%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP175^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n019.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 08:56:09 AM UTC 2026
% Result : Theorem 3.27s 1.67s
% Output : Refutation 3.27s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 3
% Syntax : Number of formulae : 18 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 68 ( 18 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 188 ( 10 ~; 3 |; 8 &; 164 @)
% ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Number of types : 77 ( 76 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 222 ( 219 usr; 21 con; 0-3 aty)
% Number of variables : 10 ( 0 ^; 10 !; 0 ?; 10 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_Product_prod_b_b: $tType ).
thf(type_def_6,type,
b: $tType ).
thf(type_def_7,type,
set_b: $tType ).
thf(type_def_9,type,
set_Pr1463111367_set_b: $tType ).
thf(type_def_10,type,
product_prod_b_b: $tType ).
thf(type_def_11,type,
set_Pr1912306449_a_nat: $tType ).
thf(type_def_12,type,
set_Pr2074180935_a_nat: $tType ).
thf(type_def_13,type,
set_Pr2007082183od_b_b: $tType ).
thf(type_def_14,type,
produc598811825_a_nat: $tType ).
thf(type_def_15,type,
set_se480333169_a_nat: $tType ).
thf(type_def_16,type,
set_se668355239od_b_b: $tType ).
thf(type_def_17,type,
set_set_b: $tType ).
thf(type_def_18,type,
a: $tType ).
thf(type_def_19,type,
labele553847424_a_nat: $tType ).
thf(type_def_20,type,
labele1586099759_a_nat: $tType ).
thf(type_def_21,type,
labele488176869od_b_b: $tType ).
thf(type_def_22,type,
labeled_graph_a_b: $tType ).
thf(type_def_23,type,
allegorical_term_a: $tType ).
thf(type_def_24,type,
labeled_graph_a_nat: $tType ).
thf(type_def_25,type,
set_Pr2106913242_nat_b: $tType ).
thf(type_def_26,type,
set_Pr1914572181_nat_b: $tType ).
thf(type_def_27,type,
set_Pr21610864_a_nat: $tType ).
thf(type_def_28,type,
set_Pr1900477680od_b_b: $tType ).
thf(type_def_29,type,
set_la1108361502_a_nat: $tType ).
thf(type_def_30,type,
set_Pr1370504391_a_nat: $tType ).
thf(type_def_31,type,
set_Pr1870121716_a_nat: $tType ).
thf(type_def_32,type,
set_Pr1223907581_a_nat: $tType ).
thf(type_def_33,type,
set_Pr471017666_nat_b: $tType ).
thf(type_def_34,type,
set_Pr2028709629od_b_b: $tType ).
thf(type_def_35,type,
set_Pr1549208893od_b_b: $tType ).
thf(type_def_36,type,
set_Pr357419743_b_b_b: $tType ).
thf(type_def_37,type,
set_Pr1391123143od_b_b: $tType ).
thf(type_def_38,type,
set_Pr1271557767od_b_b: $tType ).
thf(type_def_39,type,
set_Pr2010644300_a_nat: $tType ).
thf(type_def_40,type,
set_Pr2054759953_a_nat: $tType ).
thf(type_def_41,type,
set_Pr1626978257_a_nat: $tType ).
thf(type_def_42,type,
set_Pr352593785_a_nat: $tType ).
thf(type_def_43,type,
set_Pr2072721159_a_nat: $tType ).
thf(type_def_44,type,
set_Pr621705391od_b_b: $tType ).
thf(type_def_45,type,
set_Pr1229594311od_b_b: $tType ).
thf(type_def_46,type,
set_Pr406163591od_b_b: $tType ).
thf(type_def_47,type,
set_Pr454079687od_b_b: $tType ).
thf(type_def_48,type,
set_Pr963356295od_b_b: $tType ).
thf(type_def_49,type,
set_Pr1045675335_a_nat: $tType ).
thf(type_def_50,type,
produc297983550_a_nat: $tType ).
thf(type_def_51,type,
produc1248251020_nat_b: $tType ).
thf(type_def_52,type,
produc1099860071_a_nat: $tType ).
thf(type_def_53,type,
produc1428649885_a_nat: $tType ).
thf(type_def_54,type,
produc1734972647_a_nat: $tType ).
thf(type_def_55,type,
produc588599965_a_nat: $tType ).
thf(type_def_56,type,
produc759315249_a_nat: $tType ).
thf(type_def_57,type,
produc1168037607od_b_b: $tType ).
thf(type_def_58,type,
produc130774173od_b_b: $tType ).
thf(type_def_59,type,
produc1529836775od_b_b: $tType ).
thf(type_def_60,type,
produc300546791od_b_b: $tType ).
thf(type_def_61,type,
produc1230257585_a_nat: $tType ).
thf(type_def_62,type,
produc1554726631od_b_b: $tType ).
thf(type_def_63,type,
produc1921243879od_b_b: $tType ).
thf(type_def_64,type,
produc1063300186_a_nat: $tType ).
thf(type_def_65,type,
produc1133639312od_b_b: $tType ).
thf(type_def_66,type,
produc401211613od_b_b: $tType ).
thf(type_def_67,type,
produc1614181008od_b_b: $tType ).
thf(type_def_68,type,
produc1114535079od_b_b: $tType ).
thf(type_def_69,type,
produc776320369_a_nat: $tType ).
thf(type_def_70,type,
produc1942261264_a_nat: $tType ).
thf(type_def_71,type,
produc2106791463_a_nat: $tType ).
thf(type_def_72,type,
produc1053464231od_b_b: $tType ).
thf(type_def_73,type,
produc197046951od_b_b: $tType ).
thf(type_def_74,type,
produc2112523263_b_b_b: $tType ).
thf(type_def_75,type,
produc1869574247_a_nat: $tType ).
thf(type_def_76,type,
produc51693287od_b_b: $tType ).
thf(type_def_77,type,
produc150548199_set_b: $tType ).
thf(type_def_78,type,
produc126142486_a_nat: $tType ).
thf(type_def_79,type,
produc207618585_a_nat: $tType ).
thf(type_def_80,type,
produc255402447od_b_b: $tType ).
thf(type_def_81,type,
set_Pr773647559od_b_b: $tType ).
thf(func_def_0,type,
set_Pr1223907581_a_nat: $tType ).
thf(func_def_1,type,
produc588599965_a_nat: $tType ).
thf(func_def_2,type,
set_Pr1370504391_a_nat: $tType ).
thf(func_def_3,type,
produc1734972647_a_nat: $tType ).
thf(func_def_4,type,
produc1428649885_a_nat: $tType ).
thf(func_def_5,type,
set_Pr2072721159_a_nat: $tType ).
thf(func_def_6,type,
produc2106791463_a_nat: $tType ).
thf(func_def_7,type,
set_Pr2054759953_a_nat: $tType ).
thf(func_def_8,type,
set_Pr2028709629od_b_b: $tType ).
thf(func_def_9,type,
set_Pr21610864_a_nat: $tType ).
thf(func_def_10,type,
set_Pr1045675335_a_nat: $tType ).
thf(func_def_11,type,
produc1230257585_a_nat: $tType ).
thf(func_def_12,type,
produc130774173od_b_b: $tType ).
thf(func_def_13,type,
produc1942261264_a_nat: $tType ).
thf(func_def_14,type,
produc1869574247_a_nat: $tType ).
thf(func_def_15,type,
set_Pr1229594311od_b_b: $tType ).
thf(func_def_16,type,
set_Pr1391123143od_b_b: $tType ).
thf(func_def_17,type,
set_Pr2074180935_a_nat: $tType ).
thf(func_def_18,type,
produc1554726631od_b_b: $tType ).
thf(func_def_19,type,
produc300546791od_b_b: $tType ).
thf(func_def_20,type,
produc1099860071_a_nat: $tType ).
thf(func_def_21,type,
set_Pr1626978257_a_nat: $tType ).
thf(func_def_22,type,
set_Pr1549208893od_b_b: $tType ).
thf(func_def_23,type,
produc776320369_a_nat: $tType ).
thf(func_def_24,type,
produc401211613od_b_b: $tType ).
thf(func_def_25,type,
set_Pr406163591od_b_b: $tType ).
thf(func_def_26,type,
set_Pr1271557767od_b_b: $tType ).
thf(func_def_27,type,
set_Pr454079687od_b_b: $tType ).
thf(func_def_28,type,
produc1053464231od_b_b: $tType ).
thf(func_def_29,type,
produc1114535079od_b_b: $tType ).
thf(func_def_30,type,
produc1921243879od_b_b: $tType ).
thf(func_def_31,type,
set_Pr1870121716_a_nat: $tType ).
thf(func_def_32,type,
produc297983550_a_nat: $tType ).
thf(func_def_33,type,
produc759315249_a_nat: $tType ).
thf(func_def_34,type,
set_Pr963356295od_b_b: $tType ).
thf(func_def_35,type,
set_Pr352593785_a_nat: $tType ).
thf(func_def_36,type,
set_Pr1914572181_nat_b: $tType ).
thf(func_def_37,type,
labele1586099759_a_nat: $tType ).
thf(func_def_38,type,
produc1529836775od_b_b: $tType ).
thf(func_def_39,type,
produc197046951od_b_b: $tType ).
thf(func_def_40,type,
produc207618585_a_nat: $tType ).
thf(func_def_41,type,
produc1063300186_a_nat: $tType ).
thf(func_def_42,type,
set_se480333169_a_nat: $tType ).
thf(func_def_43,type,
set_Pr1912306449_a_nat: $tType ).
thf(func_def_44,type,
set_Pr773647559od_b_b: $tType ).
thf(func_def_45,type,
produc1614181008od_b_b: $tType ).
thf(func_def_46,type,
produc598811825_a_nat: $tType ).
thf(func_def_47,type,
produc51693287od_b_b: $tType ).
thf(func_def_48,type,
set_Pr2007082183od_b_b: $tType ).
thf(func_def_49,type,
produc1168037607od_b_b: $tType ).
thf(func_def_50,type,
set_Pr2010644300_a_nat: $tType ).
thf(func_def_51,type,
set_Pr471017666_nat_b: $tType ).
thf(func_def_52,type,
labele553847424_a_nat: $tType ).
thf(func_def_53,type,
produc126142486_a_nat: $tType ).
thf(func_def_54,type,
produc1248251020_nat_b: $tType ).
thf(func_def_55,type,
set_Pr621705391od_b_b: $tType ).
thf(func_def_56,type,
set_Pr1900477680od_b_b: $tType ).
thf(func_def_57,type,
set_Pr357419743_b_b_b: $tType ).
thf(func_def_58,type,
set_Pr1463111367_set_b: $tType ).
thf(func_def_59,type,
labele488176869od_b_b: $tType ).
thf(func_def_60,type,
produc255402447od_b_b: $tType ).
thf(func_def_61,type,
produc1133639312od_b_b: $tType ).
thf(func_def_62,type,
produc2112523263_b_b_b: $tType ).
thf(func_def_63,type,
set_la1108361502_a_nat: $tType ).
thf(func_def_64,type,
set_se668355239od_b_b: $tType ).
thf(func_def_65,type,
produc150548199_set_b: $tType ).
thf(func_def_66,type,
set_Pr2106913242_nat_b: $tType ).
thf(func_def_67,type,
labeled_graph_a_nat: $tType ).
thf(func_def_68,type,
set_Product_prod_b_b: $tType ).
thf(func_def_69,type,
allegorical_term_a: $tType ).
thf(func_def_70,type,
labeled_graph_a_b: $tType ).
thf(func_def_71,type,
product_prod_b_b: $tType ).
thf(func_def_72,type,
set_set_b: $tType ).
thf(func_def_73,type,
set_b: $tType ).
thf(func_def_74,type,
b: $tType ).
thf(func_def_75,type,
a: $tType ).
thf(func_def_76,type,
bNF_Gr345950468_set_b: set_Product_prod_b_b > ( b > set_b ) > set_Pr1463111367_set_b ).
thf(func_def_77,type,
bNF_Gr1693995344od_b_b: ( b > product_prod_b_b ) > set_b > product_prod_b_b ).
thf(func_def_78,type,
equiv_1778431934od_b_b: set_Product_prod_b_b > ( b > product_prod_b_b ) > $o ).
thf(func_def_79,type,
equiv_281634663_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o ).
thf(func_def_80,type,
equiv_1125628061od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_81,type,
equiv_equiv_b: set_b > set_Product_prod_b_b > $o ).
thf(func_def_82,type,
equiv_1873824970_a_nat: set_Pr2074180935_a_nat > produc598811825_a_nat > set_Pr1912306449_a_nat ).
thf(func_def_83,type,
equiv_270653002od_b_b: set_Pr2007082183od_b_b > product_prod_b_b > set_Product_prod_b_b ).
thf(func_def_84,type,
equiv_proj_b_b: set_Product_prod_b_b > b > set_b ).
thf(func_def_85,type,
equiv_1338033912_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > set_se480333169_a_nat ).
thf(func_def_86,type,
equiv_1530858158od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > set_se668355239od_b_b ).
thf(func_def_87,type,
equiv_quotient_b: set_b > set_Product_prod_b_b > set_set_b ).
thf(func_def_88,type,
getRel1657740669_a_nat: a > labele553847424_a_nat > set_Pr1912306449_a_nat ).
thf(func_def_89,type,
getRel1248627292_a_nat: a > labele1586099759_a_nat > set_Pr2074180935_a_nat ).
thf(func_def_90,type,
getRel1183868946od_b_b: a > labele488176869od_b_b > set_Pr2007082183od_b_b ).
thf(func_def_91,type,
getRel_a_b: a > labeled_graph_a_b > set_Product_prod_b_b ).
thf(func_def_92,type,
semantics_a_b: labeled_graph_a_b > allegorical_term_a > set_Product_prod_b_b ).
thf(func_def_93,type,
graph_914848022_nat_b: labeled_graph_a_nat > labeled_graph_a_b > set_Pr2106913242_nat_b > $o ).
thf(func_def_94,type,
graph_23275493_nat_b: labele1586099759_a_nat > labeled_graph_a_b > set_Pr1914572181_nat_b > $o ).
thf(func_def_95,type,
graph_1322245399_a_b_b: labeled_graph_a_b > labeled_graph_a_b > set_Product_prod_b_b > $o ).
thf(func_def_96,type,
graph_union_a_b: labeled_graph_a_b > labeled_graph_a_b > labeled_graph_a_b ).
thf(func_def_97,type,
labele1171788421_a_nat: set_Pr21610864_a_nat > set_Pr1912306449_a_nat > labele1586099759_a_nat ).
thf(func_def_98,type,
labeled_LG_a_b: set_Pr1900477680od_b_b > set_b > labeled_graph_a_b ).
thf(func_def_99,type,
labeled_edges_a_b: labeled_graph_a_b > set_Pr1900477680od_b_b ).
thf(func_def_100,type,
labele953802518_a_nat: labele553847424_a_nat > set_la1108361502_a_nat ).
thf(func_def_101,type,
labele198153475_a_nat: labele1586099759_a_nat > set_Pr1912306449_a_nat ).
thf(func_def_102,type,
labele240831545od_b_b: labele488176869od_b_b > set_Product_prod_b_b ).
thf(func_def_103,type,
labeled_vertices_a_b: labeled_graph_a_b > set_b ).
thf(func_def_104,type,
on_tri232494965_a_nat: set_Pr1912306449_a_nat > set_Pr1370504391_a_nat ).
thf(func_def_105,type,
on_tri1766439058_a_nat: set_Pr1870121716_a_nat > set_Pr1223907581_a_nat ).
thf(func_def_106,type,
on_tri1706508026od_b_b: set_Pr471017666_nat_b > set_Pr2028709629od_b_b ).
thf(func_def_107,type,
on_tri1993240081at_b_a: set_Pr471017666_nat_b > set_Pr1549208893od_b_b ).
thf(func_def_108,type,
on_tri97791909od_b_b: set_Pr357419743_b_b_b > set_Pr1391123143od_b_b ).
thf(func_def_109,type,
on_tri1764442556_b_b_a: set_Pr357419743_b_b_b > set_Pr1271557767od_b_b ).
thf(func_def_110,type,
on_tri1837931688od_b_b: set_Pr2010644300_a_nat > set_Pr2054759953_a_nat ).
thf(func_def_111,type,
on_tri2117183935_nat_a: set_Pr2010644300_a_nat > set_Pr1626978257_a_nat ).
thf(func_def_112,type,
on_tri1794757026_nat_a: set_Pr352593785_a_nat > set_Pr2072721159_a_nat ).
thf(func_def_113,type,
on_tri927008213od_b_b: set_Pr621705391od_b_b > set_Pr1229594311od_b_b ).
thf(func_def_114,type,
on_tri2003683308_b_b_a: set_Pr621705391od_b_b > set_Pr406163591od_b_b ).
thf(func_def_115,type,
on_tri111564221od_b_b: set_Product_prod_b_b > set_Pr454079687od_b_b ).
thf(func_def_116,type,
on_triple_b_b_a: set_Product_prod_b_b > set_Pr963356295od_b_b ).
thf(func_def_117,type,
restri1808752350_a_nat: labele553847424_a_nat > labele553847424_a_nat ).
thf(func_def_118,type,
restri147273531_a_nat: labele1586099759_a_nat > labele1586099759_a_nat ).
thf(func_def_119,type,
restri431340145od_b_b: labele488176869od_b_b > labele488176869od_b_b ).
thf(func_def_120,type,
restrict_a_b: labeled_graph_a_b > labeled_graph_a_b ).
thf(func_def_121,type,
idempo848975736_a_nat: set_Pr1912306449_a_nat > $o ).
thf(func_def_122,type,
idempo365306145_a_nat: set_Pr2074180935_a_nat > $o ).
thf(func_def_123,type,
idempo371692375od_b_b: set_Pr2007082183od_b_b > $o ).
thf(func_def_124,type,
idempotent_b: set_Product_prod_b_b > $o ).
thf(func_def_125,type,
order_1770871027_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat > $o ).
thf(func_def_126,type,
order_1458491494_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o ).
thf(func_def_127,type,
order_653178932r_on_b: set_b > set_Product_prod_b_b > $o ).
thf(func_def_128,type,
order_989282449_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat > $o ).
thf(func_def_129,type,
order_145089608_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o ).
thf(func_def_130,type,
order_265664679od_b_b: set_Pr1900477680od_b_b > set_Pr963356295od_b_b > $o ).
thf(func_def_131,type,
order_1195372542od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_132,type,
order_640573096_a_nat: set_se480333169_a_nat > set_Pr1045675335_a_nat > $o ).
thf(func_def_133,type,
order_preorder_on_b: set_b > set_Product_prod_b_b > $o ).
thf(func_def_134,type,
bot_bo186023328_nat_o: produc598811825_a_nat > $o ).
thf(func_def_135,type,
bot_bo1302443370_b_b_o: product_prod_b_b > $o ).
thf(func_def_136,type,
bot_bo1119737266_a_nat: set_la1108361502_a_nat ).
thf(func_def_137,type,
bot_bo2066531197_a_nat: set_Pr1912306449_a_nat ).
thf(func_def_138,type,
bot_bo1803599795_a_nat: set_Pr2074180935_a_nat ).
thf(func_def_139,type,
bot_bo1936722433_nat_b: set_Pr1914572181_nat_b ).
thf(func_def_140,type,
bot_bo896299827_a_nat: set_Pr1370504391_a_nat ).
thf(func_def_141,type,
bot_bo1856908531od_b_b: set_Pr963356295od_b_b ).
thf(func_def_142,type,
bot_bo1174586163od_b_b: set_Pr2007082183od_b_b ).
thf(func_def_143,type,
bot_bo2142478259_a_nat: set_Pr1045675335_a_nat ).
thf(func_def_144,type,
bot_bo1463793628_a_nat: set_Pr21610864_a_nat ).
thf(func_def_145,type,
bot_bo519661404od_b_b: set_Pr1900477680od_b_b ).
thf(func_def_146,type,
bot_bo1343651123od_b_b: set_Product_prod_b_b ).
thf(func_def_147,type,
bot_bo1746593245_a_nat: set_se480333169_a_nat ).
thf(func_def_148,type,
bot_bot_set_b: set_b ).
thf(func_def_149,type,
ord_le1374266033_a_nat: set_Pr1912306449_a_nat > set_Pr1912306449_a_nat > $o ).
thf(func_def_150,type,
ord_le1484132922_nat_b: set_Pr2106913242_nat_b > set_Pr2106913242_nat_b > $o ).
thf(func_def_151,type,
ord_le2142668391_set_b: set_Pr1463111367_set_b > set_Pr1463111367_set_b > $o ).
thf(func_def_152,type,
ord_le887919760od_b_b: set_Pr1900477680od_b_b > set_Pr1900477680od_b_b > $o ).
thf(func_def_153,type,
ord_le1036320359od_b_b: set_Product_prod_b_b > set_Product_prod_b_b > $o ).
thf(func_def_154,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(func_def_155,type,
produc2000589545_a_nat: labeled_graph_a_nat > labeled_graph_a_nat > produc598811825_a_nat ).
thf(func_def_156,type,
produc2092742640_a_nat: labeled_graph_a_nat > produc598811825_a_nat > produc297983550_a_nat ).
thf(func_def_157,type,
produc1600512702_nat_b: labeled_graph_a_nat > b > produc1248251020_nat_b ).
thf(func_def_158,type,
produc410388695_a_nat: produc598811825_a_nat > produc598811825_a_nat > produc1099860071_a_nat ).
thf(func_def_159,type,
produc639906189_a_nat: produc598811825_a_nat > produc1099860071_a_nat > produc1428649885_a_nat ).
thf(func_def_160,type,
produc404908887_a_nat: produc1099860071_a_nat > produc1099860071_a_nat > produc1734972647_a_nat ).
thf(func_def_161,type,
produc1645191821_a_nat: produc1099860071_a_nat > produc1428649885_a_nat > produc588599965_a_nat ).
thf(func_def_162,type,
produc926922637od_b_b: produc759315249_a_nat > produc1168037607od_b_b > produc130774173od_b_b ).
thf(func_def_163,type,
produc885982551od_b_b: produc1529836775od_b_b > produc1168037607od_b_b > produc300546791od_b_b ).
thf(func_def_164,type,
produc1783652257_a_nat: produc1168037607od_b_b > produc759315249_a_nat > produc1230257585_a_nat ).
thf(func_def_165,type,
produc962089303od_b_b: produc1168037607od_b_b > produc1529836775od_b_b > produc1554726631od_b_b ).
thf(func_def_166,type,
produc1403229015od_b_b: produc1168037607od_b_b > produc1168037607od_b_b > produc1921243879od_b_b ).
thf(func_def_167,type,
produc179745357od_b_b: produc1063300186_a_nat > produc1133639312od_b_b > produc401211613od_b_b ).
thf(func_def_168,type,
produc1116925335od_b_b: produc1614181008od_b_b > produc1133639312od_b_b > produc1114535079od_b_b ).
thf(func_def_169,type,
produc966067681_a_nat: produc1133639312od_b_b > produc1063300186_a_nat > produc776320369_a_nat ).
thf(func_def_170,type,
produc797892631_a_nat: produc1133639312od_b_b > produc1942261264_a_nat > produc2106791463_a_nat ).
thf(func_def_171,type,
produc789452695od_b_b: produc1133639312od_b_b > produc1614181008od_b_b > produc1053464231od_b_b ).
thf(func_def_172,type,
produc1037687191od_b_b: produc1133639312od_b_b > produc1133639312od_b_b > produc197046951od_b_b ).
thf(func_def_173,type,
produc90796321_a_nat: product_prod_b_b > produc598811825_a_nat > produc759315249_a_nat ).
thf(func_def_174,type,
produc1850071895od_b_b: product_prod_b_b > produc1168037607od_b_b > produc1529836775od_b_b ).
thf(func_def_175,type,
produc546497367od_b_b: product_prod_b_b > product_prod_b_b > produc1168037607od_b_b ).
thf(func_def_176,type,
produc1001682799_b_b_b: product_prod_b_b > b > produc2112523263_b_b_b ).
thf(func_def_177,type,
produc1767291095_a_nat: set_Pr1912306449_a_nat > set_Pr1912306449_a_nat > produc1869574247_a_nat ).
thf(func_def_178,type,
produc1027169111od_b_b: set_Product_prod_b_b > set_Product_prod_b_b > produc51693287od_b_b ).
thf(func_def_179,type,
produc1403713879_set_b: set_b > set_b > produc150548199_set_b ).
thf(func_def_180,type,
produc1110849866_a_nat: a > produc598811825_a_nat > produc1063300186_a_nat ).
thf(func_def_181,type,
produc1904940160_a_nat: a > produc1099860071_a_nat > produc1942261264_a_nat ).
thf(func_def_182,type,
produc690925568od_b_b: a > produc1168037607od_b_b > produc1614181008od_b_b ).
thf(func_def_183,type,
produc2135284224od_b_b: a > product_prod_b_b > produc1133639312od_b_b ).
thf(func_def_184,type,
produc1311571856_a_nat: b > labeled_graph_a_nat > produc126142486_a_nat ).
thf(func_def_185,type,
produc255168265_a_nat: b > produc598811825_a_nat > produc207618585_a_nat ).
thf(func_def_186,type,
produc1257047359od_b_b: b > product_prod_b_b > produc255402447od_b_b ).
thf(func_def_187,type,
product_Pair_b_b: b > b > product_prod_b_b ).
thf(func_def_188,type,
id_on_1401723275_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat ).
thf(func_def_189,type,
id_on_1649591758_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat ).
thf(func_def_190,type,
id_on_2019020932od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b ).
thf(func_def_191,type,
id_on_set_b: set_set_b > set_Pr1463111367_set_b ).
thf(func_def_192,type,
id_on_b: set_b > set_Product_prod_b_b ).
thf(func_def_193,type,
refl_o1018941609_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat > $o ).
thf(func_def_194,type,
refl_o206400048_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o ).
thf(func_def_195,type,
refl_o2093460070_a_nat: set_Pr2074180935_a_nat > set_Pr1370504391_a_nat > $o ).
thf(func_def_196,type,
refl_o1941611494od_b_b: set_Pr2007082183od_b_b > set_Pr454079687od_b_b > $o ).
thf(func_def_197,type,
refl_o686103183od_b_b: set_Pr1900477680od_b_b > set_Pr963356295od_b_b > $o ).
thf(func_def_198,type,
refl_o2134473190od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_199,type,
refl_o730481808_a_nat: set_se480333169_a_nat > set_Pr1045675335_a_nat > $o ).
thf(func_def_200,type,
refl_o1899808198od_b_b: set_se668355239od_b_b > set_Pr773647559od_b_b > $o ).
thf(func_def_201,type,
refl_on_set_b: set_set_b > set_Pr1463111367_set_b > $o ).
thf(func_def_202,type,
refl_on_b: set_b > set_Product_prod_b_b > $o ).
thf(func_def_203,type,
sym_b: set_Product_prod_b_b > $o ).
thf(func_def_204,type,
total_6094180_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat > $o ).
thf(func_def_205,type,
total_2145009205_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o ).
thf(func_def_206,type,
total_178717291od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_207,type,
total_on_b: set_b > set_Product_prod_b_b > $o ).
thf(func_def_208,type,
trans_1002055497_a_nat: set_Pr1912306449_a_nat > $o ).
thf(func_def_209,type,
trans_444496784_a_nat: set_Pr2074180935_a_nat > $o ).
thf(func_def_210,type,
trans_1808857926od_b_b: set_Pr2007082183od_b_b > $o ).
thf(func_def_211,type,
trans_b: set_Product_prod_b_b > $o ).
thf(func_def_212,type,
conseq1058228089_nat_b: set_Pr1912306449_a_nat > labeled_graph_a_b > $o ).
thf(func_def_213,type,
extensible_a_nat_b: produc598811825_a_nat > labeled_graph_a_b > set_Pr2106913242_nat_b > $o ).
thf(func_def_214,type,
fin_ma603645735_nat_b: produc598811825_a_nat > labeled_graph_a_b > $o ).
thf(func_def_215,type,
mainta535344367_a_nat: produc598811825_a_nat > labele553847424_a_nat > $o ).
thf(func_def_216,type,
mainta1627462506_a_nat: produc598811825_a_nat > labele1586099759_a_nat > $o ).
thf(func_def_217,type,
mainta552012832od_b_b: produc598811825_a_nat > labele488176869od_b_b > $o ).
thf(func_def_218,type,
maintained_a_nat_b: produc598811825_a_nat > labeled_graph_a_b > $o ).
thf(func_def_219,type,
collec978177436_a_nat: ( produc598811825_a_nat > $o ) > set_Pr1912306449_a_nat ).
thf(func_def_220,type,
collec1231052114od_b_b: ( produc1168037607od_b_b > $o ) > set_Pr2007082183od_b_b ).
thf(func_def_221,type,
collec1481886546od_b_b: ( product_prod_b_b > $o ) > set_Product_prod_b_b ).
thf(func_def_222,type,
collec971438076_a_nat: ( set_Pr1912306449_a_nat > $o ) > set_se480333169_a_nat ).
thf(func_def_223,type,
collec1543202610od_b_b: ( set_Product_prod_b_b > $o ) > set_se668355239od_b_b ).
thf(func_def_224,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(func_def_225,type,
collect_b: ( b > $o ) > set_b ).
thf(func_def_226,type,
insert1337853464_a_nat: labeled_graph_a_nat > set_la1108361502_a_nat > set_la1108361502_a_nat ).
thf(func_def_227,type,
insert2058869377_a_nat: produc598811825_a_nat > set_Pr1912306449_a_nat > set_Pr1912306449_a_nat ).
thf(func_def_228,type,
insert354391607_a_nat: produc1099860071_a_nat > set_Pr2074180935_a_nat > set_Pr2074180935_a_nat ).
thf(func_def_229,type,
insert1772835511_a_nat: produc1734972647_a_nat > set_Pr1370504391_a_nat > set_Pr1370504391_a_nat ).
thf(func_def_230,type,
insert428421239od_b_b: produc197046951od_b_b > set_Pr963356295od_b_b > set_Pr963356295od_b_b ).
thf(func_def_231,type,
insert1098249399od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > set_Pr2007082183od_b_b ).
thf(func_def_232,type,
insert885128759_a_nat: produc1869574247_a_nat > set_Pr1045675335_a_nat > set_Pr1045675335_a_nat ).
thf(func_def_233,type,
insert1655137376od_b_b: produc1133639312od_b_b > set_Pr1900477680od_b_b > set_Pr1900477680od_b_b ).
thf(func_def_234,type,
insert1952693431od_b_b: product_prod_b_b > set_Product_prod_b_b > set_Product_prod_b_b ).
thf(func_def_235,type,
insert1220735201_a_nat: set_Pr1912306449_a_nat > set_se480333169_a_nat > set_se480333169_a_nat ).
thf(func_def_236,type,
insert_b: b > set_b > set_b ).
thf(func_def_237,type,
is_emp1506632323_a_nat: set_Pr1912306449_a_nat > $o ).
thf(func_def_238,type,
is_sin1787449765_a_nat: set_Pr1912306449_a_nat > $o ).
thf(func_def_239,type,
is_sin1713057243od_b_b: set_Product_prod_b_b > $o ).
thf(func_def_240,type,
the_el537460006_a_nat: set_Pr1912306449_a_nat > produc598811825_a_nat ).
thf(func_def_241,type,
standa414753934rule_a: a > a > produc598811825_a_nat ).
thf(func_def_242,type,
standa879863265rule_a: produc598811825_a_nat ).
thf(func_def_243,type,
standa1329480012rule_a: a > produc598811825_a_nat ).
thf(func_def_244,type,
standa709637829rule_a: a > produc598811825_a_nat ).
thf(func_def_245,type,
standa390703484rule_a: a > produc598811825_a_nat ).
thf(func_def_246,type,
same_fst_b_b: ( b > $o ) > ( b > set_Product_prod_b_b ) > set_Pr2007082183od_b_b ).
thf(func_def_247,type,
member154916863_a_nat: labeled_graph_a_nat > set_la1108361502_a_nat > $o ).
thf(func_def_248,type,
member327031898_a_nat: produc598811825_a_nat > set_Pr1912306449_a_nat > $o ).
thf(func_def_249,type,
member2103088853_a_nat: produc297983550_a_nat > set_Pr1870121716_a_nat > $o ).
thf(func_def_250,type,
member1885627811_nat_b: produc1248251020_nat_b > set_Pr471017666_nat_b > $o ).
thf(func_def_251,type,
member1542696848_a_nat: produc1099860071_a_nat > set_Pr2074180935_a_nat > $o ).
thf(func_def_252,type,
member121756432_a_nat: produc1734972647_a_nat > set_Pr1370504391_a_nat > $o ).
thf(func_def_253,type,
member428378182_a_nat: produc588599965_a_nat > set_Pr1223907581_a_nat > $o ).
