%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP176^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n017.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 08:56:10 AM UTC 2026
% Result : Theorem 3.98s 1.92s
% Output : Refutation 3.98s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 2
% Syntax : Number of formulae : 10 ( 10 unt; 0 typ; 0 def)
% Number of atoms : 24 ( 9 equ; 0 cnn)
% Maximal formula atoms : 1 ( 2 avg)
% Number of connectives : 156 ( 6 ~; 0 |; 0 &; 150 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 2 ( 2 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 145 ( 143 usr; 12 con; 0-4 aty)
% Number of variables : 14 ( 14 ^; 0 !; 0 ?; 14 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_Pr2109276827od_b_b: $tType ).
thf(type_def_6,type,
set_Product_prod_b_b: $tType ).
thf(type_def_7,type,
produc1213276845od_b_b: $tType ).
thf(type_def_8,type,
b: $tType ).
thf(type_def_9,type,
set_b: $tType ).
thf(type_def_11,type,
set_Pr2007082183od_b_b: $tType ).
thf(type_def_12,type,
product_prod_b_b: $tType ).
thf(type_def_13,type,
set_Pr1324126435od_b_b: $tType ).
thf(type_def_14,type,
standard_Constant_a: $tType ).
thf(type_def_15,type,
labele1159362096nt_a_b: $tType ).
thf(type_def_16,type,
labele1835558936od_b_b: $tType ).
thf(type_def_17,type,
set_Pr357419743_b_b_b: $tType ).
thf(type_def_18,type,
set_Pr1021918435od_b_b: $tType ).
thf(type_def_19,type,
set_St761939237tant_a: $tType ).
thf(type_def_20,type,
produc970027067od_b_b: $tType ).
thf(type_def_21,type,
produc1990339951od_b_b: $tType ).
thf(type_def_22,type,
produc276146823_b_b_b: $tType ).
thf(type_def_23,type,
produc1367125253tant_a: $tType ).
thf(type_def_24,type,
produc1052326083od_b_b: $tType ).
thf(type_def_25,type,
produc1168037607od_b_b: $tType ).
thf(type_def_26,type,
produc2112523263_b_b_b: $tType ).
thf(type_def_27,type,
produc1605346267od_b_b: $tType ).
thf(type_def_28,type,
produc255402447od_b_b: $tType ).
thf(type_def_29,type,
set_Pr2134561957od_b_b: $tType ).
thf(type_def_30,type,
set_Pr1906739389_b_b_b: $tType ).
thf(type_def_31,type,
set_Pr1071216121od_b_b: $tType ).
thf(type_def_32,type,
set_Pr2060523537od_b_b: $tType ).
thf(type_def_33,type,
set_Pr621705391od_b_b: $tType ).
thf(type_def_34,type,
set_Pr306736955tant_a: $tType ).
thf(func_def_0,type,
set_Pr2109276827od_b_b: $tType ).
thf(func_def_1,type,
produc970027067od_b_b: $tType ).
thf(func_def_2,type,
set_Pr1071216121od_b_b: $tType ).
thf(func_def_3,type,
set_Pr2134561957od_b_b: $tType ).
thf(func_def_4,type,
set_Pr1021918435od_b_b: $tType ).
thf(func_def_5,type,
produc1052326083od_b_b: $tType ).
thf(func_def_6,type,
produc1990339951od_b_b: $tType ).
thf(func_def_7,type,
set_Pr2060523537od_b_b: $tType ).
thf(func_def_8,type,
set_Pr1906739389_b_b_b: $tType ).
thf(func_def_9,type,
produc1605346267od_b_b: $tType ).
thf(func_def_10,type,
produc276146823_b_b_b: $tType ).
thf(func_def_11,type,
set_Pr306736955tant_a: $tType ).
thf(func_def_12,type,
set_Pr1324126435od_b_b: $tType ).
thf(func_def_13,type,
labele1835558936od_b_b: $tType ).
thf(func_def_14,type,
set_Pr2007082183od_b_b: $tType ).
thf(func_def_15,type,
produc1367125253tant_a: $tType ).
thf(func_def_16,type,
produc1213276845od_b_b: $tType ).
thf(func_def_17,type,
produc1168037607od_b_b: $tType ).
thf(func_def_18,type,
labele1159362096nt_a_b: $tType ).
thf(func_def_19,type,
set_Pr621705391od_b_b: $tType ).
thf(func_def_20,type,
set_Pr357419743_b_b_b: $tType ).
thf(func_def_21,type,
produc255402447od_b_b: $tType ).
thf(func_def_22,type,
produc2112523263_b_b_b: $tType ).
thf(func_def_23,type,
set_St761939237tant_a: $tType ).
thf(func_def_24,type,
set_Product_prod_b_b: $tType ).
thf(func_def_25,type,
standard_Constant_a: $tType ).
thf(func_def_26,type,
product_prod_b_b: $tType ).
thf(func_def_27,type,
set_b: $tType ).
thf(func_def_28,type,
b: $tType ).
thf(func_def_29,type,
equiv_1629197578_set_b: set_Pr2109276827od_b_b > set_Product_prod_b_b > ( produc1213276845od_b_b > b > set_b ) > $o ).
thf(func_def_30,type,
equiv_1036235914_set_b: set_Pr2007082183od_b_b > set_Product_prod_b_b > ( product_prod_b_b > b > set_b ) > $o ).
thf(func_def_31,type,
equiv_167626658_set_b: set_Product_prod_b_b > set_Product_prod_b_b > ( b > b > set_b ) > $o ).
thf(func_def_32,type,
equiv_1821124534_set_b: set_Product_prod_b_b > ( b > set_b ) > $o ).
thf(func_def_33,type,
equiv_2048442231od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o ).
thf(func_def_34,type,
equiv_1125628061od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_35,type,
equiv_equiv_b: set_b > set_Product_prod_b_b > $o ).
thf(func_def_36,type,
equiv_proj_b_b: set_Product_prod_b_b > b > set_b ).
thf(func_def_37,type,
hilber46179633od_b_b: ( produc1213276845od_b_b > $o ) > produc1213276845od_b_b ).
thf(func_def_38,type,
hilber620807523od_b_b: ( product_prod_b_b > $o ) > product_prod_b_b ).
thf(func_def_39,type,
hilbert_Eps_b: ( b > $o ) > b ).
thf(func_def_40,type,
if_b: $o > b > b > b ).
thf(func_def_41,type,
standard_S_Idt_a: standard_Constant_a ).
thf(func_def_42,type,
getRel904497637nt_a_b: standard_Constant_a > labele1159362096nt_a_b > set_Product_prod_b_b ).
thf(func_def_43,type,
graph_792023598_b_b_b: labele1835558936od_b_b > labele1159362096nt_a_b > set_Pr357419743_b_b_b > $o ).
thf(func_def_44,type,
graph_222606102_a_b_b: labele1159362096nt_a_b > labele1159362096nt_a_b > set_Product_prod_b_b > $o ).
thf(func_def_45,type,
labele170595300od_b_b: set_Pr1021918435od_b_b > set_Product_prod_b_b > labele1835558936od_b_b ).
thf(func_def_46,type,
labele1230159100nt_a_b: set_Pr1324126435od_b_b > set_b > labele1159362096nt_a_b ).
thf(func_def_47,type,
labele377702064od_b_b: ( set_Pr1324126435od_b_b > set_b > set_Pr1324126435od_b_b ) > labele1159362096nt_a_b > set_Pr1324126435od_b_b ).
thf(func_def_48,type,
labele1287464530_set_b: ( set_Pr1324126435od_b_b > set_b > set_b ) > labele1159362096nt_a_b > set_b ).
thf(func_def_49,type,
labele1741081071nt_a_b: labele1159362096nt_a_b > set_Pr1324126435od_b_b ).
thf(func_def_50,type,
labele1424214014nt_a_b: labele1159362096nt_a_b > set_b ).
thf(func_def_51,type,
restri446606278nt_a_b: labele1159362096nt_a_b > labele1159362096nt_a_b ).
thf(func_def_52,type,
order_609163036od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_53,type,
order_653178932r_on_b: set_b > set_Product_prod_b_b > $o ).
thf(func_def_54,type,
bot_bo1734638456_b_b_o: standard_Constant_a > product_prod_b_b > $o ).
thf(func_def_55,type,
bot_bo1167600078_b_b_o: produc1213276845od_b_b > $o ).
thf(func_def_56,type,
bot_bo1302443370_b_b_o: product_prod_b_b > $o ).
thf(func_def_57,type,
bot_bot_b_b_o: b > b > $o ).
thf(func_def_58,type,
bot_bot_b_o: b > $o ).
thf(func_def_59,type,
bot_bo1160111033tant_a: set_St761939237tant_a ).
thf(func_def_60,type,
bot_bo1964903287od_b_b: set_Pr1021918435od_b_b ).
thf(func_def_61,type,
bot_bo1664927607od_b_b: set_Pr1324126435od_b_b ).
thf(func_def_62,type,
bot_bo1174586163od_b_b: set_Pr2007082183od_b_b ).
thf(func_def_63,type,
bot_bo1124087115_b_b_b: set_Pr357419743_b_b_b ).
thf(func_def_64,type,
bot_bo1343651123od_b_b: set_Product_prod_b_b ).
thf(func_def_65,type,
bot_bot_set_b: set_b ).
thf(func_def_66,type,
ord_le123089219od_b_b: set_Pr1324126435od_b_b > set_Pr1324126435od_b_b > $o ).
thf(func_def_67,type,
ord_le1036320359od_b_b: set_Product_prod_b_b > set_Product_prod_b_b > $o ).
thf(func_def_68,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(func_def_69,type,
produc1432590431od_b_b: standard_Constant_a > product_prod_b_b > produc1213276845od_b_b ).
thf(func_def_70,type,
produc449289715od_b_b: produc1213276845od_b_b > produc1213276845od_b_b > produc970027067od_b_b ).
thf(func_def_71,type,
produc1597690145od_b_b: produc1213276845od_b_b > product_prod_b_b > produc1990339951od_b_b ).
thf(func_def_72,type,
produc1580777273_b_b_b: produc1213276845od_b_b > b > produc276146823_b_b_b ).
thf(func_def_73,type,
produc1977323903tant_a: product_prod_b_b > standard_Constant_a > produc1367125253tant_a ).
thf(func_def_74,type,
produc732676669od_b_b: product_prod_b_b > produc1213276845od_b_b > produc1052326083od_b_b ).
thf(func_def_75,type,
produc546497367od_b_b: product_prod_b_b > product_prod_b_b > produc1168037607od_b_b ).
thf(func_def_76,type,
produc1001682799_b_b_b: product_prod_b_b > b > produc2112523263_b_b_b ).
thf(func_def_77,type,
produc800495189od_b_b: b > produc1213276845od_b_b > produc1605346267od_b_b ).
thf(func_def_78,type,
produc1257047359od_b_b: b > product_prod_b_b > produc255402447od_b_b ).
thf(func_def_79,type,
product_Pair_b_b: b > b > product_prod_b_b ).
thf(func_def_80,type,
image_1916422435od_b_b: set_Pr1324126435od_b_b > set_St761939237tant_a > set_Product_prod_b_b ).
thf(func_def_81,type,
image_763305007od_b_b: set_Pr2109276827od_b_b > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b ).
thf(func_def_82,type,
image_1397930469od_b_b: set_Pr2134561957od_b_b > set_Pr1324126435od_b_b > set_Product_prod_b_b ).
thf(func_def_83,type,
image_1764625405_b_b_b: set_Pr1906739389_b_b_b > set_Pr1324126435od_b_b > set_b ).
thf(func_def_84,type,
image_532916993od_b_b: set_Pr1071216121od_b_b > set_Product_prod_b_b > set_Pr1324126435od_b_b ).
thf(func_def_85,type,
image_1928024979od_b_b: set_Pr2007082183od_b_b > set_Product_prod_b_b > set_Product_prod_b_b ).
thf(func_def_86,type,
image_921732011_b_b_b: set_Pr357419743_b_b_b > set_Product_prod_b_b > set_b ).
thf(func_def_87,type,
image_984343321od_b_b: set_Pr2060523537od_b_b > set_b > set_Pr1324126435od_b_b ).
thf(func_def_88,type,
image_1177096571od_b_b: set_Pr621705391od_b_b > set_b > set_Product_prod_b_b ).
thf(func_def_89,type,
image_b_b: set_Product_prod_b_b > set_b > set_b ).
thf(func_def_90,type,
conver1398856763od_b_b: set_Pr1324126435od_b_b > set_Pr306736955tant_a ).
thf(func_def_91,type,
conver1943590235tant_a: set_Pr306736955tant_a > set_Pr1324126435od_b_b ).
thf(func_def_92,type,
converse_b_b: set_Product_prod_b_b > set_Product_prod_b_b ).
thf(func_def_93,type,
refl_o1921098222od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o ).
thf(func_def_94,type,
refl_o2134473190od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_95,type,
refl_on_b: set_b > set_Product_prod_b_b > $o ).
thf(func_def_96,type,
total_771970601od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o ).
thf(func_def_97,type,
total_178717291od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_98,type,
total_on_b: set_b > set_Product_prod_b_b > $o ).
thf(func_def_99,type,
collec279084418od_b_b: ( produc1213276845od_b_b > $o ) > set_Pr1324126435od_b_b ).
thf(func_def_100,type,
collec1481886546od_b_b: ( product_prod_b_b > $o ) > set_Product_prod_b_b ).
thf(func_def_101,type,
collect_b: ( b > $o ) > set_b ).
thf(func_def_102,type,
insert1909710879tant_a: standard_Constant_a > set_St761939237tant_a > set_St761939237tant_a ).
thf(func_def_103,type,
insert2037698781od_b_b: produc1213276845od_b_b > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b ).
thf(func_def_104,type,
insert1098249399od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > set_Pr2007082183od_b_b ).
thf(func_def_105,type,
insert1952693431od_b_b: product_prod_b_b > set_Product_prod_b_b > set_Product_prod_b_b ).
thf(func_def_106,type,
insert_b: b > set_b > set_b ).
thf(func_def_107,type,
is_sin1465183929od_b_b: set_Pr1324126435od_b_b > $o ).
thf(func_def_108,type,
is_sin1713057243od_b_b: set_Product_prod_b_b > $o ).
thf(func_def_109,type,
is_singleton_b: set_b > $o ).
thf(func_def_110,type,
the_el329196508od_b_b: set_Product_prod_b_b > product_prod_b_b ).
thf(func_def_111,type,
the_elem_b: set_b > b ).
thf(func_def_112,type,
member1632892294tant_a: standard_Constant_a > set_St761939237tant_a > $o ).
thf(func_def_113,type,
member1516365892od_b_b: produc1213276845od_b_b > set_Pr1324126435od_b_b > $o ).
thf(func_def_114,type,
member1086024932od_b_b: produc970027067od_b_b > set_Pr2109276827od_b_b > $o ).
thf(func_def_115,type,
member942707974od_b_b: produc1990339951od_b_b > set_Pr2134561957od_b_b > $o ).
thf(func_def_116,type,
member1417789726_b_b_b: produc276146823_b_b_b > set_Pr1906739389_b_b_b > $o ).
thf(func_def_117,type,
member1670214300tant_a: produc1367125253tant_a > set_Pr306736955tant_a > $o ).
thf(func_def_118,type,
member4694106od_b_b: produc1052326083od_b_b > set_Pr1071216121od_b_b > $o ).
thf(func_def_119,type,
member1652792080od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_120,type,
member351494440_b_b_b: produc2112523263_b_b_b > set_Pr357419743_b_b_b > $o ).
thf(func_def_121,type,
member599505522od_b_b: produc1605346267od_b_b > set_Pr2060523537od_b_b > $o ).
thf(func_def_122,type,
member641857272od_b_b: produc255402447od_b_b > set_Pr621705391od_b_b > $o ).
thf(func_def_123,type,
member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(func_def_124,type,
member_b: b > set_b > $o ).
thf(func_def_125,type,
g: labele1159362096nt_a_b ).
thf(func_def_126,type,
p: b > b > $o ).
thf(func_def_127,type,
f: b > b ).
thf(func_def_128,type,
x: b ).
thf(func_def_129,type,
y: b ).
thf(func_def_145,type,
sP0: set_b > b > b > set_b > $o ).
thf(func_def_146,type,
sP1: set_b > b > b > set_b > $o ).
thf(func_def_147,type,
sK2: product_prod_b_b > b ).
thf(func_def_148,type,
sK3: product_prod_b_b > b ).
thf(func_def_149,type,
sK4: ( b > $o ) > ( b > $o ) > b ).
thf(func_def_150,type,
sK5: ( product_prod_b_b > $o ) > b ).
thf(func_def_151,type,
sK6: ( product_prod_b_b > $o ) > b ).
thf(func_def_152,type,
sK7: set_Product_prod_b_b > product_prod_b_b ).
thf(func_def_153,type,
sK8: b > set_b > set_Product_prod_b_b > b ).
thf(func_def_154,type,
sK9: ( b > $o ) > ( b > $o ) > b ).
thf(func_def_155,type,
sK10: set_b > b > b > set_b > set_b ).
thf(func_def_156,type,
sK11: b > ( b > $o ) > b ).
thf(func_def_157,type,
sK12: set_Product_prod_b_b > b > set_b > b ).
thf(func_def_158,type,
sK13: set_Product_prod_b_b > product_prod_b_b ).
thf(func_def_159,type,
sK14: ( produc1213276845od_b_b > $o ) > b ).
thf(func_def_160,type,
sK15: ( produc1213276845od_b_b > $o ) > standard_Constant_a ).
thf(func_def_161,type,
sK16: ( produc1213276845od_b_b > $o ) > b ).
thf(func_def_162,type,
sK17: set_Product_prod_b_b > product_prod_b_b ).
thf(func_def_163,type,
sK18: b > set_b > set_b ).
thf(func_def_164,type,
sK19: ( product_prod_b_b > $o ) > b ).
thf(func_def_165,type,
sK20: ( product_prod_b_b > $o ) > b ).
thf(func_def_166,type,
sK21: set_b > b ).
thf(func_def_167,type,
sK22: ( b > $o ) > ( b > $o ) > b ).
thf(func_def_168,type,
sK23: set_b > b > set_b ).
thf(func_def_169,type,
sK24: product_prod_b_b > b ).
thf(func_def_170,type,
sK25: product_prod_b_b > b ).
thf(func_def_171,type,
sK26: ( b > $o ) > b > b ).
thf(func_def_172,type,
sK27: produc1213276845od_b_b > standard_Constant_a ).
thf(func_def_173,type,
sK28: produc1213276845od_b_b > product_prod_b_b ).
thf(func_def_174,type,
sK29: set_b > b ).
thf(func_def_175,type,
sK30: ( produc1213276845od_b_b > $o ) > product_prod_b_b ).
thf(func_def_176,type,
sK31: ( produc1213276845od_b_b > $o ) > standard_Constant_a ).
thf(func_def_177,type,
sK32: set_b > b ).
thf(func_def_178,type,
sK33: produc1213276845od_b_b > b ).
thf(func_def_179,type,
sK34: produc1213276845od_b_b > b ).
thf(func_def_180,type,
sK35: produc1213276845od_b_b > standard_Constant_a ).
thf(func_def_181,type,
sK36: ( produc1213276845od_b_b > $o ) > product_prod_b_b ).
thf(func_def_182,type,
sK37: ( produc1213276845od_b_b > $o ) > standard_Constant_a ).
thf(func_def_183,type,
sK38: ( b > $o ) > b ).
thf(func_def_184,type,
sK39: produc1213276845od_b_b > product_prod_b_b ).
thf(func_def_185,type,
sK40: produc1213276845od_b_b > standard_Constant_a ).
thf(func_def_187,type,
ph42:
!>[X0: $tType] : X0 ).
thf(f1307,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1306]) ).
thf(f1306,plain,
( ( hilbert_Eps_b
@ ^ [Y0: b] : ( member_b @ Y0 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) )
!= ( hilbert_Eps_b
@ ^ [Y0: b] : ( member_b @ Y0 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1218,f1165]) ).
thf(f1165,plain,
( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) )
= ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) )
= ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ),
file('/export/starexec/sandbox/tmp/tmp.4auU2Smifg/DTF2THF_20997.p',fact_0__092_060open_062getRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_AgetRel_AS__Idt_AG_A_096_096_A_123y_125_092_060close_062) ).
thf(f1218,plain,
( ( hilbert_Eps_b
@ ^ [Y0: b] : ( member_b @ Y0 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) )
!= ( hilbert_Eps_b
@ ^ [Y0: b] : ( member_b @ Y0 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f966]) ).
thf(f966,plain,
( ( hilbert_Eps_b
@ ^ [Y0: b] : ( member_b @ Y0 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) )
!= ( hilbert_Eps_b
@ ^ [Y0: b] : ( member_b @ Y0 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) ) ),
inference(flattening,[status(thm)],[f429]) ).
thf(f429,plain,
( ( hilbert_Eps_b
@ ^ [Y0: b] : ( member_b @ Y0 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) )
!= ( hilbert_Eps_b
@ ^ [Y0: b] : ( member_b @ Y0 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f428]) ).
thf(f428,plain,
( ( hilbert_Eps_b
@ ^ [X0: b] : ( member_b @ X0 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) )
!= ( hilbert_Eps_b
@ ^ [X1: b] : ( member_b @ X1 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) ) ),
inference(rectify,[status(thm)],[f360]) ).
thf(f360,negated_conjecture,
( ( hilbert_Eps_b
@ ^ [X2: b] : ( member_b @ X2 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) )
!= ( hilbert_Eps_b
@ ^ [X2: b] : ( member_b @ X2 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f359]) ).
thf(f359,conjecture,
( ( hilbert_Eps_b
@ ^ [X2: b] : ( member_b @ X2 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) )
= ( hilbert_Eps_b
@ ^ [X2: b] : ( member_b @ X2 @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.4auU2Smifg/DTF2THF_20997.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP176^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n017.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:27:26 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : ITP176^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.46 % Domain : Interactive Theorem Proving
% 0.19/0.46 % Problem : Sledgehammer StandardRules problem prob_305__5389912_1
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Des21]
% 0.19/0.46 % Names : StandardRules/prob_305__5389912_1 [Des21]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.19/0.46 % Syntax : Number of formulae : 489 ( 184 unt; 130 typ; 0 def)
% 0.19/0.46 % Number of atoms : 970 ( 382 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 6 ( 2 avg)
% 0.19/0.46 % Number of connectives : 2891 ( 117 ~; 17 |; 75 &;2345 @)
% 0.19/0.46 % ( 0 <=>; 337 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 14 ( 6 avg)
% 0.19/0.46 % Number of types : 30 ( 29 usr)
% 0.19/0.46 % Number of type conns : 243 ( 243 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 104 ( 101 usr; 13 con; 0-3 aty)
% 0.19/0.46 % Number of variables : 978 ( 93 ^; 855 !; 30 ?; 978 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:37:50.745
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % Could-be-implicit typings (29)
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
% 0.19/0.46 set_Pr2109276827od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 produc970027067od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
% 0.19/0.46 set_Pr1071216121od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 set_Pr2134561957od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
% 0.19/0.46 set_Pr1021918435od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 produc1052326083od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 produc1990339951od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
% 0.19/0.46 set_Pr2060523537od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mtf__b_J_J,type,
% 0.19/0.46 set_Pr1906739389_b_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 produc1605346267od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mtf__b_J,type,
% 0.19/0.46 produc276146823_b_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J_J,type,
% 0.19/0.46 set_Pr306736955tant_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 set_Pr1324126435od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 labele1835558936od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 set_Pr2007082183od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J,type,
% 0.19/0.46 produc1367125253tant_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 produc1213276845od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 produc1168037607od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__LabeledGraphs__Olabeled____graph_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mtf__b_J,type,
% 0.19/0.46 labele1159362096nt_a_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 set_Pr621705391od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mtf__b_J_J,type,
% 0.19/0.46 set_Pr357419743_b_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 produc255402447od_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mtf__b_J,type,
% 0.19/0.46 produc2112523263_b_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J,type,
% 0.19/0.46 set_St761939237tant_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 set_Product_prod_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
% 0.19/0.46 standard_Constant_a: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
% 0.19/0.46 product_prod_b_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_Itf__b_J,type,
% 0.19/0.46 set_b: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_tf__b,type,
% 0.19/0.46 b: $tType ).
% 0.19/0.46
% 0.19/0.46 % Explicit typings (101)
% 0.19/0.46 thf(sy_c_Equiv__Relations_Ocongruent2_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_001tf__b_001t__Set__Oset_Itf__b_J,type,
% 0.19/0.46 equiv_1629197578_set_b: set_Pr2109276827od_b_b > set_Product_prod_b_b > ( produc1213276845od_b_b > b > set_b ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Equiv__Relations_Ocongruent2_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b_001t__Set__Oset_Itf__b_J,type,
% 0.19/0.46 equiv_1036235914_set_b: set_Pr2007082183od_b_b > set_Product_prod_b_b > ( product_prod_b_b > b > set_b ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Equiv__Relations_Ocongruent2_001tf__b_001tf__b_001t__Set__Oset_Itf__b_J,type,
% 0.19/0.46 equiv_167626658_set_b: set_Product_prod_b_b > set_Product_prod_b_b > ( b > b > set_b ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Equiv__Relations_Ocongruent_001tf__b_001t__Set__Oset_Itf__b_J,type,
% 0.19/0.46 equiv_1821124534_set_b: set_Product_prod_b_b > ( b > set_b ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 equiv_2048442231od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
% 0.19/0.46 equiv_1125628061od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Equiv__Relations_Oequiv_001tf__b,type,
% 0.19/0.46 equiv_equiv_b: set_b > set_Product_prod_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Equiv__Relations_Oproj_001tf__b_001tf__b,type,
% 0.19/0.46 equiv_proj_b_b: set_Product_prod_b_b > b > set_b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Hilbert__Choice_OEps_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 hilber46179633od_b_b: ( produc1213276845od_b_b > $o ) > produc1213276845od_b_b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Hilbert__Choice_OEps_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
% 0.19/0.46 hilber620807523od_b_b: ( product_prod_b_b > $o ) > product_prod_b_b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Hilbert__Choice_OEps_001tf__b,type,
% 0.19/0.46 hilbert_Eps_b: ( b > $o ) > b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_If_001tf__b,type,
% 0.19/0.46 if_b: $o > b > b > b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_LabeledGraphSemantics_OStandard__Constant_OS__Idt_001tf__a,type,
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% 0.19/0.46 thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 is_sin1465183929od_b_b: set_Pr1324126435od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
% 0.19/0.46 is_sin1713057243od_b_b: set_Product_prod_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Ois__singleton_001tf__b,type,
% 0.19/0.46 is_singleton_b: set_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
% 0.19/0.46 the_el329196508od_b_b: set_Product_prod_b_b > product_prod_b_b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Othe__elem_001tf__b,type,
% 0.19/0.46 the_elem_b: set_b > b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__LabeledGraphSemantics__OStandard____Constant_Itf__a_J,type,
% 0.19/0.46 member1632892294tant_a: standard_Constant_a > set_St761939237tant_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 member1516365892od_b_b: produc1213276845od_b_b > set_Pr1324126435od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 member1086024932od_b_b: produc970027067od_b_b > set_Pr2109276827od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 member942707974od_b_b: produc1990339951od_b_b > set_Pr2134561957od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mtf__b_J,type,
% 0.19/0.46 member1417789726_b_b_b: produc276146823_b_b_b > set_Pr1906739389_b_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_J,type,
% 0.19/0.46 member1670214300tant_a: produc1367125253tant_a > set_Pr306736955tant_a > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 member4694106od_b_b: produc1052326083od_b_b > set_Pr1071216121od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 member1652792080od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mtf__b_J,type,
% 0.19/0.46 member351494440_b_b_b: produc2112523263_b_b_b > set_Pr357419743_b_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 0.19/0.46 member599505522od_b_b: produc1605346267od_b_b > set_Pr2060523537od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.19/0.46 member641857272od_b_b: produc255402447od_b_b > set_Pr621705391od_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
% 0.19/0.46 member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001tf__b,type,
% 0.19/0.46 member_b: b > set_b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_G,type,
% 0.19/0.46 g: labele1159362096nt_a_b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_P____,type,
% 0.19/0.46 p: b > b > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_f____,type,
% 0.19/0.46 f: b > b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_x____,type,
% 0.19/0.46 x: b ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_y____,type,
% 0.19/0.46 y: b ).
% 0.19/0.46
% 0.19/0.46 % Relevant facts (355)
% 0.19/0.46 thf(fact_0__092_060open_062getRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_AgetRel_AS__Idt_AG_A_096_096_A_123y_125_092_060close_062,axiom,
% 0.19/0.46 ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) )
% 0.19/0.46 = ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>getRel S_Idt G `` {x} = getRel S_Idt G `` {y}\<close>
% 0.19/0.46 thf(fact_1_assms_I1_J,axiom,
% 0.19/0.46 ( g
% 0.19/0.46 = ( restri446606278nt_a_b @ g ) ) ).
% 0.19/0.46
% 0.19/0.46 % assms(1)
% 0.19/0.46 thf(fact_2_P__eq,axiom,
% 0.19/0.46 ! [X: b] :
% 0.19/0.46 ( ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X @ bot_bot_set_b ) )
% 0.19/0.46 != bot_bot_set_b )
% 0.19/0.46 => ( ( hilbert_Eps_b @ ( p @ ( hilbert_Eps_b @ ( p @ X ) ) ) )
% 0.19/0.46 = ( hilbert_Eps_b @ ( p @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % P_eq
% 0.19/0.46 thf(fact_3_P,axiom,
% 0.19/0.46 ( p
% 0.19/0.46 = ( ^ [X2: b,Y: b] : ( member_b @ Y @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % P
% 0.19/0.46 thf(fact_4__092_060open_062_Ix_M_Ay_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
% 0.19/0.46 member1285940496od_b_b @ ( product_Pair_b_b @ x @ y ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>(x, y) \<in> getRel S_Idt G\<close>
% 0.19/0.46 thf(fact_5__092_060open_062f_A_092_060equiv_062_A_092_060lambda_062x_O_Aif_AgetRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_A_123_125_Athen_Ax_Aelse_AEps_A_IP_Ax_J_092_060close_062,axiom,
% 0.19/0.46 ( f
% 0.19/0.46 = ( ^ [X2: b] :
% 0.19/0.46 ( if_b
% 0.19/0.46 @ ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
% 0.19/0.46 = bot_bot_set_b )
% 0.19/0.46 @ X2
% 0.19/0.46 @ ( hilbert_Eps_b @ ( p @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>f \<equiv> \<lambda>x. if getRel S_Idt G `` {x} = {} then x else Eps (P x)\<close>
% 0.19/0.46 thf(fact_6_singleton__conv,axiom,
% 0.19/0.46 ! [A: product_prod_b_b] :
% 0.19/0.46 ( ( collec1481886546od_b_b
% 0.19/0.46 @ ^ [X2: product_prod_b_b] : ( X2 = A ) )
% 0.19/0.46 = ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % singleton_conv
% 0.19/0.46 thf(fact_7_singleton__conv,axiom,
% 0.19/0.46 ! [A: b] :
% 0.19/0.46 ( ( collect_b
% 0.19/0.46 @ ^ [X2: b] : ( X2 = A ) )
% 0.19/0.46 = ( insert_b @ A @ bot_bot_set_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % singleton_conv
% 0.19/0.46 thf(fact_8_singleton__conv2,axiom,
% 0.19/0.46 ! [A: product_prod_b_b] :
% 0.19/0.46 ( ( collec1481886546od_b_b
% 0.19/0.46 @ ( ^ [Y2: product_prod_b_b,Z: product_prod_b_b] : ( Y2 = Z )
% 0.19/0.46 @ A ) )
% 0.19/0.46 = ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % singleton_conv2
% 0.19/0.46 thf(fact_9_singleton__conv2,axiom,
% 0.19/0.46 ! [A: b] :
% 0.19/0.46 ( ( collect_b
% 0.19/0.46 @ ( ^ [Y2: b,Z: b] : ( Y2 = Z )
% 0.19/0.46 @ A ) )
% 0.19/0.46 = ( insert_b @ A @ bot_bot_set_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % singleton_conv2
% 0.19/0.46 thf(fact_10_tuple__disj,axiom,
% 0.19/0.46 ! [X: product_prod_b_b,Z2: product_prod_b_b] :
% 0.19/0.46 ( ( collec1481886546od_b_b
% 0.19/0.46 @ ^ [Y: product_prod_b_b] :
% 0.19/0.46 ( ( Y = X )
% 0.19/0.46 | ( Y = Z2 ) ) )
% 0.19/0.46 = ( insert1952693431od_b_b @ X @ ( insert1952693431od_b_b @ Z2 @ bot_bo1343651123od_b_b ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tuple_disj
% 0.19/0.46 thf(fact_11_tuple__disj,axiom,
% 0.19/0.46 ! [X: b,Z2: b] :
% 0.19/0.46 ( ( collect_b
% 0.19/0.46 @ ^ [Y: b] :
% 0.19/0.46 ( ( Y = X )
% 0.19/0.46 | ( Y = Z2 ) ) )
% 0.19/0.46 = ( insert_b @ X @ ( insert_b @ Z2 @ bot_bot_set_b ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tuple_disj
% 0.19/0.46 thf(fact_12_Image__empty1,axiom,
% 0.19/0.46 ! [X3: set_b] :
% 0.19/0.46 ( ( image_b_b @ bot_bo1343651123od_b_b @ X3 )
% 0.19/0.46 = bot_bot_set_b ) ).
% 0.19/0.46
% 0.19/0.46 % Image_empty1
% 0.19/0.46 thf(fact_13_Image__empty2,axiom,
% 0.19/0.46 ! [R: set_Product_prod_b_b] :
% 0.19/0.46 ( ( image_b_b @ R @ bot_bot_set_b )
% 0.19/0.46 = bot_bot_set_b ) ).
% 0.19/0.46
% 0.19/0.46 % Image_empty2
% 0.19/0.46 thf(fact_14_Image__empty2,axiom,
% 0.19/0.46 ! [R: set_Pr621705391od_b_b] :
% 0.19/0.46 ( ( image_1177096571od_b_b @ R @ bot_bot_set_b )
% 0.19/0.46 = bot_bo1343651123od_b_b ) ).
% 0.19/0.46
% 0.19/0.46 % Image_empty2
% 0.19/0.46 thf(fact_15_Image__empty2,axiom,
% 0.19/0.46 ! [R: set_Pr357419743_b_b_b] :
% 0.19/0.46 ( ( image_921732011_b_b_b @ R @ bot_bo1343651123od_b_b )
% 0.19/0.46 = bot_bot_set_b ) ).
% 0.19/0.46
% 0.19/0.46 % Image_empty2
% 0.19/0.46 thf(fact_16_Image__empty2,axiom,
% 0.19/0.46 ! [R: set_Pr2007082183od_b_b] :
% 0.19/0.46 ( ( image_1928024979od_b_b @ R @ bot_bo1343651123od_b_b )
% 0.19/0.46 = bot_bo1343651123od_b_b ) ).
% 0.19/0.46
% 0.19/0.46 % Image_empty2
% 0.19/0.46 thf(fact_17_singletonI,axiom,
% 0.19/0.46 ! [A: produc1213276845od_b_b] : ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % singletonI
% 0.19/0.46 thf(fact_18_singletonI,axiom,
% 0.19/0.46 ! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % singletonI
% 0.19/0.46 thf(fact_19_singletonI,axiom,
% 0.19/0.46 ! [A: product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % singletonI
% 0.19/0.46 thf(fact_20_some__equality,axiom,
% 0.19/0.46 ! [P: b > $o,A: b] :
% 0.19/0.46 ( ( P @ A )
% 0.19/0.46 => ( ! [X4: b] :
% 0.19/0.46 ( ( P @ X4 )
% 0.19/0.46 => ( X4 = A ) )
% 0.19/0.46 => ( ( hilbert_Eps_b @ P )
% 0.19/0.46 = A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % some_equality
% 0.19/0.46 thf(fact_21_some__eq__trivial,axiom,
% 0.19/0.46 ! [X: b] :
% 0.19/0.46 ( ( hilbert_Eps_b
% 0.19/0.46 @ ^ [Y: b] : ( Y = X ) )
% 0.19/0.46 = X ) ).
