 = ( ^ [X4: produc1572603623od_a_a,B5: set_Pr1948701895od_a_a,P3: set_Pr1295299783od_a_a] :
% 0.28/0.47 ( collec1635618130od_a_a
% 0.28/0.47 @ ^ [Y4: produc1572603623od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ Y4 @ B5 )
% 0.28/0.47 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % no_better_than_def
% 0.28/0.47 thf(fact_72_no__better__than__def,axiom,
% 0.28/0.47 ( prefer1676310729than_a
% 0.28/0.47 = ( ^ [X4: a,B5: set_a,P3: set_Product_prod_a_a] :
% 0.28/0.47 ( collect_a
% 0.28/0.47 @ ^ [Y4: a] :
% 0.28/0.47 ( ( member_a @ Y4 @ B5 )
% 0.28/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % no_better_than_def
% 0.28/0.47 thf(fact_73_no__better__than__def,axiom,
% 0.28/0.47 ( prefer1498087410od_a_a
% 0.28/0.47 = ( ^ [X4: product_prod_a_a,B5: set_Product_prod_a_a,P3: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( collec645855634od_a_a
% 0.28/0.47 @ ^ [Y4: product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ Y4 @ B5 )
% 0.28/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % no_better_than_def
% 0.28/0.47 thf(fact_74_as__good__as__def,axiom,
% 0.28/0.47 ( prefer1867335523od_a_a
% 0.28/0.47 = ( ^ [X4: produc1572603623od_a_a,B5: set_Pr1948701895od_a_a,P3: set_Pr1295299783od_a_a] :
% 0.28/0.47 ( collec1635618130od_a_a
% 0.28/0.47 @ ^ [Y4: produc1572603623od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ Y4 @ B5 )
% 0.28/0.47 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ P3 )
% 0.28/0.47 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % as_good_as_def
% 0.28/0.47 thf(fact_75_as__good__as__def,axiom,
% 0.28/0.47 ( prefer436369274d_as_a
% 0.28/0.47 = ( ^ [X4: a,B5: set_a,P3: set_Product_prod_a_a] :
% 0.28/0.47 ( collect_a
% 0.28/0.47 @ ^ [Y4: a] :
% 0.28/0.47 ( ( member_a @ Y4 @ B5 )
% 0.28/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ P3 )
% 0.28/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % as_good_as_def
% 0.28/0.47 thf(fact_76_as__good__as__def,axiom,
% 0.28/0.47 ( prefer74720675od_a_a
% 0.28/0.47 = ( ^ [X4: product_prod_a_a,B5: set_Product_prod_a_a,P3: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( collec645855634od_a_a
% 0.28/0.47 @ ^ [Y4: product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ Y4 @ B5 )
% 0.28/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ P3 )
% 0.28/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % as_good_as_def
% 0.28/0.47 thf(fact_77_pred__equals__eq2,axiom,
% 0.28/0.47 ! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 0.28/0.47 ( ( ( ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R2 ) )
% 0.28/0.47 = ( ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ S3 ) ) )
% 0.28/0.47 = ( R2 = S3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % pred_equals_eq2
% 0.28/0.47 thf(fact_78_pred__equals__eq2,axiom,
% 0.28/0.47 ! [R2: set_Pr1948701895od_a_a,S3: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( ( ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R2 ) )
% 0.28/0.47 = ( ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ S3 ) ) )
% 0.28/0.47 = ( R2 = S3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % pred_equals_eq2
% 0.28/0.47 thf(fact_79_preference__def,axiom,
% 0.28/0.47 ( prefer1409946857od_a_a
% 0.28/0.47 = ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
% 0.28/0.47 ( ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
% 0.28/0.47 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.28/0.47 => ( member2057358096od_a_a @ X4 @ Carrier2 ) )
% 0.28/0.47 & ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
% 0.28/0.47 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.28/0.47 => ( member2057358096od_a_a @ Y4 @ Carrier2 ) )
% 0.28/0.47 & ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference_def
% 0.28/0.47 thf(fact_80_preference__def,axiom,
% 0.28/0.47 ( prefer1292084992ence_a
% 0.28/0.47 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 0.28/0.47 ( ! [X4: a,Y4: a] :
% 0.28/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.28/0.47 => ( member_a @ X4 @ Carrier2 ) )
% 0.28/0.47 & ! [X4: a,Y4: a] :
% 0.28/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.28/0.47 => ( member_a @ Y4 @ Carrier2 ) )
% 0.28/0.47 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference_def
% 0.28/0.47 thf(fact_81_preference__def,axiom,
% 0.28/0.47 ( prefer1338892073od_a_a
% 0.28/0.47 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.28/0.47 => ( member449909584od_a_a @ X4 @ Carrier2 ) )
% 0.28/0.47 & ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.28/0.47 => ( member449909584od_a_a @ Y4 @ Carrier2 ) )
% 0.28/0.47 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference_def
% 0.28/0.47 thf(fact_82_preference_Ointro,axiom,
% 0.28/0.47 ! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.28/0.47 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
% 0.28/0.47 => ( member2057358096od_a_a @ X3 @ Carrier ) )
% 0.28/0.47 => ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.28/0.47 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
% 0.28/0.47 => ( member2057358096od_a_a @ Y3 @ Carrier ) )
% 0.28/0.47 => ( ( order_491428350od_a_a @ Carrier @ Relation )
% 0.28/0.47 => ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference.intro
% 0.28/0.47 thf(fact_83_preference_Ointro,axiom,
% 0.28/0.47 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 0.28/0.47 ( ! [X3: a,Y3: a] :
% 0.28/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
% 0.28/0.47 => ( member_a @ X3 @ Carrier ) )
% 0.28/0.47 => ( ! [X3: a,Y3: a] :
% 0.28/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
% 0.28/0.47 => ( member_a @ Y3 @ Carrier ) )
% 0.28/0.47 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 0.28/0.47 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference.intro
% 0.28/0.47 thf(fact_84_preference_Ointro,axiom,
% 0.28/0.47 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 0.28/0.47 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
% 0.28/0.47 => ( member449909584od_a_a @ X3 @ Carrier ) )
% 0.28/0.47 => ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
% 0.28/0.47 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 0.28/0.47 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 0.28/0.47 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % preference.intro
% 0.28/0.47 thf(fact_85_pair__in__swap__image,axiom,
% 0.28/0.47 ! [Y: a,X: a,A4: set_Product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( image_794773445od_a_a @ product_swap_a_a @ A4 ) )
% 0.28/0.47 = ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ A4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % pair_in_swap_image
% 0.28/0.47 thf(fact_86_pair__in__swap__image,axiom,
% 0.28/0.47 ! [Y: product_prod_a_a,X: product_prod_a_a,A4: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ ( image_236793797od_a_a @ produc1641853175od_a_a @ A4 ) )
% 0.28/0.47 = ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ A4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % pair_in_swap_image
% 0.28/0.47 thf(fact_87_strict__contour__is__diff,axiom,
% 0.28/0.47 ! [B: set_Pr1948701895od_a_a,Y: produc1572603623od_a_a,Pr: set_Pr1295299783od_a_a] :
% 0.28/0.47 ( ( collec1635618130od_a_a
% 0.28/0.47 @ ^ [A5: produc1572603623od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ A5 @ B )
% 0.28/0.47 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A5 @ Y ) @ Pr )
% 0.28/0.47 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ A5 ) @ Pr ) ) )
% 0.28/0.47 = ( minus_1034916142od_a_a @ ( prefer55414563od_a_a @ Y @ B @ Pr ) @ ( prefer1867335523od_a_a @ Y @ B @ Pr ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % strict_contour_is_diff
% 0.28/0.47 thf(fact_88_strict__contour__is__diff,axiom,
% 0.28/0.47 ! [B: set_a,Y: a,Pr: set_Product_prod_a_a] :
% 0.28/0.47 ( ( collect_a
% 0.28/0.47 @ ^ [A5: a] :
% 0.28/0.47 ( ( member_a @ A5 @ B )
% 0.28/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ Y ) @ Pr )
% 0.28/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ A5 ) @ Pr ) ) )
% 0.28/0.47 = ( minus_minus_set_a @ ( prefer161218362good_a @ Y @ B @ Pr ) @ ( prefer436369274d_as_a @ Y @ B @ Pr ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % strict_contour_is_diff
% 0.28/0.47 thf(fact_89_strict__contour__is__diff,axiom,
% 0.28/0.47 ! [B: set_Product_prod_a_a,Y: product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( collec645855634od_a_a
% 0.28/0.47 @ ^ [A5: product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ A5 @ B )
% 0.28/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A5 @ Y ) @ Pr )
% 0.28/0.47 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ A5 ) @ Pr ) ) )
% 0.28/0.47 = ( minus_1507102574od_a_a @ ( prefer477097315od_a_a @ Y @ B @ Pr ) @ ( prefer74720675od_a_a @ Y @ B @ Pr ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % strict_contour_is_diff
% 0.28/0.47 thf(fact_90_mem__Collect__eq,axiom,
% 0.28/0.47 ! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
% 0.28/0.47 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
% 0.28/0.47 = ( P2 @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 % mem_Collect_eq
% 0.28/0.47 thf(fact_91_mem__Collect__eq,axiom,
% 0.28/0.47 ! [A: a,P2: a > $o] :
% 0.28/0.47 ( ( member_a @ A @ ( collect_a @ P2 ) )
% 0.28/0.47 = ( P2 @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 % mem_Collect_eq
% 0.28/0.47 thf(fact_92_mem__Collect__eq,axiom,
% 0.28/0.47 ! [A: produc1572603623od_a_a,P2: produc1572603623od_a_a > $o] :
% 0.28/0.47 ( ( member2057358096od_a_a @ A @ ( collec1635618130od_a_a @ P2 ) )
% 0.28/0.47 = ( P2 @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 % mem_Collect_eq
% 0.28/0.47 thf(fact_93_Collect__mem__eq,axiom,
% 0.28/0.47 ! [A4: set_Product_prod_a_a] :
% 0.28/0.47 ( ( collec645855634od_a_a
% 0.28/0.47 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A4 ) )
% 0.28/0.47 = A4 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mem_eq
% 0.28/0.47 thf(fact_94_Collect__mem__eq,axiom,
% 0.28/0.47 ! [A4: set_a] :
% 0.28/0.47 ( ( collect_a
% 0.28/0.47 @ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
% 0.28/0.47 = A4 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mem_eq
% 0.28/0.47 thf(fact_95_Collect__mem__eq,axiom,
% 0.28/0.47 ! [A4: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( collec1635618130od_a_a
% 0.28/0.47 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A4 ) )
% 0.28/0.47 = A4 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mem_eq
% 0.28/0.47 thf(fact_96_DiffI,axiom,
% 0.28/0.47 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ C2 @ A4 )
% 0.28/0.47 => ( ~ ( member449909584od_a_a @ C2 @ B )
% 0.28/0.47 => ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffI
% 0.28/0.47 thf(fact_97_DiffI,axiom,
% 0.28/0.47 ! [C2: a,A4: set_a,B: set_a] :
% 0.28/0.47 ( ( member_a @ C2 @ A4 )
% 0.28/0.47 => ( ~ ( member_a @ C2 @ B )
% 0.28/0.47 => ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffI
% 0.28/0.47 thf(fact_98_DiffI,axiom,
% 0.28/0.47 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ C2 @ A4 )
% 0.28/0.47 => ( ~ ( member2057358096od_a_a @ C2 @ B )
% 0.28/0.47 => ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffI
% 0.28/0.47 thf(fact_99_image__eqI,axiom,
% 0.28/0.47 ! [B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,A4: set_Product_prod_a_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.47 => ( member449909584od_a_a @ B2 @ ( image_794773445od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % image_eqI
% 0.28/0.47 thf(fact_100_image__eqI,axiom,
% 0.28/0.47 ! [B2: a,F: product_prod_a_a > a,X: product_prod_a_a,A4: set_Product_prod_a_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.47 => ( member_a @ B2 @ ( image_340790172_a_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % image_eqI
% 0.28/0.47 thf(fact_101_image__eqI,axiom,
% 0.28/0.47 ! [B2: produc1572603623od_a_a,F: product_prod_a_a > produc1572603623od_a_a,X: product_prod_a_a,A4: set_Product_prod_a_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.47 => ( member2057358096od_a_a @ B2 @ ( image_1765377413od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % image_eqI
% 0.28/0.47 thf(fact_102_image__eqI,axiom,
% 0.28/0.47 ! [B2: product_prod_a_a,F: a > product_prod_a_a,X: a,A4: set_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( ( member_a @ X @ A4 )
% 0.28/0.47 => ( member449909584od_a_a @ B2 @ ( image_1725938350od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % image_eqI
% 0.28/0.47 thf(fact_103_image__eqI,axiom,
% 0.28/0.47 ! [B2: a,F: a > a,X: a,A4: set_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( ( member_a @ X @ A4 )
% 0.28/0.47 => ( member_a @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % image_eqI
% 0.28/0.47 thf(fact_104_image__eqI,axiom,
% 0.28/0.47 ! [B2: produc1572603623od_a_a,F: a > produc1572603623od_a_a,X: a,A4: set_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( ( member_a @ X @ A4 )
% 0.28/0.47 => ( member2057358096od_a_a @ B2 @ ( image_1763923566od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % image_eqI
% 0.28/0.47 thf(fact_105_image__eqI,axiom,
% 0.28/0.47 ! [B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a,X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.47 => ( member449909584od_a_a @ B2 @ ( image_734753285od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % image_eqI
% 0.28/0.47 thf(fact_106_image__eqI,axiom,
% 0.28/0.47 ! [B2: a,F: produc1572603623od_a_a > a,X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.47 => ( member_a @ B2 @ ( image_1105229276_a_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % image_eqI
% 0.28/0.47 thf(fact_107_image__eqI,axiom,
% 0.28/0.47 ! [B2: produc1572603623od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a,X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.47 => ( member2057358096od_a_a @ B2 @ ( image_236793797od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % image_eqI
% 0.28/0.47 thf(fact_108_Diff__iff,axiom,
% 0.28/0.47 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
% 0.28/0.47 = ( ( member449909584od_a_a @ C2 @ A4 )
% 0.28/0.47 & ~ ( member449909584od_a_a @ C2 @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Diff_iff
% 0.28/0.47 thf(fact_109_Diff__iff,axiom,
% 0.28/0.47 ! [C2: a,A4: set_a,B: set_a] :
% 0.28/0.47 ( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
% 0.28/0.47 = ( ( member_a @ C2 @ A4 )
% 0.28/0.47 & ~ ( member_a @ C2 @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Diff_iff
% 0.28/0.47 thf(fact_110_Diff__iff,axiom,
% 0.28/0.47 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
% 0.28/0.47 = ( ( member2057358096od_a_a @ C2 @ A4 )
% 0.28/0.47 & ~ ( member2057358096od_a_a @ C2 @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Diff_iff
% 0.28/0.47 thf(fact_111_minus__set__def,axiom,
% 0.28/0.47 ( minus_1507102574od_a_a
% 0.28/0.47 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 0.28/0.47 ( collec645855634od_a_a
% 0.28/0.47 @ ( minus_1797478127_a_a_o
% 0.28/0.47 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A6 )
% 0.28/0.47 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minus_set_def
% 0.28/0.47 thf(fact_112_minus__set__def,axiom,
% 0.28/0.47 ( minus_minus_set_a
% 0.28/0.47 = ( ^ [A6: set_a,B5: set_a] :
% 0.28/0.47 ( collect_a
% 0.28/0.47 @ ( minus_minus_a_o
% 0.28/0.47 @ ^ [X4: a] : ( member_a @ X4 @ A6 )
% 0.28/0.47 @ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minus_set_def
% 0.28/0.47 thf(fact_113_minus__set__def,axiom,
% 0.28/0.47 ( minus_1034916142od_a_a
% 0.28/0.47 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( collec1635618130od_a_a
% 0.28/0.47 @ ( minus_107310383_a_a_o
% 0.28/0.47 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A6 )
% 0.28/0.47 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % minus_set_def
% 0.28/0.47 thf(fact_114_rev__image__eqI,axiom,
% 0.28/0.47 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( member449909584od_a_a @ B2 @ ( image_794773445od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_image_eqI
% 0.28/0.47 thf(fact_115_rev__image__eqI,axiom,
% 0.28/0.47 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,B2: a,F: product_prod_a_a > a] :
% 0.28/0.47 ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( member_a @ B2 @ ( image_340790172_a_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_image_eqI
% 0.28/0.47 thf(fact_116_rev__image__eqI,axiom,
% 0.28/0.47 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,B2: produc1572603623od_a_a,F: product_prod_a_a > produc1572603623od_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( member2057358096od_a_a @ B2 @ ( image_1765377413od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_image_eqI
% 0.28/0.47 thf(fact_117_rev__image__eqI,axiom,
% 0.28/0.47 ! [X: a,A4: set_a,B2: product_prod_a_a,F: a > product_prod_a_a] :
% 0.28/0.47 ( ( member_a @ X @ A4 )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( member449909584od_a_a @ B2 @ ( image_1725938350od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_image_eqI
% 0.28/0.47 thf(fact_118_rev__image__eqI,axiom,
% 0.28/0.47 ! [X: a,A4: set_a,B2: a,F: a > a] :
% 0.28/0.47 ( ( member_a @ X @ A4 )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( member_a @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_image_eqI
% 0.28/0.47 thf(fact_119_rev__image__eqI,axiom,
% 0.28/0.47 ! [X: a,A4: set_a,B2: produc1572603623od_a_a,F: a > produc1572603623od_a_a] :
% 0.28/0.47 ( ( member_a @ X @ A4 )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( member2057358096od_a_a @ B2 @ ( image_1763923566od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_image_eqI
% 0.28/0.47 thf(fact_120_rev__image__eqI,axiom,
% 0.28/0.47 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( member449909584od_a_a @ B2 @ ( image_734753285od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_image_eqI
% 0.28/0.47 thf(fact_121_rev__image__eqI,axiom,
% 0.28/0.47 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: a,F: produc1572603623od_a_a > a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( member_a @ B2 @ ( image_1105229276_a_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_image_eqI
% 0.28/0.47 thf(fact_122_rev__image__eqI,axiom,
% 0.28/0.47 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: produc1572603623od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 = ( F @ X ) )
% 0.28/0.47 => ( member2057358096od_a_a @ B2 @ ( image_236793797od_a_a @ F @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % rev_image_eqI
% 0.28/0.47 thf(fact_123_imageI,axiom,
% 0.28/0.47 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.47 => ( member449909584od_a_a @ ( F @ X ) @ ( image_794773445od_a_a @ F @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imageI
% 0.28/0.47 thf(fact_124_imageI,axiom,
% 0.28/0.47 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > a] :
% 0.28/0.47 ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.47 => ( member_a @ ( F @ X ) @ ( image_340790172_a_a_a @ F @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imageI
% 0.28/0.47 thf(fact_125_imageI,axiom,
% 0.28/0.47 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > produc1572603623od_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.47 => ( member2057358096od_a_a @ ( F @ X ) @ ( image_1765377413od_a_a @ F @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imageI
% 0.28/0.47 thf(fact_126_imageI,axiom,
% 0.28/0.47 ! [X: a,A4: set_a,F: a > product_prod_a_a] :
% 0.28/0.47 ( ( member_a @ X @ A4 )
% 0.28/0.47 => ( member449909584od_a_a @ ( F @ X ) @ ( image_1725938350od_a_a @ F @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imageI
% 0.28/0.47 thf(fact_127_imageI,axiom,
% 0.28/0.47 ! [X: a,A4: set_a,F: a > a] :
% 0.28/0.47 ( ( member_a @ X @ A4 )
% 0.28/0.47 => ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imageI
% 0.28/0.47 thf(fact_128_imageI,axiom,
% 0.28/0.47 ! [X: a,A4: set_a,F: a > produc1572603623od_a_a] :
% 0.28/0.47 ( ( member_a @ X @ A4 )
% 0.28/0.47 => ( member2057358096od_a_a @ ( F @ X ) @ ( image_1763923566od_a_a @ F @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imageI
% 0.28/0.47 thf(fact_129_imageI,axiom,
% 0.28/0.47 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > product_prod_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.47 => ( member449909584od_a_a @ ( F @ X ) @ ( image_734753285od_a_a @ F @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imageI
% 0.28/0.47 thf(fact_130_imageI,axiom,
% 0.28/0.47 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.47 => ( member_a @ ( F @ X ) @ ( image_1105229276_a_a_a @ F @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imageI
% 0.28/0.47 thf(fact_131_imageI,axiom,
% 0.28/0.47 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.47 => ( member2057358096od_a_a @ ( F @ X ) @ ( image_236793797od_a_a @ F @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imageI
% 0.28/0.47 thf(fact_132_DiffD2,axiom,
% 0.28/0.47 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
% 0.28/0.47 => ~ ( member449909584od_a_a @ C2 @ B ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffD2
% 0.28/0.47 thf(fact_133_DiffD2,axiom,
% 0.28/0.47 ! [C2: a,A4: set_a,B: set_a] :
% 0.28/0.47 ( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
% 0.28/0.47 => ~ ( member_a @ C2 @ B ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffD2
% 0.28/0.47 thf(fact_134_DiffD2,axiom,
% 0.28/0.47 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
% 0.28/0.47 => ~ ( member2057358096od_a_a @ C2 @ B ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffD2
% 0.28/0.47 thf(fact_135_DiffD1,axiom,
% 0.28/0.47 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
% 0.28/0.47 => ( member449909584od_a_a @ C2 @ A4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffD1
% 0.28/0.47 thf(fact_136_DiffD1,axiom,
% 0.28/0.47 ! [C2: a,A4: set_a,B: set_a] :
% 0.28/0.47 ( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
% 0.28/0.47 => ( member_a @ C2 @ A4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffD1
% 0.28/0.47 thf(fact_137_DiffD1,axiom,
% 0.28/0.47 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
% 0.28/0.47 => ( member2057358096od_a_a @ C2 @ A4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffD1
% 0.28/0.47 thf(fact_138_DiffE,axiom,
% 0.28/0.47 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
% 0.28/0.47 => ~ ( ( member449909584od_a_a @ C2 @ A4 )
% 0.28/0.47 => ( member449909584od_a_a @ C2 @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffE
% 0.28/0.47 thf(fact_139_DiffE,axiom,
% 0.28/0.47 ! [C2: a,A4: set_a,B: set_a] :
% 0.28/0.47 ( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
% 0.28/0.47 => ~ ( ( member_a @ C2 @ A4 )
% 0.28/0.47 => ( member_a @ C2 @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffE
% 0.28/0.47 thf(fact_140_DiffE,axiom,
% 0.28/0.47 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.47 ( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
% 0.28/0.47 => ~ ( ( member2057358096od_a_a @ C2 @ A4 )
% 0.28/0.47 => ( member2057358096od_a_a @ C2 @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % DiffE
% 0.28/0.47 thf(fact_141_imageE,axiom,
% 0.28/0.47 ! [B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,A4: set_Product_prod_a_a] :
% 0.28/0.47 ( ( member449909584od_a_a @ B2 @ ( image_794773445od_a_a @ F @ A4 ) )
% 0.28/0.47 => ~ ! [X3: product_prod_a_a] :
% 0.28/0.47 ( ( B2
% 0.28/0.47 = ( F @ X3 ) )
% 0.28/0.47 => ~ ( member449909584od_a_a @ X3 @ A4 ) ) ) ).
% 0.28/0.47
% 0.28/0.48 % imageE
% 0.28/0.48 thf(fact_142_imageE,axiom,
% 0.28/0.48 ! [B2: product_prod_a_a,F: a > product_prod_a_a,A4: set_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ B2 @ ( image_1725938350od_a_a @ F @ A4 ) )
% 0.28/0.48 => ~ ! [X3: a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X3 ) )
% 0.28/0.48 => ~ ( member_a @ X3 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % imageE
% 0.28/0.48 thf(fact_143_imageE,axiom,
% 0.28/0.48 ! [B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a,A4: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ B2 @ ( image_734753285od_a_a @ F @ A4 ) )
% 0.28/0.48 => ~ ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X3 ) )
% 0.28/0.48 => ~ ( member2057358096od_a_a @ X3 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % imageE
% 0.28/0.48 thf(fact_144_imageE,axiom,
% 0.28/0.48 ! [B2: a,F: product_prod_a_a > a,A4: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member_a @ B2 @ ( image_340790172_a_a_a @ F @ A4 ) )
% 0.28/0.48 => ~ ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X3 ) )
% 0.28/0.48 => ~ ( member449909584od_a_a @ X3 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % imageE
% 0.28/0.48 thf(fact_145_imageE,axiom,
% 0.28/0.48 ! [B2: a,F: a > a,A4: set_a] :
% 0.28/0.48 ( ( member_a @ B2 @ ( image_a_a @ F @ A4 ) )
% 0.28/0.48 => ~ ! [X3: a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X3 ) )
% 0.28/0.48 => ~ ( member_a @ X3 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % imageE
% 0.28/0.48 thf(fact_146_imageE,axiom,
% 0.28/0.48 ! [B2: a,F: produc1572603623od_a_a > a,A4: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member_a @ B2 @ ( image_1105229276_a_a_a @ F @ A4 ) )
% 0.28/0.48 => ~ ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X3 ) )
% 0.28/0.48 => ~ ( member2057358096od_a_a @ X3 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % imageE
% 0.28/0.48 thf(fact_147_imageE,axiom,
% 0.28/0.48 ! [B2: produc1572603623od_a_a,F: product_prod_a_a > produc1572603623od_a_a,A4: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ B2 @ ( image_1765377413od_a_a @ F @ A4 ) )
% 0.28/0.48 => ~ ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X3 ) )
% 0.28/0.48 => ~ ( member449909584od_a_a @ X3 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % imageE
% 0.28/0.48 thf(fact_148_imageE,axiom,
% 0.28/0.48 ! [B2: produc1572603623od_a_a,F: a > produc1572603623od_a_a,A4: set_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ B2 @ ( image_1763923566od_a_a @ F @ A4 ) )
% 0.28/0.48 => ~ ! [X3: a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X3 ) )
% 0.28/0.48 => ~ ( member_a @ X3 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % imageE
% 0.28/0.48 thf(fact_149_imageE,axiom,
% 0.28/0.48 ! [B2: produc1572603623od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ B2 @ ( image_236793797od_a_a @ F @ A4 ) )
% 0.28/0.48 => ~ ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X3 ) )
% 0.28/0.48 => ~ ( member2057358096od_a_a @ X3 @ A4 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % imageE
% 0.28/0.48 thf(fact_150_Compr__image__eq,axiom,
% 0.28/0.48 ! [F: product_prod_a_a > product_prod_a_a,A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 0.28/0.48 ( ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ ( image_794773445od_a_a @ F @ A4 ) )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 = ( image_794773445od_a_a @ F
% 0.28/0.48 @ ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Compr_image_eq
% 0.28/0.48 thf(fact_151_Compr__image__eq,axiom,
% 0.28/0.48 ! [F: a > product_prod_a_a,A4: set_a,P2: product_prod_a_a > $o] :
% 0.28/0.48 ( ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ ( image_1725938350od_a_a @ F @ A4 ) )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 = ( image_1725938350od_a_a @ F
% 0.28/0.48 @ ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Compr_image_eq
% 0.28/0.48 thf(fact_152_Compr__image__eq,axiom,
% 0.28/0.48 ! [F: produc1572603623od_a_a > product_prod_a_a,A4: set_Pr1948701895od_a_a,P2: product_prod_a_a > $o] :
% 0.28/0.48 ( ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ ( image_734753285od_a_a @ F @ A4 ) )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 = ( image_734753285od_a_a @ F
% 0.28/0.48 @ ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Compr_image_eq
% 0.28/0.48 thf(fact_153_Compr__image__eq,axiom,
% 0.28/0.48 ! [F: product_prod_a_a > a,A4: set_Product_prod_a_a,P2: a > $o] :
% 0.28/0.48 ( ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ ( image_340790172_a_a_a @ F @ A4 ) )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 = ( image_340790172_a_a_a @ F
% 0.28/0.48 @ ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Compr_image_eq
% 0.28/0.48 thf(fact_154_Compr__image__eq,axiom,
% 0.28/0.48 ! [F: a > a,A4: set_a,P2: a > $o] :
% 0.28/0.48 ( ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ ( image_a_a @ F @ A4 ) )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 = ( image_a_a @ F
% 0.28/0.48 @ ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Compr_image_eq
% 0.28/0.48 thf(fact_155_Compr__image__eq,axiom,
% 0.28/0.48 ! [F: produc1572603623od_a_a > a,A4: set_Pr1948701895od_a_a,P2: a > $o] :
% 0.28/0.48 ( ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ ( image_1105229276_a_a_a @ F @ A4 ) )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 = ( image_1105229276_a_a_a @ F
% 0.28/0.48 @ ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Compr_image_eq
% 0.28/0.48 thf(fact_156_Compr__image__eq,axiom,
% 0.28/0.48 ! [F: product_prod_a_a > produc1572603623od_a_a,A4: set_Product_prod_a_a,P2: produc1572603623od_a_a > $o] :
% 0.28/0.48 ( ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ ( image_1765377413od_a_a @ F @ A4 ) )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 = ( image_1765377413od_a_a @ F
% 0.28/0.48 @ ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Compr_image_eq
% 0.28/0.48 thf(fact_157_Compr__image__eq,axiom,
% 0.28/0.48 ! [F: a > produc1572603623od_a_a,A4: set_a,P2: produc1572603623od_a_a > $o] :
% 0.28/0.48 ( ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ ( image_1763923566od_a_a @ F @ A4 ) )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 = ( image_1763923566od_a_a @ F
% 0.28/0.48 @ ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Compr_image_eq
% 0.28/0.48 thf(fact_158_Compr__image__eq,axiom,
% 0.28/0.48 ! [F: produc1572603623od_a_a > produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 0.28/0.48 ( ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ ( image_236793797od_a_a @ F @ A4 ) )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 = ( image_236793797od_a_a @ F
% 0.28/0.48 @ ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % Compr_image_eq
% 0.28/0.48 thf(fact_159_set__diff__eq,axiom,
% 0.28/0.48 ( minus_1507102574od_a_a
% 0.28/0.48 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 0.28/0.48 ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ A6 )
% 0.28/0.48 & ~ ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % set_diff_eq
% 0.28/0.48 thf(fact_160_set__diff__eq,axiom,
% 0.28/0.48 ( minus_minus_set_a
% 0.28/0.48 = ( ^ [A6: set_a,B5: set_a] :
% 0.28/0.48 ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ A6 )
% 0.28/0.48 & ~ ( member_a @ X4 @ B5 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % set_diff_eq
% 0.28/0.48 thf(fact_161_set__diff__eq,axiom,
% 0.28/0.48 ( minus_1034916142od_a_a
% 0.28/0.48 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ A6 )
% 0.28/0.48 & ~ ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % set_diff_eq
% 0.28/0.48 thf(fact_162_UNIV__I,axiom,
% 0.28/0.48 ! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ top_to398455383od_a_a ) ).
% 0.28/0.48
% 0.28/0.48 % UNIV_I
% 0.28/0.48 thf(fact_163_UNIV__I,axiom,
% 0.28/0.48 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 0.28/0.48
% 0.28/0.48 % UNIV_I
% 0.28/0.48 thf(fact_164_UNIV__I,axiom,
% 0.28/0.48 ! [X: produc1572603623od_a_a] : ( member2057358096od_a_a @ X @ top_to52711959od_a_a ) ).
% 0.28/0.48
% 0.28/0.48 % UNIV_I
% 0.28/0.48 thf(fact_165_UNIV__witness,axiom,
% 0.28/0.48 ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ top_to398455383od_a_a ) ).
% 0.28/0.48
% 0.28/0.48 % UNIV_witness
% 0.28/0.48 thf(fact_166_UNIV__witness,axiom,
% 0.28/0.48 ? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% 0.28/0.48
% 0.28/0.48 % UNIV_witness
% 0.28/0.48 thf(fact_167_UNIV__witness,axiom,
% 0.28/0.48 ? [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ top_to52711959od_a_a ) ).
% 0.28/0.48
% 0.28/0.48 % UNIV_witness
% 0.28/0.48 thf(fact_168_UNIV__eq__I,axiom,
% 0.28/0.48 ! [A4: set_Product_prod_a_a] :
% 0.28/0.48 ( ! [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( top_to398455383od_a_a = A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % UNIV_eq_I
% 0.28/0.48 thf(fact_169_UNIV__eq__I,axiom,
% 0.28/0.48 ! [A4: set_a] :
% 0.28/0.48 ( ! [X3: a] : ( member_a @ X3 @ A4 )
% 0.28/0.48 => ( top_top_set_a = A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % UNIV_eq_I
% 0.28/0.48 thf(fact_170_UNIV__eq__I,axiom,
% 0.28/0.48 ! [A4: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ! [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( top_to52711959od_a_a = A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % UNIV_eq_I
% 0.28/0.48 thf(fact_171_iso__tuple__UNIV__I,axiom,
% 0.28/0.48 ! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ top_to398455383od_a_a ) ).