thf(func_def_254,type,
member670041670od_b_b: produc130774173od_b_b > set_Pr2028709629od_b_b > $o ).
thf(func_def_255,type,
member178477328od_b_b: produc300546791od_b_b > set_Pr1391123143od_b_b > $o ).
thf(func_def_256,type,
member1769525082_a_nat: produc1230257585_a_nat > set_Pr2054759953_a_nat > $o ).
thf(func_def_257,type,
member1432657168od_b_b: produc1554726631od_b_b > set_Pr1229594311od_b_b > $o ).
thf(func_def_258,type,
member1829919504od_b_b: produc1921243879od_b_b > set_Pr454079687od_b_b > $o ).
thf(func_def_259,type,
member442676358od_b_b: produc401211613od_b_b > set_Pr1549208893od_b_b > $o ).
thf(func_def_260,type,
member1023210704od_b_b: produc1114535079od_b_b > set_Pr1271557767od_b_b > $o ).
thf(func_def_261,type,
member817785114_a_nat: produc776320369_a_nat > set_Pr1626978257_a_nat > $o ).
thf(func_def_262,type,
member1379268944_a_nat: produc2106791463_a_nat > set_Pr2072721159_a_nat > $o ).
thf(func_def_263,type,
member962139856od_b_b: produc1053464231od_b_b > set_Pr406163591od_b_b > $o ).
thf(func_def_264,type,
member1694913744od_b_b: produc197046951od_b_b > set_Pr963356295od_b_b > $o ).
thf(func_def_265,type,
member1652792080od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > $o ).
thf(func_def_266,type,
member351494440_b_b_b: produc2112523263_b_b_b > set_Pr357419743_b_b_b > $o ).
thf(func_def_267,type,
member1072844688_a_nat: produc1869574247_a_nat > set_Pr1045675335_a_nat > $o ).
thf(func_def_268,type,
member1009730320od_b_b: produc51693287od_b_b > set_Pr773647559od_b_b > $o ).
thf(func_def_269,type,
member1199885584_set_b: produc150548199_set_b > set_Pr1463111367_set_b > $o ).
thf(func_def_270,type,
member763519277_a_nat: produc126142486_a_nat > set_Pr2010644300_a_nat > $o ).
thf(func_def_271,type,
member692218050_a_nat: produc207618585_a_nat > set_Pr352593785_a_nat > $o ).
thf(func_def_272,type,
member641857272od_b_b: produc255402447od_b_b > set_Pr621705391od_b_b > $o ).
thf(func_def_273,type,
member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
thf(func_def_274,type,
member101535418_a_nat: set_Pr1912306449_a_nat > set_se480333169_a_nat > $o ).
thf(func_def_275,type,
member1050118384od_b_b: set_Product_prod_b_b > set_se668355239od_b_b > $o ).
thf(func_def_276,type,
member_set_b: set_b > set_set_b > $o ).
thf(func_def_277,type,
member_b: b > set_b > $o ).
thf(func_def_278,type,
g: labeled_graph_a_b ).
thf(func_def_279,type,
i: a ).
thf(func_def_280,type,
l: a ).
thf(func_def_281,type,
y: b ).
thf(func_def_282,type,
z: b ).
thf(func_def_298,type,
sK0: ( product_prod_b_b > $o ) > b ).
thf(func_def_299,type,
sK1: ( product_prod_b_b > $o ) > b ).
thf(func_def_300,type,
sK2: product_prod_b_b > b ).
thf(func_def_301,type,
sK3: product_prod_b_b > b ).
thf(func_def_302,type,
sK4: ( product_prod_b_b > $o ) > b ).
thf(func_def_303,type,
sK5: ( product_prod_b_b > $o ) > b ).
thf(func_def_304,type,
sK6: product_prod_b_b > b ).
thf(func_def_305,type,
sK7: product_prod_b_b > b ).
thf(func_def_306,type,
sK8: set_Product_prod_b_b > b ).
thf(func_def_307,type,
sK9: set_Product_prod_b_b > b ).
thf(func_def_308,type,
sK10: set_Product_prod_b_b > b ).
thf(func_def_310,type,
ph12:
!>[X0: $tType] : X0 ).
thf(f1074,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1070,f1056]) ).
thf(f1056,plain,
( ( member1285940496od_b_b @ ( product_Pair_b_b @ y @ z ) @ ( getRel_a_b @ i @ g ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f638]) ).
thf(f638,plain,
( ( member1285940496od_b_b @ ( product_Pair_b_b @ y @ z ) @ ( getRel_a_b @ i @ g ) )
= $true ),
inference(fool_elimination,[status(thm)],[f637]) ).
thf(f637,plain,
member1285940496od_b_b @ ( product_Pair_b_b @ y @ z ) @ ( getRel_a_b @ i @ g ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
member1285940496od_b_b @ ( product_Pair_b_b @ y @ z ) @ ( getRel_a_b @ i @ g ),
file('/export/starexec/sandbox2/tmp/tmp.2AfYOHPK5t/DTF2THF_16406.p',fact_1_aI) ).
thf(f1070,plain,
( ( member1285940496od_b_b @ ( product_Pair_b_b @ y @ z ) @ ( getRel_a_b @ i @ g ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1067]) ).
thf(f1067,plain,
( ( ( member1285940496od_b_b @ ( product_Pair_b_b @ y @ z ) @ ( getRel_a_b @ i @ g ) )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1024,f1050]) ).
thf(f1050,plain,
! [X0: b,X1: b] :
( ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X0 @ X1 ) @ ( getRel_a_b @ i @ g ) )
= $true )
| ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X1 @ X0 ) @ ( getRel_a_b @ i @ g ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f994]) ).
thf(f994,plain,
( ( ( trans_b @ ( getRel_a_b @ i @ g ) )
= $true )
& ( ( refl_on_b @ ( labeled_vertices_a_b @ g ) @ ( getRel_a_b @ i @ g ) )
= $true )
& ! [X0: b,X1: b] :
( ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X1 @ X0 ) @ ( getRel_a_b @ i @ g ) )
!= $true )
| ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X0 @ X1 ) @ ( getRel_a_b @ i @ g ) )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f928]) ).
thf(f928,plain,
( ( ( refl_on_b @ ( labeled_vertices_a_b @ g ) @ ( getRel_a_b @ i @ g ) )
= $true )
& ! [X0: b,X1: b] :
( ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X1 @ X0 ) @ ( getRel_a_b @ i @ g ) )
= $true )
=> ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X0 @ X1 ) @ ( getRel_a_b @ i @ g ) )
= $true ) )
& ( ( trans_b @ ( getRel_a_b @ i @ g ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f927]) ).
thf(f927,plain,
( ( trans_b @ ( getRel_a_b @ i @ g ) )
& ! [X0: b,X1: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X1 @ X0 ) @ ( getRel_a_b @ i @ g ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X0 @ X1 ) @ ( getRel_a_b @ i @ g ) ) )
& ( refl_on_b @ ( labeled_vertices_a_b @ g ) @ ( getRel_a_b @ i @ g ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( trans_b @ ( getRel_a_b @ i @ g ) )
& ! [X1: b,X0: b] :
( ( member1285940496od_b_b @ ( product_Pair_b_b @ X0 @ X1 ) @ ( getRel_a_b @ i @ g ) )
=> ( member1285940496od_b_b @ ( product_Pair_b_b @ X1 @ X0 ) @ ( getRel_a_b @ i @ g ) ) )
& ( refl_on_b @ ( labeled_vertices_a_b @ g ) @ ( getRel_a_b @ i @ g ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.2AfYOHPK5t/DTF2THF_16406.p',fact_0__092_060open_062refl__on_A_Ivertices_AG_J_A_IgetRel_AI_AG_J_A_092_060and_062_A_I_092_060forall_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_AgetRel_AI_AG_A_092_060longrightarrow_062_A_Iy_M_Ax_J_A_092_060in_062_AgetRel_AI_AG_J_A_092_060and_062_Atrans_A_IgetRel_AI_AG_J_092_060close_062) ).
thf(f1024,plain,
( ( member1285940496od_b_b @ ( product_Pair_b_b @ z @ y ) @ ( getRel_a_b @ i @ g ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f979]) ).
thf(f979,plain,
( ( member1285940496od_b_b @ ( product_Pair_b_b @ z @ y ) @ ( getRel_a_b @ i @ g ) )
!= $true ),
inference(flattening,[status(thm)],[f398]) ).
thf(f398,plain,
( ( member1285940496od_b_b @ ( product_Pair_b_b @ z @ y ) @ ( getRel_a_b @ i @ g ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f397]) ).
thf(f397,plain,
~ ( member1285940496od_b_b @ ( product_Pair_b_b @ z @ y ) @ ( getRel_a_b @ i @ g ) ),
inference(rectify,[status(thm)],[f357]) ).
thf(f357,negated_conjecture,
~ ( member1285940496od_b_b @ ( product_Pair_b_b @ z @ y ) @ ( getRel_a_b @ i @ g ) ),
inference(negated_conjecture,[status(cth)],[f356]) ).
thf(f356,conjecture,
member1285940496od_b_b @ ( product_Pair_b_b @ z @ y ) @ ( getRel_a_b @ i @ g ),
file('/export/starexec/sandbox2/tmp/tmp.2AfYOHPK5t/DTF2THF_16406.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP175^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n019.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:28:54 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.47 thf(spec,logic,$$dhol).
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % File : ITP175^1 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.47 % Domain : Interactive Theorem Proving
% 0.20/0.47 % Problem : Sledgehammer StandardRules problem prob_201__5389132_1
% 0.20/0.47 % Version : Especial.
% 0.20/0.47 % English :
% 0.20/0.47
% 0.20/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.47 % Source : [Des21]
% 0.20/0.47 % Names : StandardRules/prob_201__5389132_1 [Des21]
% 0.20/0.47
% 0.20/0.47 % Status : Theorem
% 0.20/0.47 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.20/0.47 % Syntax : Number of formulae : 639 ( 162 unt; 283 typ; 0 def)
% 0.20/0.47 % Number of atoms : 976 ( 325 equ; 0 cnn)
% 0.20/0.47 % Maximal formula atoms : 7 ( 2 avg)
% 0.20/0.47 % Number of connectives : 3193 ( 85 ~; 8 |; 70 &;2641 @)
% 0.20/0.47 % ( 0 <=>; 389 =>; 0 <=; 0 <~>)
% 0.20/0.47 % Maximal formula depth : 17 ( 7 avg)
% 0.20/0.47 % Number of types : 77 ( 76 usr)
% 0.20/0.47 % Number of type conns : 370 ( 370 >; 0 *; 0 +; 0 <<)
% 0.20/0.47 % Number of symbols : 208 ( 207 usr; 19 con; 0-3 aty)
% 0.20/0.47 % Number of variables : 1046 ( 33 ^; 996 !; 17 ?;1046 :)
% 0.20/0.47 % SPC : TH0_THM_EQU_NAR
% 0.20/0.47
% 0.20/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:37:20.179
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 % Could-be-implicit typings (76)
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% 0.20/0.47 member641857272od_b_b: produc255402447od_b_b > set_Pr621705391od_b_b > $o ).
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% 0.20/0.47 member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J,type,
% 0.20/0.47 member101535418_a_nat: set_Pr1912306449_a_nat > set_se480333169_a_nat > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
% 0.20/0.47 member1050118384od_b_b: set_Product_prod_b_b > set_se668355239od_b_b > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
% 0.20/0.47 member_set_b: set_b > set_set_b > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_c_member_001tf__b,type,
% 0.20/0.47 member_b: b > set_b > $o ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_G,type,
% 0.20/0.47 g: labeled_graph_a_b ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_I,type,
% 0.20/0.47 i: a ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_l,type,
% 0.20/0.47 l: a ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_y____,type,
% 0.20/0.47 y: b ).
% 0.20/0.47
% 0.20/0.47 thf(sy_v_z____,type,
% 0.20/0.47 z: b ).
% 0.20/0.47
% 0.20/0.47 % Relevant facts (355)
% 0.20/0.47 thf(fact_0__092_060open_062refl__on_A_Ivertices_AG_J_A_IgetRel_AI_AG_J_A_092_060and_062_A_I_092_060forall_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_AgetRel_AI_AG_A_092_060longrightarrow_062_A_Iy_M_Ax_J_A_092_060in_062_AgetRel_AI_AG_J_A_092_060and_062_Atrans_A_IgetRel_AI_AG_J_092_060close_062,axiom,
% 0.20/0.47 ( ( refl_on_b @ ( labeled_vertices_a_b @ g ) @ ( getRel_a_b @ i @ g ) )
% 0.20/0.47 & ! [X: b,Y: b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X @ Y ) @ ( getRel_a_b @ i @ g ) )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ Y @ X ) @ ( getRel_a_b @ i @ g ) ) )
% 0.20/0.47 & ( trans_b @ ( getRel_a_b @ i @ g ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % \<open>refl_on (vertices G) (getRel I G) \<and> (\<forall>x y. (x, y) \<in> getRel I G \<longrightarrow> (y, x) \<in> getRel I G) \<and> trans (getRel I G)\<close>
% 0.20/0.47 thf(fact_1_aI,axiom,
% 0.20/0.47 member1285940496od_b_b @ ( product_Pair_b_b @ y @ z ) @ ( getRel_a_b @ i @ g ) ).
% 0.20/0.47
% 0.20/0.47 % aI
% 0.20/0.47 thf(fact_2_g,axiom,
% 0.20/0.47 ( g
% 0.20/0.47 = ( restrict_a_b @ g ) ) ).
% 0.20/0.47
% 0.20/0.47 % g
% 0.20/0.47 thf(fact_3_m,axiom,
% 0.20/0.47 maintained_a_nat_b @ ( standa414753934rule_a @ i @ l ) @ g ).
% 0.20/0.47
% 0.20/0.47 % m
% 0.20/0.47 thf(fact_4_eq,axiom,
% 0.20/0.47 equiv_equiv_b @ ( labeled_vertices_a_b @ g ) @ ( getRel_a_b @ i @ g ) ).
% 0.20/0.47
% 0.20/0.47 % eq
% 0.20/0.47 thf(fact_5_prod_Oinject,axiom,
% 0.20/0.47 ! [X1: labeled_graph_a_nat,X2: labeled_graph_a_nat,Y1: labeled_graph_a_nat,Y2: labeled_graph_a_nat] :
% 0.20/0.47 ( ( ( produc2000589545_a_nat @ X1 @ X2 )
% 0.20/0.47 = ( produc2000589545_a_nat @ Y1 @ Y2 ) )
% 0.20/0.47 = ( ( X1 = Y1 )
% 0.20/0.47 & ( X2 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.inject
% 0.20/0.47 thf(fact_6_prod_Oinject,axiom,
% 0.20/0.47 ! [X1: product_prod_b_b,X2: product_prod_b_b,Y1: product_prod_b_b,Y2: product_prod_b_b] :
% 0.20/0.47 ( ( ( produc546497367od_b_b @ X1 @ X2 )
% 0.20/0.47 = ( produc546497367od_b_b @ Y1 @ Y2 ) )
% 0.20/0.47 = ( ( X1 = Y1 )
% 0.20/0.47 & ( X2 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.inject
% 0.20/0.47 thf(fact_7_prod_Oinject,axiom,
% 0.20/0.47 ! [X1: produc598811825_a_nat,X2: produc598811825_a_nat,Y1: produc598811825_a_nat,Y2: produc598811825_a_nat] :
% 0.20/0.47 ( ( ( produc410388695_a_nat @ X1 @ X2 )
% 0.20/0.47 = ( produc410388695_a_nat @ Y1 @ Y2 ) )
% 0.20/0.47 = ( ( X1 = Y1 )
% 0.20/0.47 & ( X2 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.inject
% 0.20/0.47 thf(fact_8_prod_Oinject,axiom,
% 0.20/0.47 ! [X1: a,X2: product_prod_b_b,Y1: a,Y2: product_prod_b_b] :
% 0.20/0.47 ( ( ( produc2135284224od_b_b @ X1 @ X2 )
% 0.20/0.47 = ( produc2135284224od_b_b @ Y1 @ Y2 ) )
% 0.20/0.47 = ( ( X1 = Y1 )
% 0.20/0.47 & ( X2 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.inject
% 0.20/0.47 thf(fact_9_prod_Oinject,axiom,
% 0.20/0.47 ! [X1: b,X2: b,Y1: b,Y2: b] :
% 0.20/0.47 ( ( ( product_Pair_b_b @ X1 @ X2 )
% 0.20/0.47 = ( product_Pair_b_b @ Y1 @ Y2 ) )
% 0.20/0.47 = ( ( X1 = Y1 )
% 0.20/0.47 & ( X2 = Y2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod.inject
% 0.20/0.47 thf(fact_10_old_Oprod_Oinject,axiom,
% 0.20/0.47 ! [A: labeled_graph_a_nat,B: labeled_graph_a_nat,A2: labeled_graph_a_nat,B2: labeled_graph_a_nat] :
% 0.20/0.47 ( ( ( produc2000589545_a_nat @ A @ B )
% 0.20/0.47 = ( produc2000589545_a_nat @ A2 @ B2 ) )
% 0.20/0.47 = ( ( A = A2 )
% 0.20/0.47 & ( B = B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inject
% 0.20/0.47 thf(fact_11_old_Oprod_Oinject,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,B: product_prod_b_b,A2: product_prod_b_b,B2: product_prod_b_b] :
% 0.20/0.47 ( ( ( produc546497367od_b_b @ A @ B )
% 0.20/0.47 = ( produc546497367od_b_b @ A2 @ B2 ) )
% 0.20/0.47 = ( ( A = A2 )
% 0.20/0.47 & ( B = B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inject
% 0.20/0.47 thf(fact_12_old_Oprod_Oinject,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,A2: produc598811825_a_nat,B2: produc598811825_a_nat] :
% 0.20/0.47 ( ( ( produc410388695_a_nat @ A @ B )
% 0.20/0.47 = ( produc410388695_a_nat @ A2 @ B2 ) )
% 0.20/0.47 = ( ( A = A2 )
% 0.20/0.47 & ( B = B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inject
% 0.20/0.47 thf(fact_13_old_Oprod_Oinject,axiom,
% 0.20/0.47 ! [A: a,B: product_prod_b_b,A2: a,B2: product_prod_b_b] :
% 0.20/0.47 ( ( ( produc2135284224od_b_b @ A @ B )
% 0.20/0.47 = ( produc2135284224od_b_b @ A2 @ B2 ) )
% 0.20/0.47 = ( ( A = A2 )
% 0.20/0.47 & ( B = B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inject
% 0.20/0.47 thf(fact_14_old_Oprod_Oinject,axiom,
% 0.20/0.47 ! [A: b,B: b,A2: b,B2: b] :
% 0.20/0.47 ( ( ( product_Pair_b_b @ A @ B )
% 0.20/0.47 = ( product_Pair_b_b @ A2 @ B2 ) )
% 0.20/0.47 = ( ( A = A2 )
% 0.20/0.47 & ( B = B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inject
% 0.20/0.47 thf(fact_15_surj__pair,axiom,
% 0.20/0.47 ! [P: produc598811825_a_nat] :
% 0.20/0.47 ? [X3: labeled_graph_a_nat,Y3: labeled_graph_a_nat] :
% 0.20/0.47 ( P
% 0.20/0.47 = ( produc2000589545_a_nat @ X3 @ Y3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_pair
% 0.20/0.47 thf(fact_16_surj__pair,axiom,
% 0.20/0.47 ! [P: produc1168037607od_b_b] :
% 0.20/0.47 ? [X3: product_prod_b_b,Y3: product_prod_b_b] :
% 0.20/0.47 ( P
% 0.20/0.47 = ( produc546497367od_b_b @ X3 @ Y3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_pair
% 0.20/0.47 thf(fact_17_surj__pair,axiom,
% 0.20/0.47 ! [P: produc1099860071_a_nat] :
% 0.20/0.47 ? [X3: produc598811825_a_nat,Y3: produc598811825_a_nat] :
% 0.20/0.47 ( P
% 0.20/0.47 = ( produc410388695_a_nat @ X3 @ Y3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_pair
% 0.20/0.47 thf(fact_18_surj__pair,axiom,
% 0.20/0.47 ! [P: produc1133639312od_b_b] :
% 0.20/0.47 ? [X3: a,Y3: product_prod_b_b] :
% 0.20/0.47 ( P
% 0.20/0.47 = ( produc2135284224od_b_b @ X3 @ Y3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_pair
% 0.20/0.47 thf(fact_19_surj__pair,axiom,
% 0.20/0.47 ! [P: product_prod_b_b] :
% 0.20/0.47 ? [X3: b,Y3: b] :
% 0.20/0.47 ( P
% 0.20/0.47 = ( product_Pair_b_b @ X3 @ Y3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % surj_pair
% 0.20/0.47 thf(fact_20_prod__cases,axiom,
% 0.20/0.47 ! [P2: produc598811825_a_nat > $o,P: produc598811825_a_nat] :
% 0.20/0.47 ( ! [A3: labeled_graph_a_nat,B3: labeled_graph_a_nat] : ( P2 @ ( produc2000589545_a_nat @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ P ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases
% 0.20/0.47 thf(fact_21_prod__cases,axiom,
% 0.20/0.47 ! [P2: produc1168037607od_b_b > $o,P: produc1168037607od_b_b] :
% 0.20/0.47 ( ! [A3: product_prod_b_b,B3: product_prod_b_b] : ( P2 @ ( produc546497367od_b_b @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ P ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases
% 0.20/0.47 thf(fact_22_prod__cases,axiom,
% 0.20/0.47 ! [P2: produc1099860071_a_nat > $o,P: produc1099860071_a_nat] :
% 0.20/0.47 ( ! [A3: produc598811825_a_nat,B3: produc598811825_a_nat] : ( P2 @ ( produc410388695_a_nat @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ P ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases
% 0.20/0.47 thf(fact_23_prod__cases,axiom,
% 0.20/0.47 ! [P2: produc1133639312od_b_b > $o,P: produc1133639312od_b_b] :
% 0.20/0.47 ( ! [A3: a,B3: product_prod_b_b] : ( P2 @ ( produc2135284224od_b_b @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ P ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases
% 0.20/0.47 thf(fact_24_prod__cases,axiom,
% 0.20/0.47 ! [P2: product_prod_b_b > $o,P: product_prod_b_b] :
% 0.20/0.47 ( ! [A3: b,B3: b] : ( P2 @ ( product_Pair_b_b @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ P ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases
% 0.20/0.47 thf(fact_25_Pair__inject,axiom,
% 0.20/0.47 ! [A: labeled_graph_a_nat,B: labeled_graph_a_nat,A2: labeled_graph_a_nat,B2: labeled_graph_a_nat] :
% 0.20/0.47 ( ( ( produc2000589545_a_nat @ A @ B )
% 0.20/0.47 = ( produc2000589545_a_nat @ A2 @ B2 ) )
% 0.20/0.47 => ~ ( ( A = A2 )
% 0.20/0.47 => ( B != B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Pair_inject
% 0.20/0.47 thf(fact_26_Pair__inject,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,B: product_prod_b_b,A2: product_prod_b_b,B2: product_prod_b_b] :
% 0.20/0.47 ( ( ( produc546497367od_b_b @ A @ B )
% 0.20/0.47 = ( produc546497367od_b_b @ A2 @ B2 ) )
% 0.20/0.47 => ~ ( ( A = A2 )
% 0.20/0.47 => ( B != B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Pair_inject
% 0.20/0.47 thf(fact_27_Pair__inject,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,A2: produc598811825_a_nat,B2: produc598811825_a_nat] :
% 0.20/0.47 ( ( ( produc410388695_a_nat @ A @ B )
% 0.20/0.47 = ( produc410388695_a_nat @ A2 @ B2 ) )
% 0.20/0.47 => ~ ( ( A = A2 )
% 0.20/0.47 => ( B != B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Pair_inject
% 0.20/0.47 thf(fact_28_Pair__inject,axiom,
% 0.20/0.47 ! [A: a,B: product_prod_b_b,A2: a,B2: product_prod_b_b] :
% 0.20/0.47 ( ( ( produc2135284224od_b_b @ A @ B )
% 0.20/0.47 = ( produc2135284224od_b_b @ A2 @ B2 ) )
% 0.20/0.47 => ~ ( ( A = A2 )
% 0.20/0.47 => ( B != B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Pair_inject
% 0.20/0.47 thf(fact_29_Pair__inject,axiom,
% 0.20/0.47 ! [A: b,B: b,A2: b,B2: b] :
% 0.20/0.47 ( ( ( product_Pair_b_b @ A @ B )
% 0.20/0.47 = ( product_Pair_b_b @ A2 @ B2 ) )
% 0.20/0.47 => ~ ( ( A = A2 )
% 0.20/0.47 => ( B != B2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Pair_inject
% 0.20/0.47 thf(fact_30_prod__cases3,axiom,
% 0.20/0.47 ! [Y4: produc1168037607od_b_b] :
% 0.20/0.47 ~ ! [A3: product_prod_b_b,B3: b,C: b] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( produc546497367od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases3
% 0.20/0.47 thf(fact_31_prod__cases3,axiom,
% 0.20/0.47 ! [Y4: produc1099860071_a_nat] :
% 0.20/0.47 ~ ! [A3: produc598811825_a_nat,B3: labeled_graph_a_nat,C: labeled_graph_a_nat] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( produc410388695_a_nat @ A3 @ ( produc2000589545_a_nat @ B3 @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases3
% 0.20/0.47 thf(fact_32_prod__cases3,axiom,
% 0.20/0.47 ! [Y4: produc1133639312od_b_b] :
% 0.20/0.47 ~ ! [A3: a,B3: b,C: b] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( produc2135284224od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_cases3
% 0.20/0.47 thf(fact_33_old_Oprod_Oinducts,axiom,
% 0.20/0.47 ! [P2: produc598811825_a_nat > $o,Prod: produc598811825_a_nat] :
% 0.20/0.47 ( ! [A3: labeled_graph_a_nat,B3: labeled_graph_a_nat] : ( P2 @ ( produc2000589545_a_nat @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ Prod ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inducts
% 0.20/0.47 thf(fact_34_old_Oprod_Oinducts,axiom,
% 0.20/0.47 ! [P2: produc1168037607od_b_b > $o,Prod: produc1168037607od_b_b] :
% 0.20/0.47 ( ! [A3: product_prod_b_b,B3: product_prod_b_b] : ( P2 @ ( produc546497367od_b_b @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ Prod ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inducts
% 0.20/0.47 thf(fact_35_old_Oprod_Oinducts,axiom,
% 0.20/0.47 ! [P2: produc1099860071_a_nat > $o,Prod: produc1099860071_a_nat] :
% 0.20/0.47 ( ! [A3: produc598811825_a_nat,B3: produc598811825_a_nat] : ( P2 @ ( produc410388695_a_nat @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ Prod ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inducts
% 0.20/0.47 thf(fact_36_old_Oprod_Oinducts,axiom,
% 0.20/0.47 ! [P2: produc1133639312od_b_b > $o,Prod: produc1133639312od_b_b] :
% 0.20/0.47 ( ! [A3: a,B3: product_prod_b_b] : ( P2 @ ( produc2135284224od_b_b @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ Prod ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inducts
% 0.20/0.47 thf(fact_37_old_Oprod_Oinducts,axiom,
% 0.20/0.47 ! [P2: product_prod_b_b > $o,Prod: product_prod_b_b] :
% 0.20/0.47 ( ! [A3: b,B3: b] : ( P2 @ ( product_Pair_b_b @ A3 @ B3 ) )
% 0.20/0.47 => ( P2 @ Prod ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.inducts
% 0.20/0.47 thf(fact_38_old_Oprod_Oexhaust,axiom,
% 0.20/0.47 ! [Y4: produc598811825_a_nat] :
% 0.20/0.47 ~ ! [A3: labeled_graph_a_nat,B3: labeled_graph_a_nat] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( produc2000589545_a_nat @ A3 @ B3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.exhaust
% 0.20/0.47 thf(fact_39_old_Oprod_Oexhaust,axiom,
% 0.20/0.47 ! [Y4: produc1168037607od_b_b] :
% 0.20/0.47 ~ ! [A3: product_prod_b_b,B3: product_prod_b_b] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( produc546497367od_b_b @ A3 @ B3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.exhaust
% 0.20/0.47 thf(fact_40_old_Oprod_Oexhaust,axiom,
% 0.20/0.47 ! [Y4: produc1099860071_a_nat] :
% 0.20/0.47 ~ ! [A3: produc598811825_a_nat,B3: produc598811825_a_nat] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( produc410388695_a_nat @ A3 @ B3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.exhaust
% 0.20/0.47 thf(fact_41_old_Oprod_Oexhaust,axiom,
% 0.20/0.47 ! [Y4: produc1133639312od_b_b] :
% 0.20/0.47 ~ ! [A3: a,B3: product_prod_b_b] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( produc2135284224od_b_b @ A3 @ B3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.exhaust
% 0.20/0.47 thf(fact_42_old_Oprod_Oexhaust,axiom,
% 0.20/0.47 ! [Y4: product_prod_b_b] :
% 0.20/0.47 ~ ! [A3: b,B3: b] :
% 0.20/0.47 ( Y4
% 0.20/0.47 != ( product_Pair_b_b @ A3 @ B3 ) ) ).
% 0.20/0.47
% 0.20/0.47 % old.prod.exhaust
% 0.20/0.47 thf(fact_43_prod__induct3,axiom,
% 0.20/0.47 ! [P2: produc1168037607od_b_b > $o,X4: produc1168037607od_b_b] :
% 0.20/0.47 ( ! [A3: product_prod_b_b,B3: b,C: b] : ( P2 @ ( produc546497367od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) )
% 0.20/0.47 => ( P2 @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct3
% 0.20/0.47 thf(fact_44_prod__induct3,axiom,
% 0.20/0.47 ! [P2: produc1099860071_a_nat > $o,X4: produc1099860071_a_nat] :
% 0.20/0.47 ( ! [A3: produc598811825_a_nat,B3: labeled_graph_a_nat,C: labeled_graph_a_nat] : ( P2 @ ( produc410388695_a_nat @ A3 @ ( produc2000589545_a_nat @ B3 @ C ) ) )
% 0.20/0.47 => ( P2 @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct3
% 0.20/0.47 thf(fact_45_prod__induct3,axiom,
% 0.20/0.47 ! [P2: produc1133639312od_b_b > $o,X4: produc1133639312od_b_b] :
% 0.20/0.47 ( ! [A3: a,B3: b,C: b] : ( P2 @ ( produc2135284224od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) )
% 0.20/0.47 => ( P2 @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % prod_induct3
% 0.20/0.47 thf(fact_46_vertices__restrict,axiom,
% 0.20/0.47 ! [G: labeled_graph_a_b] :
% 0.20/0.47 ( ( labeled_vertices_a_b @ ( restrict_a_b @ G ) )
% 0.20/0.47 = ( labeled_vertices_a_b @ G ) ) ).
% 0.20/0.47
% 0.20/0.47 % vertices_restrict
% 0.20/0.47 thf(fact_47_getRel__dom_I1_J,axiom,
% 0.20/0.47 ! [G: labeled_graph_a_b,A: b,B: b,L: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restrict_a_b @ G ) )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel_a_b @ L @ G ) )
% 0.20/0.47 => ( member_b @ A @ ( labeled_vertices_a_b @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % getRel_dom(1)
% 0.20/0.47 thf(fact_48_getRel__dom_I2_J,axiom,
% 0.20/0.47 ! [G: labeled_graph_a_b,A: b,B: b,L: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restrict_a_b @ G ) )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel_a_b @ L @ G ) )
% 0.20/0.47 => ( member_b @ B @ ( labeled_vertices_a_b @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % getRel_dom(2)
% 0.20/0.47 thf(fact_49_transitive__rule,axiom,
% 0.20/0.47 ! [G: labele553847424_a_nat,L: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restri1808752350_a_nat @ G ) )
% 0.20/0.47 => ( ( mainta535344367_a_nat @ ( standa390703484rule_a @ L ) @ G )
% 0.20/0.47 => ( trans_1002055497_a_nat @ ( getRel1657740669_a_nat @ L @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transitive_rule
% 0.20/0.47 thf(fact_50_transitive__rule,axiom,
% 0.20/0.47 ! [G: labeled_graph_a_b,L: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restrict_a_b @ G ) )
% 0.20/0.47 => ( ( maintained_a_nat_b @ ( standa390703484rule_a @ L ) @ G )
% 0.20/0.47 => ( trans_b @ ( getRel_a_b @ L @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transitive_rule
% 0.20/0.47 thf(fact_51_maintained__refl,axiom,
% 0.20/0.47 ! [R: labeled_graph_a_nat,G: labeled_graph_a_b] : ( maintained_a_nat_b @ ( produc2000589545_a_nat @ R @ R ) @ G ) ).