% 0.19/0.46
% 0.19/0.46 % some_eq_trivial
% 0.19/0.46 thf(fact_22_empty__Collect__eq,axiom,
% 0.19/0.46 ! [P: b > $o] :
% 0.19/0.46 ( ( bot_bot_set_b
% 0.19/0.46 = ( collect_b @ P ) )
% 0.19/0.46 = ( ! [X2: b] :
% 0.19/0.46 ~ ( P @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % empty_Collect_eq
% 0.19/0.46 thf(fact_23_empty__Collect__eq,axiom,
% 0.19/0.46 ! [P: product_prod_b_b > $o] :
% 0.19/0.46 ( ( bot_bo1343651123od_b_b
% 0.19/0.46 = ( collec1481886546od_b_b @ P ) )
% 0.19/0.46 = ( ! [X2: product_prod_b_b] :
% 0.19/0.46 ~ ( P @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % empty_Collect_eq
% 0.19/0.46 thf(fact_24_Collect__empty__eq,axiom,
% 0.19/0.46 ! [P: b > $o] :
% 0.19/0.46 ( ( ( collect_b @ P )
% 0.19/0.46 = bot_bot_set_b )
% 0.19/0.46 = ( ! [X2: b] :
% 0.19/0.46 ~ ( P @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_empty_eq
% 0.19/0.46 thf(fact_25_Collect__empty__eq,axiom,
% 0.19/0.46 ! [P: product_prod_b_b > $o] :
% 0.19/0.46 ( ( ( collec1481886546od_b_b @ P )
% 0.19/0.46 = bot_bo1343651123od_b_b )
% 0.19/0.46 = ( ! [X2: product_prod_b_b] :
% 0.19/0.46 ~ ( P @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_empty_eq
% 0.19/0.46 thf(fact_26_all__not__in__conv,axiom,
% 0.19/0.46 ! [A2: set_Pr1324126435od_b_b] :
% 0.19/0.46 ( ( ! [X2: produc1213276845od_b_b] :
% 0.19/0.46 ~ ( member1516365892od_b_b @ X2 @ A2 ) )
% 0.19/0.46 = ( A2 = bot_bo1664927607od_b_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % all_not_in_conv
% 0.19/0.46 thf(fact_27_all__not__in__conv,axiom,
% 0.19/0.46 ! [A2: set_b] :
% 0.19/0.46 ( ( ! [X2: b] :
% 0.19/0.46 ~ ( member_b @ X2 @ A2 ) )
% 0.19/0.46 = ( A2 = bot_bot_set_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % all_not_in_conv
% 0.19/0.46 thf(fact_28_all__not__in__conv,axiom,
% 0.19/0.46 ! [A2: set_Product_prod_b_b] :
% 0.19/0.46 ( ( ! [X2: product_prod_b_b] :
% 0.19/0.46 ~ ( member1285940496od_b_b @ X2 @ A2 ) )
% 0.19/0.46 = ( A2 = bot_bo1343651123od_b_b ) ) ).
% 0.19/0.46
% 0.19/0.46 % all_not_in_conv
% 0.19/0.46 thf(fact_29_empty__iff,axiom,
% 0.19/0.46 ! [C: produc1213276845od_b_b] :
% 0.19/0.46 ~ ( member1516365892od_b_b @ C @ bot_bo1664927607od_b_b ) ).
% 0.19/0.46
% 0.19/0.46 % empty_iff
% 0.19/0.46 thf(fact_30_empty__iff,axiom,
% 0.19/0.46 ! [C: b] :
% 0.19/0.46 ~ ( member_b @ C @ bot_bot_set_b ) ).
% 0.19/0.46
% 0.19/0.46 % empty_iff
% 0.19/0.46 thf(fact_31_empty__iff,axiom,
% 0.19/0.46 ! [C: product_prod_b_b] :
% 0.19/0.46 ~ ( member1285940496od_b_b @ C @ bot_bo1343651123od_b_b ) ).
% 0.19/0.46
% 0.19/0.46 % empty_iff
% 0.19/0.46 thf(fact_32_insert__absorb2,axiom,
% 0.19/0.46 ! [X: b,A2: set_b] :
% 0.19/0.46 ( ( insert_b @ X @ ( insert_b @ X @ A2 ) )
% 0.19/0.46 = ( insert_b @ X @ A2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_absorb2
% 0.19/0.46 thf(fact_33_insert__iff,axiom,
% 0.19/0.46 ! [A: b,B: b,A2: set_b] :
% 0.19/0.46 ( ( member_b @ A @ ( insert_b @ B @ A2 ) )
% 0.19/0.46 = ( ( A = B )
% 0.19/0.46 | ( member_b @ A @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_iff
% 0.19/0.46 thf(fact_34_insert__iff,axiom,
% 0.19/0.46 ! [A: product_prod_b_b,B: product_prod_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.46 ( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A2 ) )
% 0.19/0.46 = ( ( A = B )
% 0.19/0.46 | ( member1285940496od_b_b @ A @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_iff
% 0.19/0.46 thf(fact_35_insert__iff,axiom,
% 0.19/0.46 ! [A: produc1213276845od_b_b,B: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.46 ( ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ A2 ) )
% 0.19/0.46 = ( ( A = B )
% 0.19/0.46 | ( member1516365892od_b_b @ A @ A2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insert_iff
% 0.19/0.46 thf(fact_36_insertCI,axiom,
% 0.19/0.46 ! [A: b,B2: set_b,B: b] :
% 0.19/0.46 ( ( ~ ( member_b @ A @ B2 )
% 0.19/0.46 => ( A = B ) )
% 0.19/0.46 => ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insertCI
% 0.19/0.46 thf(fact_37_insertCI,axiom,
% 0.19/0.46 ! [A: product_prod_b_b,B2: set_Product_prod_b_b,B: product_prod_b_b] :
% 0.19/0.46 ( ( ~ ( member1285940496od_b_b @ A @ B2 )
% 0.19/0.46 => ( A = B ) )
% 0.19/0.46 => ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insertCI
% 0.19/0.46 thf(fact_38_insertCI,axiom,
% 0.19/0.46 ! [A: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b] :
% 0.19/0.46 ( ( ~ ( member1516365892od_b_b @ A @ B2 )
% 0.19/0.46 => ( A = B ) )
% 0.19/0.46 => ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ B2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % insertCI
% 0.19/0.46 thf(fact_39_some__sym__eq__trivial,axiom,
% 0.19/0.46 ! [X: b] :
% 0.19/0.46 ( ( hilbert_Eps_b
% 0.19/0.46 @ ( ^ [Y2: b,Z: b] : ( Y2 = Z )
% 0.19/0.46 @ X ) )
% 0.19/0.46 = X ) ).
% 0.19/0.46
% 0.19/0.46 % some_sym_eq_trivial
% 0.19/0.46 thf(fact_40_ImageI,axiom,
% 0.19/0.46 ! [A: b,B: product_prod_b_b,R2: set_Pr621705391od_b_b,A2: set_b] :
% 0.19/0.46 ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member_b @ A @ A2 )
% 0.19/0.46 => ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_41_ImageI,axiom,
% 0.19/0.46 ! [A: b,B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b,A2: set_b] :
% 0.19/0.46 ( ( member599505522od_b_b @ ( produc800495189od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member_b @ A @ A2 )
% 0.19/0.46 => ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_42_ImageI,axiom,
% 0.19/0.46 ! [A: product_prod_b_b,B: b,R2: set_Pr357419743_b_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.46 ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.46 => ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_43_ImageI,axiom,
% 0.19/0.46 ! [A: product_prod_b_b,B: product_prod_b_b,R2: set_Pr2007082183od_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.46 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.46 => ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_44_ImageI,axiom,
% 0.19/0.46 ! [A: product_prod_b_b,B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.46 ( ( member4694106od_b_b @ ( produc732676669od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.46 => ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_45_ImageI,axiom,
% 0.19/0.46 ! [A: produc1213276845od_b_b,B: b,R2: set_Pr1906739389_b_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.46 ( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.46 => ( member_b @ B @ ( image_1764625405_b_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_46_ImageI,axiom,
% 0.19/0.46 ! [A: produc1213276845od_b_b,B: product_prod_b_b,R2: set_Pr2134561957od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.46 ( ( member942707974od_b_b @ ( produc1597690145od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.46 => ( member1285940496od_b_b @ B @ ( image_1397930469od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_47_ImageI,axiom,
% 0.19/0.46 ! [A: produc1213276845od_b_b,B: produc1213276845od_b_b,R2: set_Pr2109276827od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.46 ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.46 => ( member1516365892od_b_b @ B @ ( image_763305007od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_48_ImageI,axiom,
% 0.19/0.46 ! [A: b,B: b,R2: set_Product_prod_b_b,A2: set_b] :
% 0.19/0.46 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member_b @ A @ A2 )
% 0.19/0.46 => ( member_b @ B @ ( image_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_49_ImageI,axiom,
% 0.19/0.46 ! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A2: set_St761939237tant_a] :
% 0.19/0.46 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( ( member1632892294tant_a @ A @ A2 )
% 0.19/0.46 => ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ImageI
% 0.19/0.46 thf(fact_50_f,axiom,
% 0.19/0.46 ( f
% 0.19/0.46 = ( ^ [X2: b] :
% 0.19/0.46 ( if_b
% 0.19/0.46 @ ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
% 0.19/0.46 = bot_bot_set_b )
% 0.19/0.46 @ X2
% 0.19/0.46 @ ( hilbert_Eps_b @ ( p @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % f
% 0.19/0.46 thf(fact_51_Image__singleton__iff,axiom,
% 0.19/0.46 ! [B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A: standard_Constant_a] :
% 0.19/0.46 ( ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) ) )
% 0.19/0.46 = ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Image_singleton_iff
% 0.19/0.46 thf(fact_52_Image__singleton__iff,axiom,
% 0.19/0.46 ! [B: product_prod_b_b,R2: set_Pr621705391od_b_b,A: b] :
% 0.19/0.46 ( ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
% 0.19/0.46 = ( member641857272od_b_b @ ( produc1257047359od_b_b @ A @ B ) @ R2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Image_singleton_iff
% 0.19/0.46 thf(fact_53_Image__singleton__iff,axiom,
% 0.19/0.46 ! [B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b,A: b] :
% 0.19/0.46 ( ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
% 0.19/0.46 = ( member599505522od_b_b @ ( produc800495189od_b_b @ A @ B ) @ R2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Image_singleton_iff
% 0.19/0.46 thf(fact_54_Image__singleton__iff,axiom,
% 0.19/0.46 ! [B: b,R2: set_Product_prod_b_b,A: b] :
% 0.19/0.46 ( ( member_b @ B @ ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
% 0.19/0.46 = ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Image_singleton_iff
% 0.19/0.46 thf(fact_55_Image__singleton__iff,axiom,
% 0.19/0.46 ! [B: b,R2: set_Pr357419743_b_b_b,A: product_prod_b_b] :
% 0.19/0.46 ( ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 0.19/0.46 = ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B ) @ R2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Image_singleton_iff
% 0.19/0.46 thf(fact_56_Image__singleton__iff,axiom,
% 0.19/0.46 ! [B: product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
% 0.19/0.46 ( ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 0.19/0.46 = ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Image_singleton_iff
% 0.19/0.46 thf(fact_57_Image__singleton__iff,axiom,
% 0.19/0.46 ! [B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b,A: product_prod_b_b] :
% 0.19/0.46 ( ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 0.19/0.46 = ( member4694106od_b_b @ ( produc732676669od_b_b @ A @ B ) @ R2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Image_singleton_iff
% 0.19/0.46 thf(fact_58_pred__equals__eq2,axiom,
% 0.19/0.46 ! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
% 0.19/0.46 ( ( ( ^ [X2: b,Y: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R ) )
% 0.19/0.46 = ( ^ [X2: b,Y: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ S ) ) )
% 0.19/0.46 = ( R = S ) ) ).
% 0.19/0.46
% 0.19/0.46 % pred_equals_eq2
% 0.19/0.46 thf(fact_59_pred__equals__eq2,axiom,
% 0.19/0.46 ! [R: set_Pr1324126435od_b_b,S: set_Pr1324126435od_b_b] :
% 0.19/0.46 ( ( ( ^ [X2: standard_Constant_a,Y: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ Y ) @ R ) )
% 0.19/0.46 = ( ^ [X2: standard_Constant_a,Y: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ Y ) @ S ) ) )
% 0.19/0.46 = ( R = S ) ) ).
% 0.19/0.46
% 0.19/0.46 % pred_equals_eq2
% 0.19/0.46 thf(fact_60_rev__ImageI,axiom,
% 0.19/0.46 ! [A: b,A2: set_b,B: product_prod_b_b,R2: set_Pr621705391od_b_b] :
% 0.19/0.46 ( ( member_b @ A @ A2 )
% 0.19/0.46 => ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_ImageI
% 0.19/0.46 thf(fact_61_rev__ImageI,axiom,
% 0.19/0.46 ! [A: b,A2: set_b,B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b] :
% 0.19/0.46 ( ( member_b @ A @ A2 )
% 0.19/0.46 => ( ( member599505522od_b_b @ ( produc800495189od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_ImageI
% 0.19/0.46 thf(fact_62_rev__ImageI,axiom,
% 0.19/0.46 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: b,R2: set_Pr357419743_b_b_b] :
% 0.19/0.46 ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.46 => ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_ImageI
% 0.19/0.46 thf(fact_63_rev__ImageI,axiom,
% 0.19/0.46 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: product_prod_b_b,R2: set_Pr2007082183od_b_b] :
% 0.19/0.46 ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.46 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_ImageI
% 0.19/0.46 thf(fact_64_rev__ImageI,axiom,
% 0.19/0.46 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b] :
% 0.19/0.46 ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.46 => ( ( member4694106od_b_b @ ( produc732676669od_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_ImageI
% 0.19/0.46 thf(fact_65_rev__ImageI,axiom,
% 0.19/0.46 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: b,R2: set_Pr1906739389_b_b_b] :
% 0.19/0.46 ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.46 => ( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ A @ B ) @ R2 )
% 0.19/0.46 => ( member_b @ B @ ( image_1764625405_b_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % rev_ImageI
% 0.19/0.46 thf(fact_66_rev__ImageI,axiom,
% 0.19/0.46 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: product_prod_b_b,R2: set_Pr2134561957od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.47 => ( ( member942707974od_b_b @ ( produc1597690145od_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( member1285940496od_b_b @ B @ ( image_1397930469od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_ImageI
% 0.19/0.47 thf(fact_67_rev__ImageI,axiom,
% 0.19/0.47 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b,R2: set_Pr2109276827od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.47 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( member1516365892od_b_b @ B @ ( image_763305007od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_ImageI
% 0.19/0.47 thf(fact_68_rev__ImageI,axiom,
% 0.19/0.47 ! [A: b,A2: set_b,B: b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member_b @ A @ A2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( member_b @ B @ ( image_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_ImageI
% 0.19/0.47 thf(fact_69_rev__ImageI,axiom,
% 0.19/0.47 ! [A: standard_Constant_a,A2: set_St761939237tant_a,B: product_prod_b_b,R2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1632892294tant_a @ A @ A2 )
% 0.19/0.47 => ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_ImageI
% 0.19/0.47 thf(fact_70_Image__iff,axiom,
% 0.19/0.47 ! [B: b,R2: set_Product_prod_b_b,A2: set_b] :
% 0.19/0.47 ( ( member_b @ B @ ( image_b_b @ R2 @ A2 ) )
% 0.19/0.47 = ( ? [X2: b] :
% 0.19/0.47 ( ( member_b @ X2 @ A2 )
% 0.19/0.47 & ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ B ) @ R2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Image_iff
% 0.19/0.47 thf(fact_71_Image__iff,axiom,
% 0.19/0.47 ! [B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A2: set_St761939237tant_a] :
% 0.19/0.47 ( ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) )
% 0.19/0.47 = ( ? [X2: standard_Constant_a] :
% 0.19/0.47 ( ( member1632892294tant_a @ X2 @ A2 )
% 0.19/0.47 & ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ B ) @ R2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Image_iff
% 0.19/0.47 thf(fact_72_ImageE,axiom,
% 0.19/0.47 ! [B: b,R2: set_Pr357419743_b_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: product_prod_b_b] :
% 0.19/0.47 ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member1285940496od_b_b @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_73_ImageE,axiom,
% 0.19/0.47 ! [B: b,R2: set_Pr1906739389_b_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member_b @ B @ ( image_1764625405_b_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member1516365892od_b_b @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_74_ImageE,axiom,
% 0.19/0.47 ! [B: product_prod_b_b,R2: set_Pr621705391od_b_b,A2: set_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: b] :
% 0.19/0.47 ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member_b @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_75_ImageE,axiom,
% 0.19/0.47 ! [B: product_prod_b_b,R2: set_Pr2007082183od_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: product_prod_b_b] :
% 0.19/0.47 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member1285940496od_b_b @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_76_ImageE,axiom,
% 0.19/0.47 ! [B: product_prod_b_b,R2: set_Pr2134561957od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ B @ ( image_1397930469od_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member942707974od_b_b @ ( produc1597690145od_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member1516365892od_b_b @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_77_ImageE,axiom,
% 0.19/0.47 ! [B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b,A2: set_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: b] :
% 0.19/0.47 ( ( member599505522od_b_b @ ( produc800495189od_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member_b @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_78_ImageE,axiom,
% 0.19/0.47 ! [B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: product_prod_b_b] :
% 0.19/0.47 ( ( member4694106od_b_b @ ( produc732676669od_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member1285940496od_b_b @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_79_ImageE,axiom,
% 0.19/0.47 ! [B: produc1213276845od_b_b,R2: set_Pr2109276827od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ B @ ( image_763305007od_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member1516365892od_b_b @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_80_ImageE,axiom,
% 0.19/0.47 ! [B: b,R2: set_Product_prod_b_b,A2: set_b] :
% 0.19/0.47 ( ( member_b @ B @ ( image_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member_b @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_81_ImageE,axiom,
% 0.19/0.47 ! [B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A2: set_St761939237tant_a] :
% 0.19/0.47 ( ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) )
% 0.19/0.47 => ~ ! [X4: standard_Constant_a] :
% 0.19/0.47 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ B ) @ R2 )
% 0.19/0.47 => ~ ( member1632892294tant_a @ X4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ImageE
% 0.19/0.47 thf(fact_82_Image__singleton,axiom,
% 0.19/0.47 ! [R2: set_Pr1324126435od_b_b,A: standard_Constant_a] :
% 0.19/0.47 ( ( image_1916422435od_b_b @ R2 @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) )
% 0.19/0.47 = ( collec1481886546od_b_b
% 0.19/0.47 @ ^ [B3: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B3 ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Image_singleton
% 0.19/0.47 thf(fact_83_Image__singleton,axiom,
% 0.19/0.47 ! [R2: set_Product_prod_b_b,A: b] :
% 0.19/0.47 ( ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) )
% 0.19/0.47 = ( collect_b
% 0.19/0.47 @ ^ [B3: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Image_singleton
% 0.19/0.47 thf(fact_84_Image__singleton,axiom,
% 0.19/0.47 ! [R2: set_Pr357419743_b_b_b,A: product_prod_b_b] :
% 0.19/0.47 ( ( image_921732011_b_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 = ( collect_b
% 0.19/0.47 @ ^ [B3: b] : ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B3 ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Image_singleton
% 0.19/0.47 thf(fact_85_ex__in__conv,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( ? [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ A2 ) )
% 0.19/0.47 = ( A2 != bot_bo1664927607od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % ex_in_conv
% 0.19/0.47 thf(fact_86_ex__in__conv,axiom,
% 0.19/0.47 ! [A2: set_b] :
% 0.19/0.47 ( ( ? [X2: b] : ( member_b @ X2 @ A2 ) )
% 0.19/0.47 = ( A2 != bot_bot_set_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % ex_in_conv
% 0.19/0.47 thf(fact_87_ex__in__conv,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( ? [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ A2 ) )
% 0.19/0.47 = ( A2 != bot_bo1343651123od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % ex_in_conv
% 0.19/0.47 thf(fact_88_equals0I,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ! [Y3: produc1213276845od_b_b] :
% 0.19/0.47 ~ ( member1516365892od_b_b @ Y3 @ A2 )
% 0.19/0.47 => ( A2 = bot_bo1664927607od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0I
% 0.19/0.47 thf(fact_89_equals0I,axiom,
% 0.19/0.47 ! [A2: set_b] :
% 0.19/0.47 ( ! [Y3: b] :
% 0.19/0.47 ~ ( member_b @ Y3 @ A2 )
% 0.19/0.47 => ( A2 = bot_bot_set_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0I
% 0.19/0.47 thf(fact_90_equals0I,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ! [Y3: product_prod_b_b] :
% 0.19/0.47 ~ ( member1285940496od_b_b @ Y3 @ A2 )
% 0.19/0.47 => ( A2 = bot_bo1343651123od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0I
% 0.19/0.47 thf(fact_91_equals0D,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,A: produc1213276845od_b_b] :
% 0.19/0.47 ( ( A2 = bot_bo1664927607od_b_b )
% 0.19/0.47 => ~ ( member1516365892od_b_b @ A @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0D
% 0.19/0.47 thf(fact_92_equals0D,axiom,
% 0.19/0.47 ! [A2: set_b,A: b] :
% 0.19/0.47 ( ( A2 = bot_bot_set_b )
% 0.19/0.47 => ~ ( member_b @ A @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0D
% 0.19/0.47 thf(fact_93_equals0D,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,A: product_prod_b_b] :
% 0.19/0.47 ( ( A2 = bot_bo1343651123od_b_b )
% 0.19/0.47 => ~ ( member1285940496od_b_b @ A @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0D
% 0.19/0.47 thf(fact_94_emptyE,axiom,
% 0.19/0.47 ! [A: produc1213276845od_b_b] :
% 0.19/0.47 ~ ( member1516365892od_b_b @ A @ bot_bo1664927607od_b_b ) ).
% 0.19/0.47
% 0.19/0.47 % emptyE
% 0.19/0.47 thf(fact_95_emptyE,axiom,
% 0.19/0.47 ! [A: b] :
% 0.19/0.47 ~ ( member_b @ A @ bot_bot_set_b ) ).
% 0.19/0.47
% 0.19/0.47 % emptyE
% 0.19/0.47 thf(fact_96_emptyE,axiom,
% 0.19/0.47 ! [A: product_prod_b_b] :
% 0.19/0.47 ~ ( member1285940496od_b_b @ A @ bot_bo1343651123od_b_b ) ).
% 0.19/0.47
% 0.19/0.47 % emptyE
% 0.19/0.47 thf(fact_97_mk__disjoint__insert,axiom,
% 0.19/0.47 ! [A: b,A2: set_b] :
% 0.19/0.47 ( ( member_b @ A @ A2 )
% 0.19/0.47 => ? [B4: set_b] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert_b @ A @ B4 ) )
% 0.19/0.47 & ~ ( member_b @ A @ B4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_disjoint_insert
% 0.19/0.47 thf(fact_98_mk__disjoint__insert,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.47 => ? [B4: set_Product_prod_b_b] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert1952693431od_b_b @ A @ B4 ) )
% 0.19/0.47 & ~ ( member1285940496od_b_b @ A @ B4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_disjoint_insert
% 0.19/0.47 thf(fact_99_mk__disjoint__insert,axiom,
% 0.19/0.47 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.47 => ? [B4: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert2037698781od_b_b @ A @ B4 ) )
% 0.19/0.47 & ~ ( member1516365892od_b_b @ A @ B4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_disjoint_insert
% 0.19/0.47 thf(fact_100_insert__commute,axiom,
% 0.19/0.47 ! [X: b,Y4: b,A2: set_b] :
% 0.19/0.47 ( ( insert_b @ X @ ( insert_b @ Y4 @ A2 ) )
% 0.19/0.47 = ( insert_b @ Y4 @ ( insert_b @ X @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_commute
% 0.19/0.47 thf(fact_101_insert__eq__iff,axiom,
% 0.19/0.47 ! [A: b,A2: set_b,B: b,B2: set_b] :
% 0.19/0.47 ( ~ ( member_b @ A @ A2 )
% 0.19/0.47 => ( ~ ( member_b @ B @ B2 )
% 0.19/0.47 => ( ( ( insert_b @ A @ A2 )
% 0.19/0.47 = ( insert_b @ B @ B2 ) )
% 0.19/0.47 = ( ( ( A = B )
% 0.19/0.47 => ( A2 = B2 ) )
% 0.19/0.47 & ( ( A != B )
% 0.19/0.47 => ? [C2: set_b] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert_b @ B @ C2 ) )
% 0.19/0.47 & ~ ( member_b @ B @ C2 )
% 0.19/0.47 & ( B2
% 0.19/0.47 = ( insert_b @ A @ C2 ) )
% 0.19/0.47 & ~ ( member_b @ A @ C2 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_eq_iff
% 0.19/0.47 thf(fact_102_insert__eq__iff,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: product_prod_b_b,B2: set_Product_prod_b_b] :
% 0.19/0.47 ( ~ ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.47 => ( ~ ( member1285940496od_b_b @ B @ B2 )
% 0.19/0.47 => ( ( ( insert1952693431od_b_b @ A @ A2 )
% 0.19/0.47 = ( insert1952693431od_b_b @ B @ B2 ) )
% 0.19/0.47 = ( ( ( A = B )
% 0.19/0.47 => ( A2 = B2 ) )
% 0.19/0.47 & ( ( A != B )
% 0.19/0.47 => ? [C2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert1952693431od_b_b @ B @ C2 ) )
% 0.19/0.47 & ~ ( member1285940496od_b_b @ B @ C2 )
% 0.19/0.47 & ( B2
% 0.19/0.47 = ( insert1952693431od_b_b @ A @ C2 ) )
% 0.19/0.47 & ~ ( member1285940496od_b_b @ A @ C2 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_eq_iff
% 0.19/0.47 thf(fact_103_insert__eq__iff,axiom,
% 0.19/0.47 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ~ ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.47 => ( ~ ( member1516365892od_b_b @ B @ B2 )
% 0.19/0.47 => ( ( ( insert2037698781od_b_b @ A @ A2 )
% 0.19/0.47 = ( insert2037698781od_b_b @ B @ B2 ) )
% 0.19/0.47 = ( ( ( A = B )
% 0.19/0.47 => ( A2 = B2 ) )
% 0.19/0.47 & ( ( A != B )
% 0.19/0.47 => ? [C2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert2037698781od_b_b @ B @ C2 ) )
% 0.19/0.47 & ~ ( member1516365892od_b_b @ B @ C2 )
% 0.19/0.47 & ( B2
% 0.19/0.47 = ( insert2037698781od_b_b @ A @ C2 ) )
% 0.19/0.47 & ~ ( member1516365892od_b_b @ A @ C2 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_eq_iff
% 0.19/0.47 thf(fact_104_insert__absorb,axiom,
% 0.19/0.47 ! [A: b,A2: set_b] :
% 0.19/0.47 ( ( member_b @ A @ A2 )
% 0.19/0.47 => ( ( insert_b @ A @ A2 )
% 0.19/0.47 = A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_absorb
% 0.19/0.47 thf(fact_105_insert__absorb,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.47 => ( ( insert1952693431od_b_b @ A @ A2 )
% 0.19/0.47 = A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_absorb
% 0.19/0.47 thf(fact_106_insert__absorb,axiom,
% 0.19/0.47 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.47 => ( ( insert2037698781od_b_b @ A @ A2 )
% 0.19/0.47 = A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_absorb
% 0.19/0.47 thf(fact_107_insert__ident,axiom,
% 0.19/0.47 ! [X: b,A2: set_b,B2: set_b] :
% 0.19/0.47 ( ~ ( member_b @ X @ A2 )
% 0.19/0.47 => ( ~ ( member_b @ X @ B2 )
% 0.19/0.47 => ( ( ( insert_b @ X @ A2 )
% 0.19/0.47 = ( insert_b @ X @ B2 ) )
% 0.19/0.47 = ( A2 = B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_ident
% 0.19/0.47 thf(fact_108_insert__ident,axiom,
% 0.19/0.47 ! [X: product_prod_b_b,A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
% 0.19/0.47 ( ~ ( member1285940496od_b_b @ X @ A2 )
% 0.19/0.47 => ( ~ ( member1285940496od_b_b @ X @ B2 )
% 0.19/0.47 => ( ( ( insert1952693431od_b_b @ X @ A2 )
% 0.19/0.47 = ( insert1952693431od_b_b @ X @ B2 ) )
% 0.19/0.47 = ( A2 = B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_ident
% 0.19/0.47 thf(fact_109_insert__ident,axiom,
% 0.19/0.47 ! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ~ ( member1516365892od_b_b @ X @ A2 )
% 0.19/0.47 => ( ~ ( member1516365892od_b_b @ X @ B2 )
% 0.19/0.47 => ( ( ( insert2037698781od_b_b @ X @ A2 )
% 0.19/0.47 = ( insert2037698781od_b_b @ X @ B2 ) )
% 0.19/0.47 = ( A2 = B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_ident
% 0.19/0.47 thf(fact_110_Set_Oset__insert,axiom,
% 0.19/0.47 ! [X: b,A2: set_b] :
% 0.19/0.47 ( ( member_b @ X @ A2 )
% 0.19/0.47 => ~ ! [B4: set_b] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert_b @ X @ B4 ) )
% 0.19/0.47 => ( member_b @ X @ B4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Set.set_insert
% 0.19/0.47 thf(fact_111_Set_Oset__insert,axiom,
% 0.19/0.47 ! [X: product_prod_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ X @ A2 )
% 0.19/0.47 => ~ ! [B4: set_Product_prod_b_b] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert1952693431od_b_b @ X @ B4 ) )
% 0.19/0.47 => ( member1285940496od_b_b @ X @ B4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Set.set_insert
% 0.19/0.47 thf(fact_112_Set_Oset__insert,axiom,
% 0.19/0.47 ! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ X @ A2 )
% 0.19/0.47 => ~ ! [B4: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( insert2037698781od_b_b @ X @ B4 ) )
% 0.19/0.47 => ( member1516365892od_b_b @ X @ B4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Set.set_insert
% 0.19/0.47 thf(fact_113_insertI2,axiom,
% 0.19/0.47 ! [A: b,B2: set_b,B: b] :
% 0.19/0.47 ( ( member_b @ A @ B2 )
% 0.19/0.47 => ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI2
% 0.19/0.47 thf(fact_114_insertI2,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,B2: set_Product_prod_b_b,B: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ A @ B2 )
% 0.19/0.47 => ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI2
% 0.19/0.47 thf(fact_115_insertI2,axiom,
% 0.19/0.47 ! [A: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ A @ B2 )
% 0.19/0.47 => ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI2
% 0.19/0.47 thf(fact_116_insertI1,axiom,
% 0.19/0.47 ! [A: b,B2: set_b] : ( member_b @ A @ ( insert_b @ A @ B2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI1
% 0.19/0.47 thf(fact_117_insertI1,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,B2: set_Product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ B2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI1
% 0.19/0.47 thf(fact_118_insertI1,axiom,
% 0.19/0.47 ! [A: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b] : ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ A @ B2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI1
% 0.19/0.47 thf(fact_119_insertE,axiom,
% 0.19/0.47 ! [A: b,B: b,A2: set_b] :
% 0.19/0.47 ( ( member_b @ A @ ( insert_b @ B @ A2 ) )
% 0.19/0.47 => ( ( A != B )
% 0.19/0.47 => ( member_b @ A @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertE
% 0.19/0.47 thf(fact_120_insertE,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,B: product_prod_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A2 ) )
% 0.19/0.47 => ( ( A != B )
% 0.19/0.47 => ( member1285940496od_b_b @ A @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertE
% 0.19/0.47 thf(fact_121_insertE,axiom,
% 0.19/0.47 ! [A: produc1213276845od_b_b,B: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ A2 ) )
% 0.19/0.47 => ( ( A != B )
% 0.19/0.47 => ( member1516365892od_b_b @ A @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertE
% 0.19/0.47 thf(fact_122_some__eq__imp,axiom,
% 0.19/0.47 ! [P: b > $o,A: b,B: b] :
% 0.19/0.47 ( ( ( hilbert_Eps_b @ P )
% 0.19/0.47 = A )
% 0.19/0.47 => ( ( P @ B )
% 0.19/0.47 => ( P @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % some_eq_imp
% 0.19/0.47 thf(fact_123_tfl__some,axiom,
% 0.19/0.47 ! [P2: b > $o,X5: b] :
% 0.19/0.47 ( ( P2 @ X5 )
% 0.19/0.47 => ( P2 @ ( hilbert_Eps_b @ P2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % tfl_some
% 0.19/0.47 thf(fact_124_mem__Collect__eq,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,P: product_prod_b_b > $o] :
% 0.19/0.47 ( ( member1285940496od_b_b @ A @ ( collec1481886546od_b_b @ P ) )
% 0.19/0.47 = ( P @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 % mem_Collect_eq
% 0.19/0.47 thf(fact_125_mem__Collect__eq,axiom,
% 0.19/0.47 ! [A: produc1213276845od_b_b,P: produc1213276845od_b_b > $o] :
% 0.19/0.47 ( ( member1516365892od_b_b @ A @ ( collec279084418od_b_b @ P ) )
% 0.19/0.47 = ( P @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 % mem_Collect_eq
% 0.19/0.47 thf(fact_126_mem__Collect__eq,axiom,
% 0.19/0.47 ! [A: b,P: b > $o] :
% 0.19/0.47 ( ( member_b @ A @ ( collect_b @ P ) )
% 0.19/0.47 = ( P @ A ) ) ).