% 0.28/0.48
% 0.28/0.48 % iso_tuple_UNIV_I
% 0.28/0.48 thf(fact_172_iso__tuple__UNIV__I,axiom,
% 0.28/0.48 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 0.28/0.48
% 0.28/0.48 % iso_tuple_UNIV_I
% 0.28/0.48 thf(fact_173_iso__tuple__UNIV__I,axiom,
% 0.28/0.48 ! [X: produc1572603623od_a_a] : ( member2057358096od_a_a @ X @ top_to52711959od_a_a ) ).
% 0.28/0.48
% 0.28/0.48 % iso_tuple_UNIV_I
% 0.28/0.48 thf(fact_174_same__fstI,axiom,
% 0.28/0.48 ! [P2: a > $o,X: a,Y5: a,Y: a,R2: a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( P2 @ X )
% 0.28/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y5 @ Y ) @ ( R2 @ X ) )
% 0.28/0.48 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ ( product_Pair_a_a @ X @ Y ) ) @ ( same_fst_a_a @ P2 @ R2 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % same_fstI
% 0.28/0.48 thf(fact_175_same__fstI,axiom,
% 0.28/0.48 ! [P2: product_prod_a_a > $o,X: product_prod_a_a,Y5: product_prod_a_a,Y: product_prod_a_a,R2: product_prod_a_a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( P2 @ X )
% 0.28/0.48 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y5 @ Y ) @ ( R2 @ X ) )
% 0.28/0.48 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ ( produc1474507607od_a_a @ X @ Y5 ) @ ( produc1474507607od_a_a @ X @ Y ) ) @ ( same_f1138477100od_a_a @ P2 @ R2 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % same_fstI
% 0.28/0.48 thf(fact_176_map__prod__simp,axiom,
% 0.28/0.48 ! [F: a > a,G: a > a,A: a,B2: a] :
% 0.28/0.48 ( ( produc700022774_a_a_a @ F @ G @ ( product_Pair_a_a @ A @ B2 ) )
% 0.28/0.48 = ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % map_prod_simp
% 0.28/0.48 thf(fact_177_map__prod__simp,axiom,
% 0.28/0.48 ! [F: a > product_prod_a_a,G: a > product_prod_a_a,A: a,B2: a] :
% 0.28/0.48 ( ( produc2012818120od_a_a @ F @ G @ ( product_Pair_a_a @ A @ B2 ) )
% 0.28/0.48 = ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % map_prod_simp
% 0.28/0.48 thf(fact_178_map__prod__simp,axiom,
% 0.28/0.48 ! [F: product_prod_a_a > a,G: product_prod_a_a > a,A: product_prod_a_a,B2: product_prod_a_a] :
% 0.28/0.48 ( ( produc363290532_a_a_a @ F @ G @ ( produc1474507607od_a_a @ A @ B2 ) )
% 0.28/0.48 = ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % map_prod_simp
% 0.28/0.48 thf(fact_179_map__prod__simp,axiom,
% 0.28/0.48 ! [F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a,A: product_prod_a_a,B2: product_prod_a_a] :
% 0.28/0.48 ( ( produc1779756150od_a_a @ F @ G @ ( produc1474507607od_a_a @ A @ B2 ) )
% 0.28/0.48 = ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % map_prod_simp
% 0.28/0.48 thf(fact_180_map__prod__imageI,axiom,
% 0.28/0.48 ! [A: a,B2: a,R2: set_Product_prod_a_a,F: a > a,G: a > a] :
% 0.28/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ R2 )
% 0.28/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F @ G ) @ R2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % map_prod_imageI
% 0.28/0.48 thf(fact_181_map__prod__imageI,axiom,
% 0.28/0.48 ! [A: a,B2: a,R2: set_Product_prod_a_a,F: a > product_prod_a_a,G: a > product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ R2 )
% 0.28/0.48 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_1765377413od_a_a @ ( produc2012818120od_a_a @ F @ G ) @ R2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % map_prod_imageI
% 0.28/0.48 thf(fact_182_map__prod__imageI,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: product_prod_a_a,R2: set_Pr1948701895od_a_a,F: product_prod_a_a > a,G: product_prod_a_a > a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ R2 )
% 0.28/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_734753285od_a_a @ ( produc363290532_a_a_a @ F @ G ) @ R2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % map_prod_imageI
% 0.28/0.48 thf(fact_183_map__prod__imageI,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: product_prod_a_a,R2: set_Pr1948701895od_a_a,F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ R2 )
% 0.28/0.48 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_236793797od_a_a @ ( produc1779756150od_a_a @ F @ G ) @ R2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % map_prod_imageI
% 0.28/0.48 thf(fact_184_top__empty__eq,axiom,
% 0.28/0.48 ( top_to1483957574_a_a_o
% 0.28/0.48 = ( ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ top_to398455383od_a_a ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % top_empty_eq
% 0.28/0.48 thf(fact_185_top__empty__eq,axiom,
% 0.28/0.48 ( top_top_a_o
% 0.28/0.48 = ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % top_empty_eq
% 0.28/0.48 thf(fact_186_top__empty__eq,axiom,
% 0.28/0.48 ( top_to2011536774_a_a_o
% 0.28/0.48 = ( ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ top_to52711959od_a_a ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % top_empty_eq
% 0.28/0.48 thf(fact_187_fst__eqD,axiom,
% 0.28/0.48 ! [X: a,Y: a,A: a] :
% 0.28/0.48 ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 0.28/0.48 = A )
% 0.28/0.48 => ( X = A ) ) ).
% 0.28/0.48
% 0.28/0.48 % fst_eqD
% 0.28/0.48 thf(fact_188_fst__eqD,axiom,
% 0.28/0.48 ! [X: product_prod_a_a,Y: product_prod_a_a,A: product_prod_a_a] :
% 0.28/0.48 ( ( ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
% 0.28/0.48 = A )
% 0.28/0.48 => ( X = A ) ) ).
% 0.28/0.48
% 0.28/0.48 % fst_eqD
% 0.28/0.48 thf(fact_189_fst__conv,axiom,
% 0.28/0.48 ! [X1: a,X2: a] :
% 0.28/0.48 ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
% 0.28/0.48 = X1 ) ).
% 0.28/0.48
% 0.28/0.48 % fst_conv
% 0.28/0.48 thf(fact_190_fst__conv,axiom,
% 0.28/0.48 ! [X1: product_prod_a_a,X2: product_prod_a_a] :
% 0.28/0.48 ( ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ X1 @ X2 ) )
% 0.28/0.48 = X1 ) ).
% 0.28/0.48
% 0.28/0.48 % fst_conv
% 0.28/0.48 thf(fact_191_prod__fun__imageE,axiom,
% 0.28/0.48 ! [C2: product_prod_a_a,F: a > a,G: a > a,R2: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ C2 @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F @ G ) @ R2 ) )
% 0.28/0.48 => ~ ! [X3: a,Y3: a] :
% 0.28/0.48 ( ( C2
% 0.28/0.48 = ( product_Pair_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 0.28/0.48 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prod_fun_imageE
% 0.28/0.48 thf(fact_192_prod__fun__imageE,axiom,
% 0.28/0.48 ! [C2: product_prod_a_a,F: product_prod_a_a > a,G: product_prod_a_a > a,R2: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ C2 @ ( image_734753285od_a_a @ ( produc363290532_a_a_a @ F @ G ) @ R2 ) )
% 0.28/0.48 => ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.28/0.48 ( ( C2
% 0.28/0.48 = ( product_Pair_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 0.28/0.48 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prod_fun_imageE
% 0.28/0.48 thf(fact_193_prod__fun__imageE,axiom,
% 0.28/0.48 ! [C2: produc1572603623od_a_a,F: a > product_prod_a_a,G: a > product_prod_a_a,R2: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ C2 @ ( image_1765377413od_a_a @ ( produc2012818120od_a_a @ F @ G ) @ R2 ) )
% 0.28/0.48 => ~ ! [X3: a,Y3: a] :
% 0.28/0.48 ( ( C2
% 0.28/0.48 = ( produc1474507607od_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 0.28/0.48 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prod_fun_imageE
% 0.28/0.48 thf(fact_194_prod__fun__imageE,axiom,
% 0.28/0.48 ! [C2: produc1572603623od_a_a,F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a,R2: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ C2 @ ( image_236793797od_a_a @ ( produc1779756150od_a_a @ F @ G ) @ R2 ) )
% 0.28/0.48 => ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.28/0.48 ( ( C2
% 0.28/0.48 = ( produc1474507607od_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 0.28/0.48 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prod_fun_imageE
% 0.28/0.48 thf(fact_195_top__empty__eq2,axiom,
% 0.28/0.48 ( top_top_a_a_o
% 0.28/0.48 = ( ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ top_to398455383od_a_a ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % top_empty_eq2
% 0.28/0.48 thf(fact_196_top__empty__eq2,axiom,
% 0.28/0.48 ( top_to993969022_a_a_o
% 0.28/0.48 = ( ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ top_to52711959od_a_a ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % top_empty_eq2
% 0.28/0.48 thf(fact_197_eq__fst__iff,axiom,
% 0.28/0.48 ! [A: a,P: product_prod_a_a] :
% 0.28/0.48 ( ( A
% 0.28/0.48 = ( product_fst_a_a @ P ) )
% 0.28/0.48 = ( ? [B6: a] :
% 0.28/0.48 ( P
% 0.28/0.48 = ( product_Pair_a_a @ A @ B6 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % eq_fst_iff
% 0.28/0.48 thf(fact_198_eq__fst__iff,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,P: produc1572603623od_a_a] :
% 0.28/0.48 ( ( A
% 0.28/0.48 = ( produc2115494211od_a_a @ P ) )
% 0.28/0.48 = ( ? [B6: product_prod_a_a] :
% 0.28/0.48 ( P
% 0.28/0.48 = ( produc1474507607od_a_a @ A @ B6 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % eq_fst_iff
% 0.28/0.48 thf(fact_199_fstI,axiom,
% 0.28/0.48 ! [X: product_prod_a_a,Y: a,Z: a] :
% 0.28/0.48 ( ( X
% 0.28/0.48 = ( product_Pair_a_a @ Y @ Z ) )
% 0.28/0.48 => ( ( product_fst_a_a @ X )
% 0.28/0.48 = Y ) ) ).
% 0.28/0.48
% 0.28/0.48 % fstI
% 0.28/0.48 thf(fact_200_fstI,axiom,
% 0.28/0.48 ! [X: produc1572603623od_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.28/0.48 ( ( X
% 0.28/0.48 = ( produc1474507607od_a_a @ Y @ Z ) )
% 0.28/0.48 => ( ( produc2115494211od_a_a @ X )
% 0.28/0.48 = Y ) ) ).
% 0.28/0.48
% 0.28/0.48 % fstI
% 0.28/0.48 thf(fact_201_prod_Ocollapse,axiom,
% 0.28/0.48 ! [Prod: product_prod_a_a] :
% 0.28/0.48 ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
% 0.28/0.48 = Prod ) ).
% 0.28/0.48
% 0.28/0.48 % prod.collapse
% 0.28/0.48 thf(fact_202_prod_Ocollapse,axiom,
% 0.28/0.48 ! [Prod: produc1572603623od_a_a] :
% 0.28/0.48 ( ( produc1474507607od_a_a @ ( produc2115494211od_a_a @ Prod ) @ ( produc368344197od_a_a @ Prod ) )
% 0.28/0.48 = Prod ) ).
% 0.28/0.48
% 0.28/0.48 % prod.collapse
% 0.28/0.48 thf(fact_203_eq__snd__iff,axiom,
% 0.28/0.48 ! [B2: a,P: product_prod_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( product_snd_a_a @ P ) )
% 0.28/0.48 = ( ? [A5: a] :
% 0.28/0.48 ( P
% 0.28/0.48 = ( product_Pair_a_a @ A5 @ B2 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % eq_snd_iff
% 0.28/0.48 thf(fact_204_eq__snd__iff,axiom,
% 0.28/0.48 ! [B2: product_prod_a_a,P: produc1572603623od_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( produc368344197od_a_a @ P ) )
% 0.28/0.48 = ( ? [A5: product_prod_a_a] :
% 0.28/0.48 ( P
% 0.28/0.48 = ( produc1474507607od_a_a @ A5 @ B2 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % eq_snd_iff
% 0.28/0.48 thf(fact_205_sndI,axiom,
% 0.28/0.48 ! [X: product_prod_a_a,Y: a,Z: a] :
% 0.28/0.48 ( ( X
% 0.28/0.48 = ( product_Pair_a_a @ Y @ Z ) )
% 0.28/0.48 => ( ( product_snd_a_a @ X )
% 0.28/0.48 = Z ) ) ).
% 0.28/0.48
% 0.28/0.48 % sndI
% 0.28/0.48 thf(fact_206_sndI,axiom,
% 0.28/0.48 ! [X: produc1572603623od_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.28/0.48 ( ( X
% 0.28/0.48 = ( produc1474507607od_a_a @ Y @ Z ) )
% 0.28/0.48 => ( ( produc368344197od_a_a @ X )
% 0.28/0.48 = Z ) ) ).
% 0.28/0.48
% 0.28/0.48 % sndI
% 0.28/0.48 thf(fact_207_snd__eqD,axiom,
% 0.28/0.48 ! [X: a,Y: a,A: a] :
% 0.28/0.48 ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 0.28/0.48 = A )
% 0.28/0.48 => ( Y = A ) ) ).
% 0.28/0.48
% 0.28/0.48 % snd_eqD
% 0.28/0.48 thf(fact_208_snd__eqD,axiom,
% 0.28/0.48 ! [X: product_prod_a_a,Y: product_prod_a_a,A: product_prod_a_a] :
% 0.28/0.48 ( ( ( produc368344197od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
% 0.28/0.48 = A )
% 0.28/0.48 => ( Y = A ) ) ).
% 0.28/0.48
% 0.28/0.48 % snd_eqD
% 0.28/0.48 thf(fact_209_snd__conv,axiom,
% 0.28/0.48 ! [X1: a,X2: a] :
% 0.28/0.48 ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
% 0.28/0.48 = X2 ) ).
% 0.28/0.48
% 0.28/0.48 % snd_conv
% 0.28/0.48 thf(fact_210_snd__conv,axiom,
% 0.28/0.48 ! [X1: product_prod_a_a,X2: product_prod_a_a] :
% 0.28/0.48 ( ( produc368344197od_a_a @ ( produc1474507607od_a_a @ X1 @ X2 ) )
% 0.28/0.48 = X2 ) ).
% 0.28/0.48
% 0.28/0.48 % snd_conv
% 0.28/0.48 thf(fact_211_surjective__pairing,axiom,
% 0.28/0.48 ! [T: product_prod_a_a] :
% 0.28/0.48 ( T
% 0.28/0.48 = ( product_Pair_a_a @ ( product_fst_a_a @ T ) @ ( product_snd_a_a @ T ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % surjective_pairing
% 0.28/0.48 thf(fact_212_surjective__pairing,axiom,
% 0.28/0.48 ! [T: produc1572603623od_a_a] :
% 0.28/0.48 ( T
% 0.28/0.48 = ( produc1474507607od_a_a @ ( produc2115494211od_a_a @ T ) @ ( produc368344197od_a_a @ T ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % surjective_pairing
% 0.28/0.48 thf(fact_213_prod_Oexhaust__sel,axiom,
% 0.28/0.48 ! [Prod: product_prod_a_a] :
% 0.28/0.48 ( Prod
% 0.28/0.48 = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prod.exhaust_sel
% 0.28/0.48 thf(fact_214_prod_Oexhaust__sel,axiom,
% 0.28/0.48 ! [Prod: produc1572603623od_a_a] :
% 0.28/0.48 ( Prod
% 0.28/0.48 = ( produc1474507607od_a_a @ ( produc2115494211od_a_a @ Prod ) @ ( produc368344197od_a_a @ Prod ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prod.exhaust_sel
% 0.28/0.48 thf(fact_215_prod_Oswap__def,axiom,
% 0.28/0.48 ( product_swap_a_a
% 0.28/0.48 = ( ^ [P4: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P4 ) @ ( product_fst_a_a @ P4 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prod.swap_def
% 0.28/0.48 thf(fact_216_prod_Oswap__def,axiom,
% 0.28/0.48 ( produc1641853175od_a_a
% 0.28/0.48 = ( ^ [P4: produc1572603623od_a_a] : ( produc1474507607od_a_a @ ( produc368344197od_a_a @ P4 ) @ ( produc2115494211od_a_a @ P4 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prod.swap_def
% 0.28/0.48 thf(fact_217_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.28/0.48 ! [P2: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
% 0.28/0.48 ( ( P2 @ X @ Y )
% 0.28/0.48 => ( ( A
% 0.28/0.48 = ( product_Pair_a_a @ X @ Y ) )
% 0.28/0.48 => ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % BNF_Greatest_Fixpoint.subst_Pair
% 0.28/0.48 thf(fact_218_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.28/0.48 ! [P2: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a,Y: product_prod_a_a,A: produc1572603623od_a_a] :
% 0.28/0.48 ( ( P2 @ X @ Y )
% 0.28/0.48 => ( ( A
% 0.28/0.48 = ( produc1474507607od_a_a @ X @ Y ) )
% 0.28/0.48 => ( P2 @ ( produc2115494211od_a_a @ A ) @ ( produc368344197od_a_a @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % BNF_Greatest_Fixpoint.subst_Pair
% 0.28/0.48 thf(fact_219_conjI__realizer,axiom,
% 0.28/0.48 ! [P2: a > $o,P: a,Q: a > $o,Q2: a] :
% 0.28/0.48 ( ( P2 @ P )
% 0.28/0.48 => ( ( Q @ Q2 )
% 0.28/0.48 => ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) )
% 0.28/0.48 & ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % conjI_realizer
% 0.28/0.48 thf(fact_220_conjI__realizer,axiom,
% 0.28/0.48 ! [P2: product_prod_a_a > $o,P: product_prod_a_a,Q: product_prod_a_a > $o,Q2: product_prod_a_a] :
% 0.28/0.48 ( ( P2 @ P )
% 0.28/0.48 => ( ( Q @ Q2 )
% 0.28/0.48 => ( ( P2 @ ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ P @ Q2 ) ) )
% 0.28/0.48 & ( Q @ ( produc368344197od_a_a @ ( produc1474507607od_a_a @ P @ Q2 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % conjI_realizer
% 0.28/0.48 thf(fact_221_exI__realizer,axiom,
% 0.28/0.48 ! [P2: a > a > $o,Y: a,X: a] :
% 0.28/0.48 ( ( P2 @ Y @ X )
% 0.28/0.48 => ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % exI_realizer
% 0.28/0.48 thf(fact_222_exI__realizer,axiom,
% 0.28/0.48 ! [P2: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a,X: product_prod_a_a] :
% 0.28/0.48 ( ( P2 @ Y @ X )
% 0.28/0.48 => ( P2 @ ( produc368344197od_a_a @ ( produc1474507607od_a_a @ X @ Y ) ) @ ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ X @ Y ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % exI_realizer
% 0.28/0.48 thf(fact_223_image2__eqI,axiom,
% 0.28/0.48 ! [B2: a,F: product_prod_a_a > a,X: product_prod_a_a,C2: a,G: product_prod_a_a > a,A4: set_Product_prod_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X ) )
% 0.28/0.48 => ( ( C2
% 0.28/0.48 = ( G @ X ) )
% 0.28/0.48 => ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ C2 ) @ ( bNF_Gr1853143117_a_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % image2_eqI
% 0.28/0.48 thf(fact_224_image2__eqI,axiom,
% 0.28/0.48 ! [B2: a,F: a > a,X: a,C2: a,G: a > a,A4: set_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X ) )
% 0.28/0.48 => ( ( C2
% 0.28/0.48 = ( G @ X ) )
% 0.28/0.48 => ( ( member_a @ X @ A4 )
% 0.28/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ C2 ) @ ( bNF_Gr423046308_a_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % image2_eqI
% 0.28/0.48 thf(fact_225_image2__eqI,axiom,
% 0.28/0.48 ! [B2: a,F: produc1572603623od_a_a > a,X: produc1572603623od_a_a,C2: a,G: produc1572603623od_a_a > a,A4: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X ) )
% 0.28/0.48 => ( ( C2
% 0.28/0.48 = ( G @ X ) )
% 0.28/0.48 => ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ C2 ) @ ( bNF_Gr356189709_a_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % image2_eqI
% 0.28/0.48 thf(fact_226_image2__eqI,axiom,
% 0.28/0.48 ! [B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,C2: product_prod_a_a,G: product_prod_a_a > product_prod_a_a,A4: set_Product_prod_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X ) )
% 0.28/0.48 => ( ( C2
% 0.28/0.48 = ( G @ X ) )
% 0.28/0.48 => ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.48 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ C2 ) @ ( bNF_Gr200450189od_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % image2_eqI
% 0.28/0.48 thf(fact_227_image2__eqI,axiom,
% 0.28/0.48 ! [B2: product_prod_a_a,F: a > product_prod_a_a,X: a,C2: product_prod_a_a,G: a > product_prod_a_a,A4: set_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X ) )
% 0.28/0.48 => ( ( C2
% 0.28/0.48 = ( G @ X ) )
% 0.28/0.48 => ( ( member_a @ X @ A4 )
% 0.28/0.48 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ C2 ) @ ( bNF_Gr1715025636od_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % image2_eqI
% 0.28/0.48 thf(fact_228_image2__eqI,axiom,
% 0.28/0.48 ! [B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a,X: produc1572603623od_a_a,C2: product_prod_a_a,G: produc1572603623od_a_a > product_prod_a_a,A4: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( B2
% 0.28/0.48 = ( F @ X ) )
% 0.28/0.48 => ( ( C2
% 0.28/0.48 = ( G @ X ) )
% 0.28/0.48 => ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.48 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ C2 ) @ ( bNF_Gr533818957od_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % image2_eqI
% 0.28/0.48 thf(fact_229_in__lex__prod,axiom,
% 0.28/0.48 ! [A: a,B2: product_prod_a_a,A2: a,B3: product_prod_a_a,R: set_Product_prod_a_a,S: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member1361298640od_a_a @ ( produc563767703od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc1299253312od_a_a @ A2 @ B3 ) ) @ ( lex_pr1321960536od_a_a @ R @ S ) )
% 0.28/0.48 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R )
% 0.28/0.48 | ( ( A = A2 )
% 0.28/0.48 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % in_lex_prod
% 0.28/0.48 thf(fact_230_in__lex__prod,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: a,A2: product_prod_a_a,B3: a,R: set_Pr1948701895od_a_a,S: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member470802768_a_a_a @ ( produc2054708503_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc2061588782_a_a_a @ A2 @ B3 ) ) @ ( lex_pr2084296006_a_a_a @ R @ S ) )
% 0.28/0.48 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A2 ) @ R )
% 0.28/0.48 | ( ( A = A2 )
% 0.28/0.48 & ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % in_lex_prod
% 0.28/0.48 thf(fact_231_in__lex__prod,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a,R: set_Pr1948701895od_a_a,S: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc1474507607od_a_a @ A2 @ B3 ) ) @ ( lex_pr942790511od_a_a @ R @ S ) )
% 0.28/0.48 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A2 ) @ R )
% 0.28/0.48 | ( ( A = A2 )
% 0.28/0.48 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % in_lex_prod
% 0.28/0.48 thf(fact_232_in__lex__prod,axiom,
% 0.28/0.48 ! [A: a,B2: a,A2: a,B3: a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Pair_a_a @ A2 @ B3 ) ) @ ( lex_prod_a_a @ R @ S ) )
% 0.28/0.48 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R )
% 0.28/0.48 | ( ( A = A2 )
% 0.28/0.48 & ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % in_lex_prod
% 0.28/0.48 thf(fact_233_mem__Sigma__iff,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
% 0.28/0.48 ( ( member1403138919_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc1282482655_a_a_a @ A4 @ B ) )
% 0.28/0.48 = ( ( member449909584od_a_a @ A @ A4 )
% 0.28/0.48 & ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Sigma_iff
% 0.28/0.48 thf(fact_234_mem__Sigma__iff,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member122808528od_a_a @ ( produc1555606807od_a_a @ A @ B2 ) @ ( produc1227942600od_a_a @ A4 @ B ) )
% 0.28/0.48 = ( ( member449909584od_a_a @ A @ A4 )
% 0.28/0.48 & ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Sigma_iff
% 0.28/0.48 thf(fact_235_mem__Sigma__iff,axiom,
% 0.28/0.48 ! [A: a,B2: product_prod_a_a,A4: set_a,B: a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member160619001od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc520147185od_a_a @ A4 @ B ) )
% 0.28/0.48 = ( ( member_a @ A @ A4 )
% 0.28/0.48 & ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Sigma_iff
% 0.28/0.48 thf(fact_236_mem__Sigma__iff,axiom,
% 0.28/0.48 ! [A: a,B2: produc1572603623od_a_a,A4: set_a,B: a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member448091577od_a_a @ ( produc1095491584od_a_a @ A @ B2 ) @ ( produc663418545od_a_a @ A4 @ B ) )
% 0.28/0.48 = ( ( member_a @ A @ A4 )
% 0.28/0.48 & ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Sigma_iff
% 0.28/0.48 thf(fact_237_mem__Sigma__iff,axiom,
% 0.28/0.48 ! [A: produc1572603623od_a_a,B2: product_prod_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member1545452880od_a_a @ ( produc524982679od_a_a @ A @ B2 ) @ ( produc197318472od_a_a @ A4 @ B ) )
% 0.28/0.48 = ( ( member2057358096od_a_a @ A @ A4 )
% 0.28/0.48 & ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Sigma_iff
% 0.28/0.48 thf(fact_238_mem__Sigma__iff,axiom,
% 0.28/0.48 ! [A: produc1572603623od_a_a,B2: a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_a] :
% 0.28/0.48 ( ( member575366567_a_a_a @ ( produc436797294_a_a_a @ A @ B2 ) @ ( produc4724255_a_a_a @ A4 @ B ) )
% 0.28/0.48 = ( ( member2057358096od_a_a @ A @ A4 )
% 0.28/0.48 & ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Sigma_iff
% 0.28/0.48 thf(fact_239_mem__Sigma__iff,axiom,
% 0.28/0.48 ! [A: produc1572603623od_a_a,B2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B2 ) @ ( produc1460477704od_a_a @ A4 @ B ) )
% 0.28/0.48 = ( ( member2057358096od_a_a @ A @ A4 )
% 0.28/0.48 & ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Sigma_iff
% 0.28/0.48 thf(fact_240_mem__Sigma__iff,axiom,
% 0.28/0.48 ! [A: a,B2: a,A4: set_a,B: a > set_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
% 0.28/0.48 = ( ( member_a @ A @ A4 )
% 0.28/0.48 & ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Sigma_iff
% 0.28/0.48 thf(fact_241_mem__Sigma__iff,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
% 0.28/0.48 = ( ( member449909584od_a_a @ A @ A4 )
% 0.28/0.48 & ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Sigma_iff
% 0.28/0.48 thf(fact_242_SigmaI,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,A4: set_Product_prod_a_a,B2: a,B: product_prod_a_a > set_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ A @ A4 )
% 0.28/0.48 => ( ( member_a @ B2 @ ( B @ A ) )
% 0.28/0.48 => ( member1403138919_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc1282482655_a_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaI
% 0.28/0.48 thf(fact_243_SigmaI,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,A4: set_Product_prod_a_a,B2: produc1572603623od_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ A @ A4 )
% 0.28/0.48 => ( ( member2057358096od_a_a @ B2 @ ( B @ A ) )
% 0.28/0.48 => ( member122808528od_a_a @ ( produc1555606807od_a_a @ A @ B2 ) @ ( produc1227942600od_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaI
% 0.28/0.48 thf(fact_244_SigmaI,axiom,
% 0.28/0.48 ! [A: a,A4: set_a,B2: product_prod_a_a,B: a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member_a @ A @ A4 )
% 0.28/0.48 => ( ( member449909584od_a_a @ B2 @ ( B @ A ) )
% 0.28/0.48 => ( member160619001od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc520147185od_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaI
% 0.28/0.48 thf(fact_245_SigmaI,axiom,
% 0.28/0.48 ! [A: a,A4: set_a,B2: produc1572603623od_a_a,B: a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member_a @ A @ A4 )
% 0.28/0.48 => ( ( member2057358096od_a_a @ B2 @ ( B @ A ) )
% 0.28/0.48 => ( member448091577od_a_a @ ( produc1095491584od_a_a @ A @ B2 ) @ ( produc663418545od_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaI
% 0.28/0.48 thf(fact_246_SigmaI,axiom,
% 0.28/0.48 ! [A: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: product_prod_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ A @ A4 )
% 0.28/0.48 => ( ( member449909584od_a_a @ B2 @ ( B @ A ) )
% 0.28/0.48 => ( member1545452880od_a_a @ ( produc524982679od_a_a @ A @ B2 ) @ ( produc197318472od_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaI
% 0.28/0.48 thf(fact_247_SigmaI,axiom,
% 0.28/0.48 ! [A: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: a,B: produc1572603623od_a_a > set_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ A @ A4 )
% 0.28/0.48 => ( ( member_a @ B2 @ ( B @ A ) )
% 0.28/0.48 => ( member575366567_a_a_a @ ( produc436797294_a_a_a @ A @ B2 ) @ ( produc4724255_a_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaI
% 0.28/0.48 thf(fact_248_SigmaI,axiom,
% 0.28/0.48 ! [A: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: produc1572603623od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ A @ A4 )
% 0.28/0.48 => ( ( member2057358096od_a_a @ B2 @ ( B @ A ) )
% 0.28/0.48 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B2 ) @ ( produc1460477704od_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaI
% 0.28/0.48 thf(fact_249_SigmaI,axiom,
% 0.28/0.48 ! [A: a,A4: set_a,B2: a,B: a > set_a] :
% 0.28/0.48 ( ( member_a @ A @ A4 )
% 0.28/0.48 => ( ( member_a @ B2 @ ( B @ A ) )
% 0.28/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaI
% 0.28/0.48 thf(fact_250_SigmaI,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,A4: set_Product_prod_a_a,B2: product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ A @ A4 )
% 0.28/0.48 => ( ( member449909584od_a_a @ B2 @ ( B @ A ) )
% 0.28/0.48 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaI
% 0.28/0.48 thf(fact_251_SigmaE2,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
% 0.28/0.48 ( ( member1403138919_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc1282482655_a_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ( ( member449909584od_a_a @ A @ A4 )
% 0.28/0.48 => ~ ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE2
% 0.28/0.48 thf(fact_252_SigmaE2,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member122808528od_a_a @ ( produc1555606807od_a_a @ A @ B2 ) @ ( produc1227942600od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ( ( member449909584od_a_a @ A @ A4 )
% 0.28/0.48 => ~ ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE2
% 0.28/0.48 thf(fact_253_SigmaE2,axiom,
% 0.28/0.48 ! [A: a,B2: product_prod_a_a,A4: set_a,B: a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member160619001od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc520147185od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ( ( member_a @ A @ A4 )
% 0.28/0.48 => ~ ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE2
% 0.28/0.48 thf(fact_254_SigmaE2,axiom,
% 0.28/0.48 ! [A: a,B2: produc1572603623od_a_a,A4: set_a,B: a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member448091577od_a_a @ ( produc1095491584od_a_a @ A @ B2 ) @ ( produc663418545od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ( ( member_a @ A @ A4 )
% 0.28/0.48 => ~ ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE2
% 0.28/0.48 thf(fact_255_SigmaE2,axiom,
% 0.28/0.48 ! [A: produc1572603623od_a_a,B2: product_prod_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member1545452880od_a_a @ ( produc524982679od_a_a @ A @ B2 ) @ ( produc197318472od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ( ( member2057358096od_a_a @ A @ A4 )
% 0.28/0.48 => ~ ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE2
% 0.28/0.48 thf(fact_256_SigmaE2,axiom,
% 0.28/0.48 ! [A: produc1572603623od_a_a,B2: a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_a] :
% 0.28/0.48 ( ( member575366567_a_a_a @ ( produc436797294_a_a_a @ A @ B2 ) @ ( produc4724255_a_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ( ( member2057358096od_a_a @ A @ A4 )
% 0.28/0.48 => ~ ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE2
% 0.28/0.48 thf(fact_257_SigmaE2,axiom,
% 0.28/0.48 ! [A: produc1572603623od_a_a,B2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B2 ) @ ( produc1460477704od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ( ( member2057358096od_a_a @ A @ A4 )