% 0.20/0.47
% 0.20/0.47 % maintained_refl
% 0.20/0.47 thf(fact_52_reflexivity__rule,axiom,
% 0.20/0.47 ! [G: labele488176869od_b_b,L: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restri431340145od_b_b @ G ) )
% 0.20/0.47 => ( ( mainta552012832od_b_b @ ( standa1329480012rule_a @ L ) @ G )
% 0.20/0.47 => ( refl_o2134473190od_b_b @ ( labele240831545od_b_b @ G ) @ ( getRel1183868946od_b_b @ L @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflexivity_rule
% 0.20/0.47 thf(fact_53_reflexivity__rule,axiom,
% 0.20/0.47 ! [G: labele1586099759_a_nat,L: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restri147273531_a_nat @ G ) )
% 0.20/0.47 => ( ( mainta1627462506_a_nat @ ( standa1329480012rule_a @ L ) @ G )
% 0.20/0.47 => ( refl_o206400048_a_nat @ ( labele198153475_a_nat @ G ) @ ( getRel1248627292_a_nat @ L @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflexivity_rule
% 0.20/0.47 thf(fact_54_reflexivity__rule,axiom,
% 0.20/0.47 ! [G: labele553847424_a_nat,L: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restri1808752350_a_nat @ G ) )
% 0.20/0.47 => ( ( mainta535344367_a_nat @ ( standa1329480012rule_a @ L ) @ G )
% 0.20/0.47 => ( refl_o1018941609_a_nat @ ( labele953802518_a_nat @ G ) @ ( getRel1657740669_a_nat @ L @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflexivity_rule
% 0.20/0.47 thf(fact_55_reflexivity__rule,axiom,
% 0.20/0.47 ! [G: labeled_graph_a_b,L: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restrict_a_b @ G ) )
% 0.20/0.47 => ( ( maintained_a_nat_b @ ( standa1329480012rule_a @ L ) @ G )
% 0.20/0.47 => ( refl_on_b @ ( labeled_vertices_a_b @ G ) @ ( getRel_a_b @ L @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % reflexivity_rule
% 0.20/0.47 thf(fact_56_restrict__idemp,axiom,
% 0.20/0.47 ! [X4: labeled_graph_a_b] :
% 0.20/0.47 ( ( restrict_a_b @ ( restrict_a_b @ X4 ) )
% 0.20/0.47 = ( restrict_a_b @ X4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % restrict_idemp
% 0.20/0.47 thf(fact_57_refl__on__domain,axiom,
% 0.20/0.47 ! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,A: produc1168037607od_b_b,B: produc1168037607od_b_b] :
% 0.20/0.47 ( ( refl_o1941611494od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ A @ B ) @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ A @ A4 )
% 0.20/0.47 & ( member1652792080od_b_b @ B @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_domain
% 0.20/0.47 thf(fact_58_refl__on__domain,axiom,
% 0.20/0.47 ! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,A: set_Pr1912306449_a_nat,B: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( refl_o730481808_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member1072844688_a_nat @ ( produc1767291095_a_nat @ A @ B ) @ R2 )
% 0.20/0.47 => ( ( member101535418_a_nat @ A @ A4 )
% 0.20/0.47 & ( member101535418_a_nat @ B @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_domain
% 0.20/0.47 thf(fact_59_refl__on__domain,axiom,
% 0.20/0.47 ! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
% 0.20/0.47 ( ( refl_o1899808198od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1009730320od_b_b @ ( produc1027169111od_b_b @ A @ B ) @ R2 )
% 0.20/0.47 => ( ( member1050118384od_b_b @ A @ A4 )
% 0.20/0.47 & ( member1050118384od_b_b @ B @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_domain
% 0.20/0.47 thf(fact_60_refl__on__domain,axiom,
% 0.20/0.47 ! [A4: set_set_b,R2: set_Pr1463111367_set_b,A: set_b,B: set_b] :
% 0.20/0.47 ( ( refl_on_set_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1199885584_set_b @ ( produc1403713879_set_b @ A @ B ) @ R2 )
% 0.20/0.47 => ( ( member_set_b @ A @ A4 )
% 0.20/0.47 & ( member_set_b @ B @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_domain
% 0.20/0.47 thf(fact_61_refl__on__domain,axiom,
% 0.20/0.47 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,A: labeled_graph_a_nat,B: labeled_graph_a_nat] :
% 0.20/0.47 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ A @ B ) @ R2 )
% 0.20/0.47 => ( ( member154916863_a_nat @ A @ A4 )
% 0.20/0.47 & ( member154916863_a_nat @ B @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_domain
% 0.20/0.47 thf(fact_62_refl__on__domain,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,A: produc598811825_a_nat,B: produc598811825_a_nat] :
% 0.20/0.47 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ A @ B ) @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ A @ A4 )
% 0.20/0.47 & ( member327031898_a_nat @ B @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_domain
% 0.20/0.47 thf(fact_63_refl__on__domain,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
% 0.20/0.47 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ A @ A4 )
% 0.20/0.47 & ( member1285940496od_b_b @ B @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_domain
% 0.20/0.47 thf(fact_64_refl__on__domain,axiom,
% 0.20/0.47 ! [A4: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
% 0.20/0.47 ( ( refl_on_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 0.20/0.47 => ( ( member_b @ A @ A4 )
% 0.20/0.47 & ( member_b @ B @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_domain
% 0.20/0.47 thf(fact_65_refl__onD2,axiom,
% 0.20/0.47 ! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,X4: produc1168037607od_b_b,Y4: produc1168037607od_b_b] :
% 0.20/0.47 ( ( refl_o1941611494od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member1652792080od_b_b @ Y4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD2
% 0.20/0.47 thf(fact_66_refl__onD2,axiom,
% 0.20/0.47 ! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,X4: set_Pr1912306449_a_nat,Y4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( refl_o730481808_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member1072844688_a_nat @ ( produc1767291095_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member101535418_a_nat @ Y4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD2
% 0.20/0.47 thf(fact_67_refl__onD2,axiom,
% 0.20/0.47 ! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,X4: set_Product_prod_b_b,Y4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( refl_o1899808198od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1009730320od_b_b @ ( produc1027169111od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member1050118384od_b_b @ Y4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD2
% 0.20/0.47 thf(fact_68_refl__onD2,axiom,
% 0.20/0.47 ! [A4: set_set_b,R2: set_Pr1463111367_set_b,X4: set_b,Y4: set_b] :
% 0.20/0.47 ( ( refl_on_set_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1199885584_set_b @ ( produc1403713879_set_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member_set_b @ Y4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD2
% 0.20/0.47 thf(fact_69_refl__onD2,axiom,
% 0.20/0.47 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat] :
% 0.20/0.47 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member154916863_a_nat @ Y4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD2
% 0.20/0.47 thf(fact_70_refl__onD2,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 0.20/0.47 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member327031898_a_nat @ Y4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD2
% 0.20/0.47 thf(fact_71_refl__onD2,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 0.20/0.47 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member1285940496od_b_b @ Y4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD2
% 0.20/0.47 thf(fact_72_refl__onD2,axiom,
% 0.20/0.47 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
% 0.20/0.47 ( ( refl_on_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member_b @ Y4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD2
% 0.20/0.47 thf(fact_73_transD,axiom,
% 0.20/0.47 ! [R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat,Z: labeled_graph_a_nat] :
% 0.20/0.47 ( ( trans_1002055497_a_nat @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transD
% 0.20/0.47 thf(fact_74_transD,axiom,
% 0.20/0.47 ! [R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b,Z: product_prod_b_b] :
% 0.20/0.47 ( ( trans_1808857926od_b_b @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transD
% 0.20/0.47 thf(fact_75_transD,axiom,
% 0.20/0.47 ! [R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat,Z: produc598811825_a_nat] :
% 0.20/0.47 ( ( trans_444496784_a_nat @ R2 )
% 0.20/0.47 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transD
% 0.20/0.47 thf(fact_76_transD,axiom,
% 0.20/0.47 ! [R2: set_Product_prod_b_b,X4: b,Y4: b,Z: b] :
% 0.20/0.47 ( ( trans_b @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transD
% 0.20/0.47 thf(fact_77_transE,axiom,
% 0.20/0.47 ! [R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat,Z: labeled_graph_a_nat] :
% 0.20/0.47 ( ( trans_1002055497_a_nat @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transE
% 0.20/0.47 thf(fact_78_transE,axiom,
% 0.20/0.47 ! [R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b,Z: product_prod_b_b] :
% 0.20/0.47 ( ( trans_1808857926od_b_b @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transE
% 0.20/0.47 thf(fact_79_transE,axiom,
% 0.20/0.47 ! [R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat,Z: produc598811825_a_nat] :
% 0.20/0.47 ( ( trans_444496784_a_nat @ R2 )
% 0.20/0.47 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transE
% 0.20/0.47 thf(fact_80_transE,axiom,
% 0.20/0.47 ! [R2: set_Product_prod_b_b,X4: b,Y4: b,Z: b] :
% 0.20/0.47 ( ( trans_b @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ R2 )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % transE
% 0.20/0.47 thf(fact_81_transI,axiom,
% 0.20/0.47 ! [R2: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ! [X3: labeled_graph_a_nat,Y3: labeled_graph_a_nat,Z2: labeled_graph_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X3 @ Y3 ) @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y3 @ Z2 ) @ R2 )
% 0.20/0.47 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ X3 @ Z2 ) @ R2 ) ) )
% 0.20/0.47 => ( trans_1002055497_a_nat @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % transI
% 0.20/0.47 thf(fact_82_transI,axiom,
% 0.20/0.47 ! [R2: set_Pr2007082183od_b_b] :
% 0.20/0.47 ( ! [X3: product_prod_b_b,Y3: product_prod_b_b,Z2: product_prod_b_b] :
% 0.20/0.47 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X3 @ Y3 ) @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ Z2 ) @ R2 )
% 0.20/0.47 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ X3 @ Z2 ) @ R2 ) ) )
% 0.20/0.47 => ( trans_1808857926od_b_b @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % transI
% 0.20/0.47 thf(fact_83_transI,axiom,
% 0.20/0.47 ! [R2: set_Pr2074180935_a_nat] :
% 0.20/0.47 ( ! [X3: produc598811825_a_nat,Y3: produc598811825_a_nat,Z2: produc598811825_a_nat] :
% 0.20/0.47 ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X3 @ Y3 ) @ R2 )
% 0.20/0.47 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y3 @ Z2 ) @ R2 )
% 0.20/0.47 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X3 @ Z2 ) @ R2 ) ) )
% 0.20/0.47 => ( trans_444496784_a_nat @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % transI
% 0.20/0.47 thf(fact_84_transI,axiom,
% 0.20/0.47 ! [R2: set_Product_prod_b_b] :
% 0.20/0.47 ( ! [X3: b,Y3: b,Z2: b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ Z2 ) @ R2 )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Z2 ) @ R2 ) ) )
% 0.20/0.47 => ( trans_b @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % transI
% 0.20/0.47 thf(fact_85_Relation_Otrans__def,axiom,
% 0.20/0.47 ( trans_1002055497_a_nat
% 0.20/0.47 = ( ^ [R3: set_Pr1912306449_a_nat] :
% 0.20/0.47 ! [X5: labeled_graph_a_nat,Y5: labeled_graph_a_nat,Z3: labeled_graph_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X5 @ Y5 ) @ R3 )
% 0.20/0.47 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y5 @ Z3 ) @ R3 )
% 0.20/0.47 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Relation.trans_def
% 0.20/0.47 thf(fact_86_Relation_Otrans__def,axiom,
% 0.20/0.47 ( trans_1808857926od_b_b
% 0.20/0.47 = ( ^ [R3: set_Pr2007082183od_b_b] :
% 0.20/0.47 ! [X5: product_prod_b_b,Y5: product_prod_b_b,Z3: product_prod_b_b] :
% 0.20/0.47 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X5 @ Y5 ) @ R3 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y5 @ Z3 ) @ R3 )
% 0.20/0.47 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Relation.trans_def
% 0.20/0.47 thf(fact_87_Relation_Otrans__def,axiom,
% 0.20/0.47 ( trans_444496784_a_nat
% 0.20/0.47 = ( ^ [R3: set_Pr2074180935_a_nat] :
% 0.20/0.47 ! [X5: produc598811825_a_nat,Y5: produc598811825_a_nat,Z3: produc598811825_a_nat] :
% 0.20/0.47 ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X5 @ Y5 ) @ R3 )
% 0.20/0.47 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y5 @ Z3 ) @ R3 )
% 0.20/0.47 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Relation.trans_def
% 0.20/0.47 thf(fact_88_Relation_Otrans__def,axiom,
% 0.20/0.47 ( trans_b
% 0.20/0.47 = ( ^ [R3: set_Product_prod_b_b] :
% 0.20/0.47 ! [X5: b,Y5: b,Z3: b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Y5 ) @ R3 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y5 @ Z3 ) @ R3 )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Relation.trans_def
% 0.20/0.47 thf(fact_89_refl__onD,axiom,
% 0.20/0.47 ! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,A: produc1168037607od_b_b] :
% 0.20/0.47 ( ( refl_o1941611494od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ A @ A4 )
% 0.20/0.47 => ( member1829919504od_b_b @ ( produc1403229015od_b_b @ A @ A ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD
% 0.20/0.47 thf(fact_90_refl__onD,axiom,
% 0.20/0.47 ! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,A: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( refl_o730481808_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member101535418_a_nat @ A @ A4 )
% 0.20/0.47 => ( member1072844688_a_nat @ ( produc1767291095_a_nat @ A @ A ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD
% 0.20/0.47 thf(fact_91_refl__onD,axiom,
% 0.20/0.47 ! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,A: set_Product_prod_b_b] :
% 0.20/0.47 ( ( refl_o1899808198od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1050118384od_b_b @ A @ A4 )
% 0.20/0.47 => ( member1009730320od_b_b @ ( produc1027169111od_b_b @ A @ A ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD
% 0.20/0.47 thf(fact_92_refl__onD,axiom,
% 0.20/0.47 ! [A4: set_set_b,R2: set_Pr1463111367_set_b,A: set_b] :
% 0.20/0.47 ( ( refl_on_set_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member_set_b @ A @ A4 )
% 0.20/0.47 => ( member1199885584_set_b @ ( produc1403713879_set_b @ A @ A ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD
% 0.20/0.47 thf(fact_93_refl__onD,axiom,
% 0.20/0.47 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,A: labeled_graph_a_nat] :
% 0.20/0.47 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member154916863_a_nat @ A @ A4 )
% 0.20/0.47 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ A @ A ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD
% 0.20/0.47 thf(fact_94_refl__onD,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,A: produc598811825_a_nat] :
% 0.20/0.47 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ A @ A4 )
% 0.20/0.47 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ A @ A ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD
% 0.20/0.47 thf(fact_95_refl__onD,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
% 0.20/0.47 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ A @ A4 )
% 0.20/0.47 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ A ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD
% 0.20/0.47 thf(fact_96_refl__onD,axiom,
% 0.20/0.47 ! [A4: set_b,R2: set_Product_prod_b_b,A: b] :
% 0.20/0.47 ( ( refl_on_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member_b @ A @ A4 )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ A ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD
% 0.20/0.47 thf(fact_97_refl__onD1,axiom,
% 0.20/0.47 ! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,X4: produc1168037607od_b_b,Y4: produc1168037607od_b_b] :
% 0.20/0.47 ( ( refl_o1941611494od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member1652792080od_b_b @ X4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD1
% 0.20/0.47 thf(fact_98_refl__onD1,axiom,
% 0.20/0.47 ! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,X4: set_Pr1912306449_a_nat,Y4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( refl_o730481808_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member1072844688_a_nat @ ( produc1767291095_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member101535418_a_nat @ X4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD1
% 0.20/0.47 thf(fact_99_refl__onD1,axiom,
% 0.20/0.47 ! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,X4: set_Product_prod_b_b,Y4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( refl_o1899808198od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1009730320od_b_b @ ( produc1027169111od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member1050118384od_b_b @ X4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD1
% 0.20/0.47 thf(fact_100_refl__onD1,axiom,
% 0.20/0.47 ! [A4: set_set_b,R2: set_Pr1463111367_set_b,X4: set_b,Y4: set_b] :
% 0.20/0.47 ( ( refl_on_set_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1199885584_set_b @ ( produc1403713879_set_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member_set_b @ X4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD1
% 0.20/0.47 thf(fact_101_refl__onD1,axiom,
% 0.20/0.47 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat] :
% 0.20/0.47 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member154916863_a_nat @ X4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD1
% 0.20/0.47 thf(fact_102_refl__onD1,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 0.20/0.47 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member327031898_a_nat @ X4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD1
% 0.20/0.47 thf(fact_103_refl__onD1,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 0.20/0.47 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member1285940496od_b_b @ X4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD1
% 0.20/0.47 thf(fact_104_refl__onD1,axiom,
% 0.20/0.47 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
% 0.20/0.47 ( ( refl_on_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.47 => ( member_b @ X4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_onD1
% 0.20/0.47 thf(fact_105_mA,axiom,
% 0.20/0.47 ! [X: produc598811825_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ X @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ i ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ i ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ i ) @ bot_bo2066531197_a_nat ) ) ) )
% 0.20/0.47 => ( maintained_a_nat_b @ X @ g ) ) ).
% 0.20/0.47
% 0.20/0.47 % mA
% 0.20/0.47 thf(fact_106_preorder__on__def,axiom,
% 0.20/0.47 ( order_1195372542od_b_b
% 0.20/0.47 = ( ^ [A5: set_Product_prod_b_b,R3: set_Pr2007082183od_b_b] :
% 0.20/0.47 ( ( refl_o2134473190od_b_b @ A5 @ R3 )
% 0.20/0.47 & ( trans_1808857926od_b_b @ R3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_def
% 0.20/0.47 thf(fact_107_preorder__on__def,axiom,
% 0.20/0.47 ( order_145089608_a_nat
% 0.20/0.47 = ( ^ [A5: set_Pr1912306449_a_nat,R3: set_Pr2074180935_a_nat] :
% 0.20/0.47 ( ( refl_o206400048_a_nat @ A5 @ R3 )
% 0.20/0.47 & ( trans_444496784_a_nat @ R3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_def
% 0.20/0.47 thf(fact_108_preorder__on__def,axiom,
% 0.20/0.47 ( order_989282449_a_nat
% 0.20/0.47 = ( ^ [A5: set_la1108361502_a_nat,R3: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( refl_o1018941609_a_nat @ A5 @ R3 )
% 0.20/0.47 & ( trans_1002055497_a_nat @ R3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_def
% 0.20/0.47 thf(fact_109_preorder__on__def,axiom,
% 0.20/0.47 ( order_preorder_on_b
% 0.20/0.47 = ( ^ [A5: set_b,R3: set_Product_prod_b_b] :
% 0.20/0.47 ( ( refl_on_b @ A5 @ R3 )
% 0.20/0.47 & ( trans_b @ R3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_def
% 0.20/0.47 thf(fact_110_on__triple,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: b,V4: b,R: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1694913744od_b_b @ ( produc1037687191od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_triple_b_b_a @ R ) )
% 0.20/0.47 = ( ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member1285940496od_b_b @ ( product_Pair_b_b @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_111_on__triple,axiom,
% 0.20/0.47 ! [L1: a,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: a,V3: b,V4: b,R: set_Pr471017666_nat_b] :
% 0.20/0.47 ( ( member442676358od_b_b @ ( produc179745357od_b_b @ ( produc1110849866_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1993240081at_b_a @ R ) )
% 0.20/0.47 = ( ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_112_on__triple,axiom,
% 0.20/0.47 ! [L1: a,V1: product_prod_b_b,V2: product_prod_b_b,L2: a,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
% 0.20/0.47 ( ( member1023210704od_b_b @ ( produc1116925335od_b_b @ ( produc690925568od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1764442556_b_b_a @ R ) )
% 0.20/0.47 = ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_113_on__triple,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: b,V4: b,R: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri111564221od_b_b @ R ) )
% 0.20/0.47 = ( ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member1285940496od_b_b @ ( product_Pair_b_b @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_114_on__triple,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 0.20/0.47 ( ( member817785114_a_nat @ ( produc966067681_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1110849866_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri2117183935_nat_a @ R ) )
% 0.20/0.47 = ( ( member763519277_a_nat @ ( produc1311571856_a_nat @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member763519277_a_nat @ ( produc1311571856_a_nat @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_115_on__triple,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 0.20/0.47 ( ( member962139856od_b_b @ ( produc789452695od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc690925568od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri2003683308_b_b_a @ R ) )
% 0.20/0.47 = ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member641857272od_b_b @ ( produc1257047359od_b_b @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_116_on__triple,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr471017666_nat_b] :
% 0.20/0.47 ( ( member670041670od_b_b @ ( produc926922637od_b_b @ ( produc90796321_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1706508026od_b_b @ R ) )
% 0.20/0.47 = ( ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_117_on__triple,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: product_prod_b_b,V2: product_prod_b_b,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
% 0.20/0.47 ( ( member178477328od_b_b @ ( produc885982551od_b_b @ ( produc1850071895od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri97791909od_b_b @ R ) )
% 0.20/0.47 = ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_118_on__triple,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 0.20/0.47 ( ( member1769525082_a_nat @ ( produc1783652257_a_nat @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc90796321_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1837931688od_b_b @ R ) )
% 0.20/0.47 = ( ( member763519277_a_nat @ ( produc1311571856_a_nat @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member763519277_a_nat @ ( produc1311571856_a_nat @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_119_on__triple,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 0.20/0.47 ( ( member1432657168od_b_b @ ( produc962089303od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1850071895od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri927008213od_b_b @ R ) )
% 0.20/0.47 = ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ V1 @ V3 ) @ R )
% 0.20/0.47 & ( member641857272od_b_b @ ( produc1257047359od_b_b @ V2 @ V4 ) @ R )
% 0.20/0.47 & ( L1 = L2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_triple
% 0.20/0.47 thf(fact_120_maintained__then__fin__maintained,axiom,
% 0.20/0.47 ! [R: produc598811825_a_nat,G: labeled_graph_a_b] :
% 0.20/0.47 ( ( maintained_a_nat_b @ R @ G )
% 0.20/0.47 => ( fin_ma603645735_nat_b @ R @ G ) ) ).
% 0.20/0.47
% 0.20/0.47 % maintained_then_fin_maintained
% 0.20/0.47 thf(fact_121_refl__trans__impl__idempotent,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b] :
% 0.20/0.47 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( trans_1808857926od_b_b @ R2 )
% 0.20/0.47 => ( idempo371692375od_b_b @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_trans_impl_idempotent
% 0.20/0.47 thf(fact_122_refl__trans__impl__idempotent,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat] :
% 0.20/0.47 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( trans_444496784_a_nat @ R2 )
% 0.20/0.47 => ( idempo365306145_a_nat @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_trans_impl_idempotent
% 0.20/0.47 thf(fact_123_refl__trans__impl__idempotent,axiom,
% 0.20/0.47 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( trans_1002055497_a_nat @ R2 )
% 0.20/0.47 => ( idempo848975736_a_nat @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_trans_impl_idempotent
% 0.20/0.47 thf(fact_124_refl__trans__impl__idempotent,axiom,
% 0.20/0.47 ! [A4: set_b,R2: set_Product_prod_b_b] :
% 0.20/0.47 ( ( refl_on_b @ A4 @ R2 )
% 0.20/0.47 => ( ( trans_b @ R2 )
% 0.20/0.47 => ( idempotent_b @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_trans_impl_idempotent
% 0.20/0.47 thf(fact_125_equivalence,axiom,
% 0.20/0.47 ! [G: labele1586099759_a_nat,I: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restri147273531_a_nat @ G ) )
% 0.20/0.47 => ( ! [X3: produc598811825_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ X3 @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ I ) @ bot_bo2066531197_a_nat ) ) ) )
% 0.20/0.47 => ( mainta1627462506_a_nat @ X3 @ G ) )
% 0.20/0.47 => ( equiv_281634663_a_nat @ ( labele198153475_a_nat @ G ) @ ( getRel1248627292_a_nat @ I @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equivalence
% 0.20/0.47 thf(fact_126_equivalence,axiom,
% 0.20/0.47 ! [G: labele488176869od_b_b,I: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restri431340145od_b_b @ G ) )
% 0.20/0.47 => ( ! [X3: produc598811825_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ X3 @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ I ) @ bot_bo2066531197_a_nat ) ) ) )
% 0.20/0.47 => ( mainta552012832od_b_b @ X3 @ G ) )
% 0.20/0.47 => ( equiv_1125628061od_b_b @ ( labele240831545od_b_b @ G ) @ ( getRel1183868946od_b_b @ I @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equivalence
% 0.20/0.47 thf(fact_127_equivalence,axiom,
% 0.20/0.47 ! [G: labeled_graph_a_b,I: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restrict_a_b @ G ) )
% 0.20/0.47 => ( ! [X3: produc598811825_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ X3 @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ I ) @ bot_bo2066531197_a_nat ) ) ) )
% 0.20/0.47 => ( maintained_a_nat_b @ X3 @ G ) )
% 0.20/0.47 => ( equiv_equiv_b @ ( labeled_vertices_a_b @ G ) @ ( getRel_a_b @ I @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equivalence
% 0.20/0.47 thf(fact_128_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: produc1168037607od_b_b,P2: produc1168037607od_b_b > $o] :
% 0.20/0.47 ( ( member1652792080od_b_b @ A @ ( collec1231052114od_b_b @ P2 ) )
% 0.20/0.47 = ( P2 @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_129_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: set_Pr1912306449_a_nat,P2: set_Pr1912306449_a_nat > $o] :
% 0.20/0.47 ( ( member101535418_a_nat @ A @ ( collec971438076_a_nat @ P2 ) )
% 0.20/0.47 = ( P2 @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_130_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: set_Product_prod_b_b,P2: set_Product_prod_b_b > $o] :
% 0.20/0.47 ( ( member1050118384od_b_b @ A @ ( collec1543202610od_b_b @ P2 ) )
% 0.20/0.47 = ( P2 @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_131_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: set_b,P2: set_b > $o] :
% 0.20/0.47 ( ( member_set_b @ A @ ( collect_set_b @ P2 ) )
% 0.20/0.47 = ( P2 @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_132_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: b,P2: b > $o] :
% 0.20/0.47 ( ( member_b @ A @ ( collect_b @ P2 ) )
% 0.20/0.47 = ( P2 @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_133_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,P2: produc598811825_a_nat > $o] :
% 0.20/0.47 ( ( member327031898_a_nat @ A @ ( collec978177436_a_nat @ P2 ) )
% 0.20/0.47 = ( P2 @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_134_mem__Collect__eq,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,P2: product_prod_b_b > $o] :
% 0.20/0.47 ( ( member1285940496od_b_b @ A @ ( collec1481886546od_b_b @ P2 ) )
% 0.20/0.47 = ( P2 @ A ) ) ).
% 0.20/0.47
% 0.20/0.47 % mem_Collect_eq
% 0.20/0.47 thf(fact_135_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A4: set_Pr2007082183od_b_b] :
% 0.20/0.47 ( ( collec1231052114od_b_b
% 0.20/0.47 @ ^ [X5: produc1168037607od_b_b] : ( member1652792080od_b_b @ X5 @ A4 ) )
% 0.20/0.47 = A4 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_136_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A4: set_se480333169_a_nat] :
% 0.20/0.47 ( ( collec971438076_a_nat
% 0.20/0.47 @ ^ [X5: set_Pr1912306449_a_nat] : ( member101535418_a_nat @ X5 @ A4 ) )
% 0.20/0.47 = A4 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_137_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A4: set_se668355239od_b_b] :
% 0.20/0.47 ( ( collec1543202610od_b_b
% 0.20/0.47 @ ^ [X5: set_Product_prod_b_b] : ( member1050118384od_b_b @ X5 @ A4 ) )
% 0.20/0.47 = A4 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_138_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A4: set_set_b] :
% 0.20/0.47 ( ( collect_set_b
% 0.20/0.47 @ ^ [X5: set_b] : ( member_set_b @ X5 @ A4 ) )
% 0.20/0.47 = A4 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_139_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A4: set_b] :
% 0.20/0.47 ( ( collect_b
% 0.20/0.47 @ ^ [X5: b] : ( member_b @ X5 @ A4 ) )
% 0.20/0.47 = A4 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_140_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( collec978177436_a_nat
% 0.20/0.47 @ ^ [X5: produc598811825_a_nat] : ( member327031898_a_nat @ X5 @ A4 ) )
% 0.20/0.47 = A4 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_141_Collect__mem__eq,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( collec1481886546od_b_b
% 0.20/0.47 @ ^ [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ A4 ) )
% 0.20/0.47 = A4 ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_mem_eq
% 0.20/0.47 thf(fact_142_Collect__cong,axiom,
% 0.20/0.47 ! [P2: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
% 0.20/0.47 ( ! [X3: product_prod_b_b] :
% 0.20/0.47 ( ( P2 @ X3 )
% 0.20/0.47 = ( Q @ X3 ) )
% 0.20/0.47 => ( ( collec1481886546od_b_b @ P2 )
% 0.20/0.47 = ( collec1481886546od_b_b @ Q ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_cong
% 0.20/0.47 thf(fact_143_Collect__cong,axiom,
% 0.20/0.47 ! [P2: produc598811825_a_nat > $o,Q: produc598811825_a_nat > $o] :
% 0.20/0.47 ( ! [X3: produc598811825_a_nat] :
% 0.20/0.47 ( ( P2 @ X3 )
% 0.20/0.47 = ( Q @ X3 ) )
% 0.20/0.47 => ( ( collec978177436_a_nat @ P2 )
% 0.20/0.47 = ( collec978177436_a_nat @ Q ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_cong
% 0.20/0.47 thf(fact_144_semantics__in__vertices_I1_J,axiom,
% 0.20/0.47 ! [A4: labeled_graph_a_b,A: b,B: b,E: allegorical_term_a] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( restrict_a_b @ A4 ) )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ A4 @ E ) )
% 0.20/0.47 => ( member_b @ A @ ( labeled_vertices_a_b @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % semantics_in_vertices(1)
% 0.20/0.47 thf(fact_145_semantics__in__vertices_I2_J,axiom,
% 0.20/0.47 ! [A4: labeled_graph_a_b,A: b,B: b,E: allegorical_term_a] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( restrict_a_b @ A4 ) )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ A4 @ E ) )
% 0.20/0.47 => ( member_b @ B @ ( labeled_vertices_a_b @ A4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % semantics_in_vertices(2)
% 0.20/0.47 thf(fact_146_refl__on__singleton,axiom,
% 0.20/0.47 ! [X4: produc1099860071_a_nat] : ( refl_o2093460070_a_nat @ ( insert354391607_a_nat @ X4 @ bot_bo1803599795_a_nat ) @ ( insert1772835511_a_nat @ ( produc404908887_a_nat @ X4 @ X4 ) @ bot_bo896299827_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_singleton
% 0.20/0.47 thf(fact_147_refl__on__singleton,axiom,
% 0.20/0.47 ! [X4: product_prod_b_b] : ( refl_o2134473190od_b_b @ ( insert1952693431od_b_b @ X4 @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X4 @ X4 ) @ bot_bo1174586163od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_singleton
% 0.20/0.47 thf(fact_148_refl__on__singleton,axiom,
% 0.20/0.47 ! [X4: produc1133639312od_b_b] : ( refl_o686103183od_b_b @ ( insert1655137376od_b_b @ X4 @ bot_bo519661404od_b_b ) @ ( insert428421239od_b_b @ ( produc1037687191od_b_b @ X4 @ X4 ) @ bot_bo1856908531od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_singleton
% 0.20/0.47 thf(fact_149_refl__on__singleton,axiom,
% 0.20/0.47 ! [X4: set_Pr1912306449_a_nat] : ( refl_o730481808_a_nat @ ( insert1220735201_a_nat @ X4 @ bot_bo1746593245_a_nat ) @ ( insert885128759_a_nat @ ( produc1767291095_a_nat @ X4 @ X4 ) @ bot_bo2142478259_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_singleton
% 0.20/0.47 thf(fact_150_refl__on__singleton,axiom,
% 0.20/0.47 ! [X4: b] : ( refl_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X4 @ X4 ) @ bot_bo1343651123od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_singleton
% 0.20/0.47 thf(fact_151_refl__on__singleton,axiom,
% 0.20/0.47 ! [X4: labeled_graph_a_nat] : ( refl_o1018941609_a_nat @ ( insert1337853464_a_nat @ X4 @ bot_bo1119737266_a_nat ) @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ X4 @ X4 ) @ bot_bo2066531197_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_singleton
% 0.20/0.47 thf(fact_152_refl__on__singleton,axiom,
% 0.20/0.47 ! [X4: produc598811825_a_nat] : ( refl_o206400048_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ X4 @ X4 ) @ bot_bo1803599795_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % refl_on_singleton
% 0.20/0.47 thf(fact_153_preorder__on__empty,axiom,
% 0.20/0.47 order_265664679od_b_b @ bot_bo519661404od_b_b @ bot_bo1856908531od_b_b ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_empty
% 0.20/0.47 thf(fact_154_preorder__on__empty,axiom,
% 0.20/0.47 order_640573096_a_nat @ bot_bo1746593245_a_nat @ bot_bo2142478259_a_nat ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_empty
% 0.20/0.47 thf(fact_155_preorder__on__empty,axiom,
% 0.20/0.47 order_preorder_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_empty
% 0.20/0.47 thf(fact_156_preorder__on__empty,axiom,
% 0.20/0.47 order_989282449_a_nat @ bot_bo1119737266_a_nat @ bot_bo2066531197_a_nat ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_empty
% 0.20/0.47 thf(fact_157_preorder__on__empty,axiom,
% 0.20/0.47 order_145089608_a_nat @ bot_bo2066531197_a_nat @ bot_bo1803599795_a_nat ).