% 0.19/0.47
% 0.19/0.47 % mem_Collect_eq
% 0.19/0.47 thf(fact_127_Collect__mem__eq,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( collec1481886546od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ A2 ) )
% 0.19/0.47 = A2 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mem_eq
% 0.19/0.47 thf(fact_128_Collect__mem__eq,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( collec279084418od_b_b
% 0.19/0.47 @ ^ [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ A2 ) )
% 0.19/0.47 = A2 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mem_eq
% 0.19/0.47 thf(fact_129_Collect__mem__eq,axiom,
% 0.19/0.47 ! [A2: set_b] :
% 0.19/0.47 ( ( collect_b
% 0.19/0.47 @ ^ [X2: b] : ( member_b @ X2 @ A2 ) )
% 0.19/0.47 = A2 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mem_eq
% 0.19/0.47 thf(fact_130_Collect__cong,axiom,
% 0.19/0.47 ! [P: b > $o,Q: b > $o] :
% 0.19/0.47 ( ! [X4: b] :
% 0.19/0.47 ( ( P @ X4 )
% 0.19/0.47 = ( Q @ X4 ) )
% 0.19/0.47 => ( ( collect_b @ P )
% 0.19/0.47 = ( collect_b @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_cong
% 0.19/0.47 thf(fact_131_Eps__cong,axiom,
% 0.19/0.47 ! [P: b > $o,Q: b > $o] :
% 0.19/0.47 ( ! [X4: b] :
% 0.19/0.47 ( ( P @ X4 )
% 0.19/0.47 = ( Q @ X4 ) )
% 0.19/0.47 => ( ( hilbert_Eps_b @ P )
% 0.19/0.47 = ( hilbert_Eps_b @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Eps_cong
% 0.19/0.47 thf(fact_132_someI,axiom,
% 0.19/0.47 ! [P: b > $o,X: b] :
% 0.19/0.47 ( ( P @ X )
% 0.19/0.47 => ( P @ ( hilbert_Eps_b @ P ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % someI
% 0.19/0.47 thf(fact_133_empty__def,axiom,
% 0.19/0.47 ( bot_bot_set_b
% 0.19/0.47 = ( collect_b
% 0.19/0.47 @ ^ [X2: b] : $false ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_def
% 0.19/0.47 thf(fact_134_empty__def,axiom,
% 0.19/0.47 ( bot_bo1343651123od_b_b
% 0.19/0.47 = ( collec1481886546od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] : $false ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_def
% 0.19/0.47 thf(fact_135_insert__Collect,axiom,
% 0.19/0.47 ! [A: b,P: b > $o] :
% 0.19/0.47 ( ( insert_b @ A @ ( collect_b @ P ) )
% 0.19/0.47 = ( collect_b
% 0.19/0.47 @ ^ [U: b] :
% 0.19/0.47 ( ( U != A )
% 0.19/0.47 => ( P @ U ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_Collect
% 0.19/0.47 thf(fact_136_insert__compr,axiom,
% 0.19/0.47 ( insert1952693431od_b_b
% 0.19/0.47 = ( ^ [A3: product_prod_b_b,B5: set_Product_prod_b_b] :
% 0.19/0.47 ( collec1481886546od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] :
% 0.19/0.47 ( ( X2 = A3 )
% 0.19/0.47 | ( member1285940496od_b_b @ X2 @ B5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_compr
% 0.19/0.47 thf(fact_137_insert__compr,axiom,
% 0.19/0.47 ( insert2037698781od_b_b
% 0.19/0.47 = ( ^ [A3: produc1213276845od_b_b,B5: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( collec279084418od_b_b
% 0.19/0.47 @ ^ [X2: produc1213276845od_b_b] :
% 0.19/0.47 ( ( X2 = A3 )
% 0.19/0.47 | ( member1516365892od_b_b @ X2 @ B5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_compr
% 0.19/0.47 thf(fact_138_insert__compr,axiom,
% 0.19/0.47 ( insert_b
% 0.19/0.47 = ( ^ [A3: b,B5: set_b] :
% 0.19/0.47 ( collect_b
% 0.19/0.47 @ ^ [X2: b] :
% 0.19/0.47 ( ( X2 = A3 )
% 0.19/0.47 | ( member_b @ X2 @ B5 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_compr
% 0.19/0.47 thf(fact_139_some1__equality,axiom,
% 0.19/0.47 ! [P: b > $o,A: b] :
% 0.19/0.47 ( ? [X5: b] :
% 0.19/0.47 ( ( P @ X5 )
% 0.19/0.47 & ! [Y3: b] :
% 0.19/0.47 ( ( P @ Y3 )
% 0.19/0.47 => ( Y3 = X5 ) ) )
% 0.19/0.47 => ( ( P @ A )
% 0.19/0.47 => ( ( hilbert_Eps_b @ P )
% 0.19/0.47 = A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % some1_equality
% 0.19/0.47 thf(fact_140_some__eq__ex,axiom,
% 0.19/0.47 ! [P: b > $o] :
% 0.19/0.47 ( ( P @ ( hilbert_Eps_b @ P ) )
% 0.19/0.47 = ( ? [X6: b] : ( P @ X6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % some_eq_ex
% 0.19/0.47 thf(fact_141_someI2__bex,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
% 0.19/0.47 ( ? [X5: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ X5 @ A2 )
% 0.19/0.47 & ( P @ X5 ) )
% 0.19/0.47 => ( ! [X4: product_prod_b_b] :
% 0.19/0.47 ( ( ( member1285940496od_b_b @ X4 @ A2 )
% 0.19/0.47 & ( P @ X4 ) )
% 0.19/0.47 => ( Q @ X4 ) )
% 0.19/0.47 => ( Q
% 0.19/0.47 @ ( hilber620807523od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ X2 @ A2 )
% 0.19/0.47 & ( P @ X2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % someI2_bex
% 0.19/0.47 thf(fact_142_someI2__bex,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,P: produc1213276845od_b_b > $o,Q: produc1213276845od_b_b > $o] :
% 0.19/0.47 ( ? [X5: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ X5 @ A2 )
% 0.19/0.47 & ( P @ X5 ) )
% 0.19/0.47 => ( ! [X4: produc1213276845od_b_b] :
% 0.19/0.47 ( ( ( member1516365892od_b_b @ X4 @ A2 )
% 0.19/0.47 & ( P @ X4 ) )
% 0.19/0.47 => ( Q @ X4 ) )
% 0.19/0.47 => ( Q
% 0.19/0.47 @ ( hilber46179633od_b_b
% 0.19/0.47 @ ^ [X2: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ X2 @ A2 )
% 0.19/0.47 & ( P @ X2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % someI2_bex
% 0.19/0.47 thf(fact_143_someI2__bex,axiom,
% 0.19/0.47 ! [A2: set_b,P: b > $o,Q: b > $o] :
% 0.19/0.47 ( ? [X5: b] :
% 0.19/0.47 ( ( member_b @ X5 @ A2 )
% 0.19/0.47 & ( P @ X5 ) )
% 0.19/0.47 => ( ! [X4: b] :
% 0.19/0.47 ( ( ( member_b @ X4 @ A2 )
% 0.19/0.47 & ( P @ X4 ) )
% 0.19/0.47 => ( Q @ X4 ) )
% 0.19/0.47 => ( Q
% 0.19/0.47 @ ( hilbert_Eps_b
% 0.19/0.47 @ ^ [X2: b] :
% 0.19/0.47 ( ( member_b @ X2 @ A2 )
% 0.19/0.47 & ( P @ X2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % someI2_bex
% 0.19/0.47 thf(fact_144_someI2__ex,axiom,
% 0.19/0.47 ! [P: b > $o,Q: b > $o] :
% 0.19/0.47 ( ? [X_1: b] : ( P @ X_1 )
% 0.19/0.47 => ( ! [X4: b] :
% 0.19/0.47 ( ( P @ X4 )
% 0.19/0.47 => ( Q @ X4 ) )
% 0.19/0.47 => ( Q @ ( hilbert_Eps_b @ P ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % someI2_ex
% 0.19/0.47 thf(fact_145_someI__ex,axiom,
% 0.19/0.47 ! [P: b > $o] :
% 0.19/0.47 ( ? [X_1: b] : ( P @ X_1 )
% 0.19/0.47 => ( P @ ( hilbert_Eps_b @ P ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % someI_ex
% 0.19/0.47 thf(fact_146_someI2,axiom,
% 0.19/0.47 ! [P: b > $o,A: b,Q: b > $o] :
% 0.19/0.47 ( ( P @ A )
% 0.19/0.47 => ( ! [X4: b] :
% 0.19/0.47 ( ( P @ X4 )
% 0.19/0.47 => ( Q @ X4 ) )
% 0.19/0.47 => ( Q @ ( hilbert_Eps_b @ P ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % someI2
% 0.19/0.47 thf(fact_147_singleton__inject,axiom,
% 0.19/0.47 ! [A: b,B: b] :
% 0.19/0.47 ( ( ( insert_b @ A @ bot_bot_set_b )
% 0.19/0.47 = ( insert_b @ B @ bot_bot_set_b ) )
% 0.19/0.47 => ( A = B ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_inject
% 0.19/0.47 thf(fact_148_singleton__inject,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,B: product_prod_b_b] :
% 0.19/0.47 ( ( ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b )
% 0.19/0.47 = ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 => ( A = B ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_inject
% 0.19/0.47 thf(fact_149_insert__not__empty,axiom,
% 0.19/0.47 ! [A: b,A2: set_b] :
% 0.19/0.47 ( ( insert_b @ A @ A2 )
% 0.19/0.47 != bot_bot_set_b ) ).
% 0.19/0.47
% 0.19/0.47 % insert_not_empty
% 0.19/0.47 thf(fact_150_insert__not__empty,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( insert1952693431od_b_b @ A @ A2 )
% 0.19/0.47 != bot_bo1343651123od_b_b ) ).
% 0.19/0.47
% 0.19/0.47 % insert_not_empty
% 0.19/0.47 thf(fact_151_doubleton__eq__iff,axiom,
% 0.19/0.47 ! [A: b,B: b,C: b,D: b] :
% 0.19/0.47 ( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
% 0.19/0.47 = ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
% 0.19/0.47 = ( ( ( A = C )
% 0.19/0.47 & ( B = D ) )
% 0.19/0.47 | ( ( A = D )
% 0.19/0.47 & ( B = C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % doubleton_eq_iff
% 0.19/0.47 thf(fact_152_doubleton__eq__iff,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,B: product_prod_b_b,C: product_prod_b_b,D: product_prod_b_b] :
% 0.19/0.47 ( ( ( insert1952693431od_b_b @ A @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 = ( insert1952693431od_b_b @ C @ ( insert1952693431od_b_b @ D @ bot_bo1343651123od_b_b ) ) )
% 0.19/0.47 = ( ( ( A = C )
% 0.19/0.47 & ( B = D ) )
% 0.19/0.47 | ( ( A = D )
% 0.19/0.47 & ( B = C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % doubleton_eq_iff
% 0.19/0.47 thf(fact_153_singleton__iff,axiom,
% 0.19/0.47 ! [B: produc1213276845od_b_b,A: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ B @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) )
% 0.19/0.47 = ( B = A ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_iff
% 0.19/0.47 thf(fact_154_singleton__iff,axiom,
% 0.19/0.47 ! [B: b,A: b] :
% 0.19/0.47 ( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
% 0.19/0.47 = ( B = A ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_iff
% 0.19/0.47 thf(fact_155_singleton__iff,axiom,
% 0.19/0.47 ! [B: product_prod_b_b,A: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 = ( B = A ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_iff
% 0.19/0.47 thf(fact_156_singletonD,axiom,
% 0.19/0.47 ! [B: produc1213276845od_b_b,A: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ B @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) )
% 0.19/0.47 => ( B = A ) ) ).
% 0.19/0.47
% 0.19/0.47 % singletonD
% 0.19/0.47 thf(fact_157_singletonD,axiom,
% 0.19/0.47 ! [B: b,A: b] :
% 0.19/0.47 ( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
% 0.19/0.47 => ( B = A ) ) ).
% 0.19/0.47
% 0.19/0.47 % singletonD
% 0.19/0.47 thf(fact_158_singletonD,axiom,
% 0.19/0.47 ! [B: product_prod_b_b,A: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 => ( B = A ) ) ).
% 0.19/0.47
% 0.19/0.47 % singletonD
% 0.19/0.47 thf(fact_159_Collect__conv__if2,axiom,
% 0.19/0.47 ! [P: b > $o,A: b] :
% 0.19/0.47 ( ( ( P @ A )
% 0.19/0.47 => ( ( collect_b
% 0.19/0.47 @ ^ [X2: b] :
% 0.19/0.47 ( ( A = X2 )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 = ( insert_b @ A @ bot_bot_set_b ) ) )
% 0.19/0.47 & ( ~ ( P @ A )
% 0.19/0.47 => ( ( collect_b
% 0.19/0.47 @ ^ [X2: b] :
% 0.19/0.47 ( ( A = X2 )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 = bot_bot_set_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_conv_if2
% 0.19/0.47 thf(fact_160_Collect__conv__if2,axiom,
% 0.19/0.47 ! [P: product_prod_b_b > $o,A: product_prod_b_b] :
% 0.19/0.47 ( ( ( P @ A )
% 0.19/0.47 => ( ( collec1481886546od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] :
% 0.19/0.47 ( ( A = X2 )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 = ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 0.19/0.47 & ( ~ ( P @ A )
% 0.19/0.47 => ( ( collec1481886546od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] :
% 0.19/0.47 ( ( A = X2 )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 = bot_bo1343651123od_b_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_conv_if2
% 0.19/0.47 thf(fact_161_Collect__conv__if,axiom,
% 0.19/0.47 ! [P: b > $o,A: b] :
% 0.19/0.47 ( ( ( P @ A )
% 0.19/0.47 => ( ( collect_b
% 0.19/0.47 @ ^ [X2: b] :
% 0.19/0.47 ( ( X2 = A )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 = ( insert_b @ A @ bot_bot_set_b ) ) )
% 0.19/0.47 & ( ~ ( P @ A )
% 0.19/0.47 => ( ( collect_b
% 0.19/0.47 @ ^ [X2: b] :
% 0.19/0.47 ( ( X2 = A )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 = bot_bot_set_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_conv_if
% 0.19/0.47 thf(fact_162_Collect__conv__if,axiom,
% 0.19/0.47 ! [P: product_prod_b_b > $o,A: product_prod_b_b] :
% 0.19/0.47 ( ( ( P @ A )
% 0.19/0.47 => ( ( collec1481886546od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] :
% 0.19/0.47 ( ( X2 = A )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 = ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 0.19/0.47 & ( ~ ( P @ A )
% 0.19/0.47 => ( ( collec1481886546od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] :
% 0.19/0.47 ( ( X2 = A )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 = bot_bo1343651123od_b_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_conv_if
% 0.19/0.47 thf(fact_163_some__in__eq,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b
% 0.19/0.47 @ ( hilber46179633od_b_b
% 0.19/0.47 @ ^ [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ A2 ) )
% 0.19/0.47 @ A2 )
% 0.19/0.47 = ( A2 != bot_bo1664927607od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % some_in_eq
% 0.19/0.47 thf(fact_164_some__in__eq,axiom,
% 0.19/0.47 ! [A2: set_b] :
% 0.19/0.47 ( ( member_b
% 0.19/0.47 @ ( hilbert_Eps_b
% 0.19/0.47 @ ^ [X2: b] : ( member_b @ X2 @ A2 ) )
% 0.19/0.47 @ A2 )
% 0.19/0.47 = ( A2 != bot_bot_set_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % some_in_eq
% 0.19/0.47 thf(fact_165_some__in__eq,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b
% 0.19/0.47 @ ( hilber620807523od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ A2 ) )
% 0.19/0.47 @ A2 )
% 0.19/0.47 = ( A2 != bot_bo1343651123od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % some_in_eq
% 0.19/0.47 thf(fact_166_congr_I1_J,axiom,
% 0.19/0.47 ! [X: b,Y4: b,L: standard_Constant_a] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ ( getRel904497637nt_a_b @ L @ g ) )
% 0.19/0.47 => ( equiv_1821124534_set_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g )
% 0.19/0.47 @ ^ [V: b] : ( image_b_b @ ( getRel904497637nt_a_b @ L @ g ) @ ( insert_b @ V @ bot_bot_set_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % congr(1)
% 0.19/0.47 thf(fact_167_vert__P,axiom,
% 0.19/0.47 ! [X: b] :
% 0.19/0.47 ( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
% 0.19/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ ( hilbert_Eps_b @ ( p @ X ) ) ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % vert_P
% 0.19/0.47 thf(fact_168__092_060open_062_IS__Idt_M_Ax_M_Ay_J_A_092_060in_062_Aedges_AG_092_060close_062,axiom,
% 0.19/0.47 member1516365892od_b_b @ ( produc1432590431od_b_b @ standard_S_Idt_a @ ( product_Pair_b_b @ x @ y ) ) @ ( labele1741081071nt_a_b @ g ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>(S_Idt, x, y) \<in> edges G\<close>
% 0.19/0.47 thf(fact_169_restrict__idemp,axiom,
% 0.19/0.47 ! [X: labele1159362096nt_a_b] :
% 0.19/0.47 ( ( restri446606278nt_a_b @ ( restri446606278nt_a_b @ X ) )
% 0.19/0.47 = ( restri446606278nt_a_b @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % restrict_idemp
% 0.19/0.47 thf(fact_170_r2,axiom,
% 0.19/0.47 ! [X: b] :
% 0.19/0.47 ( ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X @ bot_bot_set_b ) )
% 0.19/0.47 = bot_bot_set_b )
% 0.19/0.47 = ( ~ ( member_b @ X @ ( labele1424214014nt_a_b @ g ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % r2
% 0.19/0.47 thf(fact_171_r1,axiom,
% 0.19/0.47 ! [X: b] :
% 0.19/0.47 ( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
% 0.19/0.47 = ( p @ X @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % r1
% 0.19/0.47 thf(fact_172_congr_I2_J,axiom,
% 0.19/0.47 ! [X: b,Y4: b,L: standard_Constant_a] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ ( getRel904497637nt_a_b @ L @ g ) )
% 0.19/0.47 => ( equiv_1821124534_set_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g )
% 0.19/0.47 @ ^ [V: b] : ( image_b_b @ ( converse_b_b @ ( getRel904497637nt_a_b @ L @ g ) ) @ ( insert_b @ V @ bot_bot_set_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % congr(2)
% 0.19/0.47 thf(fact_173_the__elem__eq,axiom,
% 0.19/0.47 ! [X: b] :
% 0.19/0.47 ( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
% 0.19/0.47 = X ) ).
% 0.19/0.47
% 0.19/0.47 % the_elem_eq
% 0.19/0.47 thf(fact_174_the__elem__eq,axiom,
% 0.19/0.47 ! [X: product_prod_b_b] :
% 0.19/0.47 ( ( the_el329196508od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 = X ) ).
% 0.19/0.47
% 0.19/0.47 % the_elem_eq
% 0.19/0.47 thf(fact_175_prod_Oinject,axiom,
% 0.19/0.47 ! [X1: b,X22: b,Y1: b,Y22: b] :
% 0.19/0.47 ( ( ( product_Pair_b_b @ X1 @ X22 )
% 0.19/0.47 = ( product_Pair_b_b @ Y1 @ Y22 ) )
% 0.19/0.47 = ( ( X1 = Y1 )
% 0.19/0.47 & ( X22 = Y22 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod.inject
% 0.19/0.47 thf(fact_176_prod_Oinject,axiom,
% 0.19/0.47 ! [X1: standard_Constant_a,X22: product_prod_b_b,Y1: standard_Constant_a,Y22: product_prod_b_b] :
% 0.19/0.47 ( ( ( produc1432590431od_b_b @ X1 @ X22 )
% 0.19/0.47 = ( produc1432590431od_b_b @ Y1 @ Y22 ) )
% 0.19/0.47 = ( ( X1 = Y1 )
% 0.19/0.47 & ( X22 = Y22 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod.inject
% 0.19/0.47 thf(fact_177_old_Oprod_Oinject,axiom,
% 0.19/0.47 ! [A: b,B: b,A4: b,B6: b] :
% 0.19/0.47 ( ( ( product_Pair_b_b @ A @ B )
% 0.19/0.47 = ( product_Pair_b_b @ A4 @ B6 ) )
% 0.19/0.47 = ( ( A = A4 )
% 0.19/0.47 & ( B = B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.inject
% 0.19/0.47 thf(fact_178_old_Oprod_Oinject,axiom,
% 0.19/0.47 ! [A: standard_Constant_a,B: product_prod_b_b,A4: standard_Constant_a,B6: product_prod_b_b] :
% 0.19/0.47 ( ( ( produc1432590431od_b_b @ A @ B )
% 0.19/0.47 = ( produc1432590431od_b_b @ A4 @ B6 ) )
% 0.19/0.47 = ( ( A = A4 )
% 0.19/0.47 & ( B = B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.inject
% 0.19/0.47 thf(fact_179_converse__inject,axiom,
% 0.19/0.47 ! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( ( converse_b_b @ R2 )
% 0.19/0.47 = ( converse_b_b @ S2 ) )
% 0.19/0.47 = ( R2 = S2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse_inject
% 0.19/0.47 thf(fact_180_converse__converse,axiom,
% 0.19/0.47 ! [R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( converse_b_b @ ( converse_b_b @ R2 ) )
% 0.19/0.47 = R2 ) ).
% 0.19/0.47
% 0.19/0.47 % converse_converse
% 0.19/0.47 thf(fact_181_converse__iff,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,B: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A @ B ) @ ( conver1398856763od_b_b @ R2 ) )
% 0.19/0.47 = ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B @ A ) @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse_iff
% 0.19/0.47 thf(fact_182_converse__iff,axiom,
% 0.19/0.47 ! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr306736955tant_a] :
% 0.19/0.47 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ ( conver1943590235tant_a @ R2 ) )
% 0.19/0.47 = ( member1670214300tant_a @ ( produc1977323903tant_a @ B @ A ) @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse_iff
% 0.19/0.47 thf(fact_183_converse__iff,axiom,
% 0.19/0.47 ! [A: b,B: b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( converse_b_b @ R2 ) )
% 0.19/0.47 = ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse_iff
% 0.19/0.47 thf(fact_184_vertices__restrict,axiom,
% 0.19/0.47 ! [G: labele1159362096nt_a_b] :
% 0.19/0.47 ( ( labele1424214014nt_a_b @ ( restri446606278nt_a_b @ G ) )
% 0.19/0.47 = ( labele1424214014nt_a_b @ G ) ) ).
% 0.19/0.47
% 0.19/0.47 % vertices_restrict
% 0.19/0.47 thf(fact_185_converse__empty,axiom,
% 0.19/0.47 ( ( converse_b_b @ bot_bo1343651123od_b_b )
% 0.19/0.47 = bot_bo1343651123od_b_b ) ).
% 0.19/0.47
% 0.19/0.47 % converse_empty
% 0.19/0.47 thf(fact_186__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
% 0.19/0.47 ! [X: b] :
% 0.19/0.47 ( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
% 0.19/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ X ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>\<And>x. x \<in> vertices G \<Longrightarrow> (x, x) \<in> getRel S_Idt G\<close>
% 0.19/0.47 thf(fact_187_labeled__graph_Oexpand,axiom,
% 0.19/0.47 ! [Labeled_graph: labele1159362096nt_a_b,Labeled_graph2: labele1159362096nt_a_b] :
% 0.19/0.47 ( ( ( ( labele1741081071nt_a_b @ Labeled_graph )
% 0.19/0.47 = ( labele1741081071nt_a_b @ Labeled_graph2 ) )
% 0.19/0.47 & ( ( labele1424214014nt_a_b @ Labeled_graph )
% 0.19/0.47 = ( labele1424214014nt_a_b @ Labeled_graph2 ) ) )
% 0.19/0.47 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % labeled_graph.expand
% 0.19/0.47 thf(fact_188_converse_Oinducts,axiom,
% 0.19/0.47 ! [X1: product_prod_b_b,X22: standard_Constant_a,R2: set_Pr1324126435od_b_b,P: product_prod_b_b > standard_Constant_a > $o] :
% 0.19/0.47 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ X1 @ X22 ) @ ( conver1398856763od_b_b @ R2 ) )
% 0.19/0.47 => ( ! [A5: standard_Constant_a,B7: product_prod_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A5 @ B7 ) @ R2 )
% 0.19/0.47 => ( P @ B7 @ A5 ) )
% 0.19/0.47 => ( P @ X1 @ X22 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.inducts
% 0.19/0.47 thf(fact_189_converse_Oinducts,axiom,
% 0.19/0.47 ! [X1: standard_Constant_a,X22: product_prod_b_b,R2: set_Pr306736955tant_a,P: standard_Constant_a > product_prod_b_b > $o] :
% 0.19/0.47 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X1 @ X22 ) @ ( conver1943590235tant_a @ R2 ) )
% 0.19/0.47 => ( ! [A5: product_prod_b_b,B7: standard_Constant_a] :
% 0.19/0.47 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A5 @ B7 ) @ R2 )
% 0.19/0.47 => ( P @ B7 @ A5 ) )
% 0.19/0.47 => ( P @ X1 @ X22 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.inducts
% 0.19/0.47 thf(fact_190_converse_Oinducts,axiom,
% 0.19/0.47 ! [X1: b,X22: b,R2: set_Product_prod_b_b,P: b > b > $o] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X1 @ X22 ) @ ( converse_b_b @ R2 ) )
% 0.19/0.47 => ( ! [A5: b,B7: b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A5 @ B7 ) @ R2 )
% 0.19/0.47 => ( P @ B7 @ A5 ) )
% 0.19/0.47 => ( P @ X1 @ X22 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.inducts
% 0.19/0.47 thf(fact_191_converse_Ointros,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,B: standard_Constant_a,R2: set_Pr306736955tant_a] :
% 0.19/0.47 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A @ B ) @ R2 )
% 0.19/0.47 => ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B @ A ) @ ( conver1943590235tant_a @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.intros
% 0.19/0.47 thf(fact_192_converse_Ointros,axiom,
% 0.19/0.47 ! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( member1670214300tant_a @ ( produc1977323903tant_a @ B @ A ) @ ( conver1398856763od_b_b @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.intros
% 0.19/0.47 thf(fact_193_converse_Ointros,axiom,
% 0.19/0.47 ! [A: b,B: b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ ( converse_b_b @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.intros
% 0.19/0.47 thf(fact_194_converse_Osimps,axiom,
% 0.19/0.47 ! [A1: product_prod_b_b,A22: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A1 @ A22 ) @ ( conver1398856763od_b_b @ R2 ) )
% 0.19/0.47 = ( ? [A3: standard_Constant_a,B3: product_prod_b_b] :
% 0.19/0.47 ( ( A1 = B3 )
% 0.19/0.47 & ( A22 = A3 )
% 0.19/0.47 & ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A3 @ B3 ) @ R2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.simps
% 0.19/0.47 thf(fact_195_converse_Osimps,axiom,
% 0.19/0.47 ! [A1: standard_Constant_a,A22: product_prod_b_b,R2: set_Pr306736955tant_a] :
% 0.19/0.47 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( conver1943590235tant_a @ R2 ) )
% 0.19/0.47 = ( ? [A3: product_prod_b_b,B3: standard_Constant_a] :
% 0.19/0.47 ( ( A1 = B3 )
% 0.19/0.47 & ( A22 = A3 )
% 0.19/0.47 & ( member1670214300tant_a @ ( produc1977323903tant_a @ A3 @ B3 ) @ R2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.simps
% 0.19/0.47 thf(fact_196_converse_Osimps,axiom,
% 0.19/0.47 ! [A1: b,A22: b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A1 @ A22 ) @ ( converse_b_b @ R2 ) )
% 0.19/0.47 = ( ? [A3: b,B3: b] :
% 0.19/0.47 ( ( A1 = B3 )
% 0.19/0.47 & ( A22 = A3 )
% 0.19/0.47 & ( member1285940496od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ R2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.simps
% 0.19/0.47 thf(fact_197_converse_Ocases,axiom,
% 0.19/0.47 ! [A1: product_prod_b_b,A22: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A1 @ A22 ) @ ( conver1398856763od_b_b @ R2 ) )
% 0.19/0.47 => ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A22 @ A1 ) @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.cases
% 0.19/0.47 thf(fact_198_converse_Ocases,axiom,
% 0.19/0.47 ! [A1: standard_Constant_a,A22: product_prod_b_b,R2: set_Pr306736955tant_a] :
% 0.19/0.47 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( conver1943590235tant_a @ R2 ) )
% 0.19/0.47 => ( member1670214300tant_a @ ( produc1977323903tant_a @ A22 @ A1 ) @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.cases
% 0.19/0.47 thf(fact_199_converse_Ocases,axiom,
% 0.19/0.47 ! [A1: b,A22: b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A1 @ A22 ) @ ( converse_b_b @ R2 ) )
% 0.19/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A22 @ A1 ) @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse.cases
% 0.19/0.47 thf(fact_200_converseE,axiom,
% 0.19/0.47 ! [Yx: produc1367125253tant_a,R2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1670214300tant_a @ Yx @ ( conver1398856763od_b_b @ R2 ) )
% 0.19/0.47 => ~ ! [X4: standard_Constant_a,Y3: product_prod_b_b] :
% 0.19/0.47 ( ( Yx
% 0.19/0.47 = ( produc1977323903tant_a @ Y3 @ X4 ) )
% 0.19/0.47 => ~ ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converseE
% 0.19/0.47 thf(fact_201_converseE,axiom,
% 0.19/0.47 ! [Yx: produc1213276845od_b_b,R2: set_Pr306736955tant_a] :
% 0.19/0.47 ( ( member1516365892od_b_b @ Yx @ ( conver1943590235tant_a @ R2 ) )
% 0.19/0.47 => ~ ! [X4: product_prod_b_b,Y3: standard_Constant_a] :
% 0.19/0.47 ( ( Yx
% 0.19/0.47 = ( produc1432590431od_b_b @ Y3 @ X4 ) )
% 0.19/0.47 => ~ ( member1670214300tant_a @ ( produc1977323903tant_a @ X4 @ Y3 ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converseE
% 0.19/0.47 thf(fact_202_converseE,axiom,
% 0.19/0.47 ! [Yx: product_prod_b_b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ Yx @ ( converse_b_b @ R2 ) )
% 0.19/0.47 => ~ ! [X4: b,Y3: b] :
% 0.19/0.47 ( ( Yx
% 0.19/0.47 = ( product_Pair_b_b @ Y3 @ X4 ) )
% 0.19/0.47 => ~ ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % converseE
% 0.19/0.47 thf(fact_203_converseD,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,B: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A @ B ) @ ( conver1398856763od_b_b @ R2 ) )
% 0.19/0.47 => ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B @ A ) @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converseD
% 0.19/0.47 thf(fact_204_converseD,axiom,
% 0.19/0.47 ! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr306736955tant_a] :
% 0.19/0.47 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ ( conver1943590235tant_a @ R2 ) )
% 0.19/0.47 => ( member1670214300tant_a @ ( produc1977323903tant_a @ B @ A ) @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converseD
% 0.19/0.47 thf(fact_205_converseD,axiom,
% 0.19/0.47 ! [A: b,B: b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( converse_b_b @ R2 ) )
% 0.19/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converseD
% 0.19/0.47 thf(fact_206_bot__set__def,axiom,
% 0.19/0.47 ( bot_bot_set_b
% 0.19/0.47 = ( collect_b @ bot_bot_b_o ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_set_def
% 0.19/0.47 thf(fact_207_bot__set__def,axiom,
% 0.19/0.47 ( bot_bo1343651123od_b_b
% 0.19/0.47 = ( collec1481886546od_b_b @ bot_bo1302443370_b_b_o ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_set_def
% 0.19/0.47 thf(fact_208_bot__empty__eq,axiom,
% 0.19/0.47 ( bot_bo1167600078_b_b_o
% 0.19/0.47 = ( ^ [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ bot_bo1664927607od_b_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_empty_eq
% 0.19/0.47 thf(fact_209_bot__empty__eq,axiom,
% 0.19/0.47 ( bot_bot_b_o
% 0.19/0.47 = ( ^ [X2: b] : ( member_b @ X2 @ bot_bot_set_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_empty_eq
% 0.19/0.47 thf(fact_210_bot__empty__eq,axiom,
% 0.19/0.47 ( bot_bo1302443370_b_b_o
% 0.19/0.47 = ( ^ [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ bot_bo1343651123od_b_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_empty_eq
% 0.19/0.47 thf(fact_211_bot__empty__eq2,axiom,
% 0.19/0.47 ( bot_bo1734638456_b_b_o
% 0.19/0.47 = ( ^ [X2: standard_Constant_a,Y: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ Y ) @ bot_bo1664927607od_b_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_empty_eq2
% 0.19/0.47 thf(fact_212_bot__empty__eq2,axiom,
% 0.19/0.47 ( bot_bot_b_b_o
% 0.19/0.47 = ( ^ [X2: b,Y: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ bot_bo1343651123od_b_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_empty_eq2
% 0.19/0.47 thf(fact_213_old_Oprod_Oinducts,axiom,
% 0.19/0.47 ! [P: product_prod_b_b > $o,Prod: product_prod_b_b] :
% 0.19/0.47 ( ! [A5: b,B7: b] : ( P @ ( product_Pair_b_b @ A5 @ B7 ) )
% 0.19/0.47 => ( P @ Prod ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.inducts
% 0.19/0.47 thf(fact_214_old_Oprod_Oinducts,axiom,
% 0.19/0.47 ! [P: produc1213276845od_b_b > $o,Prod: produc1213276845od_b_b] :
% 0.19/0.47 ( ! [A5: standard_Constant_a,B7: product_prod_b_b] : ( P @ ( produc1432590431od_b_b @ A5 @ B7 ) )
% 0.19/0.47 => ( P @ Prod ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.inducts
% 0.19/0.47 thf(fact_215_old_Oprod_Oexhaust,axiom,
% 0.19/0.47 ! [Y4: product_prod_b_b] :
% 0.19/0.47 ~ ! [A5: b,B7: b] :
% 0.19/0.47 ( Y4
% 0.19/0.47 != ( product_Pair_b_b @ A5 @ B7 ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.exhaust
% 0.19/0.47 thf(fact_216_old_Oprod_Oexhaust,axiom,
% 0.19/0.47 ! [Y4: produc1213276845od_b_b] :
% 0.19/0.47 ~ ! [A5: standard_Constant_a,B7: product_prod_b_b] :
% 0.19/0.47 ( Y4
% 0.19/0.47 != ( produc1432590431od_b_b @ A5 @ B7 ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.exhaust
% 0.19/0.47 thf(fact_217_prod__induct3,axiom,
% 0.19/0.47 ! [P: produc1213276845od_b_b > $o,X: produc1213276845od_b_b] :
% 0.19/0.47 ( ! [A5: standard_Constant_a,B7: b,C3: b] : ( P @ ( produc1432590431od_b_b @ A5 @ ( product_Pair_b_b @ B7 @ C3 ) ) )
% 0.19/0.47 => ( P @ X ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct3
% 0.19/0.47 thf(fact_218_prod__cases3,axiom,
% 0.19/0.47 ! [Y4: produc1213276845od_b_b] :
% 0.19/0.47 ~ ! [A5: standard_Constant_a,B7: b,C3: b] :
% 0.19/0.47 ( Y4
% 0.19/0.47 != ( produc1432590431od_b_b @ A5 @ ( product_Pair_b_b @ B7 @ C3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases3
% 0.19/0.47 thf(fact_219_Pair__inject,axiom,
% 0.19/0.47 ! [A: b,B: b,A4: b,B6: b] :
% 0.19/0.47 ( ( ( product_Pair_b_b @ A @ B )
% 0.19/0.47 = ( product_Pair_b_b @ A4 @ B6 ) )
% 0.19/0.47 => ~ ( ( A = A4 )
% 0.19/0.47 => ( B != B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pair_inject
% 0.19/0.47 thf(fact_220_Pair__inject,axiom,
% 0.19/0.47 ! [A: standard_Constant_a,B: product_prod_b_b,A4: standard_Constant_a,B6: product_prod_b_b] :
% 0.19/0.47 ( ( ( produc1432590431od_b_b @ A @ B )
% 0.19/0.47 = ( produc1432590431od_b_b @ A4 @ B6 ) )
% 0.19/0.47 => ~ ( ( A = A4 )
% 0.19/0.47 => ( B != B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pair_inject
% 0.19/0.47 thf(fact_221_prod__cases,axiom,
% 0.19/0.47 ! [P: product_prod_b_b > $o,P3: product_prod_b_b] :
% 0.19/0.47 ( ! [A5: b,B7: b] : ( P @ ( product_Pair_b_b @ A5 @ B7 ) )
% 0.19/0.47 => ( P @ P3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases
% 0.19/0.47 thf(fact_222_prod__cases,axiom,
% 0.19/0.47 ! [P: produc1213276845od_b_b > $o,P3: produc1213276845od_b_b] :
% 0.19/0.47 ( ! [A5: standard_Constant_a,B7: product_prod_b_b] : ( P @ ( produc1432590431od_b_b @ A5 @ B7 ) )
% 0.19/0.47 => ( P @ P3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases
% 0.19/0.47 thf(fact_223_surj__pair,axiom,
% 0.19/0.47 ! [P3: product_prod_b_b] :
% 0.19/0.47 ? [X4: b,Y3: b] :
% 0.19/0.47 ( P3
% 0.19/0.47 = ( product_Pair_b_b @ X4 @ Y3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % surj_pair
% 0.19/0.47 thf(fact_224_surj__pair,axiom,
% 0.19/0.47 ! [P3: produc1213276845od_b_b] :
% 0.19/0.47 ? [X4: standard_Constant_a,Y3: product_prod_b_b] :
% 0.19/0.47 ( P3
% 0.19/0.47 = ( produc1432590431od_b_b @ X4 @ Y3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % surj_pair
% 0.19/0.47 thf(fact_225_local_Orefl,axiom,
% 0.19/0.47 refl_on_b @ ( labele1424214014nt_a_b @ g ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% 0.19/0.47
% 0.19/0.47 % local.refl
% 0.19/0.47 thf(fact_226_equiv,axiom,
% 0.19/0.47 equiv_equiv_b @ ( labele1424214014nt_a_b @ g ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% 0.19/0.47
% 0.19/0.47 % equiv
% 0.19/0.47 thf(fact_227_getRel__dom_I1_J,axiom,
% 0.19/0.47 ! [G: labele1159362096nt_a_b,A: b,B: b,L: standard_Constant_a] :
% 0.19/0.47 ( ( G
% 0.19/0.47 = ( restri446606278nt_a_b @ G ) )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel904497637nt_a_b @ L @ G ) )
% 0.19/0.47 => ( member_b @ A @ ( labele1424214014nt_a_b @ G ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % getRel_dom(1)
% 0.19/0.47 thf(fact_228_getRel__dom_I2_J,axiom,
% 0.19/0.47 ! [G: labele1159362096nt_a_b,A: b,B: b,L: standard_Constant_a] :
% 0.19/0.47 ( ( G
% 0.19/0.47 = ( restri446606278nt_a_b @ G ) )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel904497637nt_a_b @ L @ G ) )
% 0.19/0.47 => ( member_b @ B @ ( labele1424214014nt_a_b @ G ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % getRel_dom(2)
% 0.19/0.47 thf(fact_229_is__singleton__the__elem,axiom,
% 0.19/0.47 ( is_singleton_b
% 0.19/0.47 = ( ^ [A6: set_b] :
% 0.19/0.47 ( A6
% 0.19/0.47 = ( insert_b @ ( the_elem_b @ A6 ) @ bot_bot_set_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singleton_the_elem
% 0.19/0.47 thf(fact_230_is__singleton__the__elem,axiom,
% 0.19/0.47 ( is_sin1713057243od_b_b
% 0.19/0.47 = ( ^ [A6: set_Product_prod_b_b] :
% 0.19/0.47 ( A6
% 0.19/0.47 = ( insert1952693431od_b_b @ ( the_el329196508od_b_b @ A6 ) @ bot_bo1343651123od_b_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singleton_the_elem
% 0.19/0.47 thf(fact_231_refl__on__converse,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( refl_on_b @ A2 @ ( converse_b_b @ R2 ) )
% 0.19/0.47 = ( refl_on_b @ A2 @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_on_converse
% 0.19/0.47 thf(fact_232_is__singletonI,axiom,
% 0.19/0.47 ! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonI
% 0.19/0.47 thf(fact_233_is__singletonI,axiom,
% 0.19/0.47 ! [X: product_prod_b_b] : ( is_sin1713057243od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonI
% 0.19/0.47 thf(fact_234_refl__onD,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
% 0.19/0.47 ( ( refl_o2134473190od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.47 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ A ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_onD
% 0.19/0.47 thf(fact_235_refl__onD,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b] :
% 0.19/0.47 ( ( refl_o1921098222od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.47 => ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ A ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_onD
% 0.19/0.47 thf(fact_236_refl__onD,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b,A: b] :
% 0.19/0.47 ( ( refl_on_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member_b @ A @ A2 )
% 0.19/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ A ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_onD
% 0.19/0.47 thf(fact_237_refl__onD1,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
% 0.19/0.47 ( ( refl_o2134473190od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 )
% 0.19/0.47 => ( member1285940496od_b_b @ X @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_onD1
% 0.19/0.47 thf(fact_238_refl__onD1,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
% 0.19/0.47 ( ( refl_o1921098222od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 )
% 0.19/0.47 => ( member1516365892od_b_b @ X @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_onD1
% 0.19/0.47 thf(fact_239_refl__onD1,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
% 0.19/0.47 ( ( refl_on_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 )
% 0.19/0.47 => ( member_b @ X @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_onD1
% 0.19/0.47 thf(fact_240_refl__onD2,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
% 0.19/0.47 ( ( refl_o2134473190od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 )
% 0.19/0.47 => ( member1285940496od_b_b @ Y4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_onD2
% 0.19/0.47 thf(fact_241_refl__onD2,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
% 0.19/0.47 ( ( refl_o1921098222od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 )
% 0.19/0.47 => ( member1516365892od_b_b @ Y4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_onD2
% 0.19/0.47 thf(fact_242_refl__onD2,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
% 0.19/0.47 ( ( refl_on_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 )
% 0.19/0.47 => ( member_b @ Y4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_onD2
% 0.19/0.47 thf(fact_243_refl__on__empty,axiom,
% 0.19/0.47 refl_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% 0.19/0.47
% 0.19/0.47 % refl_on_empty
% 0.19/0.47 thf(fact_244_refl__on__empty,axiom,
% 0.19/0.47 refl_o2134473190od_b_b @ bot_bo1343651123od_b_b @ bot_bo1174586163od_b_b ).