% 0.28/0.48 => ~ ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE2
% 0.28/0.48 thf(fact_258_SigmaE2,axiom,
% 0.28/0.48 ! [A: a,B2: a,A4: set_a,B: a > set_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ( ( member_a @ A @ A4 )
% 0.28/0.48 => ~ ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE2
% 0.28/0.48 thf(fact_259_SigmaE2,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ( ( member449909584od_a_a @ A @ A4 )
% 0.28/0.48 => ~ ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE2
% 0.28/0.48 thf(fact_260_SigmaD2,axiom,
% 0.28/0.48 ! [A: a,B2: a,A4: set_a,B: a > set_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
% 0.28/0.48 => ( member_a @ B2 @ ( B @ A ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaD2
% 0.28/0.48 thf(fact_261_SigmaD2,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
% 0.28/0.48 => ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaD2
% 0.28/0.48 thf(fact_262_SigmaD1,axiom,
% 0.28/0.48 ! [A: a,B2: a,A4: set_a,B: a > set_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
% 0.28/0.48 => ( member_a @ A @ A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaD1
% 0.28/0.48 thf(fact_263_SigmaD1,axiom,
% 0.28/0.48 ! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
% 0.28/0.48 => ( member449909584od_a_a @ A @ A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaD1
% 0.28/0.48 thf(fact_264_SigmaE,axiom,
% 0.28/0.48 ! [C2: produc1016684094_a_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
% 0.28/0.48 ( ( member1403138919_a_a_a @ C2 @ ( produc1282482655_a_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X3 @ A4 )
% 0.28/0.48 => ! [Y3: a] :
% 0.28/0.48 ( ( member_a @ Y3 @ ( B @ X3 ) )
% 0.28/0.48 => ( C2
% 0.28/0.48 != ( produc2061588782_a_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE
% 0.28/0.48 thf(fact_265_SigmaE,axiom,
% 0.28/0.48 ! [C2: produc772425383od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member122808528od_a_a @ C2 @ ( produc1227942600od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X3 @ A4 )
% 0.28/0.48 => ! [Y3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ Y3 @ ( B @ X3 ) )
% 0.28/0.48 => ( C2
% 0.28/0.48 != ( produc1555606807od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE
% 0.28/0.48 thf(fact_266_SigmaE,axiom,
% 0.28/0.48 ! [C2: produc1921647824od_a_a,A4: set_a,B: a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member160619001od_a_a @ C2 @ ( produc520147185od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ! [X3: a] :
% 0.28/0.48 ( ( member_a @ X3 @ A4 )
% 0.28/0.48 => ! [Y3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ Y3 @ ( B @ X3 ) )
% 0.28/0.48 => ( C2
% 0.28/0.48 != ( produc1299253312od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE
% 0.28/0.48 thf(fact_267_SigmaE,axiom,
% 0.28/0.48 ! [C2: produc1555800720od_a_a,A4: set_a,B: a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member448091577od_a_a @ C2 @ ( produc663418545od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ! [X3: a] :
% 0.28/0.48 ( ( member_a @ X3 @ A4 )
% 0.28/0.48 => ! [Y3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ Y3 @ ( B @ X3 ) )
% 0.28/0.48 => ( C2
% 0.28/0.48 != ( produc1095491584od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE
% 0.28/0.48 thf(fact_268_SigmaE,axiom,
% 0.28/0.48 ! [C2: produc47586087od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member1545452880od_a_a @ C2 @ ( produc197318472od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X3 @ A4 )
% 0.28/0.48 => ! [Y3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ Y3 @ ( B @ X3 ) )
% 0.28/0.48 => ( C2
% 0.28/0.48 != ( produc524982679od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE
% 0.28/0.48 thf(fact_269_SigmaE,axiom,
% 0.28/0.48 ! [C2: produc1683075710_a_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_a] :
% 0.28/0.48 ( ( member575366567_a_a_a @ C2 @ ( produc4724255_a_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X3 @ A4 )
% 0.28/0.48 => ! [Y3: a] :
% 0.28/0.48 ( ( member_a @ Y3 @ ( B @ X3 ) )
% 0.28/0.48 => ( C2
% 0.28/0.48 != ( produc436797294_a_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE
% 0.28/0.48 thf(fact_270_SigmaE,axiom,
% 0.28/0.48 ! [C2: produc1990712039od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member1899387664od_a_a @ C2 @ ( produc1460477704od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X3 @ A4 )
% 0.28/0.48 => ! [Y3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ Y3 @ ( B @ X3 ) )
% 0.28/0.48 => ( C2
% 0.28/0.48 != ( produc1935643479od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE
% 0.28/0.48 thf(fact_271_SigmaE,axiom,
% 0.28/0.48 ! [C2: product_prod_a_a,A4: set_a,B: a > set_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ C2 @ ( product_Sigma_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ! [X3: a] :
% 0.28/0.48 ( ( member_a @ X3 @ A4 )
% 0.28/0.48 => ! [Y3: a] :
% 0.28/0.48 ( ( member_a @ Y3 @ ( B @ X3 ) )
% 0.28/0.48 => ( C2
% 0.28/0.48 != ( product_Pair_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE
% 0.28/0.48 thf(fact_272_SigmaE,axiom,
% 0.28/0.48 ! [C2: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ C2 @ ( produc304751368od_a_a @ A4 @ B ) )
% 0.28/0.48 => ~ ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X3 @ A4 )
% 0.28/0.48 => ! [Y3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ Y3 @ ( B @ X3 ) )
% 0.28/0.48 => ( C2
% 0.28/0.48 != ( produc1474507607od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % SigmaE
% 0.28/0.48 thf(fact_273_mem__Times__iff,axiom,
% 0.28/0.48 ! [X: produc1016684094_a_a_a,A4: set_Product_prod_a_a,B: set_a] :
% 0.28/0.48 ( ( member1403138919_a_a_a @ X
% 0.28/0.48 @ ( produc1282482655_a_a_a @ A4
% 0.28/0.48 @ ^ [Uu: product_prod_a_a] : B ) )
% 0.28/0.48 = ( ( member449909584od_a_a @ ( produc1805832986_a_a_a @ X ) @ A4 )
% 0.28/0.48 & ( member_a @ ( produc1954021468_a_a_a @ X ) @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Times_iff
% 0.28/0.48 thf(fact_274_mem__Times__iff,axiom,
% 0.28/0.48 ! [X: produc772425383od_a_a,A4: set_Product_prod_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member122808528od_a_a @ X
% 0.28/0.48 @ ( produc1227942600od_a_a @ A4
% 0.28/0.48 @ ^ [Uu: product_prod_a_a] : B ) )
% 0.28/0.48 = ( ( member449909584od_a_a @ ( produc1775747843od_a_a @ X ) @ A4 )
% 0.28/0.48 & ( member2057358096od_a_a @ ( produc1055472197od_a_a @ X ) @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Times_iff
% 0.28/0.48 thf(fact_275_mem__Times__iff,axiom,
% 0.28/0.48 ! [X: produc1921647824od_a_a,A4: set_a,B: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member160619001od_a_a @ X
% 0.28/0.48 @ ( produc520147185od_a_a @ A4
% 0.28/0.48 @ ^ [Uu: a] : B ) )
% 0.28/0.48 = ( ( member_a @ ( produc1043497516od_a_a @ X ) @ A4 )
% 0.28/0.48 & ( member449909584od_a_a @ ( produc1191685998od_a_a @ X ) @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Times_iff
% 0.28/0.48 thf(fact_276_mem__Times__iff,axiom,
% 0.28/0.48 ! [X: produc1555800720od_a_a,A4: set_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member448091577od_a_a @ X
% 0.28/0.48 @ ( produc663418545od_a_a @ A4
% 0.28/0.48 @ ^ [Uu: a] : B ) )
% 0.28/0.48 = ( ( member_a @ ( produc530563564od_a_a @ X ) @ A4 )
% 0.28/0.48 & ( member2057358096od_a_a @ ( produc921029422od_a_a @ X ) @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Times_iff
% 0.28/0.48 thf(fact_277_mem__Times__iff,axiom,
% 0.28/0.48 ! [X: produc47586087od_a_a,A4: set_Pr1948701895od_a_a,B: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member1545452880od_a_a @ X
% 0.28/0.48 @ ( produc197318472od_a_a @ A4
% 0.28/0.48 @ ^ [Uu: produc1572603623od_a_a] : B ) )
% 0.28/0.48 = ( ( member2057358096od_a_a @ ( produc745123715od_a_a @ X ) @ A4 )
% 0.28/0.48 & ( member449909584od_a_a @ ( produc24848069od_a_a @ X ) @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Times_iff
% 0.28/0.48 thf(fact_278_mem__Times__iff,axiom,
% 0.28/0.48 ! [X: produc1683075710_a_a_a,A4: set_Pr1948701895od_a_a,B: set_a] :
% 0.28/0.48 ( ( member575366567_a_a_a @ X
% 0.28/0.48 @ ( produc4724255_a_a_a @ A4
% 0.28/0.48 @ ^ [Uu: produc1572603623od_a_a] : B ) )
% 0.28/0.48 = ( ( member2057358096od_a_a @ ( produc2019352922_a_a_a @ X ) @ A4 )
% 0.28/0.48 & ( member_a @ ( produc262335132_a_a_a @ X ) @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Times_iff
% 0.28/0.48 thf(fact_279_mem__Times__iff,axiom,
% 0.28/0.48 ! [X: produc1990712039od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( member1899387664od_a_a @ X
% 0.28/0.48 @ ( produc1460477704od_a_a @ A4
% 0.28/0.48 @ ^ [Uu: produc1572603623od_a_a] : B ) )
% 0.28/0.48 = ( ( member2057358096od_a_a @ ( produc303632707od_a_a @ X ) @ A4 )
% 0.28/0.48 & ( member2057358096od_a_a @ ( produc1441227909od_a_a @ X ) @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Times_iff
% 0.28/0.48 thf(fact_280_mem__Times__iff,axiom,
% 0.28/0.48 ! [X: product_prod_a_a,A4: set_a,B: set_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X
% 0.28/0.48 @ ( product_Sigma_a_a @ A4
% 0.28/0.48 @ ^ [Uu: a] : B ) )
% 0.28/0.48 = ( ( member_a @ ( product_fst_a_a @ X ) @ A4 )
% 0.28/0.48 & ( member_a @ ( product_snd_a_a @ X ) @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Times_iff
% 0.28/0.48 thf(fact_281_mem__Times__iff,axiom,
% 0.28/0.48 ! [X: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X
% 0.28/0.48 @ ( produc304751368od_a_a @ A4
% 0.28/0.48 @ ^ [Uu: product_prod_a_a] : B ) )
% 0.28/0.48 = ( ( member449909584od_a_a @ ( produc2115494211od_a_a @ X ) @ A4 )
% 0.28/0.48 & ( member449909584od_a_a @ ( produc368344197od_a_a @ X ) @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % mem_Times_iff
% 0.28/0.48 thf(fact_282_member__product,axiom,
% 0.28/0.48 ! [X: product_prod_a_a,A4: set_a,B: set_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X @ ( product_product_a_a @ A4 @ B ) )
% 0.28/0.48 = ( member449909584od_a_a @ X
% 0.28/0.48 @ ( product_Sigma_a_a @ A4
% 0.28/0.48 @ ^ [Uu: a] : B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % member_product
% 0.28/0.48 thf(fact_283_member__product,axiom,
% 0.28/0.48 ! [X: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X @ ( produc905674258od_a_a @ A4 @ B ) )
% 0.28/0.48 = ( member2057358096od_a_a @ X
% 0.28/0.48 @ ( produc304751368od_a_a @ A4
% 0.28/0.48 @ ^ [Uu: product_prod_a_a] : B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % member_product
% 0.28/0.48 thf(fact_284_subsetI,axiom,
% 0.28/0.48 ! [A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.48 ( ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( member449909584od_a_a @ X3 @ B ) )
% 0.28/0.48 => ( ord_le1824328871od_a_a @ A4 @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % subsetI
% 0.28/0.48 thf(fact_285_subsetI,axiom,
% 0.28/0.48 ! [A4: set_a,B: set_a] :
% 0.28/0.48 ( ! [X3: a] :
% 0.28/0.48 ( ( member_a @ X3 @ A4 )
% 0.28/0.48 => ( member_a @ X3 @ B ) )
% 0.28/0.48 => ( ord_less_eq_set_a @ A4 @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % subsetI
% 0.28/0.48 thf(fact_286_subsetI,axiom,
% 0.28/0.48 ! [A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( member2057358096od_a_a @ X3 @ B ) )
% 0.28/0.48 => ( ord_le456379495od_a_a @ A4 @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % subsetI
% 0.28/0.48 thf(fact_287_image__subsetI,axiom,
% 0.28/0.48 ! [A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.48 ( ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( member449909584od_a_a @ ( F @ X3 ) @ B ) )
% 0.28/0.48 => ( ord_le1824328871od_a_a @ ( image_794773445od_a_a @ F @ A4 ) @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % image_subsetI
% 0.28/0.48 thf(fact_288_image__subsetI,axiom,
% 0.28/0.48 ! [A4: set_Product_prod_a_a,F: product_prod_a_a > a,B: set_a] :
% 0.28/0.48 ( ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( member_a @ ( F @ X3 ) @ B ) )
% 0.28/0.48 => ( ord_less_eq_set_a @ ( image_340790172_a_a_a @ F @ A4 ) @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % image_subsetI
% 0.28/0.48 thf(fact_289_image__subsetI,axiom,
% 0.28/0.48 ! [A4: set_Product_prod_a_a,F: product_prod_a_a > produc1572603623od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( member2057358096od_a_a @ ( F @ X3 ) @ B ) )
% 0.28/0.48 => ( ord_le456379495od_a_a @ ( image_1765377413od_a_a @ F @ A4 ) @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % image_subsetI
% 0.28/0.48 thf(fact_290_image__subsetI,axiom,
% 0.28/0.48 ! [A4: set_a,F: a > product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.48 ( ! [X3: a] :
% 0.28/0.48 ( ( member_a @ X3 @ A4 )
% 0.28/0.48 => ( member449909584od_a_a @ ( F @ X3 ) @ B ) )
% 0.28/0.48 => ( ord_le1824328871od_a_a @ ( image_1725938350od_a_a @ F @ A4 ) @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % image_subsetI
% 0.28/0.48 thf(fact_291_image__subsetI,axiom,
% 0.28/0.48 ! [A4: set_a,F: a > a,B: set_a] :
% 0.28/0.48 ( ! [X3: a] :
% 0.28/0.48 ( ( member_a @ X3 @ A4 )
% 0.28/0.48 => ( member_a @ ( F @ X3 ) @ B ) )
% 0.28/0.48 => ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % image_subsetI
% 0.28/0.48 thf(fact_292_image__subsetI,axiom,
% 0.28/0.48 ! [A4: set_a,F: a > produc1572603623od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ! [X3: a] :
% 0.28/0.48 ( ( member_a @ X3 @ A4 )
% 0.28/0.48 => ( member2057358096od_a_a @ ( F @ X3 ) @ B ) )
% 0.28/0.48 => ( ord_le456379495od_a_a @ ( image_1763923566od_a_a @ F @ A4 ) @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % image_subsetI
% 0.28/0.48 thf(fact_293_image__subsetI,axiom,
% 0.28/0.48 ! [A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > product_prod_a_a,B: set_Product_prod_a_a] :
% 0.28/0.48 ( ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( member449909584od_a_a @ ( F @ X3 ) @ B ) )
% 0.28/0.48 => ( ord_le1824328871od_a_a @ ( image_734753285od_a_a @ F @ A4 ) @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % image_subsetI
% 0.28/0.48 thf(fact_294_image__subsetI,axiom,
% 0.28/0.48 ! [A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > a,B: set_a] :
% 0.28/0.48 ( ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( member_a @ ( F @ X3 ) @ B ) )
% 0.28/0.48 => ( ord_less_eq_set_a @ ( image_1105229276_a_a_a @ F @ A4 ) @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % image_subsetI
% 0.28/0.48 thf(fact_295_image__subsetI,axiom,
% 0.28/0.48 ! [A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a,B: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X3 @ A4 )
% 0.28/0.48 => ( member2057358096od_a_a @ ( F @ X3 ) @ B ) )
% 0.28/0.48 => ( ord_le456379495od_a_a @ ( image_236793797od_a_a @ F @ A4 ) @ B ) ) ).
% 0.28/0.48
% 0.28/0.48 % image_subsetI
% 0.28/0.48 thf(fact_296_subrelI,axiom,
% 0.28/0.48 ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 0.28/0.48 ( ! [X3: a,Y3: a] :
% 0.28/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
% 0.28/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S ) )
% 0.28/0.48 => ( ord_le1824328871od_a_a @ R @ S ) ) ).
% 0.28/0.48
% 0.28/0.48 % subrelI
% 0.28/0.48 thf(fact_297_subrelI,axiom,
% 0.28/0.48 ! [R: set_Pr1948701895od_a_a,S: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R )
% 0.28/0.48 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ S ) )
% 0.28/0.48 => ( ord_le456379495od_a_a @ R @ S ) ) ).
% 0.28/0.48
% 0.28/0.48 % subrelI
% 0.28/0.48 thf(fact_298_subset__iff,axiom,
% 0.28/0.48 ( ord_le1824328871od_a_a
% 0.28/0.48 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 0.28/0.48 ! [T2: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ T2 @ A6 )
% 0.28/0.48 => ( member449909584od_a_a @ T2 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_iff
% 0.28/0.48 thf(fact_299_subset__iff,axiom,
% 0.28/0.48 ( ord_less_eq_set_a
% 0.28/0.48 = ( ^ [A6: set_a,B5: set_a] :
% 0.28/0.48 ! [T2: a] :
% 0.28/0.48 ( ( member_a @ T2 @ A6 )
% 0.28/0.48 => ( member_a @ T2 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_iff
% 0.28/0.48 thf(fact_300_subset__iff,axiom,
% 0.28/0.48 ( ord_le456379495od_a_a
% 0.28/0.48 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 0.28/0.48 ! [T2: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ T2 @ A6 )
% 0.28/0.48 => ( member2057358096od_a_a @ T2 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_iff
% 0.28/0.48 thf(fact_301_subset__eq,axiom,
% 0.28/0.48 ( ord_le1824328871od_a_a
% 0.28/0.48 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 0.28/0.48 ! [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ A6 )
% 0.28/0.48 => ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_eq
% 0.28/0.48 thf(fact_302_subset__eq,axiom,
% 0.28/0.48 ( ord_less_eq_set_a
% 0.28/0.48 = ( ^ [A6: set_a,B5: set_a] :
% 0.28/0.48 ! [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ A6 )
% 0.28/0.48 => ( member_a @ X4 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_eq
% 0.28/0.48 thf(fact_303_subset__eq,axiom,
% 0.28/0.48 ( ord_le456379495od_a_a
% 0.28/0.48 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 0.28/0.48 ! [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ A6 )
% 0.28/0.48 => ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_eq
% 0.28/0.48 thf(fact_304_subsetD,axiom,
% 0.28/0.48 ! [A4: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a] :
% 0.28/0.48 ( ( ord_le1824328871od_a_a @ A4 @ B )
% 0.28/0.48 => ( ( member449909584od_a_a @ C2 @ A4 )
% 0.28/0.48 => ( member449909584od_a_a @ C2 @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subsetD
% 0.28/0.48 thf(fact_305_subsetD,axiom,
% 0.28/0.48 ! [A4: set_a,B: set_a,C2: a] :
% 0.28/0.48 ( ( ord_less_eq_set_a @ A4 @ B )
% 0.28/0.48 => ( ( member_a @ C2 @ A4 )
% 0.28/0.48 => ( member_a @ C2 @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subsetD
% 0.28/0.48 thf(fact_306_subsetD,axiom,
% 0.28/0.48 ! [A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a,C2: produc1572603623od_a_a] :
% 0.28/0.48 ( ( ord_le456379495od_a_a @ A4 @ B )
% 0.28/0.48 => ( ( member2057358096od_a_a @ C2 @ A4 )
% 0.28/0.48 => ( member2057358096od_a_a @ C2 @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subsetD
% 0.28/0.48 thf(fact_307_in__mono,axiom,
% 0.28/0.48 ! [A4: set_Product_prod_a_a,B: set_Product_prod_a_a,X: product_prod_a_a] :
% 0.28/0.48 ( ( ord_le1824328871od_a_a @ A4 @ B )
% 0.28/0.48 => ( ( member449909584od_a_a @ X @ A4 )
% 0.28/0.48 => ( member449909584od_a_a @ X @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % in_mono
% 0.28/0.48 thf(fact_308_in__mono,axiom,
% 0.28/0.48 ! [A4: set_a,B: set_a,X: a] :
% 0.28/0.48 ( ( ord_less_eq_set_a @ A4 @ B )
% 0.28/0.48 => ( ( member_a @ X @ A4 )
% 0.28/0.48 => ( member_a @ X @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % in_mono
% 0.28/0.48 thf(fact_309_in__mono,axiom,
% 0.28/0.48 ! [A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a,X: produc1572603623od_a_a] :
% 0.28/0.48 ( ( ord_le456379495od_a_a @ A4 @ B )
% 0.28/0.48 => ( ( member2057358096od_a_a @ X @ A4 )
% 0.28/0.48 => ( member2057358096od_a_a @ X @ B ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % in_mono
% 0.28/0.48 thf(fact_310_prop__restrict,axiom,
% 0.28/0.48 ! [X: product_prod_a_a,Z2: set_Product_prod_a_a,X5: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 0.28/0.48 ( ( member449909584od_a_a @ X @ Z2 )
% 0.28/0.48 => ( ( ord_le1824328871od_a_a @ Z2
% 0.28/0.48 @ ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ X5 )
% 0.28/0.48 & ( P2 @ X4 ) ) ) )
% 0.28/0.48 => ( P2 @ X ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prop_restrict
% 0.28/0.48 thf(fact_311_prop__restrict,axiom,
% 0.28/0.48 ! [X: a,Z2: set_a,X5: set_a,P2: a > $o] :
% 0.28/0.48 ( ( member_a @ X @ Z2 )
% 0.28/0.48 => ( ( ord_less_eq_set_a @ Z2
% 0.28/0.48 @ ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ X5 )
% 0.28/0.48 & ( P2 @ X4 ) ) ) )
% 0.28/0.48 => ( P2 @ X ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prop_restrict
% 0.28/0.48 thf(fact_312_prop__restrict,axiom,
% 0.28/0.48 ! [X: produc1572603623od_a_a,Z2: set_Pr1948701895od_a_a,X5: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X @ Z2 )
% 0.28/0.48 => ( ( ord_le456379495od_a_a @ Z2
% 0.28/0.48 @ ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ X5 )
% 0.28/0.48 & ( P2 @ X4 ) ) ) )
% 0.28/0.48 => ( P2 @ X ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % prop_restrict
% 0.28/0.48 thf(fact_313_Collect__restrict,axiom,
% 0.28/0.48 ! [X5: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 0.28/0.48 ( ord_le1824328871od_a_a
% 0.28/0.48 @ ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ X5 )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 @ X5 ) ).
% 0.28/0.48
% 0.28/0.48 % Collect_restrict
% 0.28/0.48 thf(fact_314_Collect__restrict,axiom,
% 0.28/0.48 ! [X5: set_a,P2: a > $o] :
% 0.28/0.48 ( ord_less_eq_set_a
% 0.28/0.48 @ ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ X5 )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 @ X5 ) ).
% 0.28/0.48
% 0.28/0.48 % Collect_restrict
% 0.28/0.48 thf(fact_315_Collect__restrict,axiom,
% 0.28/0.48 ! [X5: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 0.28/0.48 ( ord_le456379495od_a_a
% 0.28/0.48 @ ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ X5 )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 @ X5 ) ).
% 0.28/0.48
% 0.28/0.48 % Collect_restrict
% 0.28/0.48 thf(fact_316_Collect__subset,axiom,
% 0.28/0.48 ! [A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 0.28/0.48 ( ord_le1824328871od_a_a
% 0.28/0.48 @ ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 @ A4 ) ).
% 0.28/0.48
% 0.28/0.48 % Collect_subset
% 0.28/0.48 thf(fact_317_Collect__subset,axiom,
% 0.28/0.48 ! [A4: set_a,P2: a > $o] :
% 0.28/0.48 ( ord_less_eq_set_a
% 0.28/0.48 @ ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 @ A4 ) ).
% 0.28/0.48
% 0.28/0.48 % Collect_subset
% 0.28/0.48 thf(fact_318_Collect__subset,axiom,
% 0.28/0.48 ! [A4: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 0.28/0.48 ( ord_le456379495od_a_a
% 0.28/0.48 @ ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ X4 ) ) )
% 0.28/0.48 @ A4 ) ).
% 0.28/0.48
% 0.28/0.48 % Collect_subset
% 0.28/0.48 thf(fact_319_subset__Collect__iff,axiom,
% 0.28/0.48 ! [B: set_Product_prod_a_a,A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 0.28/0.48 ( ( ord_le1824328871od_a_a @ B @ A4 )
% 0.28/0.48 => ( ( ord_le1824328871od_a_a @ B
% 0.28/0.48 @ ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ X4 ) ) ) )
% 0.28/0.48 = ( ! [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ B )
% 0.28/0.48 => ( P2 @ X4 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_Collect_iff
% 0.28/0.48 thf(fact_320_subset__Collect__iff,axiom,
% 0.28/0.48 ! [B: set_a,A4: set_a,P2: a > $o] :
% 0.28/0.48 ( ( ord_less_eq_set_a @ B @ A4 )
% 0.28/0.48 => ( ( ord_less_eq_set_a @ B
% 0.28/0.48 @ ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ X4 ) ) ) )
% 0.28/0.48 = ( ! [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ B )
% 0.28/0.48 => ( P2 @ X4 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_Collect_iff
% 0.28/0.48 thf(fact_321_subset__Collect__iff,axiom,
% 0.28/0.48 ! [B: set_Pr1948701895od_a_a,A4: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 0.28/0.48 ( ( ord_le456379495od_a_a @ B @ A4 )
% 0.28/0.48 => ( ( ord_le456379495od_a_a @ B
% 0.28/0.48 @ ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ X4 ) ) ) )
% 0.28/0.48 = ( ! [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ B )
% 0.28/0.48 => ( P2 @ X4 ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_Collect_iff
% 0.28/0.48 thf(fact_322_subset__CollectI,axiom,
% 0.28/0.48 ! [B: set_Product_prod_a_a,A4: set_Product_prod_a_a,Q: product_prod_a_a > $o,P2: product_prod_a_a > $o] :
% 0.28/0.48 ( ( ord_le1824328871od_a_a @ B @ A4 )
% 0.28/0.48 => ( ! [X3: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X3 @ B )
% 0.28/0.48 => ( ( Q @ X3 )
% 0.28/0.48 => ( P2 @ X3 ) ) )
% 0.28/0.48 => ( ord_le1824328871od_a_a
% 0.28/0.48 @ ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ B )
% 0.28/0.48 & ( Q @ X4 ) ) )
% 0.28/0.48 @ ( collec645855634od_a_a
% 0.28/0.48 @ ^ [X4: product_prod_a_a] :
% 0.28/0.48 ( ( member449909584od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_CollectI
% 0.28/0.48 thf(fact_323_subset__CollectI,axiom,
% 0.28/0.48 ! [B: set_a,A4: set_a,Q: a > $o,P2: a > $o] :
% 0.28/0.48 ( ( ord_less_eq_set_a @ B @ A4 )
% 0.28/0.48 => ( ! [X3: a] :
% 0.28/0.48 ( ( member_a @ X3 @ B )
% 0.28/0.48 => ( ( Q @ X3 )
% 0.28/0.48 => ( P2 @ X3 ) ) )
% 0.28/0.48 => ( ord_less_eq_set_a
% 0.28/0.48 @ ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ B )
% 0.28/0.48 & ( Q @ X4 ) ) )
% 0.28/0.48 @ ( collect_a
% 0.28/0.48 @ ^ [X4: a] :
% 0.28/0.48 ( ( member_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_CollectI
% 0.28/0.48 thf(fact_324_subset__CollectI,axiom,
% 0.28/0.48 ! [B: set_Pr1948701895od_a_a,A4: set_Pr1948701895od_a_a,Q: produc1572603623od_a_a > $o,P2: produc1572603623od_a_a > $o] :
% 0.28/0.48 ( ( ord_le456379495od_a_a @ B @ A4 )
% 0.28/0.48 => ( ! [X3: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X3 @ B )
% 0.28/0.48 => ( ( Q @ X3 )
% 0.28/0.48 => ( P2 @ X3 ) ) )
% 0.28/0.48 => ( ord_le456379495od_a_a
% 0.28/0.48 @ ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ B )
% 0.28/0.48 & ( Q @ X4 ) ) )
% 0.28/0.48 @ ( collec1635618130od_a_a
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ X4 @ A4 )
% 0.28/0.48 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % subset_CollectI
% 0.28/0.48 thf(fact_325_pred__subset__eq,axiom,
% 0.28/0.48 ! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 0.28/0.48 ( ( ord_le1347718902_a_a_o
% 0.28/0.48 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ R2 )
% 0.28/0.48 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ S3 ) )
% 0.28/0.48 = ( ord_le1824328871od_a_a @ R2 @ S3 ) ) ).
% 0.28/0.48
% 0.28/0.48 % pred_subset_eq
% 0.28/0.48 thf(fact_326_pred__subset__eq,axiom,
% 0.28/0.48 ! [R2: set_a,S3: set_a] :
% 0.28/0.48 ( ( ord_less_eq_a_o
% 0.28/0.48 @ ^ [X4: a] : ( member_a @ X4 @ R2 )
% 0.28/0.48 @ ^ [X4: a] : ( member_a @ X4 @ S3 ) )
% 0.28/0.48 = ( ord_less_eq_set_a @ R2 @ S3 ) ) ).
% 0.28/0.48
% 0.28/0.48 % pred_subset_eq
% 0.28/0.48 thf(fact_327_pred__subset__eq,axiom,
% 0.28/0.48 ! [R2: set_Pr1948701895od_a_a,S3: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( ord_le1525442358_a_a_o
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ R2 )
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ S3 ) )
% 0.28/0.48 = ( ord_le456379495od_a_a @ R2 @ S3 ) ) ).
% 0.28/0.48
% 0.28/0.48 % pred_subset_eq
% 0.28/0.48 thf(fact_328_less__eq__set__def,axiom,
% 0.28/0.48 ( ord_le1824328871od_a_a
% 0.28/0.48 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 0.28/0.48 ( ord_le1347718902_a_a_o
% 0.28/0.48 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A6 )
% 0.28/0.48 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % less_eq_set_def
% 0.28/0.48 thf(fact_329_less__eq__set__def,axiom,
% 0.28/0.48 ( ord_less_eq_set_a
% 0.28/0.48 = ( ^ [A6: set_a,B5: set_a] :
% 0.28/0.48 ( ord_less_eq_a_o
% 0.28/0.48 @ ^ [X4: a] : ( member_a @ X4 @ A6 )
% 0.28/0.48 @ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % less_eq_set_def
% 0.28/0.48 thf(fact_330_less__eq__set__def,axiom,
% 0.28/0.48 ( ord_le456379495od_a_a
% 0.28/0.48 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ord_le1525442358_a_a_o
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A6 )
% 0.28/0.48 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 % less_eq_set_def
% 0.28/0.48 thf(fact_331_pred__subset__eq2,axiom,
% 0.28/0.48 ! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 0.28/0.48 ( ( ord_less_eq_a_a_o
% 0.28/0.48 @ ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R2 )
% 0.28/0.48 @ ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ S3 ) )
% 0.28/0.48 = ( ord_le1824328871od_a_a @ R2 @ S3 ) ) ).
% 0.28/0.48
% 0.28/0.48 % pred_subset_eq2
% 0.28/0.48 thf(fact_332_pred__subset__eq2,axiom,
% 0.28/0.48 ! [R2: set_Pr1948701895od_a_a,S3: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( ord_le252190158_a_a_o
% 0.28/0.48 @ ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R2 )
% 0.28/0.48 @ ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ S3 ) )
% 0.28/0.48 = ( ord_le456379495od_a_a @ R2 @ S3 ) ) ).
% 0.28/0.48
% 0.28/0.48 % pred_subset_eq2
% 0.28/0.48 thf(fact_333_GrD1,axiom,
% 0.28/0.48 ! [X: a,Fx: a,A4: set_a,F: a > a] :
% 0.28/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Fx ) @ ( bNF_Gr_a_a @ A4 @ F ) )
% 0.28/0.48 => ( member_a @ X @ A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % GrD1
% 0.28/0.48 thf(fact_334_GrD1,axiom,
% 0.28/0.48 ! [X: product_prod_a_a,Fx: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Fx ) @ ( bNF_Gr1372473186od_a_a @ A4 @ F ) )
% 0.28/0.48 => ( member449909584od_a_a @ X @ A4 ) ) ).
% 0.28/0.48
% 0.28/0.48 % GrD1
% 0.28/0.48 thf(fact_335_GrD2,axiom,
% 0.28/0.48 ! [X: a,Fx: a,A4: set_a,F: a > a] :
% 0.28/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Fx ) @ ( bNF_Gr_a_a @ A4 @ F ) )
% 0.28/0.48 => ( ( F @ X )
% 0.28/0.48 = Fx ) ) ).
% 0.28/0.48
% 0.28/0.48 % GrD2
% 0.28/0.48 thf(fact_336_GrD2,axiom,
% 0.28/0.48 ! [X: product_prod_a_a,Fx: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
% 0.28/0.48 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Fx ) @ ( bNF_Gr1372473186od_a_a @ A4 @ F ) )
% 0.28/0.48 => ( ( F @ X )
% 0.28/0.48 = Fx ) ) ).