% 0.20/0.47
% 0.20/0.47 % preorder_on_empty
% 0.20/0.47 thf(fact_158_refl__on__empty,axiom,
% 0.20/0.47 refl_o2134473190od_b_b @ bot_bo1343651123od_b_b @ bot_bo1174586163od_b_b ).
% 0.20/0.47
% 0.20/0.47 % refl_on_empty
% 0.20/0.47 thf(fact_159_refl__on__empty,axiom,
% 0.20/0.47 refl_o686103183od_b_b @ bot_bo519661404od_b_b @ bot_bo1856908531od_b_b ).
% 0.20/0.47
% 0.20/0.47 % refl_on_empty
% 0.20/0.47 thf(fact_160_refl__on__empty,axiom,
% 0.20/0.47 refl_o730481808_a_nat @ bot_bo1746593245_a_nat @ bot_bo2142478259_a_nat ).
% 0.20/0.47
% 0.20/0.47 % refl_on_empty
% 0.20/0.47 thf(fact_161_refl__on__empty,axiom,
% 0.20/0.47 refl_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% 0.20/0.47
% 0.20/0.47 % refl_on_empty
% 0.20/0.47 thf(fact_162_refl__on__empty,axiom,
% 0.20/0.47 refl_o1018941609_a_nat @ bot_bo1119737266_a_nat @ bot_bo2066531197_a_nat ).
% 0.20/0.47
% 0.20/0.47 % refl_on_empty
% 0.20/0.47 thf(fact_163_refl__on__empty,axiom,
% 0.20/0.47 refl_o206400048_a_nat @ bot_bo2066531197_a_nat @ bot_bo1803599795_a_nat ).
% 0.20/0.47
% 0.20/0.47 % refl_on_empty
% 0.20/0.47 thf(fact_164_trans__singleton,axiom,
% 0.20/0.47 ! [A: product_prod_b_b] : ( trans_1808857926od_b_b @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ A @ A ) @ bot_bo1174586163od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % trans_singleton
% 0.20/0.47 thf(fact_165_trans__singleton,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat] : ( trans_444496784_a_nat @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ A @ A ) @ bot_bo1803599795_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % trans_singleton
% 0.20/0.47 thf(fact_166_trans__singleton,axiom,
% 0.20/0.47 ! [A: b] : ( trans_b @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ A @ A ) @ bot_bo1343651123od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % trans_singleton
% 0.20/0.47 thf(fact_167_trans__singleton,axiom,
% 0.20/0.47 ! [A: labeled_graph_a_nat] : ( trans_1002055497_a_nat @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ A @ A ) @ bot_bo2066531197_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % trans_singleton
% 0.20/0.47 thf(fact_168_idempotent__impl__trans,axiom,
% 0.20/0.47 ! [R2: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( idempo848975736_a_nat @ R2 )
% 0.20/0.47 => ( trans_1002055497_a_nat @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % idempotent_impl_trans
% 0.20/0.47 thf(fact_169_idempotent__impl__trans,axiom,
% 0.20/0.47 ! [R2: set_Product_prod_b_b] :
% 0.20/0.47 ( ( idempotent_b @ R2 )
% 0.20/0.47 => ( trans_b @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % idempotent_impl_trans
% 0.20/0.47 thf(fact_170_trans__empty,axiom,
% 0.20/0.47 trans_b @ bot_bo1343651123od_b_b ).
% 0.20/0.47
% 0.20/0.47 % trans_empty
% 0.20/0.47 thf(fact_171_trans__empty,axiom,
% 0.20/0.47 trans_1002055497_a_nat @ bot_bo2066531197_a_nat ).
% 0.20/0.47
% 0.20/0.47 % trans_empty
% 0.20/0.47 thf(fact_172_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: b,V4: b,R: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1694913744od_b_b @ ( produc1037687191od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_triple_b_b_a @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_173_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: a,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: a,V3: b,V4: b,R: set_Pr471017666_nat_b] :
% 0.20/0.47 ( ( member442676358od_b_b @ ( produc179745357od_b_b @ ( produc1110849866_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1993240081at_b_a @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_174_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: a,V1: product_prod_b_b,V2: product_prod_b_b,L2: a,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
% 0.20/0.47 ( ( member1023210704od_b_b @ ( produc1116925335od_b_b @ ( produc690925568od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1764442556_b_b_a @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_175_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: b,V4: b,R: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri111564221od_b_b @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_176_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 0.20/0.47 ( ( member817785114_a_nat @ ( produc966067681_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1110849866_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri2117183935_nat_a @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_177_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 0.20/0.47 ( ( member962139856od_b_b @ ( produc789452695od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc690925568od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri2003683308_b_b_a @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_178_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr471017666_nat_b] :
% 0.20/0.47 ( ( member670041670od_b_b @ ( produc926922637od_b_b @ ( produc90796321_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1706508026od_b_b @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_179_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: product_prod_b_b,V2: product_prod_b_b,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
% 0.20/0.47 ( ( member178477328od_b_b @ ( produc885982551od_b_b @ ( produc1850071895od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri97791909od_b_b @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_180_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 0.20/0.47 ( ( member1769525082_a_nat @ ( produc1783652257_a_nat @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc90796321_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1837931688od_b_b @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_181_on__tripleD_I1_J,axiom,
% 0.20/0.47 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 0.20/0.47 ( ( member1432657168od_b_b @ ( produc962089303od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1850071895od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri927008213od_b_b @ R ) )
% 0.20/0.47 => ( L2 = L1 ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(1)
% 0.20/0.47 thf(fact_182_on__tripleD_I2_J,axiom,
% 0.20/0.47 ! [L1: produc598811825_a_nat,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: produc598811825_a_nat,V3: produc598811825_a_nat,V4: produc598811825_a_nat,R: set_Pr1870121716_a_nat] :
% 0.20/0.47 ( ( member428378182_a_nat @ ( produc1645191821_a_nat @ ( produc410388695_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc639906189_a_nat @ L2 @ ( produc410388695_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1766439058_a_nat @ R ) )
% 0.20/0.47 => ( member2103088853_a_nat @ ( produc2092742640_a_nat @ V1 @ V3 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(2)
% 0.20/0.47 thf(fact_183_on__tripleD_I2_J,axiom,
% 0.20/0.47 ! [L1: produc598811825_a_nat,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: produc598811825_a_nat,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( member121756432_a_nat @ ( produc404908887_a_nat @ ( produc410388695_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc410388695_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri232494965_a_nat @ R ) )
% 0.20/0.47 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ V1 @ V3 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(2)
% 0.20/0.47 thf(fact_184_on__tripleD_I2_J,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 0.20/0.47 ( ( member817785114_a_nat @ ( produc966067681_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1110849866_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri2117183935_nat_a @ R ) )
% 0.20/0.47 => ( member763519277_a_nat @ ( produc1311571856_a_nat @ V1 @ V3 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(2)
% 0.20/0.47 thf(fact_185_on__tripleD_I2_J,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 0.20/0.47 ( ( member962139856od_b_b @ ( produc789452695od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc690925568od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri2003683308_b_b_a @ R ) )
% 0.20/0.47 => ( member641857272od_b_b @ ( produc1257047359od_b_b @ V1 @ V3 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(2)
% 0.20/0.47 thf(fact_186_on__tripleD_I2_J,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: produc598811825_a_nat,V4: produc598811825_a_nat,R: set_Pr352593785_a_nat] :
% 0.20/0.47 ( ( member1379268944_a_nat @ ( produc797892631_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1904940160_a_nat @ L2 @ ( produc410388695_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1794757026_nat_a @ R ) )
% 0.20/0.47 => ( member692218050_a_nat @ ( produc255168265_a_nat @ V1 @ V3 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(2)
% 0.20/0.47 thf(fact_187_on__tripleD_I2_J,axiom,
% 0.20/0.47 ! [L1: a,V1: b,V2: b,L2: a,V3: b,V4: b,R: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1694913744od_b_b @ ( produc1037687191od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_triple_b_b_a @ R ) )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V3 ) @ R ) ) ).
% 0.20/0.47
% 0.20/0.47 % on_tripleD(2)
% 0.20/0.47 thf(fact_188_singletonI,axiom,
% 0.20/0.47 ! [A: product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % singletonI
% 0.20/0.47 thf(fact_189_singletonI,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat] : ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % singletonI
% 0.20/0.47 thf(fact_190_nonempty__rule,axiom,
% 0.20/0.47 ! [G: labeled_graph_a_b] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restrict_a_b @ G ) )
% 0.20/0.47 => ( ( maintained_a_nat_b @ standa879863265rule_a @ G )
% 0.20/0.47 = ( ( labeled_vertices_a_b @ G )
% 0.20/0.47 != bot_bot_set_b ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % nonempty_rule
% 0.20/0.47 thf(fact_191_insertCI,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,B4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
% 0.20/0.47 ( ( ~ ( member327031898_a_nat @ A @ B4 )
% 0.20/0.47 => ( A = B ) )
% 0.20/0.47 => ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertCI
% 0.20/0.47 thf(fact_192_insertCI,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,B4: set_Product_prod_b_b,B: product_prod_b_b] :
% 0.20/0.47 ( ( ~ ( member1285940496od_b_b @ A @ B4 )
% 0.20/0.47 => ( A = B ) )
% 0.20/0.47 => ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertCI
% 0.20/0.47 thf(fact_193_insert__iff,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ A4 ) )
% 0.20/0.47 = ( ( A = B )
% 0.20/0.47 | ( member327031898_a_nat @ A @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_iff
% 0.20/0.47 thf(fact_194_insert__iff,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,B: product_prod_b_b,A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A4 ) )
% 0.20/0.47 = ( ( A = B )
% 0.20/0.47 | ( member1285940496od_b_b @ A @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_iff
% 0.20/0.47 thf(fact_195_insert__absorb2,axiom,
% 0.20/0.47 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ X4 @ A4 ) )
% 0.20/0.47 = ( insert2058869377_a_nat @ X4 @ A4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_absorb2
% 0.20/0.47 thf(fact_196_empty__iff,axiom,
% 0.20/0.47 ! [C2: product_prod_b_b] :
% 0.20/0.47 ~ ( member1285940496od_b_b @ C2 @ bot_bo1343651123od_b_b ) ).
% 0.20/0.47
% 0.20/0.47 % empty_iff
% 0.20/0.47 thf(fact_197_empty__iff,axiom,
% 0.20/0.47 ! [C2: produc598811825_a_nat] :
% 0.20/0.47 ~ ( member327031898_a_nat @ C2 @ bot_bo2066531197_a_nat ) ).
% 0.20/0.47
% 0.20/0.47 % empty_iff
% 0.20/0.47 thf(fact_198_all__not__in__conv,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( ! [X5: product_prod_b_b] :
% 0.20/0.47 ~ ( member1285940496od_b_b @ X5 @ A4 ) )
% 0.20/0.47 = ( A4 = bot_bo1343651123od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % all_not_in_conv
% 0.20/0.47 thf(fact_199_all__not__in__conv,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( ! [X5: produc598811825_a_nat] :
% 0.20/0.47 ~ ( member327031898_a_nat @ X5 @ A4 ) )
% 0.20/0.47 = ( A4 = bot_bo2066531197_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % all_not_in_conv
% 0.20/0.47 thf(fact_200_Collect__empty__eq,axiom,
% 0.20/0.47 ! [P2: produc598811825_a_nat > $o] :
% 0.20/0.47 ( ( ( collec978177436_a_nat @ P2 )
% 0.20/0.47 = bot_bo2066531197_a_nat )
% 0.20/0.47 = ( ! [X5: produc598811825_a_nat] :
% 0.20/0.47 ~ ( P2 @ X5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_empty_eq
% 0.20/0.47 thf(fact_201_empty__Collect__eq,axiom,
% 0.20/0.47 ! [P2: produc598811825_a_nat > $o] :
% 0.20/0.47 ( ( bot_bo2066531197_a_nat
% 0.20/0.47 = ( collec978177436_a_nat @ P2 ) )
% 0.20/0.47 = ( ! [X5: produc598811825_a_nat] :
% 0.20/0.47 ~ ( P2 @ X5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % empty_Collect_eq
% 0.20/0.47 thf(fact_202_ex__in__conv,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( ? [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ A4 ) )
% 0.20/0.47 = ( A4 != bot_bo1343651123od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % ex_in_conv
% 0.20/0.47 thf(fact_203_ex__in__conv,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( ? [X5: produc598811825_a_nat] : ( member327031898_a_nat @ X5 @ A4 ) )
% 0.20/0.47 = ( A4 != bot_bo2066531197_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % ex_in_conv
% 0.20/0.47 thf(fact_204_equals0I,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ! [Y3: product_prod_b_b] :
% 0.20/0.47 ~ ( member1285940496od_b_b @ Y3 @ A4 )
% 0.20/0.47 => ( A4 = bot_bo1343651123od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % equals0I
% 0.20/0.47 thf(fact_205_equals0I,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ! [Y3: produc598811825_a_nat] :
% 0.20/0.47 ~ ( member327031898_a_nat @ Y3 @ A4 )
% 0.20/0.47 => ( A4 = bot_bo2066531197_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % equals0I
% 0.20/0.47 thf(fact_206_equals0D,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b,A: product_prod_b_b] :
% 0.20/0.47 ( ( A4 = bot_bo1343651123od_b_b )
% 0.20/0.47 => ~ ( member1285940496od_b_b @ A @ A4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % equals0D
% 0.20/0.47 thf(fact_207_equals0D,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat,A: produc598811825_a_nat] :
% 0.20/0.47 ( ( A4 = bot_bo2066531197_a_nat )
% 0.20/0.47 => ~ ( member327031898_a_nat @ A @ A4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % equals0D
% 0.20/0.47 thf(fact_208_emptyE,axiom,
% 0.20/0.47 ! [A: product_prod_b_b] :
% 0.20/0.47 ~ ( member1285940496od_b_b @ A @ bot_bo1343651123od_b_b ) ).
% 0.20/0.47
% 0.20/0.47 % emptyE
% 0.20/0.47 thf(fact_209_emptyE,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat] :
% 0.20/0.47 ~ ( member327031898_a_nat @ A @ bot_bo2066531197_a_nat ) ).
% 0.20/0.47
% 0.20/0.47 % emptyE
% 0.20/0.47 thf(fact_210_mk__disjoint__insert,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ A @ A4 )
% 0.20/0.47 => ? [B5: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( insert2058869377_a_nat @ A @ B5 ) )
% 0.20/0.47 & ~ ( member327031898_a_nat @ A @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mk_disjoint_insert
% 0.20/0.47 thf(fact_211_mk__disjoint__insert,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ A @ A4 )
% 0.20/0.47 => ? [B5: set_Product_prod_b_b] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( insert1952693431od_b_b @ A @ B5 ) )
% 0.20/0.47 & ~ ( member1285940496od_b_b @ A @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % mk_disjoint_insert
% 0.20/0.47 thf(fact_212_insert__commute,axiom,
% 0.20/0.47 ! [X4: produc598811825_a_nat,Y4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ Y4 @ A4 ) )
% 0.20/0.47 = ( insert2058869377_a_nat @ Y4 @ ( insert2058869377_a_nat @ X4 @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_commute
% 0.20/0.47 thf(fact_213_insert__eq__iff,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B: produc598811825_a_nat,B4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ~ ( member327031898_a_nat @ A @ A4 )
% 0.20/0.47 => ( ~ ( member327031898_a_nat @ B @ B4 )
% 0.20/0.47 => ( ( ( insert2058869377_a_nat @ A @ A4 )
% 0.20/0.47 = ( insert2058869377_a_nat @ B @ B4 ) )
% 0.20/0.47 = ( ( ( A = B )
% 0.20/0.47 => ( A4 = B4 ) )
% 0.20/0.47 & ( ( A != B )
% 0.20/0.47 => ? [C3: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( insert2058869377_a_nat @ B @ C3 ) )
% 0.20/0.47 & ~ ( member327031898_a_nat @ B @ C3 )
% 0.20/0.47 & ( B4
% 0.20/0.47 = ( insert2058869377_a_nat @ A @ C3 ) )
% 0.20/0.47 & ~ ( member327031898_a_nat @ A @ C3 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_eq_iff
% 0.20/0.47 thf(fact_214_insert__eq__iff,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,A4: set_Product_prod_b_b,B: product_prod_b_b,B4: set_Product_prod_b_b] :
% 0.20/0.47 ( ~ ( member1285940496od_b_b @ A @ A4 )
% 0.20/0.47 => ( ~ ( member1285940496od_b_b @ B @ B4 )
% 0.20/0.47 => ( ( ( insert1952693431od_b_b @ A @ A4 )
% 0.20/0.47 = ( insert1952693431od_b_b @ B @ B4 ) )
% 0.20/0.47 = ( ( ( A = B )
% 0.20/0.47 => ( A4 = B4 ) )
% 0.20/0.47 & ( ( A != B )
% 0.20/0.47 => ? [C3: set_Product_prod_b_b] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( insert1952693431od_b_b @ B @ C3 ) )
% 0.20/0.47 & ~ ( member1285940496od_b_b @ B @ C3 )
% 0.20/0.47 & ( B4
% 0.20/0.47 = ( insert1952693431od_b_b @ A @ C3 ) )
% 0.20/0.47 & ~ ( member1285940496od_b_b @ A @ C3 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_eq_iff
% 0.20/0.47 thf(fact_215_insert__absorb,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ A @ A4 )
% 0.20/0.47 => ( ( insert2058869377_a_nat @ A @ A4 )
% 0.20/0.47 = A4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_absorb
% 0.20/0.47 thf(fact_216_insert__absorb,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ A @ A4 )
% 0.20/0.47 => ( ( insert1952693431od_b_b @ A @ A4 )
% 0.20/0.47 = A4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_absorb
% 0.20/0.47 thf(fact_217_insert__ident,axiom,
% 0.20/0.47 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ~ ( member327031898_a_nat @ X4 @ A4 )
% 0.20/0.47 => ( ~ ( member327031898_a_nat @ X4 @ B4 )
% 0.20/0.47 => ( ( ( insert2058869377_a_nat @ X4 @ A4 )
% 0.20/0.47 = ( insert2058869377_a_nat @ X4 @ B4 ) )
% 0.20/0.47 = ( A4 = B4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_ident
% 0.20/0.47 thf(fact_218_insert__ident,axiom,
% 0.20/0.47 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
% 0.20/0.47 ( ~ ( member1285940496od_b_b @ X4 @ A4 )
% 0.20/0.47 => ( ~ ( member1285940496od_b_b @ X4 @ B4 )
% 0.20/0.47 => ( ( ( insert1952693431od_b_b @ X4 @ A4 )
% 0.20/0.47 = ( insert1952693431od_b_b @ X4 @ B4 ) )
% 0.20/0.47 = ( A4 = B4 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insert_ident
% 0.20/0.47 thf(fact_219_Set_Oset__insert,axiom,
% 0.20/0.47 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ X4 @ A4 )
% 0.20/0.47 => ~ ! [B5: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( insert2058869377_a_nat @ X4 @ B5 ) )
% 0.20/0.47 => ( member327031898_a_nat @ X4 @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Set.set_insert
% 0.20/0.47 thf(fact_220_Set_Oset__insert,axiom,
% 0.20/0.47 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ X4 @ A4 )
% 0.20/0.47 => ~ ! [B5: set_Product_prod_b_b] :
% 0.20/0.47 ( ( A4
% 0.20/0.47 = ( insert1952693431od_b_b @ X4 @ B5 ) )
% 0.20/0.47 => ( member1285940496od_b_b @ X4 @ B5 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Set.set_insert
% 0.20/0.47 thf(fact_221_insertI2,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,B4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ A @ B4 )
% 0.20/0.47 => ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertI2
% 0.20/0.47 thf(fact_222_insertI2,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,B4: set_Product_prod_b_b,B: product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ A @ B4 )
% 0.20/0.47 => ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertI2
% 0.20/0.47 thf(fact_223_insertI1,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,B4: set_Pr1912306449_a_nat] : ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ A @ B4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertI1
% 0.20/0.47 thf(fact_224_insertI1,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,B4: set_Product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ B4 ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertI1
% 0.20/0.47 thf(fact_225_insertE,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ A4 ) )
% 0.20/0.47 => ( ( A != B )
% 0.20/0.47 => ( member327031898_a_nat @ A @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertE
% 0.20/0.47 thf(fact_226_insertE,axiom,
% 0.20/0.47 ! [A: product_prod_b_b,B: product_prod_b_b,A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A4 ) )
% 0.20/0.47 => ( ( A != B )
% 0.20/0.47 => ( member1285940496od_b_b @ A @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % insertE
% 0.20/0.47 thf(fact_227_singleton__inject,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,B: produc598811825_a_nat] :
% 0.20/0.47 ( ( ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat )
% 0.20/0.47 = ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) )
% 0.20/0.47 => ( A = B ) ) ).
% 0.20/0.47
% 0.20/0.47 % singleton_inject
% 0.20/0.47 thf(fact_228_insert__not__empty,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( insert2058869377_a_nat @ A @ A4 )
% 0.20/0.47 != bot_bo2066531197_a_nat ) ).
% 0.20/0.47
% 0.20/0.47 % insert_not_empty
% 0.20/0.47 thf(fact_229_doubleton__eq__iff,axiom,
% 0.20/0.47 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,C2: produc598811825_a_nat,D: produc598811825_a_nat] :
% 0.20/0.47 ( ( ( insert2058869377_a_nat @ A @ ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) )
% 0.20/0.47 = ( insert2058869377_a_nat @ C2 @ ( insert2058869377_a_nat @ D @ bot_bo2066531197_a_nat ) ) )
% 0.20/0.47 = ( ( ( A = C2 )
% 0.20/0.47 & ( B = D ) )
% 0.20/0.47 | ( ( A = D )
% 0.20/0.47 & ( B = C2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % doubleton_eq_iff
% 0.20/0.47 thf(fact_230_singleton__iff,axiom,
% 0.20/0.47 ! [B: product_prod_b_b,A: product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 0.20/0.47 = ( B = A ) ) ).
% 0.20/0.47
% 0.20/0.47 % singleton_iff
% 0.20/0.47 thf(fact_231_singleton__iff,axiom,
% 0.20/0.47 ! [B: produc598811825_a_nat,A: produc598811825_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ B @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) )
% 0.20/0.47 = ( B = A ) ) ).
% 0.20/0.47
% 0.20/0.47 % singleton_iff
% 0.20/0.47 thf(fact_232_singletonD,axiom,
% 0.20/0.47 ! [B: product_prod_b_b,A: product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 0.20/0.47 => ( B = A ) ) ).
% 0.20/0.47
% 0.20/0.47 % singletonD
% 0.20/0.47 thf(fact_233_singletonD,axiom,
% 0.20/0.47 ! [B: produc598811825_a_nat,A: produc598811825_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ B @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) )
% 0.20/0.47 => ( B = A ) ) ).
% 0.20/0.47
% 0.20/0.47 % singletonD
% 0.20/0.47 thf(fact_234_the__elem__eq,axiom,
% 0.20/0.47 ! [X4: produc598811825_a_nat] :
% 0.20/0.47 ( ( the_el537460006_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) )
% 0.20/0.47 = X4 ) ).
% 0.20/0.47
% 0.20/0.47 % the_elem_eq
% 0.20/0.47 thf(fact_235_graph__single,axiom,
% 0.20/0.47 ! [A: a,B: b,C2: b] :
% 0.20/0.47 ( ( labeled_LG_a_b @ ( insert1655137376od_b_b @ ( produc2135284224od_b_b @ A @ ( product_Pair_b_b @ B @ C2 ) ) @ bot_bo519661404od_b_b ) @ ( insert_b @ B @ ( insert_b @ C2 @ bot_bot_set_b ) ) )
% 0.20/0.47 = ( restrict_a_b @ ( labeled_LG_a_b @ ( insert1655137376od_b_b @ ( produc2135284224od_b_b @ A @ ( product_Pair_b_b @ B @ C2 ) ) @ bot_bo519661404od_b_b ) @ ( insert_b @ B @ ( insert_b @ C2 @ bot_bot_set_b ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_single
% 0.20/0.47 thf(fact_236_is__singletonI,axiom,
% 0.20/0.47 ! [X4: produc598811825_a_nat] : ( is_sin1787449765_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singletonI
% 0.20/0.47 thf(fact_237_linear__order__on__singleton,axiom,
% 0.20/0.47 ! [X4: b] : ( order_653178932r_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X4 @ X4 ) @ bot_bo1343651123od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % linear_order_on_singleton
% 0.20/0.47 thf(fact_238_linear__order__on__singleton,axiom,
% 0.20/0.47 ! [X4: labeled_graph_a_nat] : ( order_1770871027_a_nat @ ( insert1337853464_a_nat @ X4 @ bot_bo1119737266_a_nat ) @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ X4 @ X4 ) @ bot_bo2066531197_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % linear_order_on_singleton
% 0.20/0.47 thf(fact_239_linear__order__on__singleton,axiom,
% 0.20/0.47 ! [X4: produc598811825_a_nat] : ( order_1458491494_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ X4 @ X4 ) @ bot_bo1803599795_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % linear_order_on_singleton
% 0.20/0.47 thf(fact_240_total__on__singleton,axiom,
% 0.20/0.47 ! [X4: b] : ( total_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X4 @ X4 ) @ bot_bo1343651123od_b_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_on_singleton
% 0.20/0.47 thf(fact_241_total__on__singleton,axiom,
% 0.20/0.47 ! [X4: labeled_graph_a_nat] : ( total_6094180_a_nat @ ( insert1337853464_a_nat @ X4 @ bot_bo1119737266_a_nat ) @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ X4 @ X4 ) @ bot_bo2066531197_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_on_singleton
% 0.20/0.47 thf(fact_242_total__on__singleton,axiom,
% 0.20/0.47 ! [X4: produc598811825_a_nat] : ( total_2145009205_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ X4 @ X4 ) @ bot_bo1803599795_a_nat ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_on_singleton
% 0.20/0.47 thf(fact_243_symmetry__rule,axiom,
% 0.20/0.47 ! [G: labeled_graph_a_b,L: a] :
% 0.20/0.47 ( ( G
% 0.20/0.47 = ( restrict_a_b @ G ) )
% 0.20/0.47 => ( ( maintained_a_nat_b @ ( standa709637829rule_a @ L ) @ G )
% 0.20/0.47 => ( sym_b @ ( getRel_a_b @ L @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % symmetry_rule
% 0.20/0.47 thf(fact_244_graph__empty__e,axiom,
% 0.20/0.47 ! [V: set_b] :
% 0.20/0.47 ( ( labeled_LG_a_b @ bot_bo519661404od_b_b @ V )
% 0.20/0.47 = ( restrict_a_b @ ( labeled_LG_a_b @ bot_bo519661404od_b_b @ V ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_empty_e
% 0.20/0.47 thf(fact_245_bot__empty__eq,axiom,
% 0.20/0.47 ( bot_bo1302443370_b_b_o
% 0.20/0.47 = ( ^ [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ bot_bo1343651123od_b_b ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_empty_eq
% 0.20/0.47 thf(fact_246_bot__empty__eq,axiom,
% 0.20/0.47 ( bot_bo186023328_nat_o
% 0.20/0.47 = ( ^ [X5: produc598811825_a_nat] : ( member327031898_a_nat @ X5 @ bot_bo2066531197_a_nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_empty_eq
% 0.20/0.47 thf(fact_247_bot__set__def,axiom,
% 0.20/0.47 ( bot_bo2066531197_a_nat
% 0.20/0.47 = ( collec978177436_a_nat @ bot_bo186023328_nat_o ) ) ).
% 0.20/0.47
% 0.20/0.47 % bot_set_def
% 0.20/0.47 thf(fact_248_sym__def,axiom,
% 0.20/0.47 ( sym_b
% 0.20/0.47 = ( ^ [R3: set_Product_prod_b_b] :
% 0.20/0.47 ! [X5: b,Y5: b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Y5 ) @ R3 )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ Y5 @ X5 ) @ R3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sym_def
% 0.20/0.47 thf(fact_249_symI,axiom,
% 0.20/0.47 ! [R2: set_Product_prod_b_b] :
% 0.20/0.47 ( ! [A3: b,B3: b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ R2 )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ B3 @ A3 ) @ R2 ) )
% 0.20/0.47 => ( sym_b @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % symI
% 0.20/0.47 thf(fact_250_symE,axiom,
% 0.20/0.47 ! [R2: set_Product_prod_b_b,B: b,A: b] :
% 0.20/0.47 ( ( sym_b @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % symE
% 0.20/0.47 thf(fact_251_symD,axiom,
% 0.20/0.47 ! [R2: set_Product_prod_b_b,B: b,A: b] :
% 0.20/0.47 ( ( sym_b @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 )
% 0.20/0.47 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % symD
% 0.20/0.47 thf(fact_252_total__on__def,axiom,
% 0.20/0.47 ( total_on_b
% 0.20/0.47 = ( ^ [A5: set_b,R3: set_Product_prod_b_b] :
% 0.20/0.47 ! [X5: b] :
% 0.20/0.47 ( ( member_b @ X5 @ A5 )
% 0.20/0.47 => ! [Y5: b] :
% 0.20/0.47 ( ( member_b @ Y5 @ A5 )
% 0.20/0.47 => ( ( X5 != Y5 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Y5 ) @ R3 )
% 0.20/0.47 | ( member1285940496od_b_b @ ( product_Pair_b_b @ Y5 @ X5 ) @ R3 ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_on_def
% 0.20/0.47 thf(fact_253_total__onI,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b] :
% 0.20/0.47 ( ! [X3: product_prod_b_b,Y3: product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ X3 @ A4 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ Y3 @ A4 )
% 0.20/0.47 => ( ( X3 != Y3 )
% 0.20/0.47 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X3 @ Y3 ) @ R2 )
% 0.20/0.47 | ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.20/0.47 => ( total_178717291od_b_b @ A4 @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_onI
% 0.20/0.47 thf(fact_254_total__onI,axiom,
% 0.20/0.47 ! [A4: set_b,R2: set_Product_prod_b_b] :
% 0.20/0.47 ( ! [X3: b,Y3: b] :
% 0.20/0.47 ( ( member_b @ X3 @ A4 )
% 0.20/0.47 => ( ( member_b @ Y3 @ A4 )
% 0.20/0.47 => ( ( X3 != Y3 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R2 )
% 0.20/0.47 | ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.20/0.47 => ( total_on_b @ A4 @ R2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % total_onI
% 0.20/0.47 thf(fact_255_total__on__empty,axiom,
% 0.20/0.47 ! [R2: set_Pr2074180935_a_nat] : ( total_2145009205_a_nat @ bot_bo2066531197_a_nat @ R2 ) ).
% 0.20/0.47
% 0.20/0.47 % total_on_empty
% 0.20/0.47 thf(fact_256_labeled__graph_Osel_I2_J,axiom,
% 0.20/0.47 ! [X1: set_Pr1900477680od_b_b,X2: set_b] :
% 0.20/0.47 ( ( labeled_vertices_a_b @ ( labeled_LG_a_b @ X1 @ X2 ) )
% 0.20/0.47 = X2 ) ).