% 0.19/0.47
% 0.19/0.47 % refl_on_empty
% 0.19/0.47 thf(fact_245_is__singletonI_H,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( A2 != bot_bo1664927607od_b_b )
% 0.19/0.47 => ( ! [X4: produc1213276845od_b_b,Y3: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ X4 @ A2 )
% 0.19/0.47 => ( ( member1516365892od_b_b @ Y3 @ A2 )
% 0.19/0.47 => ( X4 = Y3 ) ) )
% 0.19/0.47 => ( is_sin1465183929od_b_b @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonI'
% 0.19/0.47 thf(fact_246_is__singletonI_H,axiom,
% 0.19/0.47 ! [A2: set_b] :
% 0.19/0.47 ( ( A2 != bot_bot_set_b )
% 0.19/0.47 => ( ! [X4: b,Y3: b] :
% 0.19/0.47 ( ( member_b @ X4 @ A2 )
% 0.19/0.47 => ( ( member_b @ Y3 @ A2 )
% 0.19/0.47 => ( X4 = Y3 ) ) )
% 0.19/0.47 => ( is_singleton_b @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonI'
% 0.19/0.47 thf(fact_247_is__singletonI_H,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( A2 != bot_bo1343651123od_b_b )
% 0.19/0.47 => ( ! [X4: product_prod_b_b,Y3: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ X4 @ A2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ Y3 @ A2 )
% 0.19/0.47 => ( X4 = Y3 ) ) )
% 0.19/0.47 => ( is_sin1713057243od_b_b @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonI'
% 0.19/0.47 thf(fact_248_refl__on__singleton,axiom,
% 0.19/0.47 ! [X: b] : ( refl_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_on_singleton
% 0.19/0.47 thf(fact_249_refl__on__singleton,axiom,
% 0.19/0.47 ! [X: product_prod_b_b] : ( refl_o2134473190od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X @ X ) @ bot_bo1174586163od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_on_singleton
% 0.19/0.47 thf(fact_250_is__singleton__def,axiom,
% 0.19/0.47 ( is_singleton_b
% 0.19/0.47 = ( ^ [A6: set_b] :
% 0.19/0.47 ? [X2: b] :
% 0.19/0.47 ( A6
% 0.19/0.47 = ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singleton_def
% 0.19/0.47 thf(fact_251_is__singleton__def,axiom,
% 0.19/0.47 ( is_sin1713057243od_b_b
% 0.19/0.47 = ( ^ [A6: set_Product_prod_b_b] :
% 0.19/0.47 ? [X2: product_prod_b_b] :
% 0.19/0.47 ( A6
% 0.19/0.47 = ( insert1952693431od_b_b @ X2 @ bot_bo1343651123od_b_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singleton_def
% 0.19/0.47 thf(fact_252_is__singletonE,axiom,
% 0.19/0.47 ! [A2: set_b] :
% 0.19/0.47 ( ( is_singleton_b @ A2 )
% 0.19/0.47 => ~ ! [X4: b] :
% 0.19/0.47 ( A2
% 0.19/0.47 != ( insert_b @ X4 @ bot_bot_set_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonE
% 0.19/0.47 thf(fact_253_is__singletonE,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( is_sin1713057243od_b_b @ A2 )
% 0.19/0.47 => ~ ! [X4: product_prod_b_b] :
% 0.19/0.47 ( A2
% 0.19/0.47 != ( insert1952693431od_b_b @ X4 @ bot_bo1343651123od_b_b ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonE
% 0.19/0.47 thf(fact_254_equiv__class__eq__iff,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
% 0.19/0.47 ( ( equiv_2048442231od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 )
% 0.19/0.47 = ( ( ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ X @ bot_bo1664927607od_b_b ) )
% 0.19/0.47 = ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ Y4 @ bot_bo1664927607od_b_b ) ) )
% 0.19/0.47 & ( member1516365892od_b_b @ X @ A2 )
% 0.19/0.47 & ( member1516365892od_b_b @ Y4 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_eq_iff
% 0.19/0.47 thf(fact_255_equiv__class__eq__iff,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
% 0.19/0.47 ( ( equiv_equiv_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 )
% 0.19/0.47 = ( ( ( image_b_b @ R2 @ ( insert_b @ X @ bot_bot_set_b ) )
% 0.19/0.47 = ( image_b_b @ R2 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) )
% 0.19/0.47 & ( member_b @ X @ A2 )
% 0.19/0.47 & ( member_b @ Y4 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_eq_iff
% 0.19/0.47 thf(fact_256_equiv__class__eq__iff,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
% 0.19/0.47 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 )
% 0.19/0.47 = ( ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 = ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ Y4 @ bot_bo1343651123od_b_b ) ) )
% 0.19/0.47 & ( member1285940496od_b_b @ X @ A2 )
% 0.19/0.47 & ( member1285940496od_b_b @ Y4 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_eq_iff
% 0.19/0.47 thf(fact_257_eq__equiv__class__iff,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
% 0.19/0.47 ( ( equiv_2048442231od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1516365892od_b_b @ X @ A2 )
% 0.19/0.47 => ( ( member1516365892od_b_b @ Y4 @ A2 )
% 0.19/0.47 => ( ( ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ X @ bot_bo1664927607od_b_b ) )
% 0.19/0.47 = ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ Y4 @ bot_bo1664927607od_b_b ) ) )
% 0.19/0.47 = ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_equiv_class_iff
% 0.19/0.47 thf(fact_258_eq__equiv__class__iff,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
% 0.19/0.47 ( ( equiv_equiv_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member_b @ X @ A2 )
% 0.19/0.47 => ( ( member_b @ Y4 @ A2 )
% 0.19/0.47 => ( ( ( image_b_b @ R2 @ ( insert_b @ X @ bot_bot_set_b ) )
% 0.19/0.47 = ( image_b_b @ R2 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) )
% 0.19/0.47 = ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_equiv_class_iff
% 0.19/0.47 thf(fact_259_eq__equiv__class__iff,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
% 0.19/0.47 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ X @ A2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ Y4 @ A2 )
% 0.19/0.47 => ( ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 = ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ Y4 @ bot_bo1343651123od_b_b ) ) )
% 0.19/0.47 = ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_equiv_class_iff
% 0.19/0.47 thf(fact_260_equiv__class__eq,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
% 0.19/0.47 ( ( equiv_equiv_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) )
% 0.19/0.47 = ( image_b_b @ R2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_eq
% 0.19/0.47 thf(fact_261_equiv__class__eq,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
% 0.19/0.47 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 = ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_eq
% 0.19/0.47 thf(fact_262_eq__equiv__class,axiom,
% 0.19/0.47 ! [R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b,B: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) )
% 0.19/0.47 = ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ B @ bot_bo1664927607od_b_b ) ) )
% 0.19/0.47 => ( ( equiv_2048442231od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1516365892od_b_b @ B @ A2 )
% 0.19/0.47 => ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_equiv_class
% 0.19/0.47 thf(fact_263_eq__equiv__class,axiom,
% 0.19/0.47 ! [R2: set_Product_prod_b_b,A: b,B: b,A2: set_b] :
% 0.19/0.47 ( ( ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) )
% 0.19/0.47 = ( image_b_b @ R2 @ ( insert_b @ B @ bot_bot_set_b ) ) )
% 0.19/0.47 => ( ( equiv_equiv_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member_b @ B @ A2 )
% 0.19/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_equiv_class
% 0.19/0.47 thf(fact_264_eq__equiv__class,axiom,
% 0.19/0.47 ! [R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 = ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) )
% 0.19/0.47 => ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ B @ A2 )
% 0.19/0.47 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_equiv_class
% 0.19/0.47 thf(fact_265_congruentI,axiom,
% 0.19/0.47 ! [R2: set_Product_prod_b_b,F: b > set_b] :
% 0.19/0.47 ( ! [Y3: b,Z3: b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ Z3 ) @ R2 )
% 0.19/0.47 => ( ( F @ Y3 )
% 0.19/0.47 = ( F @ Z3 ) ) )
% 0.19/0.47 => ( equiv_1821124534_set_b @ R2 @ F ) ) ).
% 0.19/0.47
% 0.19/0.47 % congruentI
% 0.19/0.47 thf(fact_266_congruentD,axiom,
% 0.19/0.47 ! [R2: set_Product_prod_b_b,F: b > set_b,Y4: b,Z2: b] :
% 0.19/0.47 ( ( equiv_1821124534_set_b @ R2 @ F )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z2 ) @ R2 )
% 0.19/0.47 => ( ( F @ Y4 )
% 0.19/0.47 = ( F @ Z2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % congruentD
% 0.19/0.47 thf(fact_267_equiv__class__self,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b] :
% 0.19/0.47 ( ( equiv_2048442231od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.47 => ( member1516365892od_b_b @ A @ ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_self
% 0.19/0.47 thf(fact_268_equiv__class__self,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b,A: b] :
% 0.19/0.47 ( ( equiv_equiv_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member_b @ A @ A2 )
% 0.19/0.47 => ( member_b @ A @ ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_self
% 0.19/0.47 thf(fact_269_equiv__class__self,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
% 0.19/0.47 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.47 => ( member1285940496od_b_b @ A @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_self
% 0.19/0.47 thf(fact_270_Collect__empty__eq__bot,axiom,
% 0.19/0.47 ! [P: b > $o] :
% 0.19/0.47 ( ( ( collect_b @ P )
% 0.19/0.47 = bot_bot_set_b )
% 0.19/0.47 = ( P = bot_bot_b_o ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_empty_eq_bot
% 0.19/0.47 thf(fact_271_Collect__empty__eq__bot,axiom,
% 0.19/0.47 ! [P: product_prod_b_b > $o] :
% 0.19/0.47 ( ( ( collec1481886546od_b_b @ P )
% 0.19/0.47 = bot_bo1343651123od_b_b )
% 0.19/0.47 = ( P = bot_bo1302443370_b_b_o ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_empty_eq_bot
% 0.19/0.47 thf(fact_272_graph__single,axiom,
% 0.19/0.47 ! [A: standard_Constant_a,B: b,C: b] :
% 0.19/0.47 ( ( labele1230159100nt_a_b @ ( insert2037698781od_b_b @ ( produc1432590431od_b_b @ A @ ( product_Pair_b_b @ B @ C ) ) @ bot_bo1664927607od_b_b ) @ ( insert_b @ B @ ( insert_b @ C @ bot_bot_set_b ) ) )
% 0.19/0.47 = ( restri446606278nt_a_b @ ( labele1230159100nt_a_b @ ( insert2037698781od_b_b @ ( produc1432590431od_b_b @ A @ ( product_Pair_b_b @ B @ C ) ) @ bot_bo1664927607od_b_b ) @ ( insert_b @ B @ ( insert_b @ C @ bot_bot_set_b ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % graph_single
% 0.19/0.47 thf(fact_273_refl__on__domain,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
% 0.19/0.47 ( ( refl_o2134473190od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.47 & ( member1285940496od_b_b @ B @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_on_domain
% 0.19/0.47 thf(fact_274_refl__on__domain,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b,B: produc1213276845od_b_b] :
% 0.19/0.47 ( ( refl_o1921098222od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.47 & ( member1516365892od_b_b @ B @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_on_domain
% 0.19/0.47 thf(fact_275_refl__on__domain,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
% 0.19/0.47 ( ( refl_on_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( ( member_b @ A @ A2 )
% 0.19/0.47 & ( member_b @ B @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % refl_on_domain
% 0.19/0.47 thf(fact_276_total__on__singleton,axiom,
% 0.19/0.47 ! [X: b] : ( total_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % total_on_singleton
% 0.19/0.47 thf(fact_277_total__on__singleton,axiom,
% 0.19/0.47 ! [X: product_prod_b_b] : ( total_178717291od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X @ X ) @ bot_bo1174586163od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % total_on_singleton
% 0.19/0.47 thf(fact_278_total__on__converse,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( total_on_b @ A2 @ ( converse_b_b @ R2 ) )
% 0.19/0.47 = ( total_on_b @ A2 @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % total_on_converse
% 0.19/0.47 thf(fact_279_labeled__graph_Ocollapse,axiom,
% 0.19/0.47 ! [Labeled_graph: labele1159362096nt_a_b] :
% 0.19/0.47 ( ( labele1230159100nt_a_b @ ( labele1741081071nt_a_b @ Labeled_graph ) @ ( labele1424214014nt_a_b @ Labeled_graph ) )
% 0.19/0.47 = Labeled_graph ) ).
% 0.19/0.47
% 0.19/0.47 % labeled_graph.collapse
% 0.19/0.47 thf(fact_280_graph__empty__e,axiom,
% 0.19/0.47 ! [V2: set_b] :
% 0.19/0.47 ( ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ V2 )
% 0.19/0.47 = ( restri446606278nt_a_b @ ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ V2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % graph_empty_e
% 0.19/0.47 thf(fact_281_total__on__def,axiom,
% 0.19/0.47 ( total_on_b
% 0.19/0.47 = ( ^ [A6: set_b,R3: set_Product_prod_b_b] :
% 0.19/0.47 ! [X2: b] :
% 0.19/0.47 ( ( member_b @ X2 @ A6 )
% 0.19/0.47 => ! [Y: b] :
% 0.19/0.47 ( ( member_b @ Y @ A6 )
% 0.19/0.47 => ( ( X2 != Y )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R3 )
% 0.19/0.47 | ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ X2 ) @ R3 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % total_on_def
% 0.19/0.47 thf(fact_282_total__onI,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b] :
% 0.19/0.47 ( ! [X4: product_prod_b_b,Y3: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ X4 @ A2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ Y3 @ A2 )
% 0.19/0.47 => ( ( X4 != Y3 )
% 0.19/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y3 ) @ R2 )
% 0.19/0.47 | ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 0.19/0.47 => ( total_178717291od_b_b @ A2 @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % total_onI
% 0.19/0.47 thf(fact_283_total__onI,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b] :
% 0.19/0.47 ( ! [X4: produc1213276845od_b_b,Y3: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ X4 @ A2 )
% 0.19/0.47 => ( ( member1516365892od_b_b @ Y3 @ A2 )
% 0.19/0.47 => ( ( X4 != Y3 )
% 0.19/0.47 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X4 @ Y3 ) @ R2 )
% 0.19/0.47 | ( member1086024932od_b_b @ ( produc449289715od_b_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 0.19/0.47 => ( total_771970601od_b_b @ A2 @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % total_onI
% 0.19/0.47 thf(fact_284_total__onI,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ! [X4: b,Y3: b] :
% 0.19/0.47 ( ( member_b @ X4 @ A2 )
% 0.19/0.47 => ( ( member_b @ Y3 @ A2 )
% 0.19/0.47 => ( ( X4 != Y3 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R2 )
% 0.19/0.47 | ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 0.19/0.47 => ( total_on_b @ A2 @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % total_onI
% 0.19/0.47 thf(fact_285_total__on__empty,axiom,
% 0.19/0.47 ! [R2: set_Product_prod_b_b] : ( total_on_b @ bot_bot_set_b @ R2 ) ).
% 0.19/0.47
% 0.19/0.47 % total_on_empty
% 0.19/0.47 thf(fact_286_total__on__empty,axiom,
% 0.19/0.47 ! [R2: set_Pr2007082183od_b_b] : ( total_178717291od_b_b @ bot_bo1343651123od_b_b @ R2 ) ).
% 0.19/0.47
% 0.19/0.47 % total_on_empty
% 0.19/0.47 thf(fact_287_labeled__graph_Osel_I2_J,axiom,
% 0.19/0.47 ! [X1: set_Pr1324126435od_b_b,X22: set_b] :
% 0.19/0.47 ( ( labele1424214014nt_a_b @ ( labele1230159100nt_a_b @ X1 @ X22 ) )
% 0.19/0.47 = X22 ) ).
% 0.19/0.47
% 0.19/0.47 % labeled_graph.sel(2)
% 0.19/0.47 thf(fact_288_labeled__graph_Osel_I1_J,axiom,
% 0.19/0.47 ! [X1: set_Pr1324126435od_b_b,X22: set_b] :
% 0.19/0.47 ( ( labele1741081071nt_a_b @ ( labele1230159100nt_a_b @ X1 @ X22 ) )
% 0.19/0.47 = X1 ) ).
% 0.19/0.47
% 0.19/0.47 % labeled_graph.sel(1)
% 0.19/0.47 thf(fact_289_labeled__graph_Oexhaust__sel,axiom,
% 0.19/0.47 ! [Labeled_graph: labele1159362096nt_a_b] :
% 0.19/0.47 ( Labeled_graph
% 0.19/0.47 = ( labele1230159100nt_a_b @ ( labele1741081071nt_a_b @ Labeled_graph ) @ ( labele1424214014nt_a_b @ Labeled_graph ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % labeled_graph.exhaust_sel
% 0.19/0.47 thf(fact_290_linear__order__on__singleton,axiom,
% 0.19/0.47 ! [X: b] : ( order_653178932r_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % linear_order_on_singleton
% 0.19/0.47 thf(fact_291_linear__order__on__singleton,axiom,
% 0.19/0.47 ! [X: product_prod_b_b] : ( order_609163036od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X @ X ) @ bot_bo1174586163od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % linear_order_on_singleton
% 0.19/0.47 thf(fact_292_graph__homomorphism__empty,axiom,
% 0.19/0.47 ! [G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
% 0.19/0.47 ( ( graph_222606102_a_b_b @ ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ bot_bot_set_b ) @ G @ F )
% 0.19/0.47 = ( ( F = bot_bo1343651123od_b_b )
% 0.19/0.47 & ( G
% 0.19/0.47 = ( restri446606278nt_a_b @ G ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % graph_homomorphism_empty
% 0.19/0.47 thf(fact_293_graph__homomorphism__empty,axiom,
% 0.19/0.47 ! [G: labele1159362096nt_a_b,F: set_Pr357419743_b_b_b] :
% 0.19/0.47 ( ( graph_792023598_b_b_b @ ( labele170595300od_b_b @ bot_bo1964903287od_b_b @ bot_bo1343651123od_b_b ) @ G @ F )
% 0.19/0.47 = ( ( F = bot_bo1124087115_b_b_b )
% 0.19/0.47 & ( G
% 0.19/0.47 = ( restri446606278nt_a_b @ G ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % graph_homomorphism_empty
% 0.19/0.47 thf(fact_294_linear__order__on__converse,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( order_653178932r_on_b @ A2 @ ( converse_b_b @ R2 ) )
% 0.19/0.47 = ( order_653178932r_on_b @ A2 @ R2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % linear_order_on_converse
% 0.19/0.47 thf(fact_295_edges__def,axiom,
% 0.19/0.47 ( labele1741081071nt_a_b
% 0.19/0.47 = ( labele377702064od_b_b
% 0.19/0.47 @ ^ [X12: set_Pr1324126435od_b_b,X23: set_b] : X12 ) ) ).
% 0.19/0.47
% 0.19/0.47 % edges_def
% 0.19/0.47 thf(fact_296_vertices__def,axiom,
% 0.19/0.47 ( labele1424214014nt_a_b
% 0.19/0.47 = ( labele1287464530_set_b
% 0.19/0.47 @ ^ [X12: set_Pr1324126435od_b_b,X23: set_b] : X23 ) ) ).
% 0.19/0.47
% 0.19/0.47 % vertices_def
% 0.19/0.47 thf(fact_297_lnear__order__on__empty,axiom,
% 0.19/0.47 order_653178932r_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% 0.19/0.47
% 0.19/0.47 % lnear_order_on_empty
% 0.19/0.47 thf(fact_298_lnear__order__on__empty,axiom,
% 0.19/0.47 order_609163036od_b_b @ bot_bo1343651123od_b_b @ bot_bo1174586163od_b_b ).
% 0.19/0.47
% 0.19/0.47 % lnear_order_on_empty
% 0.19/0.47 thf(fact_299_congruent2__implies__congruent,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R1: set_Pr2007082183od_b_b,R22: set_Product_prod_b_b,F: product_prod_b_b > b > set_b,A: product_prod_b_b] :
% 0.19/0.47 ( ( equiv_1125628061od_b_b @ A2 @ R1 )
% 0.19/0.47 => ( ( equiv_1036235914_set_b @ R1 @ R22 @ F )
% 0.19/0.47 => ( ( member1285940496od_b_b @ A @ A2 )
% 0.19/0.47 => ( equiv_1821124534_set_b @ R22 @ ( F @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % congruent2_implies_congruent
% 0.19/0.47 thf(fact_300_congruent2__implies__congruent,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,R1: set_Pr2109276827od_b_b,R22: set_Product_prod_b_b,F: produc1213276845od_b_b > b > set_b,A: produc1213276845od_b_b] :
% 0.19/0.47 ( ( equiv_2048442231od_b_b @ A2 @ R1 )
% 0.19/0.47 => ( ( equiv_1629197578_set_b @ R1 @ R22 @ F )
% 0.19/0.47 => ( ( member1516365892od_b_b @ A @ A2 )
% 0.19/0.47 => ( equiv_1821124534_set_b @ R22 @ ( F @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % congruent2_implies_congruent
% 0.19/0.47 thf(fact_301_congruent2__implies__congruent,axiom,
% 0.19/0.47 ! [A2: set_b,R1: set_Product_prod_b_b,R22: set_Product_prod_b_b,F: b > b > set_b,A: b] :
% 0.19/0.47 ( ( equiv_equiv_b @ A2 @ R1 )
% 0.19/0.47 => ( ( equiv_167626658_set_b @ R1 @ R22 @ F )
% 0.19/0.47 => ( ( member_b @ A @ A2 )
% 0.19/0.47 => ( equiv_1821124534_set_b @ R22 @ ( F @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % congruent2_implies_congruent
% 0.19/0.47 thf(fact_302_proj__def,axiom,
% 0.19/0.47 ( equiv_proj_b_b
% 0.19/0.47 = ( ^ [R3: set_Product_prod_b_b,X2: b] : ( image_b_b @ R3 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % proj_def
% 0.19/0.47 thf(fact_303_equiv__class__subset,axiom,
% 0.19/0.47 ! [A2: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
% 0.19/0.47 ( ( equiv_equiv_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( ord_less_eq_set_b @ ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ ( image_b_b @ R2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_subset
% 0.19/0.47 thf(fact_304_equiv__class__subset,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
% 0.19/0.47 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 0.19/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 0.19/0.47 => ( ord_le1036320359od_b_b @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equiv_class_subset
% 0.19/0.47 thf(fact_305_subsetI,axiom,
% 0.19/0.47 ! [A2: set_b,B2: set_b] :
% 0.19/0.47 ( ! [X4: b] :
% 0.19/0.47 ( ( member_b @ X4 @ A2 )
% 0.19/0.47 => ( member_b @ X4 @ B2 ) )
% 0.19/0.47 => ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetI
% 0.19/0.47 thf(fact_306_subsetI,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
% 0.19/0.47 ( ! [X4: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ X4 @ A2 )
% 0.19/0.47 => ( member1285940496od_b_b @ X4 @ B2 ) )
% 0.19/0.47 => ( ord_le1036320359od_b_b @ A2 @ B2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetI
% 0.19/0.47 thf(fact_307_subsetI,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ! [X4: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ X4 @ A2 )
% 0.19/0.47 => ( member1516365892od_b_b @ X4 @ B2 ) )
% 0.19/0.47 => ( ord_le123089219od_b_b @ A2 @ B2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetI
% 0.19/0.47 thf(fact_308_converse__mono,axiom,
% 0.19/0.47 ! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( ord_le1036320359od_b_b @ ( converse_b_b @ R2 ) @ ( converse_b_b @ S2 ) )
% 0.19/0.47 = ( ord_le1036320359od_b_b @ R2 @ S2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % converse_mono
% 0.19/0.47 thf(fact_309_empty__subsetI,axiom,
% 0.19/0.47 ! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% 0.19/0.47
% 0.19/0.47 % empty_subsetI
% 0.19/0.47 thf(fact_310_empty__subsetI,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b] : ( ord_le1036320359od_b_b @ bot_bo1343651123od_b_b @ A2 ) ).
% 0.19/0.47
% 0.19/0.47 % empty_subsetI
% 0.19/0.47 thf(fact_311_subset__empty,axiom,
% 0.19/0.47 ! [A2: set_b] :
% 0.19/0.47 ( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
% 0.19/0.47 = ( A2 = bot_bot_set_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_empty
% 0.19/0.47 thf(fact_312_subset__empty,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( ord_le1036320359od_b_b @ A2 @ bot_bo1343651123od_b_b )
% 0.19/0.47 = ( A2 = bot_bo1343651123od_b_b ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_empty
% 0.19/0.47 thf(fact_313_insert__subset,axiom,
% 0.19/0.47 ! [X: b,A2: set_b,B2: set_b] :
% 0.19/0.47 ( ( ord_less_eq_set_b @ ( insert_b @ X @ A2 ) @ B2 )
% 0.19/0.47 = ( ( member_b @ X @ B2 )
% 0.19/0.47 & ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_subset
% 0.19/0.47 thf(fact_314_insert__subset,axiom,
% 0.19/0.47 ! [X: product_prod_b_b,A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X @ A2 ) @ B2 )
% 0.19/0.47 = ( ( member1285940496od_b_b @ X @ B2 )
% 0.19/0.47 & ( ord_le1036320359od_b_b @ A2 @ B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_subset
% 0.19/0.47 thf(fact_315_insert__subset,axiom,
% 0.19/0.47 ! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
% 0.19/0.47 ( ( ord_le123089219od_b_b @ ( insert2037698781od_b_b @ X @ A2 ) @ B2 )
% 0.19/0.47 = ( ( member1516365892od_b_b @ X @ B2 )
% 0.19/0.47 & ( ord_le123089219od_b_b @ A2 @ B2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_subset
% 0.19/0.47 thf(fact_316_singleton__insert__inj__eq_H,axiom,
% 0.19/0.47 ! [A: b,A2: set_b,B: b] :
% 0.19/0.47 ( ( ( insert_b @ A @ A2 )
% 0.19/0.47 = ( insert_b @ B @ bot_bot_set_b ) )
% 0.19/0.47 = ( ( A = B )
% 0.19/0.47 & ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_insert_inj_eq'
% 0.19/0.47 thf(fact_317_singleton__insert__inj__eq_H,axiom,
% 0.19/0.47 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: product_prod_b_b] :
% 0.19/0.47 ( ( ( insert1952693431od_b_b @ A @ A2 )
% 0.19/0.47 = ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) )
% 0.19/0.47 = ( ( A = B )
% 0.19/0.47 & ( ord_le1036320359od_b_b @ A2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_insert_inj_eq'
% 0.19/0.47 thf(fact_318_singleton__insert__inj__eq,axiom,
% 0.19/0.47 ! [B: b,A: b,A2: set_b] :
% 0.19/0.47 ( ( ( insert_b @ B @ bot_bot_set_b )
% 0.19/0.47 = ( insert_b @ A @ A2 ) )
% 0.19/0.47 = ( ( A = B )
% 0.19/0.47 & ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_insert_inj_eq
% 0.19/0.47 thf(fact_319_singleton__insert__inj__eq,axiom,
% 0.19/0.47 ! [B: product_prod_b_b,A: product_prod_b_b,A2: set_Product_prod_b_b] :
% 0.19/0.47 ( ( ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b )
% 0.19/0.47 = ( insert1952693431od_b_b @ A @ A2 ) )
% 0.19/0.47 = ( ( A = B )
% 0.19/0.47 & ( ord_le1036320359od_b_b @ A2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_insert_inj_eq
% 0.19/0.47 thf(fact_320_Image__mono,axiom,
% 0.19/0.47 ! [R4: set_Product_prod_b_b,R2: set_Product_prod_b_b,A7: set_b,A2: set_b] :
% 0.19/0.47 ( ( ord_le1036320359od_b_b @ R4 @ R2 )
% 0.19/0.47 => ( ( ord_less_eq_set_b @ A7 @ A2 )
% 0.19/0.47 => ( ord_less_eq_set_b @ ( image_b_b @ R4 @ A7 ) @ ( image_b_b @ R2 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Image_mono
% 0.19/0.47 thf(fact_321_Collect__subset,axiom,
% 0.19/0.47 ! [A2: set_Product_prod_b_b,P: product_prod_b_b > $o] :
% 0.19/0.47 ( ord_le1036320359od_b_b
% 0.19/0.47 @ ( collec1481886546od_b_b
% 0.19/0.47 @ ^ [X2: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ X2 @ A2 )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 @ A2 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_subset
% 0.19/0.47 thf(fact_322_Collect__subset,axiom,
% 0.19/0.47 ! [A2: set_Pr1324126435od_b_b,P: produc1213276845od_b_b > $o] :
% 0.19/0.47 ( ord_le123089219od_b_b
% 0.19/0.47 @ ( collec279084418od_b_b
% 0.19/0.47 @ ^ [X2: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ X2 @ A2 )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 @ A2 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_subset
% 0.19/0.47 thf(fact_323_Collect__subset,axiom,
% 0.19/0.47 ! [A2: set_b,P: b > $o] :
% 0.19/0.47 ( ord_less_eq_set_b
% 0.19/0.47 @ ( collect_b
% 0.19/0.47 @ ^ [X2: b] :
% 0.19/0.47 ( ( member_b @ X2 @ A2 )
% 0.19/0.47 & ( P @ X2 ) ) )
% 0.19/0.47 @ A2 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_subset
% 0.19/0.47 thf(fact_324_Collect__mono__iff,axiom,
% 0.19/0.47 ! [P: b > $o,Q: b > $o] :
% 0.19/0.47 ( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
% 0.19/0.47 = ( ! [X2: b] :
% 0.19/0.47 ( ( P @ X2 )
% 0.19/0.47 => ( Q @ X2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono_iff
% 0.19/0.47 thf(fact_325_Collect__mono,axiom,
% 0.19/0.47 ! [P: b > $o,Q: b > $o] :
% 0.19/0.47 ( ! [X4: b] :
% 0.19/0.47 ( ( P @ X4 )
% 0.19/0.47 => ( Q @ X4 ) )
% 0.19/0.47 => ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono
% 0.19/0.47 thf(fact_326_subset__iff,axiom,
% 0.19/0.47 ( ord_less_eq_set_b
% 0.19/0.47 = ( ^ [A6: set_b,B5: set_b] :
% 0.19/0.47 ! [T: b] :
% 0.19/0.47 ( ( member_b @ T @ A6 )
% 0.19/0.47 => ( member_b @ T @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_iff
% 0.19/0.47 thf(fact_327_subset__iff,axiom,
% 0.19/0.47 ( ord_le1036320359od_b_b
% 0.19/0.47 = ( ^ [A6: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
% 0.19/0.47 ! [T: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ T @ A6 )
% 0.19/0.47 => ( member1285940496od_b_b @ T @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_iff
% 0.19/0.47 thf(fact_328_subset__iff,axiom,
% 0.19/0.47 ( ord_le123089219od_b_b
% 0.19/0.47 = ( ^ [A6: set_Pr1324126435od_b_b,B5: set_Pr1324126435od_b_b] :
% 0.19/0.47 ! [T: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ T @ A6 )
% 0.19/0.47 => ( member1516365892od_b_b @ T @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_iff
% 0.19/0.47 thf(fact_329_subset__eq,axiom,
% 0.19/0.47 ( ord_less_eq_set_b
% 0.19/0.47 = ( ^ [A6: set_b,B5: set_b] :
% 0.19/0.47 ! [X2: b] :
% 0.19/0.47 ( ( member_b @ X2 @ A6 )
% 0.19/0.47 => ( member_b @ X2 @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_eq
% 0.19/0.47 thf(fact_330_subset__eq,axiom,
% 0.19/0.47 ( ord_le1036320359od_b_b
% 0.19/0.47 = ( ^ [A6: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
% 0.19/0.47 ! [X2: product_prod_b_b] :
% 0.19/0.47 ( ( member1285940496od_b_b @ X2 @ A6 )
% 0.19/0.47 => ( member1285940496od_b_b @ X2 @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_eq
% 0.19/0.47 thf(fact_331_subset__eq,axiom,
% 0.19/0.47 ( ord_le123089219od_b_b
% 0.19/0.47 = ( ^ [A6: set_Pr1324126435od_b_b,B5: set_Pr1324126435od_b_b] :
% 0.19/0.47 ! [X2: produc1213276845od_b_b] :
% 0.19/0.47 ( ( member1516365892od_b_b @ X2 @ A6 )
% 0.19/0.47 => ( member1516365892od_b_b @ X2 @ B5 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_eq
% 0.31/0.47 thf(fact_332_subsetD,axiom,
% 0.31/0.47 ! [A2: set_b,B2: set_b,C: b] :
% 0.31/0.47 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 0.31/0.47 => ( ( member_b @ C @ A2 )
% 0.31/0.47 => ( member_b @ C @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetD
% 0.31/0.47 thf(fact_333_subsetD,axiom,
% 0.31/0.47 ! [A2: set_Product_prod_b_b,B2: set_Product_prod_b_b,C: product_prod_b_b] :
% 0.31/0.47 ( ( ord_le1036320359od_b_b @ A2 @ B2 )
% 0.31/0.47 => ( ( member1285940496od_b_b @ C @ A2 )
% 0.31/0.47 => ( member1285940496od_b_b @ C @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetD
% 0.31/0.47 thf(fact_334_subsetD,axiom,
% 0.31/0.47 ! [A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b,C: produc1213276845od_b_b] :
% 0.31/0.47 ( ( ord_le123089219od_b_b @ A2 @ B2 )
% 0.31/0.47 => ( ( member1516365892od_b_b @ C @ A2 )
% 0.31/0.47 => ( member1516365892od_b_b @ C @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetD
% 0.31/0.47 thf(fact_335_in__mono,axiom,
% 0.31/0.47 ! [A2: set_b,B2: set_b,X: b] :
% 0.31/0.47 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 0.31/0.47 => ( ( member_b @ X @ A2 )
% 0.31/0.47 => ( member_b @ X @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % in_mono
% 0.31/0.47 thf(fact_336_in__mono,axiom,
% 0.31/0.47 ! [A2: set_Product_prod_b_b,B2: set_Product_prod_b_b,X: product_prod_b_b] :
% 0.31/0.47 ( ( ord_le1036320359od_b_b @ A2 @ B2 )
% 0.31/0.47 => ( ( member1285940496od_b_b @ X @ A2 )
% 0.31/0.47 => ( member1285940496od_b_b @ X @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % in_mono
% 0.31/0.47 thf(fact_337_in__mono,axiom,
% 0.31/0.47 ! [A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b,X: produc1213276845od_b_b] :
% 0.31/0.47 ( ( ord_le123089219od_b_b @ A2 @ B2 )
% 0.31/0.47 => ( ( member1516365892od_b_b @ X @ A2 )
% 0.31/0.47 => ( member1516365892od_b_b @ X @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % in_mono
% 0.31/0.47 thf(fact_338_converse__subset__swap,axiom,
% 0.31/0.47 ! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
% 0.31/0.47 ( ( ord_le1036320359od_b_b @ R2 @ ( converse_b_b @ S2 ) )
% 0.31/0.47 = ( ord_le1036320359od_b_b @ ( converse_b_b @ R2 ) @ S2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % converse_subset_swap
% 0.31/0.47 thf(fact_339_subrelI,axiom,
% 0.31/0.47 ! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
% 0.31/0.47 ( ! [X4: b,Y3: b] :
% 0.31/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R2 )
% 0.31/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ S2 ) )
% 0.31/0.47 => ( ord_le1036320359od_b_b @ R2 @ S2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subrelI
% 0.31/0.47 thf(fact_340_subrelI,axiom,
% 0.31/0.47 ! [R2: set_Pr1324126435od_b_b,S2: set_Pr1324126435od_b_b] :
% 0.31/0.47 ( ! [X4: standard_Constant_a,Y3: product_prod_b_b] :
% 0.31/0.47 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ R2 )
% 0.31/0.47 => ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ S2 ) )
% 0.31/0.47 => ( ord_le123089219od_b_b @ R2 @ S2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subrelI
% 0.31/0.47 thf(fact_341_bot_Oextremum__uniqueI,axiom,
% 0.31/0.47 ! [A: set_b] :
% 0.31/0.47 ( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
% 0.31/0.47 => ( A = bot_bot_set_b ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot.extremum_uniqueI
% 0.31/0.47 thf(fact_342_bot_Oextremum__uniqueI,axiom,
% 0.31/0.47 ! [A: set_Product_prod_b_b] :
% 0.31/0.47 ( ( ord_le1036320359od_b_b @ A @ bot_bo1343651123od_b_b )
% 0.31/0.47 => ( A = bot_bo1343651123od_b_b ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot.extremum_uniqueI
% 0.31/0.47 thf(fact_343_bot_Oextremum__unique,axiom,
% 0.31/0.47 ! [A: set_b] :
% 0.31/0.47 ( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
% 0.31/0.47 = ( A = bot_bot_set_b ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot.extremum_unique
% 0.31/0.47 thf(fact_344_bot_Oextremum__unique,axiom,
% 0.31/0.47 ! [A: set_Product_prod_b_b] :
% 0.31/0.47 ( ( ord_le1036320359od_b_b @ A @ bot_bo1343651123od_b_b )
% 0.31/0.47 = ( A = bot_bo1343651123od_b_b ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot.extremum_unique
% 0.31/0.47 thf(fact_345_bot_Oextremum,axiom,
% 0.31/0.47 ! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% 0.31/0.47
% 0.31/0.47 % bot.extremum
% 0.31/0.47 thf(fact_346_bot_Oextremum,axiom,
% 0.31/0.47 ! [A: set_Product_prod_b_b] : ( ord_le1036320359od_b_b @ bot_bo1343651123od_b_b @ A ) ).