% 0.28/0.48
% 0.28/0.48 % GrD2
% 0.28/0.48 thf(fact_337_all__not__in__conv,axiom,
% 0.28/0.48 ! [A4: set_Product_prod_a_a] :
% 0.28/0.48 ( ( ! [X4: product_prod_a_a] :
% 0.28/0.48 ~ ( member449909584od_a_a @ X4 @ A4 ) )
% 0.28/0.48 = ( A4 = bot_bo2131659635od_a_a ) ) ).
% 0.28/0.48
% 0.28/0.48 % all_not_in_conv
% 0.28/0.48 thf(fact_338_all__not__in__conv,axiom,
% 0.28/0.48 ! [A4: set_a] :
% 0.28/0.48 ( ( ! [X4: a] :
% 0.28/0.48 ~ ( member_a @ X4 @ A4 ) )
% 0.28/0.48 = ( A4 = bot_bot_set_a ) ) ).
% 0.28/0.48
% 0.28/0.48 % all_not_in_conv
% 0.28/0.48 thf(fact_339_all__not__in__conv,axiom,
% 0.28/0.48 ! [A4: set_Pr1948701895od_a_a] :
% 0.28/0.48 ( ( ! [X4: produc1572603623od_a_a] :
% 0.28/0.48 ~ ( member2057358096od_a_a @ X4 @ A4 ) )
% 0.28/0.48 = ( A4 = bot_bo1116205875od_a_a ) ) ).
% 0.28/0.48
% 0.28/0.48 % all_not_in_conv
% 0.28/0.48 thf(fact_340_empty__iff,axiom,
% 0.28/0.48 ! [C2: product_prod_a_a] :
% 0.28/0.48 ~ ( member449909584od_a_a @ C2 @ bot_bo2131659635od_a_a ) ).
% 0.28/0.48
% 0.28/0.48 % empty_iff
% 0.28/0.48 thf(fact_341_empty__iff,axiom,
% 0.28/0.48 ! [C2: a] :
% 0.28/0.48 ~ ( member_a @ C2 @ bot_bot_set_a ) ).
% 0.28/0.48
% 0.28/0.48 % empty_iff
% 0.28/0.48 thf(fact_342_empty__iff,axiom,
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% 0.28/0.48 % equals0D
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% 0.28/0.48 % equals0D
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% 3.00/1.55 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.d5SnneaLK1/DTF2DTF13046.p
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% 3.00/1.59 member1545452880od_a_a: produc47586087od_a_a > set_Pr173461255od_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mtf__a_J,type,
% 3.00/1.59 member575366567_a_a_a: produc1683075710_a_a_a > set_Pr6137822_a_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 3.00/1.59 member470802768_a_a_a: produc1120419623_a_a_a > set_Pr729215751_a_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 3.00/1.59 member1424209616od_a_a: produc1546279079od_a_a > set_Pr660101255od_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.00/1.59 member1361298640od_a_a: produc2010915495od_a_a > set_Pr231806599od_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.00/1.59 member122808528od_a_a: produc772425383od_a_a > set_Pr1194146439od_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.00/1.59 member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 3.00/1.59 member1403138919_a_a_a: produc1016684094_a_a_a > set_Pr1689873822_a_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 3.00/1.59 member448091577od_a_a: produc1555800720od_a_a > set_Pr195866352od_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.00/1.59 member160619001od_a_a: produc1921647824od_a_a > set_Pr681306928od_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.00/1.59 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_c_member_001tf__a,type,
% 3.00/1.59 member_a: a > set_a > $o ).
% 3.00/1.59
% 3.00/1.59 thf(sy_v_B,type,
% 3.00/1.59 b: set_a ).
% 3.00/1.59
% 3.00/1.59 thf(sy_v_Pr,type,
% 3.00/1.59 pr: set_Product_prod_a_a ).
% 3.00/1.59
% 3.00/1.59 thf(sy_v_y,type,
% 3.00/1.59 y: a ).
% 3.00/1.59
% 3.00/1.59 thf(sy_v_z,type,
% 3.00/1.59 z: a ).
% 3.00/1.59
% 3.00/1.59 thf(fact_0_assms,axiom,
% 3.00/1.59 member_a @ z @ ( prefer161218362good_a @ y @ b @ pr ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_1_at__lst__asgd__ge,axiom,
% 3.00/1.59 ! [X: produc2010915495od_a_a,Y: produc2010915495od_a_a,B: set_Pr231806599od_a_a,Pr: set_Pr532616391od_a_a] :
% 3.00/1.59 ( ( member1361298640od_a_a @ X @ ( prefer468349667od_a_a @ Y @ B @ Pr ) )
% 3.00/1.59 => ( member180906768od_a_a @ ( produc539587415od_a_a @ X @ Y ) @ Pr ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_2_at__lst__asgd__ge,axiom,
% 3.00/1.59 ! [X: produc772425383od_a_a,Y: produc772425383od_a_a,B: set_Pr1194146439od_a_a,Pr: set_Pr2081559751od_a_a] :
% 3.00/1.59 ( ( member122808528od_a_a @ X @ ( prefer1377343203od_a_a @ Y @ B @ Pr ) )
% 3.00/1.59 => ( member547056400od_a_a @ ( produc852390743od_a_a @ X @ Y ) @ Pr ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_3_at__lst__asgd__ge,axiom,
% 3.00/1.59 ! [X: produc1016684094_a_a_a,Y: produc1016684094_a_a_a,B: set_Pr1689873822_a_a_a,Pr: set_Pr729215751_a_a_a] :
% 3.00/1.59 ( ( member1403138919_a_a_a @ X @ ( prefer1944486522_a_a_a @ Y @ B @ Pr ) )
% 3.00/1.59 => ( member470802768_a_a_a @ ( produc2054708503_a_a_a @ X @ Y ) @ Pr ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_4_at__lst__asgd__ge,axiom,
% 3.00/1.59 ! [X: produc1555800720od_a_a,Y: produc1555800720od_a_a,B: set_Pr195866352od_a_a,Pr: set_Pr660101255od_a_a] :
% 3.00/1.59 ( ( member448091577od_a_a @ X @ ( prefer445484748od_a_a @ Y @ B @ Pr ) )
% 3.00/1.59 => ( member1424209616od_a_a @ ( produc1573596055od_a_a @ X @ Y ) @ Pr ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_5_at__lst__asgd__ge,axiom,
% 3.00/1.59 ! [X: produc1921647824od_a_a,Y: produc1921647824od_a_a,B: set_Pr681306928od_a_a,Pr: set_Pr231806599od_a_a] :
% 3.00/1.59 ( ( member160619001od_a_a @ X @ ( prefer701966604od_a_a @ Y @ B @ Pr ) )
% 3.00/1.59 => ( member1361298640od_a_a @ ( produc563767703od_a_a @ X @ Y ) @ Pr ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_6_at__lst__asgd__ge,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,Y: produc1572603623od_a_a,B: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ B @ Pr ) )
% 3.00/1.59 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Pr ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_7_at__lst__asgd__ge,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,Y: product_prod_a_a,B: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B @ Pr ) )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_8_at__lst__asgd__ge,axiom,
% 3.00/1.59 ! [X: a,Y: a,B: set_a,Pr: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member_a @ X @ ( prefer161218362good_a @ Y @ B @ Pr ) )
% 3.00/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_9_prod_Oinject,axiom,
% 3.00/1.59 ! [X1: produc1921647824od_a_a,X2: produc1921647824od_a_a,Y1: produc1921647824od_a_a,Y2: produc1921647824od_a_a] :
% 3.00/1.59 ( ( ( produc563767703od_a_a @ X1 @ X2 )
% 3.00/1.59 = ( produc563767703od_a_a @ Y1 @ Y2 ) )
% 3.00/1.59 = ( ( X1 = Y1 )
% 3.00/1.59 & ( X2 = Y2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_10_prod_Oinject,axiom,
% 3.00/1.59 ! [X1: product_prod_a_a,X2: produc1572603623od_a_a,Y1: product_prod_a_a,Y2: produc1572603623od_a_a] :
% 3.00/1.59 ( ( ( produc1555606807od_a_a @ X1 @ X2 )
% 3.00/1.59 = ( produc1555606807od_a_a @ Y1 @ Y2 ) )
% 3.00/1.59 = ( ( X1 = Y1 )
% 3.00/1.59 & ( X2 = Y2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_11_prod_Oinject,axiom,
% 3.00/1.59 ! [X1: product_prod_a_a,X2: a,Y1: product_prod_a_a,Y2: a] :
% 3.00/1.59 ( ( ( produc2061588782_a_a_a @ X1 @ X2 )
% 3.00/1.59 = ( produc2061588782_a_a_a @ Y1 @ Y2 ) )
% 3.00/1.59 = ( ( X1 = Y1 )
% 3.00/1.59 & ( X2 = Y2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_12_prod_Oinject,axiom,
% 3.00/1.59 ! [X1: a,X2: produc1572603623od_a_a,Y1: a,Y2: produc1572603623od_a_a] :
% 3.00/1.59 ( ( ( produc1095491584od_a_a @ X1 @ X2 )
% 3.00/1.59 = ( produc1095491584od_a_a @ Y1 @ Y2 ) )
% 3.00/1.59 = ( ( X1 = Y1 )
% 3.00/1.59 & ( X2 = Y2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_13_prod_Oinject,axiom,
% 3.00/1.59 ! [X1: a,X2: product_prod_a_a,Y1: a,Y2: product_prod_a_a] :
% 3.00/1.59 ( ( ( produc1299253312od_a_a @ X1 @ X2 )
% 3.00/1.59 = ( produc1299253312od_a_a @ Y1 @ Y2 ) )
% 3.00/1.59 = ( ( X1 = Y1 )
% 3.00/1.59 & ( X2 = Y2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_14_prod_Oinject,axiom,
% 3.00/1.59 ! [X1: product_prod_a_a,X2: product_prod_a_a,Y1: product_prod_a_a,Y2: product_prod_a_a] :
% 3.00/1.59 ( ( ( produc1474507607od_a_a @ X1 @ X2 )
% 3.00/1.59 = ( produc1474507607od_a_a @ Y1 @ Y2 ) )
% 3.00/1.59 = ( ( X1 = Y1 )
% 3.00/1.59 & ( X2 = Y2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_15_prod_Oinject,axiom,
% 3.00/1.59 ! [X1: a,X2: a,Y1: a,Y2: a] :
% 3.00/1.59 ( ( ( product_Pair_a_a @ X1 @ X2 )
% 3.00/1.59 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 3.00/1.59 = ( ( X1 = Y1 )
% 3.00/1.59 & ( X2 = Y2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_16_old_Oprod_Oinject,axiom,
% 3.00/1.59 ! [A: produc1921647824od_a_a,B2: produc1921647824od_a_a,A2: produc1921647824od_a_a,B3: produc1921647824od_a_a] :
% 3.00/1.59 ( ( ( produc563767703od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc563767703od_a_a @ A2 @ B3 ) )
% 3.00/1.59 = ( ( A = A2 )
% 3.00/1.59 & ( B2 = B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_17_old_Oprod_Oinject,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: produc1572603623od_a_a,A2: product_prod_a_a,B3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( ( produc1555606807od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc1555606807od_a_a @ A2 @ B3 ) )
% 3.00/1.59 = ( ( A = A2 )
% 3.00/1.59 & ( B2 = B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_18_old_Oprod_Oinject,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: a,A2: product_prod_a_a,B3: a] :
% 3.00/1.59 ( ( ( produc2061588782_a_a_a @ A @ B2 )
% 3.00/1.59 = ( produc2061588782_a_a_a @ A2 @ B3 ) )
% 3.00/1.59 = ( ( A = A2 )
% 3.00/1.59 & ( B2 = B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_19_old_Oprod_Oinject,axiom,
% 3.00/1.59 ! [A: a,B2: produc1572603623od_a_a,A2: a,B3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( ( produc1095491584od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc1095491584od_a_a @ A2 @ B3 ) )
% 3.00/1.59 = ( ( A = A2 )
% 3.00/1.59 & ( B2 = B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_20_old_Oprod_Oinject,axiom,
% 3.00/1.59 ! [A: a,B2: product_prod_a_a,A2: a,B3: product_prod_a_a] :
% 3.00/1.59 ( ( ( produc1299253312od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc1299253312od_a_a @ A2 @ B3 ) )
% 3.00/1.59 = ( ( A = A2 )
% 3.00/1.59 & ( B2 = B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_21_old_Oprod_Oinject,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
% 3.00/1.59 ( ( ( produc1474507607od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc1474507607od_a_a @ A2 @ B3 ) )
% 3.00/1.59 = ( ( A = A2 )
% 3.00/1.59 & ( B2 = B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_22_old_Oprod_Oinject,axiom,
% 3.00/1.59 ! [A: a,B2: a,A2: a,B3: a] :
% 3.00/1.59 ( ( ( product_Pair_a_a @ A @ B2 )
% 3.00/1.59 = ( product_Pair_a_a @ A2 @ B3 ) )
% 3.00/1.59 = ( ( A = A2 )
% 3.00/1.59 & ( B2 = B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_23_surj__pair,axiom,
% 3.00/1.59 ! [P: produc2010915495od_a_a] :
% 3.00/1.59 ? [X3: produc1921647824od_a_a,Y3: produc1921647824od_a_a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( produc563767703od_a_a @ X3 @ Y3 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_24_surj__pair,axiom,
% 3.00/1.59 ! [P: produc772425383od_a_a] :
% 3.00/1.59 ? [X3: product_prod_a_a,Y3: produc1572603623od_a_a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( produc1555606807od_a_a @ X3 @ Y3 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_25_surj__pair,axiom,
% 3.00/1.59 ! [P: produc1016684094_a_a_a] :
% 3.00/1.59 ? [X3: product_prod_a_a,Y3: a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( produc2061588782_a_a_a @ X3 @ Y3 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_26_surj__pair,axiom,
% 3.00/1.59 ! [P: produc1555800720od_a_a] :
% 3.00/1.59 ? [X3: a,Y3: produc1572603623od_a_a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( produc1095491584od_a_a @ X3 @ Y3 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_27_surj__pair,axiom,
% 3.00/1.59 ! [P: produc1921647824od_a_a] :
% 3.00/1.59 ? [X3: a,Y3: product_prod_a_a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( produc1299253312od_a_a @ X3 @ Y3 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_28_surj__pair,axiom,
% 3.00/1.59 ! [P: produc1572603623od_a_a] :
% 3.00/1.59 ? [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( produc1474507607od_a_a @ X3 @ Y3 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_29_surj__pair,axiom,
% 3.00/1.59 ! [P: product_prod_a_a] :
% 3.00/1.59 ? [X3: a,Y3: a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_30_prod__cases,axiom,
% 3.00/1.59 ! [P2: produc2010915495od_a_a > $o,P: produc2010915495od_a_a] :
% 3.00/1.59 ( ! [A3: produc1921647824od_a_a,B4: produc1921647824od_a_a] : ( P2 @ ( produc563767703od_a_a @ A3 @ B4 ) )
% 3.00/1.59 => ( P2 @ P ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_31_prod__cases,axiom,
% 3.00/1.59 ! [P2: produc772425383od_a_a > $o,P: produc772425383od_a_a] :
% 3.00/1.59 ( ! [A3: product_prod_a_a,B4: produc1572603623od_a_a] : ( P2 @ ( produc1555606807od_a_a @ A3 @ B4 ) )
% 3.00/1.59 => ( P2 @ P ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_32_prod__cases,axiom,
% 3.00/1.59 ! [P2: produc1016684094_a_a_a > $o,P: produc1016684094_a_a_a] :
% 3.00/1.59 ( ! [A3: product_prod_a_a,B4: a] : ( P2 @ ( produc2061588782_a_a_a @ A3 @ B4 ) )
% 3.00/1.59 => ( P2 @ P ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_33_prod__cases,axiom,
% 3.00/1.59 ! [P2: produc1555800720od_a_a > $o,P: produc1555800720od_a_a] :
% 3.00/1.59 ( ! [A3: a,B4: produc1572603623od_a_a] : ( P2 @ ( produc1095491584od_a_a @ A3 @ B4 ) )
% 3.00/1.59 => ( P2 @ P ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_34_prod__cases,axiom,
% 3.00/1.59 ! [P2: produc1921647824od_a_a > $o,P: produc1921647824od_a_a] :
% 3.00/1.59 ( ! [A3: a,B4: product_prod_a_a] : ( P2 @ ( produc1299253312od_a_a @ A3 @ B4 ) )
% 3.00/1.59 => ( P2 @ P ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_35_prod__cases,axiom,
% 3.00/1.59 ! [P2: produc1572603623od_a_a > $o,P: produc1572603623od_a_a] :
% 3.00/1.59 ( ! [A3: product_prod_a_a,B4: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B4 ) )
% 3.00/1.59 => ( P2 @ P ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_36_prod__cases,axiom,
% 3.00/1.59 ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
% 3.00/1.59 ( ! [A3: a,B4: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B4 ) )
% 3.00/1.59 => ( P2 @ P ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_37_Pair__inject,axiom,
% 3.00/1.59 ! [A: produc1921647824od_a_a,B2: produc1921647824od_a_a,A2: produc1921647824od_a_a,B3: produc1921647824od_a_a] :
% 3.00/1.59 ( ( ( produc563767703od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc563767703od_a_a @ A2 @ B3 ) )
% 3.00/1.59 => ~ ( ( A = A2 )
% 3.00/1.59 => ( B2 != B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_38_Pair__inject,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: produc1572603623od_a_a,A2: product_prod_a_a,B3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( ( produc1555606807od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc1555606807od_a_a @ A2 @ B3 ) )
% 3.00/1.59 => ~ ( ( A = A2 )
% 3.00/1.59 => ( B2 != B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_39_Pair__inject,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: a,A2: product_prod_a_a,B3: a] :
% 3.00/1.59 ( ( ( produc2061588782_a_a_a @ A @ B2 )
% 3.00/1.59 = ( produc2061588782_a_a_a @ A2 @ B3 ) )
% 3.00/1.59 => ~ ( ( A = A2 )
% 3.00/1.59 => ( B2 != B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_40_Pair__inject,axiom,
% 3.00/1.59 ! [A: a,B2: produc1572603623od_a_a,A2: a,B3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( ( produc1095491584od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc1095491584od_a_a @ A2 @ B3 ) )
% 3.00/1.59 => ~ ( ( A = A2 )
% 3.00/1.59 => ( B2 != B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_41_Pair__inject,axiom,
% 3.00/1.59 ! [A: a,B2: product_prod_a_a,A2: a,B3: product_prod_a_a] :
% 3.00/1.59 ( ( ( produc1299253312od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc1299253312od_a_a @ A2 @ B3 ) )
% 3.00/1.59 => ~ ( ( A = A2 )
% 3.00/1.59 => ( B2 != B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_42_Pair__inject,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a] :
% 3.00/1.59 ( ( ( produc1474507607od_a_a @ A @ B2 )
% 3.00/1.59 = ( produc1474507607od_a_a @ A2 @ B3 ) )
% 3.00/1.59 => ~ ( ( A = A2 )
% 3.00/1.59 => ( B2 != B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_43_Pair__inject,axiom,
% 3.00/1.59 ! [A: a,B2: a,A2: a,B3: a] :
% 3.00/1.59 ( ( ( product_Pair_a_a @ A @ B2 )
% 3.00/1.59 = ( product_Pair_a_a @ A2 @ B3 ) )
% 3.00/1.59 => ~ ( ( A = A2 )
% 3.00/1.59 => ( B2 != B3 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_44_prod__cases3,axiom,
% 3.00/1.59 ! [Y: produc772425383od_a_a] :
% 3.00/1.59 ~ ! [A3: product_prod_a_a,B4: product_prod_a_a,C: product_prod_a_a] :
% 3.00/1.59 ( Y
% 3.00/1.59 != ( produc1555606807od_a_a @ A3 @ ( produc1474507607od_a_a @ B4 @ C ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_45_prod__cases3,axiom,
% 3.00/1.59 ! [Y: produc1555800720od_a_a] :
% 3.00/1.59 ~ ! [A3: a,B4: product_prod_a_a,C: product_prod_a_a] :
% 3.00/1.59 ( Y
% 3.00/1.59 != ( produc1095491584od_a_a @ A3 @ ( produc1474507607od_a_a @ B4 @ C ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_46_prod__cases3,axiom,
% 3.00/1.59 ! [Y: produc1921647824od_a_a] :
% 3.00/1.59 ~ ! [A3: a,B4: a,C: a] :
% 3.00/1.59 ( Y
% 3.00/1.59 != ( produc1299253312od_a_a @ A3 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_47_prod__cases3,axiom,
% 3.00/1.59 ! [Y: produc1572603623od_a_a] :
% 3.00/1.59 ~ ! [A3: product_prod_a_a,B4: a,C: a] :
% 3.00/1.59 ( Y
% 3.00/1.59 != ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B4 @ C ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_48_prod__induct3,axiom,
% 3.00/1.59 ! [P2: produc1572603623od_a_a > $o,X: produc1572603623od_a_a] :
% 3.00/1.59 ( ! [A3: product_prod_a_a,B4: a,C: a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B4 @ C ) ) )
% 3.00/1.59 => ( P2 @ X ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_49_old_Oprod_Oinducts,axiom,
% 3.00/1.59 ! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 3.00/1.59 ( ! [A3: a,B4: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B4 ) )
% 3.00/1.59 => ( P2 @ Prod ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_50_old_Oprod_Oinducts,axiom,
% 3.00/1.59 ! [P2: produc1572603623od_a_a > $o,Prod: produc1572603623od_a_a] :
% 3.00/1.59 ( ! [A3: product_prod_a_a,B4: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B4 ) )
% 3.00/1.59 => ( P2 @ Prod ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_51_old_Oprod_Oexhaust,axiom,
% 3.00/1.59 ! [Y: product_prod_a_a] :
% 3.00/1.59 ~ ! [A3: a,B4: a] :
% 3.00/1.59 ( Y
% 3.00/1.59 != ( product_Pair_a_a @ A3 @ B4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_52_old_Oprod_Oexhaust,axiom,
% 3.00/1.59 ! [Y: produc1572603623od_a_a] :
% 3.00/1.59 ~ ! [A3: product_prod_a_a,B4: product_prod_a_a] :
% 3.00/1.59 ( Y
% 3.00/1.59 != ( produc1474507607od_a_a @ A3 @ B4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_53_at__least__as__good__def,axiom,
% 3.00/1.59 ( prefer55414563od_a_a
% 3.00/1.59 = ( ^ [X4: produc1572603623od_a_a,B5: set_Pr1948701895od_a_a,P3: set_Pr1295299783od_a_a] :
% 3.00/1.59 ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [Y4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ Y4 @ B5 )
% 3.00/1.59 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_54_at__least__as__good__def,axiom,
% 3.00/1.59 ( prefer161218362good_a
% 3.00/1.59 = ( ^ [X4: a,B5: set_a,P3: set_Product_prod_a_a] :
% 3.00/1.59 ( collect_a
% 3.00/1.59 @ ^ [Y4: a] :
% 3.00/1.59 ( ( member_a @ Y4 @ B5 )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_55_at__least__as__good__def,axiom,
% 3.00/1.59 ( prefer477097315od_a_a
% 3.00/1.59 = ( ^ [X4: product_prod_a_a,B5: set_Product_prod_a_a,P3: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( collec645855634od_a_a
% 3.00/1.59 @ ^ [Y4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ Y4 @ B5 )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_56_ssubst__Pair__rhs,axiom,
% 3.00/1.59 ! [R: a,S: a,R2: set_Product_prod_a_a,S2: a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ R2 )
% 3.00/1.59 => ( ( S2 = S )
% 3.00/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S2 ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_57_ssubst__Pair__rhs,axiom,
% 3.00/1.59 ! [R: product_prod_a_a,S: product_prod_a_a,R2: set_Pr1948701895od_a_a,S2: product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R @ S ) @ R2 )
% 3.00/1.59 => ( ( S2 = S )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R @ S2 ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_58_curryI,axiom,
% 3.00/1.59 ! [F: product_prod_a_a > $o,A: a,B2: a] :
% 3.00/1.59 ( ( F @ ( product_Pair_a_a @ A @ B2 ) )
% 3.00/1.59 => ( product_curry_a_a_o @ F @ A @ B2 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_59_curryI,axiom,
% 3.00/1.59 ! [F: produc1572603623od_a_a > $o,A: product_prod_a_a,B2: product_prod_a_a] :
% 3.00/1.59 ( ( F @ ( produc1474507607od_a_a @ A @ B2 ) )
% 3.00/1.59 => ( produc94438221_a_a_o @ F @ A @ B2 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_60_preference_Oindiff__trans,axiom,
% 3.00/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 3.00/1.59 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 3.00/1.59 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 3.00/1.59 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 3.00/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_61_preference_Oindiff__trans,axiom,
% 3.00/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 3.00/1.59 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 3.00/1.59 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 3.00/1.59 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 3.00/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_62_preference_Onot__outside,axiom,
% 3.00/1.59 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 3.00/1.59 ( ( prefer1409946857od_a_a @ Carrier @ Relation )
% 3.00/1.59 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 3.00/1.59 => ( member2057358096od_a_a @ X @ Carrier ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_63_preference_Onot__outside,axiom,
% 3.00/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.00/1.59 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 3.00/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.00/1.59 => ( member_a @ X @ Carrier ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_64_preference_Onot__outside,axiom,
% 3.00/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.00/1.59 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 3.00/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.00/1.59 => ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_65_swap__simp,axiom,
% 3.00/1.59 ! [X: a,Y: a] :
% 3.00/1.59 ( ( product_swap_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 3.00/1.59 = ( product_Pair_a_a @ Y @ X ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_66_swap__simp,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,Y: product_prod_a_a] :
% 3.00/1.59 ( ( produc1641853175od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
% 3.00/1.59 = ( produc1474507607od_a_a @ Y @ X ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_67_curryD,axiom,
% 3.00/1.59 ! [F: product_prod_a_a > $o,A: a,B2: a] :
% 3.00/1.59 ( ( product_curry_a_a_o @ F @ A @ B2 )
% 3.00/1.59 => ( F @ ( product_Pair_a_a @ A @ B2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_68_curryD,axiom,
% 3.00/1.59 ! [F: produc1572603623od_a_a > $o,A: product_prod_a_a,B2: product_prod_a_a] :
% 3.00/1.59 ( ( produc94438221_a_a_o @ F @ A @ B2 )
% 3.00/1.59 => ( F @ ( produc1474507607od_a_a @ A @ B2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_69_curryE,axiom,
% 3.00/1.59 ! [F: product_prod_a_a > $o,A: a,B2: a] :
% 3.00/1.59 ( ( product_curry_a_a_o @ F @ A @ B2 )
% 3.00/1.59 => ( F @ ( product_Pair_a_a @ A @ B2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_70_curryE,axiom,
% 3.00/1.59 ! [F: produc1572603623od_a_a > $o,A: product_prod_a_a,B2: product_prod_a_a] :
% 3.00/1.59 ( ( produc94438221_a_a_o @ F @ A @ B2 )
% 3.00/1.59 => ( F @ ( produc1474507607od_a_a @ A @ B2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_71_no__better__than__def,axiom,
% 3.00/1.59 ( prefer926864306od_a_a
% 3.00/1.59 = ( ^ [X4: produc1572603623od_a_a,B5: set_Pr1948701895od_a_a,P3: set_Pr1295299783od_a_a] :
% 3.00/1.59 ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [Y4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ Y4 @ B5 )
% 3.00/1.59 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_72_no__better__than__def,axiom,
% 3.00/1.59 ( prefer1676310729than_a
% 3.00/1.59 = ( ^ [X4: a,B5: set_a,P3: set_Product_prod_a_a] :
% 3.00/1.59 ( collect_a
% 3.00/1.59 @ ^ [Y4: a] :
% 3.00/1.59 ( ( member_a @ Y4 @ B5 )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_73_no__better__than__def,axiom,
% 3.00/1.59 ( prefer1498087410od_a_a
% 3.00/1.59 = ( ^ [X4: product_prod_a_a,B5: set_Product_prod_a_a,P3: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( collec645855634od_a_a
% 3.00/1.59 @ ^ [Y4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ Y4 @ B5 )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ P3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_74_as__good__as__def,axiom,
% 3.00/1.59 ( prefer1867335523od_a_a
% 3.00/1.59 = ( ^ [X4: produc1572603623od_a_a,B5: set_Pr1948701895od_a_a,P3: set_Pr1295299783od_a_a] :
% 3.00/1.59 ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [Y4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ Y4 @ B5 )
% 3.00/1.59 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ P3 )
% 3.00/1.59 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_75_as__good__as__def,axiom,
% 3.00/1.59 ( prefer436369274d_as_a
% 3.00/1.59 = ( ^ [X4: a,B5: set_a,P3: set_Product_prod_a_a] :
% 3.00/1.59 ( collect_a
% 3.00/1.59 @ ^ [Y4: a] :
% 3.00/1.59 ( ( member_a @ Y4 @ B5 )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ P3 )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_76_as__good__as__def,axiom,
% 3.00/1.59 ( prefer74720675od_a_a
% 3.00/1.59 = ( ^ [X4: product_prod_a_a,B5: set_Product_prod_a_a,P3: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( collec645855634od_a_a
% 3.00/1.59 @ ^ [Y4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ Y4 @ B5 )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ P3 )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ X4 ) @ P3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_77_pred__equals__eq2,axiom,
% 3.00/1.59 ! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 3.00/1.59 ( ( ( ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R2 ) )
% 3.00/1.59 = ( ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ S3 ) ) )
% 3.00/1.59 = ( R2 = S3 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_78_pred__equals__eq2,axiom,
% 3.00/1.59 ! [R2: set_Pr1948701895od_a_a,S3: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( ( ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R2 ) )
% 3.00/1.59 = ( ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ S3 ) ) )
% 3.00/1.59 = ( R2 = S3 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_79_preference__def,axiom,
% 3.00/1.59 ( prefer1409946857od_a_a
% 3.00/1.59 = ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
% 3.00/1.59 ( ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
% 3.00/1.59 => ( member2057358096od_a_a @ X4 @ Carrier2 ) )
% 3.00/1.59 & ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
% 3.00/1.59 => ( member2057358096od_a_a @ Y4 @ Carrier2 ) )
% 3.00/1.59 & ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_80_preference__def,axiom,
% 3.00/1.59 ( prefer1292084992ence_a
% 3.00/1.59 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 3.00/1.59 ( ! [X4: a,Y4: a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
% 3.00/1.59 => ( member_a @ X4 @ Carrier2 ) )
% 3.00/1.59 & ! [X4: a,Y4: a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
% 3.00/1.59 => ( member_a @ Y4 @ Carrier2 ) )
% 3.00/1.59 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_81_preference__def,axiom,
% 3.00/1.59 ( prefer1338892073od_a_a
% 3.00/1.59 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
% 3.00/1.59 => ( member449909584od_a_a @ X4 @ Carrier2 ) )
% 3.00/1.59 & ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
% 3.00/1.59 => ( member449909584od_a_a @ Y4 @ Carrier2 ) )
% 3.00/1.59 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_82_preference_Ointro,axiom,
% 3.00/1.59 ! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
% 3.00/1.59 => ( member2057358096od_a_a @ X3 @ Carrier ) )
% 3.00/1.59 => ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
% 3.00/1.59 => ( member2057358096od_a_a @ Y3 @ Carrier ) )
% 3.00/1.59 => ( ( order_491428350od_a_a @ Carrier @ Relation )
% 3.00/1.59 => ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_83_preference_Ointro,axiom,
% 3.00/1.59 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 3.00/1.59 ( ! [X3: a,Y3: a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
% 3.00/1.59 => ( member_a @ X3 @ Carrier ) )
% 3.00/1.59 => ( ! [X3: a,Y3: a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
% 3.00/1.59 => ( member_a @ Y3 @ Carrier ) )
% 3.00/1.59 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 3.00/1.59 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_84_preference_Ointro,axiom,
% 3.00/1.59 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 3.00/1.59 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
% 3.00/1.59 => ( member449909584od_a_a @ X3 @ Carrier ) )
% 3.00/1.59 => ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
% 3.00/1.59 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 3.00/1.59 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 3.00/1.59 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_85_pair__in__swap__image,axiom,
% 3.00/1.59 ! [Y: a,X: a,A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( image_794773445od_a_a @ product_swap_a_a @ A4 ) )
% 3.00/1.59 = ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_86_pair__in__swap__image,axiom,
% 3.00/1.59 ! [Y: product_prod_a_a,X: product_prod_a_a,A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ ( image_236793797od_a_a @ produc1641853175od_a_a @ A4 ) )
% 3.00/1.59 = ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_87_strict__contour__is__diff,axiom,
% 3.00/1.59 ! [B: set_Pr1948701895od_a_a,Y: produc1572603623od_a_a,Pr: set_Pr1295299783od_a_a] :
% 3.00/1.59 ( ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [A5: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ A5 @ B )
% 3.00/1.59 & ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A5 @ Y ) @ Pr )
% 3.00/1.59 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ A5 ) @ Pr ) ) )
% 3.00/1.59 = ( minus_1034916142od_a_a @ ( prefer55414563od_a_a @ Y @ B @ Pr ) @ ( prefer1867335523od_a_a @ Y @ B @ Pr ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_88_strict__contour__is__diff,axiom,
% 3.00/1.59 ! [B: set_a,Y: a,Pr: set_Product_prod_a_a] :
% 3.00/1.59 ( ( collect_a
% 3.00/1.59 @ ^ [A5: a] :
% 3.00/1.59 ( ( member_a @ A5 @ B )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ A5 @ Y ) @ Pr )
% 3.00/1.59 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ A5 ) @ Pr ) ) )
% 3.00/1.59 = ( minus_minus_set_a @ ( prefer161218362good_a @ Y @ B @ Pr ) @ ( prefer436369274d_as_a @ Y @ B @ Pr ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_89_strict__contour__is__diff,axiom,
% 3.00/1.59 ! [B: set_Product_prod_a_a,Y: product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( collec645855634od_a_a
% 3.00/1.59 @ ^ [A5: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ A5 @ B )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A5 @ Y ) @ Pr )
% 3.00/1.59 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ A5 ) @ Pr ) ) )
% 3.00/1.59 = ( minus_1507102574od_a_a @ ( prefer477097315od_a_a @ Y @ B @ Pr ) @ ( prefer74720675od_a_a @ Y @ B @ Pr ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_90_mem__Collect__eq,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
% 3.00/1.59 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
% 3.00/1.59 = ( P2 @ A ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_91_mem__Collect__eq,axiom,
% 3.00/1.59 ! [A: a,P2: a > $o] :
% 3.00/1.59 ( ( member_a @ A @ ( collect_a @ P2 ) )
% 3.00/1.59 = ( P2 @ A ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_92_mem__Collect__eq,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,P2: produc1572603623od_a_a > $o] :
% 3.00/1.59 ( ( member2057358096od_a_a @ A @ ( collec1635618130od_a_a @ P2 ) )
% 3.00/1.59 = ( P2 @ A ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_93_Collect__mem__eq,axiom,
% 3.00/1.59 ! [A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A4 ) )
% 3.00/1.59 = A4 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_94_Collect__mem__eq,axiom,
% 3.00/1.59 ! [A4: set_a] :
% 3.00/1.59 ( ( collect_a
% 3.00/1.59 @ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
% 3.00/1.59 = A4 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_95_Collect__mem__eq,axiom,
% 3.00/1.59 ! [A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A4 ) )
% 3.00/1.59 = A4 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_96_DiffI,axiom,
% 3.00/1.59 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ C2 @ A4 )
% 3.00/1.59 => ( ~ ( member449909584od_a_a @ C2 @ B )