% 0.20/0.47
% 0.20/0.47 % labeled_graph.sel(2)
% 0.20/0.47 thf(fact_257_is__singleton__the__elem,axiom,
% 0.20/0.47 ( is_sin1787449765_a_nat
% 0.20/0.47 = ( ^ [A5: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( A5
% 0.20/0.47 = ( insert2058869377_a_nat @ ( the_el537460006_a_nat @ A5 ) @ bot_bo2066531197_a_nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singleton_the_elem
% 0.20/0.47 thf(fact_258_is__singletonI_H,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b] :
% 0.20/0.47 ( ( A4 != bot_bo1343651123od_b_b )
% 0.20/0.47 => ( ! [X3: product_prod_b_b,Y3: product_prod_b_b] :
% 0.20/0.47 ( ( member1285940496od_b_b @ X3 @ A4 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ Y3 @ A4 )
% 0.20/0.47 => ( X3 = Y3 ) ) )
% 0.20/0.47 => ( is_sin1713057243od_b_b @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singletonI'
% 0.20/0.47 thf(fact_259_is__singletonI_H,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( A4 != bot_bo2066531197_a_nat )
% 0.20/0.47 => ( ! [X3: produc598811825_a_nat,Y3: produc598811825_a_nat] :
% 0.20/0.47 ( ( member327031898_a_nat @ X3 @ A4 )
% 0.20/0.47 => ( ( member327031898_a_nat @ Y3 @ A4 )
% 0.20/0.47 => ( X3 = Y3 ) ) )
% 0.20/0.47 => ( is_sin1787449765_a_nat @ A4 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singletonI'
% 0.20/0.47 thf(fact_260_lnear__order__on__empty,axiom,
% 0.20/0.47 order_1770871027_a_nat @ bot_bo1119737266_a_nat @ bot_bo2066531197_a_nat ).
% 0.20/0.47
% 0.20/0.47 % lnear_order_on_empty
% 0.20/0.47 thf(fact_261_lnear__order__on__empty,axiom,
% 0.20/0.47 order_1458491494_a_nat @ bot_bo2066531197_a_nat @ bot_bo1803599795_a_nat ).
% 0.20/0.47
% 0.20/0.47 % lnear_order_on_empty
% 0.20/0.47 thf(fact_262_is__singletonE,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat] :
% 0.20/0.47 ( ( is_sin1787449765_a_nat @ A4 )
% 0.20/0.47 => ~ ! [X3: produc598811825_a_nat] :
% 0.20/0.47 ( A4
% 0.20/0.47 != ( insert2058869377_a_nat @ X3 @ bot_bo2066531197_a_nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singletonE
% 0.20/0.47 thf(fact_263_is__singleton__def,axiom,
% 0.20/0.47 ( is_sin1787449765_a_nat
% 0.20/0.47 = ( ^ [A5: set_Pr1912306449_a_nat] :
% 0.20/0.47 ? [X5: produc598811825_a_nat] :
% 0.20/0.47 ( A5
% 0.20/0.47 = ( insert2058869377_a_nat @ X5 @ bot_bo2066531197_a_nat ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % is_singleton_def
% 0.20/0.47 thf(fact_264_equivE,axiom,
% 0.20/0.47 ! [A4: set_b,R2: set_Product_prod_b_b] :
% 0.20/0.47 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.47 => ~ ( ( refl_on_b @ A4 @ R2 )
% 0.20/0.47 => ( ( sym_b @ R2 )
% 0.20/0.47 => ~ ( trans_b @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equivE
% 0.20/0.47 thf(fact_265_equivI,axiom,
% 0.20/0.47 ! [A4: set_b,R2: set_Product_prod_b_b] :
% 0.20/0.47 ( ( refl_on_b @ A4 @ R2 )
% 0.20/0.47 => ( ( sym_b @ R2 )
% 0.20/0.47 => ( ( trans_b @ R2 )
% 0.20/0.47 => ( equiv_equiv_b @ A4 @ R2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equivI
% 0.20/0.47 thf(fact_266_equiv__def,axiom,
% 0.20/0.47 ( equiv_equiv_b
% 0.20/0.47 = ( ^ [A5: set_b,R3: set_Product_prod_b_b] :
% 0.20/0.47 ( ( refl_on_b @ A5 @ R3 )
% 0.20/0.47 & ( sym_b @ R3 )
% 0.20/0.47 & ( trans_b @ R3 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % equiv_def
% 0.20/0.47 thf(fact_267_graph__homomorphism__empty,axiom,
% 0.20/0.47 ! [G: labeled_graph_a_b,F: set_Pr1914572181_nat_b] :
% 0.20/0.47 ( ( graph_23275493_nat_b @ ( labele1171788421_a_nat @ bot_bo1463793628_a_nat @ bot_bo2066531197_a_nat ) @ G @ F )
% 0.20/0.47 = ( ( F = bot_bo1936722433_nat_b )
% 0.20/0.47 & ( G
% 0.20/0.47 = ( restrict_a_b @ G ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % graph_homomorphism_empty
% 0.20/0.47 thf(fact_268_Collect__empty__eq__bot,axiom,
% 0.20/0.47 ! [P2: produc598811825_a_nat > $o] :
% 0.20/0.47 ( ( ( collec978177436_a_nat @ P2 )
% 0.20/0.47 = bot_bo2066531197_a_nat )
% 0.20/0.47 = ( P2 = bot_bo186023328_nat_o ) ) ).
% 0.20/0.47
% 0.20/0.47 % Collect_empty_eq_bot
% 0.20/0.47 thf(fact_269_maintainedI,axiom,
% 0.20/0.47 ! [A4: labeled_graph_a_nat,G: labeled_graph_a_b,B4: labeled_graph_a_nat] :
% 0.20/0.47 ( ! [F2: set_Pr2106913242_nat_b] :
% 0.20/0.47 ( ( graph_914848022_nat_b @ A4 @ G @ F2 )
% 0.20/0.47 => ( extensible_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G @ F2 ) )
% 0.20/0.47 => ( maintained_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G ) ) ).
% 0.20/0.47
% 0.20/0.47 % maintainedI
% 0.20/0.47 thf(fact_270_maintainedD,axiom,
% 0.20/0.47 ! [A4: labeled_graph_a_nat,B4: labeled_graph_a_nat,G: labeled_graph_a_b,F: set_Pr2106913242_nat_b] :
% 0.20/0.47 ( ( maintained_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G )
% 0.20/0.47 => ( ( graph_914848022_nat_b @ A4 @ G @ F )
% 0.20/0.47 => ( extensible_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G @ F ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % maintainedD
% 0.20/0.47 thf(fact_271_Set_Ois__empty__def,axiom,
% 0.20/0.47 ( is_emp1506632323_a_nat
% 0.20/0.47 = ( ^ [A5: set_Pr1912306449_a_nat] : ( A5 = bot_bo2066531197_a_nat ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % Set.is_empty_def
% 0.20/0.47 thf(fact_272_eq__equiv__class__iff2,axiom,
% 0.20/0.47 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 0.20/0.47 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ X4 @ A4 )
% 0.20/0.47 => ( ( member1285940496od_b_b @ Y4 @ A4 )
% 0.20/0.47 => ( ( ( equiv_1530858158od_b_b @ ( insert1952693431od_b_b @ X4 @ bot_bo1343651123od_b_b ) @ R2 )
% 0.20/0.47 = ( equiv_1530858158od_b_b @ ( insert1952693431od_b_b @ Y4 @ bot_bo1343651123od_b_b ) @ R2 ) )
% 0.20/0.47 = ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_equiv_class_iff2
% 0.20/0.47 thf(fact_273_eq__equiv__class__iff2,axiom,
% 0.20/0.47 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
% 0.20/0.47 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.47 => ( ( member_b @ X4 @ A4 )
% 0.20/0.47 => ( ( member_b @ Y4 @ A4 )
% 0.20/0.47 => ( ( ( equiv_quotient_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ R2 )
% 0.20/0.47 = ( equiv_quotient_b @ ( insert_b @ Y4 @ bot_bot_set_b ) @ R2 ) )
% 0.20/0.47 = ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % eq_equiv_class_iff2
% 0.20/0.47 thf(fact_274_eq__equiv__class__iff2,axiom,
% 0.20/0.47 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 0.20/0.47 ( ( equiv_281634663_a_nat @ A4 @ R2 )
% 0.20/0.47 => ( ( member327031898_a_nat @ X4 @ A4 )
% 0.20/0.47 => ( ( member327031898_a_nat @ Y4 @ A4 )
% 0.20/0.47 => ( ( ( equiv_1338033912_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ R2 )
% 0.20/0.47 = ( equiv_1338033912_a_nat @ ( insert2058869377_a_nat @ Y4 @ bot_bo2066531197_a_nat ) @ R2 ) )
% 0.20/0.48 = ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % eq_equiv_class_iff2
% 0.20/0.48 thf(fact_275_quotient__is__empty2,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat] :
% 0.20/0.48 ( ( bot_bo1746593245_a_nat
% 0.20/0.48 = ( equiv_1338033912_a_nat @ A4 @ R2 ) )
% 0.20/0.48 = ( A4 = bot_bo2066531197_a_nat ) ) ).
% 0.20/0.48
% 0.20/0.48 % quotient_is_empty2
% 0.20/0.48 thf(fact_276_quotient__is__empty,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat] :
% 0.20/0.48 ( ( ( equiv_1338033912_a_nat @ A4 @ R2 )
% 0.20/0.48 = bot_bo1746593245_a_nat )
% 0.20/0.48 = ( A4 = bot_bo2066531197_a_nat ) ) ).
% 0.20/0.48
% 0.20/0.48 % quotient_is_empty
% 0.20/0.48 thf(fact_277_quotient__empty,axiom,
% 0.20/0.48 ! [R2: set_Pr2074180935_a_nat] :
% 0.20/0.48 ( ( equiv_1338033912_a_nat @ bot_bo2066531197_a_nat @ R2 )
% 0.20/0.48 = bot_bo1746593245_a_nat ) ).
% 0.20/0.48
% 0.20/0.48 % quotient_empty
% 0.20/0.48 thf(fact_278_in__quotient__imp__closed,axiom,
% 0.20/0.48 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X6: set_Product_prod_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 0.20/0.48 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member1050118384od_b_b @ X6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member1285940496od_b_b @ X4 @ X6 )
% 0.20/0.48 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.48 => ( member1285940496od_b_b @ Y4 @ X6 ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % in_quotient_imp_closed
% 0.20/0.48 thf(fact_279_in__quotient__imp__closed,axiom,
% 0.20/0.48 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,X4: b,Y4: b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member_b @ X4 @ X6 )
% 0.20/0.48 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.48 => ( member_b @ Y4 @ X6 ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % in_quotient_imp_closed
% 0.20/0.48 thf(fact_280_quotient__eq__iff,axiom,
% 0.20/0.48 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X6: set_Product_prod_b_b,Y6: set_Product_prod_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 0.20/0.48 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member1050118384od_b_b @ X6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member1050118384od_b_b @ Y6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member1285940496od_b_b @ X4 @ X6 )
% 0.20/0.48 => ( ( member1285940496od_b_b @ Y4 @ Y6 )
% 0.20/0.48 => ( ( X6 = Y6 )
% 0.20/0.48 = ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % quotient_eq_iff
% 0.20/0.48 thf(fact_281_quotient__eq__iff,axiom,
% 0.20/0.48 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,Y6: set_b,X4: b,Y4: b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member_set_b @ Y6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member_b @ X4 @ X6 )
% 0.20/0.48 => ( ( member_b @ Y4 @ Y6 )
% 0.20/0.48 => ( ( X6 = Y6 )
% 0.20/0.48 = ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % quotient_eq_iff
% 0.20/0.48 thf(fact_282_quotient__eqI,axiom,
% 0.20/0.48 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X6: set_Product_prod_b_b,Y6: set_Product_prod_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 0.20/0.48 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member1050118384od_b_b @ X6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member1050118384od_b_b @ Y6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member1285940496od_b_b @ X4 @ X6 )
% 0.20/0.48 => ( ( member1285940496od_b_b @ Y4 @ Y6 )
% 0.20/0.48 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.48 => ( X6 = Y6 ) ) ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % quotient_eqI
% 0.20/0.48 thf(fact_283_quotient__eqI,axiom,
% 0.20/0.48 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,Y6: set_b,X4: b,Y4: b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member_set_b @ Y6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( member_b @ X4 @ X6 )
% 0.20/0.48 => ( ( member_b @ Y4 @ Y6 )
% 0.20/0.48 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 0.20/0.48 => ( X6 = Y6 ) ) ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % quotient_eqI
% 0.20/0.48 thf(fact_284_in__quotient__imp__non__empty,axiom,
% 0.20/0.48 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 => ( X6 != bot_bot_set_b ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % in_quotient_imp_non_empty
% 0.20/0.48 thf(fact_285_in__quotient__imp__non__empty,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X6: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ( equiv_281634663_a_nat @ A4 @ R2 )
% 0.20/0.48 => ( ( member101535418_a_nat @ X6 @ ( equiv_1338033912_a_nat @ A4 @ R2 ) )
% 0.20/0.48 => ( X6 != bot_bo2066531197_a_nat ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % in_quotient_imp_non_empty
% 0.20/0.48 thf(fact_286_in__quotient__imp__in__rel,axiom,
% 0.20/0.48 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,X4: b,Y4: b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ( ord_less_eq_set_b @ ( insert_b @ X4 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) @ X6 )
% 0.20/0.48 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % in_quotient_imp_in_rel
% 0.20/0.48 thf(fact_287_in__quotient__imp__in__rel,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X6: set_Pr1912306449_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 0.20/0.48 ( ( equiv_281634663_a_nat @ A4 @ R2 )
% 0.20/0.48 => ( ( member101535418_a_nat @ X6 @ ( equiv_1338033912_a_nat @ A4 @ R2 ) )
% 0.20/0.48 => ( ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ Y4 @ bot_bo2066531197_a_nat ) ) @ X6 )
% 0.20/0.48 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % in_quotient_imp_in_rel
% 0.20/0.48 thf(fact_288_same__fstI,axiom,
% 0.20/0.48 ! [P2: b > $o,X4: b,Y7: b,Y4: b,R: b > set_Product_prod_b_b] :
% 0.20/0.48 ( ( P2 @ X4 )
% 0.20/0.48 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y7 @ Y4 ) @ ( R @ X4 ) )
% 0.20/0.48 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ ( product_Pair_b_b @ X4 @ Y7 ) @ ( product_Pair_b_b @ X4 @ Y4 ) ) @ ( same_fst_b_b @ P2 @ R ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % same_fstI
% 0.20/0.48 thf(fact_289_getRel__subgraph,axiom,
% 0.20/0.48 ! [Y4: b,Z: b,L: a,G: labeled_graph_a_b,G2: labeled_graph_a_b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ ( getRel_a_b @ L @ G ) )
% 0.20/0.48 => ( ( graph_1322245399_a_b_b @ G @ G2 @ ( id_on_b @ ( labeled_vertices_a_b @ G ) ) )
% 0.20/0.48 => ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ ( getRel_a_b @ L @ G2 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % getRel_subgraph
% 0.20/0.48 thf(fact_290_subsetI,axiom,
% 0.20/0.48 ! [A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
% 0.20/0.48 ( ! [X3: product_prod_b_b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ X3 @ A4 )
% 0.20/0.48 => ( member1285940496od_b_b @ X3 @ B4 ) )
% 0.20/0.48 => ( ord_le1036320359od_b_b @ A4 @ B4 ) ) ).
% 0.20/0.48
% 0.20/0.48 % subsetI
% 0.20/0.48 thf(fact_291_empty__subsetI,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat] : ( ord_le1374266033_a_nat @ bot_bo2066531197_a_nat @ A4 ) ).
% 0.20/0.48
% 0.20/0.48 % empty_subsetI
% 0.20/0.48 thf(fact_292_subset__empty,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ( ord_le1374266033_a_nat @ A4 @ bot_bo2066531197_a_nat )
% 0.20/0.48 = ( A4 = bot_bo2066531197_a_nat ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_empty
% 0.20/0.48 thf(fact_293_insert__subset,axiom,
% 0.20/0.48 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ A4 ) @ B4 )
% 0.20/0.48 = ( ( member327031898_a_nat @ X4 @ B4 )
% 0.20/0.48 & ( ord_le1374266033_a_nat @ A4 @ B4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % insert_subset
% 0.20/0.48 thf(fact_294_insert__subset,axiom,
% 0.20/0.48 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
% 0.20/0.48 ( ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X4 @ A4 ) @ B4 )
% 0.20/0.48 = ( ( member1285940496od_b_b @ X4 @ B4 )
% 0.20/0.48 & ( ord_le1036320359od_b_b @ A4 @ B4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % insert_subset
% 0.20/0.48 thf(fact_295_Id__onI,axiom,
% 0.20/0.48 ! [A: product_prod_b_b,A4: set_Product_prod_b_b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ A @ A4 )
% 0.20/0.48 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ A ) @ ( id_on_2019020932od_b_b @ A4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % Id_onI
% 0.20/0.48 thf(fact_296_Id__onI,axiom,
% 0.20/0.48 ! [A: b,A4: set_b] :
% 0.20/0.48 ( ( member_b @ A @ A4 )
% 0.20/0.48 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ A ) @ ( id_on_b @ A4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % Id_onI
% 0.20/0.48 thf(fact_297_singleton__insert__inj__eq,axiom,
% 0.20/0.48 ! [B: produc598811825_a_nat,A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ( ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat )
% 0.20/0.48 = ( insert2058869377_a_nat @ A @ A4 ) )
% 0.20/0.48 = ( ( A = B )
% 0.20/0.48 & ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % singleton_insert_inj_eq
% 0.20/0.48 thf(fact_298_singleton__insert__inj__eq_H,axiom,
% 0.20/0.48 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
% 0.20/0.48 ( ( ( insert2058869377_a_nat @ A @ A4 )
% 0.20/0.48 = ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) )
% 0.20/0.48 = ( ( A = B )
% 0.20/0.48 & ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % singleton_insert_inj_eq'
% 0.20/0.48 thf(fact_299_Id__on__empty,axiom,
% 0.20/0.48 ( ( id_on_1401723275_a_nat @ bot_bo1119737266_a_nat )
% 0.20/0.48 = bot_bo2066531197_a_nat ) ).
% 0.20/0.48
% 0.20/0.48 % Id_on_empty
% 0.20/0.48 thf(fact_300_Id__on__empty,axiom,
% 0.20/0.48 ( ( id_on_1649591758_a_nat @ bot_bo2066531197_a_nat )
% 0.20/0.48 = bot_bo1803599795_a_nat ) ).
% 0.20/0.48
% 0.20/0.48 % Id_on_empty
% 0.20/0.48 thf(fact_301_subgraph__refl,axiom,
% 0.20/0.48 ! [G: labeled_graph_a_b] :
% 0.20/0.48 ( ( graph_1322245399_a_b_b @ G @ G @ ( id_on_b @ ( labeled_vertices_a_b @ G ) ) )
% 0.20/0.48 = ( G
% 0.20/0.48 = ( restrict_a_b @ G ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subgraph_refl
% 0.20/0.48 thf(fact_302_subgraph__restrict,axiom,
% 0.20/0.48 ! [G: labeled_graph_a_b] :
% 0.20/0.48 ( ( graph_1322245399_a_b_b @ G @ ( restrict_a_b @ G ) @ ( id_on_b @ ( labeled_vertices_a_b @ G ) ) )
% 0.20/0.48 = ( G
% 0.20/0.48 = ( restrict_a_b @ G ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subgraph_restrict
% 0.20/0.48 thf(fact_303_graph__homomorphism__Id,axiom,
% 0.20/0.48 ! [A: labeled_graph_a_b] : ( graph_1322245399_a_b_b @ ( restrict_a_b @ A ) @ ( restrict_a_b @ A ) @ ( id_on_b @ ( labeled_vertices_a_b @ A ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % graph_homomorphism_Id
% 0.20/0.48 thf(fact_304_subgraph__subset_I1_J,axiom,
% 0.20/0.48 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 0.20/0.48 ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 0.20/0.48 => ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subgraph_subset(1)
% 0.20/0.48 thf(fact_305_Id__onE,axiom,
% 0.20/0.48 ! [C2: produc1168037607od_b_b,A4: set_Product_prod_b_b] :
% 0.20/0.48 ( ( member1652792080od_b_b @ C2 @ ( id_on_2019020932od_b_b @ A4 ) )
% 0.20/0.48 => ~ ! [X3: product_prod_b_b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ X3 @ A4 )
% 0.20/0.48 => ( C2
% 0.20/0.48 != ( produc546497367od_b_b @ X3 @ X3 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % Id_onE
% 0.20/0.48 thf(fact_306_Id__onE,axiom,
% 0.20/0.48 ! [C2: product_prod_b_b,A4: set_b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ C2 @ ( id_on_b @ A4 ) )
% 0.20/0.48 => ~ ! [X3: b] :
% 0.20/0.48 ( ( member_b @ X3 @ A4 )
% 0.20/0.48 => ( C2
% 0.20/0.48 != ( product_Pair_b_b @ X3 @ X3 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % Id_onE
% 0.20/0.48 thf(fact_307_Id__on__eqI,axiom,
% 0.20/0.48 ! [A: product_prod_b_b,B: product_prod_b_b,A4: set_Product_prod_b_b] :
% 0.20/0.48 ( ( A = B )
% 0.20/0.48 => ( ( member1285940496od_b_b @ A @ A4 )
% 0.20/0.48 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ ( id_on_2019020932od_b_b @ A4 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % Id_on_eqI
% 0.20/0.48 thf(fact_308_Id__on__eqI,axiom,
% 0.20/0.48 ! [A: b,B: b,A4: set_b] :
% 0.20/0.48 ( ( A = B )
% 0.20/0.48 => ( ( member_b @ A @ A4 )
% 0.20/0.48 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( id_on_b @ A4 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % Id_on_eqI
% 0.20/0.48 thf(fact_309_Id__on__iff,axiom,
% 0.20/0.48 ! [X4: product_prod_b_b,Y4: product_prod_b_b,A4: set_Product_prod_b_b] :
% 0.20/0.48 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ ( id_on_2019020932od_b_b @ A4 ) )
% 0.20/0.48 = ( ( X4 = Y4 )
% 0.20/0.48 & ( member1285940496od_b_b @ X4 @ A4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % Id_on_iff
% 0.20/0.48 thf(fact_310_Id__on__iff,axiom,
% 0.20/0.48 ! [X4: b,Y4: b,A4: set_b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ ( id_on_b @ A4 ) )
% 0.20/0.48 = ( ( X4 = Y4 )
% 0.20/0.48 & ( member_b @ X4 @ A4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % Id_on_iff
% 0.20/0.48 thf(fact_311_bot_Oextremum__uniqueI,axiom,
% 0.20/0.48 ! [A: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ( ord_le1374266033_a_nat @ A @ bot_bo2066531197_a_nat )
% 0.20/0.48 => ( A = bot_bo2066531197_a_nat ) ) ).
% 0.20/0.48
% 0.20/0.48 % bot.extremum_uniqueI
% 0.20/0.48 thf(fact_312_bot_Oextremum__unique,axiom,
% 0.20/0.48 ! [A: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ( ord_le1374266033_a_nat @ A @ bot_bo2066531197_a_nat )
% 0.20/0.48 = ( A = bot_bo2066531197_a_nat ) ) ).
% 0.20/0.48
% 0.20/0.48 % bot.extremum_unique
% 0.20/0.48 thf(fact_313_bot_Oextremum,axiom,
% 0.20/0.48 ! [A: set_Pr1912306449_a_nat] : ( ord_le1374266033_a_nat @ bot_bo2066531197_a_nat @ A ) ).
% 0.20/0.48
% 0.20/0.48 % bot.extremum
% 0.20/0.48 thf(fact_314_subrelI,axiom,
% 0.20/0.48 ! [R2: set_Product_prod_b_b,S: set_Product_prod_b_b] :
% 0.20/0.48 ( ! [X3: b,Y3: b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R2 )
% 0.20/0.48 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ S ) )
% 0.20/0.48 => ( ord_le1036320359od_b_b @ R2 @ S ) ) ).
% 0.20/0.48
% 0.20/0.48 % subrelI
% 0.20/0.48 thf(fact_315_insert__mono,axiom,
% 0.20/0.48 ! [C4: set_Pr1912306449_a_nat,D2: set_Pr1912306449_a_nat,A: produc598811825_a_nat] :
% 0.20/0.48 ( ( ord_le1374266033_a_nat @ C4 @ D2 )
% 0.20/0.48 => ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ A @ C4 ) @ ( insert2058869377_a_nat @ A @ D2 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % insert_mono
% 0.20/0.48 thf(fact_316_subset__insert,axiom,
% 0.20/0.48 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ~ ( member327031898_a_nat @ X4 @ A4 )
% 0.20/0.48 => ( ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ X4 @ B4 ) )
% 0.20/0.48 = ( ord_le1374266033_a_nat @ A4 @ B4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_insert
% 0.20/0.48 thf(fact_317_subset__insert,axiom,
% 0.20/0.48 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
% 0.20/0.48 ( ~ ( member1285940496od_b_b @ X4 @ A4 )
% 0.20/0.48 => ( ( ord_le1036320359od_b_b @ A4 @ ( insert1952693431od_b_b @ X4 @ B4 ) )
% 0.20/0.48 = ( ord_le1036320359od_b_b @ A4 @ B4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_insert
% 0.20/0.48 thf(fact_318_subset__insertI,axiom,
% 0.20/0.48 ! [B4: set_Pr1912306449_a_nat,A: produc598811825_a_nat] : ( ord_le1374266033_a_nat @ B4 @ ( insert2058869377_a_nat @ A @ B4 ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_insertI
% 0.20/0.48 thf(fact_319_subset__insertI2,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
% 0.20/0.48 ( ( ord_le1374266033_a_nat @ A4 @ B4 )
% 0.20/0.48 => ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ B @ B4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_insertI2
% 0.20/0.48 thf(fact_320_trans__Id__on,axiom,
% 0.20/0.48 ! [A4: set_b] : ( trans_b @ ( id_on_b @ A4 ) ) ).
% 0.20/0.48
% 0.20/0.48 % trans_Id_on
% 0.20/0.48 thf(fact_321_refl__on__Id__on,axiom,
% 0.20/0.48 ! [A4: set_b] : ( refl_on_b @ A4 @ ( id_on_b @ A4 ) ) ).
% 0.20/0.48
% 0.20/0.48 % refl_on_Id_on
% 0.20/0.48 thf(fact_322_subset__iff,axiom,
% 0.20/0.48 ( ord_le1036320359od_b_b
% 0.20/0.48 = ( ^ [A5: set_Product_prod_b_b,B6: set_Product_prod_b_b] :
% 0.20/0.48 ! [T: product_prod_b_b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ T @ A5 )
% 0.20/0.48 => ( member1285940496od_b_b @ T @ B6 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_iff
% 0.20/0.48 thf(fact_323_subset__eq,axiom,
% 0.20/0.48 ( ord_le1036320359od_b_b
% 0.20/0.48 = ( ^ [A5: set_Product_prod_b_b,B6: set_Product_prod_b_b] :
% 0.20/0.48 ! [X5: product_prod_b_b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ X5 @ A5 )
% 0.20/0.48 => ( member1285940496od_b_b @ X5 @ B6 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_eq
% 0.20/0.48 thf(fact_324_subsetD,axiom,
% 0.20/0.48 ! [A4: set_Product_prod_b_b,B4: set_Product_prod_b_b,C2: product_prod_b_b] :
% 0.20/0.48 ( ( ord_le1036320359od_b_b @ A4 @ B4 )
% 0.20/0.48 => ( ( member1285940496od_b_b @ C2 @ A4 )
% 0.20/0.48 => ( member1285940496od_b_b @ C2 @ B4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subsetD
% 0.20/0.48 thf(fact_325_in__mono,axiom,
% 0.20/0.48 ! [A4: set_Product_prod_b_b,B4: set_Product_prod_b_b,X4: product_prod_b_b] :
% 0.20/0.48 ( ( ord_le1036320359od_b_b @ A4 @ B4 )
% 0.20/0.48 => ( ( member1285940496od_b_b @ X4 @ A4 )
% 0.20/0.48 => ( member1285940496od_b_b @ X4 @ B4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % in_mono
% 0.20/0.48 thf(fact_326_subset__singletonD,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat,X4: produc598811825_a_nat] :
% 0.20/0.48 ( ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) )
% 0.20/0.48 => ( ( A4 = bot_bo2066531197_a_nat )
% 0.20/0.48 | ( A4
% 0.20/0.48 = ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_singletonD
% 0.20/0.48 thf(fact_327_subset__singleton__iff,axiom,
% 0.20/0.48 ! [X6: set_Pr1912306449_a_nat,A: produc598811825_a_nat] :
% 0.20/0.48 ( ( ord_le1374266033_a_nat @ X6 @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) )
% 0.20/0.48 = ( ( X6 = bot_bo2066531197_a_nat )
% 0.20/0.48 | ( X6
% 0.20/0.48 = ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_singleton_iff
% 0.20/0.48 thf(fact_328_subgraph__trans,axiom,
% 0.20/0.48 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b,G_3: labeled_graph_a_b] :
% 0.20/0.48 ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 0.20/0.48 => ( ( graph_1322245399_a_b_b @ G_2 @ G_3 @ ( id_on_b @ ( labeled_vertices_a_b @ G_2 ) ) )
% 0.20/0.48 => ( graph_1322245399_a_b_b @ G_1 @ G_3 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subgraph_trans
% 0.20/0.48 thf(fact_329_in__quotient__imp__subset,axiom,
% 0.20/0.48 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 => ( ord_less_eq_set_b @ X6 @ A4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % in_quotient_imp_subset
% 0.20/0.48 thf(fact_330_maintainedD2,axiom,
% 0.20/0.48 ! [A4: labeled_graph_a_nat,B4: labeled_graph_a_nat,G: labeled_graph_a_b,F: set_Pr2106913242_nat_b] :
% 0.20/0.48 ( ( maintained_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G )
% 0.20/0.48 => ( ( graph_914848022_nat_b @ A4 @ G @ F )
% 0.20/0.48 => ~ ! [G3: set_Pr2106913242_nat_b] :
% 0.20/0.48 ( ( graph_914848022_nat_b @ B4 @ G @ G3 )
% 0.20/0.48 => ~ ( ord_le1484132922_nat_b @ F @ G3 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % maintainedD2
% 0.20/0.48 thf(fact_331_subgraph__semantics,axiom,
% 0.20/0.48 ! [A4: labeled_graph_a_b,B4: labeled_graph_a_b,A: b,B: b,E: allegorical_term_a] :
% 0.20/0.48 ( ( graph_1322245399_a_b_b @ A4 @ B4 @ ( id_on_b @ ( labeled_vertices_a_b @ A4 ) ) )
% 0.20/0.48 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ A4 @ E ) )
% 0.20/0.48 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ B4 @ E ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subgraph_semantics
% 0.20/0.48 thf(fact_332_proj__iff,axiom,
% 0.20/0.48 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.48 => ( ( ord_less_eq_set_b @ ( insert_b @ X4 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) @ A4 )
% 0.20/0.48 => ( ( ( equiv_proj_b_b @ R2 @ X4 )
% 0.20/0.48 = ( equiv_proj_b_b @ R2 @ Y4 ) )
% 0.20/0.48 = ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % proj_iff
% 0.20/0.48 thf(fact_333_proj__iff,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 0.20/0.48 ( ( equiv_281634663_a_nat @ A4 @ R2 )
% 0.20/0.48 => ( ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ Y4 @ bot_bo2066531197_a_nat ) ) @ A4 )
% 0.20/0.48 => ( ( ( equiv_1873824970_a_nat @ R2 @ X4 )
% 0.20/0.48 = ( equiv_1873824970_a_nat @ R2 @ Y4 ) )
% 0.20/0.48 = ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % proj_iff
% 0.20/0.48 thf(fact_334_consequence__graphD_I1_J,axiom,
% 0.20/0.48 ! [Rs: set_Pr1912306449_a_nat,G: labeled_graph_a_b,R: produc598811825_a_nat] :
% 0.20/0.48 ( ( conseq1058228089_nat_b @ Rs @ G )
% 0.20/0.48 => ( ( member327031898_a_nat @ R @ Rs )
% 0.20/0.48 => ( maintained_a_nat_b @ R @ G ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % consequence_graphD(1)
% 0.20/0.48 thf(fact_335_proj__in__iff,axiom,
% 0.20/0.48 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b] :
% 0.20/0.48 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member1050118384od_b_b @ ( equiv_270653002od_b_b @ R2 @ X4 ) @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 0.20/0.48 = ( member1285940496od_b_b @ X4 @ A4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % proj_in_iff
% 0.20/0.48 thf(fact_336_proj__in__iff,axiom,
% 0.20/0.48 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.48 => ( ( member_set_b @ ( equiv_proj_b_b @ R2 @ X4 ) @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 = ( member_b @ X4 @ A4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % proj_in_iff
% 0.20/0.48 thf(fact_337_less__by__empty,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ( A4 = bot_bo2066531197_a_nat )
% 0.20/0.48 => ( ord_le1374266033_a_nat @ A4 @ B4 ) ) ).