% 0.31/0.47
% 0.31/0.47 % bot.extremum
% 0.31/0.47 thf(fact_347_insert__mono,axiom,
% 0.31/0.47 ! [C4: set_b,D2: set_b,A: b] :
% 0.31/0.47 ( ( ord_less_eq_set_b @ C4 @ D2 )
% 0.31/0.47 => ( ord_less_eq_set_b @ ( insert_b @ A @ C4 ) @ ( insert_b @ A @ D2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % insert_mono
% 0.31/0.47 thf(fact_348_subset__insert,axiom,
% 0.31/0.47 ! [X: b,A2: set_b,B2: set_b] :
% 0.31/0.47 ( ~ ( member_b @ X @ A2 )
% 0.31/0.47 => ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
% 0.31/0.47 = ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_insert
% 0.31/0.47 thf(fact_349_subset__insert,axiom,
% 0.31/0.47 ! [X: product_prod_b_b,A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
% 0.31/0.47 ( ~ ( member1285940496od_b_b @ X @ A2 )
% 0.31/0.47 => ( ( ord_le1036320359od_b_b @ A2 @ ( insert1952693431od_b_b @ X @ B2 ) )
% 0.31/0.47 = ( ord_le1036320359od_b_b @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_insert
% 0.31/0.47 thf(fact_350_subset__insert,axiom,
% 0.31/0.47 ! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
% 0.31/0.47 ( ~ ( member1516365892od_b_b @ X @ A2 )
% 0.31/0.47 => ( ( ord_le123089219od_b_b @ A2 @ ( insert2037698781od_b_b @ X @ B2 ) )
% 0.31/0.47 = ( ord_le123089219od_b_b @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_insert
% 0.31/0.47 thf(fact_351_subset__insertI,axiom,
% 0.31/0.47 ! [B2: set_b,A: b] : ( ord_less_eq_set_b @ B2 @ ( insert_b @ A @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_insertI
% 0.31/0.47 thf(fact_352_subset__insertI2,axiom,
% 0.31/0.47 ! [A2: set_b,B2: set_b,B: b] :
% 0.31/0.47 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 0.31/0.47 => ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_insertI2
% 0.31/0.47 thf(fact_353_subset__singleton__iff,axiom,
% 0.31/0.47 ! [X3: set_b,A: b] :
% 0.31/0.47 ( ( ord_less_eq_set_b @ X3 @ ( insert_b @ A @ bot_bot_set_b ) )
% 0.31/0.47 = ( ( X3 = bot_bot_set_b )
% 0.31/0.47 | ( X3
% 0.31/0.47 = ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_singleton_iff
% 0.31/0.47 thf(fact_354_subset__singleton__iff,axiom,
% 0.31/0.47 ! [X3: set_Product_prod_b_b,A: product_prod_b_b] :
% 0.31/0.47 ( ( ord_le1036320359od_b_b @ X3 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 0.31/0.47 = ( ( X3 = bot_bo1343651123od_b_b )
% 0.31/0.47 | ( X3
% 0.31/0.47 = ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_singleton_iff
% 0.31/0.47
% 0.31/0.47 % Helper facts (3)
% 0.31/0.47 thf(help_If_3_1_If_001tf__b_T,axiom,
% 0.31/0.47 ! [P: $o] :
% 0.31/0.47 ( ( P = $true )
% 0.31/0.47 | ( P = $false ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_2_1_If_001tf__b_T,axiom,
% 0.31/0.47 ! [X: b,Y4: b] :
% 0.31/0.47 ( ( if_b @ $false @ X @ Y4 )
% 0.31/0.47 = Y4 ) ).
% 0.31/0.47
% 0.31/0.47 thf(help_If_1_1_If_001tf__b_T,axiom,
% 0.31/0.47 ! [X: b,Y4: b] :
% 0.31/0.47 ( ( if_b @ $true @ X @ Y4 )
% 0.31/0.47 = X ) ).
% 0.31/0.47
% 0.31/0.47 % Conjectures (1)
% 0.31/0.47 thf(conj_0,conjecture,
% 0.31/0.47 ( ( hilbert_Eps_b
% 0.31/0.47 @ ^ [Y: b] : ( member_b @ Y @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) )
% 0.31/0.47 = ( hilbert_Eps_b
% 0.31/0.47 @ ^ [Y: b] : ( member_b @ Y @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 3.75/1.76 ---- Embedded in THF ---
% 3.75/1.76 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.75/1.76 %%% Generated on Tue Feb 24 08:27:27 EST 2026
% 3.75/1.76 %%% using '$$dhol' embedding, version 1.3.0.
% 3.75/1.76 %%% Logic specification used:
% 3.75/1.76 %%% thf(spec, logic, $$dhol).
% 3.75/1.76
% 3.75/1.76 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.4auU2Smifg/DTF2DTF20997.p
% See solution above
% 3.75/1.76 ------------------------
% 3.92/1.82 ---- Cleaned THF ---
% 3.92/1.83 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
% 3.92/1.83 set_Pr2109276827od_b_b: $tType ).
% 3.92/1.83
% 3.92/1.83 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 3.92/1.83 produc970027067od_b_b: $tType ).
% 3.92/1.83
% 3.92/1.83 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
% 3.92/1.83 set_Pr1071216121od_b_b: $tType ).
% 3.92/1.83
% 3.92/1.83 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 3.92/1.83 set_Pr2134561957od_b_b: $tType ).
% 3.92/1.83
% 3.92/1.83 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
% 3.92/1.83 set_Pr1021918435od_b_b: $tType ).
% 3.92/1.83
% 3.92/1.83 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
% 3.92/1.83 produc1052326083od_b_b: $tType ).
% 3.92/1.83
% 3.92/1.83 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 3.92/1.83 produc1990339951od_b_b: $tType ).
% 3.92/1.83
% 3.92/1.83 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_It__LabeledGraphSemantics__OStandard____Constant_Itf__a_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J_J,type,
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% 3.92/1.83 = bot_bot_set_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_13_Image__empty2,axiom,
% 3.92/1.83 ! [R: set_Product_prod_b_b] :
% 3.92/1.83 ( ( image_b_b @ R @ bot_bot_set_b )
% 3.92/1.83 = bot_bot_set_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_14_Image__empty2,axiom,
% 3.92/1.83 ! [R: set_Pr621705391od_b_b] :
% 3.92/1.83 ( ( image_1177096571od_b_b @ R @ bot_bot_set_b )
% 3.92/1.83 = bot_bo1343651123od_b_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_15_Image__empty2,axiom,
% 3.92/1.83 ! [R: set_Pr357419743_b_b_b] :
% 3.92/1.83 ( ( image_921732011_b_b_b @ R @ bot_bo1343651123od_b_b )
% 3.92/1.83 = bot_bot_set_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_16_Image__empty2,axiom,
% 3.92/1.83 ! [R: set_Pr2007082183od_b_b] :
% 3.92/1.83 ( ( image_1928024979od_b_b @ R @ bot_bo1343651123od_b_b )
% 3.92/1.83 = bot_bo1343651123od_b_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_17_singletonI,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b] : ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_18_singletonI,axiom,
% 3.92/1.83 ! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_19_singletonI,axiom,
% 3.92/1.83 ! [A: product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_20_some__equality,axiom,
% 3.92/1.83 ! [P: b > $o,A: b] :
% 3.92/1.83 ( ( P @ A )
% 3.92/1.83 => ( ! [X4: b] :
% 3.92/1.83 ( ( P @ X4 )
% 3.92/1.83 => ( X4 = A ) )
% 3.92/1.83 => ( ( hilbert_Eps_b @ P )
% 3.92/1.83 = A ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_21_some__eq__trivial,axiom,
% 3.92/1.83 ! [X: b] :
% 3.92/1.83 ( ( hilbert_Eps_b
% 3.92/1.83 @ ^ [Y: b] : ( Y = X ) )
% 3.92/1.83 = X ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_22_empty__Collect__eq,axiom,
% 3.92/1.83 ! [P: b > $o] :
% 3.92/1.83 ( ( bot_bot_set_b
% 3.92/1.83 = ( collect_b @ P ) )
% 3.92/1.83 = ( ! [X2: b] :
% 3.92/1.83 ~ ( P @ X2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_23_empty__Collect__eq,axiom,
% 3.92/1.83 ! [P: product_prod_b_b > $o] :
% 3.92/1.83 ( ( bot_bo1343651123od_b_b
% 3.92/1.83 = ( collec1481886546od_b_b @ P ) )
% 3.92/1.83 = ( ! [X2: product_prod_b_b] :
% 3.92/1.83 ~ ( P @ X2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_24_Collect__empty__eq,axiom,
% 3.92/1.83 ! [P: b > $o] :
% 3.92/1.83 ( ( ( collect_b @ P )
% 3.92/1.83 = bot_bot_set_b )
% 3.92/1.83 = ( ! [X2: b] :
% 3.92/1.83 ~ ( P @ X2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_25_Collect__empty__eq,axiom,
% 3.92/1.83 ! [P: product_prod_b_b > $o] :
% 3.92/1.83 ( ( ( collec1481886546od_b_b @ P )
% 3.92/1.83 = bot_bo1343651123od_b_b )
% 3.92/1.83 = ( ! [X2: product_prod_b_b] :
% 3.92/1.83 ~ ( P @ X2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_26_all__not__in__conv,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( ! [X2: produc1213276845od_b_b] :
% 3.92/1.83 ~ ( member1516365892od_b_b @ X2 @ A2 ) )
% 3.92/1.83 = ( A2 = bot_bo1664927607od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_27_all__not__in__conv,axiom,
% 3.92/1.83 ! [A2: set_b] :
% 3.92/1.83 ( ( ! [X2: b] :
% 3.92/1.83 ~ ( member_b @ X2 @ A2 ) )
% 3.92/1.83 = ( A2 = bot_bot_set_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_28_all__not__in__conv,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ! [X2: product_prod_b_b] :
% 3.92/1.83 ~ ( member1285940496od_b_b @ X2 @ A2 ) )
% 3.92/1.83 = ( A2 = bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_29_empty__iff,axiom,
% 3.92/1.83 ! [C: produc1213276845od_b_b] :
% 3.92/1.83 ~ ( member1516365892od_b_b @ C @ bot_bo1664927607od_b_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_30_empty__iff,axiom,
% 3.92/1.83 ! [C: b] :
% 3.92/1.83 ~ ( member_b @ C @ bot_bot_set_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_31_empty__iff,axiom,
% 3.92/1.83 ! [C: product_prod_b_b] :
% 3.92/1.83 ~ ( member1285940496od_b_b @ C @ bot_bo1343651123od_b_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_32_insert__absorb2,axiom,
% 3.92/1.83 ! [X: b,A2: set_b] :
% 3.92/1.83 ( ( insert_b @ X @ ( insert_b @ X @ A2 ) )
% 3.92/1.83 = ( insert_b @ X @ A2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_33_insert__iff,axiom,
% 3.92/1.83 ! [A: b,B: b,A2: set_b] :
% 3.92/1.83 ( ( member_b @ A @ ( insert_b @ B @ A2 ) )
% 3.92/1.83 = ( ( A = B )
% 3.92/1.83 | ( member_b @ A @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_34_insert__iff,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: product_prod_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A2 ) )
% 3.92/1.83 = ( ( A = B )
% 3.92/1.83 | ( member1285940496od_b_b @ A @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_35_insert__iff,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,B: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ A2 ) )
% 3.92/1.83 = ( ( A = B )
% 3.92/1.83 | ( member1516365892od_b_b @ A @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_36_insertCI,axiom,
% 3.92/1.83 ! [A: b,B2: set_b,B: b] :
% 3.92/1.83 ( ( ~ ( member_b @ A @ B2 )
% 3.92/1.83 => ( A = B ) )
% 3.92/1.83 => ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_37_insertCI,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B2: set_Product_prod_b_b,B: product_prod_b_b] :
% 3.92/1.83 ( ( ~ ( member1285940496od_b_b @ A @ B2 )
% 3.92/1.83 => ( A = B ) )
% 3.92/1.83 => ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_38_insertCI,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b] :
% 3.92/1.83 ( ( ~ ( member1516365892od_b_b @ A @ B2 )
% 3.92/1.83 => ( A = B ) )
% 3.92/1.83 => ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_39_some__sym__eq__trivial,axiom,
% 3.92/1.83 ! [X: b] :
% 3.92/1.83 ( ( hilbert_Eps_b
% 3.92/1.83 @ ( ^ [Y2: b,Z: b] : ( Y2 = Z )
% 3.92/1.83 @ X ) )
% 3.92/1.83 = X ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_40_ImageI,axiom,
% 3.92/1.83 ! [A: b,B: product_prod_b_b,R2: set_Pr621705391od_b_b,A2: set_b] :
% 3.92/1.83 ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_41_ImageI,axiom,
% 3.92/1.83 ! [A: b,B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b,A2: set_b] :
% 3.92/1.83 ( ( member599505522od_b_b @ ( produc800495189od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_42_ImageI,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: b,R2: set_Pr357419743_b_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_43_ImageI,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: product_prod_b_b,R2: set_Pr2007082183od_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_44_ImageI,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member4694106od_b_b @ ( produc732676669od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_45_ImageI,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,B: b,R2: set_Pr1906739389_b_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( member_b @ B @ ( image_1764625405_b_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_46_ImageI,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,B: product_prod_b_b,R2: set_Pr2134561957od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member942707974od_b_b @ ( produc1597690145od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ B @ ( image_1397930469od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_47_ImageI,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,B: produc1213276845od_b_b,R2: set_Pr2109276827od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( member1516365892od_b_b @ B @ ( image_763305007od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_48_ImageI,axiom,
% 3.92/1.83 ! [A: b,B: b,R2: set_Product_prod_b_b,A2: set_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( member_b @ B @ ( image_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_49_ImageI,axiom,
% 3.92/1.83 ! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A2: set_St761939237tant_a] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member1632892294tant_a @ A @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_50_f,axiom,
% 3.92/1.83 ( f
% 3.92/1.83 = ( ^ [X2: b] :
% 3.92/1.83 ( if_b
% 3.92/1.83 @ ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
% 3.92/1.83 = bot_bot_set_b )
% 3.92/1.83 @ X2
% 3.92/1.83 @ ( hilbert_Eps_b @ ( p @ X2 ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_51_Image__singleton__iff,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A: standard_Constant_a] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) ) )
% 3.92/1.83 = ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_52_Image__singleton__iff,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,R2: set_Pr621705391od_b_b,A: b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
% 3.92/1.83 = ( member641857272od_b_b @ ( produc1257047359od_b_b @ A @ B ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_53_Image__singleton__iff,axiom,
% 3.92/1.83 ! [B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b,A: b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
% 3.92/1.83 = ( member599505522od_b_b @ ( produc800495189od_b_b @ A @ B ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_54_Image__singleton__iff,axiom,
% 3.92/1.83 ! [B: b,R2: set_Product_prod_b_b,A: b] :
% 3.92/1.83 ( ( member_b @ B @ ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
% 3.92/1.83 = ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_55_Image__singleton__iff,axiom,
% 3.92/1.83 ! [B: b,R2: set_Pr357419743_b_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 3.92/1.83 = ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_56_Image__singleton__iff,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 3.92/1.83 = ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_57_Image__singleton__iff,axiom,
% 3.92/1.83 ! [B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 3.92/1.83 = ( member4694106od_b_b @ ( produc732676669od_b_b @ A @ B ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_58_pred__equals__eq2,axiom,
% 3.92/1.83 ! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ( ^ [X2: b,Y: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R ) )
% 3.92/1.83 = ( ^ [X2: b,Y: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ S ) ) )
% 3.92/1.83 = ( R = S ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_59_pred__equals__eq2,axiom,
% 3.92/1.83 ! [R: set_Pr1324126435od_b_b,S: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( ( ^ [X2: standard_Constant_a,Y: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ Y ) @ R ) )
% 3.92/1.83 = ( ^ [X2: standard_Constant_a,Y: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ Y ) @ S ) ) )
% 3.92/1.83 = ( R = S ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_60_rev__ImageI,axiom,
% 3.92/1.83 ! [A: b,A2: set_b,B: product_prod_b_b,R2: set_Pr621705391od_b_b] :
% 3.92/1.83 ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_61_rev__ImageI,axiom,
% 3.92/1.83 ! [A: b,A2: set_b,B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b] :
% 3.92/1.83 ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( ( member599505522od_b_b @ ( produc800495189od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_62_rev__ImageI,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: b,R2: set_Pr357419743_b_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_63_rev__ImageI,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: product_prod_b_b,R2: set_Pr2007082183od_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_64_rev__ImageI,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( ( member4694106od_b_b @ ( produc732676669od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_65_rev__ImageI,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: b,R2: set_Pr1906739389_b_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member_b @ B @ ( image_1764625405_b_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_66_rev__ImageI,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: product_prod_b_b,R2: set_Pr2134561957od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( ( member942707974od_b_b @ ( produc1597690145od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1285940496od_b_b @ B @ ( image_1397930469od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_67_rev__ImageI,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b,R2: set_Pr2109276827od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1516365892od_b_b @ B @ ( image_763305007od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_68_rev__ImageI,axiom,
% 3.92/1.83 ! [A: b,A2: set_b,B: b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member_b @ B @ ( image_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_69_rev__ImageI,axiom,
% 3.92/1.83 ! [A: standard_Constant_a,A2: set_St761939237tant_a,B: product_prod_b_b,R2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1632892294tant_a @ A @ A2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_70_Image__iff,axiom,
% 3.92/1.83 ! [B: b,R2: set_Product_prod_b_b,A2: set_b] :
% 3.92/1.83 ( ( member_b @ B @ ( image_b_b @ R2 @ A2 ) )
% 3.92/1.83 = ( ? [X2: b] :
% 3.92/1.83 ( ( member_b @ X2 @ A2 )
% 3.92/1.83 & ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ B ) @ R2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_71_Image__iff,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A2: set_St761939237tant_a] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) )
% 3.92/1.83 = ( ? [X2: standard_Constant_a] :
% 3.92/1.83 ( ( member1632892294tant_a @ X2 @ A2 )
% 3.92/1.83 & ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ B ) @ R2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_72_ImageE,axiom,
% 3.92/1.83 ! [B: b,R2: set_Pr357419743_b_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member_b @ B @ ( image_921732011_b_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: product_prod_b_b] :
% 3.92/1.83 ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member1285940496od_b_b @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_73_ImageE,axiom,
% 3.92/1.83 ! [B: b,R2: set_Pr1906739389_b_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member_b @ B @ ( image_1764625405_b_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1417789726_b_b_b @ ( produc1580777273_b_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member1516365892od_b_b @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_74_ImageE,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,R2: set_Pr621705391od_b_b,A2: set_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( image_1177096571od_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: b] :
% 3.92/1.83 ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member_b @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_75_ImageE,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,R2: set_Pr2007082183od_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( image_1928024979od_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: product_prod_b_b] :
% 3.92/1.83 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member1285940496od_b_b @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_76_ImageE,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,R2: set_Pr2134561957od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( image_1397930469od_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member942707974od_b_b @ ( produc1597690145od_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member1516365892od_b_b @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_77_ImageE,axiom,
% 3.92/1.83 ! [B: produc1213276845od_b_b,R2: set_Pr2060523537od_b_b,A2: set_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ B @ ( image_984343321od_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: b] :
% 3.92/1.83 ( ( member599505522od_b_b @ ( produc800495189od_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member_b @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_78_ImageE,axiom,
% 3.92/1.83 ! [B: produc1213276845od_b_b,R2: set_Pr1071216121od_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ B @ ( image_532916993od_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: product_prod_b_b] :
% 3.92/1.83 ( ( member4694106od_b_b @ ( produc732676669od_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member1285940496od_b_b @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_79_ImageE,axiom,
% 3.92/1.83 ! [B: produc1213276845od_b_b,R2: set_Pr2109276827od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ B @ ( image_763305007od_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member1516365892od_b_b @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_80_ImageE,axiom,
% 3.92/1.83 ! [B: b,R2: set_Product_prod_b_b,A2: set_b] :
% 3.92/1.83 ( ( member_b @ B @ ( image_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member_b @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_81_ImageE,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,R2: set_Pr1324126435od_b_b,A2: set_St761939237tant_a] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( image_1916422435od_b_b @ R2 @ A2 ) )
% 3.92/1.83 => ~ ! [X4: standard_Constant_a] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ B ) @ R2 )
% 3.92/1.83 => ~ ( member1632892294tant_a @ X4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_82_Image__singleton,axiom,
% 3.92/1.83 ! [R2: set_Pr1324126435od_b_b,A: standard_Constant_a] :
% 3.92/1.83 ( ( image_1916422435od_b_b @ R2 @ ( insert1909710879tant_a @ A @ bot_bo1160111033tant_a ) )
% 3.92/1.83 = ( collec1481886546od_b_b
% 3.92/1.83 @ ^ [B3: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B3 ) @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_83_Image__singleton,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b,A: b] :
% 3.92/1.83 ( ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) )
% 3.92/1.83 = ( collect_b
% 3.92/1.83 @ ^ [B3: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B3 ) @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_84_Image__singleton,axiom,
% 3.92/1.83 ! [R2: set_Pr357419743_b_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( image_921732011_b_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = ( collect_b
% 3.92/1.83 @ ^ [B3: b] : ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ A @ B3 ) @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_85_ex__in__conv,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( ? [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ A2 ) )
% 3.92/1.83 = ( A2 != bot_bo1664927607od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_86_ex__in__conv,axiom,
% 3.92/1.83 ! [A2: set_b] :
% 3.92/1.83 ( ( ? [X2: b] : ( member_b @ X2 @ A2 ) )
% 3.92/1.83 = ( A2 != bot_bot_set_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_87_ex__in__conv,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ? [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ A2 ) )
% 3.92/1.83 = ( A2 != bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_88_equals0I,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ! [Y3: produc1213276845od_b_b] :
% 3.92/1.83 ~ ( member1516365892od_b_b @ Y3 @ A2 )
% 3.92/1.83 => ( A2 = bot_bo1664927607od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_89_equals0I,axiom,
% 3.92/1.83 ! [A2: set_b] :
% 3.92/1.83 ( ! [Y3: b] :
% 3.92/1.83 ~ ( member_b @ Y3 @ A2 )
% 3.92/1.83 => ( A2 = bot_bot_set_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_90_equals0I,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ! [Y3: product_prod_b_b] :
% 3.92/1.83 ~ ( member1285940496od_b_b @ Y3 @ A2 )
% 3.92/1.83 => ( A2 = bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_91_equals0D,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,A: produc1213276845od_b_b] :
% 3.92/1.83 ( ( A2 = bot_bo1664927607od_b_b )
% 3.92/1.83 => ~ ( member1516365892od_b_b @ A @ A2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_92_equals0D,axiom,
% 3.92/1.83 ! [A2: set_b,A: b] :
% 3.92/1.83 ( ( A2 = bot_bot_set_b )
% 3.92/1.83 => ~ ( member_b @ A @ A2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_93_equals0D,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( A2 = bot_bo1343651123od_b_b )
% 3.92/1.83 => ~ ( member1285940496od_b_b @ A @ A2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_94_emptyE,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b] :
% 3.92/1.83 ~ ( member1516365892od_b_b @ A @ bot_bo1664927607od_b_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_95_emptyE,axiom,
% 3.92/1.83 ! [A: b] :
% 3.92/1.83 ~ ( member_b @ A @ bot_bot_set_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_96_emptyE,axiom,
% 3.92/1.83 ! [A: product_prod_b_b] :
% 3.92/1.83 ~ ( member1285940496od_b_b @ A @ bot_bo1343651123od_b_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_97_mk__disjoint__insert,axiom,
% 3.92/1.83 ! [A: b,A2: set_b] :
% 3.92/1.83 ( ( member_b @ A @ A2 )
% 3.92/1.83 => ? [B4: set_b] :
% 3.92/1.83 ( ( A2
% 3.92/1.83 = ( insert_b @ A @ B4 ) )
% 3.92/1.83 & ~ ( member_b @ A @ B4 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_98_mk__disjoint__insert,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ? [B4: set_Product_prod_b_b] :
% 3.92/1.83 ( ( A2
% 3.92/1.83 = ( insert1952693431od_b_b @ A @ B4 ) )
% 3.92/1.83 & ~ ( member1285940496od_b_b @ A @ B4 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_99_mk__disjoint__insert,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ? [B4: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( A2
% 3.92/1.83 = ( insert2037698781od_b_b @ A @ B4 ) )
% 3.92/1.83 & ~ ( member1516365892od_b_b @ A @ B4 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_100_insert__commute,axiom,
% 3.92/1.83 ! [X: b,Y4: b,A2: set_b] :
% 3.92/1.83 ( ( insert_b @ X @ ( insert_b @ Y4 @ A2 ) )
% 3.92/1.83 = ( insert_b @ Y4 @ ( insert_b @ X @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_101_insert__eq__iff,axiom,
% 3.92/1.83 ! [A: b,A2: set_b,B: b,B2: set_b] :
% 3.92/1.83 ( ~ ( member_b @ A @ A2 )
% 3.92/1.83 => ( ~ ( member_b @ B @ B2 )
% 3.92/1.83 => ( ( ( insert_b @ A @ A2 )
% 3.92/1.83 = ( insert_b @ B @ B2 ) )
% 3.92/1.83 = ( ( ( A = B )
% 3.92/1.83 => ( A2 = B2 ) )
% 3.92/1.83 & ( ( A != B )
% 3.92/1.83 => ? [C2: set_b] :
% 3.92/1.83 ( ( A2
% 3.92/1.83 = ( insert_b @ B @ C2 ) )
% 3.92/1.83 & ~ ( member_b @ B @ C2 )
% 3.92/1.83 & ( B2
% 3.92/1.83 = ( insert_b @ A @ C2 ) )
% 3.92/1.83 & ~ ( member_b @ A @ C2 ) ) ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_102_insert__eq__iff,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: product_prod_b_b,B2: set_Product_prod_b_b] :
% 3.92/1.83 ( ~ ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( ~ ( member1285940496od_b_b @ B @ B2 )
% 3.92/1.83 => ( ( ( insert1952693431od_b_b @ A @ A2 )
% 3.92/1.83 = ( insert1952693431od_b_b @ B @ B2 ) )
% 3.92/1.83 = ( ( ( A = B )
% 3.92/1.83 => ( A2 = B2 ) )
% 3.92/1.83 & ( ( A != B )
% 3.92/1.83 => ? [C2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( A2
% 3.92/1.83 = ( insert1952693431od_b_b @ B @ C2 ) )
% 3.92/1.83 & ~ ( member1285940496od_b_b @ B @ C2 )
% 3.92/1.83 & ( B2
% 3.92/1.83 = ( insert1952693431od_b_b @ A @ C2 ) )
% 3.92/1.83 & ~ ( member1285940496od_b_b @ A @ C2 ) ) ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_103_insert__eq__iff,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ~ ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( ~ ( member1516365892od_b_b @ B @ B2 )
% 3.92/1.83 => ( ( ( insert2037698781od_b_b @ A @ A2 )
% 3.92/1.83 = ( insert2037698781od_b_b @ B @ B2 ) )
% 3.92/1.83 = ( ( ( A = B )
% 3.92/1.83 => ( A2 = B2 ) )
% 3.92/1.83 & ( ( A != B )
% 3.92/1.83 => ? [C2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( A2
% 3.92/1.83 = ( insert2037698781od_b_b @ B @ C2 ) )
% 3.92/1.83 & ~ ( member1516365892od_b_b @ B @ C2 )
% 3.92/1.83 & ( B2
% 3.92/1.83 = ( insert2037698781od_b_b @ A @ C2 ) )
% 3.92/1.83 & ~ ( member1516365892od_b_b @ A @ C2 ) ) ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_104_insert__absorb,axiom,
% 3.92/1.83 ! [A: b,A2: set_b] :
% 3.92/1.83 ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( ( insert_b @ A @ A2 )
% 3.92/1.83 = A2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_105_insert__absorb,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( ( insert1952693431od_b_b @ A @ A2 )
% 3.92/1.83 = A2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_106_insert__absorb,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( ( insert2037698781od_b_b @ A @ A2 )
% 3.92/1.83 = A2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_107_insert__ident,axiom,
% 3.92/1.83 ! [X: b,A2: set_b,B2: set_b] :
% 3.92/1.83 ( ~ ( member_b @ X @ A2 )
% 3.92/1.83 => ( ~ ( member_b @ X @ B2 )
% 3.92/1.83 => ( ( ( insert_b @ X @ A2 )
% 3.92/1.83 = ( insert_b @ X @ B2 ) )
% 3.92/1.83 = ( A2 = B2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_108_insert__ident,axiom,
% 3.92/1.83 ! [X: product_prod_b_b,A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
% 3.92/1.83 ( ~ ( member1285940496od_b_b @ X @ A2 )
% 3.92/1.83 => ( ~ ( member1285940496od_b_b @ X @ B2 )
% 3.92/1.83 => ( ( ( insert1952693431od_b_b @ X @ A2 )
% 3.92/1.83 = ( insert1952693431od_b_b @ X @ B2 ) )
% 3.92/1.83 = ( A2 = B2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_109_insert__ident,axiom,
% 3.92/1.83 ! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ~ ( member1516365892od_b_b @ X @ A2 )
% 3.92/1.83 => ( ~ ( member1516365892od_b_b @ X @ B2 )
% 3.92/1.83 => ( ( ( insert2037698781od_b_b @ X @ A2 )
% 3.92/1.83 = ( insert2037698781od_b_b @ X @ B2 ) )
% 3.92/1.83 = ( A2 = B2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_110_Set_Oset__insert,axiom,
% 3.92/1.83 ! [X: b,A2: set_b] :
% 3.92/1.83 ( ( member_b @ X @ A2 )
% 3.92/1.83 => ~ ! [B4: set_b] :
% 3.92/1.83 ( ( A2
% 3.92/1.83 = ( insert_b @ X @ B4 ) )
% 3.92/1.83 => ( member_b @ X @ B4 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_111_Set_Oset__insert,axiom,
% 3.92/1.83 ! [X: product_prod_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ X @ A2 )
% 3.92/1.83 => ~ ! [B4: set_Product_prod_b_b] :
% 3.92/1.83 ( ( A2
% 3.92/1.83 = ( insert1952693431od_b_b @ X @ B4 ) )
% 3.92/1.83 => ( member1285940496od_b_b @ X @ B4 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_112_Set_Oset__insert,axiom,
% 3.92/1.83 ! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ X @ A2 )
% 3.92/1.83 => ~ ! [B4: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( A2
% 3.92/1.83 = ( insert2037698781od_b_b @ X @ B4 ) )
% 3.92/1.83 => ( member1516365892od_b_b @ X @ B4 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_113_insertI2,axiom,
% 3.92/1.83 ! [A: b,B2: set_b,B: b] :
% 3.92/1.83 ( ( member_b @ A @ B2 )
% 3.92/1.83 => ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_114_insertI2,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B2: set_Product_prod_b_b,B: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ A @ B2 )
% 3.92/1.83 => ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_115_insertI2,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b,B: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ A @ B2 )
% 3.92/1.83 => ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_116_insertI1,axiom,
% 3.92/1.83 ! [A: b,B2: set_b] : ( member_b @ A @ ( insert_b @ A @ B2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_117_insertI1,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B2: set_Product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ B2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_118_insertI1,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,B2: set_Pr1324126435od_b_b] : ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ A @ B2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_119_insertE,axiom,
% 3.92/1.83 ! [A: b,B: b,A2: set_b] :
% 3.92/1.83 ( ( member_b @ A @ ( insert_b @ B @ A2 ) )
% 3.92/1.83 => ( ( A != B )
% 3.92/1.83 => ( member_b @ A @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_120_insertE,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: product_prod_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A2 ) )
% 3.92/1.83 => ( ( A != B )
% 3.92/1.83 => ( member1285940496od_b_b @ A @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_121_insertE,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,B: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ A @ ( insert2037698781od_b_b @ B @ A2 ) )
% 3.92/1.83 => ( ( A != B )
% 3.92/1.83 => ( member1516365892od_b_b @ A @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_122_some__eq__imp,axiom,
% 3.92/1.83 ! [P: b > $o,A: b,B: b] :
% 3.92/1.83 ( ( ( hilbert_Eps_b @ P )
% 3.92/1.83 = A )
% 3.92/1.83 => ( ( P @ B )
% 3.92/1.83 => ( P @ A ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_123_tfl__some,axiom,
% 3.92/1.83 ! [P2: b > $o,X5: b] :
% 3.92/1.83 ( ( P2 @ X5 )
% 3.92/1.83 => ( P2 @ ( hilbert_Eps_b @ P2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_124_mem__Collect__eq,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,P: product_prod_b_b > $o] :
% 3.92/1.83 ( ( member1285940496od_b_b @ A @ ( collec1481886546od_b_b @ P ) )
% 3.92/1.83 = ( P @ A ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_125_mem__Collect__eq,axiom,