% 3.00/1.59 => ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_97_DiffI,axiom,
% 3.00/1.59 ! [C2: a,A4: set_a,B: set_a] :
% 3.00/1.59 ( ( member_a @ C2 @ A4 )
% 3.00/1.59 => ( ~ ( member_a @ C2 @ B )
% 3.00/1.59 => ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_98_DiffI,axiom,
% 3.00/1.59 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ C2 @ A4 )
% 3.00/1.59 => ( ~ ( member2057358096od_a_a @ C2 @ B )
% 3.00/1.59 => ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_99_image__eqI,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ B2 @ ( image_794773445od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_100_image__eqI,axiom,
% 3.00/1.59 ! [B2: a,F: product_prod_a_a > a,X: product_prod_a_a,A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( member_a @ B2 @ ( image_340790172_a_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_101_image__eqI,axiom,
% 3.00/1.59 ! [B2: produc1572603623od_a_a,F: product_prod_a_a > produc1572603623od_a_a,X: product_prod_a_a,A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ B2 @ ( image_1765377413od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_102_image__eqI,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,F: a > product_prod_a_a,X: a,A4: set_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ B2 @ ( image_1725938350od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_103_image__eqI,axiom,
% 3.00/1.59 ! [B2: a,F: a > a,X: a,A4: set_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( member_a @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_104_image__eqI,axiom,
% 3.00/1.59 ! [B2: produc1572603623od_a_a,F: a > produc1572603623od_a_a,X: a,A4: set_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ B2 @ ( image_1763923566od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_105_image__eqI,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a,X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ B2 @ ( image_734753285od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_106_image__eqI,axiom,
% 3.00/1.59 ! [B2: a,F: produc1572603623od_a_a > a,X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( member_a @ B2 @ ( image_1105229276_a_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_107_image__eqI,axiom,
% 3.00/1.59 ! [B2: produc1572603623od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a,X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ B2 @ ( image_236793797od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_108_Diff__iff,axiom,
% 3.00/1.59 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member449909584od_a_a @ C2 @ A4 )
% 3.00/1.59 & ~ ( member449909584od_a_a @ C2 @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_109_Diff__iff,axiom,
% 3.00/1.59 ! [C2: a,A4: set_a,B: set_a] :
% 3.00/1.59 ( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member_a @ C2 @ A4 )
% 3.00/1.59 & ~ ( member_a @ C2 @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_110_Diff__iff,axiom,
% 3.00/1.59 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member2057358096od_a_a @ C2 @ A4 )
% 3.00/1.59 & ~ ( member2057358096od_a_a @ C2 @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_111_minus__set__def,axiom,
% 3.00/1.59 ( minus_1507102574od_a_a
% 3.00/1.59 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 3.00/1.59 ( collec645855634od_a_a
% 3.00/1.59 @ ( minus_1797478127_a_a_o
% 3.00/1.59 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A6 )
% 3.00/1.59 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_112_minus__set__def,axiom,
% 3.00/1.59 ( minus_minus_set_a
% 3.00/1.59 = ( ^ [A6: set_a,B5: set_a] :
% 3.00/1.59 ( collect_a
% 3.00/1.59 @ ( minus_minus_a_o
% 3.00/1.59 @ ^ [X4: a] : ( member_a @ X4 @ A6 )
% 3.00/1.59 @ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_113_minus__set__def,axiom,
% 3.00/1.59 ( minus_1034916142od_a_a
% 3.00/1.59 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( collec1635618130od_a_a
% 3.00/1.59 @ ( minus_107310383_a_a_o
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A6 )
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_114_rev__image__eqI,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( member449909584od_a_a @ B2 @ ( image_794773445od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_115_rev__image__eqI,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,B2: a,F: product_prod_a_a > a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( member_a @ B2 @ ( image_340790172_a_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_116_rev__image__eqI,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,B2: produc1572603623od_a_a,F: product_prod_a_a > produc1572603623od_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( member2057358096od_a_a @ B2 @ ( image_1765377413od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_117_rev__image__eqI,axiom,
% 3.00/1.59 ! [X: a,A4: set_a,B2: product_prod_a_a,F: a > product_prod_a_a] :
% 3.00/1.59 ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( member449909584od_a_a @ B2 @ ( image_1725938350od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_118_rev__image__eqI,axiom,
% 3.00/1.59 ! [X: a,A4: set_a,B2: a,F: a > a] :
% 3.00/1.59 ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( member_a @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_119_rev__image__eqI,axiom,
% 3.00/1.59 ! [X: a,A4: set_a,B2: produc1572603623od_a_a,F: a > produc1572603623od_a_a] :
% 3.00/1.59 ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( member2057358096od_a_a @ B2 @ ( image_1763923566od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_120_rev__image__eqI,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( member449909584od_a_a @ B2 @ ( image_734753285od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_121_rev__image__eqI,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: a,F: produc1572603623od_a_a > a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( member_a @ B2 @ ( image_1105229276_a_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_122_rev__image__eqI,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: produc1572603623od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( member2057358096od_a_a @ B2 @ ( image_236793797od_a_a @ F @ A4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_123_imageI,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ ( F @ X ) @ ( image_794773445od_a_a @ F @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_124_imageI,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( member_a @ ( F @ X ) @ ( image_340790172_a_a_a @ F @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_125_imageI,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > produc1572603623od_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( F @ X ) @ ( image_1765377413od_a_a @ F @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_126_imageI,axiom,
% 3.00/1.59 ! [X: a,A4: set_a,F: a > product_prod_a_a] :
% 3.00/1.59 ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ ( F @ X ) @ ( image_1725938350od_a_a @ F @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_127_imageI,axiom,
% 3.00/1.59 ! [X: a,A4: set_a,F: a > a] :
% 3.00/1.59 ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_128_imageI,axiom,
% 3.00/1.59 ! [X: a,A4: set_a,F: a > produc1572603623od_a_a] :
% 3.00/1.59 ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( F @ X ) @ ( image_1763923566od_a_a @ F @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_129_imageI,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ ( F @ X ) @ ( image_734753285od_a_a @ F @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_130_imageI,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( member_a @ ( F @ X ) @ ( image_1105229276_a_a_a @ F @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_131_imageI,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( F @ X ) @ ( image_236793797od_a_a @ F @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_132_DiffD2,axiom,
% 3.00/1.59 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( member449909584od_a_a @ C2 @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_133_DiffD2,axiom,
% 3.00/1.59 ! [C2: a,A4: set_a,B: set_a] :
% 3.00/1.59 ( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( member_a @ C2 @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_134_DiffD2,axiom,
% 3.00/1.59 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( member2057358096od_a_a @ C2 @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_135_DiffD1,axiom,
% 3.00/1.59 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
% 3.00/1.59 => ( member449909584od_a_a @ C2 @ A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_136_DiffD1,axiom,
% 3.00/1.59 ! [C2: a,A4: set_a,B: set_a] :
% 3.00/1.59 ( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
% 3.00/1.59 => ( member_a @ C2 @ A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_137_DiffD1,axiom,
% 3.00/1.59 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
% 3.00/1.59 => ( member2057358096od_a_a @ C2 @ A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_138_DiffE,axiom,
% 3.00/1.59 ! [C2: product_prod_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ C2 @ ( minus_1507102574od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member449909584od_a_a @ C2 @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ C2 @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_139_DiffE,axiom,
% 3.00/1.59 ! [C2: a,A4: set_a,B: set_a] :
% 3.00/1.59 ( ( member_a @ C2 @ ( minus_minus_set_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member_a @ C2 @ A4 )
% 3.00/1.59 => ( member_a @ C2 @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_140_DiffE,axiom,
% 3.00/1.59 ! [C2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ C2 @ ( minus_1034916142od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member2057358096od_a_a @ C2 @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ C2 @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_141_imageE,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ B2 @ ( image_794773445od_a_a @ F @ A4 ) )
% 3.00/1.59 => ~ ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X3 ) )
% 3.00/1.59 => ~ ( member449909584od_a_a @ X3 @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_142_imageE,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,F: a > product_prod_a_a,A4: set_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ B2 @ ( image_1725938350od_a_a @ F @ A4 ) )
% 3.00/1.59 => ~ ! [X3: a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X3 ) )
% 3.00/1.59 => ~ ( member_a @ X3 @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_143_imageE,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a,A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ B2 @ ( image_734753285od_a_a @ F @ A4 ) )
% 3.00/1.59 => ~ ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X3 ) )
% 3.00/1.59 => ~ ( member2057358096od_a_a @ X3 @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_144_imageE,axiom,
% 3.00/1.59 ! [B2: a,F: product_prod_a_a > a,A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member_a @ B2 @ ( image_340790172_a_a_a @ F @ A4 ) )
% 3.00/1.59 => ~ ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X3 ) )
% 3.00/1.59 => ~ ( member449909584od_a_a @ X3 @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_145_imageE,axiom,
% 3.00/1.59 ! [B2: a,F: a > a,A4: set_a] :
% 3.00/1.59 ( ( member_a @ B2 @ ( image_a_a @ F @ A4 ) )
% 3.00/1.59 => ~ ! [X3: a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X3 ) )
% 3.00/1.59 => ~ ( member_a @ X3 @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_146_imageE,axiom,
% 3.00/1.59 ! [B2: a,F: produc1572603623od_a_a > a,A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member_a @ B2 @ ( image_1105229276_a_a_a @ F @ A4 ) )
% 3.00/1.59 => ~ ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X3 ) )
% 3.00/1.59 => ~ ( member2057358096od_a_a @ X3 @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_147_imageE,axiom,
% 3.00/1.59 ! [B2: produc1572603623od_a_a,F: product_prod_a_a > produc1572603623od_a_a,A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ B2 @ ( image_1765377413od_a_a @ F @ A4 ) )
% 3.00/1.59 => ~ ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X3 ) )
% 3.00/1.59 => ~ ( member449909584od_a_a @ X3 @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_148_imageE,axiom,
% 3.00/1.59 ! [B2: produc1572603623od_a_a,F: a > produc1572603623od_a_a,A4: set_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ B2 @ ( image_1763923566od_a_a @ F @ A4 ) )
% 3.00/1.59 => ~ ! [X3: a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X3 ) )
% 3.00/1.59 => ~ ( member_a @ X3 @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_149_imageE,axiom,
% 3.00/1.59 ! [B2: produc1572603623od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a,A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ B2 @ ( image_236793797od_a_a @ F @ A4 ) )
% 3.00/1.59 => ~ ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X3 ) )
% 3.00/1.59 => ~ ( member2057358096od_a_a @ X3 @ A4 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_150_Compr__image__eq,axiom,
% 3.00/1.59 ! [F: product_prod_a_a > product_prod_a_a,A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 3.00/1.59 ( ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ ( image_794773445od_a_a @ F @ A4 ) )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 = ( image_794773445od_a_a @ F
% 3.00/1.59 @ ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ A4 )
% 3.00/1.59 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_151_Compr__image__eq,axiom,
% 3.00/1.59 ! [F: a > product_prod_a_a,A4: set_a,P2: product_prod_a_a > $o] :
% 3.00/1.59 ( ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ ( image_1725938350od_a_a @ F @ A4 ) )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 = ( image_1725938350od_a_a @ F
% 3.00/1.59 @ ( collect_a
% 3.00/1.59 @ ^ [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ A4 )
% 3.00/1.59 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_152_Compr__image__eq,axiom,
% 3.00/1.59 ! [F: produc1572603623od_a_a > product_prod_a_a,A4: set_Pr1948701895od_a_a,P2: product_prod_a_a > $o] :
% 3.00/1.59 ( ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ ( image_734753285od_a_a @ F @ A4 ) )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 = ( image_734753285od_a_a @ F
% 3.00/1.59 @ ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ A4 )
% 3.00/1.59 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_153_Compr__image__eq,axiom,
% 3.00/1.59 ! [F: product_prod_a_a > a,A4: set_Product_prod_a_a,P2: a > $o] :
% 3.00/1.59 ( ( collect_a
% 3.00/1.59 @ ^ [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ ( image_340790172_a_a_a @ F @ A4 ) )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 = ( image_340790172_a_a_a @ F
% 3.00/1.59 @ ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ A4 )
% 3.00/1.59 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_154_Compr__image__eq,axiom,
% 3.00/1.59 ! [F: a > a,A4: set_a,P2: a > $o] :
% 3.00/1.59 ( ( collect_a
% 3.00/1.59 @ ^ [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ ( image_a_a @ F @ A4 ) )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 = ( image_a_a @ F
% 3.00/1.59 @ ( collect_a
% 3.00/1.59 @ ^ [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ A4 )
% 3.00/1.59 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_155_Compr__image__eq,axiom,
% 3.00/1.59 ! [F: produc1572603623od_a_a > a,A4: set_Pr1948701895od_a_a,P2: a > $o] :
% 3.00/1.59 ( ( collect_a
% 3.00/1.59 @ ^ [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ ( image_1105229276_a_a_a @ F @ A4 ) )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 = ( image_1105229276_a_a_a @ F
% 3.00/1.59 @ ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ A4 )
% 3.00/1.59 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_156_Compr__image__eq,axiom,
% 3.00/1.59 ! [F: product_prod_a_a > produc1572603623od_a_a,A4: set_Product_prod_a_a,P2: produc1572603623od_a_a > $o] :
% 3.00/1.59 ( ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ ( image_1765377413od_a_a @ F @ A4 ) )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 = ( image_1765377413od_a_a @ F
% 3.00/1.59 @ ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ A4 )
% 3.00/1.59 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_157_Compr__image__eq,axiom,
% 3.00/1.59 ! [F: a > produc1572603623od_a_a,A4: set_a,P2: produc1572603623od_a_a > $o] :
% 3.00/1.59 ( ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ ( image_1763923566od_a_a @ F @ A4 ) )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 = ( image_1763923566od_a_a @ F
% 3.00/1.59 @ ( collect_a
% 3.00/1.59 @ ^ [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ A4 )
% 3.00/1.59 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_158_Compr__image__eq,axiom,
% 3.00/1.59 ! [F: produc1572603623od_a_a > produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 3.00/1.59 ( ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ ( image_236793797od_a_a @ F @ A4 ) )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 = ( image_236793797od_a_a @ F
% 3.00/1.59 @ ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ A4 )
% 3.00/1.59 & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_159_set__diff__eq,axiom,
% 3.00/1.59 ( minus_1507102574od_a_a
% 3.00/1.59 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 3.00/1.59 ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ A6 )
% 3.00/1.59 & ~ ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_160_set__diff__eq,axiom,
% 3.00/1.59 ( minus_minus_set_a
% 3.00/1.59 = ( ^ [A6: set_a,B5: set_a] :
% 3.00/1.59 ( collect_a
% 3.00/1.59 @ ^ [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ A6 )
% 3.00/1.59 & ~ ( member_a @ X4 @ B5 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_161_set__diff__eq,axiom,
% 3.00/1.59 ( minus_1034916142od_a_a
% 3.00/1.59 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ A6 )
% 3.00/1.59 & ~ ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_162_UNIV__I,axiom,
% 3.00/1.59 ! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ top_to398455383od_a_a ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_163_UNIV__I,axiom,
% 3.00/1.59 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_164_UNIV__I,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a] : ( member2057358096od_a_a @ X @ top_to52711959od_a_a ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_165_UNIV__witness,axiom,
% 3.00/1.59 ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ top_to398455383od_a_a ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_166_UNIV__witness,axiom,
% 3.00/1.59 ? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_167_UNIV__witness,axiom,
% 3.00/1.59 ? [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ top_to52711959od_a_a ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_168_UNIV__eq__I,axiom,
% 3.00/1.59 ! [A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ! [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( top_to398455383od_a_a = A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_169_UNIV__eq__I,axiom,
% 3.00/1.59 ! [A4: set_a] :
% 3.00/1.59 ( ! [X3: a] : ( member_a @ X3 @ A4 )
% 3.00/1.59 => ( top_top_set_a = A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_170_UNIV__eq__I,axiom,
% 3.00/1.59 ! [A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ! [X3: produc1572603623od_a_a] : ( member2057358096od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( top_to52711959od_a_a = A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_171_iso__tuple__UNIV__I,axiom,
% 3.00/1.59 ! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ top_to398455383od_a_a ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_172_iso__tuple__UNIV__I,axiom,
% 3.00/1.59 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_173_iso__tuple__UNIV__I,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a] : ( member2057358096od_a_a @ X @ top_to52711959od_a_a ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_174_same__fstI,axiom,
% 3.00/1.59 ! [P2: a > $o,X: a,Y5: a,Y: a,R2: a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( P2 @ X )
% 3.00/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y5 @ Y ) @ ( R2 @ X ) )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ ( product_Pair_a_a @ X @ Y ) ) @ ( same_fst_a_a @ P2 @ R2 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_175_same__fstI,axiom,
% 3.00/1.59 ! [P2: product_prod_a_a > $o,X: product_prod_a_a,Y5: product_prod_a_a,Y: product_prod_a_a,R2: product_prod_a_a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( P2 @ X )
% 3.00/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y5 @ Y ) @ ( R2 @ X ) )
% 3.00/1.59 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ ( produc1474507607od_a_a @ X @ Y5 ) @ ( produc1474507607od_a_a @ X @ Y ) ) @ ( same_f1138477100od_a_a @ P2 @ R2 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_176_map__prod__simp,axiom,
% 3.00/1.59 ! [F: a > a,G: a > a,A: a,B2: a] :
% 3.00/1.59 ( ( produc700022774_a_a_a @ F @ G @ ( product_Pair_a_a @ A @ B2 ) )
% 3.00/1.59 = ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_177_map__prod__simp,axiom,
% 3.00/1.59 ! [F: a > product_prod_a_a,G: a > product_prod_a_a,A: a,B2: a] :
% 3.00/1.59 ( ( produc2012818120od_a_a @ F @ G @ ( product_Pair_a_a @ A @ B2 ) )
% 3.00/1.59 = ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_178_map__prod__simp,axiom,
% 3.00/1.59 ! [F: product_prod_a_a > a,G: product_prod_a_a > a,A: product_prod_a_a,B2: product_prod_a_a] :
% 3.00/1.59 ( ( produc363290532_a_a_a @ F @ G @ ( produc1474507607od_a_a @ A @ B2 ) )
% 3.00/1.59 = ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_179_map__prod__simp,axiom,
% 3.00/1.59 ! [F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a,A: product_prod_a_a,B2: product_prod_a_a] :
% 3.00/1.59 ( ( produc1779756150od_a_a @ F @ G @ ( produc1474507607od_a_a @ A @ B2 ) )
% 3.00/1.59 = ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_180_map__prod__imageI,axiom,
% 3.00/1.59 ! [A: a,B2: a,R2: set_Product_prod_a_a,F: a > a,G: a > a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ R2 )
% 3.00/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F @ G ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_181_map__prod__imageI,axiom,
% 3.00/1.59 ! [A: a,B2: a,R2: set_Product_prod_a_a,F: a > product_prod_a_a,G: a > product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ R2 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_1765377413od_a_a @ ( produc2012818120od_a_a @ F @ G ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_182_map__prod__imageI,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: product_prod_a_a,R2: set_Pr1948701895od_a_a,F: product_prod_a_a > a,G: product_prod_a_a > a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ R2 )
% 3.00/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_734753285od_a_a @ ( produc363290532_a_a_a @ F @ G ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_183_map__prod__imageI,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: product_prod_a_a,R2: set_Pr1948701895od_a_a,F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ R2 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_236793797od_a_a @ ( produc1779756150od_a_a @ F @ G ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_184_top__empty__eq,axiom,
% 3.00/1.59 ( top_to1483957574_a_a_o
% 3.00/1.59 = ( ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ top_to398455383od_a_a ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_185_top__empty__eq,axiom,
% 3.00/1.59 ( top_top_a_o
% 3.00/1.59 = ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_186_top__empty__eq,axiom,
% 3.00/1.59 ( top_to2011536774_a_a_o
% 3.00/1.59 = ( ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ top_to52711959od_a_a ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_187_fst__eqD,axiom,
% 3.00/1.59 ! [X: a,Y: a,A: a] :
% 3.00/1.59 ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 3.00/1.59 = A )
% 3.00/1.59 => ( X = A ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_188_fst__eqD,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,Y: product_prod_a_a,A: product_prod_a_a] :
% 3.00/1.59 ( ( ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
% 3.00/1.59 = A )
% 3.00/1.59 => ( X = A ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_189_fst__conv,axiom,
% 3.00/1.59 ! [X1: a,X2: a] :
% 3.00/1.59 ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
% 3.00/1.59 = X1 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_190_fst__conv,axiom,
% 3.00/1.59 ! [X1: product_prod_a_a,X2: product_prod_a_a] :
% 3.00/1.59 ( ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ X1 @ X2 ) )
% 3.00/1.59 = X1 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_191_prod__fun__imageE,axiom,
% 3.00/1.59 ! [C2: product_prod_a_a,F: a > a,G: a > a,R2: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ C2 @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F @ G ) @ R2 ) )
% 3.00/1.59 => ~ ! [X3: a,Y3: a] :
% 3.00/1.59 ( ( C2
% 3.00/1.59 = ( product_Pair_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 3.00/1.59 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_192_prod__fun__imageE,axiom,
% 3.00/1.59 ! [C2: product_prod_a_a,F: product_prod_a_a > a,G: product_prod_a_a > a,R2: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ C2 @ ( image_734753285od_a_a @ ( produc363290532_a_a_a @ F @ G ) @ R2 ) )
% 3.00/1.59 => ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 3.00/1.59 ( ( C2
% 3.00/1.59 = ( product_Pair_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 3.00/1.59 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_193_prod__fun__imageE,axiom,
% 3.00/1.59 ! [C2: produc1572603623od_a_a,F: a > product_prod_a_a,G: a > product_prod_a_a,R2: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ C2 @ ( image_1765377413od_a_a @ ( produc2012818120od_a_a @ F @ G ) @ R2 ) )
% 3.00/1.59 => ~ ! [X3: a,Y3: a] :
% 3.00/1.59 ( ( C2
% 3.00/1.59 = ( produc1474507607od_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 3.00/1.59 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_194_prod__fun__imageE,axiom,
% 3.00/1.59 ! [C2: produc1572603623od_a_a,F: product_prod_a_a > product_prod_a_a,G: product_prod_a_a > product_prod_a_a,R2: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ C2 @ ( image_236793797od_a_a @ ( produc1779756150od_a_a @ F @ G ) @ R2 ) )
% 3.00/1.59 => ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 3.00/1.59 ( ( C2
% 3.00/1.59 = ( produc1474507607od_a_a @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 3.00/1.59 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_195_top__empty__eq2,axiom,
% 3.00/1.59 ( top_top_a_a_o
% 3.00/1.59 = ( ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ top_to398455383od_a_a ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_196_top__empty__eq2,axiom,
% 3.00/1.59 ( top_to993969022_a_a_o
% 3.00/1.59 = ( ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ top_to52711959od_a_a ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_197_eq__fst__iff,axiom,
% 3.00/1.59 ! [A: a,P: product_prod_a_a] :
% 3.00/1.59 ( ( A
% 3.00/1.59 = ( product_fst_a_a @ P ) )
% 3.00/1.59 = ( ? [B6: a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( product_Pair_a_a @ A @ B6 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_198_eq__fst__iff,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,P: produc1572603623od_a_a] :
% 3.00/1.59 ( ( A
% 3.00/1.59 = ( produc2115494211od_a_a @ P ) )
% 3.00/1.59 = ( ? [B6: product_prod_a_a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( produc1474507607od_a_a @ A @ B6 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_199_fstI,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,Y: a,Z: a] :
% 3.00/1.59 ( ( X
% 3.00/1.59 = ( product_Pair_a_a @ Y @ Z ) )
% 3.00/1.59 => ( ( product_fst_a_a @ X )
% 3.00/1.59 = Y ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_200_fstI,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 3.00/1.59 ( ( X
% 3.00/1.59 = ( produc1474507607od_a_a @ Y @ Z ) )
% 3.00/1.59 => ( ( produc2115494211od_a_a @ X )
% 3.00/1.59 = Y ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_201_prod_Ocollapse,axiom,
% 3.00/1.59 ! [Prod: product_prod_a_a] :
% 3.00/1.59 ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
% 3.00/1.59 = Prod ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_202_prod_Ocollapse,axiom,
% 3.00/1.59 ! [Prod: produc1572603623od_a_a] :
% 3.00/1.59 ( ( produc1474507607od_a_a @ ( produc2115494211od_a_a @ Prod ) @ ( produc368344197od_a_a @ Prod ) )
% 3.00/1.59 = Prod ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_203_eq__snd__iff,axiom,
% 3.00/1.59 ! [B2: a,P: product_prod_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( product_snd_a_a @ P ) )
% 3.00/1.59 = ( ? [A5: a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( product_Pair_a_a @ A5 @ B2 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_204_eq__snd__iff,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,P: produc1572603623od_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( produc368344197od_a_a @ P ) )
% 3.00/1.59 = ( ? [A5: product_prod_a_a] :
% 3.00/1.59 ( P
% 3.00/1.59 = ( produc1474507607od_a_a @ A5 @ B2 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_205_sndI,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,Y: a,Z: a] :
% 3.00/1.59 ( ( X
% 3.00/1.59 = ( product_Pair_a_a @ Y @ Z ) )
% 3.00/1.59 => ( ( product_snd_a_a @ X )
% 3.00/1.59 = Z ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_206_sndI,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 3.00/1.59 ( ( X
% 3.00/1.59 = ( produc1474507607od_a_a @ Y @ Z ) )
% 3.00/1.59 => ( ( produc368344197od_a_a @ X )
% 3.00/1.59 = Z ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_207_snd__eqD,axiom,
% 3.00/1.59 ! [X: a,Y: a,A: a] :
% 3.00/1.59 ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 3.00/1.59 = A )
% 3.00/1.59 => ( Y = A ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_208_snd__eqD,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,Y: product_prod_a_a,A: product_prod_a_a] :
% 3.00/1.59 ( ( ( produc368344197od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
% 3.00/1.59 = A )
% 3.00/1.59 => ( Y = A ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_209_snd__conv,axiom,
% 3.00/1.59 ! [X1: a,X2: a] :
% 3.00/1.59 ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
% 3.00/1.59 = X2 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_210_snd__conv,axiom,
% 3.00/1.59 ! [X1: product_prod_a_a,X2: product_prod_a_a] :
% 3.00/1.59 ( ( produc368344197od_a_a @ ( produc1474507607od_a_a @ X1 @ X2 ) )
% 3.00/1.59 = X2 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_211_surjective__pairing,axiom,
% 3.00/1.59 ! [T: product_prod_a_a] :
% 3.00/1.59 ( T
% 3.00/1.59 = ( product_Pair_a_a @ ( product_fst_a_a @ T ) @ ( product_snd_a_a @ T ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_212_surjective__pairing,axiom,
% 3.00/1.59 ! [T: produc1572603623od_a_a] :
% 3.00/1.59 ( T
% 3.00/1.59 = ( produc1474507607od_a_a @ ( produc2115494211od_a_a @ T ) @ ( produc368344197od_a_a @ T ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_213_prod_Oexhaust__sel,axiom,
% 3.00/1.59 ! [Prod: product_prod_a_a] :
% 3.00/1.59 ( Prod
% 3.00/1.59 = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_214_prod_Oexhaust__sel,axiom,
% 3.00/1.59 ! [Prod: produc1572603623od_a_a] :
% 3.00/1.59 ( Prod
% 3.00/1.59 = ( produc1474507607od_a_a @ ( produc2115494211od_a_a @ Prod ) @ ( produc368344197od_a_a @ Prod ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_215_prod_Oswap__def,axiom,
% 3.00/1.59 ( product_swap_a_a
% 3.00/1.59 = ( ^ [P4: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P4 ) @ ( product_fst_a_a @ P4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_216_prod_Oswap__def,axiom,
% 3.00/1.59 ( produc1641853175od_a_a
% 3.00/1.59 = ( ^ [P4: produc1572603623od_a_a] : ( produc1474507607od_a_a @ ( produc368344197od_a_a @ P4 ) @ ( produc2115494211od_a_a @ P4 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_217_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 3.00/1.59 ! [P2: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
% 3.00/1.59 ( ( P2 @ X @ Y )
% 3.00/1.59 => ( ( A
% 3.00/1.59 = ( product_Pair_a_a @ X @ Y ) )
% 3.00/1.59 => ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_218_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 3.00/1.59 ! [P2: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a,Y: product_prod_a_a,A: produc1572603623od_a_a] :
% 3.00/1.59 ( ( P2 @ X @ Y )
% 3.00/1.59 => ( ( A
% 3.00/1.59 = ( produc1474507607od_a_a @ X @ Y ) )
% 3.00/1.59 => ( P2 @ ( produc2115494211od_a_a @ A ) @ ( produc368344197od_a_a @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_219_conjI__realizer,axiom,
% 3.00/1.59 ! [P2: a > $o,P: a,Q: a > $o,Q2: a] :
% 3.00/1.59 ( ( P2 @ P )
% 3.00/1.59 => ( ( Q @ Q2 )
% 3.00/1.59 => ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) )
% 3.00/1.59 & ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_220_conjI__realizer,axiom,
% 3.00/1.59 ! [P2: product_prod_a_a > $o,P: product_prod_a_a,Q: product_prod_a_a > $o,Q2: product_prod_a_a] :