% 0.20/0.48
% 0.20/0.48 % less_by_empty
% 0.20/0.48 thf(fact_338_insert__subsetI,axiom,
% 0.20/0.48 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,X6: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ( member327031898_a_nat @ X4 @ A4 )
% 0.20/0.48 => ( ( ord_le1374266033_a_nat @ X6 @ A4 )
% 0.20/0.48 => ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ X6 ) @ A4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % insert_subsetI
% 0.20/0.48 thf(fact_339_insert__subsetI,axiom,
% 0.20/0.48 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b,X6: set_Product_prod_b_b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ X4 @ A4 )
% 0.20/0.48 => ( ( ord_le1036320359od_b_b @ X6 @ A4 )
% 0.20/0.48 => ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X4 @ X6 ) @ A4 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % insert_subsetI
% 0.20/0.48 thf(fact_340_ssubst__Pair__rhs,axiom,
% 0.20/0.48 ! [R2: b,S: b,R: set_Product_prod_b_b,S2: b] :
% 0.20/0.48 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ R2 @ S ) @ R )
% 0.20/0.48 => ( ( S2 = S )
% 0.20/0.48 => ( member1285940496od_b_b @ ( product_Pair_b_b @ R2 @ S2 ) @ R ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % ssubst_Pair_rhs
% 0.20/0.48 thf(fact_341_subset__emptyI,axiom,
% 0.20/0.48 ! [A4: set_Product_prod_b_b] :
% 0.20/0.48 ( ! [X3: product_prod_b_b] :
% 0.20/0.48 ~ ( member1285940496od_b_b @ X3 @ A4 )
% 0.20/0.48 => ( ord_le1036320359od_b_b @ A4 @ bot_bo1343651123od_b_b ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_emptyI
% 0.20/0.48 thf(fact_342_subset__emptyI,axiom,
% 0.20/0.48 ! [A4: set_Pr1912306449_a_nat] :
% 0.20/0.48 ( ! [X3: produc598811825_a_nat] :
% 0.20/0.48 ~ ( member327031898_a_nat @ X3 @ A4 )
% 0.20/0.48 => ( ord_le1374266033_a_nat @ A4 @ bot_bo2066531197_a_nat ) ) ).
% 0.20/0.48
% 0.20/0.48 % subset_emptyI
% 0.20/0.48 thf(fact_343_subgraph__def,axiom,
% 0.20/0.48 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 0.20/0.48 ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 0.20/0.48 = ( ( G_1
% 0.20/0.48 = ( restrict_a_b @ G_1 ) )
% 0.20/0.48 & ( G_2
% 0.20/0.48 = ( restrict_a_b @ G_2 ) )
% 0.20/0.48 & ( ( graph_union_a_b @ G_1 @ G_2 )
% 0.20/0.48 = G_2 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subgraph_def
% 0.20/0.48 thf(fact_344_graph__union__preserves__restrict,axiom,
% 0.20/0.48 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 0.20/0.48 ( ( G_1
% 0.20/0.48 = ( restrict_a_b @ G_1 ) )
% 0.20/0.48 => ( ( G_2
% 0.20/0.48 = ( restrict_a_b @ G_2 ) )
% 0.20/0.48 => ( ( graph_union_a_b @ G_1 @ G_2 )
% 0.20/0.48 = ( restrict_a_b @ ( graph_union_a_b @ G_1 @ G_2 ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % graph_union_preserves_restrict
% 0.20/0.48 thf(fact_345_univ__preserves,axiom,
% 0.20/0.48 ! [A4: set_b,R2: set_Product_prod_b_b,F: b > product_prod_b_b,B4: set_Product_prod_b_b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R2 )
% 0.20/0.48 => ( ( equiv_1778431934od_b_b @ R2 @ F )
% 0.20/0.48 => ( ! [X3: b] :
% 0.20/0.48 ( ( member_b @ X3 @ A4 )
% 0.20/0.48 => ( member1285940496od_b_b @ ( F @ X3 ) @ B4 ) )
% 0.20/0.48 => ! [X: set_b] :
% 0.20/0.48 ( ( member_set_b @ X @ ( equiv_quotient_b @ A4 @ R2 ) )
% 0.20/0.48 => ( member1285940496od_b_b @ ( bNF_Gr1693995344od_b_b @ F @ X ) @ B4 ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % univ_preserves
% 0.20/0.48 thf(fact_346_relImage__proj,axiom,
% 0.20/0.48 ! [A4: set_b,R: set_Product_prod_b_b] :
% 0.20/0.48 ( ( equiv_equiv_b @ A4 @ R )
% 0.20/0.48 => ( ord_le2142668391_set_b @ ( bNF_Gr345950468_set_b @ R @ ( equiv_proj_b_b @ R ) ) @ ( id_on_set_b @ ( equiv_quotient_b @ A4 @ R ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % relImage_proj
% 0.20/0.48 thf(fact_347_subgraph__def2,axiom,
% 0.20/0.48 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 0.20/0.48 ( ( G_1
% 0.20/0.48 = ( restrict_a_b @ G_1 ) )
% 0.20/0.48 => ( ( G_2
% 0.20/0.48 = ( restrict_a_b @ G_2 ) )
% 0.20/0.48 => ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 0.20/0.48 = ( ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) )
% 0.20/0.48 & ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G_1 ) @ ( labeled_edges_a_b @ G_2 ) ) ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subgraph_def2
% 0.20/0.48 thf(fact_348_labeled__graph_Ocollapse,axiom,
% 0.20/0.48 ! [Labeled_graph: labeled_graph_a_b] :
% 0.20/0.48 ( ( labeled_LG_a_b @ ( labeled_edges_a_b @ Labeled_graph ) @ ( labeled_vertices_a_b @ Labeled_graph ) )
% 0.20/0.48 = Labeled_graph ) ).
% 0.20/0.48
% 0.20/0.48 % labeled_graph.collapse
% 0.20/0.48 thf(fact_349_graph__unionI,axiom,
% 0.20/0.48 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 0.20/0.48 ( ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G_1 ) @ ( labeled_edges_a_b @ G_2 ) )
% 0.20/0.48 => ( ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) )
% 0.20/0.48 => ( ( graph_union_a_b @ G_1 @ G_2 )
% 0.20/0.48 = G_2 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % graph_unionI
% 0.20/0.48 thf(fact_350_restrict__subsD,axiom,
% 0.20/0.48 ! [G: labeled_graph_a_b] :
% 0.20/0.48 ( ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G ) @ ( labeled_edges_a_b @ ( restrict_a_b @ G ) ) )
% 0.20/0.48 => ( G
% 0.20/0.48 = ( restrict_a_b @ G ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % restrict_subsD
% 0.20/0.48 thf(fact_351_labeled__graph_Oexhaust__sel,axiom,
% 0.20/0.48 ! [Labeled_graph: labeled_graph_a_b] :
% 0.20/0.48 ( Labeled_graph
% 0.20/0.48 = ( labeled_LG_a_b @ ( labeled_edges_a_b @ Labeled_graph ) @ ( labeled_vertices_a_b @ Labeled_graph ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % labeled_graph.exhaust_sel
% 0.20/0.48 thf(fact_352_labeled__graph_Oexpand,axiom,
% 0.20/0.48 ! [Labeled_graph: labeled_graph_a_b,Labeled_graph2: labeled_graph_a_b] :
% 0.20/0.48 ( ( ( ( labeled_edges_a_b @ Labeled_graph )
% 0.20/0.48 = ( labeled_edges_a_b @ Labeled_graph2 ) )
% 0.20/0.48 & ( ( labeled_vertices_a_b @ Labeled_graph )
% 0.20/0.48 = ( labeled_vertices_a_b @ Labeled_graph2 ) ) )
% 0.20/0.48 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 0.20/0.48
% 0.20/0.48 % labeled_graph.expand
% 0.20/0.48 thf(fact_353_graph__union__iff,axiom,
% 0.20/0.48 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 0.20/0.48 ( ( ( graph_union_a_b @ G_1 @ G_2 )
% 0.20/0.48 = G_2 )
% 0.20/0.48 = ( ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G_1 ) @ ( labeled_edges_a_b @ G_2 ) )
% 0.20/0.48 & ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % graph_union_iff
% 0.20/0.48 thf(fact_354_subgraph__subset_I2_J,axiom,
% 0.20/0.48 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 0.20/0.48 ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 0.20/0.48 => ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ ( restrict_a_b @ G_1 ) ) @ ( labeled_edges_a_b @ G_2 ) ) ) ).
% 0.20/0.48
% 0.20/0.48 % subgraph_subset(2)
% 0.20/0.48
% 0.20/0.48 % Conjectures (1)
% 0.20/0.48 thf(conj_0,conjecture,
% 0.20/0.48 member1285940496od_b_b @ ( product_Pair_b_b @ z @ y ) @ ( getRel_a_b @ i @ g ) ).
% 0.20/0.48
% 0.20/0.48 %------------------------------------------------------------------------------
% 0.20/0.48 ------------------------
% 3.06/1.56 ---- Embedded in THF ---
% 3.06/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.06/1.56 %%% Generated on Tue Feb 24 08:28:55 EST 2026
% 3.06/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 3.06/1.56 %%% Logic specification used:
% 3.06/1.56 %%% thf(spec, logic, $$dhol).
% 3.06/1.56
% 3.06/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.2AfYOHPK5t/DTF2DTF16406.p
% See solution above
% 3.06/1.56 ------------------------
% 3.27/1.62 ---- Cleaned THF ---
% 3.27/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J,type,
% 3.27/1.62 set_Pr1223907581_a_nat: $tType ).
% 3.27/1.62
% 3.27/1.62 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
% 3.27/1.62 produc588599965_a_nat: $tType ).
% 3.27/1.62
% 3.27/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
% 3.27/1.62 set_Pr1370504391_a_nat: $tType ).
% 3.27/1.62
% 3.27/1.62 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
% 3.27/1.62 produc1734972647_a_nat: $tType ).
% 3.27/1.62
% 3.27/1.62 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
% 3.27/1.62 produc1428649885_a_nat: $tType ).
% 3.27/1.62
% 3.27/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J_J_J,type,
% 3.27/1.62 set_Pr2072721159_a_nat: $tType ).
% 3.27/1.62
% 3.27/1.62 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
% 3.27/1.62 produc2106791463_a_nat: $tType ).
% 3.27/1.62
% 3.27/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
% 3.27/1.62 set_Pr2054759953_a_nat: $tType ).
% 3.27/1.62
% 3.27/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__b_Mtf__b_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_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,
% 3.27/1.62 set_Pr2028709629od_b_b: $tType ).
% 3.27/1.62
% 3.27/1.62 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_Mt__LabeledGraphs__Olabeled____graph_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
% 3.27/1.62 set_Pr21610864_a_nat: $tType ).
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% 3.27/1.62
% 3.27/1.62 thf(fact_6_prod_Oinject,axiom,
% 3.27/1.62 ! [X1: product_prod_b_b,X2: product_prod_b_b,Y1: product_prod_b_b,Y2: product_prod_b_b] :
% 3.27/1.62 ( ( ( produc546497367od_b_b @ X1 @ X2 )
% 3.27/1.62 = ( produc546497367od_b_b @ Y1 @ Y2 ) )
% 3.27/1.62 = ( ( X1 = Y1 )
% 3.27/1.62 & ( X2 = Y2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_7_prod_Oinject,axiom,
% 3.27/1.62 ! [X1: produc598811825_a_nat,X2: produc598811825_a_nat,Y1: produc598811825_a_nat,Y2: produc598811825_a_nat] :
% 3.27/1.62 ( ( ( produc410388695_a_nat @ X1 @ X2 )
% 3.27/1.62 = ( produc410388695_a_nat @ Y1 @ Y2 ) )
% 3.27/1.62 = ( ( X1 = Y1 )
% 3.27/1.62 & ( X2 = Y2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_8_prod_Oinject,axiom,
% 3.27/1.62 ! [X1: a,X2: product_prod_b_b,Y1: a,Y2: product_prod_b_b] :
% 3.27/1.62 ( ( ( produc2135284224od_b_b @ X1 @ X2 )
% 3.27/1.62 = ( produc2135284224od_b_b @ Y1 @ Y2 ) )
% 3.27/1.62 = ( ( X1 = Y1 )
% 3.27/1.62 & ( X2 = Y2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_9_prod_Oinject,axiom,
% 3.27/1.62 ! [X1: b,X2: b,Y1: b,Y2: b] :
% 3.27/1.62 ( ( ( product_Pair_b_b @ X1 @ X2 )
% 3.27/1.62 = ( product_Pair_b_b @ Y1 @ Y2 ) )
% 3.27/1.62 = ( ( X1 = Y1 )
% 3.27/1.62 & ( X2 = Y2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_10_old_Oprod_Oinject,axiom,
% 3.27/1.62 ! [A: labeled_graph_a_nat,B: labeled_graph_a_nat,A2: labeled_graph_a_nat,B2: labeled_graph_a_nat] :
% 3.27/1.62 ( ( ( produc2000589545_a_nat @ A @ B )
% 3.27/1.62 = ( produc2000589545_a_nat @ A2 @ B2 ) )
% 3.27/1.62 = ( ( A = A2 )
% 3.27/1.62 & ( B = B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_11_old_Oprod_Oinject,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,B: product_prod_b_b,A2: product_prod_b_b,B2: product_prod_b_b] :
% 3.27/1.62 ( ( ( produc546497367od_b_b @ A @ B )
% 3.27/1.62 = ( produc546497367od_b_b @ A2 @ B2 ) )
% 3.27/1.62 = ( ( A = A2 )
% 3.27/1.62 & ( B = B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_12_old_Oprod_Oinject,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,A2: produc598811825_a_nat,B2: produc598811825_a_nat] :
% 3.27/1.62 ( ( ( produc410388695_a_nat @ A @ B )
% 3.27/1.62 = ( produc410388695_a_nat @ A2 @ B2 ) )
% 3.27/1.62 = ( ( A = A2 )
% 3.27/1.62 & ( B = B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_13_old_Oprod_Oinject,axiom,
% 3.27/1.62 ! [A: a,B: product_prod_b_b,A2: a,B2: product_prod_b_b] :
% 3.27/1.62 ( ( ( produc2135284224od_b_b @ A @ B )
% 3.27/1.62 = ( produc2135284224od_b_b @ A2 @ B2 ) )
% 3.27/1.62 = ( ( A = A2 )
% 3.27/1.62 & ( B = B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_14_old_Oprod_Oinject,axiom,
% 3.27/1.62 ! [A: b,B: b,A2: b,B2: b] :
% 3.27/1.62 ( ( ( product_Pair_b_b @ A @ B )
% 3.27/1.62 = ( product_Pair_b_b @ A2 @ B2 ) )
% 3.27/1.62 = ( ( A = A2 )
% 3.27/1.62 & ( B = B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_15_surj__pair,axiom,
% 3.27/1.62 ! [P: produc598811825_a_nat] :
% 3.27/1.62 ? [X3: labeled_graph_a_nat,Y3: labeled_graph_a_nat] :
% 3.27/1.62 ( P
% 3.27/1.62 = ( produc2000589545_a_nat @ X3 @ Y3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_16_surj__pair,axiom,
% 3.27/1.62 ! [P: produc1168037607od_b_b] :
% 3.27/1.62 ? [X3: product_prod_b_b,Y3: product_prod_b_b] :
% 3.27/1.62 ( P
% 3.27/1.62 = ( produc546497367od_b_b @ X3 @ Y3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_17_surj__pair,axiom,
% 3.27/1.62 ! [P: produc1099860071_a_nat] :
% 3.27/1.62 ? [X3: produc598811825_a_nat,Y3: produc598811825_a_nat] :
% 3.27/1.62 ( P
% 3.27/1.62 = ( produc410388695_a_nat @ X3 @ Y3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_18_surj__pair,axiom,
% 3.27/1.62 ! [P: produc1133639312od_b_b] :
% 3.27/1.62 ? [X3: a,Y3: product_prod_b_b] :
% 3.27/1.62 ( P
% 3.27/1.62 = ( produc2135284224od_b_b @ X3 @ Y3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_19_surj__pair,axiom,
% 3.27/1.62 ! [P: product_prod_b_b] :
% 3.27/1.62 ? [X3: b,Y3: b] :
% 3.27/1.62 ( P
% 3.27/1.62 = ( product_Pair_b_b @ X3 @ Y3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_20_prod__cases,axiom,
% 3.27/1.62 ! [P2: produc598811825_a_nat > $o,P: produc598811825_a_nat] :
% 3.27/1.62 ( ! [A3: labeled_graph_a_nat,B3: labeled_graph_a_nat] : ( P2 @ ( produc2000589545_a_nat @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ P ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_21_prod__cases,axiom,
% 3.27/1.62 ! [P2: produc1168037607od_b_b > $o,P: produc1168037607od_b_b] :
% 3.27/1.62 ( ! [A3: product_prod_b_b,B3: product_prod_b_b] : ( P2 @ ( produc546497367od_b_b @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ P ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_22_prod__cases,axiom,
% 3.27/1.62 ! [P2: produc1099860071_a_nat > $o,P: produc1099860071_a_nat] :
% 3.27/1.62 ( ! [A3: produc598811825_a_nat,B3: produc598811825_a_nat] : ( P2 @ ( produc410388695_a_nat @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ P ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_23_prod__cases,axiom,
% 3.27/1.62 ! [P2: produc1133639312od_b_b > $o,P: produc1133639312od_b_b] :
% 3.27/1.62 ( ! [A3: a,B3: product_prod_b_b] : ( P2 @ ( produc2135284224od_b_b @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ P ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_24_prod__cases,axiom,
% 3.27/1.62 ! [P2: product_prod_b_b > $o,P: product_prod_b_b] :
% 3.27/1.62 ( ! [A3: b,B3: b] : ( P2 @ ( product_Pair_b_b @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ P ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_25_Pair__inject,axiom,
% 3.27/1.62 ! [A: labeled_graph_a_nat,B: labeled_graph_a_nat,A2: labeled_graph_a_nat,B2: labeled_graph_a_nat] :
% 3.27/1.62 ( ( ( produc2000589545_a_nat @ A @ B )
% 3.27/1.62 = ( produc2000589545_a_nat @ A2 @ B2 ) )
% 3.27/1.62 => ~ ( ( A = A2 )
% 3.27/1.62 => ( B != B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_26_Pair__inject,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,B: product_prod_b_b,A2: product_prod_b_b,B2: product_prod_b_b] :
% 3.27/1.62 ( ( ( produc546497367od_b_b @ A @ B )
% 3.27/1.62 = ( produc546497367od_b_b @ A2 @ B2 ) )
% 3.27/1.62 => ~ ( ( A = A2 )
% 3.27/1.62 => ( B != B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_27_Pair__inject,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,A2: produc598811825_a_nat,B2: produc598811825_a_nat] :
% 3.27/1.62 ( ( ( produc410388695_a_nat @ A @ B )
% 3.27/1.62 = ( produc410388695_a_nat @ A2 @ B2 ) )
% 3.27/1.62 => ~ ( ( A = A2 )
% 3.27/1.62 => ( B != B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_28_Pair__inject,axiom,
% 3.27/1.62 ! [A: a,B: product_prod_b_b,A2: a,B2: product_prod_b_b] :
% 3.27/1.62 ( ( ( produc2135284224od_b_b @ A @ B )
% 3.27/1.62 = ( produc2135284224od_b_b @ A2 @ B2 ) )
% 3.27/1.62 => ~ ( ( A = A2 )
% 3.27/1.62 => ( B != B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_29_Pair__inject,axiom,
% 3.27/1.62 ! [A: b,B: b,A2: b,B2: b] :
% 3.27/1.62 ( ( ( product_Pair_b_b @ A @ B )
% 3.27/1.62 = ( product_Pair_b_b @ A2 @ B2 ) )
% 3.27/1.62 => ~ ( ( A = A2 )
% 3.27/1.62 => ( B != B2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_30_prod__cases3,axiom,
% 3.27/1.62 ! [Y4: produc1168037607od_b_b] :
% 3.27/1.62 ~ ! [A3: product_prod_b_b,B3: b,C: b] :
% 3.27/1.62 ( Y4
% 3.27/1.62 != ( produc546497367od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_31_prod__cases3,axiom,
% 3.27/1.62 ! [Y4: produc1099860071_a_nat] :
% 3.27/1.62 ~ ! [A3: produc598811825_a_nat,B3: labeled_graph_a_nat,C: labeled_graph_a_nat] :
% 3.27/1.62 ( Y4
% 3.27/1.62 != ( produc410388695_a_nat @ A3 @ ( produc2000589545_a_nat @ B3 @ C ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_32_prod__cases3,axiom,
% 3.27/1.62 ! [Y4: produc1133639312od_b_b] :
% 3.27/1.62 ~ ! [A3: a,B3: b,C: b] :
% 3.27/1.62 ( Y4
% 3.27/1.62 != ( produc2135284224od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_33_old_Oprod_Oinducts,axiom,
% 3.27/1.62 ! [P2: produc598811825_a_nat > $o,Prod: produc598811825_a_nat] :
% 3.27/1.62 ( ! [A3: labeled_graph_a_nat,B3: labeled_graph_a_nat] : ( P2 @ ( produc2000589545_a_nat @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ Prod ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_34_old_Oprod_Oinducts,axiom,
% 3.27/1.62 ! [P2: produc1168037607od_b_b > $o,Prod: produc1168037607od_b_b] :
% 3.27/1.62 ( ! [A3: product_prod_b_b,B3: product_prod_b_b] : ( P2 @ ( produc546497367od_b_b @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ Prod ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_35_old_Oprod_Oinducts,axiom,
% 3.27/1.62 ! [P2: produc1099860071_a_nat > $o,Prod: produc1099860071_a_nat] :
% 3.27/1.62 ( ! [A3: produc598811825_a_nat,B3: produc598811825_a_nat] : ( P2 @ ( produc410388695_a_nat @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ Prod ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_36_old_Oprod_Oinducts,axiom,
% 3.27/1.62 ! [P2: produc1133639312od_b_b > $o,Prod: produc1133639312od_b_b] :
% 3.27/1.62 ( ! [A3: a,B3: product_prod_b_b] : ( P2 @ ( produc2135284224od_b_b @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ Prod ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_37_old_Oprod_Oinducts,axiom,
% 3.27/1.62 ! [P2: product_prod_b_b > $o,Prod: product_prod_b_b] :
% 3.27/1.62 ( ! [A3: b,B3: b] : ( P2 @ ( product_Pair_b_b @ A3 @ B3 ) )
% 3.27/1.62 => ( P2 @ Prod ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_38_old_Oprod_Oexhaust,axiom,
% 3.27/1.62 ! [Y4: produc598811825_a_nat] :
% 3.27/1.62 ~ ! [A3: labeled_graph_a_nat,B3: labeled_graph_a_nat] :
% 3.27/1.62 ( Y4
% 3.27/1.62 != ( produc2000589545_a_nat @ A3 @ B3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_39_old_Oprod_Oexhaust,axiom,
% 3.27/1.62 ! [Y4: produc1168037607od_b_b] :
% 3.27/1.62 ~ ! [A3: product_prod_b_b,B3: product_prod_b_b] :
% 3.27/1.62 ( Y4
% 3.27/1.62 != ( produc546497367od_b_b @ A3 @ B3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_40_old_Oprod_Oexhaust,axiom,
% 3.27/1.62 ! [Y4: produc1099860071_a_nat] :
% 3.27/1.62 ~ ! [A3: produc598811825_a_nat,B3: produc598811825_a_nat] :
% 3.27/1.62 ( Y4
% 3.27/1.62 != ( produc410388695_a_nat @ A3 @ B3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_41_old_Oprod_Oexhaust,axiom,
% 3.27/1.62 ! [Y4: produc1133639312od_b_b] :
% 3.27/1.62 ~ ! [A3: a,B3: product_prod_b_b] :
% 3.27/1.62 ( Y4
% 3.27/1.62 != ( produc2135284224od_b_b @ A3 @ B3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_42_old_Oprod_Oexhaust,axiom,
% 3.27/1.62 ! [Y4: product_prod_b_b] :
% 3.27/1.62 ~ ! [A3: b,B3: b] :
% 3.27/1.62 ( Y4
% 3.27/1.62 != ( product_Pair_b_b @ A3 @ B3 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_43_prod__induct3,axiom,
% 3.27/1.62 ! [P2: produc1168037607od_b_b > $o,X4: produc1168037607od_b_b] :
% 3.27/1.62 ( ! [A3: product_prod_b_b,B3: b,C: b] : ( P2 @ ( produc546497367od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) )
% 3.27/1.62 => ( P2 @ X4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_44_prod__induct3,axiom,
% 3.27/1.62 ! [P2: produc1099860071_a_nat > $o,X4: produc1099860071_a_nat] :
% 3.27/1.62 ( ! [A3: produc598811825_a_nat,B3: labeled_graph_a_nat,C: labeled_graph_a_nat] : ( P2 @ ( produc410388695_a_nat @ A3 @ ( produc2000589545_a_nat @ B3 @ C ) ) )
% 3.27/1.62 => ( P2 @ X4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_45_prod__induct3,axiom,
% 3.27/1.62 ! [P2: produc1133639312od_b_b > $o,X4: produc1133639312od_b_b] :
% 3.27/1.62 ( ! [A3: a,B3: b,C: b] : ( P2 @ ( produc2135284224od_b_b @ A3 @ ( product_Pair_b_b @ B3 @ C ) ) )
% 3.27/1.62 => ( P2 @ X4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_46_vertices__restrict,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b] :
% 3.27/1.62 ( ( labeled_vertices_a_b @ ( restrict_a_b @ G ) )
% 3.27/1.62 = ( labeled_vertices_a_b @ G ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_47_getRel__dom_I1_J,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b,A: b,B: b,L: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restrict_a_b @ G ) )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel_a_b @ L @ G ) )
% 3.27/1.62 => ( member_b @ A @ ( labeled_vertices_a_b @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_48_getRel__dom_I2_J,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b,A: b,B: b,L: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restrict_a_b @ G ) )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( getRel_a_b @ L @ G ) )
% 3.27/1.62 => ( member_b @ B @ ( labeled_vertices_a_b @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_49_transitive__rule,axiom,
% 3.27/1.62 ! [G: labele553847424_a_nat,L: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restri1808752350_a_nat @ G ) )
% 3.27/1.62 => ( ( mainta535344367_a_nat @ ( standa390703484rule_a @ L ) @ G )
% 3.27/1.62 => ( trans_1002055497_a_nat @ ( getRel1657740669_a_nat @ L @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_50_transitive__rule,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b,L: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restrict_a_b @ G ) )
% 3.27/1.62 => ( ( maintained_a_nat_b @ ( standa390703484rule_a @ L ) @ G )
% 3.27/1.62 => ( trans_b @ ( getRel_a_b @ L @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_51_maintained__refl,axiom,
% 3.27/1.62 ! [R: labeled_graph_a_nat,G: labeled_graph_a_b] : ( maintained_a_nat_b @ ( produc2000589545_a_nat @ R @ R ) @ G ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_52_reflexivity__rule,axiom,
% 3.27/1.62 ! [G: labele488176869od_b_b,L: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restri431340145od_b_b @ G ) )
% 3.27/1.62 => ( ( mainta552012832od_b_b @ ( standa1329480012rule_a @ L ) @ G )
% 3.27/1.62 => ( refl_o2134473190od_b_b @ ( labele240831545od_b_b @ G ) @ ( getRel1183868946od_b_b @ L @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_53_reflexivity__rule,axiom,
% 3.27/1.62 ! [G: labele1586099759_a_nat,L: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restri147273531_a_nat @ G ) )
% 3.27/1.62 => ( ( mainta1627462506_a_nat @ ( standa1329480012rule_a @ L ) @ G )
% 3.27/1.62 => ( refl_o206400048_a_nat @ ( labele198153475_a_nat @ G ) @ ( getRel1248627292_a_nat @ L @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_54_reflexivity__rule,axiom,
% 3.27/1.62 ! [G: labele553847424_a_nat,L: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restri1808752350_a_nat @ G ) )
% 3.27/1.62 => ( ( mainta535344367_a_nat @ ( standa1329480012rule_a @ L ) @ G )
% 3.27/1.62 => ( refl_o1018941609_a_nat @ ( labele953802518_a_nat @ G ) @ ( getRel1657740669_a_nat @ L @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_55_reflexivity__rule,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b,L: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restrict_a_b @ G ) )
% 3.27/1.62 => ( ( maintained_a_nat_b @ ( standa1329480012rule_a @ L ) @ G )
% 3.27/1.62 => ( refl_on_b @ ( labeled_vertices_a_b @ G ) @ ( getRel_a_b @ L @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_56_restrict__idemp,axiom,
% 3.27/1.62 ! [X4: labeled_graph_a_b] :
% 3.27/1.62 ( ( restrict_a_b @ ( restrict_a_b @ X4 ) )
% 3.27/1.62 = ( restrict_a_b @ X4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_57_refl__on__domain,axiom,
% 3.27/1.62 ! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,A: produc1168037607od_b_b,B: produc1168037607od_b_b] :
% 3.27/1.62 ( ( refl_o1941611494od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ A @ B ) @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ A @ A4 )
% 3.27/1.62 & ( member1652792080od_b_b @ B @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_58_refl__on__domain,axiom,
% 3.27/1.62 ! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,A: set_Pr1912306449_a_nat,B: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( refl_o730481808_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member1072844688_a_nat @ ( produc1767291095_a_nat @ A @ B ) @ R2 )
% 3.27/1.62 => ( ( member101535418_a_nat @ A @ A4 )
% 3.27/1.62 & ( member101535418_a_nat @ B @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_59_refl__on__domain,axiom,
% 3.27/1.62 ! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
% 3.27/1.62 ( ( refl_o1899808198od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1009730320od_b_b @ ( produc1027169111od_b_b @ A @ B ) @ R2 )
% 3.27/1.62 => ( ( member1050118384od_b_b @ A @ A4 )
% 3.27/1.62 & ( member1050118384od_b_b @ B @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_60_refl__on__domain,axiom,
% 3.27/1.62 ! [A4: set_set_b,R2: set_Pr1463111367_set_b,A: set_b,B: set_b] :
% 3.27/1.62 ( ( refl_on_set_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1199885584_set_b @ ( produc1403713879_set_b @ A @ B ) @ R2 )
% 3.27/1.62 => ( ( member_set_b @ A @ A4 )
% 3.27/1.62 & ( member_set_b @ B @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_61_refl__on__domain,axiom,
% 3.27/1.62 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,A: labeled_graph_a_nat,B: labeled_graph_a_nat] :
% 3.27/1.62 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ A @ B ) @ R2 )
% 3.27/1.62 => ( ( member154916863_a_nat @ A @ A4 )
% 3.27/1.62 & ( member154916863_a_nat @ B @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_62_refl__on__domain,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,A: produc598811825_a_nat,B: produc598811825_a_nat] :
% 3.27/1.62 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ A @ B ) @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ A @ A4 )
% 3.27/1.62 & ( member327031898_a_nat @ B @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_63_refl__on__domain,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b,B: product_prod_b_b] :
% 3.27/1.62 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ A @ A4 )
% 3.27/1.62 & ( member1285940496od_b_b @ B @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_64_refl__on__domain,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,A: b,B: b] :
% 3.27/1.62 ( ( refl_on_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 )
% 3.27/1.62 => ( ( member_b @ A @ A4 )
% 3.27/1.62 & ( member_b @ B @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_65_refl__onD2,axiom,
% 3.27/1.62 ! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,X4: produc1168037607od_b_b,Y4: produc1168037607od_b_b] :
% 3.27/1.62 ( ( refl_o1941611494od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member1652792080od_b_b @ Y4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_66_refl__onD2,axiom,
% 3.27/1.62 ! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,X4: set_Pr1912306449_a_nat,Y4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( refl_o730481808_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member1072844688_a_nat @ ( produc1767291095_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member101535418_a_nat @ Y4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_67_refl__onD2,axiom,
% 3.27/1.62 ! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,X4: set_Product_prod_b_b,Y4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( refl_o1899808198od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1009730320od_b_b @ ( produc1027169111od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member1050118384od_b_b @ Y4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_68_refl__onD2,axiom,
% 3.27/1.62 ! [A4: set_set_b,R2: set_Pr1463111367_set_b,X4: set_b,Y4: set_b] :
% 3.27/1.62 ( ( refl_on_set_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1199885584_set_b @ ( produc1403713879_set_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member_set_b @ Y4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_69_refl__onD2,axiom,
% 3.27/1.62 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat] :
% 3.27/1.62 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member154916863_a_nat @ Y4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_70_refl__onD2,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 3.27/1.62 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member327031898_a_nat @ Y4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_71_refl__onD2,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 3.27/1.62 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ Y4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_72_refl__onD2,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
% 3.27/1.62 ( ( refl_on_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member_b @ Y4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_73_transD,axiom,