% 3.92/1.83 ! [A: produc1213276845od_b_b,P: produc1213276845od_b_b > $o] :
% 3.92/1.83 ( ( member1516365892od_b_b @ A @ ( collec279084418od_b_b @ P ) )
% 3.92/1.83 = ( P @ A ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_126_mem__Collect__eq,axiom,
% 3.92/1.83 ! [A: b,P: b > $o] :
% 3.92/1.83 ( ( member_b @ A @ ( collect_b @ P ) )
% 3.92/1.83 = ( P @ A ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_127_Collect__mem__eq,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( collec1481886546od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ A2 ) )
% 3.92/1.83 = A2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_128_Collect__mem__eq,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( collec279084418od_b_b
% 3.92/1.83 @ ^ [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ A2 ) )
% 3.92/1.83 = A2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_129_Collect__mem__eq,axiom,
% 3.92/1.83 ! [A2: set_b] :
% 3.92/1.83 ( ( collect_b
% 3.92/1.83 @ ^ [X2: b] : ( member_b @ X2 @ A2 ) )
% 3.92/1.83 = A2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_130_Collect__cong,axiom,
% 3.92/1.83 ! [P: b > $o,Q: b > $o] :
% 3.92/1.83 ( ! [X4: b] :
% 3.92/1.83 ( ( P @ X4 )
% 3.92/1.83 = ( Q @ X4 ) )
% 3.92/1.83 => ( ( collect_b @ P )
% 3.92/1.83 = ( collect_b @ Q ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_131_Eps__cong,axiom,
% 3.92/1.83 ! [P: b > $o,Q: b > $o] :
% 3.92/1.83 ( ! [X4: b] :
% 3.92/1.83 ( ( P @ X4 )
% 3.92/1.83 = ( Q @ X4 ) )
% 3.92/1.83 => ( ( hilbert_Eps_b @ P )
% 3.92/1.83 = ( hilbert_Eps_b @ Q ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_132_someI,axiom,
% 3.92/1.83 ! [P: b > $o,X: b] :
% 3.92/1.83 ( ( P @ X )
% 3.92/1.83 => ( P @ ( hilbert_Eps_b @ P ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_133_empty__def,axiom,
% 3.92/1.83 ( bot_bot_set_b
% 3.92/1.83 = ( collect_b
% 3.92/1.83 @ ^ [X2: b] : $false ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_134_empty__def,axiom,
% 3.92/1.83 ( bot_bo1343651123od_b_b
% 3.92/1.83 = ( collec1481886546od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] : $false ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_135_insert__Collect,axiom,
% 3.92/1.83 ! [A: b,P: b > $o] :
% 3.92/1.83 ( ( insert_b @ A @ ( collect_b @ P ) )
% 3.92/1.83 = ( collect_b
% 3.92/1.83 @ ^ [U: b] :
% 3.92/1.83 ( ( U != A )
% 3.92/1.83 => ( P @ U ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_136_insert__compr,axiom,
% 3.92/1.83 ( insert1952693431od_b_b
% 3.92/1.83 = ( ^ [A3: product_prod_b_b,B5: set_Product_prod_b_b] :
% 3.92/1.83 ( collec1481886546od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] :
% 3.92/1.83 ( ( X2 = A3 )
% 3.92/1.83 | ( member1285940496od_b_b @ X2 @ B5 ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_137_insert__compr,axiom,
% 3.92/1.83 ( insert2037698781od_b_b
% 3.92/1.83 = ( ^ [A3: produc1213276845od_b_b,B5: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( collec279084418od_b_b
% 3.92/1.83 @ ^ [X2: produc1213276845od_b_b] :
% 3.92/1.83 ( ( X2 = A3 )
% 3.92/1.83 | ( member1516365892od_b_b @ X2 @ B5 ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_138_insert__compr,axiom,
% 3.92/1.83 ( insert_b
% 3.92/1.83 = ( ^ [A3: b,B5: set_b] :
% 3.92/1.83 ( collect_b
% 3.92/1.83 @ ^ [X2: b] :
% 3.92/1.83 ( ( X2 = A3 )
% 3.92/1.83 | ( member_b @ X2 @ B5 ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_139_some1__equality,axiom,
% 3.92/1.83 ! [P: b > $o,A: b] :
% 3.92/1.83 ( ? [X5: b] :
% 3.92/1.83 ( ( P @ X5 )
% 3.92/1.83 & ! [Y3: b] :
% 3.92/1.83 ( ( P @ Y3 )
% 3.92/1.83 => ( Y3 = X5 ) ) )
% 3.92/1.83 => ( ( P @ A )
% 3.92/1.83 => ( ( hilbert_Eps_b @ P )
% 3.92/1.83 = A ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_140_some__eq__ex,axiom,
% 3.92/1.83 ! [P: b > $o] :
% 3.92/1.83 ( ( P @ ( hilbert_Eps_b @ P ) )
% 3.92/1.83 = ( ? [X6: b] : ( P @ X6 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_141_someI2__bex,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
% 3.92/1.83 ( ? [X5: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ X5 @ A2 )
% 3.92/1.83 & ( P @ X5 ) )
% 3.92/1.83 => ( ! [X4: product_prod_b_b] :
% 3.92/1.83 ( ( ( member1285940496od_b_b @ X4 @ A2 )
% 3.92/1.83 & ( P @ X4 ) )
% 3.92/1.83 => ( Q @ X4 ) )
% 3.92/1.83 => ( Q
% 3.92/1.83 @ ( hilber620807523od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ X2 @ A2 )
% 3.92/1.83 & ( P @ X2 ) ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_142_someI2__bex,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,P: produc1213276845od_b_b > $o,Q: produc1213276845od_b_b > $o] :
% 3.92/1.83 ( ? [X5: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ X5 @ A2 )
% 3.92/1.83 & ( P @ X5 ) )
% 3.92/1.83 => ( ! [X4: produc1213276845od_b_b] :
% 3.92/1.83 ( ( ( member1516365892od_b_b @ X4 @ A2 )
% 3.92/1.83 & ( P @ X4 ) )
% 3.92/1.83 => ( Q @ X4 ) )
% 3.92/1.83 => ( Q
% 3.92/1.83 @ ( hilber46179633od_b_b
% 3.92/1.83 @ ^ [X2: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ X2 @ A2 )
% 3.92/1.83 & ( P @ X2 ) ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_143_someI2__bex,axiom,
% 3.92/1.83 ! [A2: set_b,P: b > $o,Q: b > $o] :
% 3.92/1.83 ( ? [X5: b] :
% 3.92/1.83 ( ( member_b @ X5 @ A2 )
% 3.92/1.83 & ( P @ X5 ) )
% 3.92/1.83 => ( ! [X4: b] :
% 3.92/1.83 ( ( ( member_b @ X4 @ A2 )
% 3.92/1.83 & ( P @ X4 ) )
% 3.92/1.83 => ( Q @ X4 ) )
% 3.92/1.83 => ( Q
% 3.92/1.83 @ ( hilbert_Eps_b
% 3.92/1.83 @ ^ [X2: b] :
% 3.92/1.83 ( ( member_b @ X2 @ A2 )
% 3.92/1.83 & ( P @ X2 ) ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_144_someI2__ex,axiom,
% 3.92/1.83 ! [P: b > $o,Q: b > $o] :
% 3.92/1.83 ( ? [X_1: b] : ( P @ X_1 )
% 3.92/1.83 => ( ! [X4: b] :
% 3.92/1.83 ( ( P @ X4 )
% 3.92/1.83 => ( Q @ X4 ) )
% 3.92/1.83 => ( Q @ ( hilbert_Eps_b @ P ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_145_someI__ex,axiom,
% 3.92/1.83 ! [P: b > $o] :
% 3.92/1.83 ( ? [X_1: b] : ( P @ X_1 )
% 3.92/1.83 => ( P @ ( hilbert_Eps_b @ P ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_146_someI2,axiom,
% 3.92/1.83 ! [P: b > $o,A: b,Q: b > $o] :
% 3.92/1.83 ( ( P @ A )
% 3.92/1.83 => ( ! [X4: b] :
% 3.92/1.83 ( ( P @ X4 )
% 3.92/1.83 => ( Q @ X4 ) )
% 3.92/1.83 => ( Q @ ( hilbert_Eps_b @ P ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_147_singleton__inject,axiom,
% 3.92/1.83 ! [A: b,B: b] :
% 3.92/1.83 ( ( ( insert_b @ A @ bot_bot_set_b )
% 3.92/1.83 = ( insert_b @ B @ bot_bot_set_b ) )
% 3.92/1.83 => ( A = B ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_148_singleton__inject,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: product_prod_b_b] :
% 3.92/1.83 ( ( ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b )
% 3.92/1.83 = ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 => ( A = B ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_149_insert__not__empty,axiom,
% 3.92/1.83 ! [A: b,A2: set_b] :
% 3.92/1.83 ( ( insert_b @ A @ A2 )
% 3.92/1.83 != bot_bot_set_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_150_insert__not__empty,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( insert1952693431od_b_b @ A @ A2 )
% 3.92/1.83 != bot_bo1343651123od_b_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_151_doubleton__eq__iff,axiom,
% 3.92/1.83 ! [A: b,B: b,C: b,D: b] :
% 3.92/1.83 ( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
% 3.92/1.83 = ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
% 3.92/1.83 = ( ( ( A = C )
% 3.92/1.83 & ( B = D ) )
% 3.92/1.83 | ( ( A = D )
% 3.92/1.83 & ( B = C ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_152_doubleton__eq__iff,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: product_prod_b_b,C: product_prod_b_b,D: product_prod_b_b] :
% 3.92/1.83 ( ( ( insert1952693431od_b_b @ A @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = ( insert1952693431od_b_b @ C @ ( insert1952693431od_b_b @ D @ bot_bo1343651123od_b_b ) ) )
% 3.92/1.83 = ( ( ( A = C )
% 3.92/1.83 & ( B = D ) )
% 3.92/1.83 | ( ( A = D )
% 3.92/1.83 & ( B = C ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_153_singleton__iff,axiom,
% 3.92/1.83 ! [B: produc1213276845od_b_b,A: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ B @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) )
% 3.92/1.83 = ( B = A ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_154_singleton__iff,axiom,
% 3.92/1.83 ! [B: b,A: b] :
% 3.92/1.83 ( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
% 3.92/1.83 = ( B = A ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_155_singleton__iff,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = ( B = A ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_156_singletonD,axiom,
% 3.92/1.83 ! [B: produc1213276845od_b_b,A: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ B @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) )
% 3.92/1.83 => ( B = A ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_157_singletonD,axiom,
% 3.92/1.83 ! [B: b,A: b] :
% 3.92/1.83 ( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
% 3.92/1.83 => ( B = A ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_158_singletonD,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 => ( B = A ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_159_Collect__conv__if2,axiom,
% 3.92/1.83 ! [P: b > $o,A: b] :
% 3.92/1.83 ( ( ( P @ A )
% 3.92/1.83 => ( ( collect_b
% 3.92/1.83 @ ^ [X2: b] :
% 3.92/1.83 ( ( A = X2 )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 = ( insert_b @ A @ bot_bot_set_b ) ) )
% 3.92/1.83 & ( ~ ( P @ A )
% 3.92/1.83 => ( ( collect_b
% 3.92/1.83 @ ^ [X2: b] :
% 3.92/1.83 ( ( A = X2 )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 = bot_bot_set_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_160_Collect__conv__if2,axiom,
% 3.92/1.83 ! [P: product_prod_b_b > $o,A: product_prod_b_b] :
% 3.92/1.83 ( ( ( P @ A )
% 3.92/1.83 => ( ( collec1481886546od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] :
% 3.92/1.83 ( ( A = X2 )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 = ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 3.92/1.83 & ( ~ ( P @ A )
% 3.92/1.83 => ( ( collec1481886546od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] :
% 3.92/1.83 ( ( A = X2 )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 = bot_bo1343651123od_b_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_161_Collect__conv__if,axiom,
% 3.92/1.83 ! [P: b > $o,A: b] :
% 3.92/1.83 ( ( ( P @ A )
% 3.92/1.83 => ( ( collect_b
% 3.92/1.83 @ ^ [X2: b] :
% 3.92/1.83 ( ( X2 = A )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 = ( insert_b @ A @ bot_bot_set_b ) ) )
% 3.92/1.83 & ( ~ ( P @ A )
% 3.92/1.83 => ( ( collect_b
% 3.92/1.83 @ ^ [X2: b] :
% 3.92/1.83 ( ( X2 = A )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 = bot_bot_set_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_162_Collect__conv__if,axiom,
% 3.92/1.83 ! [P: product_prod_b_b > $o,A: product_prod_b_b] :
% 3.92/1.83 ( ( ( P @ A )
% 3.92/1.83 => ( ( collec1481886546od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] :
% 3.92/1.83 ( ( X2 = A )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 = ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) )
% 3.92/1.83 & ( ~ ( P @ A )
% 3.92/1.83 => ( ( collec1481886546od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] :
% 3.92/1.83 ( ( X2 = A )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 = bot_bo1343651123od_b_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_163_some__in__eq,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b
% 3.92/1.83 @ ( hilber46179633od_b_b
% 3.92/1.83 @ ^ [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ A2 ) )
% 3.92/1.83 @ A2 )
% 3.92/1.83 = ( A2 != bot_bo1664927607od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_164_some__in__eq,axiom,
% 3.92/1.83 ! [A2: set_b] :
% 3.92/1.83 ( ( member_b
% 3.92/1.83 @ ( hilbert_Eps_b
% 3.92/1.83 @ ^ [X2: b] : ( member_b @ X2 @ A2 ) )
% 3.92/1.83 @ A2 )
% 3.92/1.83 = ( A2 != bot_bot_set_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_165_some__in__eq,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b
% 3.92/1.83 @ ( hilber620807523od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ A2 ) )
% 3.92/1.83 @ A2 )
% 3.92/1.83 = ( A2 != bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_166_congr_I1_J,axiom,
% 3.92/1.83 ! [X: b,Y4: b,L: standard_Constant_a] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ ( getRel904497637nt_a_b @ L @ g ) )
% 3.92/1.83 => ( equiv_1821124534_set_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g )
% 3.92/1.83 @ ^ [V: b] : ( image_b_b @ ( getRel904497637nt_a_b @ L @ g ) @ ( insert_b @ V @ bot_bot_set_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_167_vert__P,axiom,
% 3.92/1.83 ! [X: b] :
% 3.92/1.83 ( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
% 3.92/1.83 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ ( hilbert_Eps_b @ ( p @ X ) ) ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_168__092_060open_062_IS__Idt_M_Ax_M_Ay_J_A_092_060in_062_Aedges_AG_092_060close_062,axiom,
% 3.92/1.83 member1516365892od_b_b @ ( produc1432590431od_b_b @ standard_S_Idt_a @ ( product_Pair_b_b @ x @ y ) ) @ ( labele1741081071nt_a_b @ g ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_169_restrict__idemp,axiom,
% 3.92/1.83 ! [X: labele1159362096nt_a_b] :
% 3.92/1.83 ( ( restri446606278nt_a_b @ ( restri446606278nt_a_b @ X ) )
% 3.92/1.83 = ( restri446606278nt_a_b @ X ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_170_r2,axiom,
% 3.92/1.83 ! [X: b] :
% 3.92/1.83 ( ( ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ X @ bot_bot_set_b ) )
% 3.92/1.83 = bot_bot_set_b )
% 3.92/1.83 = ( ~ ( member_b @ X @ ( labele1424214014nt_a_b @ g ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_171_r1,axiom,
% 3.92/1.83 ! [X: b] :
% 3.92/1.83 ( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
% 3.92/1.83 = ( p @ X @ X ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_172_congr_I2_J,axiom,
% 3.92/1.83 ! [X: b,Y4: b,L: standard_Constant_a] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ ( getRel904497637nt_a_b @ L @ g ) )
% 3.92/1.83 => ( equiv_1821124534_set_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g )
% 3.92/1.83 @ ^ [V: b] : ( image_b_b @ ( converse_b_b @ ( getRel904497637nt_a_b @ L @ g ) ) @ ( insert_b @ V @ bot_bot_set_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_173_the__elem__eq,axiom,
% 3.92/1.83 ! [X: b] :
% 3.92/1.83 ( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
% 3.92/1.83 = X ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_174_the__elem__eq,axiom,
% 3.92/1.83 ! [X: product_prod_b_b] :
% 3.92/1.83 ( ( the_el329196508od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = X ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_175_prod_Oinject,axiom,
% 3.92/1.83 ! [X1: b,X22: b,Y1: b,Y22: b] :
% 3.92/1.83 ( ( ( product_Pair_b_b @ X1 @ X22 )
% 3.92/1.83 = ( product_Pair_b_b @ Y1 @ Y22 ) )
% 3.92/1.83 = ( ( X1 = Y1 )
% 3.92/1.83 & ( X22 = Y22 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_176_prod_Oinject,axiom,
% 3.92/1.83 ! [X1: standard_Constant_a,X22: product_prod_b_b,Y1: standard_Constant_a,Y22: product_prod_b_b] :
% 3.92/1.83 ( ( ( produc1432590431od_b_b @ X1 @ X22 )
% 3.92/1.83 = ( produc1432590431od_b_b @ Y1 @ Y22 ) )
% 3.92/1.83 = ( ( X1 = Y1 )
% 3.92/1.83 & ( X22 = Y22 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_177_old_Oprod_Oinject,axiom,
% 3.92/1.83 ! [A: b,B: b,A4: b,B6: b] :
% 3.92/1.83 ( ( ( product_Pair_b_b @ A @ B )
% 3.92/1.83 = ( product_Pair_b_b @ A4 @ B6 ) )
% 3.92/1.83 = ( ( A = A4 )
% 3.92/1.83 & ( B = B6 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_178_old_Oprod_Oinject,axiom,
% 3.92/1.83 ! [A: standard_Constant_a,B: product_prod_b_b,A4: standard_Constant_a,B6: product_prod_b_b] :
% 3.92/1.83 ( ( ( produc1432590431od_b_b @ A @ B )
% 3.92/1.83 = ( produc1432590431od_b_b @ A4 @ B6 ) )
% 3.92/1.83 = ( ( A = A4 )
% 3.92/1.83 & ( B = B6 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_179_converse__inject,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ( converse_b_b @ R2 )
% 3.92/1.83 = ( converse_b_b @ S2 ) )
% 3.92/1.83 = ( R2 = S2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_180_converse__converse,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( converse_b_b @ ( converse_b_b @ R2 ) )
% 3.92/1.83 = R2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_181_converse__iff,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A @ B ) @ ( conver1398856763od_b_b @ R2 ) )
% 3.92/1.83 = ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B @ A ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_182_converse__iff,axiom,
% 3.92/1.83 ! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr306736955tant_a] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ ( conver1943590235tant_a @ R2 ) )
% 3.92/1.83 = ( member1670214300tant_a @ ( produc1977323903tant_a @ B @ A ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_183_converse__iff,axiom,
% 3.92/1.83 ! [A: b,B: b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( converse_b_b @ R2 ) )
% 3.92/1.83 = ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_184_vertices__restrict,axiom,
% 3.92/1.83 ! [G: labele1159362096nt_a_b] :
% 3.92/1.83 ( ( labele1424214014nt_a_b @ ( restri446606278nt_a_b @ G ) )
% 3.92/1.83 = ( labele1424214014nt_a_b @ G ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_185_converse__empty,axiom,
% 3.92/1.83 ( ( converse_b_b @ bot_bo1343651123od_b_b )
% 3.92/1.83 = bot_bo1343651123od_b_b ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_186__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Avertices_AG_A_092_060Longrightarrow_062_A_Ix_M_Ax_J_A_092_060in_062_AgetRel_AS__Idt_AG_092_060close_062,axiom,
% 3.92/1.83 ! [X: b] :
% 3.92/1.83 ( ( member_b @ X @ ( labele1424214014nt_a_b @ g ) )
% 3.92/1.83 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ X ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_187_labeled__graph_Oexpand,axiom,
% 3.92/1.83 ! [Labeled_graph: labele1159362096nt_a_b,Labeled_graph2: labele1159362096nt_a_b] :
% 3.92/1.83 ( ( ( ( labele1741081071nt_a_b @ Labeled_graph )
% 3.92/1.83 = ( labele1741081071nt_a_b @ Labeled_graph2 ) )
% 3.92/1.83 & ( ( labele1424214014nt_a_b @ Labeled_graph )
% 3.92/1.83 = ( labele1424214014nt_a_b @ Labeled_graph2 ) ) )
% 3.92/1.83 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_188_converse_Oinducts,axiom,
% 3.92/1.83 ! [X1: product_prod_b_b,X22: standard_Constant_a,R2: set_Pr1324126435od_b_b,P: product_prod_b_b > standard_Constant_a > $o] :
% 3.92/1.83 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ X1 @ X22 ) @ ( conver1398856763od_b_b @ R2 ) )
% 3.92/1.83 => ( ! [A5: standard_Constant_a,B7: product_prod_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A5 @ B7 ) @ R2 )
% 3.92/1.83 => ( P @ B7 @ A5 ) )
% 3.92/1.83 => ( P @ X1 @ X22 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_189_converse_Oinducts,axiom,
% 3.92/1.83 ! [X1: standard_Constant_a,X22: product_prod_b_b,R2: set_Pr306736955tant_a,P: standard_Constant_a > product_prod_b_b > $o] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X1 @ X22 ) @ ( conver1943590235tant_a @ R2 ) )
% 3.92/1.83 => ( ! [A5: product_prod_b_b,B7: standard_Constant_a] :
% 3.92/1.83 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A5 @ B7 ) @ R2 )
% 3.92/1.83 => ( P @ B7 @ A5 ) )
% 3.92/1.83 => ( P @ X1 @ X22 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_190_converse_Oinducts,axiom,
% 3.92/1.83 ! [X1: b,X22: b,R2: set_Product_prod_b_b,P: b > b > $o] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X1 @ X22 ) @ ( converse_b_b @ R2 ) )
% 3.92/1.83 => ( ! [A5: b,B7: b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A5 @ B7 ) @ R2 )
% 3.92/1.83 => ( P @ B7 @ A5 ) )
% 3.92/1.83 => ( P @ X1 @ X22 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_191_converse_Ointros,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: standard_Constant_a,R2: set_Pr306736955tant_a] :
% 3.92/1.83 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B @ A ) @ ( conver1943590235tant_a @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_192_converse_Ointros,axiom,
% 3.92/1.83 ! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1670214300tant_a @ ( produc1977323903tant_a @ B @ A ) @ ( conver1398856763od_b_b @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_193_converse_Ointros,axiom,
% 3.92/1.83 ! [A: b,B: b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ ( converse_b_b @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_194_converse_Osimps,axiom,
% 3.92/1.83 ! [A1: product_prod_b_b,A22: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A1 @ A22 ) @ ( conver1398856763od_b_b @ R2 ) )
% 3.92/1.83 = ( ? [A3: standard_Constant_a,B3: product_prod_b_b] :
% 3.92/1.83 ( ( A1 = B3 )
% 3.92/1.83 & ( A22 = A3 )
% 3.92/1.83 & ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A3 @ B3 ) @ R2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_195_converse_Osimps,axiom,
% 3.92/1.83 ! [A1: standard_Constant_a,A22: product_prod_b_b,R2: set_Pr306736955tant_a] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( conver1943590235tant_a @ R2 ) )
% 3.92/1.83 = ( ? [A3: product_prod_b_b,B3: standard_Constant_a] :
% 3.92/1.83 ( ( A1 = B3 )
% 3.92/1.83 & ( A22 = A3 )
% 3.92/1.83 & ( member1670214300tant_a @ ( produc1977323903tant_a @ A3 @ B3 ) @ R2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_196_converse_Osimps,axiom,
% 3.92/1.83 ! [A1: b,A22: b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A1 @ A22 ) @ ( converse_b_b @ R2 ) )
% 3.92/1.83 = ( ? [A3: b,B3: b] :
% 3.92/1.83 ( ( A1 = B3 )
% 3.92/1.83 & ( A22 = A3 )
% 3.92/1.83 & ( member1285940496od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ R2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_197_converse_Ocases,axiom,
% 3.92/1.83 ! [A1: product_prod_b_b,A22: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A1 @ A22 ) @ ( conver1398856763od_b_b @ R2 ) )
% 3.92/1.83 => ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A22 @ A1 ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_198_converse_Ocases,axiom,
% 3.92/1.83 ! [A1: standard_Constant_a,A22: product_prod_b_b,R2: set_Pr306736955tant_a] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A1 @ A22 ) @ ( conver1943590235tant_a @ R2 ) )
% 3.92/1.83 => ( member1670214300tant_a @ ( produc1977323903tant_a @ A22 @ A1 ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_199_converse_Ocases,axiom,
% 3.92/1.83 ! [A1: b,A22: b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A1 @ A22 ) @ ( converse_b_b @ R2 ) )
% 3.92/1.83 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A22 @ A1 ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_200_converseE,axiom,
% 3.92/1.83 ! [Yx: produc1367125253tant_a,R2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1670214300tant_a @ Yx @ ( conver1398856763od_b_b @ R2 ) )
% 3.92/1.83 => ~ ! [X4: standard_Constant_a,Y3: product_prod_b_b] :
% 3.92/1.83 ( ( Yx
% 3.92/1.83 = ( produc1977323903tant_a @ Y3 @ X4 ) )
% 3.92/1.83 => ~ ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_201_converseE,axiom,
% 3.92/1.83 ! [Yx: produc1213276845od_b_b,R2: set_Pr306736955tant_a] :
% 3.92/1.83 ( ( member1516365892od_b_b @ Yx @ ( conver1943590235tant_a @ R2 ) )
% 3.92/1.83 => ~ ! [X4: product_prod_b_b,Y3: standard_Constant_a] :
% 3.92/1.83 ( ( Yx
% 3.92/1.83 = ( produc1432590431od_b_b @ Y3 @ X4 ) )
% 3.92/1.83 => ~ ( member1670214300tant_a @ ( produc1977323903tant_a @ X4 @ Y3 ) @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_202_converseE,axiom,
% 3.92/1.83 ! [Yx: product_prod_b_b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ Yx @ ( converse_b_b @ R2 ) )
% 3.92/1.83 => ~ ! [X4: b,Y3: b] :
% 3.92/1.83 ( ( Yx
% 3.92/1.83 = ( product_Pair_b_b @ Y3 @ X4 ) )
% 3.92/1.83 => ~ ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_203_converseD,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,B: standard_Constant_a,R2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( member1670214300tant_a @ ( produc1977323903tant_a @ A @ B ) @ ( conver1398856763od_b_b @ R2 ) )
% 3.92/1.83 => ( member1516365892od_b_b @ ( produc1432590431od_b_b @ B @ A ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_204_converseD,axiom,
% 3.92/1.83 ! [A: standard_Constant_a,B: product_prod_b_b,R2: set_Pr306736955tant_a] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ A @ B ) @ ( conver1943590235tant_a @ R2 ) )
% 3.92/1.83 => ( member1670214300tant_a @ ( produc1977323903tant_a @ B @ A ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_205_converseD,axiom,
% 3.92/1.83 ! [A: b,B: b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( converse_b_b @ R2 ) )
% 3.92/1.83 => ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_206_bot__set__def,axiom,
% 3.92/1.83 ( bot_bot_set_b
% 3.92/1.83 = ( collect_b @ bot_bot_b_o ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_207_bot__set__def,axiom,
% 3.92/1.83 ( bot_bo1343651123od_b_b
% 3.92/1.83 = ( collec1481886546od_b_b @ bot_bo1302443370_b_b_o ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_208_bot__empty__eq,axiom,
% 3.92/1.83 ( bot_bo1167600078_b_b_o
% 3.92/1.83 = ( ^ [X2: produc1213276845od_b_b] : ( member1516365892od_b_b @ X2 @ bot_bo1664927607od_b_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_209_bot__empty__eq,axiom,
% 3.92/1.83 ( bot_bot_b_o
% 3.92/1.83 = ( ^ [X2: b] : ( member_b @ X2 @ bot_bot_set_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_210_bot__empty__eq,axiom,
% 3.92/1.83 ( bot_bo1302443370_b_b_o
% 3.92/1.83 = ( ^ [X2: product_prod_b_b] : ( member1285940496od_b_b @ X2 @ bot_bo1343651123od_b_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_211_bot__empty__eq2,axiom,
% 3.92/1.83 ( bot_bo1734638456_b_b_o
% 3.92/1.83 = ( ^ [X2: standard_Constant_a,Y: product_prod_b_b] : ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X2 @ Y ) @ bot_bo1664927607od_b_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_212_bot__empty__eq2,axiom,
% 3.92/1.83 ( bot_bot_b_b_o
% 3.92/1.83 = ( ^ [X2: b,Y: b] : ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ bot_bo1343651123od_b_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_213_old_Oprod_Oinducts,axiom,
% 3.92/1.83 ! [P: product_prod_b_b > $o,Prod: product_prod_b_b] :
% 3.92/1.83 ( ! [A5: b,B7: b] : ( P @ ( product_Pair_b_b @ A5 @ B7 ) )
% 3.92/1.83 => ( P @ Prod ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_214_old_Oprod_Oinducts,axiom,
% 3.92/1.83 ! [P: produc1213276845od_b_b > $o,Prod: produc1213276845od_b_b] :
% 3.92/1.83 ( ! [A5: standard_Constant_a,B7: product_prod_b_b] : ( P @ ( produc1432590431od_b_b @ A5 @ B7 ) )
% 3.92/1.83 => ( P @ Prod ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_215_old_Oprod_Oexhaust,axiom,
% 3.92/1.83 ! [Y4: product_prod_b_b] :
% 3.92/1.83 ~ ! [A5: b,B7: b] :
% 3.92/1.83 ( Y4
% 3.92/1.83 != ( product_Pair_b_b @ A5 @ B7 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_216_old_Oprod_Oexhaust,axiom,
% 3.92/1.83 ! [Y4: produc1213276845od_b_b] :
% 3.92/1.83 ~ ! [A5: standard_Constant_a,B7: product_prod_b_b] :
% 3.92/1.83 ( Y4
% 3.92/1.83 != ( produc1432590431od_b_b @ A5 @ B7 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_217_prod__induct3,axiom,
% 3.92/1.83 ! [P: produc1213276845od_b_b > $o,X: produc1213276845od_b_b] :
% 3.92/1.83 ( ! [A5: standard_Constant_a,B7: b,C3: b] : ( P @ ( produc1432590431od_b_b @ A5 @ ( product_Pair_b_b @ B7 @ C3 ) ) )
% 3.92/1.83 => ( P @ X ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_218_prod__cases3,axiom,
% 3.92/1.83 ! [Y4: produc1213276845od_b_b] :
% 3.92/1.83 ~ ! [A5: standard_Constant_a,B7: b,C3: b] :
% 3.92/1.83 ( Y4
% 3.92/1.83 != ( produc1432590431od_b_b @ A5 @ ( product_Pair_b_b @ B7 @ C3 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_219_Pair__inject,axiom,
% 3.92/1.83 ! [A: b,B: b,A4: b,B6: b] :
% 3.92/1.83 ( ( ( product_Pair_b_b @ A @ B )
% 3.92/1.83 = ( product_Pair_b_b @ A4 @ B6 ) )
% 3.92/1.83 => ~ ( ( A = A4 )
% 3.92/1.83 => ( B != B6 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_220_Pair__inject,axiom,
% 3.92/1.83 ! [A: standard_Constant_a,B: product_prod_b_b,A4: standard_Constant_a,B6: product_prod_b_b] :
% 3.92/1.83 ( ( ( produc1432590431od_b_b @ A @ B )
% 3.92/1.83 = ( produc1432590431od_b_b @ A4 @ B6 ) )
% 3.92/1.83 => ~ ( ( A = A4 )
% 3.92/1.83 => ( B != B6 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_221_prod__cases,axiom,
% 3.92/1.83 ! [P: product_prod_b_b > $o,P3: product_prod_b_b] :
% 3.92/1.83 ( ! [A5: b,B7: b] : ( P @ ( product_Pair_b_b @ A5 @ B7 ) )
% 3.92/1.83 => ( P @ P3 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_222_prod__cases,axiom,
% 3.92/1.83 ! [P: produc1213276845od_b_b > $o,P3: produc1213276845od_b_b] :
% 3.92/1.83 ( ! [A5: standard_Constant_a,B7: product_prod_b_b] : ( P @ ( produc1432590431od_b_b @ A5 @ B7 ) )
% 3.92/1.83 => ( P @ P3 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_223_surj__pair,axiom,
% 3.92/1.83 ! [P3: product_prod_b_b] :
% 3.92/1.83 ? [X4: b,Y3: b] :
% 3.92/1.83 ( P3
% 3.92/1.83 = ( product_Pair_b_b @ X4 @ Y3 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_224_surj__pair,axiom,
% 3.92/1.83 ! [P3: produc1213276845od_b_b] :
% 3.92/1.83 ? [X4: standard_Constant_a,Y3: product_prod_b_b] :
% 3.92/1.83 ( P3
% 3.92/1.83 = ( produc1432590431od_b_b @ X4 @ Y3 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_225_local_Orefl,axiom,
% 3.92/1.83 refl_on_b @ ( labele1424214014nt_a_b @ g ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_226_equiv,axiom,
% 3.92/1.83 equiv_equiv_b @ ( labele1424214014nt_a_b @ g ) @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_227_getRel__dom_I1_J,axiom,
% 3.92/1.83 ! [G: labele1159362096nt_a_b,A: b,B: b,L: standard_Constant_a] :
% 3.92/1.83 ( ( G
% 3.92/1.83 = ( restri446606278nt_a_b @ G ) )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel904497637nt_a_b @ L @ G ) )
% 3.92/1.83 => ( member_b @ A @ ( labele1424214014nt_a_b @ G ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_228_getRel__dom_I2_J,axiom,
% 3.92/1.83 ! [G: labele1159362096nt_a_b,A: b,B: b,L: standard_Constant_a] :
% 3.92/1.83 ( ( G
% 3.92/1.83 = ( restri446606278nt_a_b @ G ) )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel904497637nt_a_b @ L @ G ) )
% 3.92/1.83 => ( member_b @ B @ ( labele1424214014nt_a_b @ G ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_229_is__singleton__the__elem,axiom,
% 3.92/1.83 ( is_singleton_b
% 3.92/1.83 = ( ^ [A6: set_b] :
% 3.92/1.83 ( A6
% 3.92/1.83 = ( insert_b @ ( the_elem_b @ A6 ) @ bot_bot_set_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_230_is__singleton__the__elem,axiom,
% 3.92/1.83 ( is_sin1713057243od_b_b
% 3.92/1.83 = ( ^ [A6: set_Product_prod_b_b] :
% 3.92/1.83 ( A6
% 3.92/1.83 = ( insert1952693431od_b_b @ ( the_el329196508od_b_b @ A6 ) @ bot_bo1343651123od_b_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_231_refl__on__converse,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( refl_on_b @ A2 @ ( converse_b_b @ R2 ) )
% 3.92/1.83 = ( refl_on_b @ A2 @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_232_is__singletonI,axiom,
% 3.92/1.83 ! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_233_is__singletonI,axiom,
% 3.92/1.83 ! [X: product_prod_b_b] : ( is_sin1713057243od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_234_refl__onD,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( refl_o2134473190od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ A ) @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_235_refl__onD,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b] :
% 3.92/1.83 ( ( refl_o1921098222od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ A ) @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_236_refl__onD,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b,A: b] :
% 3.92/1.83 ( ( refl_on_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ A ) @ R2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_237_refl__onD1,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
% 3.92/1.83 ( ( refl_o2134473190od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 )
% 3.92/1.83 => ( member1285940496od_b_b @ X @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_238_refl__onD1,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
% 3.92/1.83 ( ( refl_o1921098222od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 )
% 3.92/1.83 => ( member1516365892od_b_b @ X @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_239_refl__onD1,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
% 3.92/1.83 ( ( refl_on_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 )
% 3.92/1.83 => ( member_b @ X @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_240_refl__onD2,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
% 3.92/1.83 ( ( refl_o2134473190od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 )
% 3.92/1.83 => ( member1285940496od_b_b @ Y4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_241_refl__onD2,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
% 3.92/1.83 ( ( refl_o1921098222od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 )
% 3.92/1.83 => ( member1516365892od_b_b @ Y4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_242_refl__onD2,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
% 3.92/1.83 ( ( refl_on_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 )
% 3.92/1.83 => ( member_b @ Y4 @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_243_refl__on__empty,axiom,
% 3.92/1.83 refl_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% 3.92/1.83
% 3.92/1.83 thf(fact_244_refl__on__empty,axiom,
% 3.92/1.83 refl_o2134473190od_b_b @ bot_bo1343651123od_b_b @ bot_bo1174586163od_b_b ).