% 3.00/1.59 ( ( P2 @ P )
% 3.00/1.59 => ( ( Q @ Q2 )
% 3.00/1.59 => ( ( P2 @ ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ P @ Q2 ) ) )
% 3.00/1.59 & ( Q @ ( produc368344197od_a_a @ ( produc1474507607od_a_a @ P @ Q2 ) ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_221_exI__realizer,axiom,
% 3.00/1.59 ! [P2: a > a > $o,Y: a,X: a] :
% 3.00/1.59 ( ( P2 @ Y @ X )
% 3.00/1.59 => ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_222_exI__realizer,axiom,
% 3.00/1.59 ! [P2: product_prod_a_a > product_prod_a_a > $o,Y: product_prod_a_a,X: product_prod_a_a] :
% 3.00/1.59 ( ( P2 @ Y @ X )
% 3.00/1.59 => ( P2 @ ( produc368344197od_a_a @ ( produc1474507607od_a_a @ X @ Y ) ) @ ( produc2115494211od_a_a @ ( produc1474507607od_a_a @ X @ Y ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_223_image2__eqI,axiom,
% 3.00/1.59 ! [B2: a,F: product_prod_a_a > a,X: product_prod_a_a,C2: a,G: product_prod_a_a > a,A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( C2
% 3.00/1.59 = ( G @ X ) )
% 3.00/1.59 => ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ C2 ) @ ( bNF_Gr1853143117_a_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_224_image2__eqI,axiom,
% 3.00/1.59 ! [B2: a,F: a > a,X: a,C2: a,G: a > a,A4: set_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( C2
% 3.00/1.59 = ( G @ X ) )
% 3.00/1.59 => ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ C2 ) @ ( bNF_Gr423046308_a_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_225_image2__eqI,axiom,
% 3.00/1.59 ! [B2: a,F: produc1572603623od_a_a > a,X: produc1572603623od_a_a,C2: a,G: produc1572603623od_a_a > a,A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( C2
% 3.00/1.59 = ( G @ X ) )
% 3.00/1.59 => ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ C2 ) @ ( bNF_Gr356189709_a_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_226_image2__eqI,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,C2: product_prod_a_a,G: product_prod_a_a > product_prod_a_a,A4: set_Product_prod_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( C2
% 3.00/1.59 = ( G @ X ) )
% 3.00/1.59 => ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ C2 ) @ ( bNF_Gr200450189od_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_227_image2__eqI,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,F: a > product_prod_a_a,X: a,C2: product_prod_a_a,G: a > product_prod_a_a,A4: set_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( C2
% 3.00/1.59 = ( G @ X ) )
% 3.00/1.59 => ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ C2 ) @ ( bNF_Gr1715025636od_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_228_image2__eqI,axiom,
% 3.00/1.59 ! [B2: product_prod_a_a,F: produc1572603623od_a_a > product_prod_a_a,X: produc1572603623od_a_a,C2: product_prod_a_a,G: produc1572603623od_a_a > product_prod_a_a,A4: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( B2
% 3.00/1.59 = ( F @ X ) )
% 3.00/1.59 => ( ( C2
% 3.00/1.59 = ( G @ X ) )
% 3.00/1.59 => ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ C2 ) @ ( bNF_Gr533818957od_a_a @ A4 @ F @ G ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_229_in__lex__prod,axiom,
% 3.00/1.59 ! [A: a,B2: product_prod_a_a,A2: a,B3: product_prod_a_a,R: set_Product_prod_a_a,S: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member1361298640od_a_a @ ( produc563767703od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc1299253312od_a_a @ A2 @ B3 ) ) @ ( lex_pr1321960536od_a_a @ R @ S ) )
% 3.00/1.59 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R )
% 3.00/1.59 | ( ( A = A2 )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_230_in__lex__prod,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: a,A2: product_prod_a_a,B3: a,R: set_Pr1948701895od_a_a,S: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member470802768_a_a_a @ ( produc2054708503_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc2061588782_a_a_a @ A2 @ B3 ) ) @ ( lex_pr2084296006_a_a_a @ R @ S ) )
% 3.00/1.59 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A2 ) @ R )
% 3.00/1.59 | ( ( A = A2 )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_231_in__lex__prod,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: product_prod_a_a,A2: product_prod_a_a,B3: product_prod_a_a,R: set_Pr1948701895od_a_a,S: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc1474507607od_a_a @ A2 @ B3 ) ) @ ( lex_pr942790511od_a_a @ R @ S ) )
% 3.00/1.59 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A2 ) @ R )
% 3.00/1.59 | ( ( A = A2 )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_232_in__lex__prod,axiom,
% 3.00/1.59 ! [A: a,B2: a,A2: a,B3: a,R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Pair_a_a @ A2 @ B3 ) ) @ ( lex_prod_a_a @ R @ S ) )
% 3.00/1.59 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R )
% 3.00/1.59 | ( ( A = A2 )
% 3.00/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ B2 @ B3 ) @ S ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_233_mem__Sigma__iff,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
% 3.00/1.59 ( ( member1403138919_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc1282482655_a_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member449909584od_a_a @ A @ A4 )
% 3.00/1.59 & ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_234_mem__Sigma__iff,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member122808528od_a_a @ ( produc1555606807od_a_a @ A @ B2 ) @ ( produc1227942600od_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member449909584od_a_a @ A @ A4 )
% 3.00/1.59 & ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_235_mem__Sigma__iff,axiom,
% 3.00/1.59 ! [A: a,B2: product_prod_a_a,A4: set_a,B: a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member160619001od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc520147185od_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member_a @ A @ A4 )
% 3.00/1.59 & ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_236_mem__Sigma__iff,axiom,
% 3.00/1.59 ! [A: a,B2: produc1572603623od_a_a,A4: set_a,B: a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member448091577od_a_a @ ( produc1095491584od_a_a @ A @ B2 ) @ ( produc663418545od_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member_a @ A @ A4 )
% 3.00/1.59 & ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_237_mem__Sigma__iff,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,B2: product_prod_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member1545452880od_a_a @ ( produc524982679od_a_a @ A @ B2 ) @ ( produc197318472od_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member2057358096od_a_a @ A @ A4 )
% 3.00/1.59 & ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_238_mem__Sigma__iff,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,B2: a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_a] :
% 3.00/1.59 ( ( member575366567_a_a_a @ ( produc436797294_a_a_a @ A @ B2 ) @ ( produc4724255_a_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member2057358096od_a_a @ A @ A4 )
% 3.00/1.59 & ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_239_mem__Sigma__iff,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,B2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B2 ) @ ( produc1460477704od_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member2057358096od_a_a @ A @ A4 )
% 3.00/1.59 & ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_240_mem__Sigma__iff,axiom,
% 3.00/1.59 ! [A: a,B2: a,A4: set_a,B: a > set_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member_a @ A @ A4 )
% 3.00/1.59 & ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_241_mem__Sigma__iff,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
% 3.00/1.59 = ( ( member449909584od_a_a @ A @ A4 )
% 3.00/1.59 & ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_242_SigmaI,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,A4: set_Product_prod_a_a,B2: a,B: product_prod_a_a > set_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ A @ A4 )
% 3.00/1.59 => ( ( member_a @ B2 @ ( B @ A ) )
% 3.00/1.59 => ( member1403138919_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc1282482655_a_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_243_SigmaI,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,A4: set_Product_prod_a_a,B2: produc1572603623od_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ A @ A4 )
% 3.00/1.59 => ( ( member2057358096od_a_a @ B2 @ ( B @ A ) )
% 3.00/1.59 => ( member122808528od_a_a @ ( produc1555606807od_a_a @ A @ B2 ) @ ( produc1227942600od_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_244_SigmaI,axiom,
% 3.00/1.59 ! [A: a,A4: set_a,B2: product_prod_a_a,B: a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member_a @ A @ A4 )
% 3.00/1.59 => ( ( member449909584od_a_a @ B2 @ ( B @ A ) )
% 3.00/1.59 => ( member160619001od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc520147185od_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_245_SigmaI,axiom,
% 3.00/1.59 ! [A: a,A4: set_a,B2: produc1572603623od_a_a,B: a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member_a @ A @ A4 )
% 3.00/1.59 => ( ( member2057358096od_a_a @ B2 @ ( B @ A ) )
% 3.00/1.59 => ( member448091577od_a_a @ ( produc1095491584od_a_a @ A @ B2 ) @ ( produc663418545od_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_246_SigmaI,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: product_prod_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ A @ A4 )
% 3.00/1.59 => ( ( member449909584od_a_a @ B2 @ ( B @ A ) )
% 3.00/1.59 => ( member1545452880od_a_a @ ( produc524982679od_a_a @ A @ B2 ) @ ( produc197318472od_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_247_SigmaI,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: a,B: produc1572603623od_a_a > set_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ A @ A4 )
% 3.00/1.59 => ( ( member_a @ B2 @ ( B @ A ) )
% 3.00/1.59 => ( member575366567_a_a_a @ ( produc436797294_a_a_a @ A @ B2 ) @ ( produc4724255_a_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_248_SigmaI,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B2: produc1572603623od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ A @ A4 )
% 3.00/1.59 => ( ( member2057358096od_a_a @ B2 @ ( B @ A ) )
% 3.00/1.59 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B2 ) @ ( produc1460477704od_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_249_SigmaI,axiom,
% 3.00/1.59 ! [A: a,A4: set_a,B2: a,B: a > set_a] :
% 3.00/1.59 ( ( member_a @ A @ A4 )
% 3.00/1.59 => ( ( member_a @ B2 @ ( B @ A ) )
% 3.00/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_250_SigmaI,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,A4: set_Product_prod_a_a,B2: product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ A @ A4 )
% 3.00/1.59 => ( ( member449909584od_a_a @ B2 @ ( B @ A ) )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_251_SigmaE2,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
% 3.00/1.59 ( ( member1403138919_a_a_a @ ( produc2061588782_a_a_a @ A @ B2 ) @ ( produc1282482655_a_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member449909584od_a_a @ A @ A4 )
% 3.00/1.59 => ~ ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_252_SigmaE2,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member122808528od_a_a @ ( produc1555606807od_a_a @ A @ B2 ) @ ( produc1227942600od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member449909584od_a_a @ A @ A4 )
% 3.00/1.59 => ~ ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_253_SigmaE2,axiom,
% 3.00/1.59 ! [A: a,B2: product_prod_a_a,A4: set_a,B: a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member160619001od_a_a @ ( produc1299253312od_a_a @ A @ B2 ) @ ( produc520147185od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member_a @ A @ A4 )
% 3.00/1.59 => ~ ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_254_SigmaE2,axiom,
% 3.00/1.59 ! [A: a,B2: produc1572603623od_a_a,A4: set_a,B: a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member448091577od_a_a @ ( produc1095491584od_a_a @ A @ B2 ) @ ( produc663418545od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member_a @ A @ A4 )
% 3.00/1.59 => ~ ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_255_SigmaE2,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,B2: product_prod_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member1545452880od_a_a @ ( produc524982679od_a_a @ A @ B2 ) @ ( produc197318472od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member2057358096od_a_a @ A @ A4 )
% 3.00/1.59 => ~ ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_256_SigmaE2,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,B2: a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_a] :
% 3.00/1.59 ( ( member575366567_a_a_a @ ( produc436797294_a_a_a @ A @ B2 ) @ ( produc4724255_a_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member2057358096od_a_a @ A @ A4 )
% 3.00/1.59 => ~ ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_257_SigmaE2,axiom,
% 3.00/1.59 ! [A: produc1572603623od_a_a,B2: produc1572603623od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B2 ) @ ( produc1460477704od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member2057358096od_a_a @ A @ A4 )
% 3.00/1.59 => ~ ( member2057358096od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_258_SigmaE2,axiom,
% 3.00/1.59 ! [A: a,B2: a,A4: set_a,B: a > set_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member_a @ A @ A4 )
% 3.00/1.59 => ~ ( member_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_259_SigmaE2,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ( ( member449909584od_a_a @ A @ A4 )
% 3.00/1.59 => ~ ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_260_SigmaD2,axiom,
% 3.00/1.59 ! [A: a,B2: a,A4: set_a,B: a > set_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
% 3.00/1.59 => ( member_a @ B2 @ ( B @ A ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_261_SigmaD2,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
% 3.00/1.59 => ( member449909584od_a_a @ B2 @ ( B @ A ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_262_SigmaD1,axiom,
% 3.00/1.59 ! [A: a,B2: a,A4: set_a,B: a > set_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( product_Sigma_a_a @ A4 @ B ) )
% 3.00/1.59 => ( member_a @ A @ A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_263_SigmaD1,axiom,
% 3.00/1.59 ! [A: product_prod_a_a,B2: product_prod_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B2 ) @ ( produc304751368od_a_a @ A4 @ B ) )
% 3.00/1.59 => ( member449909584od_a_a @ A @ A4 ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_264_SigmaE,axiom,
% 3.00/1.59 ! [C2: produc1016684094_a_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_a] :
% 3.00/1.59 ( ( member1403138919_a_a_a @ C2 @ ( produc1282482655_a_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X3 @ A4 )
% 3.00/1.59 => ! [Y3: a] :
% 3.00/1.59 ( ( member_a @ Y3 @ ( B @ X3 ) )
% 3.00/1.59 => ( C2
% 3.00/1.59 != ( produc2061588782_a_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_265_SigmaE,axiom,
% 3.00/1.59 ! [C2: produc772425383od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member122808528od_a_a @ C2 @ ( produc1227942600od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X3 @ A4 )
% 3.00/1.59 => ! [Y3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ Y3 @ ( B @ X3 ) )
% 3.00/1.59 => ( C2
% 3.00/1.59 != ( produc1555606807od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_266_SigmaE,axiom,
% 3.00/1.59 ! [C2: produc1921647824od_a_a,A4: set_a,B: a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member160619001od_a_a @ C2 @ ( produc520147185od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ! [X3: a] :
% 3.00/1.59 ( ( member_a @ X3 @ A4 )
% 3.00/1.59 => ! [Y3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ Y3 @ ( B @ X3 ) )
% 3.00/1.59 => ( C2
% 3.00/1.59 != ( produc1299253312od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_267_SigmaE,axiom,
% 3.00/1.59 ! [C2: produc1555800720od_a_a,A4: set_a,B: a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member448091577od_a_a @ C2 @ ( produc663418545od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ! [X3: a] :
% 3.00/1.59 ( ( member_a @ X3 @ A4 )
% 3.00/1.59 => ! [Y3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ Y3 @ ( B @ X3 ) )
% 3.00/1.59 => ( C2
% 3.00/1.59 != ( produc1095491584od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_268_SigmaE,axiom,
% 3.00/1.59 ! [C2: produc47586087od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member1545452880od_a_a @ C2 @ ( produc197318472od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X3 @ A4 )
% 3.00/1.59 => ! [Y3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ Y3 @ ( B @ X3 ) )
% 3.00/1.59 => ( C2
% 3.00/1.59 != ( produc524982679od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_269_SigmaE,axiom,
% 3.00/1.59 ! [C2: produc1683075710_a_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_a] :
% 3.00/1.59 ( ( member575366567_a_a_a @ C2 @ ( produc4724255_a_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X3 @ A4 )
% 3.00/1.59 => ! [Y3: a] :
% 3.00/1.59 ( ( member_a @ Y3 @ ( B @ X3 ) )
% 3.00/1.59 => ( C2
% 3.00/1.59 != ( produc436797294_a_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_270_SigmaE,axiom,
% 3.00/1.59 ! [C2: produc1990712039od_a_a,A4: set_Pr1948701895od_a_a,B: produc1572603623od_a_a > set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member1899387664od_a_a @ C2 @ ( produc1460477704od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X3 @ A4 )
% 3.00/1.59 => ! [Y3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ Y3 @ ( B @ X3 ) )
% 3.00/1.59 => ( C2
% 3.00/1.59 != ( produc1935643479od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_271_SigmaE,axiom,
% 3.00/1.59 ! [C2: product_prod_a_a,A4: set_a,B: a > set_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ C2 @ ( product_Sigma_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ! [X3: a] :
% 3.00/1.59 ( ( member_a @ X3 @ A4 )
% 3.00/1.59 => ! [Y3: a] :
% 3.00/1.59 ( ( member_a @ Y3 @ ( B @ X3 ) )
% 3.00/1.59 => ( C2
% 3.00/1.59 != ( product_Pair_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_272_SigmaE,axiom,
% 3.00/1.59 ! [C2: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: product_prod_a_a > set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ C2 @ ( produc304751368od_a_a @ A4 @ B ) )
% 3.00/1.59 => ~ ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X3 @ A4 )
% 3.00/1.59 => ! [Y3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ Y3 @ ( B @ X3 ) )
% 3.00/1.59 => ( C2
% 3.00/1.59 != ( produc1474507607od_a_a @ X3 @ Y3 ) ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_273_mem__Times__iff,axiom,
% 3.00/1.59 ! [X: produc1016684094_a_a_a,A4: set_Product_prod_a_a,B: set_a] :
% 3.00/1.59 ( ( member1403138919_a_a_a @ X
% 3.00/1.59 @ ( produc1282482655_a_a_a @ A4
% 3.00/1.59 @ ^ [Uu: product_prod_a_a] : B ) )
% 3.00/1.59 = ( ( member449909584od_a_a @ ( produc1805832986_a_a_a @ X ) @ A4 )
% 3.00/1.59 & ( member_a @ ( produc1954021468_a_a_a @ X ) @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_274_mem__Times__iff,axiom,
% 3.00/1.59 ! [X: produc772425383od_a_a,A4: set_Product_prod_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member122808528od_a_a @ X
% 3.00/1.59 @ ( produc1227942600od_a_a @ A4
% 3.00/1.59 @ ^ [Uu: product_prod_a_a] : B ) )
% 3.00/1.59 = ( ( member449909584od_a_a @ ( produc1775747843od_a_a @ X ) @ A4 )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1055472197od_a_a @ X ) @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_275_mem__Times__iff,axiom,
% 3.00/1.59 ! [X: produc1921647824od_a_a,A4: set_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member160619001od_a_a @ X
% 3.00/1.59 @ ( produc520147185od_a_a @ A4
% 3.00/1.59 @ ^ [Uu: a] : B ) )
% 3.00/1.59 = ( ( member_a @ ( produc1043497516od_a_a @ X ) @ A4 )
% 3.00/1.59 & ( member449909584od_a_a @ ( produc1191685998od_a_a @ X ) @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_276_mem__Times__iff,axiom,
% 3.00/1.59 ! [X: produc1555800720od_a_a,A4: set_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member448091577od_a_a @ X
% 3.00/1.59 @ ( produc663418545od_a_a @ A4
% 3.00/1.59 @ ^ [Uu: a] : B ) )
% 3.00/1.59 = ( ( member_a @ ( produc530563564od_a_a @ X ) @ A4 )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc921029422od_a_a @ X ) @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_277_mem__Times__iff,axiom,
% 3.00/1.59 ! [X: produc47586087od_a_a,A4: set_Pr1948701895od_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member1545452880od_a_a @ X
% 3.00/1.59 @ ( produc197318472od_a_a @ A4
% 3.00/1.59 @ ^ [Uu: produc1572603623od_a_a] : B ) )
% 3.00/1.59 = ( ( member2057358096od_a_a @ ( produc745123715od_a_a @ X ) @ A4 )
% 3.00/1.59 & ( member449909584od_a_a @ ( produc24848069od_a_a @ X ) @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_278_mem__Times__iff,axiom,
% 3.00/1.59 ! [X: produc1683075710_a_a_a,A4: set_Pr1948701895od_a_a,B: set_a] :
% 3.00/1.59 ( ( member575366567_a_a_a @ X
% 3.00/1.59 @ ( produc4724255_a_a_a @ A4
% 3.00/1.59 @ ^ [Uu: produc1572603623od_a_a] : B ) )
% 3.00/1.59 = ( ( member2057358096od_a_a @ ( produc2019352922_a_a_a @ X ) @ A4 )
% 3.00/1.59 & ( member_a @ ( produc262335132_a_a_a @ X ) @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_279_mem__Times__iff,axiom,
% 3.00/1.59 ! [X: produc1990712039od_a_a,A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ( member1899387664od_a_a @ X
% 3.00/1.59 @ ( produc1460477704od_a_a @ A4
% 3.00/1.59 @ ^ [Uu: produc1572603623od_a_a] : B ) )
% 3.00/1.59 = ( ( member2057358096od_a_a @ ( produc303632707od_a_a @ X ) @ A4 )
% 3.00/1.59 & ( member2057358096od_a_a @ ( produc1441227909od_a_a @ X ) @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_280_mem__Times__iff,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,A4: set_a,B: set_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X
% 3.00/1.59 @ ( product_Sigma_a_a @ A4
% 3.00/1.59 @ ^ [Uu: a] : B ) )
% 3.00/1.59 = ( ( member_a @ ( product_fst_a_a @ X ) @ A4 )
% 3.00/1.59 & ( member_a @ ( product_snd_a_a @ X ) @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_281_mem__Times__iff,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X
% 3.00/1.59 @ ( produc304751368od_a_a @ A4
% 3.00/1.59 @ ^ [Uu: product_prod_a_a] : B ) )
% 3.00/1.59 = ( ( member449909584od_a_a @ ( produc2115494211od_a_a @ X ) @ A4 )
% 3.00/1.59 & ( member449909584od_a_a @ ( produc368344197od_a_a @ X ) @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_282_member__product,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,A4: set_a,B: set_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X @ ( product_product_a_a @ A4 @ B ) )
% 3.00/1.59 = ( member449909584od_a_a @ X
% 3.00/1.59 @ ( product_Sigma_a_a @ A4
% 3.00/1.59 @ ^ [Uu: a] : B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_283_member__product,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X @ ( produc905674258od_a_a @ A4 @ B ) )
% 3.00/1.59 = ( member2057358096od_a_a @ X
% 3.00/1.59 @ ( produc304751368od_a_a @ A4
% 3.00/1.59 @ ^ [Uu: product_prod_a_a] : B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_284_subsetI,axiom,
% 3.00/1.59 ! [A4: set_Product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ X3 @ B ) )
% 3.00/1.59 => ( ord_le1824328871od_a_a @ A4 @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_285_subsetI,axiom,
% 3.00/1.59 ! [A4: set_a,B: set_a] :
% 3.00/1.59 ( ! [X3: a] :
% 3.00/1.59 ( ( member_a @ X3 @ A4 )
% 3.00/1.59 => ( member_a @ X3 @ B ) )
% 3.00/1.59 => ( ord_less_eq_set_a @ A4 @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_286_subsetI,axiom,
% 3.00/1.59 ! [A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ X3 @ B ) )
% 3.00/1.59 => ( ord_le456379495od_a_a @ A4 @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_287_image__subsetI,axiom,
% 3.00/1.59 ! [A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ ( F @ X3 ) @ B ) )
% 3.00/1.59 => ( ord_le1824328871od_a_a @ ( image_794773445od_a_a @ F @ A4 ) @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_288_image__subsetI,axiom,
% 3.00/1.59 ! [A4: set_Product_prod_a_a,F: product_prod_a_a > a,B: set_a] :
% 3.00/1.59 ( ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( member_a @ ( F @ X3 ) @ B ) )
% 3.00/1.59 => ( ord_less_eq_set_a @ ( image_340790172_a_a_a @ F @ A4 ) @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_289_image__subsetI,axiom,
% 3.00/1.59 ! [A4: set_Product_prod_a_a,F: product_prod_a_a > produc1572603623od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ! [X3: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( F @ X3 ) @ B ) )
% 3.00/1.59 => ( ord_le456379495od_a_a @ ( image_1765377413od_a_a @ F @ A4 ) @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_290_image__subsetI,axiom,
% 3.00/1.59 ! [A4: set_a,F: a > product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ! [X3: a] :
% 3.00/1.59 ( ( member_a @ X3 @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ ( F @ X3 ) @ B ) )
% 3.00/1.59 => ( ord_le1824328871od_a_a @ ( image_1725938350od_a_a @ F @ A4 ) @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_291_image__subsetI,axiom,
% 3.00/1.59 ! [A4: set_a,F: a > a,B: set_a] :
% 3.00/1.59 ( ! [X3: a] :
% 3.00/1.59 ( ( member_a @ X3 @ A4 )
% 3.00/1.59 => ( member_a @ ( F @ X3 ) @ B ) )
% 3.00/1.59 => ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_292_image__subsetI,axiom,
% 3.00/1.59 ! [A4: set_a,F: a > produc1572603623od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ! [X3: a] :
% 3.00/1.59 ( ( member_a @ X3 @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( F @ X3 ) @ B ) )
% 3.00/1.59 => ( ord_le456379495od_a_a @ ( image_1763923566od_a_a @ F @ A4 ) @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_293_image__subsetI,axiom,
% 3.00/1.59 ! [A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > product_prod_a_a,B: set_Product_prod_a_a] :
% 3.00/1.59 ( ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ ( F @ X3 ) @ B ) )
% 3.00/1.59 => ( ord_le1824328871od_a_a @ ( image_734753285od_a_a @ F @ A4 ) @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_294_image__subsetI,axiom,
% 3.00/1.59 ! [A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > a,B: set_a] :
% 3.00/1.59 ( ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( member_a @ ( F @ X3 ) @ B ) )
% 3.00/1.59 => ( ord_less_eq_set_a @ ( image_1105229276_a_a_a @ F @ A4 ) @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_295_image__subsetI,axiom,
% 3.00/1.59 ! [A4: set_Pr1948701895od_a_a,F: produc1572603623od_a_a > produc1572603623od_a_a,B: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ! [X3: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X3 @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ ( F @ X3 ) @ B ) )
% 3.00/1.59 => ( ord_le456379495od_a_a @ ( image_236793797od_a_a @ F @ A4 ) @ B ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_296_subrelI,axiom,
% 3.00/1.59 ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 3.00/1.59 ( ! [X3: a,Y3: a] :
% 3.00/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
% 3.00/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S ) )
% 3.00/1.59 => ( ord_le1824328871od_a_a @ R @ S ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_297_subrelI,axiom,
% 3.00/1.59 ! [R: set_Pr1948701895od_a_a,S: set_Pr1948701895od_a_a] :
% 3.00/1.59 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R )
% 3.00/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ S ) )
% 3.00/1.59 => ( ord_le456379495od_a_a @ R @ S ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_298_subset__iff,axiom,
% 3.00/1.59 ( ord_le1824328871od_a_a
% 3.00/1.59 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 3.00/1.59 ! [T2: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ T2 @ A6 )
% 3.00/1.59 => ( member449909584od_a_a @ T2 @ B5 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_299_subset__iff,axiom,
% 3.00/1.59 ( ord_less_eq_set_a
% 3.00/1.59 = ( ^ [A6: set_a,B5: set_a] :
% 3.00/1.59 ! [T2: a] :
% 3.00/1.59 ( ( member_a @ T2 @ A6 )
% 3.00/1.59 => ( member_a @ T2 @ B5 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_300_subset__iff,axiom,
% 3.00/1.59 ( ord_le456379495od_a_a
% 3.00/1.59 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 3.00/1.59 ! [T2: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ T2 @ A6 )
% 3.00/1.59 => ( member2057358096od_a_a @ T2 @ B5 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_301_subset__eq,axiom,
% 3.00/1.59 ( ord_le1824328871od_a_a
% 3.00/1.59 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 3.00/1.59 ! [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ A6 )
% 3.00/1.59 => ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_302_subset__eq,axiom,
% 3.00/1.59 ( ord_less_eq_set_a
% 3.00/1.59 = ( ^ [A6: set_a,B5: set_a] :
% 3.00/1.59 ! [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ A6 )
% 3.00/1.59 => ( member_a @ X4 @ B5 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_303_subset__eq,axiom,
% 3.00/1.59 ( ord_le456379495od_a_a
% 3.00/1.59 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 3.00/1.59 ! [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ A6 )
% 3.00/1.59 => ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_304_subsetD,axiom,
% 3.00/1.59 ! [A4: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a] :
% 3.00/1.59 ( ( ord_le1824328871od_a_a @ A4 @ B )
% 3.00/1.59 => ( ( member449909584od_a_a @ C2 @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ C2 @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_305_subsetD,axiom,
% 3.00/1.59 ! [A4: set_a,B: set_a,C2: a] :
% 3.00/1.59 ( ( ord_less_eq_set_a @ A4 @ B )
% 3.00/1.59 => ( ( member_a @ C2 @ A4 )
% 3.00/1.59 => ( member_a @ C2 @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_306_subsetD,axiom,
% 3.00/1.59 ! [A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a,C2: produc1572603623od_a_a] :
% 3.00/1.59 ( ( ord_le456379495od_a_a @ A4 @ B )
% 3.00/1.59 => ( ( member2057358096od_a_a @ C2 @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ C2 @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_307_in__mono,axiom,
% 3.00/1.59 ! [A4: set_Product_prod_a_a,B: set_Product_prod_a_a,X: product_prod_a_a] :
% 3.00/1.59 ( ( ord_le1824328871od_a_a @ A4 @ B )
% 3.00/1.59 => ( ( member449909584od_a_a @ X @ A4 )
% 3.00/1.59 => ( member449909584od_a_a @ X @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_308_in__mono,axiom,
% 3.00/1.59 ! [A4: set_a,B: set_a,X: a] :
% 3.00/1.59 ( ( ord_less_eq_set_a @ A4 @ B )
% 3.00/1.59 => ( ( member_a @ X @ A4 )
% 3.00/1.59 => ( member_a @ X @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_309_in__mono,axiom,
% 3.00/1.59 ! [A4: set_Pr1948701895od_a_a,B: set_Pr1948701895od_a_a,X: produc1572603623od_a_a] :
% 3.00/1.59 ( ( ord_le456379495od_a_a @ A4 @ B )
% 3.00/1.59 => ( ( member2057358096od_a_a @ X @ A4 )
% 3.00/1.59 => ( member2057358096od_a_a @ X @ B ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_310_prop__restrict,axiom,
% 3.00/1.59 ! [X: product_prod_a_a,Z2: set_Product_prod_a_a,X5: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 3.00/1.59 ( ( member449909584od_a_a @ X @ Z2 )
% 3.00/1.59 => ( ( ord_le1824328871od_a_a @ Z2
% 3.00/1.59 @ ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ X5 )
% 3.00/1.59 & ( P2 @ X4 ) ) ) )
% 3.00/1.59 => ( P2 @ X ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_311_prop__restrict,axiom,
% 3.00/1.59 ! [X: a,Z2: set_a,X5: set_a,P2: a > $o] :
% 3.00/1.59 ( ( member_a @ X @ Z2 )
% 3.00/1.59 => ( ( ord_less_eq_set_a @ Z2
% 3.00/1.59 @ ( collect_a
% 3.00/1.59 @ ^ [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ X5 )
% 3.00/1.59 & ( P2 @ X4 ) ) ) )
% 3.00/1.59 => ( P2 @ X ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_312_prop__restrict,axiom,
% 3.00/1.59 ! [X: produc1572603623od_a_a,Z2: set_Pr1948701895od_a_a,X5: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X @ Z2 )
% 3.00/1.59 => ( ( ord_le456379495od_a_a @ Z2
% 3.00/1.59 @ ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ X5 )
% 3.00/1.59 & ( P2 @ X4 ) ) ) )
% 3.00/1.59 => ( P2 @ X ) ) ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_313_Collect__restrict,axiom,
% 3.00/1.59 ! [X5: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 3.00/1.59 ( ord_le1824328871od_a_a
% 3.00/1.59 @ ( collec645855634od_a_a
% 3.00/1.59 @ ^ [X4: product_prod_a_a] :
% 3.00/1.59 ( ( member449909584od_a_a @ X4 @ X5 )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 @ X5 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_314_Collect__restrict,axiom,
% 3.00/1.59 ! [X5: set_a,P2: a > $o] :
% 3.00/1.59 ( ord_less_eq_set_a
% 3.00/1.59 @ ( collect_a
% 3.00/1.59 @ ^ [X4: a] :
% 3.00/1.59 ( ( member_a @ X4 @ X5 )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 @ X5 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_315_Collect__restrict,axiom,
% 3.00/1.59 ! [X5: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 3.00/1.59 ( ord_le456379495od_a_a
% 3.00/1.59 @ ( collec1635618130od_a_a
% 3.00/1.59 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.59 ( ( member2057358096od_a_a @ X4 @ X5 )
% 3.00/1.59 & ( P2 @ X4 ) ) )
% 3.00/1.59 @ X5 ) ).