% 3.27/1.62 ! [R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat,Z: labeled_graph_a_nat] :
% 3.27/1.62 ( ( trans_1002055497_a_nat @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y4 @ Z ) @ R2 )
% 3.27/1.62 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_74_transD,axiom,
% 3.27/1.62 ! [R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b,Z: product_prod_b_b] :
% 3.27/1.62 ( ( trans_1808857926od_b_b @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y4 @ Z ) @ R2 )
% 3.27/1.62 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_75_transD,axiom,
% 3.27/1.62 ! [R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat,Z: produc598811825_a_nat] :
% 3.27/1.62 ( ( trans_444496784_a_nat @ R2 )
% 3.27/1.62 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y4 @ Z ) @ R2 )
% 3.27/1.62 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_76_transD,axiom,
% 3.27/1.62 ! [R2: set_Product_prod_b_b,X4: b,Y4: b,Z: b] :
% 3.27/1.62 ( ( trans_b @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_77_transE,axiom,
% 3.27/1.62 ! [R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat,Z: labeled_graph_a_nat] :
% 3.27/1.62 ( ( trans_1002055497_a_nat @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y4 @ Z ) @ R2 )
% 3.27/1.62 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_78_transE,axiom,
% 3.27/1.62 ! [R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b,Z: product_prod_b_b] :
% 3.27/1.62 ( ( trans_1808857926od_b_b @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y4 @ Z ) @ R2 )
% 3.27/1.62 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_79_transE,axiom,
% 3.27/1.62 ! [R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat,Z: produc598811825_a_nat] :
% 3.27/1.62 ( ( trans_444496784_a_nat @ R2 )
% 3.27/1.62 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y4 @ Z ) @ R2 )
% 3.27/1.62 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_80_transE,axiom,
% 3.27/1.62 ! [R2: set_Product_prod_b_b,X4: b,Y4: b,Z: b] :
% 3.27/1.62 ( ( trans_b @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Z ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_81_transI,axiom,
% 3.27/1.62 ! [R2: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ! [X3: labeled_graph_a_nat,Y3: labeled_graph_a_nat,Z2: labeled_graph_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X3 @ Y3 ) @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y3 @ Z2 ) @ R2 )
% 3.27/1.62 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ X3 @ Z2 ) @ R2 ) ) )
% 3.27/1.62 => ( trans_1002055497_a_nat @ R2 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_82_transI,axiom,
% 3.27/1.62 ! [R2: set_Pr2007082183od_b_b] :
% 3.27/1.62 ( ! [X3: product_prod_b_b,Y3: product_prod_b_b,Z2: product_prod_b_b] :
% 3.27/1.62 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X3 @ Y3 ) @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ Z2 ) @ R2 )
% 3.27/1.62 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ X3 @ Z2 ) @ R2 ) ) )
% 3.27/1.62 => ( trans_1808857926od_b_b @ R2 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_83_transI,axiom,
% 3.27/1.62 ! [R2: set_Pr2074180935_a_nat] :
% 3.27/1.62 ( ! [X3: produc598811825_a_nat,Y3: produc598811825_a_nat,Z2: produc598811825_a_nat] :
% 3.27/1.62 ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X3 @ Y3 ) @ R2 )
% 3.27/1.62 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y3 @ Z2 ) @ R2 )
% 3.27/1.62 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X3 @ Z2 ) @ R2 ) ) )
% 3.27/1.62 => ( trans_444496784_a_nat @ R2 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_84_transI,axiom,
% 3.27/1.62 ! [R2: set_Product_prod_b_b] :
% 3.27/1.62 ( ! [X3: b,Y3: b,Z2: b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ Z2 ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Z2 ) @ R2 ) ) )
% 3.27/1.62 => ( trans_b @ R2 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_85_Relation_Otrans__def,axiom,
% 3.27/1.62 ( trans_1002055497_a_nat
% 3.27/1.62 = ( ^ [R3: set_Pr1912306449_a_nat] :
% 3.27/1.62 ! [X5: labeled_graph_a_nat,Y5: labeled_graph_a_nat,Z3: labeled_graph_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X5 @ Y5 ) @ R3 )
% 3.27/1.62 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ Y5 @ Z3 ) @ R3 )
% 3.27/1.62 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_86_Relation_Otrans__def,axiom,
% 3.27/1.62 ( trans_1808857926od_b_b
% 3.27/1.62 = ( ^ [R3: set_Pr2007082183od_b_b] :
% 3.27/1.62 ! [X5: product_prod_b_b,Y5: product_prod_b_b,Z3: product_prod_b_b] :
% 3.27/1.62 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X5 @ Y5 ) @ R3 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y5 @ Z3 ) @ R3 )
% 3.27/1.62 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_87_Relation_Otrans__def,axiom,
% 3.27/1.62 ( trans_444496784_a_nat
% 3.27/1.62 = ( ^ [R3: set_Pr2074180935_a_nat] :
% 3.27/1.62 ! [X5: produc598811825_a_nat,Y5: produc598811825_a_nat,Z3: produc598811825_a_nat] :
% 3.27/1.62 ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X5 @ Y5 ) @ R3 )
% 3.27/1.62 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ Y5 @ Z3 ) @ R3 )
% 3.27/1.62 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_88_Relation_Otrans__def,axiom,
% 3.27/1.62 ( trans_b
% 3.27/1.62 = ( ^ [R3: set_Product_prod_b_b] :
% 3.27/1.62 ! [X5: b,Y5: b,Z3: b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Y5 ) @ R3 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y5 @ Z3 ) @ R3 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Z3 ) @ R3 ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_89_refl__onD,axiom,
% 3.27/1.62 ! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,A: produc1168037607od_b_b] :
% 3.27/1.62 ( ( refl_o1941611494od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ A @ A4 )
% 3.27/1.62 => ( member1829919504od_b_b @ ( produc1403229015od_b_b @ A @ A ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_90_refl__onD,axiom,
% 3.27/1.62 ! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,A: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( refl_o730481808_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member101535418_a_nat @ A @ A4 )
% 3.27/1.62 => ( member1072844688_a_nat @ ( produc1767291095_a_nat @ A @ A ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_91_refl__onD,axiom,
% 3.27/1.62 ! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,A: set_Product_prod_b_b] :
% 3.27/1.62 ( ( refl_o1899808198od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1050118384od_b_b @ A @ A4 )
% 3.27/1.62 => ( member1009730320od_b_b @ ( produc1027169111od_b_b @ A @ A ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_92_refl__onD,axiom,
% 3.27/1.62 ! [A4: set_set_b,R2: set_Pr1463111367_set_b,A: set_b] :
% 3.27/1.62 ( ( refl_on_set_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member_set_b @ A @ A4 )
% 3.27/1.62 => ( member1199885584_set_b @ ( produc1403713879_set_b @ A @ A ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_93_refl__onD,axiom,
% 3.27/1.62 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,A: labeled_graph_a_nat] :
% 3.27/1.62 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member154916863_a_nat @ A @ A4 )
% 3.27/1.62 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ A @ A ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_94_refl__onD,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,A: produc598811825_a_nat] :
% 3.27/1.62 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ A @ A4 )
% 3.27/1.62 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ A @ A ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_95_refl__onD,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,A: product_prod_b_b] :
% 3.27/1.62 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ A @ A4 )
% 3.27/1.62 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ A ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_96_refl__onD,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,A: b] :
% 3.27/1.62 ( ( refl_on_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member_b @ A @ A4 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ A ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_97_refl__onD1,axiom,
% 3.27/1.62 ! [A4: set_Pr2007082183od_b_b,R2: set_Pr454079687od_b_b,X4: produc1168037607od_b_b,Y4: produc1168037607od_b_b] :
% 3.27/1.62 ( ( refl_o1941611494od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member1652792080od_b_b @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_98_refl__onD1,axiom,
% 3.27/1.62 ! [A4: set_se480333169_a_nat,R2: set_Pr1045675335_a_nat,X4: set_Pr1912306449_a_nat,Y4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( refl_o730481808_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member1072844688_a_nat @ ( produc1767291095_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member101535418_a_nat @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_99_refl__onD1,axiom,
% 3.27/1.62 ! [A4: set_se668355239od_b_b,R2: set_Pr773647559od_b_b,X4: set_Product_prod_b_b,Y4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( refl_o1899808198od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1009730320od_b_b @ ( produc1027169111od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member1050118384od_b_b @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_100_refl__onD1,axiom,
% 3.27/1.62 ! [A4: set_set_b,R2: set_Pr1463111367_set_b,X4: set_b,Y4: set_b] :
% 3.27/1.62 ( ( refl_on_set_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1199885584_set_b @ ( produc1403713879_set_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member_set_b @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_101_refl__onD1,axiom,
% 3.27/1.62 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat,X4: labeled_graph_a_nat,Y4: labeled_graph_a_nat] :
% 3.27/1.62 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ ( produc2000589545_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member154916863_a_nat @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_102_refl__onD1,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 3.27/1.62 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member327031898_a_nat @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_103_refl__onD1,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 3.27/1.62 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_104_refl__onD1,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
% 3.27/1.62 ( ( refl_on_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member_b @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_105_mA,axiom,
% 3.27/1.62 ! [X: produc598811825_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ X @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ i ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ i ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ i ) @ bot_bo2066531197_a_nat ) ) ) )
% 3.27/1.62 => ( maintained_a_nat_b @ X @ g ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_106_preorder__on__def,axiom,
% 3.27/1.62 ( order_1195372542od_b_b
% 3.27/1.62 = ( ^ [A5: set_Product_prod_b_b,R3: set_Pr2007082183od_b_b] :
% 3.27/1.62 ( ( refl_o2134473190od_b_b @ A5 @ R3 )
% 3.27/1.62 & ( trans_1808857926od_b_b @ R3 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_107_preorder__on__def,axiom,
% 3.27/1.62 ( order_145089608_a_nat
% 3.27/1.62 = ( ^ [A5: set_Pr1912306449_a_nat,R3: set_Pr2074180935_a_nat] :
% 3.27/1.62 ( ( refl_o206400048_a_nat @ A5 @ R3 )
% 3.27/1.62 & ( trans_444496784_a_nat @ R3 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_108_preorder__on__def,axiom,
% 3.27/1.62 ( order_989282449_a_nat
% 3.27/1.62 = ( ^ [A5: set_la1108361502_a_nat,R3: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( refl_o1018941609_a_nat @ A5 @ R3 )
% 3.27/1.62 & ( trans_1002055497_a_nat @ R3 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_109_preorder__on__def,axiom,
% 3.27/1.62 ( order_preorder_on_b
% 3.27/1.62 = ( ^ [A5: set_b,R3: set_Product_prod_b_b] :
% 3.27/1.62 ( ( refl_on_b @ A5 @ R3 )
% 3.27/1.62 & ( trans_b @ R3 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_110_on__triple,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: b,V4: b,R: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1694913744od_b_b @ ( produc1037687191od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_triple_b_b_a @ R ) )
% 3.27/1.62 = ( ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member1285940496od_b_b @ ( product_Pair_b_b @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_111_on__triple,axiom,
% 3.27/1.62 ! [L1: a,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: a,V3: b,V4: b,R: set_Pr471017666_nat_b] :
% 3.27/1.62 ( ( member442676358od_b_b @ ( produc179745357od_b_b @ ( produc1110849866_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1993240081at_b_a @ R ) )
% 3.27/1.62 = ( ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_112_on__triple,axiom,
% 3.27/1.62 ! [L1: a,V1: product_prod_b_b,V2: product_prod_b_b,L2: a,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
% 3.27/1.62 ( ( member1023210704od_b_b @ ( produc1116925335od_b_b @ ( produc690925568od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1764442556_b_b_a @ R ) )
% 3.27/1.62 = ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_113_on__triple,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: b,V4: b,R: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri111564221od_b_b @ R ) )
% 3.27/1.62 = ( ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member1285940496od_b_b @ ( product_Pair_b_b @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_114_on__triple,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 3.27/1.62 ( ( member817785114_a_nat @ ( produc966067681_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1110849866_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri2117183935_nat_a @ R ) )
% 3.27/1.62 = ( ( member763519277_a_nat @ ( produc1311571856_a_nat @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member763519277_a_nat @ ( produc1311571856_a_nat @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_115_on__triple,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 3.27/1.62 ( ( member962139856od_b_b @ ( produc789452695od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc690925568od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri2003683308_b_b_a @ R ) )
% 3.27/1.62 = ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member641857272od_b_b @ ( produc1257047359od_b_b @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_116_on__triple,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr471017666_nat_b] :
% 3.27/1.62 ( ( member670041670od_b_b @ ( produc926922637od_b_b @ ( produc90796321_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1706508026od_b_b @ R ) )
% 3.27/1.62 = ( ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member1885627811_nat_b @ ( produc1600512702_nat_b @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_117_on__triple,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: product_prod_b_b,V2: product_prod_b_b,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
% 3.27/1.62 ( ( member178477328od_b_b @ ( produc885982551od_b_b @ ( produc1850071895od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri97791909od_b_b @ R ) )
% 3.27/1.62 = ( ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member351494440_b_b_b @ ( produc1001682799_b_b_b @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_118_on__triple,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 3.27/1.62 ( ( member1769525082_a_nat @ ( produc1783652257_a_nat @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc90796321_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1837931688od_b_b @ R ) )
% 3.27/1.62 = ( ( member763519277_a_nat @ ( produc1311571856_a_nat @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member763519277_a_nat @ ( produc1311571856_a_nat @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_119_on__triple,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 3.27/1.62 ( ( member1432657168od_b_b @ ( produc962089303od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1850071895od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri927008213od_b_b @ R ) )
% 3.27/1.62 = ( ( member641857272od_b_b @ ( produc1257047359od_b_b @ V1 @ V3 ) @ R )
% 3.27/1.62 & ( member641857272od_b_b @ ( produc1257047359od_b_b @ V2 @ V4 ) @ R )
% 3.27/1.62 & ( L1 = L2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_120_maintained__then__fin__maintained,axiom,
% 3.27/1.62 ! [R: produc598811825_a_nat,G: labeled_graph_a_b] :
% 3.27/1.62 ( ( maintained_a_nat_b @ R @ G )
% 3.27/1.62 => ( fin_ma603645735_nat_b @ R @ G ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_121_refl__trans__impl__idempotent,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b] :
% 3.27/1.62 ( ( refl_o2134473190od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( trans_1808857926od_b_b @ R2 )
% 3.27/1.62 => ( idempo371692375od_b_b @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_122_refl__trans__impl__idempotent,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat] :
% 3.27/1.62 ( ( refl_o206400048_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( trans_444496784_a_nat @ R2 )
% 3.27/1.62 => ( idempo365306145_a_nat @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_123_refl__trans__impl__idempotent,axiom,
% 3.27/1.62 ! [A4: set_la1108361502_a_nat,R2: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( refl_o1018941609_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( trans_1002055497_a_nat @ R2 )
% 3.27/1.62 => ( idempo848975736_a_nat @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_124_refl__trans__impl__idempotent,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b] :
% 3.27/1.62 ( ( refl_on_b @ A4 @ R2 )
% 3.27/1.62 => ( ( trans_b @ R2 )
% 3.27/1.62 => ( idempotent_b @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_125_equivalence,axiom,
% 3.27/1.62 ! [G: labele1586099759_a_nat,I: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restri147273531_a_nat @ G ) )
% 3.27/1.62 => ( ! [X3: produc598811825_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ X3 @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ I ) @ bot_bo2066531197_a_nat ) ) ) )
% 3.27/1.62 => ( mainta1627462506_a_nat @ X3 @ G ) )
% 3.27/1.62 => ( equiv_281634663_a_nat @ ( labele198153475_a_nat @ G ) @ ( getRel1248627292_a_nat @ I @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_126_equivalence,axiom,
% 3.27/1.62 ! [G: labele488176869od_b_b,I: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restri431340145od_b_b @ G ) )
% 3.27/1.62 => ( ! [X3: produc598811825_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ X3 @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ I ) @ bot_bo2066531197_a_nat ) ) ) )
% 3.27/1.62 => ( mainta552012832od_b_b @ X3 @ G ) )
% 3.27/1.62 => ( equiv_1125628061od_b_b @ ( labele240831545od_b_b @ G ) @ ( getRel1183868946od_b_b @ I @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_127_equivalence,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b,I: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restrict_a_b @ G ) )
% 3.27/1.62 => ( ! [X3: produc598811825_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ X3 @ ( insert2058869377_a_nat @ ( standa1329480012rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa390703484rule_a @ I ) @ ( insert2058869377_a_nat @ ( standa709637829rule_a @ I ) @ bot_bo2066531197_a_nat ) ) ) )
% 3.27/1.62 => ( maintained_a_nat_b @ X3 @ G ) )
% 3.27/1.62 => ( equiv_equiv_b @ ( labeled_vertices_a_b @ G ) @ ( getRel_a_b @ I @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_128_mem__Collect__eq,axiom,
% 3.27/1.62 ! [A: produc1168037607od_b_b,P2: produc1168037607od_b_b > $o] :
% 3.27/1.62 ( ( member1652792080od_b_b @ A @ ( collec1231052114od_b_b @ P2 ) )
% 3.27/1.62 = ( P2 @ A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_129_mem__Collect__eq,axiom,
% 3.27/1.62 ! [A: set_Pr1912306449_a_nat,P2: set_Pr1912306449_a_nat > $o] :
% 3.27/1.62 ( ( member101535418_a_nat @ A @ ( collec971438076_a_nat @ P2 ) )
% 3.27/1.62 = ( P2 @ A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_130_mem__Collect__eq,axiom,
% 3.27/1.62 ! [A: set_Product_prod_b_b,P2: set_Product_prod_b_b > $o] :
% 3.27/1.62 ( ( member1050118384od_b_b @ A @ ( collec1543202610od_b_b @ P2 ) )
% 3.27/1.62 = ( P2 @ A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_131_mem__Collect__eq,axiom,
% 3.27/1.62 ! [A: set_b,P2: set_b > $o] :
% 3.27/1.62 ( ( member_set_b @ A @ ( collect_set_b @ P2 ) )
% 3.27/1.62 = ( P2 @ A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_132_mem__Collect__eq,axiom,
% 3.27/1.62 ! [A: b,P2: b > $o] :
% 3.27/1.62 ( ( member_b @ A @ ( collect_b @ P2 ) )
% 3.27/1.62 = ( P2 @ A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_133_mem__Collect__eq,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,P2: produc598811825_a_nat > $o] :
% 3.27/1.62 ( ( member327031898_a_nat @ A @ ( collec978177436_a_nat @ P2 ) )
% 3.27/1.62 = ( P2 @ A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_134_mem__Collect__eq,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,P2: product_prod_b_b > $o] :
% 3.27/1.62 ( ( member1285940496od_b_b @ A @ ( collec1481886546od_b_b @ P2 ) )
% 3.27/1.62 = ( P2 @ A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_135_Collect__mem__eq,axiom,
% 3.27/1.62 ! [A4: set_Pr2007082183od_b_b] :
% 3.27/1.62 ( ( collec1231052114od_b_b
% 3.27/1.62 @ ^ [X5: produc1168037607od_b_b] : ( member1652792080od_b_b @ X5 @ A4 ) )
% 3.27/1.62 = A4 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_136_Collect__mem__eq,axiom,
% 3.27/1.62 ! [A4: set_se480333169_a_nat] :
% 3.27/1.62 ( ( collec971438076_a_nat
% 3.27/1.62 @ ^ [X5: set_Pr1912306449_a_nat] : ( member101535418_a_nat @ X5 @ A4 ) )
% 3.27/1.62 = A4 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_137_Collect__mem__eq,axiom,
% 3.27/1.62 ! [A4: set_se668355239od_b_b] :
% 3.27/1.62 ( ( collec1543202610od_b_b
% 3.27/1.62 @ ^ [X5: set_Product_prod_b_b] : ( member1050118384od_b_b @ X5 @ A4 ) )
% 3.27/1.62 = A4 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_138_Collect__mem__eq,axiom,
% 3.27/1.62 ! [A4: set_set_b] :
% 3.27/1.62 ( ( collect_set_b
% 3.27/1.62 @ ^ [X5: set_b] : ( member_set_b @ X5 @ A4 ) )
% 3.27/1.62 = A4 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_139_Collect__mem__eq,axiom,
% 3.27/1.62 ! [A4: set_b] :
% 3.27/1.62 ( ( collect_b
% 3.27/1.62 @ ^ [X5: b] : ( member_b @ X5 @ A4 ) )
% 3.27/1.62 = A4 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_140_Collect__mem__eq,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( collec978177436_a_nat
% 3.27/1.62 @ ^ [X5: produc598811825_a_nat] : ( member327031898_a_nat @ X5 @ A4 ) )
% 3.27/1.62 = A4 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_141_Collect__mem__eq,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( collec1481886546od_b_b
% 3.27/1.62 @ ^ [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ A4 ) )
% 3.27/1.62 = A4 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_142_Collect__cong,axiom,
% 3.27/1.62 ! [P2: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
% 3.27/1.62 ( ! [X3: product_prod_b_b] :
% 3.27/1.62 ( ( P2 @ X3 )
% 3.27/1.62 = ( Q @ X3 ) )
% 3.27/1.62 => ( ( collec1481886546od_b_b @ P2 )
% 3.27/1.62 = ( collec1481886546od_b_b @ Q ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_143_Collect__cong,axiom,
% 3.27/1.62 ! [P2: produc598811825_a_nat > $o,Q: produc598811825_a_nat > $o] :
% 3.27/1.62 ( ! [X3: produc598811825_a_nat] :
% 3.27/1.62 ( ( P2 @ X3 )
% 3.27/1.62 = ( Q @ X3 ) )
% 3.27/1.62 => ( ( collec978177436_a_nat @ P2 )
% 3.27/1.62 = ( collec978177436_a_nat @ Q ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_144_semantics__in__vertices_I1_J,axiom,
% 3.27/1.62 ! [A4: labeled_graph_a_b,A: b,B: b,E: allegorical_term_a] :
% 3.27/1.62 ( ( A4
% 3.27/1.62 = ( restrict_a_b @ A4 ) )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ A4 @ E ) )
% 3.27/1.62 => ( member_b @ A @ ( labeled_vertices_a_b @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_145_semantics__in__vertices_I2_J,axiom,
% 3.27/1.62 ! [A4: labeled_graph_a_b,A: b,B: b,E: allegorical_term_a] :
% 3.27/1.62 ( ( A4
% 3.27/1.62 = ( restrict_a_b @ A4 ) )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ A4 @ E ) )
% 3.27/1.62 => ( member_b @ B @ ( labeled_vertices_a_b @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_146_refl__on__singleton,axiom,
% 3.27/1.62 ! [X4: produc1099860071_a_nat] : ( refl_o2093460070_a_nat @ ( insert354391607_a_nat @ X4 @ bot_bo1803599795_a_nat ) @ ( insert1772835511_a_nat @ ( produc404908887_a_nat @ X4 @ X4 ) @ bot_bo896299827_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_147_refl__on__singleton,axiom,
% 3.27/1.62 ! [X4: product_prod_b_b] : ( refl_o2134473190od_b_b @ ( insert1952693431od_b_b @ X4 @ bot_bo1343651123od_b_b ) @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ X4 @ X4 ) @ bot_bo1174586163od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_148_refl__on__singleton,axiom,
% 3.27/1.62 ! [X4: produc1133639312od_b_b] : ( refl_o686103183od_b_b @ ( insert1655137376od_b_b @ X4 @ bot_bo519661404od_b_b ) @ ( insert428421239od_b_b @ ( produc1037687191od_b_b @ X4 @ X4 ) @ bot_bo1856908531od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_149_refl__on__singleton,axiom,
% 3.27/1.62 ! [X4: set_Pr1912306449_a_nat] : ( refl_o730481808_a_nat @ ( insert1220735201_a_nat @ X4 @ bot_bo1746593245_a_nat ) @ ( insert885128759_a_nat @ ( produc1767291095_a_nat @ X4 @ X4 ) @ bot_bo2142478259_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_150_refl__on__singleton,axiom,
% 3.27/1.62 ! [X4: b] : ( refl_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X4 @ X4 ) @ bot_bo1343651123od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_151_refl__on__singleton,axiom,
% 3.27/1.62 ! [X4: labeled_graph_a_nat] : ( refl_o1018941609_a_nat @ ( insert1337853464_a_nat @ X4 @ bot_bo1119737266_a_nat ) @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ X4 @ X4 ) @ bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_152_refl__on__singleton,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat] : ( refl_o206400048_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ X4 @ X4 ) @ bot_bo1803599795_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_153_preorder__on__empty,axiom,
% 3.27/1.62 order_265664679od_b_b @ bot_bo519661404od_b_b @ bot_bo1856908531od_b_b ).
% 3.27/1.62
% 3.27/1.62 thf(fact_154_preorder__on__empty,axiom,
% 3.27/1.62 order_640573096_a_nat @ bot_bo1746593245_a_nat @ bot_bo2142478259_a_nat ).
% 3.27/1.62
% 3.27/1.62 thf(fact_155_preorder__on__empty,axiom,
% 3.27/1.62 order_preorder_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% 3.27/1.62
% 3.27/1.62 thf(fact_156_preorder__on__empty,axiom,
% 3.27/1.62 order_989282449_a_nat @ bot_bo1119737266_a_nat @ bot_bo2066531197_a_nat ).
% 3.27/1.62
% 3.27/1.62 thf(fact_157_preorder__on__empty,axiom,
% 3.27/1.62 order_145089608_a_nat @ bot_bo2066531197_a_nat @ bot_bo1803599795_a_nat ).
% 3.27/1.62
% 3.27/1.62 thf(fact_158_refl__on__empty,axiom,
% 3.27/1.62 refl_o2134473190od_b_b @ bot_bo1343651123od_b_b @ bot_bo1174586163od_b_b ).
% 3.27/1.62
% 3.27/1.62 thf(fact_159_refl__on__empty,axiom,
% 3.27/1.62 refl_o686103183od_b_b @ bot_bo519661404od_b_b @ bot_bo1856908531od_b_b ).
% 3.27/1.62
% 3.27/1.62 thf(fact_160_refl__on__empty,axiom,
% 3.27/1.62 refl_o730481808_a_nat @ bot_bo1746593245_a_nat @ bot_bo2142478259_a_nat ).
% 3.27/1.62
% 3.27/1.62 thf(fact_161_refl__on__empty,axiom,
% 3.27/1.62 refl_on_b @ bot_bot_set_b @ bot_bo1343651123od_b_b ).
% 3.27/1.62
% 3.27/1.62 thf(fact_162_refl__on__empty,axiom,
% 3.27/1.62 refl_o1018941609_a_nat @ bot_bo1119737266_a_nat @ bot_bo2066531197_a_nat ).
% 3.27/1.62
% 3.27/1.62 thf(fact_163_refl__on__empty,axiom,
% 3.27/1.62 refl_o206400048_a_nat @ bot_bo2066531197_a_nat @ bot_bo1803599795_a_nat ).
% 3.27/1.62
% 3.27/1.62 thf(fact_164_trans__singleton,axiom,
% 3.27/1.62 ! [A: product_prod_b_b] : ( trans_1808857926od_b_b @ ( insert1098249399od_b_b @ ( produc546497367od_b_b @ A @ A ) @ bot_bo1174586163od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_165_trans__singleton,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat] : ( trans_444496784_a_nat @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ A @ A ) @ bot_bo1803599795_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_166_trans__singleton,axiom,
% 3.27/1.62 ! [A: b] : ( trans_b @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ A @ A ) @ bot_bo1343651123od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_167_trans__singleton,axiom,
% 3.27/1.62 ! [A: labeled_graph_a_nat] : ( trans_1002055497_a_nat @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ A @ A ) @ bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_168_idempotent__impl__trans,axiom,
% 3.27/1.62 ! [R2: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( idempo848975736_a_nat @ R2 )
% 3.27/1.62 => ( trans_1002055497_a_nat @ R2 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_169_idempotent__impl__trans,axiom,
% 3.27/1.62 ! [R2: set_Product_prod_b_b] :
% 3.27/1.62 ( ( idempotent_b @ R2 )
% 3.27/1.62 => ( trans_b @ R2 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_170_trans__empty,axiom,
% 3.27/1.62 trans_b @ bot_bo1343651123od_b_b ).
% 3.27/1.62
% 3.27/1.62 thf(fact_171_trans__empty,axiom,
% 3.27/1.62 trans_1002055497_a_nat @ bot_bo2066531197_a_nat ).