% 3.92/1.83
% 3.92/1.83 thf(fact_245_is__singletonI_H,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( A2 != bot_bo1664927607od_b_b )
% 3.92/1.83 => ( ! [X4: produc1213276845od_b_b,Y3: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ X4 @ A2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ Y3 @ A2 )
% 3.92/1.83 => ( X4 = Y3 ) ) )
% 3.92/1.83 => ( is_sin1465183929od_b_b @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_246_is__singletonI_H,axiom,
% 3.92/1.83 ! [A2: set_b] :
% 3.92/1.83 ( ( A2 != bot_bot_set_b )
% 3.92/1.83 => ( ! [X4: b,Y3: b] :
% 3.92/1.83 ( ( member_b @ X4 @ A2 )
% 3.92/1.83 => ( ( member_b @ Y3 @ A2 )
% 3.92/1.83 => ( X4 = Y3 ) ) )
% 3.92/1.83 => ( is_singleton_b @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_247_is__singletonI_H,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( A2 != bot_bo1343651123od_b_b )
% 3.92/1.83 => ( ! [X4: product_prod_b_b,Y3: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ X4 @ A2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ Y3 @ A2 )
% 3.92/1.83 => ( X4 = Y3 ) ) )
% 3.92/1.83 => ( is_sin1713057243od_b_b @ A2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_248_refl__on__singleton,axiom,
% 3.92/1.83 ! [X: b] : ( refl_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_249_refl__on__singleton,axiom,
% 3.92/1.83 ! [X: product_prod_b_b] : ( refl_o2134473190od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X @ X ) @ bot_bo1174586163od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_250_is__singleton__def,axiom,
% 3.92/1.83 ( is_singleton_b
% 3.92/1.83 = ( ^ [A6: set_b] :
% 3.92/1.83 ? [X2: b] :
% 3.92/1.83 ( A6
% 3.92/1.83 = ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_251_is__singleton__def,axiom,
% 3.92/1.83 ( is_sin1713057243od_b_b
% 3.92/1.83 = ( ^ [A6: set_Product_prod_b_b] :
% 3.92/1.83 ? [X2: product_prod_b_b] :
% 3.92/1.83 ( A6
% 3.92/1.83 = ( insert1952693431od_b_b @ X2 @ bot_bo1343651123od_b_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_252_is__singletonE,axiom,
% 3.92/1.83 ! [A2: set_b] :
% 3.92/1.83 ( ( is_singleton_b @ A2 )
% 3.92/1.83 => ~ ! [X4: b] :
% 3.92/1.83 ( A2
% 3.92/1.83 != ( insert_b @ X4 @ bot_bot_set_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_253_is__singletonE,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( is_sin1713057243od_b_b @ A2 )
% 3.92/1.83 => ~ ! [X4: product_prod_b_b] :
% 3.92/1.83 ( A2
% 3.92/1.83 != ( insert1952693431od_b_b @ X4 @ bot_bo1343651123od_b_b ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_254_equiv__class__eq__iff,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
% 3.92/1.83 ( ( equiv_2048442231od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 )
% 3.92/1.83 = ( ( ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ X @ bot_bo1664927607od_b_b ) )
% 3.92/1.83 = ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ Y4 @ bot_bo1664927607od_b_b ) ) )
% 3.92/1.83 & ( member1516365892od_b_b @ X @ A2 )
% 3.92/1.83 & ( member1516365892od_b_b @ Y4 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_255_equiv__class__eq__iff,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
% 3.92/1.83 ( ( equiv_equiv_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 )
% 3.92/1.83 = ( ( ( image_b_b @ R2 @ ( insert_b @ X @ bot_bot_set_b ) )
% 3.92/1.83 = ( image_b_b @ R2 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) )
% 3.92/1.83 & ( member_b @ X @ A2 )
% 3.92/1.83 & ( member_b @ Y4 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_256_equiv__class__eq__iff,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
% 3.92/1.83 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 )
% 3.92/1.83 = ( ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ Y4 @ bot_bo1343651123od_b_b ) ) )
% 3.92/1.83 & ( member1285940496od_b_b @ X @ A2 )
% 3.92/1.83 & ( member1285940496od_b_b @ Y4 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_257_eq__equiv__class__iff,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,X: produc1213276845od_b_b,Y4: produc1213276845od_b_b] :
% 3.92/1.83 ( ( equiv_2048442231od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ X @ A2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ Y4 @ A2 )
% 3.92/1.83 => ( ( ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ X @ bot_bo1664927607od_b_b ) )
% 3.92/1.83 = ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ Y4 @ bot_bo1664927607od_b_b ) ) )
% 3.92/1.83 = ( member1086024932od_b_b @ ( produc449289715od_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_258_eq__equiv__class__iff,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b,X: b,Y4: b] :
% 3.92/1.83 ( ( equiv_equiv_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member_b @ X @ A2 )
% 3.92/1.83 => ( ( member_b @ Y4 @ A2 )
% 3.92/1.83 => ( ( ( image_b_b @ R2 @ ( insert_b @ X @ bot_bot_set_b ) )
% 3.92/1.83 = ( image_b_b @ R2 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) )
% 3.92/1.83 = ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_259_eq__equiv__class__iff,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X: product_prod_b_b,Y4: product_prod_b_b] :
% 3.92/1.83 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ X @ A2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ Y4 @ A2 )
% 3.92/1.83 => ( ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ Y4 @ bot_bo1343651123od_b_b ) ) )
% 3.92/1.83 = ( member1652792080od_b_b @ ( produc546497367od_b_b @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_260_equiv__class__eq,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
% 3.92/1.83 ( ( equiv_equiv_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) )
% 3.92/1.83 = ( image_b_b @ R2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_261_equiv__class__eq,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
% 3.92/1.83 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_262_eq__equiv__class,axiom,
% 3.92/1.83 ! [R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b,B: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) )
% 3.92/1.83 = ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ B @ bot_bo1664927607od_b_b ) ) )
% 3.92/1.83 => ( ( equiv_2048442231od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ B @ A2 )
% 3.92/1.83 => ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_263_eq__equiv__class,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b,A: b,B: b,A2: set_b] :
% 3.92/1.83 ( ( ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) )
% 3.92/1.83 = ( image_b_b @ R2 @ ( insert_b @ B @ bot_bot_set_b ) ) )
% 3.92/1.83 => ( ( equiv_equiv_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member_b @ B @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_264_eq__equiv__class,axiom,
% 3.92/1.83 ! [R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) )
% 3.92/1.83 => ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ B @ A2 )
% 3.92/1.83 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_265_congruentI,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b,F: b > set_b] :
% 3.92/1.83 ( ! [Y3: b,Z3: b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ Z3 ) @ R2 )
% 3.92/1.83 => ( ( F @ Y3 )
% 3.92/1.83 = ( F @ Z3 ) ) )
% 3.92/1.83 => ( equiv_1821124534_set_b @ R2 @ F ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_266_congruentD,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b,F: b > set_b,Y4: b,Z2: b] :
% 3.92/1.83 ( ( equiv_1821124534_set_b @ R2 @ F )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z2 ) @ R2 )
% 3.92/1.83 => ( ( F @ Y4 )
% 3.92/1.83 = ( F @ Z2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_267_equiv__class__self,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b] :
% 3.92/1.83 ( ( equiv_2048442231od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( member1516365892od_b_b @ A @ ( image_763305007od_b_b @ R2 @ ( insert2037698781od_b_b @ A @ bot_bo1664927607od_b_b ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_268_equiv__class__self,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b,A: b] :
% 3.92/1.83 ( ( equiv_equiv_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( member_b @ A @ ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_269_equiv__class__self,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ A @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_270_Collect__empty__eq__bot,axiom,
% 3.92/1.83 ! [P: b > $o] :
% 3.92/1.83 ( ( ( collect_b @ P )
% 3.92/1.83 = bot_bot_set_b )
% 3.92/1.83 = ( P = bot_bot_b_o ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_271_Collect__empty__eq__bot,axiom,
% 3.92/1.83 ! [P: product_prod_b_b > $o] :
% 3.92/1.83 ( ( ( collec1481886546od_b_b @ P )
% 3.92/1.83 = bot_bo1343651123od_b_b )
% 3.92/1.83 = ( P = bot_bo1302443370_b_b_o ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_272_graph__single,axiom,
% 3.92/1.83 ! [A: standard_Constant_a,B: b,C: b] :
% 3.92/1.83 ( ( labele1230159100nt_a_b @ ( insert2037698781od_b_b @ ( produc1432590431od_b_b @ A @ ( product_Pair_b_b @ B @ C ) ) @ bot_bo1664927607od_b_b ) @ ( insert_b @ B @ ( insert_b @ C @ bot_bot_set_b ) ) )
% 3.92/1.83 = ( restri446606278nt_a_b @ ( labele1230159100nt_a_b @ ( insert2037698781od_b_b @ ( produc1432590431od_b_b @ A @ ( product_Pair_b_b @ B @ C ) ) @ bot_bo1664927607od_b_b ) @ ( insert_b @ B @ ( insert_b @ C @ bot_bot_set_b ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_273_refl__on__domain,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
% 3.92/1.83 ( ( refl_o2134473190od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 & ( member1285940496od_b_b @ B @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_274_refl__on__domain,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b,A: produc1213276845od_b_b,B: produc1213276845od_b_b] :
% 3.92/1.83 ( ( refl_o1921098222od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 & ( member1516365892od_b_b @ B @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_275_refl__on__domain,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
% 3.92/1.83 ( ( refl_on_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ( member_b @ A @ A2 )
% 3.92/1.83 & ( member_b @ B @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_276_total__on__singleton,axiom,
% 3.92/1.83 ! [X: b] : ( total_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_277_total__on__singleton,axiom,
% 3.92/1.83 ! [X: product_prod_b_b] : ( total_178717291od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X @ X ) @ bot_bo1174586163od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_278_total__on__converse,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( total_on_b @ A2 @ ( converse_b_b @ R2 ) )
% 3.92/1.83 = ( total_on_b @ A2 @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_279_labeled__graph_Ocollapse,axiom,
% 3.92/1.83 ! [Labeled_graph: labele1159362096nt_a_b] :
% 3.92/1.83 ( ( labele1230159100nt_a_b @ ( labele1741081071nt_a_b @ Labeled_graph ) @ ( labele1424214014nt_a_b @ Labeled_graph ) )
% 3.92/1.83 = Labeled_graph ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_280_graph__empty__e,axiom,
% 3.92/1.83 ! [V2: set_b] :
% 3.92/1.83 ( ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ V2 )
% 3.92/1.83 = ( restri446606278nt_a_b @ ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ V2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_281_total__on__def,axiom,
% 3.92/1.83 ( total_on_b
% 3.92/1.83 = ( ^ [A6: set_b,R3: set_Product_prod_b_b] :
% 3.92/1.83 ! [X2: b] :
% 3.92/1.83 ( ( member_b @ X2 @ A6 )
% 3.92/1.83 => ! [Y: b] :
% 3.92/1.83 ( ( member_b @ Y @ A6 )
% 3.92/1.83 => ( ( X2 != Y )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X2 @ Y ) @ R3 )
% 3.92/1.83 | ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ X2 ) @ R3 ) ) ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_282_total__onI,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b] :
% 3.92/1.83 ( ! [X4: product_prod_b_b,Y3: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ X4 @ A2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ Y3 @ A2 )
% 3.92/1.83 => ( ( X4 != Y3 )
% 3.92/1.83 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y3 ) @ R2 )
% 3.92/1.83 | ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 3.92/1.83 => ( total_178717291od_b_b @ A2 @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_283_total__onI,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,R2: set_Pr2109276827od_b_b] :
% 3.92/1.83 ( ! [X4: produc1213276845od_b_b,Y3: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ X4 @ A2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ Y3 @ A2 )
% 3.92/1.83 => ( ( X4 != Y3 )
% 3.92/1.83 => ( ( member1086024932od_b_b @ ( produc449289715od_b_b @ X4 @ Y3 ) @ R2 )
% 3.92/1.83 | ( member1086024932od_b_b @ ( produc449289715od_b_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 3.92/1.83 => ( total_771970601od_b_b @ A2 @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_284_total__onI,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ! [X4: b,Y3: b] :
% 3.92/1.83 ( ( member_b @ X4 @ A2 )
% 3.92/1.83 => ( ( member_b @ Y3 @ A2 )
% 3.92/1.83 => ( ( X4 != Y3 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R2 )
% 3.92/1.83 | ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ X4 ) @ R2 ) ) ) ) )
% 3.92/1.83 => ( total_on_b @ A2 @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_285_total__on__empty,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b] : ( total_on_b @ bot_bot_set_b @ R2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_286_total__on__empty,axiom,
% 3.92/1.83 ! [R2: set_Pr2007082183od_b_b] : ( total_178717291od_b_b @ bot_bo1343651123od_b_b @ R2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_287_labeled__graph_Osel_I2_J,axiom,
% 3.92/1.83 ! [X1: set_Pr1324126435od_b_b,X22: set_b] :
% 3.92/1.83 ( ( labele1424214014nt_a_b @ ( labele1230159100nt_a_b @ X1 @ X22 ) )
% 3.92/1.83 = X22 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_288_labeled__graph_Osel_I1_J,axiom,
% 3.92/1.83 ! [X1: set_Pr1324126435od_b_b,X22: set_b] :
% 3.92/1.83 ( ( labele1741081071nt_a_b @ ( labele1230159100nt_a_b @ X1 @ X22 ) )
% 3.92/1.83 = X1 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_289_labeled__graph_Oexhaust__sel,axiom,
% 3.92/1.83 ! [Labeled_graph: labele1159362096nt_a_b] :
% 3.92/1.83 ( Labeled_graph
% 3.92/1.83 = ( labele1230159100nt_a_b @ ( labele1741081071nt_a_b @ Labeled_graph ) @ ( labele1424214014nt_a_b @ Labeled_graph ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_290_linear__order__on__singleton,axiom,
% 3.92/1.83 ! [X: b] : ( order_653178932r_on_b @ ( insert_b @ X @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X @ X ) @ bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_291_linear__order__on__singleton,axiom,
% 3.92/1.83 ! [X: product_prod_b_b] : ( order_609163036od_b_b @ ( insert1952693431od_b_b @ X @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X @ X ) @ bot_bo1174586163od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_292_graph__homomorphism__empty,axiom,
% 3.92/1.83 ! [G: labele1159362096nt_a_b,F: set_Product_prod_b_b] :
% 3.92/1.83 ( ( graph_222606102_a_b_b @ ( labele1230159100nt_a_b @ bot_bo1664927607od_b_b @ bot_bot_set_b ) @ G @ F )
% 3.92/1.83 = ( ( F = bot_bo1343651123od_b_b )
% 3.92/1.83 & ( G
% 3.92/1.83 = ( restri446606278nt_a_b @ G ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_293_graph__homomorphism__empty,axiom,
% 3.92/1.83 ! [G: labele1159362096nt_a_b,F: set_Pr357419743_b_b_b] :
% 3.92/1.83 ( ( graph_792023598_b_b_b @ ( labele170595300od_b_b @ bot_bo1964903287od_b_b @ bot_bo1343651123od_b_b ) @ G @ F )
% 3.92/1.83 = ( ( F = bot_bo1124087115_b_b_b )
% 3.92/1.83 & ( G
% 3.92/1.83 = ( restri446606278nt_a_b @ G ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_294_linear__order__on__converse,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( order_653178932r_on_b @ A2 @ ( converse_b_b @ R2 ) )
% 3.92/1.83 = ( order_653178932r_on_b @ A2 @ R2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_295_edges__def,axiom,
% 3.92/1.83 ( labele1741081071nt_a_b
% 3.92/1.83 = ( labele377702064od_b_b
% 3.92/1.83 @ ^ [X12: set_Pr1324126435od_b_b,X23: set_b] : X12 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_296_vertices__def,axiom,
% 3.92/1.83 ( labele1424214014nt_a_b
% 3.92/1.83 = ( labele1287464530_set_b
% 3.92/1.83 @ ^ [X12: set_Pr1324126435od_b_b,X23: set_b] : X23 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_297_lnear__order__on__empty,axiom,
% 3.92/1.83 order_653178932r_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% 3.92/1.83
% 3.92/1.83 thf(fact_298_lnear__order__on__empty,axiom,
% 3.92/1.83 order_609163036od_b_b @ bot_bo1343651123od_b_b @ bot_bo1174586163od_b_b ).
% 3.92/1.83
% 3.92/1.83 thf(fact_299_congruent2__implies__congruent,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R1: set_Pr2007082183od_b_b,R22: set_Product_prod_b_b,F: product_prod_b_b > b > set_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( equiv_1125628061od_b_b @ A2 @ R1 )
% 3.92/1.83 => ( ( equiv_1036235914_set_b @ R1 @ R22 @ F )
% 3.92/1.83 => ( ( member1285940496od_b_b @ A @ A2 )
% 3.92/1.83 => ( equiv_1821124534_set_b @ R22 @ ( F @ A ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_300_congruent2__implies__congruent,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,R1: set_Pr2109276827od_b_b,R22: set_Product_prod_b_b,F: produc1213276845od_b_b > b > set_b,A: produc1213276845od_b_b] :
% 3.92/1.83 ( ( equiv_2048442231od_b_b @ A2 @ R1 )
% 3.92/1.83 => ( ( equiv_1629197578_set_b @ R1 @ R22 @ F )
% 3.92/1.83 => ( ( member1516365892od_b_b @ A @ A2 )
% 3.92/1.83 => ( equiv_1821124534_set_b @ R22 @ ( F @ A ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_301_congruent2__implies__congruent,axiom,
% 3.92/1.83 ! [A2: set_b,R1: set_Product_prod_b_b,R22: set_Product_prod_b_b,F: b > b > set_b,A: b] :
% 3.92/1.83 ( ( equiv_equiv_b @ A2 @ R1 )
% 3.92/1.83 => ( ( equiv_167626658_set_b @ R1 @ R22 @ F )
% 3.92/1.83 => ( ( member_b @ A @ A2 )
% 3.92/1.83 => ( equiv_1821124534_set_b @ R22 @ ( F @ A ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_302_proj__def,axiom,
% 3.92/1.83 ( equiv_proj_b_b
% 3.92/1.83 = ( ^ [R3: set_Product_prod_b_b,X2: b] : ( image_b_b @ R3 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_303_equiv__class__subset,axiom,
% 3.92/1.83 ! [A2: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
% 3.92/1.83 ( ( equiv_equiv_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ord_less_eq_set_b @ ( image_b_b @ R2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ ( image_b_b @ R2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_304_equiv__class__subset,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
% 3.92/1.83 ( ( equiv_1125628061od_b_b @ A2 @ R2 )
% 3.92/1.83 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 3.92/1.83 => ( ord_le1036320359od_b_b @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) @ ( image_1928024979od_b_b @ R2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_305_subsetI,axiom,
% 3.92/1.83 ! [A2: set_b,B2: set_b] :
% 3.92/1.83 ( ! [X4: b] :
% 3.92/1.83 ( ( member_b @ X4 @ A2 )
% 3.92/1.83 => ( member_b @ X4 @ B2 ) )
% 3.92/1.83 => ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_306_subsetI,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
% 3.92/1.83 ( ! [X4: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ X4 @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ X4 @ B2 ) )
% 3.92/1.83 => ( ord_le1036320359od_b_b @ A2 @ B2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_307_subsetI,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ! [X4: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ X4 @ A2 )
% 3.92/1.83 => ( member1516365892od_b_b @ X4 @ B2 ) )
% 3.92/1.83 => ( ord_le123089219od_b_b @ A2 @ B2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_308_converse__mono,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ ( converse_b_b @ R2 ) @ ( converse_b_b @ S2 ) )
% 3.92/1.83 = ( ord_le1036320359od_b_b @ R2 @ S2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_309_empty__subsetI,axiom,
% 3.92/1.83 ! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_310_empty__subsetI,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b] : ( ord_le1036320359od_b_b @ bot_bo1343651123od_b_b @ A2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_311_subset__empty,axiom,
% 3.92/1.83 ! [A2: set_b] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
% 3.92/1.83 = ( A2 = bot_bot_set_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_312_subset__empty,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ A2 @ bot_bo1343651123od_b_b )
% 3.92/1.83 = ( A2 = bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_313_insert__subset,axiom,
% 3.92/1.83 ! [X: b,A2: set_b,B2: set_b] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ ( insert_b @ X @ A2 ) @ B2 )
% 3.92/1.83 = ( ( member_b @ X @ B2 )
% 3.92/1.83 & ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_314_insert__subset,axiom,
% 3.92/1.83 ! [X: product_prod_b_b,A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X @ A2 ) @ B2 )
% 3.92/1.83 = ( ( member1285940496od_b_b @ X @ B2 )
% 3.92/1.83 & ( ord_le1036320359od_b_b @ A2 @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_315_insert__subset,axiom,
% 3.92/1.83 ! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ( ord_le123089219od_b_b @ ( insert2037698781od_b_b @ X @ A2 ) @ B2 )
% 3.92/1.83 = ( ( member1516365892od_b_b @ X @ B2 )
% 3.92/1.83 & ( ord_le123089219od_b_b @ A2 @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_316_singleton__insert__inj__eq_H,axiom,
% 3.92/1.83 ! [A: b,A2: set_b,B: b] :
% 3.92/1.83 ( ( ( insert_b @ A @ A2 )
% 3.92/1.83 = ( insert_b @ B @ bot_bot_set_b ) )
% 3.92/1.83 = ( ( A = B )
% 3.92/1.83 & ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_317_singleton__insert__inj__eq_H,axiom,
% 3.92/1.83 ! [A: product_prod_b_b,A2: set_Product_prod_b_b,B: product_prod_b_b] :
% 3.92/1.83 ( ( ( insert1952693431od_b_b @ A @ A2 )
% 3.92/1.83 = ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = ( ( A = B )
% 3.92/1.83 & ( ord_le1036320359od_b_b @ A2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_318_singleton__insert__inj__eq,axiom,
% 3.92/1.83 ! [B: b,A: b,A2: set_b] :
% 3.92/1.83 ( ( ( insert_b @ B @ bot_bot_set_b )
% 3.92/1.83 = ( insert_b @ A @ A2 ) )
% 3.92/1.83 = ( ( A = B )
% 3.92/1.83 & ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_319_singleton__insert__inj__eq,axiom,
% 3.92/1.83 ! [B: product_prod_b_b,A: product_prod_b_b,A2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b )
% 3.92/1.83 = ( insert1952693431od_b_b @ A @ A2 ) )
% 3.92/1.83 = ( ( A = B )
% 3.92/1.83 & ( ord_le1036320359od_b_b @ A2 @ ( insert1952693431od_b_b @ B @ bot_bo1343651123od_b_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_320_Image__mono,axiom,
% 3.92/1.83 ! [R4: set_Product_prod_b_b,R2: set_Product_prod_b_b,A7: set_b,A2: set_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ R4 @ R2 )
% 3.92/1.83 => ( ( ord_less_eq_set_b @ A7 @ A2 )
% 3.92/1.83 => ( ord_less_eq_set_b @ ( image_b_b @ R4 @ A7 ) @ ( image_b_b @ R2 @ A2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_321_Collect__subset,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,P: product_prod_b_b > $o] :
% 3.92/1.83 ( ord_le1036320359od_b_b
% 3.92/1.83 @ ( collec1481886546od_b_b
% 3.92/1.83 @ ^ [X2: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ X2 @ A2 )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 @ A2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_322_Collect__subset,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,P: produc1213276845od_b_b > $o] :
% 3.92/1.83 ( ord_le123089219od_b_b
% 3.92/1.83 @ ( collec279084418od_b_b
% 3.92/1.83 @ ^ [X2: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ X2 @ A2 )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 @ A2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_323_Collect__subset,axiom,
% 3.92/1.83 ! [A2: set_b,P: b > $o] :
% 3.92/1.83 ( ord_less_eq_set_b
% 3.92/1.83 @ ( collect_b
% 3.92/1.83 @ ^ [X2: b] :
% 3.92/1.83 ( ( member_b @ X2 @ A2 )
% 3.92/1.83 & ( P @ X2 ) ) )
% 3.92/1.83 @ A2 ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_324_Collect__mono__iff,axiom,
% 3.92/1.83 ! [P: b > $o,Q: b > $o] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
% 3.92/1.83 = ( ! [X2: b] :
% 3.92/1.83 ( ( P @ X2 )
% 3.92/1.83 => ( Q @ X2 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_325_Collect__mono,axiom,
% 3.92/1.83 ! [P: b > $o,Q: b > $o] :
% 3.92/1.83 ( ! [X4: b] :
% 3.92/1.83 ( ( P @ X4 )
% 3.92/1.83 => ( Q @ X4 ) )
% 3.92/1.83 => ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_326_subset__iff,axiom,
% 3.92/1.83 ( ord_less_eq_set_b
% 3.92/1.83 = ( ^ [A6: set_b,B5: set_b] :
% 3.92/1.83 ! [T: b] :
% 3.92/1.83 ( ( member_b @ T @ A6 )
% 3.92/1.83 => ( member_b @ T @ B5 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_327_subset__iff,axiom,
% 3.92/1.83 ( ord_le1036320359od_b_b
% 3.92/1.83 = ( ^ [A6: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
% 3.92/1.83 ! [T: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ T @ A6 )
% 3.92/1.83 => ( member1285940496od_b_b @ T @ B5 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_328_subset__iff,axiom,
% 3.92/1.83 ( ord_le123089219od_b_b
% 3.92/1.83 = ( ^ [A6: set_Pr1324126435od_b_b,B5: set_Pr1324126435od_b_b] :
% 3.92/1.83 ! [T: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ T @ A6 )
% 3.92/1.83 => ( member1516365892od_b_b @ T @ B5 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_329_subset__eq,axiom,
% 3.92/1.83 ( ord_less_eq_set_b
% 3.92/1.83 = ( ^ [A6: set_b,B5: set_b] :
% 3.92/1.83 ! [X2: b] :
% 3.92/1.83 ( ( member_b @ X2 @ A6 )
% 3.92/1.83 => ( member_b @ X2 @ B5 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_330_subset__eq,axiom,
% 3.92/1.83 ( ord_le1036320359od_b_b
% 3.92/1.83 = ( ^ [A6: set_Product_prod_b_b,B5: set_Product_prod_b_b] :
% 3.92/1.83 ! [X2: product_prod_b_b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ X2 @ A6 )
% 3.92/1.83 => ( member1285940496od_b_b @ X2 @ B5 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_331_subset__eq,axiom,
% 3.92/1.83 ( ord_le123089219od_b_b
% 3.92/1.83 = ( ^ [A6: set_Pr1324126435od_b_b,B5: set_Pr1324126435od_b_b] :
% 3.92/1.83 ! [X2: produc1213276845od_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ X2 @ A6 )
% 3.92/1.83 => ( member1516365892od_b_b @ X2 @ B5 ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_332_subsetD,axiom,
% 3.92/1.83 ! [A2: set_b,B2: set_b,C: b] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 3.92/1.83 => ( ( member_b @ C @ A2 )
% 3.92/1.83 => ( member_b @ C @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_333_subsetD,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,B2: set_Product_prod_b_b,C: product_prod_b_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ A2 @ B2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ C @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ C @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_334_subsetD,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b,C: produc1213276845od_b_b] :
% 3.92/1.83 ( ( ord_le123089219od_b_b @ A2 @ B2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ C @ A2 )
% 3.92/1.83 => ( member1516365892od_b_b @ C @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_335_in__mono,axiom,
% 3.92/1.83 ! [A2: set_b,B2: set_b,X: b] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 3.92/1.83 => ( ( member_b @ X @ A2 )
% 3.92/1.83 => ( member_b @ X @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_336_in__mono,axiom,
% 3.92/1.83 ! [A2: set_Product_prod_b_b,B2: set_Product_prod_b_b,X: product_prod_b_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ A2 @ B2 )
% 3.92/1.83 => ( ( member1285940496od_b_b @ X @ A2 )
% 3.92/1.83 => ( member1285940496od_b_b @ X @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_337_in__mono,axiom,
% 3.92/1.83 ! [A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b,X: produc1213276845od_b_b] :
% 3.92/1.83 ( ( ord_le123089219od_b_b @ A2 @ B2 )
% 3.92/1.83 => ( ( member1516365892od_b_b @ X @ A2 )
% 3.92/1.83 => ( member1516365892od_b_b @ X @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_338_converse__subset__swap,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ R2 @ ( converse_b_b @ S2 ) )
% 3.92/1.83 = ( ord_le1036320359od_b_b @ ( converse_b_b @ R2 ) @ S2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_339_subrelI,axiom,
% 3.92/1.83 ! [R2: set_Product_prod_b_b,S2: set_Product_prod_b_b] :
% 3.92/1.83 ( ! [X4: b,Y3: b] :
% 3.92/1.83 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ R2 )
% 3.92/1.83 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y3 ) @ S2 ) )
% 3.92/1.83 => ( ord_le1036320359od_b_b @ R2 @ S2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_340_subrelI,axiom,
% 3.92/1.83 ! [R2: set_Pr1324126435od_b_b,S2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ! [X4: standard_Constant_a,Y3: product_prod_b_b] :
% 3.92/1.83 ( ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ R2 )
% 3.92/1.83 => ( member1516365892od_b_b @ ( produc1432590431od_b_b @ X4 @ Y3 ) @ S2 ) )
% 3.92/1.83 => ( ord_le123089219od_b_b @ R2 @ S2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_341_bot_Oextremum__uniqueI,axiom,
% 3.92/1.83 ! [A: set_b] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
% 3.92/1.83 => ( A = bot_bot_set_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_342_bot_Oextremum__uniqueI,axiom,
% 3.92/1.83 ! [A: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ A @ bot_bo1343651123od_b_b )
% 3.92/1.83 => ( A = bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_343_bot_Oextremum__unique,axiom,
% 3.92/1.83 ! [A: set_b] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
% 3.92/1.83 = ( A = bot_bot_set_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_344_bot_Oextremum__unique,axiom,
% 3.92/1.83 ! [A: set_Product_prod_b_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ A @ bot_bo1343651123od_b_b )
% 3.92/1.83 = ( A = bot_bo1343651123od_b_b ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_345_bot_Oextremum,axiom,
% 3.92/1.83 ! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_346_bot_Oextremum,axiom,
% 3.92/1.83 ! [A: set_Product_prod_b_b] : ( ord_le1036320359od_b_b @ bot_bo1343651123od_b_b @ A ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_347_insert__mono,axiom,
% 3.92/1.83 ! [C4: set_b,D2: set_b,A: b] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ C4 @ D2 )
% 3.92/1.83 => ( ord_less_eq_set_b @ ( insert_b @ A @ C4 ) @ ( insert_b @ A @ D2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_348_subset__insert,axiom,
% 3.92/1.83 ! [X: b,A2: set_b,B2: set_b] :
% 3.92/1.83 ( ~ ( member_b @ X @ A2 )
% 3.92/1.83 => ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
% 3.92/1.83 = ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_349_subset__insert,axiom,
% 3.92/1.83 ! [X: product_prod_b_b,A2: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
% 3.92/1.83 ( ~ ( member1285940496od_b_b @ X @ A2 )
% 3.92/1.83 => ( ( ord_le1036320359od_b_b @ A2 @ ( insert1952693431od_b_b @ X @ B2 ) )
% 3.92/1.83 = ( ord_le1036320359od_b_b @ A2 @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_350_subset__insert,axiom,
% 3.92/1.83 ! [X: produc1213276845od_b_b,A2: set_Pr1324126435od_b_b,B2: set_Pr1324126435od_b_b] :
% 3.92/1.83 ( ~ ( member1516365892od_b_b @ X @ A2 )
% 3.92/1.83 => ( ( ord_le123089219od_b_b @ A2 @ ( insert2037698781od_b_b @ X @ B2 ) )
% 3.92/1.83 = ( ord_le123089219od_b_b @ A2 @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_351_subset__insertI,axiom,
% 3.92/1.83 ! [B2: set_b,A: b] : ( ord_less_eq_set_b @ B2 @ ( insert_b @ A @ B2 ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_352_subset__insertI2,axiom,
% 3.92/1.83 ! [A2: set_b,B2: set_b,B: b] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ A2 @ B2 )
% 3.92/1.83 => ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_353_subset__singleton__iff,axiom,
% 3.92/1.83 ! [X3: set_b,A: b] :
% 3.92/1.83 ( ( ord_less_eq_set_b @ X3 @ ( insert_b @ A @ bot_bot_set_b ) )
% 3.92/1.83 = ( ( X3 = bot_bot_set_b )
% 3.92/1.83 | ( X3
% 3.92/1.83 = ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(fact_354_subset__singleton__iff,axiom,
% 3.92/1.83 ! [X3: set_Product_prod_b_b,A: product_prod_b_b] :
% 3.92/1.83 ( ( ord_le1036320359od_b_b @ X3 @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 3.92/1.83 = ( ( X3 = bot_bo1343651123od_b_b )
% 3.92/1.83 | ( X3
% 3.92/1.83 = ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(help_If_3_1_If_001tf__b_T,axiom,
% 3.92/1.83 ! [P: $o] :
% 3.92/1.83 ( ( P = $true )
% 3.92/1.83 | ( P = $false ) ) ).