% 3.00/1.59
% 3.00/1.59 thf(fact_316_Collect__subset,axiom,
% 3.00/1.59 ! [A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 3.00/1.60 ( ord_le1824328871od_a_a
% 3.00/1.60 @ ( collec645855634od_a_a
% 3.00/1.60 @ ^ [X4: product_prod_a_a] :
% 3.00/1.60 ( ( member449909584od_a_a @ X4 @ A4 )
% 3.00/1.60 & ( P2 @ X4 ) ) )
% 3.00/1.60 @ A4 ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_317_Collect__subset,axiom,
% 3.00/1.60 ! [A4: set_a,P2: a > $o] :
% 3.00/1.60 ( ord_less_eq_set_a
% 3.00/1.60 @ ( collect_a
% 3.00/1.60 @ ^ [X4: a] :
% 3.00/1.60 ( ( member_a @ X4 @ A4 )
% 3.00/1.60 & ( P2 @ X4 ) ) )
% 3.00/1.60 @ A4 ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_318_Collect__subset,axiom,
% 3.00/1.60 ! [A4: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 3.00/1.60 ( ord_le456379495od_a_a
% 3.00/1.60 @ ( collec1635618130od_a_a
% 3.00/1.60 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.60 ( ( member2057358096od_a_a @ X4 @ A4 )
% 3.00/1.60 & ( P2 @ X4 ) ) )
% 3.00/1.60 @ A4 ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_319_subset__Collect__iff,axiom,
% 3.00/1.60 ! [B: set_Product_prod_a_a,A4: set_Product_prod_a_a,P2: product_prod_a_a > $o] :
% 3.00/1.60 ( ( ord_le1824328871od_a_a @ B @ A4 )
% 3.00/1.60 => ( ( ord_le1824328871od_a_a @ B
% 3.00/1.60 @ ( collec645855634od_a_a
% 3.00/1.60 @ ^ [X4: product_prod_a_a] :
% 3.00/1.60 ( ( member449909584od_a_a @ X4 @ A4 )
% 3.00/1.60 & ( P2 @ X4 ) ) ) )
% 3.00/1.60 = ( ! [X4: product_prod_a_a] :
% 3.00/1.60 ( ( member449909584od_a_a @ X4 @ B )
% 3.00/1.60 => ( P2 @ X4 ) ) ) ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_320_subset__Collect__iff,axiom,
% 3.00/1.60 ! [B: set_a,A4: set_a,P2: a > $o] :
% 3.00/1.60 ( ( ord_less_eq_set_a @ B @ A4 )
% 3.00/1.60 => ( ( ord_less_eq_set_a @ B
% 3.00/1.60 @ ( collect_a
% 3.00/1.60 @ ^ [X4: a] :
% 3.00/1.60 ( ( member_a @ X4 @ A4 )
% 3.00/1.60 & ( P2 @ X4 ) ) ) )
% 3.00/1.60 = ( ! [X4: a] :
% 3.00/1.60 ( ( member_a @ X4 @ B )
% 3.00/1.60 => ( P2 @ X4 ) ) ) ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_321_subset__Collect__iff,axiom,
% 3.00/1.60 ! [B: set_Pr1948701895od_a_a,A4: set_Pr1948701895od_a_a,P2: produc1572603623od_a_a > $o] :
% 3.00/1.60 ( ( ord_le456379495od_a_a @ B @ A4 )
% 3.00/1.60 => ( ( ord_le456379495od_a_a @ B
% 3.00/1.60 @ ( collec1635618130od_a_a
% 3.00/1.60 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.60 ( ( member2057358096od_a_a @ X4 @ A4 )
% 3.00/1.60 & ( P2 @ X4 ) ) ) )
% 3.00/1.60 = ( ! [X4: produc1572603623od_a_a] :
% 3.00/1.60 ( ( member2057358096od_a_a @ X4 @ B )
% 3.00/1.60 => ( P2 @ X4 ) ) ) ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_322_subset__CollectI,axiom,
% 3.00/1.60 ! [B: set_Product_prod_a_a,A4: set_Product_prod_a_a,Q: product_prod_a_a > $o,P2: product_prod_a_a > $o] :
% 3.00/1.60 ( ( ord_le1824328871od_a_a @ B @ A4 )
% 3.00/1.60 => ( ! [X3: product_prod_a_a] :
% 3.00/1.60 ( ( member449909584od_a_a @ X3 @ B )
% 3.00/1.60 => ( ( Q @ X3 )
% 3.00/1.60 => ( P2 @ X3 ) ) )
% 3.00/1.60 => ( ord_le1824328871od_a_a
% 3.00/1.60 @ ( collec645855634od_a_a
% 3.00/1.60 @ ^ [X4: product_prod_a_a] :
% 3.00/1.60 ( ( member449909584od_a_a @ X4 @ B )
% 3.00/1.60 & ( Q @ X4 ) ) )
% 3.00/1.60 @ ( collec645855634od_a_a
% 3.00/1.60 @ ^ [X4: product_prod_a_a] :
% 3.00/1.60 ( ( member449909584od_a_a @ X4 @ A4 )
% 3.00/1.60 & ( P2 @ X4 ) ) ) ) ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_323_subset__CollectI,axiom,
% 3.00/1.60 ! [B: set_a,A4: set_a,Q: a > $o,P2: a > $o] :
% 3.00/1.60 ( ( ord_less_eq_set_a @ B @ A4 )
% 3.00/1.60 => ( ! [X3: a] :
% 3.00/1.60 ( ( member_a @ X3 @ B )
% 3.00/1.60 => ( ( Q @ X3 )
% 3.00/1.60 => ( P2 @ X3 ) ) )
% 3.00/1.60 => ( ord_less_eq_set_a
% 3.00/1.60 @ ( collect_a
% 3.00/1.60 @ ^ [X4: a] :
% 3.00/1.60 ( ( member_a @ X4 @ B )
% 3.00/1.60 & ( Q @ X4 ) ) )
% 3.00/1.60 @ ( collect_a
% 3.00/1.60 @ ^ [X4: a] :
% 3.00/1.60 ( ( member_a @ X4 @ A4 )
% 3.00/1.60 & ( P2 @ X4 ) ) ) ) ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_324_subset__CollectI,axiom,
% 3.00/1.60 ! [B: set_Pr1948701895od_a_a,A4: set_Pr1948701895od_a_a,Q: produc1572603623od_a_a > $o,P2: produc1572603623od_a_a > $o] :
% 3.00/1.60 ( ( ord_le456379495od_a_a @ B @ A4 )
% 3.00/1.60 => ( ! [X3: produc1572603623od_a_a] :
% 3.00/1.60 ( ( member2057358096od_a_a @ X3 @ B )
% 3.00/1.60 => ( ( Q @ X3 )
% 3.00/1.60 => ( P2 @ X3 ) ) )
% 3.00/1.60 => ( ord_le456379495od_a_a
% 3.00/1.60 @ ( collec1635618130od_a_a
% 3.00/1.60 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.60 ( ( member2057358096od_a_a @ X4 @ B )
% 3.00/1.60 & ( Q @ X4 ) ) )
% 3.00/1.60 @ ( collec1635618130od_a_a
% 3.00/1.60 @ ^ [X4: produc1572603623od_a_a] :
% 3.00/1.60 ( ( member2057358096od_a_a @ X4 @ A4 )
% 3.00/1.60 & ( P2 @ X4 ) ) ) ) ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_325_pred__subset__eq,axiom,
% 3.00/1.60 ! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 3.00/1.60 ( ( ord_le1347718902_a_a_o
% 3.00/1.60 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ R2 )
% 3.00/1.60 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ S3 ) )
% 3.00/1.60 = ( ord_le1824328871od_a_a @ R2 @ S3 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_326_pred__subset__eq,axiom,
% 3.00/1.60 ! [R2: set_a,S3: set_a] :
% 3.00/1.60 ( ( ord_less_eq_a_o
% 3.00/1.60 @ ^ [X4: a] : ( member_a @ X4 @ R2 )
% 3.00/1.60 @ ^ [X4: a] : ( member_a @ X4 @ S3 ) )
% 3.00/1.60 = ( ord_less_eq_set_a @ R2 @ S3 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_327_pred__subset__eq,axiom,
% 3.00/1.60 ! [R2: set_Pr1948701895od_a_a,S3: set_Pr1948701895od_a_a] :
% 3.00/1.60 ( ( ord_le1525442358_a_a_o
% 3.00/1.60 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ R2 )
% 3.00/1.60 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ S3 ) )
% 3.00/1.60 = ( ord_le456379495od_a_a @ R2 @ S3 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_328_less__eq__set__def,axiom,
% 3.00/1.60 ( ord_le1824328871od_a_a
% 3.00/1.60 = ( ^ [A6: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 3.00/1.60 ( ord_le1347718902_a_a_o
% 3.00/1.60 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A6 )
% 3.00/1.60 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ B5 ) ) ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_329_less__eq__set__def,axiom,
% 3.00/1.60 ( ord_less_eq_set_a
% 3.00/1.60 = ( ^ [A6: set_a,B5: set_a] :
% 3.00/1.60 ( ord_less_eq_a_o
% 3.00/1.60 @ ^ [X4: a] : ( member_a @ X4 @ A6 )
% 3.00/1.60 @ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_330_less__eq__set__def,axiom,
% 3.00/1.60 ( ord_le456379495od_a_a
% 3.00/1.60 = ( ^ [A6: set_Pr1948701895od_a_a,B5: set_Pr1948701895od_a_a] :
% 3.00/1.60 ( ord_le1525442358_a_a_o
% 3.00/1.60 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A6 )
% 3.00/1.60 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ B5 ) ) ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_331_pred__subset__eq2,axiom,
% 3.00/1.60 ! [R2: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 3.00/1.60 ( ( ord_less_eq_a_a_o
% 3.00/1.60 @ ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R2 )
% 3.00/1.60 @ ^ [X4: a,Y4: a] : ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ S3 ) )
% 3.00/1.60 = ( ord_le1824328871od_a_a @ R2 @ S3 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_332_pred__subset__eq2,axiom,
% 3.00/1.60 ! [R2: set_Pr1948701895od_a_a,S3: set_Pr1948701895od_a_a] :
% 3.00/1.60 ( ( ord_le252190158_a_a_o
% 3.00/1.60 @ ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R2 )
% 3.00/1.60 @ ^ [X4: product_prod_a_a,Y4: product_prod_a_a] : ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ S3 ) )
% 3.00/1.60 = ( ord_le456379495od_a_a @ R2 @ S3 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_333_GrD1,axiom,
% 3.00/1.60 ! [X: a,Fx: a,A4: set_a,F: a > a] :
% 3.00/1.60 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Fx ) @ ( bNF_Gr_a_a @ A4 @ F ) )
% 3.00/1.60 => ( member_a @ X @ A4 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_334_GrD1,axiom,
% 3.00/1.60 ! [X: product_prod_a_a,Fx: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
% 3.00/1.60 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Fx ) @ ( bNF_Gr1372473186od_a_a @ A4 @ F ) )
% 3.00/1.60 => ( member449909584od_a_a @ X @ A4 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_335_GrD2,axiom,
% 3.00/1.60 ! [X: a,Fx: a,A4: set_a,F: a > a] :
% 3.00/1.60 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Fx ) @ ( bNF_Gr_a_a @ A4 @ F ) )
% 3.00/1.60 => ( ( F @ X )
% 3.00/1.60 = Fx ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_336_GrD2,axiom,
% 3.00/1.60 ! [X: product_prod_a_a,Fx: product_prod_a_a,A4: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
% 3.00/1.60 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Fx ) @ ( bNF_Gr1372473186od_a_a @ A4 @ F ) )
% 3.00/1.60 => ( ( F @ X )
% 3.00/1.60 = Fx ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_337_all__not__in__conv,axiom,
% 3.00/1.60 ! [A4: set_Product_prod_a_a] :
% 3.00/1.60 ( ( ! [X4: product_prod_a_a] :
% 3.00/1.60 ~ ( member449909584od_a_a @ X4 @ A4 ) )
% 3.00/1.60 = ( A4 = bot_bo2131659635od_a_a ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_338_all__not__in__conv,axiom,
% 3.00/1.60 ! [A4: set_a] :
% 3.00/1.60 ( ( ! [X4: a] :
% 3.00/1.60 ~ ( member_a @ X4 @ A4 ) )
% 3.00/1.60 = ( A4 = bot_bot_set_a ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_339_all__not__in__conv,axiom,
% 3.00/1.60 ! [A4: set_Pr1948701895od_a_a] :
% 3.00/1.60 ( ( ! [X4: produc1572603623od_a_a] :
% 3.00/1.60 ~ ( member2057358096od_a_a @ X4 @ A4 ) )
% 3.00/1.60 = ( A4 = bot_bo1116205875od_a_a ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_340_empty__iff,axiom,
% 3.00/1.60 ! [C2: product_prod_a_a] :
% 3.00/1.60 ~ ( member449909584od_a_a @ C2 @ bot_bo2131659635od_a_a ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_341_empty__iff,axiom,
% 3.00/1.60 ! [C2: a] :
% 3.00/1.60 ~ ( member_a @ C2 @ bot_bot_set_a ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_342_empty__iff,axiom,
% 3.00/1.60 ! [C2: produc1572603623od_a_a] :
% 3.00/1.60 ~ ( member2057358096od_a_a @ C2 @ bot_bo1116205875od_a_a ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_343_ex__in__conv,axiom,
% 3.00/1.60 ! [A4: set_Product_prod_a_a] :
% 3.00/1.60 ( ( ? [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A4 ) )
% 3.00/1.60 = ( A4 != bot_bo2131659635od_a_a ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_344_ex__in__conv,axiom,
% 3.00/1.60 ! [A4: set_a] :
% 3.00/1.60 ( ( ? [X4: a] : ( member_a @ X4 @ A4 ) )
% 3.00/1.60 = ( A4 != bot_bot_set_a ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_345_ex__in__conv,axiom,
% 3.00/1.60 ! [A4: set_Pr1948701895od_a_a] :
% 3.00/1.60 ( ( ? [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A4 ) )
% 3.00/1.60 = ( A4 != bot_bo1116205875od_a_a ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_346_equals0I,axiom,
% 3.00/1.60 ! [A4: set_Product_prod_a_a] :
% 3.00/1.60 ( ! [Y3: product_prod_a_a] :
% 3.00/1.60 ~ ( member449909584od_a_a @ Y3 @ A4 )
% 3.00/1.60 => ( A4 = bot_bo2131659635od_a_a ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_347_equals0I,axiom,
% 3.00/1.60 ! [A4: set_a] :
% 3.00/1.60 ( ! [Y3: a] :
% 3.00/1.60 ~ ( member_a @ Y3 @ A4 )
% 3.00/1.60 => ( A4 = bot_bot_set_a ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_348_equals0I,axiom,
% 3.00/1.60 ! [A4: set_Pr1948701895od_a_a] :
% 3.00/1.60 ( ! [Y3: produc1572603623od_a_a] :
% 3.00/1.60 ~ ( member2057358096od_a_a @ Y3 @ A4 )
% 3.00/1.60 => ( A4 = bot_bo1116205875od_a_a ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_349_equals0D,axiom,
% 3.00/1.60 ! [A4: set_Product_prod_a_a,A: product_prod_a_a] :
% 3.00/1.60 ( ( A4 = bot_bo2131659635od_a_a )
% 3.00/1.60 => ~ ( member449909584od_a_a @ A @ A4 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_350_equals0D,axiom,
% 3.00/1.60 ! [A4: set_a,A: a] :
% 3.00/1.60 ( ( A4 = bot_bot_set_a )
% 3.00/1.60 => ~ ( member_a @ A @ A4 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_351_equals0D,axiom,
% 3.00/1.60 ! [A4: set_Pr1948701895od_a_a,A: produc1572603623od_a_a] :
% 3.00/1.60 ( ( A4 = bot_bo1116205875od_a_a )
% 3.00/1.60 => ~ ( member2057358096od_a_a @ A @ A4 ) ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_352_emptyE,axiom,
% 3.00/1.60 ! [A: product_prod_a_a] :
% 3.00/1.60 ~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_353_emptyE,axiom,
% 3.00/1.60 ! [A: a] :
% 3.00/1.60 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 3.00/1.60
% 3.00/1.60 thf(fact_354_emptyE,axiom,
% 3.00/1.60 ! [A: produc1572603623od_a_a] :
% 3.00/1.60 ~ ( member2057358096od_a_a @ A @ bot_bo1116205875od_a_a ) ).
% 3.00/1.60
% 3.00/1.60 thf(conj_0,conjecture,
% 3.00/1.60 member449909584od_a_a @ ( product_Pair_a_a @ z @ y ) @ pr ).
% 3.00/1.60 ------------------------
% 3.18/1.62 % (13221)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/183Mi)
% 3.18/1.63 % (13223)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_13046 for (2999ds/27Mi)
% 3.18/1.63 % (13222)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_13046 for (2999ds/4Mi)
% 3.18/1.63 % (13227)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/18Mi)
% 3.18/1.63 % (13224)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/2Mi)
% 3.18/1.63 % (13225)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_13046 for (2999ds/2Mi)
% 3.18/1.63 % (13226)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_13046 for (2999ds/275Mi)
% 3.18/1.63 % (13222)Instruction limit reached!
% 3.18/1.63 % (13222)------------------------------
% 3.18/1.63 % (13222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.63 % (13222)Termination reason: Unknown
% 3.18/1.63 % (13222)Termination phase: shuffling
% 3.18/1.63
% 3.18/1.63 % (13222)Memory used [KB]: 1663
% 3.18/1.63 % (13224)Instruction limit reached!
% 3.18/1.63 % (13224)------------------------------
% 3.18/1.63 % (13224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.63 % (13222)Time elapsed: 0.004 s
% 3.18/1.63 % (13222)Instructions burned: 5 (million)
% 3.18/1.63 % (13222)------------------------------
% 3.18/1.63 % (13222)------------------------------
% 3.18/1.63 % (13224)Termination reason: Unknown
% 3.18/1.63 % (13225)Instruction limit reached!
% 3.18/1.63 % (13225)------------------------------
% 3.18/1.63 % (13225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.63 % (13225)Termination reason: Unknown
% 3.18/1.63 % (13225)Termination phase: shuffling
% 3.18/1.63
% 3.18/1.63 % (13225)Memory used [KB]: 1535
% 3.18/1.63 % (13225)Time elapsed: 0.003 s
% 3.18/1.63 % (13225)Instructions burned: 2 (million)
% 3.18/1.63 % (13225)------------------------------
% 3.18/1.63 % (13225)------------------------------
% 3.18/1.63 % (13224)Termination phase: shuffling
% 3.18/1.63
% 3.18/1.63 % (13224)Memory used [KB]: 1535
% 3.18/1.63 % (13224)Time elapsed: 0.003 s
% 3.18/1.63 % (13224)Instructions burned: 2 (million)
% 3.18/1.63 % (13224)------------------------------
% 3.18/1.63 % (13224)------------------------------
% 3.18/1.63 % (13227)Instruction limit reached!
% 3.18/1.63 % (13227)------------------------------
% 3.18/1.63 % (13227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.63 % (13227)Termination reason: Unknown
% 3.18/1.63 % (13227)Termination phase: shuffling
% 3.18/1.63
% 3.18/1.63 % (13227)Memory used [KB]: 1791
% 3.18/1.63 % (13227)Time elapsed: 0.009 s
% 3.18/1.63 % (13227)Instructions burned: 18 (million)
% 3.18/1.63 % (13227)------------------------------
% 3.18/1.63 % (13227)------------------------------
% 3.18/1.64 % (13223)Instruction limit reached!
% 3.18/1.64 % (13223)------------------------------
% 3.18/1.64 % (13223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.64 % (13223)Termination reason: Unknown
% 3.18/1.64 % (13223)Termination phase: shuffling
% 3.18/1.64
% 3.18/1.64 % (13223)Memory used [KB]: 2046
% 3.18/1.64 % (13223)Time elapsed: 0.014 s
% 3.18/1.64 % (13223)Instructions burned: 27 (million)
% 3.18/1.64 % (13223)------------------------------
% 3.18/1.64 % (13223)------------------------------
% 3.18/1.64 % (13234)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_13046 for (2999ds/15Mi)
% 3.18/1.65 % (13232)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/3Mi)
% 3.18/1.65 % (13233)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_13046 for (2999ds/37Mi)
% 3.18/1.65 % (13236)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_13046 for (2999ds/3Mi)
% 3.18/1.65 % (13238)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_13046 for (2999ds/1041Mi)
% 3.18/1.65 % (13232)Instruction limit reached!
% 3.18/1.65 % (13232)------------------------------
% 3.18/1.65 % (13232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.65 % (13232)Termination reason: Unknown
% 3.18/1.65 % (13232)Termination phase: shuffling
% 3.18/1.65
% 3.18/1.65 % (13232)Memory used [KB]: 1535
% 3.18/1.65 % (13232)Time elapsed: 0.004 s
% 3.18/1.65 % (13232)Instructions burned: 4 (million)
% 3.18/1.65 % (13232)------------------------------
% 3.18/1.65 % (13232)------------------------------
% 3.18/1.65 % (13236)Instruction limit reached!
% 3.18/1.65 % (13236)------------------------------
% 3.18/1.65 % (13236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.65 % (13236)Termination reason: Unknown
% 3.18/1.65 % (13236)Termination phase: shuffling
% 3.18/1.65
% 3.18/1.65 % (13236)Memory used [KB]: 1535
% 3.18/1.65 % (13236)Time elapsed: 0.003 s
% 3.18/1.65 % (13236)Instructions burned: 3 (million)
% 3.18/1.65 % (13236)------------------------------
% 3.18/1.65 % (13236)------------------------------
% 3.18/1.65 % (13234)Instruction limit reached!
% 3.18/1.65 % (13234)------------------------------
% 3.18/1.65 % (13234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.65 % (13234)Termination reason: Unknown
% 3.18/1.65 % (13234)Termination phase: shuffling
% 3.18/1.65
% 3.18/1.65 % (13234)Memory used [KB]: 1791
% 3.18/1.65 % (13234)Time elapsed: 0.010 s
% 3.18/1.65 % (13234)Instructions burned: 15 (million)
% 3.18/1.65 % (13234)------------------------------
% 3.18/1.65 % (13234)------------------------------
% 3.18/1.66 % (13226)Refutation not found, incomplete strategy
% 3.18/1.66 % (13226)------------------------------
% 3.18/1.66 % (13226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.66 % (13226)Termination reason: Refutation not found, incomplete strategy
% 3.18/1.66
% 3.18/1.66
% 3.18/1.66 % (13226)Memory used [KB]: 6652
% 3.18/1.66 % (13226)Time elapsed: 0.033 s
% 3.18/1.66 % (13226)Instructions burned: 59 (million)
% 3.18/1.66 % (13226)------------------------------
% 3.18/1.66 % (13226)------------------------------
% 3.18/1.66 % (13241)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/16Mi)
% 3.18/1.66 % (13233)Instruction limit reached!
% 3.18/1.66 % (13233)------------------------------
% 3.18/1.66 % (13233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.66 % (13233)Termination reason: Unknown
% 3.18/1.66 % (13233)Termination phase: shuffling
% 3.18/1.66
% 3.18/1.66 % (13233)Memory used [KB]: 2302
% 3.18/1.66 % (13233)Time elapsed: 0.019 s
% 3.18/1.66 % (13233)Instructions burned: 37 (million)
% 3.18/1.66 % (13233)------------------------------
% 3.18/1.66 % (13233)------------------------------
% 3.18/1.67 % (13240)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/7Mi)
% 3.18/1.67 % (13241)Instruction limit reached!
% 3.18/1.67 % (13241)------------------------------
% 3.18/1.67 % (13241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.67 % (13241)Termination reason: Unknown
% 3.18/1.67 % (13241)Termination phase: shuffling
% 3.18/1.67
% 3.18/1.67 % (13241)Memory used [KB]: 1791
% 3.18/1.67 % (13241)Time elapsed: 0.007 s
% 3.18/1.67 % (13241)Instructions burned: 17 (million)
% 3.18/1.67 % (13241)------------------------------
% 3.18/1.67 % (13241)------------------------------
% 3.18/1.67 % (13240)Instruction limit reached!
% 3.18/1.67 % (13240)------------------------------
% 3.18/1.67 % (13240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.67 % (13240)Termination reason: Unknown
% 3.18/1.67 % (13240)Termination phase: shuffling
% 3.18/1.67
% 3.18/1.67 % (13240)Memory used [KB]: 1663
% 3.18/1.67 % (13240)Time elapsed: 0.006 s
% 3.18/1.67 % (13240)Instructions burned: 8 (million)
% 3.18/1.67 % (13240)------------------------------
% 3.18/1.67 % (13240)------------------------------
% 3.18/1.67 % (13243)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/3Mi)
% 3.18/1.67 % (13243)Instruction limit reached!
% 3.18/1.67 % (13243)------------------------------
% 3.18/1.67 % (13243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.67 % (13243)Termination reason: Unknown
% 3.18/1.67 % (13243)Termination phase: shuffling
% 3.18/1.67
% 3.18/1.67 % (13243)Memory used [KB]: 1535
% 3.18/1.67 % (13243)Time elapsed: 0.003 s
% 3.18/1.67 % (13243)Instructions burned: 3 (million)
% 3.18/1.67 % (13243)------------------------------
% 3.18/1.67 % (13243)------------------------------
% 3.18/1.68 % (13245)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/3Mi)
% 3.18/1.68 % (13248)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/4Mi)
% 3.18/1.68 % (13245)Instruction limit reached!
% 3.18/1.68 % (13245)------------------------------
% 3.18/1.68 % (13245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (13245)Termination reason: Unknown
% 3.18/1.68 % (13245)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (13245)Memory used [KB]: 1535
% 3.18/1.68 % (13245)Time elapsed: 0.004 s
% 3.18/1.68 % (13245)Instructions burned: 4 (million)
% 3.18/1.68 % (13245)------------------------------
% 3.18/1.68 % (13245)------------------------------
% 3.18/1.68 % (13246)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/7Mi)
% 3.18/1.68 % (13248)Instruction limit reached!
% 3.18/1.68 % (13248)------------------------------
% 3.18/1.68 % (13248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (13248)Termination reason: Unknown
% 3.18/1.68 % (13248)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (13248)Memory used [KB]: 1535
% 3.18/1.68 % (13248)Time elapsed: 0.003 s
% 3.18/1.68 % (13248)Instructions burned: 5 (million)
% 3.18/1.68 % (13248)------------------------------
% 3.18/1.68 % (13248)------------------------------
% 3.18/1.68 % (13246)Instruction limit reached!
% 3.18/1.68 % (13246)------------------------------
% 3.18/1.68 % (13246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.68 % (13246)Termination reason: Unknown
% 3.18/1.68 % (13246)Termination phase: shuffling
% 3.18/1.68
% 3.18/1.68 % (13246)Memory used [KB]: 1663
% 3.18/1.68 % (13246)Time elapsed: 0.005 s
% 3.18/1.68 % (13246)Instructions burned: 7 (million)
% 3.18/1.68 % (13246)------------------------------
% 3.18/1.68 % (13246)------------------------------
% 3.18/1.68 % (13247)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/3Mi)
% 3.18/1.69 % (13247)Instruction limit reached!
% 3.18/1.69 % (13247)------------------------------
% 3.18/1.69 % (13247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.69 % (13247)Termination reason: Unknown
% 3.18/1.69 % (13247)Termination phase: shuffling
% 3.18/1.69
% 3.18/1.69 % (13247)Memory used [KB]: 1535
% 3.18/1.69 % (13247)Time elapsed: 0.003 s
% 3.18/1.69 % (13247)Instructions burned: 3 (million)
% 3.18/1.69 % (13247)------------------------------
% 3.18/1.69 % (13247)------------------------------
% 3.18/1.69 % (13250)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/18Mi)
% 3.18/1.69 % (13252)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/6Mi)
% 3.18/1.69 % (13252)Instruction limit reached!
% 3.18/1.69 % (13252)------------------------------
% 3.18/1.69 % (13252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.69 % (13252)Termination reason: Unknown
% 3.18/1.69 % (13252)Termination phase: shuffling
% 3.18/1.69
% 3.18/1.69 % (13252)Memory used [KB]: 1535
% 3.18/1.69 % (13252)Time elapsed: 0.003 s
% 3.18/1.69 % (13252)Instructions burned: 6 (million)
% 3.18/1.69 % (13252)------------------------------
% 3.18/1.69 % (13252)------------------------------
% 3.18/1.70 % (13251)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/710Mi)
% 3.18/1.70 % (13221)Instruction limit reached!
% 3.18/1.70 % (13221)------------------------------
% 3.18/1.70 % (13221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.70 % (13221)Termination reason: Unknown
% 3.18/1.70 % (13221)Termination phase: Saturation
% 3.18/1.70
% 3.18/1.70 % (13221)Memory used [KB]: 7547
% 3.18/1.70 % (13221)Time elapsed: 0.080 s
% 3.18/1.70 % (13221)Instructions burned: 183 (million)
% 3.18/1.70 % (13221)------------------------------
% 3.18/1.70 % (13221)------------------------------
% 3.18/1.70 % (13250)Instruction limit reached!
% 3.18/1.70 % (13250)------------------------------
% 3.18/1.70 % (13250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.70 % (13250)Termination reason: Unknown
% 3.18/1.70 % (13250)Termination phase: shuffling
% 3.18/1.70
% 3.18/1.70 % (13250)Memory used [KB]: 1791
% 3.18/1.70 % (13250)Time elapsed: 0.012 s
% 3.18/1.70 % (13250)Instructions burned: 18 (million)
% 3.18/1.70 % (13250)------------------------------
% 3.18/1.70 % (13250)------------------------------
% 3.18/1.70 % (13253)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/902Mi)
% 3.18/1.70 % (13256)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/5Mi)
% 3.18/1.70 % (13256)Instruction limit reached!
% 3.18/1.70 % (13256)------------------------------
% 3.18/1.70 % (13256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.70 % (13256)Termination reason: Unknown
% 3.18/1.70 % (13256)Termination phase: shuffling
% 3.18/1.70
% 3.18/1.70 % (13256)Memory used [KB]: 1663
% 3.18/1.70 % (13256)Time elapsed: 0.003 s
% 3.18/1.70 % (13256)Instructions burned: 6 (million)
% 3.18/1.70 % (13256)------------------------------
% 3.18/1.70 % (13256)------------------------------
% 3.18/1.71 % (13254)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/21Mi)
% 3.18/1.71 % (13260)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/779Mi)
% 3.18/1.71 % (13257)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/6Mi)
% 3.18/1.72 % (13254)Instruction limit reached!
% 3.18/1.72 % (13254)------------------------------
% 3.18/1.72 % (13254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.72 % (13254)Termination reason: Unknown
% 3.18/1.72 % (13254)Termination phase: shuffling
% 3.18/1.72
% 3.18/1.72 % (13254)Memory used [KB]: 1918
% 3.18/1.72 % (13254)Time elapsed: 0.012 s
% 3.18/1.72 % (13254)Instructions burned: 21 (million)
% 3.18/1.72 % (13254)------------------------------
% 3.18/1.72 % (13254)------------------------------
% 3.18/1.72 % (13257)Instruction limit reached!
% 3.18/1.72 % (13257)------------------------------
% 3.18/1.72 % (13257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.72 % (13257)Termination reason: Unknown
% 3.18/1.72 % (13257)Termination phase: shuffling
% 3.18/1.72
% 3.18/1.72 % (13257)Memory used [KB]: 1663
% 3.18/1.72 % (13257)Time elapsed: 0.005 s
% 3.18/1.72 % (13257)Instructions burned: 7 (million)
% 3.18/1.72 % (13257)------------------------------
% 3.18/1.72 % (13257)------------------------------
% 3.18/1.72 % (13258)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13046 for (2999ds/377Mi)
% 3.18/1.73 % (13263)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13046 for (2998ds/879Mi)
% 3.18/1.73 % (13262)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_13046 for (2998ds/19Mi)
% 3.18/1.74 % (13262)Instruction limit reached!
% 3.18/1.74 % (13262)------------------------------
% 3.18/1.74 % (13262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.74 % (13262)Termination reason: Unknown
% 3.18/1.74 % (13262)Termination phase: shuffling
% 3.18/1.74
% 3.18/1.74 % (13262)Memory used [KB]: 1918
% 3.18/1.74 % (13262)Time elapsed: 0.010 s
% 3.18/1.74 % (13262)Instructions burned: 20 (million)
% 3.18/1.74 % (13262)------------------------------
% 3.18/1.74 % (13262)------------------------------
% 3.18/1.76 % (13265)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_13046 for (2998ds/17Mi)
% 3.18/1.77 % (13265)Instruction limit reached!
% 3.18/1.77 % (13265)------------------------------
% 3.18/1.77 % (13265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.77 % (13265)Termination reason: Unknown
% 3.18/1.77 % (13265)Termination phase: shuffling
% 3.18/1.77
% 3.18/1.77 % (13265)Memory used [KB]: 1791
% 3.18/1.77 % (13265)Time elapsed: 0.011 s
% 3.18/1.77 % (13265)Instructions burned: 18 (million)
% 3.18/1.77 % (13265)------------------------------
% 3.18/1.77 % (13265)------------------------------
% 3.18/1.79 % (13266)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13046 for (2998ds/3Mi)
% 3.18/1.79 % (13266)Instruction limit reached!
% 3.18/1.79 % (13266)------------------------------
% 3.18/1.79 % (13266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.79 % (13266)Termination reason: Unknown
% 3.18/1.79 % (13266)Termination phase: shuffling
% 3.18/1.79
% 3.18/1.79 % (13266)Memory used [KB]: 1535
% 3.18/1.79 % (13266)Time elapsed: 0.004 s
% 3.18/1.79 % (13266)Instructions burned: 4 (million)
% 3.18/1.79 % (13266)------------------------------
% 3.18/1.79 % (13266)------------------------------
% 3.18/1.80 % (13268)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_13046 for (2998ds/30Mi)
% 3.18/1.81 % (13268)Instruction limit reached!
% 3.18/1.81 % (13268)------------------------------
% 3.18/1.81 % (13268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.81 % (13268)Termination reason: Unknown
% 3.18/1.81 % (13268)Termination phase: shuffling
% 3.18/1.81
% 3.18/1.81 % (13268)Memory used [KB]: 2046
% 3.18/1.81 % (13268)Time elapsed: 0.011 s
% 3.18/1.81 % (13268)Instructions burned: 30 (million)
% 3.18/1.81 % (13268)------------------------------
% 3.18/1.81 % (13268)------------------------------
% 3.18/1.82 % (13270)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_13046 for (2998ds/127Mi)
% 3.18/1.83 % (13238)First to succeed.
% 3.18/1.84 % (13238)Refutation found. Thanks to Tanya!
% 3.18/1.84 % SZS status Theorem for DTF2THF_13046
% 3.18/1.84 % SZS output start Proof for DTF2THF_13046
% 3.18/1.84 thf(type_def_5, type, set_Product_prod_a_a: $tType).
% 3.18/1.84 thf(type_def_6, type, product_prod_a_a: $tType).
% 3.18/1.84 thf(type_def_8, type, set_Pr1948701895od_a_a: $tType).
% 3.18/1.84 thf(type_def_9, type, set_a: $tType).
% 3.18/1.84 thf(type_def_10, type, a: $tType).
% 3.18/1.84 thf(type_def_11, type, produc1572603623od_a_a: $tType).
% 3.18/1.84 thf(type_def_12, type, set_Pr1295299783od_a_a: $tType).
% 3.18/1.84 thf(type_def_13, type, produc2010915495od_a_a: $tType).
% 3.18/1.84 thf(type_def_14, type, set_Pr231806599od_a_a: $tType).
% 3.18/1.84 thf(type_def_15, type, set_Pr532616391od_a_a: $tType).
% 3.18/1.84 thf(type_def_16, type, produc772425383od_a_a: $tType).
% 3.18/1.84 thf(type_def_17, type, set_Pr1194146439od_a_a: $tType).
% 3.18/1.84 thf(type_def_18, type, set_Pr2081559751od_a_a: $tType).
% 3.18/1.84 thf(type_def_19, type, produc1016684094_a_a_a: $tType).
% 3.18/1.84 thf(type_def_20, type, set_Pr1689873822_a_a_a: $tType).
% 3.18/1.84 thf(type_def_21, type, set_Pr729215751_a_a_a: $tType).
% 3.18/1.84 thf(type_def_22, type, produc1555800720od_a_a: $tType).
% 3.18/1.84 thf(type_def_23, type, set_Pr195866352od_a_a: $tType).
% 3.18/1.84 thf(type_def_24, type, set_Pr660101255od_a_a: $tType).
% 3.18/1.84 thf(type_def_25, type, produc1921647824od_a_a: $tType).
% 3.18/1.84 thf(type_def_26, type, set_Pr681306928od_a_a: $tType).
% 3.18/1.84 thf(type_def_27, type, produc1056976615od_a_a: $tType).
% 3.18/1.84 thf(type_def_28, type, produc1423126247od_a_a: $tType).
% 3.18/1.84 thf(type_def_29, type, produc1990712039od_a_a: $tType).
% 3.18/1.84 thf(type_def_30, type, produc47586087od_a_a: $tType).
% 3.18/1.84 thf(type_def_31, type, produc1683075710_a_a_a: $tType).
% 3.18/1.84 thf(type_def_32, type, produc1120419623_a_a_a: $tType).
% 3.18/1.84 thf(type_def_33, type, produc1546279079od_a_a: $tType).
% 3.18/1.84 thf(type_def_34, type, set_Pr173461255od_a_a: $tType).
% 3.18/1.84 thf(type_def_35, type, set_Pr6137822_a_a_a: $tType).
% 3.18/1.84 thf(func_def_0, type, set_Pr2081559751od_a_a: $tType).
% 3.18/1.84 thf(func_def_1, type, set_Pr532616391od_a_a: $tType).
% 3.18/1.84 thf(func_def_2, type, produc1423126247od_a_a: $tType).
% 3.18/1.84 thf(func_def_3, type, produc1056976615od_a_a: $tType).
% 3.18/1.84 thf(func_def_4, type, set_Pr660101255od_a_a: $tType).
% 3.18/1.84 thf(func_def_5, type, produc1546279079od_a_a: $tType).
% 3.18/1.84 thf(func_def_6, type, set_Pr1295299783od_a_a: $tType).
% 3.18/1.84 thf(func_def_7, type, produc1990712039od_a_a: $tType).
% 3.18/1.84 thf(func_def_8, type, set_Pr1194146439od_a_a: $tType).
% 3.18/1.84 thf(func_def_9, type, set_Pr231806599od_a_a: $tType).
% 3.18/1.84 thf(func_def_10, type, set_Pr729215751_a_a_a: $tType).
% 3.18/1.84 thf(func_def_11, type, set_Pr173461255od_a_a: $tType).
% 3.18/1.84 thf(func_def_12, type, produc772425383od_a_a: $tType).
% 3.18/1.84 thf(func_def_13, type, produc2010915495od_a_a: $tType).
% 3.18/1.84 thf(func_def_14, type, produc1120419623_a_a_a: $tType).
% 3.18/1.84 thf(func_def_15, type, produc47586087od_a_a: $tType).
% 3.18/1.84 thf(func_def_16, type, set_Pr195866352od_a_a: $tType).
% 3.18/1.84 thf(func_def_17, type, set_Pr6137822_a_a_a: $tType).
% 3.18/1.84 thf(func_def_18, type, produc1555800720od_a_a: $tType).
% 3.18/1.84 thf(func_def_19, type, produc1683075710_a_a_a: $tType).
% 3.18/1.84 thf(func_def_20, type, set_Pr1948701895od_a_a: $tType).
% 3.18/1.84 thf(func_def_21, type, produc1572603623od_a_a: $tType).
% 3.18/1.84 thf(func_def_22, type, set_Pr681306928od_a_a: $tType).
% 3.18/1.84 thf(func_def_23, type, set_Pr1689873822_a_a_a: $tType).
% 3.18/1.84 thf(func_def_24, type, produc1921647824od_a_a: $tType).
% 3.18/1.84 thf(func_def_25, type, produc1016684094_a_a_a: $tType).
% 3.18/1.84 thf(func_def_26, type, set_Product_prod_a_a: $tType).
% 3.18/1.84 thf(func_def_27, type, product_prod_a_a: $tType).
% 3.18/1.84 thf(func_def_28, type, set_a: $tType).
% 3.18/1.84 thf(func_def_29, type, a: $tType).
% 3.18/1.84 thf(func_def_30, type, bNF_Gr1372473186od_a_a: set_Product_prod_a_a > (product_prod_a_a > product_prod_a_a) > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_31, type, bNF_Gr_a_a: set_a > (a > a) > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_32, type, bNF_Gr533818957od_a_a: set_Pr1948701895od_a_a > (produc1572603623od_a_a > product_prod_a_a) > (produc1572603623od_a_a > product_prod_a_a) > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_33, type, bNF_Gr356189709_a_a_a: set_Pr1948701895od_a_a > (produc1572603623od_a_a > a) > (produc1572603623od_a_a > a) > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_34, type, bNF_Gr200450189od_a_a: set_Product_prod_a_a > (product_prod_a_a > product_prod_a_a) > (product_prod_a_a > product_prod_a_a) > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_35, type, bNF_Gr1853143117_a_a_a: set_Product_prod_a_a > (product_prod_a_a > a) > (product_prod_a_a > a) > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_36, type, bNF_Gr1715025636od_a_a: set_a > (a > product_prod_a_a) > (a > product_prod_a_a) > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_37, type, bNF_Gr423046308_a_a_a: set_a > (a > a) > (a > a) > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_38, type, minus_107310383_a_a_o: (produc1572603623od_a_a > $o) > (produc1572603623od_a_a > $o) > produc1572603623od_a_a > $o).
% 3.18/1.84 thf(func_def_39, type, minus_1797478127_a_a_o: (product_prod_a_a > $o) > (product_prod_a_a > $o) > product_prod_a_a > $o).
% 3.18/1.84 thf(func_def_40, type, minus_minus_a_o: (a > $o) > (a > $o) > a > $o).
% 3.18/1.84 thf(func_def_41, type, minus_1034916142od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_42, type, minus_1507102574od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_43, type, minus_minus_set_a: set_a > set_a > set_a).
% 3.18/1.84 thf(func_def_44, type, order_491428350od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 3.18/1.84 thf(func_def_45, type, order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 3.18/1.84 thf(func_def_46, type, order_preorder_on_a: set_a > set_Product_prod_a_a > $o).
% 3.18/1.84 thf(func_def_47, type, bot_bo1116205875od_a_a: set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_48, type, bot_bo2131659635od_a_a: set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_49, type, bot_bot_set_a: set_a).
% 3.18/1.84 thf(func_def_50, type, ord_le1525442358_a_a_o: (produc1572603623od_a_a > $o) > (produc1572603623od_a_a > $o) > $o).
% 3.18/1.84 thf(func_def_51, type, ord_le252190158_a_a_o: (product_prod_a_a > product_prod_a_a > $o) > (product_prod_a_a > product_prod_a_a > $o) > $o).
% 3.18/1.84 thf(func_def_52, type, ord_le1347718902_a_a_o: (product_prod_a_a > $o) > (product_prod_a_a > $o) > $o).
% 3.18/1.84 thf(func_def_53, type, ord_less_eq_a_a_o: (a > a > $o) > (a > a > $o) > $o).
% 3.18/1.84 thf(func_def_54, type, ord_less_eq_a_o: (a > $o) > (a > $o) > $o).
% 3.18/1.84 thf(func_def_55, type, ord_le456379495od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > $o).
% 3.18/1.84 thf(func_def_56, type, ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 3.18/1.84 thf(func_def_57, type, ord_less_eq_set_a: set_a > set_a > $o).
% 3.18/1.84 thf(func_def_58, type, top_to2011536774_a_a_o: produc1572603623od_a_a > $o).
% 3.18/1.84 thf(func_def_59, type, top_to993969022_a_a_o: product_prod_a_a > product_prod_a_a > $o).
% 3.18/1.84 thf(func_def_60, type, top_to1483957574_a_a_o: product_prod_a_a > $o).
% 3.18/1.84 thf(func_def_61, type, top_top_a_a_o: a > a > $o).
% 3.18/1.84 thf(func_def_62, type, top_top_a_o: a > $o).
% 3.18/1.84 thf(func_def_63, type, top_to52711959od_a_a: set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_64, type, top_to398455383od_a_a: set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_65, type, top_top_set_a: set_a).
% 3.18/1.84 thf(func_def_66, type, prefer1867335523od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_67, type, prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_68, type, prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a).
% 3.18/1.84 thf(func_def_69, type, prefer468349667od_a_a: produc2010915495od_a_a > set_Pr231806599od_a_a > set_Pr532616391od_a_a > set_Pr231806599od_a_a).
% 3.18/1.84 thf(func_def_70, type, prefer1377343203od_a_a: produc772425383od_a_a > set_Pr1194146439od_a_a > set_Pr2081559751od_a_a > set_Pr1194146439od_a_a).
% 3.18/1.84 thf(func_def_71, type, prefer55414563od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_72, type, prefer1944486522_a_a_a: produc1016684094_a_a_a > set_Pr1689873822_a_a_a > set_Pr729215751_a_a_a > set_Pr1689873822_a_a_a).
% 3.18/1.84 thf(func_def_73, type, prefer445484748od_a_a: produc1555800720od_a_a > set_Pr195866352od_a_a > set_Pr660101255od_a_a > set_Pr195866352od_a_a).
% 3.18/1.84 thf(func_def_74, type, prefer701966604od_a_a: produc1921647824od_a_a > set_Pr681306928od_a_a > set_Pr231806599od_a_a > set_Pr681306928od_a_a).
% 3.18/1.84 thf(func_def_75, type, prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_76, type, prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a).
% 3.18/1.84 thf(func_def_77, type, prefer926864306od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_78, type, prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_79, type, prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a).
% 3.18/1.84 thf(func_def_80, type, prefer1409946857od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 3.18/1.84 thf(func_def_81, type, prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 3.18/1.84 thf(func_def_82, type, prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o).
% 3.18/1.84 thf(func_def_83, type, produc539587415od_a_a: produc2010915495od_a_a > produc2010915495od_a_a > produc1056976615od_a_a).
% 3.18/1.84 thf(func_def_84, type, produc852390743od_a_a: produc772425383od_a_a > produc772425383od_a_a > produc1423126247od_a_a).
% 3.18/1.84 thf(func_def_85, type, produc1935643479od_a_a: produc1572603623od_a_a > produc1572603623od_a_a > produc1990712039od_a_a).
% 3.18/1.84 thf(func_def_86, type, produc524982679od_a_a: produc1572603623od_a_a > product_prod_a_a > produc47586087od_a_a).
% 3.18/1.84 thf(func_def_87, type, produc436797294_a_a_a: produc1572603623od_a_a > a > produc1683075710_a_a_a).
% 3.18/1.84 thf(func_def_88, type, produc2054708503_a_a_a: produc1016684094_a_a_a > produc1016684094_a_a_a > produc1120419623_a_a_a).
% 3.18/1.84 thf(func_def_89, type, produc1573596055od_a_a: produc1555800720od_a_a > produc1555800720od_a_a > produc1546279079od_a_a).
% 3.18/1.84 thf(func_def_90, type, produc563767703od_a_a: produc1921647824od_a_a > produc1921647824od_a_a > produc2010915495od_a_a).
% 3.18/1.84 thf(func_def_91, type, produc1555606807od_a_a: product_prod_a_a > produc1572603623od_a_a > produc772425383od_a_a).
% 3.18/1.84 thf(func_def_92, type, produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_93, type, produc2061588782_a_a_a: product_prod_a_a > a > produc1016684094_a_a_a).
% 3.18/1.84 thf(func_def_94, type, produc1095491584od_a_a: a > produc1572603623od_a_a > produc1555800720od_a_a).
% 3.18/1.84 thf(func_def_95, type, produc1299253312od_a_a: a > product_prod_a_a > produc1921647824od_a_a).
% 3.18/1.84 thf(func_def_96, type, product_Pair_a_a: a > a > product_prod_a_a).
% 3.18/1.84 thf(func_def_97, type, produc1460477704od_a_a: set_Pr1948701895od_a_a > (produc1572603623od_a_a > set_Pr1948701895od_a_a) > set_Pr1295299783od_a_a).
% 3.18/1.84 thf(func_def_98, type, produc197318472od_a_a: set_Pr1948701895od_a_a > (produc1572603623od_a_a > set_Product_prod_a_a) > set_Pr173461255od_a_a).
% 3.18/1.84 thf(func_def_99, type, produc4724255_a_a_a: set_Pr1948701895od_a_a > (produc1572603623od_a_a > set_a) > set_Pr6137822_a_a_a).
% 3.18/1.84 thf(func_def_100, type, produc1227942600od_a_a: set_Product_prod_a_a > (product_prod_a_a > set_Pr1948701895od_a_a) > set_Pr1194146439od_a_a).
% 3.18/1.84 thf(func_def_101, type, produc304751368od_a_a: set_Product_prod_a_a > (product_prod_a_a > set_Product_prod_a_a) > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_102, type, produc1282482655_a_a_a: set_Product_prod_a_a > (product_prod_a_a > set_a) > set_Pr1689873822_a_a_a).
% 3.18/1.84 thf(func_def_103, type, produc663418545od_a_a: set_a > (a > set_Pr1948701895od_a_a) > set_Pr195866352od_a_a).
% 3.18/1.84 thf(func_def_104, type, produc520147185od_a_a: set_a > (a > set_Product_prod_a_a) > set_Pr681306928od_a_a).
% 3.18/1.84 thf(func_def_105, type, product_Sigma_a_a: set_a > (a > set_a) > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_106, type, produc94438221_a_a_o: (produc1572603623od_a_a > $o) > product_prod_a_a > product_prod_a_a > $o).
% 3.18/1.84 thf(func_def_107, type, product_curry_a_a_o: (product_prod_a_a > $o) > a > a > $o).
% 3.18/1.84 thf(func_def_108, type, produc1779756150od_a_a: (product_prod_a_a > product_prod_a_a) > (product_prod_a_a > product_prod_a_a) > produc1572603623od_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_109, type, produc363290532_a_a_a: (product_prod_a_a > a) > (product_prod_a_a > a) > produc1572603623od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_110, type, produc2012818120od_a_a: (a > product_prod_a_a) > (a > product_prod_a_a) > product_prod_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_111, type, produc700022774_a_a_a: (a > a) > (a > a) > product_prod_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_112, type, produc303632707od_a_a: produc1990712039od_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_113, type, produc745123715od_a_a: produc47586087od_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_114, type, produc2019352922_a_a_a: produc1683075710_a_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_115, type, produc1775747843od_a_a: produc772425383od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_116, type, produc2115494211od_a_a: produc1572603623od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_117, type, produc1805832986_a_a_a: produc1016684094_a_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_118, type, produc530563564od_a_a: produc1555800720od_a_a > a).
% 3.18/1.84 thf(func_def_119, type, produc1043497516od_a_a: produc1921647824od_a_a > a).
% 3.18/1.84 thf(func_def_120, type, product_fst_a_a: product_prod_a_a > a).
% 3.18/1.84 thf(func_def_121, type, produc1441227909od_a_a: produc1990712039od_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_122, type, produc24848069od_a_a: produc47586087od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_123, type, produc262335132_a_a_a: produc1683075710_a_a_a > a).
% 3.18/1.84 thf(func_def_124, type, produc1055472197od_a_a: produc772425383od_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_125, type, produc368344197od_a_a: produc1572603623od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_126, type, produc1954021468_a_a_a: produc1016684094_a_a_a > a).
% 3.18/1.84 thf(func_def_127, type, produc921029422od_a_a: produc1555800720od_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_128, type, produc1191685998od_a_a: produc1921647824od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_129, type, product_snd_a_a: product_prod_a_a > a).
% 3.18/1.84 thf(func_def_130, type, produc1641853175od_a_a: produc1572603623od_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_131, type, product_swap_a_a: product_prod_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_132, type, produc905674258od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_133, type, product_product_a_a: set_a > set_a > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_134, type, collec1635618130od_a_a: (produc1572603623od_a_a > $o) > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_135, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_136, type, collect_a: (a > $o) > set_a).
% 3.18/1.84 thf(func_def_137, type, image_236793797od_a_a: (produc1572603623od_a_a > produc1572603623od_a_a) > set_Pr1948701895od_a_a > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_138, type, image_734753285od_a_a: (produc1572603623od_a_a > product_prod_a_a) > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_139, type, image_1105229276_a_a_a: (produc1572603623od_a_a > a) > set_Pr1948701895od_a_a > set_a).
% 3.18/1.84 thf(func_def_140, type, image_1765377413od_a_a: (product_prod_a_a > produc1572603623od_a_a) > set_Product_prod_a_a > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_141, type, image_794773445od_a_a: (product_prod_a_a > product_prod_a_a) > set_Product_prod_a_a > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_142, type, image_340790172_a_a_a: (product_prod_a_a > a) > set_Product_prod_a_a > set_a).
% 3.18/1.84 thf(func_def_143, type, image_1763923566od_a_a: (a > produc1572603623od_a_a) > set_a > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_144, type, image_1725938350od_a_a: (a > product_prod_a_a) > set_a > set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_145, type, image_a_a: (a > a) > set_a > set_a).
% 3.18/1.84 thf(func_def_146, type, lex_pr942790511od_a_a: set_Pr1948701895od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a).
% 3.18/1.84 thf(func_def_147, type, lex_pr2084296006_a_a_a: set_Pr1948701895od_a_a > set_Product_prod_a_a > set_Pr729215751_a_a_a).
% 3.18/1.84 thf(func_def_148, type, lex_pr1321960536od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Pr231806599od_a_a).
% 3.18/1.84 thf(func_def_149, type, lex_prod_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_150, type, same_f1138477100od_a_a: (product_prod_a_a > $o) > (product_prod_a_a > set_Pr1948701895od_a_a) > set_Pr1295299783od_a_a).
% 3.18/1.84 thf(func_def_151, type, same_fst_a_a: (a > $o) > (a > set_Product_prod_a_a) > set_Pr1948701895od_a_a).
% 3.18/1.84 thf(func_def_152, type, member180906768od_a_a: produc1056976615od_a_a > set_Pr532616391od_a_a > $o).
% 3.18/1.84 thf(func_def_153, type, member547056400od_a_a: produc1423126247od_a_a > set_Pr2081559751od_a_a > $o).
% 3.18/1.84 thf(func_def_154, type, member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o).
% 3.18/1.84 thf(func_def_155, type, member1545452880od_a_a: produc47586087od_a_a > set_Pr173461255od_a_a > $o).
% 3.18/1.84 thf(func_def_156, type, member575366567_a_a_a: produc1683075710_a_a_a > set_Pr6137822_a_a_a > $o).
% 3.18/1.84 thf(func_def_157, type, member470802768_a_a_a: produc1120419623_a_a_a > set_Pr729215751_a_a_a > $o).
% 3.18/1.84 thf(func_def_158, type, member1424209616od_a_a: produc1546279079od_a_a > set_Pr660101255od_a_a > $o).
% 3.18/1.84 thf(func_def_159, type, member1361298640od_a_a: produc2010915495od_a_a > set_Pr231806599od_a_a > $o).
% 3.18/1.84 thf(func_def_160, type, member122808528od_a_a: produc772425383od_a_a > set_Pr1194146439od_a_a > $o).
% 3.18/1.84 thf(func_def_161, type, member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o).
% 3.18/1.84 thf(func_def_162, type, member1403138919_a_a_a: produc1016684094_a_a_a > set_Pr1689873822_a_a_a > $o).
% 3.18/1.84 thf(func_def_163, type, member448091577od_a_a: produc1555800720od_a_a > set_Pr195866352od_a_a > $o).
% 3.18/1.84 thf(func_def_164, type, member160619001od_a_a: produc1921647824od_a_a > set_Pr681306928od_a_a > $o).
% 3.18/1.84 thf(func_def_165, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 3.18/1.84 thf(func_def_166, type, member_a: a > set_a > $o).
% 3.18/1.84 thf(func_def_167, type, b: set_a).
% 3.18/1.84 thf(func_def_168, type, pr: set_Product_prod_a_a).
% 3.18/1.84 thf(func_def_169, type, y: a).
% 3.18/1.84 thf(func_def_170, type, z: a).
% 3.18/1.84 thf(func_def_190, type, ph1: !>[X0: $tType]:(X0)).
% 3.18/1.84 thf(func_def_191, type, sK2: a).
% 3.18/1.84 thf(func_def_192, type, sK3: product_prod_a_a).
% 3.18/1.84 thf(func_def_193, type, sK4: produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_194, type, sK5: product_prod_a_a > a).
% 3.18/1.84 thf(func_def_195, type, sK6: product_prod_a_a > a).
% 3.18/1.84 thf(func_def_196, type, sK7: produc1016684094_a_a_a > a).
% 3.18/1.84 thf(func_def_197, type, sK8: produc1016684094_a_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_198, type, sK9: produc1921647824od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_199, type, sK10: produc1921647824od_a_a > a).
% 3.18/1.84 thf(func_def_200, type, sK11: produc1572603623od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_201, type, sK12: produc1572603623od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_202, type, sK13: produc2010915495od_a_a > produc1921647824od_a_a).
% 3.18/1.84 thf(func_def_203, type, sK14: produc2010915495od_a_a > produc1921647824od_a_a).
% 3.18/1.84 thf(func_def_204, type, sK15: produc1555800720od_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_205, type, sK16: produc1555800720od_a_a > a).
% 3.18/1.84 thf(func_def_206, type, sK17: produc772425383od_a_a > produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_207, type, sK18: produc772425383od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_208, type, sK19: produc1572603623od_a_a).
% 3.18/1.84 thf(func_def_209, type, sK20: product_prod_a_a).
% 3.18/1.84 thf(func_def_210, type, sK21: product_prod_a_a > a).
% 3.18/1.84 thf(func_def_211, type, sK22: product_prod_a_a > a).
% 3.18/1.84 thf(func_def_212, type, sK23: produc1572603623od_a_a > product_prod_a_a).
% 3.18/1.84 thf(func_def_213, type, sK24: produc1572603623od_a_a > product_prod_a_a).
% 3.18/1.84 thf(f2153,plain,(
% 3.18/1.84 $false),
% 3.18/1.84 inference(trivial_inequality_removal,[status(thm)],[f2152])).
% 3.18/1.84 thf(f2152,plain,(
% 3.18/1.84 ($false = $true)),
% 3.18/1.84 inference(boolean_simplification,[status(thm)],[f2151])).
% 3.18/1.84 thf(f2151,plain,(
% 3.18/1.84 ($true = ((member_a @ z @ b) & $false))),
% 3.18/1.84 inference(forward_demodulation,[status(thm)],[f2147,f1608])).
% 3.18/1.84 thf(f1608,plain,(
% 3.18/1.84 ((member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ pr) = $false)),
% 3.18/1.84 inference(not_proxy_clausification,[status(thm)],[f1212])).
% 3.18/1.84 thf(f1212,plain,(
% 3.18/1.84 ((~ (member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ pr)) = $true)),
% 3.18/1.84 inference(cnf_transformation,[status(thm)],[f492])).
% 3.18/1.84 thf(f492,plain,(
% 3.18/1.84 ((~ (member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ pr)) = $true)),
% 3.18/1.84 inference(fool_elimination,[status(thm)],[f491])).
% 3.18/1.84 thf(f491,plain,(
% 3.18/1.84 ~(member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ pr)),
% 3.18/1.84 inference(rectify,[status(thm)],[f357])).
% 3.18/1.84 thf(f357,negated_conjecture,(
% 3.18/1.84 ~(member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ pr)),
% 3.18/1.84 inference(negated_conjecture,[status(cth)],[f356])).
% 3.18/1.84 thf(f356,conjecture,(
% 3.18/1.84 (member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ pr)),
% 3.18/1.84 file('/export/starexec/sandbox2/tmp/tmp.d5SnneaLK1/DTF2THF_13046.p',conj_0)).
% 3.18/1.84 thf(f2147,plain,(
% 3.18/1.84 (((member_a @ z @ b) & (member449909584od_a_a @ (product_Pair_a_a @ z @ y) @ pr)) = $true)),
% 3.18/1.84 inference(beta_eta_normalization,[status(thm)],[f2143])).
% 3.18/1.84 thf(f2143,plain,(
% 3.18/1.84 (((^[Y0 : a]: ((member_a @ Y0 @ b) & (member449909584od_a_a @ (product_Pair_a_a @ Y0 @ y) @ pr))) @ z) = $true)),
% 3.18/1.84 inference(superposition,[status(thm)],[f1509,f2126])).
% 3.18/1.84 thf(f2126,plain,(
% 3.18/1.84 ( ! [X2 : a > $o,X1 : a] : (((member_a @ X1 @ (collect_a @ X2)) = (X2 @ X1))) )),
% 3.18/1.84 inference(equality_proxy_clausification,[status(thm)],[f2122])).
% 3.18/1.84 thf(f2122,plain,(
% 3.18/1.84 ( ! [X2 : a > $o,X1 : a] : (($true = ((member_a @ X1 @ (collect_a @ X2)) = (X2 @ X1)))) )),
% 3.18/1.84 inference(beta_eta_normalization,[status(thm)],[f2121])).
% 3.18/1.84 thf(f2121,plain,(
% 3.18/1.84 ( ! [X2 : a > $o,X1 : a] : ((((^[Y0 : a > $o]: ((member_a @ X1 @ (collect_a @ Y0)) = (Y0 @ X1))) @ X2) = $true)) )),
% 3.18/1.84 inference(pi_clausification,[status(thm)],[f2108])).
% 3.18/1.84 thf(f2108,plain,(
% 3.18/1.84 ( ! [X1 : a] : (((!! @ (a > $o) @ (^[Y0 : a > $o]: ((member_a @ X1 @ (collect_a @ Y0)) = (Y0 @ X1)))) = $true)) )),
% 3.18/1.84 inference(beta_eta_normalization,[status(thm)],[f2107])).
% 3.18/1.84 thf(f2107,plain,(
% 3.18/1.84 ( ! [X1 : a] : ((((^[Y0 : a]: (!! @ (a > $o) @ (^[Y1 : a > $o]: ((member_a @ Y0 @ (collect_a @ Y1)) = (Y1 @ Y0))))) @ X1) = $true)) )),
% 3.18/1.84 inference(pi_clausification,[status(thm)],[f1166])).
% 3.18/1.84 thf(f1166,plain,(
% 3.18/1.84 ((!! @ a @ (^[Y0 : a]: (!! @ (a > $o) @ (^[Y1 : a > $o]: ((member_a @ Y0 @ (collect_a @ Y1)) = (Y1 @ Y0)))))) = $true)),
% 3.18/1.84 inference(cnf_transformation,[status(thm)],[f915])).
% 3.18/1.84 thf(f915,plain,(
% 3.18/1.84 ((!! @ a @ (^[Y0 : a]: (!! @ (a > $o) @ (^[Y1 : a > $o]: ((member_a @ Y0 @ (collect_a @ Y1)) = (Y1 @ Y0)))))) = $true)),
% 3.18/1.84 inference(fool_elimination,[status(thm)],[f914])).
% 3.18/1.84 thf(f914,plain,(
% 3.18/1.84 ! [X0 : a > $o,X1 : a] : ((member_a @ X1 @ (collect_a @ X0)) = (X0 @ X1))),
% 3.18/1.84 inference(rectify,[status(thm)],[f92])).
% 3.18/1.84 thf(f92,axiom,(
% 3.18/1.84 ! [X15 : a > $o,X8 : a] : ((member_a @ X8 @ (collect_a @ X15)) = (X15 @ X8))),
% 3.18/1.84 file('/export/starexec/sandbox2/tmp/tmp.d5SnneaLK1/DTF2THF_13046.p',fact_91_mem__Collect__eq)).
% 3.18/1.84 thf(f1509,plain,(
% 3.18/1.84 ((member_a @ z @ (collect_a @ (^[Y0 : a]: ((member_a @ Y0 @ b) & (member449909584od_a_a @ (product_Pair_a_a @ Y0 @ y) @ pr))))) = $true)),
% 3.18/1.84 inference(beta_eta_normalization,[status(thm)],[f1502])).
% 3.18/1.84 thf(f1502,plain,(
% 3.18/1.84 ((member_a @ z @ ((^[Y0 : a]: ((^[Y1 : set_a]: ((^[Y2 : set_Product_prod_a_a]: (collect_a @ (^[Y3 : a]: ((member_a @ Y3 @ Y1) & (member449909584od_a_a @ (product_Pair_a_a @ Y3 @ Y0) @ Y2))))))))) @ y @ b @ pr)) = $true)),
% 3.18/1.84 inference(definition_unfolding,[status(thm)],[f1402,f1309])).
% 3.18/1.84 thf(f1309,plain,(
% 3.18/1.84 (prefer161218362good_a = (^[Y0 : a]: ((^[Y1 : set_a]: ((^[Y2 : set_Product_prod_a_a]: (collect_a @ (^[Y3 : a]: ((member_a @ Y3 @ Y1) & (member449909584od_a_a @ (product_Pair_a_a @ Y3 @ Y0) @ Y2))))))))))),
% 3.18/1.84 inference(cnf_transformation,[status(thm)],[f532])).
% 3.18/1.84 thf(f532,plain,(
% 3.18/1.84 (prefer161218362good_a = (^[Y0 : a]: ((^[Y1 : set_a]: ((^[Y2 : set_Product_prod_a_a]: (collect_a @ (^[Y3 : a]: ((member_a @ Y3 @ Y1) & (member449909584od_a_a @ (product_Pair_a_a @ Y3 @ Y0) @ Y2))))))))))),
% 3.18/1.84 inference(fool_elimination,[status(thm)],[f531])).
% 3.18/1.84 thf(f531,plain,(
% 3.18/1.84 (prefer161218362good_a = (^[X0 : a, X1 : set_a, X2 : set_Product_prod_a_a] : (collect_a @ (^[X3 : a] : ((member449909584od_a_a @ (product_Pair_a_a @ X3 @ X0) @ X2) & (member_a @ X3 @ X1))))))),
% 3.18/1.84 inference(rectify,[status(thm)],[f55])).
% 3.18/1.84 thf(f55,axiom,(
% 3.18/1.84 (prefer161218362good_a = (^[X20 : a, X21 : set_a, X22 : set_Product_prod_a_a] : (collect_a @ (^[X23 : a] : ((member449909584od_a_a @ (product_Pair_a_a @ X23 @ X20) @ X22) & (member_a @ X23 @ X21))))))),
% 3.18/1.84 file('/export/starexec/sandbox2/tmp/tmp.d5SnneaLK1/DTF2THF_13046.p',fact_54_at__least__as__good__def)).
% 3.18/1.84 thf(f1402,plain,(
% 3.18/1.84 ((member_a @ z @ (prefer161218362good_a @ y @ b @ pr)) = $true)),
% 3.18/1.84 inference(cnf_transformation,[status(thm)],[f502])).
% 3.18/1.84 thf(f502,plain,(
% 3.18/1.84 ((member_a @ z @ (prefer161218362good_a @ y @ b @ pr)) = $true)),
% 3.18/1.84 inference(fool_elimination,[status(thm)],[f501])).
% 3.18/1.84 thf(f501,plain,(
% 3.18/1.84 (member_a @ z @ (prefer161218362good_a @ y @ b @ pr))),
% 3.18/1.84 inference(rectify,[status(thm)],[f1])).
% 3.18/1.84 thf(f1,axiom,(
% 3.18/1.84 (member_a @ z @ (prefer161218362good_a @ y @ b @ pr))),
% 3.18/1.84 file('/export/starexec/sandbox2/tmp/tmp.d5SnneaLK1/DTF2THF_13046.p',fact_0_assms)).
% 3.18/1.84 % SZS output end Proof for DTF2THF_13046
% 3.18/1.84 % (13238)------------------------------
% 3.18/1.84 % (13238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.84 % (13238)Termination reason: Refutation
% 3.18/1.84
% 3.18/1.84 % (13238)Memory used [KB]: 8443
% 3.18/1.84 % (13238)Time elapsed: 0.188 s
% 3.18/1.84 % (13238)Instructions burned: 418 (million)
% 3.18/1.84 % (13238)------------------------------
% 3.18/1.84 % (13238)------------------------------
% 3.18/1.84 % (13214)Success in time 0.224 s
%------------------------------------------------------------------------------