% 3.92/1.83
% 3.92/1.83 thf(help_If_2_1_If_001tf__b_T,axiom,
% 3.92/1.83 ! [X: b,Y4: b] :
% 3.92/1.83 ( ( if_b @ $false @ X @ Y4 )
% 3.92/1.83 = Y4 ) ).
% 3.92/1.83
% 3.92/1.83 thf(help_If_1_1_If_001tf__b_T,axiom,
% 3.92/1.83 ! [X: b,Y4: b] :
% 3.92/1.83 ( ( if_b @ $true @ X @ Y4 )
% 3.92/1.83 = X ) ).
% 3.92/1.83
% 3.92/1.83 thf(conj_0,conjecture,
% 3.92/1.83 ( ( hilbert_Eps_b
% 3.92/1.83 @ ^ [Y: b] : ( member_b @ Y @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ x @ bot_bot_set_b ) ) ) )
% 3.92/1.83 = ( hilbert_Eps_b
% 3.92/1.83 @ ^ [Y: b] : ( member_b @ Y @ ( image_b_b @ ( getRel904497637nt_a_b @ standard_S_Idt_a @ g ) @ ( insert_b @ y @ bot_bot_set_b ) ) ) ) ) ).
% 3.92/1.83 ------------------------
% 3.92/1.88 % (21114)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_20997 for (2999ds/275Mi)
% 3.92/1.88 % (21110)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_20997 for (2999ds/27Mi)
% 3.92/1.88 % (21108)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20997 for (2999ds/183Mi)
% 3.92/1.88 % (21109)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_20997 for (2999ds/4Mi)
% 3.92/1.88 % (21115)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20997 for (2999ds/18Mi)
% 3.92/1.88 % (21112)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20997 for (2999ds/2Mi)
% 3.92/1.88 % (21113)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_20997 for (2999ds/2Mi)
% 3.92/1.88 % (21112)Instruction limit reached!
% 3.92/1.88 % (21112)------------------------------
% 3.92/1.88 % (21112)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.92/1.88 % (21112)Termination reason: Unknown
% 3.92/1.88 % (21112)Termination phase: shuffling
% 3.92/1.88 % (21113)Instruction limit reached!
% 3.92/1.88 % (21113)------------------------------
% 3.92/1.88 % (21113)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.92/1.88 % (21113)Termination reason: Unknown
% 3.92/1.88 % (21113)Termination phase: shuffling
% 3.92/1.88
% 3.92/1.88 % (21113)Memory used [KB]: 1535
% 3.92/1.88 % (21113)Time elapsed: 0.004 s
% 3.92/1.88 % (21113)Instructions burned: 3 (million)
% 3.92/1.88 % (21113)------------------------------
% 3.92/1.88 % (21113)------------------------------
% 3.92/1.88
% 3.92/1.88 % (21112)Memory used [KB]: 1535
% 3.92/1.88 % (21112)Time elapsed: 0.004 s
% 3.92/1.88 % (21112)Instructions burned: 3 (million)
% 3.92/1.88 % (21112)------------------------------
% 3.92/1.88 % (21112)------------------------------
% 3.92/1.88 % (21109)Instruction limit reached!
% 3.92/1.88 % (21109)------------------------------
% 3.92/1.88 % (21109)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.92/1.88 % (21109)Termination reason: Unknown
% 3.92/1.88 % (21109)Termination phase: shuffling
% 3.92/1.88
% 3.92/1.88 % (21109)Memory used [KB]: 1535
% 3.92/1.88 % (21109)Time elapsed: 0.005 s
% 3.92/1.88 % (21109)Instructions burned: 5 (million)
% 3.92/1.88 % (21109)------------------------------
% 3.92/1.88 % (21109)------------------------------
% 3.92/1.89 % (21115)Instruction limit reached!
% 3.92/1.89 % (21115)------------------------------
% 3.92/1.89 % (21115)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.92/1.89 % (21115)Termination reason: Unknown
% 3.92/1.89 % (21115)Termination phase: shuffling
% 3.92/1.89
% 3.92/1.89 % (21115)Memory used [KB]: 1791
% 3.92/1.89 % (21115)Time elapsed: 0.012 s
% 3.92/1.89 % (21115)Instructions burned: 18 (million)
% 3.92/1.89 % (21115)------------------------------
% 3.92/1.89 % (21115)------------------------------
% 3.92/1.89 % (21110)Instruction limit reached!
% 3.92/1.89 % (21110)------------------------------
% 3.92/1.89 % (21110)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.92/1.89 % (21110)Termination reason: Unknown
% 3.92/1.89 % (21110)Termination phase: shuffling
% 3.92/1.89
% 3.92/1.89 % (21110)Memory used [KB]: 1918
% 3.92/1.89 % (21110)Time elapsed: 0.017 s
% 3.92/1.89 % (21110)Instructions burned: 27 (million)
% 3.92/1.89 % (21110)------------------------------
% 3.92/1.89 % (21110)------------------------------
% 3.98/1.90 % (21117)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_20997 for (2999ds/37Mi)
% 3.98/1.90 % (21118)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_20997 for (2999ds/15Mi)
% 3.98/1.90 % (21116)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20997 for (2999ds/3Mi)
% 3.98/1.90 % (21116)Instruction limit reached!
% 3.98/1.90 % (21116)------------------------------
% 3.98/1.90 % (21116)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.98/1.90 % (21116)Termination reason: Unknown
% 3.98/1.90 % (21116)Termination phase: shuffling
% 3.98/1.90
% 3.98/1.90 % (21116)Memory used [KB]: 1535
% 3.98/1.90 % (21116)Time elapsed: 0.004 s
% 3.98/1.90 % (21116)Instructions burned: 4 (million)
% 3.98/1.90 % (21116)------------------------------
% 3.98/1.90 % (21116)------------------------------
% 3.98/1.91 % (21119)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_20997 for (2999ds/3Mi)
% 3.98/1.91 % (21118)Instruction limit reached!
% 3.98/1.91 % (21118)------------------------------
% 3.98/1.91 % (21118)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.98/1.91 % (21118)Termination reason: Unknown
% 3.98/1.91 % (21118)Termination phase: shuffling
% 3.98/1.91
% 3.98/1.91 % (21118)Memory used [KB]: 1791
% 3.98/1.91 % (21118)Time elapsed: 0.011 s
% 3.98/1.91 % (21118)Instructions burned: 16 (million)
% 3.98/1.91 % (21118)------------------------------
% 3.98/1.91 % (21118)------------------------------
% 3.98/1.91 % (21119)Instruction limit reached!
% 3.98/1.91 % (21119)------------------------------
% 3.98/1.91 % (21119)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.98/1.91 % (21119)Termination reason: Unknown
% 3.98/1.91 % (21119)Termination phase: shuffling
% 3.98/1.91
% 3.98/1.91 % (21119)Memory used [KB]: 1535
% 3.98/1.91 % (21119)Time elapsed: 0.004 s
% 3.98/1.91 % (21119)Instructions burned: 3 (million)
% 3.98/1.91 % (21119)------------------------------
% 3.98/1.91 % (21119)------------------------------
% 3.98/1.91 % (21120)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_20997 for (2999ds/1041Mi)
% 3.98/1.92 % (21114)First to succeed.
% 3.98/1.92 % (21114)Refutation found. Thanks to Tanya!
% 3.98/1.92 % SZS status Theorem for DTF2THF_20997
% 3.98/1.92 % SZS output start Proof for DTF2THF_20997
% 3.98/1.92 thf(type_def_5, type, set_Pr2109276827od_b_b: $tType).
% 3.98/1.92 thf(type_def_6, type, set_Product_prod_b_b: $tType).
% 3.98/1.92 thf(type_def_7, type, produc1213276845od_b_b: $tType).
% 3.98/1.92 thf(type_def_8, type, b: $tType).
% 3.98/1.92 thf(type_def_9, type, set_b: $tType).
% 3.98/1.92 thf(type_def_11, type, set_Pr2007082183od_b_b: $tType).
% 3.98/1.92 thf(type_def_12, type, product_prod_b_b: $tType).
% 3.98/1.92 thf(type_def_13, type, set_Pr1324126435od_b_b: $tType).
% 3.98/1.92 thf(type_def_14, type, standard_Constant_a: $tType).
% 3.98/1.92 thf(type_def_15, type, labele1159362096nt_a_b: $tType).
% 3.98/1.92 thf(type_def_16, type, labele1835558936od_b_b: $tType).
% 3.98/1.92 thf(type_def_17, type, set_Pr357419743_b_b_b: $tType).
% 3.98/1.92 thf(type_def_18, type, set_Pr1021918435od_b_b: $tType).
% 3.98/1.92 thf(type_def_19, type, set_St761939237tant_a: $tType).
% 3.98/1.92 thf(type_def_20, type, produc970027067od_b_b: $tType).
% 3.98/1.92 thf(type_def_21, type, produc1990339951od_b_b: $tType).
% 3.98/1.92 thf(type_def_22, type, produc276146823_b_b_b: $tType).
% 3.98/1.92 thf(type_def_23, type, produc1367125253tant_a: $tType).
% 3.98/1.92 thf(type_def_24, type, produc1052326083od_b_b: $tType).
% 3.98/1.92 thf(type_def_25, type, produc1168037607od_b_b: $tType).
% 3.98/1.92 thf(type_def_26, type, produc2112523263_b_b_b: $tType).
% 3.98/1.92 thf(type_def_27, type, produc1605346267od_b_b: $tType).
% 3.98/1.92 thf(type_def_28, type, produc255402447od_b_b: $tType).
% 3.98/1.92 thf(type_def_29, type, set_Pr2134561957od_b_b: $tType).
% 3.98/1.92 thf(type_def_30, type, set_Pr1906739389_b_b_b: $tType).
% 3.98/1.92 thf(type_def_31, type, set_Pr1071216121od_b_b: $tType).
% 3.98/1.92 thf(type_def_32, type, set_Pr2060523537od_b_b: $tType).
% 3.98/1.92 thf(type_def_33, type, set_Pr621705391od_b_b: $tType).
% 3.98/1.92 thf(type_def_34, type, set_Pr306736955tant_a: $tType).
% 3.98/1.92 thf(func_def_0, type, set_Pr2109276827od_b_b: $tType).
% 3.98/1.92 thf(func_def_1, type, produc970027067od_b_b: $tType).
% 3.98/1.92 thf(func_def_2, type, set_Pr1071216121od_b_b: $tType).
% 3.98/1.92 thf(func_def_3, type, set_Pr2134561957od_b_b: $tType).
% 3.98/1.92 thf(func_def_4, type, set_Pr1021918435od_b_b: $tType).
% 3.98/1.92 thf(func_def_5, type, produc1052326083od_b_b: $tType).
% 3.98/1.92 thf(func_def_6, type, produc1990339951od_b_b: $tType).
% 3.98/1.92 thf(func_def_7, type, set_Pr2060523537od_b_b: $tType).
% 3.98/1.92 thf(func_def_8, type, set_Pr1906739389_b_b_b: $tType).
% 3.98/1.92 thf(func_def_9, type, produc1605346267od_b_b: $tType).
% 3.98/1.92 thf(func_def_10, type, produc276146823_b_b_b: $tType).
% 3.98/1.92 thf(func_def_11, type, set_Pr306736955tant_a: $tType).
% 3.98/1.92 thf(func_def_12, type, set_Pr1324126435od_b_b: $tType).
% 3.98/1.92 thf(func_def_13, type, labele1835558936od_b_b: $tType).
% 3.98/1.92 thf(func_def_14, type, set_Pr2007082183od_b_b: $tType).
% 3.98/1.92 thf(func_def_15, type, produc1367125253tant_a: $tType).
% 3.98/1.92 thf(func_def_16, type, produc1213276845od_b_b: $tType).
% 3.98/1.92 thf(func_def_17, type, produc1168037607od_b_b: $tType).
% 3.98/1.92 thf(func_def_18, type, labele1159362096nt_a_b: $tType).
% 3.98/1.92 thf(func_def_19, type, set_Pr621705391od_b_b: $tType).
% 3.98/1.92 thf(func_def_20, type, set_Pr357419743_b_b_b: $tType).
% 3.98/1.92 thf(func_def_21, type, produc255402447od_b_b: $tType).
% 3.98/1.92 thf(func_def_22, type, produc2112523263_b_b_b: $tType).
% 3.98/1.92 thf(func_def_23, type, set_St761939237tant_a: $tType).
% 3.98/1.92 thf(func_def_24, type, set_Product_prod_b_b: $tType).
% 3.98/1.92 thf(func_def_25, type, standard_Constant_a: $tType).
% 3.98/1.92 thf(func_def_26, type, product_prod_b_b: $tType).
% 3.98/1.92 thf(func_def_27, type, set_b: $tType).
% 3.98/1.92 thf(func_def_28, type, b: $tType).
% 3.98/1.92 thf(func_def_29, type, equiv_1629197578_set_b: set_Pr2109276827od_b_b > set_Product_prod_b_b > (produc1213276845od_b_b > b > set_b) > $o).
% 3.98/1.92 thf(func_def_30, type, equiv_1036235914_set_b: set_Pr2007082183od_b_b > set_Product_prod_b_b > (product_prod_b_b > b > set_b) > $o).
% 3.98/1.92 thf(func_def_31, type, equiv_167626658_set_b: set_Product_prod_b_b > set_Product_prod_b_b > (b > b > set_b) > $o).
% 3.98/1.92 thf(func_def_32, type, equiv_1821124534_set_b: set_Product_prod_b_b > (b > set_b) > $o).
% 3.98/1.92 thf(func_def_33, type, equiv_2048442231od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o).
% 3.98/1.92 thf(func_def_34, type, equiv_1125628061od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o).
% 3.98/1.92 thf(func_def_35, type, equiv_equiv_b: set_b > set_Product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_36, type, equiv_proj_b_b: set_Product_prod_b_b > b > set_b).
% 3.98/1.92 thf(func_def_37, type, hilber46179633od_b_b: (produc1213276845od_b_b > $o) > produc1213276845od_b_b).
% 3.98/1.92 thf(func_def_38, type, hilber620807523od_b_b: (product_prod_b_b > $o) > product_prod_b_b).
% 3.98/1.92 thf(func_def_39, type, hilbert_Eps_b: (b > $o) > b).
% 3.98/1.92 thf(func_def_40, type, if_b: $o > b > b > b).
% 3.98/1.92 thf(func_def_41, type, standard_S_Idt_a: standard_Constant_a).
% 3.98/1.92 thf(func_def_42, type, getRel904497637nt_a_b: standard_Constant_a > labele1159362096nt_a_b > set_Product_prod_b_b).
% 3.98/1.92 thf(func_def_43, type, graph_792023598_b_b_b: labele1835558936od_b_b > labele1159362096nt_a_b > set_Pr357419743_b_b_b > $o).
% 3.98/1.92 thf(func_def_44, type, graph_222606102_a_b_b: labele1159362096nt_a_b > labele1159362096nt_a_b > set_Product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_45, type, labele170595300od_b_b: set_Pr1021918435od_b_b > set_Product_prod_b_b > labele1835558936od_b_b).
% 3.98/1.92 thf(func_def_46, type, labele1230159100nt_a_b: set_Pr1324126435od_b_b > set_b > labele1159362096nt_a_b).
% 3.98/1.92 thf(func_def_47, type, labele377702064od_b_b: (set_Pr1324126435od_b_b > set_b > set_Pr1324126435od_b_b) > labele1159362096nt_a_b > set_Pr1324126435od_b_b).
% 3.98/1.92 thf(func_def_48, type, labele1287464530_set_b: (set_Pr1324126435od_b_b > set_b > set_b) > labele1159362096nt_a_b > set_b).
% 3.98/1.92 thf(func_def_49, type, labele1741081071nt_a_b: labele1159362096nt_a_b > set_Pr1324126435od_b_b).
% 3.98/1.92 thf(func_def_50, type, labele1424214014nt_a_b: labele1159362096nt_a_b > set_b).
% 3.98/1.92 thf(func_def_51, type, restri446606278nt_a_b: labele1159362096nt_a_b > labele1159362096nt_a_b).
% 3.98/1.92 thf(func_def_52, type, order_609163036od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o).
% 3.98/1.92 thf(func_def_53, type, order_653178932r_on_b: set_b > set_Product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_54, type, bot_bo1734638456_b_b_o: standard_Constant_a > product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_55, type, bot_bo1167600078_b_b_o: produc1213276845od_b_b > $o).
% 3.98/1.92 thf(func_def_56, type, bot_bo1302443370_b_b_o: product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_57, type, bot_bot_b_b_o: b > b > $o).
% 3.98/1.92 thf(func_def_58, type, bot_bot_b_o: b > $o).
% 3.98/1.92 thf(func_def_59, type, bot_bo1160111033tant_a: set_St761939237tant_a).
% 3.98/1.92 thf(func_def_60, type, bot_bo1964903287od_b_b: set_Pr1021918435od_b_b).
% 3.98/1.92 thf(func_def_61, type, bot_bo1664927607od_b_b: set_Pr1324126435od_b_b).
% 3.98/1.92 thf(func_def_62, type, bot_bo1174586163od_b_b: set_Pr2007082183od_b_b).
% 3.98/1.92 thf(func_def_63, type, bot_bo1124087115_b_b_b: set_Pr357419743_b_b_b).
% 3.98/1.92 thf(func_def_64, type, bot_bo1343651123od_b_b: set_Product_prod_b_b).
% 3.98/1.92 thf(func_def_65, type, bot_bot_set_b: set_b).
% 3.98/1.92 thf(func_def_66, type, ord_le123089219od_b_b: set_Pr1324126435od_b_b > set_Pr1324126435od_b_b > $o).
% 3.98/1.92 thf(func_def_67, type, ord_le1036320359od_b_b: set_Product_prod_b_b > set_Product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_68, type, ord_less_eq_set_b: set_b > set_b > $o).
% 3.98/1.92 thf(func_def_69, type, produc1432590431od_b_b: standard_Constant_a > product_prod_b_b > produc1213276845od_b_b).
% 3.98/1.92 thf(func_def_70, type, produc449289715od_b_b: produc1213276845od_b_b > produc1213276845od_b_b > produc970027067od_b_b).
% 3.98/1.92 thf(func_def_71, type, produc1597690145od_b_b: produc1213276845od_b_b > product_prod_b_b > produc1990339951od_b_b).
% 3.98/1.92 thf(func_def_72, type, produc1580777273_b_b_b: produc1213276845od_b_b > b > produc276146823_b_b_b).
% 3.98/1.92 thf(func_def_73, type, produc1977323903tant_a: product_prod_b_b > standard_Constant_a > produc1367125253tant_a).
% 3.98/1.92 thf(func_def_74, type, produc732676669od_b_b: product_prod_b_b > produc1213276845od_b_b > produc1052326083od_b_b).
% 3.98/1.92 thf(func_def_75, type, produc546497367od_b_b: product_prod_b_b > product_prod_b_b > produc1168037607od_b_b).
% 3.98/1.92 thf(func_def_76, type, produc1001682799_b_b_b: product_prod_b_b > b > produc2112523263_b_b_b).
% 3.98/1.92 thf(func_def_77, type, produc800495189od_b_b: b > produc1213276845od_b_b > produc1605346267od_b_b).
% 3.98/1.92 thf(func_def_78, type, produc1257047359od_b_b: b > product_prod_b_b > produc255402447od_b_b).
% 3.98/1.92 thf(func_def_79, type, product_Pair_b_b: b > b > product_prod_b_b).
% 3.98/1.92 thf(func_def_80, type, image_1916422435od_b_b: set_Pr1324126435od_b_b > set_St761939237tant_a > set_Product_prod_b_b).
% 3.98/1.92 thf(func_def_81, type, image_763305007od_b_b: set_Pr2109276827od_b_b > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b).
% 3.98/1.92 thf(func_def_82, type, image_1397930469od_b_b: set_Pr2134561957od_b_b > set_Pr1324126435od_b_b > set_Product_prod_b_b).
% 3.98/1.92 thf(func_def_83, type, image_1764625405_b_b_b: set_Pr1906739389_b_b_b > set_Pr1324126435od_b_b > set_b).
% 3.98/1.92 thf(func_def_84, type, image_532916993od_b_b: set_Pr1071216121od_b_b > set_Product_prod_b_b > set_Pr1324126435od_b_b).
% 3.98/1.92 thf(func_def_85, type, image_1928024979od_b_b: set_Pr2007082183od_b_b > set_Product_prod_b_b > set_Product_prod_b_b).
% 3.98/1.92 thf(func_def_86, type, image_921732011_b_b_b: set_Pr357419743_b_b_b > set_Product_prod_b_b > set_b).
% 3.98/1.92 thf(func_def_87, type, image_984343321od_b_b: set_Pr2060523537od_b_b > set_b > set_Pr1324126435od_b_b).
% 3.98/1.92 thf(func_def_88, type, image_1177096571od_b_b: set_Pr621705391od_b_b > set_b > set_Product_prod_b_b).
% 3.98/1.92 thf(func_def_89, type, image_b_b: set_Product_prod_b_b > set_b > set_b).
% 3.98/1.92 thf(func_def_90, type, conver1398856763od_b_b: set_Pr1324126435od_b_b > set_Pr306736955tant_a).
% 3.98/1.92 thf(func_def_91, type, conver1943590235tant_a: set_Pr306736955tant_a > set_Pr1324126435od_b_b).
% 3.98/1.92 thf(func_def_92, type, converse_b_b: set_Product_prod_b_b > set_Product_prod_b_b).
% 3.98/1.92 thf(func_def_93, type, refl_o1921098222od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o).
% 3.98/1.92 thf(func_def_94, type, refl_o2134473190od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o).
% 3.98/1.92 thf(func_def_95, type, refl_on_b: set_b > set_Product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_96, type, total_771970601od_b_b: set_Pr1324126435od_b_b > set_Pr2109276827od_b_b > $o).
% 3.98/1.92 thf(func_def_97, type, total_178717291od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o).
% 3.98/1.92 thf(func_def_98, type, total_on_b: set_b > set_Product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_99, type, collec279084418od_b_b: (produc1213276845od_b_b > $o) > set_Pr1324126435od_b_b).
% 3.98/1.92 thf(func_def_100, type, collec1481886546od_b_b: (product_prod_b_b > $o) > set_Product_prod_b_b).
% 3.98/1.92 thf(func_def_101, type, collect_b: (b > $o) > set_b).
% 3.98/1.92 thf(func_def_102, type, insert1909710879tant_a: standard_Constant_a > set_St761939237tant_a > set_St761939237tant_a).
% 3.98/1.92 thf(func_def_103, type, insert2037698781od_b_b: produc1213276845od_b_b > set_Pr1324126435od_b_b > set_Pr1324126435od_b_b).
% 3.98/1.92 thf(func_def_104, type, insert1098249399od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > set_Pr2007082183od_b_b).
% 3.98/1.92 thf(func_def_105, type, insert1952693431od_b_b: product_prod_b_b > set_Product_prod_b_b > set_Product_prod_b_b).
% 3.98/1.92 thf(func_def_106, type, insert_b: b > set_b > set_b).
% 3.98/1.92 thf(func_def_107, type, is_sin1465183929od_b_b: set_Pr1324126435od_b_b > $o).
% 3.98/1.92 thf(func_def_108, type, is_sin1713057243od_b_b: set_Product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_109, type, is_singleton_b: set_b > $o).
% 3.98/1.92 thf(func_def_110, type, the_el329196508od_b_b: set_Product_prod_b_b > product_prod_b_b).
% 3.98/1.92 thf(func_def_111, type, the_elem_b: set_b > b).
% 3.98/1.92 thf(func_def_112, type, member1632892294tant_a: standard_Constant_a > set_St761939237tant_a > $o).
% 3.98/1.92 thf(func_def_113, type, member1516365892od_b_b: produc1213276845od_b_b > set_Pr1324126435od_b_b > $o).
% 3.98/1.92 thf(func_def_114, type, member1086024932od_b_b: produc970027067od_b_b > set_Pr2109276827od_b_b > $o).
% 3.98/1.92 thf(func_def_115, type, member942707974od_b_b: produc1990339951od_b_b > set_Pr2134561957od_b_b > $o).
% 3.98/1.92 thf(func_def_116, type, member1417789726_b_b_b: produc276146823_b_b_b > set_Pr1906739389_b_b_b > $o).
% 3.98/1.92 thf(func_def_117, type, member1670214300tant_a: produc1367125253tant_a > set_Pr306736955tant_a > $o).
% 3.98/1.92 thf(func_def_118, type, member4694106od_b_b: produc1052326083od_b_b > set_Pr1071216121od_b_b > $o).
% 3.98/1.92 thf(func_def_119, type, member1652792080od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > $o).
% 3.98/1.92 thf(func_def_120, type, member351494440_b_b_b: produc2112523263_b_b_b > set_Pr357419743_b_b_b > $o).
% 3.98/1.92 thf(func_def_121, type, member599505522od_b_b: produc1605346267od_b_b > set_Pr2060523537od_b_b > $o).
% 3.98/1.92 thf(func_def_122, type, member641857272od_b_b: produc255402447od_b_b > set_Pr621705391od_b_b > $o).
% 3.98/1.92 thf(func_def_123, type, member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o).
% 3.98/1.92 thf(func_def_124, type, member_b: b > set_b > $o).
% 3.98/1.92 thf(func_def_125, type, g: labele1159362096nt_a_b).
% 3.98/1.92 thf(func_def_126, type, p: b > b > $o).
% 3.98/1.92 thf(func_def_127, type, f: b > b).
% 3.98/1.92 thf(func_def_128, type, x: b).
% 3.98/1.92 thf(func_def_129, type, y: b).
% 3.98/1.92 thf(func_def_145, type, sP0: set_b > b > b > set_b > $o).
% 3.98/1.92 thf(func_def_146, type, sP1: set_b > b > b > set_b > $o).
% 3.98/1.92 thf(func_def_147, type, sK2: product_prod_b_b > b).
% 3.98/1.92 thf(func_def_148, type, sK3: product_prod_b_b > b).
% 3.98/1.92 thf(func_def_149, type, sK4: (b > $o) > (b > $o) > b).
% 3.98/1.92 thf(func_def_150, type, sK5: (product_prod_b_b > $o) > b).
% 3.98/1.92 thf(func_def_151, type, sK6: (product_prod_b_b > $o) > b).
% 3.98/1.92 thf(func_def_152, type, sK7: set_Product_prod_b_b > product_prod_b_b).
% 3.98/1.92 thf(func_def_153, type, sK8: b > set_b > set_Product_prod_b_b > b).
% 3.98/1.92 thf(func_def_154, type, sK9: (b > $o) > (b > $o) > b).
% 3.98/1.92 thf(func_def_155, type, sK10: set_b > b > b > set_b > set_b).
% 3.98/1.92 thf(func_def_156, type, sK11: b > (b > $o) > b).
% 3.98/1.92 thf(func_def_157, type, sK12: set_Product_prod_b_b > b > set_b > b).
% 3.98/1.92 thf(func_def_158, type, sK13: set_Product_prod_b_b > product_prod_b_b).
% 3.98/1.92 thf(func_def_159, type, sK14: (produc1213276845od_b_b > $o) > b).
% 3.98/1.92 thf(func_def_160, type, sK15: (produc1213276845od_b_b > $o) > standard_Constant_a).
% 3.98/1.92 thf(func_def_161, type, sK16: (produc1213276845od_b_b > $o) > b).
% 3.98/1.92 thf(func_def_162, type, sK17: set_Product_prod_b_b > product_prod_b_b).
% 3.98/1.92 thf(func_def_163, type, sK18: b > set_b > set_b).
% 3.98/1.92 thf(func_def_164, type, sK19: (product_prod_b_b > $o) > b).
% 3.98/1.92 thf(func_def_165, type, sK20: (product_prod_b_b > $o) > b).
% 3.98/1.92 thf(func_def_166, type, sK21: set_b > b).
% 3.98/1.92 thf(func_def_167, type, sK22: (b > $o) > (b > $o) > b).
% 3.98/1.92 thf(func_def_168, type, sK23: set_b > b > set_b).
% 3.98/1.92 thf(func_def_169, type, sK24: product_prod_b_b > b).
% 3.98/1.92 thf(func_def_170, type, sK25: product_prod_b_b > b).
% 3.98/1.92 thf(func_def_171, type, sK26: (b > $o) > b > b).
% 3.98/1.92 thf(func_def_172, type, sK27: produc1213276845od_b_b > standard_Constant_a).
% 3.98/1.92 thf(func_def_173, type, sK28: produc1213276845od_b_b > product_prod_b_b).
% 3.98/1.92 thf(func_def_174, type, sK29: set_b > b).
% 3.98/1.92 thf(func_def_175, type, sK30: (produc1213276845od_b_b > $o) > product_prod_b_b).
% 3.98/1.92 thf(func_def_176, type, sK31: (produc1213276845od_b_b > $o) > standard_Constant_a).
% 3.98/1.92 thf(func_def_177, type, sK32: set_b > b).
% 3.98/1.92 thf(func_def_178, type, sK33: produc1213276845od_b_b > b).
% 3.98/1.92 thf(func_def_179, type, sK34: produc1213276845od_b_b > b).
% 3.98/1.92 thf(func_def_180, type, sK35: produc1213276845od_b_b > standard_Constant_a).
% 3.98/1.92 thf(func_def_181, type, sK36: (produc1213276845od_b_b > $o) > product_prod_b_b).
% 3.98/1.92 thf(func_def_182, type, sK37: (produc1213276845od_b_b > $o) > standard_Constant_a).
% 3.98/1.92 thf(func_def_183, type, sK38: (b > $o) > b).
% 3.98/1.92 thf(func_def_184, type, sK39: produc1213276845od_b_b > product_prod_b_b).
% 3.98/1.92 thf(func_def_185, type, sK40: produc1213276845od_b_b > standard_Constant_a).
% 3.98/1.92 thf(func_def_187, type, ph42: !>[X0: $tType]:(X0)).
% 3.98/1.92 thf(f1307,plain,(
% 3.98/1.92 $false),
% 3.98/1.92 inference(trivial_inequality_removal,[status(thm)],[f1306])).
% 3.98/1.92 thf(f1306,plain,(
% 3.98/1.92 ((hilbert_Eps_b @ (^[Y0 : b]: (member_b @ Y0 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b))))) != (hilbert_Eps_b @ (^[Y0 : b]: (member_b @ Y0 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b))))))),
% 3.98/1.92 inference(forward_demodulation,[status(thm)],[f1218,f1165])).
% 3.98/1.92 thf(f1165,plain,(
% 3.98/1.92 ((image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b)) = (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ y @ bot_bot_set_b)))),
% 3.98/1.92 inference(cnf_transformation,[status(thm)],[f1])).
% 3.98/1.92 thf(f1,axiom,(
% 3.98/1.92 ((image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b)) = (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ y @ bot_bot_set_b)))),
% 3.98/1.92 file('/export/starexec/sandbox/tmp/tmp.4auU2Smifg/DTF2THF_20997.p',fact_0__092_060open_062getRel_AS__Idt_AG_A_096_096_A_123x_125_A_061_AgetRel_AS__Idt_AG_A_096_096_A_123y_125_092_060close_062)).
% 3.98/1.92 thf(f1218,plain,(
% 3.98/1.92 ((hilbert_Eps_b @ (^[Y0 : b]: (member_b @ Y0 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b))))) != (hilbert_Eps_b @ (^[Y0 : b]: (member_b @ Y0 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ y @ bot_bot_set_b))))))),
% 3.98/1.92 inference(cnf_transformation,[status(thm)],[f966])).
% 3.98/1.92 thf(f966,plain,(
% 3.98/1.92 ((hilbert_Eps_b @ (^[Y0 : b]: (member_b @ Y0 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b))))) != (hilbert_Eps_b @ (^[Y0 : b]: (member_b @ Y0 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ y @ bot_bot_set_b))))))),
% 3.98/1.92 inference(flattening,[status(thm)],[f429])).
% 3.98/1.92 thf(f429,plain,(
% 3.98/1.92 ~((hilbert_Eps_b @ (^[Y0 : b]: (member_b @ Y0 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b))))) = (hilbert_Eps_b @ (^[Y0 : b]: (member_b @ Y0 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ y @ bot_bot_set_b))))))),
% 3.98/1.92 inference(fool_elimination,[status(thm)],[f428])).
% 3.98/1.92 thf(f428,plain,(
% 3.98/1.92 ~((hilbert_Eps_b @ (^[X0 : b] : (member_b @ X0 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ y @ bot_bot_set_b))))) = (hilbert_Eps_b @ (^[X1 : b] : (member_b @ X1 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b))))))),
% 3.98/1.92 inference(rectify,[status(thm)],[f360])).
% 3.98/1.92 thf(f360,negated_conjecture,(
% 3.98/1.92 ~((hilbert_Eps_b @ (^[X2 : b] : (member_b @ X2 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ y @ bot_bot_set_b))))) = (hilbert_Eps_b @ (^[X2 : b] : (member_b @ X2 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b))))))),
% 3.98/1.92 inference(negated_conjecture,[status(cth)],[f359])).
% 3.98/1.92 thf(f359,conjecture,(
% 3.98/1.92 ((hilbert_Eps_b @ (^[X2 : b] : (member_b @ X2 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ y @ bot_bot_set_b))))) = (hilbert_Eps_b @ (^[X2 : b] : (member_b @ X2 @ (image_b_b @ (getRel904497637nt_a_b @ standard_S_Idt_a @ g) @ (insert_b @ x @ bot_bot_set_b))))))),
% 3.98/1.92 file('/export/starexec/sandbox/tmp/tmp.4auU2Smifg/DTF2THF_20997.p',conj_0)).
% 3.98/1.92 % SZS output end Proof for DTF2THF_20997
% 3.98/1.92 % (21114)------------------------------
% 3.98/1.92 % (21114)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.98/1.92 % (21114)Termination reason: Refutation
% 3.98/1.92
% 3.98/1.92 % (21114)Memory used [KB]: 6780
% 3.98/1.92 % (21114)Time elapsed: 0.041 s
% 3.98/1.92 % (21114)Instructions burned: 67 (million)
% 3.98/1.92 % (21114)------------------------------
% 3.98/1.92 % (21114)------------------------------
% 3.98/1.92 % (21107)Success in time 0.081 s
%------------------------------------------------------------------------------