% 3.27/1.62
% 3.27/1.62 thf(fact_172_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: b,V4: b,R: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1694913744od_b_b @ ( produc1037687191od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_triple_b_b_a @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_173_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: a,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: a,V3: b,V4: b,R: set_Pr471017666_nat_b] :
% 3.27/1.62 ( ( member442676358od_b_b @ ( produc179745357od_b_b @ ( produc1110849866_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1993240081at_b_a @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_174_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: a,V1: product_prod_b_b,V2: product_prod_b_b,L2: a,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
% 3.27/1.62 ( ( member1023210704od_b_b @ ( produc1116925335od_b_b @ ( produc690925568od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1764442556_b_b_a @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_175_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: b,V4: b,R: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1829919504od_b_b @ ( produc1403229015od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri111564221od_b_b @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_176_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 3.27/1.62 ( ( member817785114_a_nat @ ( produc966067681_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1110849866_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri2117183935_nat_a @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_177_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 3.27/1.62 ( ( member962139856od_b_b @ ( produc789452695od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc690925568od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri2003683308_b_b_a @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_178_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr471017666_nat_b] :
% 3.27/1.62 ( ( member670041670od_b_b @ ( produc926922637od_b_b @ ( produc90796321_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri1706508026od_b_b @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_179_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: product_prod_b_b,V2: product_prod_b_b,L2: product_prod_b_b,V3: b,V4: b,R: set_Pr357419743_b_b_b] :
% 3.27/1.62 ( ( member178477328od_b_b @ ( produc885982551od_b_b @ ( produc1850071895od_b_b @ L1 @ ( produc546497367od_b_b @ V1 @ V2 ) ) @ ( produc546497367od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_tri97791909od_b_b @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_180_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 3.27/1.62 ( ( member1769525082_a_nat @ ( produc1783652257_a_nat @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc90796321_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1837931688od_b_b @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_181_on__tripleD_I1_J,axiom,
% 3.27/1.62 ! [L1: product_prod_b_b,V1: b,V2: b,L2: product_prod_b_b,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 3.27/1.62 ( ( member1432657168od_b_b @ ( produc962089303od_b_b @ ( produc546497367od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1850071895od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri927008213od_b_b @ R ) )
% 3.27/1.62 => ( L2 = L1 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_182_on__tripleD_I2_J,axiom,
% 3.27/1.62 ! [L1: produc598811825_a_nat,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: produc598811825_a_nat,V3: produc598811825_a_nat,V4: produc598811825_a_nat,R: set_Pr1870121716_a_nat] :
% 3.27/1.62 ( ( member428378182_a_nat @ ( produc1645191821_a_nat @ ( produc410388695_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc639906189_a_nat @ L2 @ ( produc410388695_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1766439058_a_nat @ R ) )
% 3.27/1.62 => ( member2103088853_a_nat @ ( produc2092742640_a_nat @ V1 @ V3 ) @ R ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_183_on__tripleD_I2_J,axiom,
% 3.27/1.62 ! [L1: produc598811825_a_nat,V1: labeled_graph_a_nat,V2: labeled_graph_a_nat,L2: produc598811825_a_nat,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( member121756432_a_nat @ ( produc404908887_a_nat @ ( produc410388695_a_nat @ L1 @ ( produc2000589545_a_nat @ V1 @ V2 ) ) @ ( produc410388695_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri232494965_a_nat @ R ) )
% 3.27/1.62 => ( member327031898_a_nat @ ( produc2000589545_a_nat @ V1 @ V3 ) @ R ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_184_on__tripleD_I2_J,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: labeled_graph_a_nat,V4: labeled_graph_a_nat,R: set_Pr2010644300_a_nat] :
% 3.27/1.62 ( ( member817785114_a_nat @ ( produc966067681_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1110849866_a_nat @ L2 @ ( produc2000589545_a_nat @ V3 @ V4 ) ) ) @ ( on_tri2117183935_nat_a @ R ) )
% 3.27/1.62 => ( member763519277_a_nat @ ( produc1311571856_a_nat @ V1 @ V3 ) @ R ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_185_on__tripleD_I2_J,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: product_prod_b_b,V4: product_prod_b_b,R: set_Pr621705391od_b_b] :
% 3.27/1.62 ( ( member962139856od_b_b @ ( produc789452695od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc690925568od_b_b @ L2 @ ( produc546497367od_b_b @ V3 @ V4 ) ) ) @ ( on_tri2003683308_b_b_a @ R ) )
% 3.27/1.62 => ( member641857272od_b_b @ ( produc1257047359od_b_b @ V1 @ V3 ) @ R ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_186_on__tripleD_I2_J,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: produc598811825_a_nat,V4: produc598811825_a_nat,R: set_Pr352593785_a_nat] :
% 3.27/1.62 ( ( member1379268944_a_nat @ ( produc797892631_a_nat @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc1904940160_a_nat @ L2 @ ( produc410388695_a_nat @ V3 @ V4 ) ) ) @ ( on_tri1794757026_nat_a @ R ) )
% 3.27/1.62 => ( member692218050_a_nat @ ( produc255168265_a_nat @ V1 @ V3 ) @ R ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_187_on__tripleD_I2_J,axiom,
% 3.27/1.62 ! [L1: a,V1: b,V2: b,L2: a,V3: b,V4: b,R: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1694913744od_b_b @ ( produc1037687191od_b_b @ ( produc2135284224od_b_b @ L1 @ ( product_Pair_b_b @ V1 @ V2 ) ) @ ( produc2135284224od_b_b @ L2 @ ( product_Pair_b_b @ V3 @ V4 ) ) ) @ ( on_triple_b_b_a @ R ) )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ V1 @ V3 ) @ R ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_188_singletonI,axiom,
% 3.27/1.62 ! [A: product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_189_singletonI,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat] : ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_190_nonempty__rule,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restrict_a_b @ G ) )
% 3.27/1.62 => ( ( maintained_a_nat_b @ standa879863265rule_a @ G )
% 3.27/1.62 = ( ( labeled_vertices_a_b @ G )
% 3.27/1.62 != bot_bot_set_b ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_191_insertCI,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,B4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
% 3.27/1.62 ( ( ~ ( member327031898_a_nat @ A @ B4 )
% 3.27/1.62 => ( A = B ) )
% 3.27/1.62 => ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ B4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_192_insertCI,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,B4: set_Product_prod_b_b,B: product_prod_b_b] :
% 3.27/1.62 ( ( ~ ( member1285940496od_b_b @ A @ B4 )
% 3.27/1.62 => ( A = B ) )
% 3.27/1.62 => ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_193_insert__iff,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ A4 ) )
% 3.27/1.62 = ( ( A = B )
% 3.27/1.62 | ( member327031898_a_nat @ A @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_194_insert__iff,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,B: product_prod_b_b,A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A4 ) )
% 3.27/1.62 = ( ( A = B )
% 3.27/1.62 | ( member1285940496od_b_b @ A @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_195_insert__absorb2,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ X4 @ A4 ) )
% 3.27/1.62 = ( insert2058869377_a_nat @ X4 @ A4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_196_empty__iff,axiom,
% 3.27/1.62 ! [C2: product_prod_b_b] :
% 3.27/1.62 ~ ( member1285940496od_b_b @ C2 @ bot_bo1343651123od_b_b ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_197_empty__iff,axiom,
% 3.27/1.62 ! [C2: produc598811825_a_nat] :
% 3.27/1.62 ~ ( member327031898_a_nat @ C2 @ bot_bo2066531197_a_nat ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_198_all__not__in__conv,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( ! [X5: product_prod_b_b] :
% 3.27/1.62 ~ ( member1285940496od_b_b @ X5 @ A4 ) )
% 3.27/1.62 = ( A4 = bot_bo1343651123od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_199_all__not__in__conv,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( ! [X5: produc598811825_a_nat] :
% 3.27/1.62 ~ ( member327031898_a_nat @ X5 @ A4 ) )
% 3.27/1.62 = ( A4 = bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_200_Collect__empty__eq,axiom,
% 3.27/1.62 ! [P2: produc598811825_a_nat > $o] :
% 3.27/1.62 ( ( ( collec978177436_a_nat @ P2 )
% 3.27/1.62 = bot_bo2066531197_a_nat )
% 3.27/1.62 = ( ! [X5: produc598811825_a_nat] :
% 3.27/1.62 ~ ( P2 @ X5 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_201_empty__Collect__eq,axiom,
% 3.27/1.62 ! [P2: produc598811825_a_nat > $o] :
% 3.27/1.62 ( ( bot_bo2066531197_a_nat
% 3.27/1.62 = ( collec978177436_a_nat @ P2 ) )
% 3.27/1.62 = ( ! [X5: produc598811825_a_nat] :
% 3.27/1.62 ~ ( P2 @ X5 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_202_ex__in__conv,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( ? [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ A4 ) )
% 3.27/1.62 = ( A4 != bot_bo1343651123od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_203_ex__in__conv,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( ? [X5: produc598811825_a_nat] : ( member327031898_a_nat @ X5 @ A4 ) )
% 3.27/1.62 = ( A4 != bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_204_equals0I,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ! [Y3: product_prod_b_b] :
% 3.27/1.62 ~ ( member1285940496od_b_b @ Y3 @ A4 )
% 3.27/1.62 => ( A4 = bot_bo1343651123od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_205_equals0I,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ! [Y3: produc598811825_a_nat] :
% 3.27/1.62 ~ ( member327031898_a_nat @ Y3 @ A4 )
% 3.27/1.62 => ( A4 = bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_206_equals0D,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,A: product_prod_b_b] :
% 3.27/1.62 ( ( A4 = bot_bo1343651123od_b_b )
% 3.27/1.62 => ~ ( member1285940496od_b_b @ A @ A4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_207_equals0D,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,A: produc598811825_a_nat] :
% 3.27/1.62 ( ( A4 = bot_bo2066531197_a_nat )
% 3.27/1.62 => ~ ( member327031898_a_nat @ A @ A4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_208_emptyE,axiom,
% 3.27/1.62 ! [A: product_prod_b_b] :
% 3.27/1.62 ~ ( member1285940496od_b_b @ A @ bot_bo1343651123od_b_b ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_209_emptyE,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat] :
% 3.27/1.62 ~ ( member327031898_a_nat @ A @ bot_bo2066531197_a_nat ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_210_mk__disjoint__insert,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ A @ A4 )
% 3.27/1.62 => ? [B5: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( A4
% 3.27/1.62 = ( insert2058869377_a_nat @ A @ B5 ) )
% 3.27/1.62 & ~ ( member327031898_a_nat @ A @ B5 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_211_mk__disjoint__insert,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ A @ A4 )
% 3.27/1.62 => ? [B5: set_Product_prod_b_b] :
% 3.27/1.62 ( ( A4
% 3.27/1.62 = ( insert1952693431od_b_b @ A @ B5 ) )
% 3.27/1.62 & ~ ( member1285940496od_b_b @ A @ B5 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_212_insert__commute,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat,Y4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ Y4 @ A4 ) )
% 3.27/1.62 = ( insert2058869377_a_nat @ Y4 @ ( insert2058869377_a_nat @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_213_insert__eq__iff,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B: produc598811825_a_nat,B4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ~ ( member327031898_a_nat @ A @ A4 )
% 3.27/1.62 => ( ~ ( member327031898_a_nat @ B @ B4 )
% 3.27/1.62 => ( ( ( insert2058869377_a_nat @ A @ A4 )
% 3.27/1.62 = ( insert2058869377_a_nat @ B @ B4 ) )
% 3.27/1.62 = ( ( ( A = B )
% 3.27/1.62 => ( A4 = B4 ) )
% 3.27/1.62 & ( ( A != B )
% 3.27/1.62 => ? [C3: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( A4
% 3.27/1.62 = ( insert2058869377_a_nat @ B @ C3 ) )
% 3.27/1.62 & ~ ( member327031898_a_nat @ B @ C3 )
% 3.27/1.62 & ( B4
% 3.27/1.62 = ( insert2058869377_a_nat @ A @ C3 ) )
% 3.27/1.62 & ~ ( member327031898_a_nat @ A @ C3 ) ) ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_214_insert__eq__iff,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,A4: set_Product_prod_b_b,B: product_prod_b_b,B4: set_Product_prod_b_b] :
% 3.27/1.62 ( ~ ( member1285940496od_b_b @ A @ A4 )
% 3.27/1.62 => ( ~ ( member1285940496od_b_b @ B @ B4 )
% 3.27/1.62 => ( ( ( insert1952693431od_b_b @ A @ A4 )
% 3.27/1.62 = ( insert1952693431od_b_b @ B @ B4 ) )
% 3.27/1.62 = ( ( ( A = B )
% 3.27/1.62 => ( A4 = B4 ) )
% 3.27/1.62 & ( ( A != B )
% 3.27/1.62 => ? [C3: set_Product_prod_b_b] :
% 3.27/1.62 ( ( A4
% 3.27/1.62 = ( insert1952693431od_b_b @ B @ C3 ) )
% 3.27/1.62 & ~ ( member1285940496od_b_b @ B @ C3 )
% 3.27/1.62 & ( B4
% 3.27/1.62 = ( insert1952693431od_b_b @ A @ C3 ) )
% 3.27/1.62 & ~ ( member1285940496od_b_b @ A @ C3 ) ) ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_215_insert__absorb,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ A @ A4 )
% 3.27/1.62 => ( ( insert2058869377_a_nat @ A @ A4 )
% 3.27/1.62 = A4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_216_insert__absorb,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ A @ A4 )
% 3.27/1.62 => ( ( insert1952693431od_b_b @ A @ A4 )
% 3.27/1.62 = A4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_217_insert__ident,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ~ ( member327031898_a_nat @ X4 @ A4 )
% 3.27/1.62 => ( ~ ( member327031898_a_nat @ X4 @ B4 )
% 3.27/1.62 => ( ( ( insert2058869377_a_nat @ X4 @ A4 )
% 3.27/1.62 = ( insert2058869377_a_nat @ X4 @ B4 ) )
% 3.27/1.62 = ( A4 = B4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_218_insert__ident,axiom,
% 3.27/1.62 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
% 3.27/1.62 ( ~ ( member1285940496od_b_b @ X4 @ A4 )
% 3.27/1.62 => ( ~ ( member1285940496od_b_b @ X4 @ B4 )
% 3.27/1.62 => ( ( ( insert1952693431od_b_b @ X4 @ A4 )
% 3.27/1.62 = ( insert1952693431od_b_b @ X4 @ B4 ) )
% 3.27/1.62 = ( A4 = B4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_219_Set_Oset__insert,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ X4 @ A4 )
% 3.27/1.62 => ~ ! [B5: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( A4
% 3.27/1.62 = ( insert2058869377_a_nat @ X4 @ B5 ) )
% 3.27/1.62 => ( member327031898_a_nat @ X4 @ B5 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_220_Set_Oset__insert,axiom,
% 3.27/1.62 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ X4 @ A4 )
% 3.27/1.62 => ~ ! [B5: set_Product_prod_b_b] :
% 3.27/1.62 ( ( A4
% 3.27/1.62 = ( insert1952693431od_b_b @ X4 @ B5 ) )
% 3.27/1.62 => ( member1285940496od_b_b @ X4 @ B5 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_221_insertI2,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,B4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ A @ B4 )
% 3.27/1.62 => ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ B4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_222_insertI2,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,B4: set_Product_prod_b_b,B: product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ A @ B4 )
% 3.27/1.62 => ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ B4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_223_insertI1,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,B4: set_Pr1912306449_a_nat] : ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ A @ B4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_224_insertI1,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,B4: set_Product_prod_b_b] : ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ A @ B4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_225_insertE,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ A @ ( insert2058869377_a_nat @ B @ A4 ) )
% 3.27/1.62 => ( ( A != B )
% 3.27/1.62 => ( member327031898_a_nat @ A @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_226_insertE,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,B: product_prod_b_b,A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ A @ ( insert1952693431od_b_b @ B @ A4 ) )
% 3.27/1.62 => ( ( A != B )
% 3.27/1.62 => ( member1285940496od_b_b @ A @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_227_singleton__inject,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,B: produc598811825_a_nat] :
% 3.27/1.62 ( ( ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat )
% 3.27/1.62 = ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) )
% 3.27/1.62 => ( A = B ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_228_insert__not__empty,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( insert2058869377_a_nat @ A @ A4 )
% 3.27/1.62 != bot_bo2066531197_a_nat ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_229_doubleton__eq__iff,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,B: produc598811825_a_nat,C2: produc598811825_a_nat,D: produc598811825_a_nat] :
% 3.27/1.62 ( ( ( insert2058869377_a_nat @ A @ ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) )
% 3.27/1.62 = ( insert2058869377_a_nat @ C2 @ ( insert2058869377_a_nat @ D @ bot_bo2066531197_a_nat ) ) )
% 3.27/1.62 = ( ( ( A = C2 )
% 3.27/1.62 & ( B = D ) )
% 3.27/1.62 | ( ( A = D )
% 3.27/1.62 & ( B = C2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_230_singleton__iff,axiom,
% 3.27/1.62 ! [B: product_prod_b_b,A: product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 3.27/1.62 = ( B = A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_231_singleton__iff,axiom,
% 3.27/1.62 ! [B: produc598811825_a_nat,A: produc598811825_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ B @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) )
% 3.27/1.62 = ( B = A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_232_singletonD,axiom,
% 3.27/1.62 ! [B: product_prod_b_b,A: product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ B @ ( insert1952693431od_b_b @ A @ bot_bo1343651123od_b_b ) )
% 3.27/1.62 => ( B = A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_233_singletonD,axiom,
% 3.27/1.62 ! [B: produc598811825_a_nat,A: produc598811825_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ B @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) )
% 3.27/1.62 => ( B = A ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_234_the__elem__eq,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat] :
% 3.27/1.62 ( ( the_el537460006_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) )
% 3.27/1.62 = X4 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_235_graph__single,axiom,
% 3.27/1.62 ! [A: a,B: b,C2: b] :
% 3.27/1.62 ( ( labeled_LG_a_b @ ( insert1655137376od_b_b @ ( produc2135284224od_b_b @ A @ ( product_Pair_b_b @ B @ C2 ) ) @ bot_bo519661404od_b_b ) @ ( insert_b @ B @ ( insert_b @ C2 @ bot_bot_set_b ) ) )
% 3.27/1.62 = ( restrict_a_b @ ( labeled_LG_a_b @ ( insert1655137376od_b_b @ ( produc2135284224od_b_b @ A @ ( product_Pair_b_b @ B @ C2 ) ) @ bot_bo519661404od_b_b ) @ ( insert_b @ B @ ( insert_b @ C2 @ bot_bot_set_b ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_236_is__singletonI,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat] : ( is_sin1787449765_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_237_linear__order__on__singleton,axiom,
% 3.27/1.62 ! [X4: b] : ( order_653178932r_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X4 @ X4 ) @ bot_bo1343651123od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_238_linear__order__on__singleton,axiom,
% 3.27/1.62 ! [X4: labeled_graph_a_nat] : ( order_1770871027_a_nat @ ( insert1337853464_a_nat @ X4 @ bot_bo1119737266_a_nat ) @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ X4 @ X4 ) @ bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_239_linear__order__on__singleton,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat] : ( order_1458491494_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ X4 @ X4 ) @ bot_bo1803599795_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_240_total__on__singleton,axiom,
% 3.27/1.62 ! [X4: b] : ( total_on_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ ( insert1952693431od_b_b @ ( product_Pair_b_b @ X4 @ X4 ) @ bot_bo1343651123od_b_b ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_241_total__on__singleton,axiom,
% 3.27/1.62 ! [X4: labeled_graph_a_nat] : ( total_6094180_a_nat @ ( insert1337853464_a_nat @ X4 @ bot_bo1119737266_a_nat ) @ ( insert2058869377_a_nat @ ( produc2000589545_a_nat @ X4 @ X4 ) @ bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_242_total__on__singleton,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat] : ( total_2145009205_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ ( insert354391607_a_nat @ ( produc410388695_a_nat @ X4 @ X4 ) @ bot_bo1803599795_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_243_symmetry__rule,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b,L: a] :
% 3.27/1.62 ( ( G
% 3.27/1.62 = ( restrict_a_b @ G ) )
% 3.27/1.62 => ( ( maintained_a_nat_b @ ( standa709637829rule_a @ L ) @ G )
% 3.27/1.62 => ( sym_b @ ( getRel_a_b @ L @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_244_graph__empty__e,axiom,
% 3.27/1.62 ! [V: set_b] :
% 3.27/1.62 ( ( labeled_LG_a_b @ bot_bo519661404od_b_b @ V )
% 3.27/1.62 = ( restrict_a_b @ ( labeled_LG_a_b @ bot_bo519661404od_b_b @ V ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_245_bot__empty__eq,axiom,
% 3.27/1.62 ( bot_bo1302443370_b_b_o
% 3.27/1.62 = ( ^ [X5: product_prod_b_b] : ( member1285940496od_b_b @ X5 @ bot_bo1343651123od_b_b ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_246_bot__empty__eq,axiom,
% 3.27/1.62 ( bot_bo186023328_nat_o
% 3.27/1.62 = ( ^ [X5: produc598811825_a_nat] : ( member327031898_a_nat @ X5 @ bot_bo2066531197_a_nat ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_247_bot__set__def,axiom,
% 3.27/1.62 ( bot_bo2066531197_a_nat
% 3.27/1.62 = ( collec978177436_a_nat @ bot_bo186023328_nat_o ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_248_sym__def,axiom,
% 3.27/1.62 ( sym_b
% 3.27/1.62 = ( ^ [R3: set_Product_prod_b_b] :
% 3.27/1.62 ! [X5: b,Y5: b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Y5 ) @ R3 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ Y5 @ X5 ) @ R3 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_249_symI,axiom,
% 3.27/1.62 ! [R2: set_Product_prod_b_b] :
% 3.27/1.62 ( ! [A3: b,B3: b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A3 @ B3 ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ B3 @ A3 ) @ R2 ) )
% 3.27/1.62 => ( sym_b @ R2 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_250_symE,axiom,
% 3.27/1.62 ! [R2: set_Product_prod_b_b,B: b,A: b] :
% 3.27/1.62 ( ( sym_b @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_251_symD,axiom,
% 3.27/1.62 ! [R2: set_Product_prod_b_b,B: b,A: b] :
% 3.27/1.62 ( ( sym_b @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ B @ A ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ R2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_252_total__on__def,axiom,
% 3.27/1.62 ( total_on_b
% 3.27/1.62 = ( ^ [A5: set_b,R3: set_Product_prod_b_b] :
% 3.27/1.62 ! [X5: b] :
% 3.27/1.62 ( ( member_b @ X5 @ A5 )
% 3.27/1.62 => ! [Y5: b] :
% 3.27/1.62 ( ( member_b @ Y5 @ A5 )
% 3.27/1.62 => ( ( X5 != Y5 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X5 @ Y5 ) @ R3 )
% 3.27/1.62 | ( member1285940496od_b_b @ ( product_Pair_b_b @ Y5 @ X5 ) @ R3 ) ) ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_253_total__onI,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b] :
% 3.27/1.62 ( ! [X3: product_prod_b_b,Y3: product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ X3 @ A4 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ Y3 @ A4 )
% 3.27/1.62 => ( ( X3 != Y3 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X3 @ Y3 ) @ R2 )
% 3.27/1.62 | ( member1652792080od_b_b @ ( produc546497367od_b_b @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 3.27/1.62 => ( total_178717291od_b_b @ A4 @ R2 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_254_total__onI,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b] :
% 3.27/1.62 ( ! [X3: b,Y3: b] :
% 3.27/1.62 ( ( member_b @ X3 @ A4 )
% 3.27/1.62 => ( ( member_b @ Y3 @ A4 )
% 3.27/1.62 => ( ( X3 != Y3 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R2 )
% 3.27/1.62 | ( member1285940496od_b_b @ ( product_Pair_b_b @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 3.27/1.62 => ( total_on_b @ A4 @ R2 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_255_total__on__empty,axiom,
% 3.27/1.62 ! [R2: set_Pr2074180935_a_nat] : ( total_2145009205_a_nat @ bot_bo2066531197_a_nat @ R2 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_256_labeled__graph_Osel_I2_J,axiom,
% 3.27/1.62 ! [X1: set_Pr1900477680od_b_b,X2: set_b] :
% 3.27/1.62 ( ( labeled_vertices_a_b @ ( labeled_LG_a_b @ X1 @ X2 ) )
% 3.27/1.62 = X2 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_257_is__singleton__the__elem,axiom,
% 3.27/1.62 ( is_sin1787449765_a_nat
% 3.27/1.62 = ( ^ [A5: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( A5
% 3.27/1.62 = ( insert2058869377_a_nat @ ( the_el537460006_a_nat @ A5 ) @ bot_bo2066531197_a_nat ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_258_is__singletonI_H,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( A4 != bot_bo1343651123od_b_b )
% 3.27/1.62 => ( ! [X3: product_prod_b_b,Y3: product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ X3 @ A4 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ Y3 @ A4 )
% 3.27/1.62 => ( X3 = Y3 ) ) )
% 3.27/1.62 => ( is_sin1713057243od_b_b @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_259_is__singletonI_H,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( A4 != bot_bo2066531197_a_nat )
% 3.27/1.62 => ( ! [X3: produc598811825_a_nat,Y3: produc598811825_a_nat] :
% 3.27/1.62 ( ( member327031898_a_nat @ X3 @ A4 )
% 3.27/1.62 => ( ( member327031898_a_nat @ Y3 @ A4 )
% 3.27/1.62 => ( X3 = Y3 ) ) )
% 3.27/1.62 => ( is_sin1787449765_a_nat @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_260_lnear__order__on__empty,axiom,
% 3.27/1.62 order_1770871027_a_nat @ bot_bo1119737266_a_nat @ bot_bo2066531197_a_nat ).
% 3.27/1.62
% 3.27/1.62 thf(fact_261_lnear__order__on__empty,axiom,
% 3.27/1.62 order_1458491494_a_nat @ bot_bo2066531197_a_nat @ bot_bo1803599795_a_nat ).
% 3.27/1.62
% 3.27/1.62 thf(fact_262_is__singletonE,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( is_sin1787449765_a_nat @ A4 )
% 3.27/1.62 => ~ ! [X3: produc598811825_a_nat] :
% 3.27/1.62 ( A4
% 3.27/1.62 != ( insert2058869377_a_nat @ X3 @ bot_bo2066531197_a_nat ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_263_is__singleton__def,axiom,
% 3.27/1.62 ( is_sin1787449765_a_nat
% 3.27/1.62 = ( ^ [A5: set_Pr1912306449_a_nat] :
% 3.27/1.62 ? [X5: produc598811825_a_nat] :
% 3.27/1.62 ( A5
% 3.27/1.62 = ( insert2058869377_a_nat @ X5 @ bot_bo2066531197_a_nat ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_264_equivE,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b] :
% 3.27/1.62 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.62 => ~ ( ( refl_on_b @ A4 @ R2 )
% 3.27/1.62 => ( ( sym_b @ R2 )
% 3.27/1.62 => ~ ( trans_b @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_265_equivI,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b] :
% 3.27/1.62 ( ( refl_on_b @ A4 @ R2 )
% 3.27/1.62 => ( ( sym_b @ R2 )
% 3.27/1.62 => ( ( trans_b @ R2 )
% 3.27/1.62 => ( equiv_equiv_b @ A4 @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_266_equiv__def,axiom,
% 3.27/1.62 ( equiv_equiv_b
% 3.27/1.62 = ( ^ [A5: set_b,R3: set_Product_prod_b_b] :
% 3.27/1.62 ( ( refl_on_b @ A5 @ R3 )
% 3.27/1.62 & ( sym_b @ R3 )
% 3.27/1.62 & ( trans_b @ R3 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_267_graph__homomorphism__empty,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b,F: set_Pr1914572181_nat_b] :
% 3.27/1.62 ( ( graph_23275493_nat_b @ ( labele1171788421_a_nat @ bot_bo1463793628_a_nat @ bot_bo2066531197_a_nat ) @ G @ F )
% 3.27/1.62 = ( ( F = bot_bo1936722433_nat_b )
% 3.27/1.62 & ( G
% 3.27/1.62 = ( restrict_a_b @ G ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_268_Collect__empty__eq__bot,axiom,
% 3.27/1.62 ! [P2: produc598811825_a_nat > $o] :
% 3.27/1.62 ( ( ( collec978177436_a_nat @ P2 )
% 3.27/1.62 = bot_bo2066531197_a_nat )
% 3.27/1.62 = ( P2 = bot_bo186023328_nat_o ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_269_maintainedI,axiom,
% 3.27/1.62 ! [A4: labeled_graph_a_nat,G: labeled_graph_a_b,B4: labeled_graph_a_nat] :
% 3.27/1.62 ( ! [F2: set_Pr2106913242_nat_b] :
% 3.27/1.62 ( ( graph_914848022_nat_b @ A4 @ G @ F2 )
% 3.27/1.62 => ( extensible_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G @ F2 ) )
% 3.27/1.62 => ( maintained_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_270_maintainedD,axiom,
% 3.27/1.62 ! [A4: labeled_graph_a_nat,B4: labeled_graph_a_nat,G: labeled_graph_a_b,F: set_Pr2106913242_nat_b] :
% 3.27/1.62 ( ( maintained_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G )
% 3.27/1.62 => ( ( graph_914848022_nat_b @ A4 @ G @ F )
% 3.27/1.62 => ( extensible_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G @ F ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_271_Set_Ois__empty__def,axiom,
% 3.27/1.62 ( is_emp1506632323_a_nat
% 3.27/1.62 = ( ^ [A5: set_Pr1912306449_a_nat] : ( A5 = bot_bo2066531197_a_nat ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_272_eq__equiv__class__iff2,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 3.27/1.62 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ X4 @ A4 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ Y4 @ A4 )
% 3.27/1.62 => ( ( ( equiv_1530858158od_b_b @ ( insert1952693431od_b_b @ X4 @ bot_bo1343651123od_b_b ) @ R2 )
% 3.27/1.62 = ( equiv_1530858158od_b_b @ ( insert1952693431od_b_b @ Y4 @ bot_bo1343651123od_b_b ) @ R2 ) )
% 3.27/1.62 = ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_273_eq__equiv__class__iff2,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
% 3.27/1.62 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member_b @ X4 @ A4 )
% 3.27/1.62 => ( ( member_b @ Y4 @ A4 )
% 3.27/1.62 => ( ( ( equiv_quotient_b @ ( insert_b @ X4 @ bot_bot_set_b ) @ R2 )
% 3.27/1.62 = ( equiv_quotient_b @ ( insert_b @ Y4 @ bot_bot_set_b ) @ R2 ) )
% 3.27/1.62 = ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_274_eq__equiv__class__iff2,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 3.27/1.62 ( ( equiv_281634663_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member327031898_a_nat @ X4 @ A4 )
% 3.27/1.62 => ( ( member327031898_a_nat @ Y4 @ A4 )
% 3.27/1.62 => ( ( ( equiv_1338033912_a_nat @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) @ R2 )
% 3.27/1.62 = ( equiv_1338033912_a_nat @ ( insert2058869377_a_nat @ Y4 @ bot_bo2066531197_a_nat ) @ R2 ) )
% 3.27/1.62 = ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_275_quotient__is__empty2,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat] :
% 3.27/1.62 ( ( bot_bo1746593245_a_nat
% 3.27/1.62 = ( equiv_1338033912_a_nat @ A4 @ R2 ) )
% 3.27/1.62 = ( A4 = bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_276_quotient__is__empty,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat] :
% 3.27/1.62 ( ( ( equiv_1338033912_a_nat @ A4 @ R2 )
% 3.27/1.62 = bot_bo1746593245_a_nat )
% 3.27/1.62 = ( A4 = bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_277_quotient__empty,axiom,
% 3.27/1.62 ! [R2: set_Pr2074180935_a_nat] :
% 3.27/1.62 ( ( equiv_1338033912_a_nat @ bot_bo2066531197_a_nat @ R2 )
% 3.27/1.62 = bot_bo1746593245_a_nat ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_278_in__quotient__imp__closed,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X6: set_Product_prod_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 3.27/1.62 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1050118384od_b_b @ X6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member1285940496od_b_b @ X4 @ X6 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ Y4 @ X6 ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_279_in__quotient__imp__closed,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,X4: b,Y4: b] :
% 3.27/1.62 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member_b @ X4 @ X6 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( member_b @ Y4 @ X6 ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_280_quotient__eq__iff,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X6: set_Product_prod_b_b,Y6: set_Product_prod_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 3.27/1.62 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1050118384od_b_b @ X6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member1050118384od_b_b @ Y6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member1285940496od_b_b @ X4 @ X6 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ Y4 @ Y6 )
% 3.27/1.62 => ( ( X6 = Y6 )
% 3.27/1.62 = ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_281_quotient__eq__iff,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,Y6: set_b,X4: b,Y4: b] :
% 3.27/1.62 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member_set_b @ Y6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member_b @ X4 @ X6 )
% 3.27/1.62 => ( ( member_b @ Y4 @ Y6 )
% 3.27/1.62 => ( ( X6 = Y6 )
% 3.27/1.62 = ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_282_quotient__eqI,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X6: set_Product_prod_b_b,Y6: set_Product_prod_b_b,X4: product_prod_b_b,Y4: product_prod_b_b] :
% 3.27/1.62 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member1050118384od_b_b @ X6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member1050118384od_b_b @ Y6 @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member1285940496od_b_b @ X4 @ X6 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ Y4 @ Y6 )
% 3.27/1.62 => ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( X6 = Y6 ) ) ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_283_quotient__eqI,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,Y6: set_b,X4: b,Y4: b] :
% 3.27/1.62 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member_set_b @ Y6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( member_b @ X4 @ X6 )
% 3.27/1.62 => ( ( member_b @ Y4 @ Y6 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 )
% 3.27/1.62 => ( X6 = Y6 ) ) ) ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_284_in__quotient__imp__non__empty,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b] :
% 3.27/1.62 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.62 => ( X6 != bot_bot_set_b ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_285_in__quotient__imp__non__empty,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X6: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( equiv_281634663_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member101535418_a_nat @ X6 @ ( equiv_1338033912_a_nat @ A4 @ R2 ) )
% 3.27/1.62 => ( X6 != bot_bo2066531197_a_nat ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_286_in__quotient__imp__in__rel,axiom,
% 3.27/1.62 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b,X4: b,Y4: b] :
% 3.27/1.62 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.62 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.62 => ( ( ord_less_eq_set_b @ ( insert_b @ X4 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) @ X6 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_287_in__quotient__imp__in__rel,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X6: set_Pr1912306449_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 3.27/1.62 ( ( equiv_281634663_a_nat @ A4 @ R2 )
% 3.27/1.62 => ( ( member101535418_a_nat @ X6 @ ( equiv_1338033912_a_nat @ A4 @ R2 ) )
% 3.27/1.62 => ( ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ Y4 @ bot_bo2066531197_a_nat ) ) @ X6 )
% 3.27/1.62 => ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_288_same__fstI,axiom,
% 3.27/1.62 ! [P2: b > $o,X4: b,Y7: b,Y4: b,R: b > set_Product_prod_b_b] :
% 3.27/1.62 ( ( P2 @ X4 )
% 3.27/1.62 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y7 @ Y4 ) @ ( R @ X4 ) )
% 3.27/1.62 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ ( product_Pair_b_b @ X4 @ Y7 ) @ ( product_Pair_b_b @ X4 @ Y4 ) ) @ ( same_fst_b_b @ P2 @ R ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_289_getRel__subgraph,axiom,
% 3.27/1.62 ! [Y4: b,Z: b,L: a,G: labeled_graph_a_b,G2: labeled_graph_a_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ ( getRel_a_b @ L @ G ) )
% 3.27/1.62 => ( ( graph_1322245399_a_b_b @ G @ G2 @ ( id_on_b @ ( labeled_vertices_a_b @ G ) ) )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ Y4 @ Z ) @ ( getRel_a_b @ L @ G2 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_290_subsetI,axiom,
% 3.27/1.62 ! [A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
% 3.27/1.62 ( ! [X3: product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ X3 @ A4 )
% 3.27/1.62 => ( member1285940496od_b_b @ X3 @ B4 ) )
% 3.27/1.62 => ( ord_le1036320359od_b_b @ A4 @ B4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_291_empty__subsetI,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat] : ( ord_le1374266033_a_nat @ bot_bo2066531197_a_nat @ A4 ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_292_subset__empty,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( ord_le1374266033_a_nat @ A4 @ bot_bo2066531197_a_nat )
% 3.27/1.62 = ( A4 = bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_293_insert__subset,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ A4 ) @ B4 )
% 3.27/1.62 = ( ( member327031898_a_nat @ X4 @ B4 )
% 3.27/1.62 & ( ord_le1374266033_a_nat @ A4 @ B4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_294_insert__subset,axiom,
% 3.27/1.62 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X4 @ A4 ) @ B4 )
% 3.27/1.62 = ( ( member1285940496od_b_b @ X4 @ B4 )
% 3.27/1.62 & ( ord_le1036320359od_b_b @ A4 @ B4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_295_Id__onI,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ A @ A4 )
% 3.27/1.62 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ A ) @ ( id_on_2019020932od_b_b @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_296_Id__onI,axiom,
% 3.27/1.62 ! [A: b,A4: set_b] :
% 3.27/1.62 ( ( member_b @ A @ A4 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ A ) @ ( id_on_b @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_297_singleton__insert__inj__eq,axiom,
% 3.27/1.62 ! [B: produc598811825_a_nat,A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat )
% 3.27/1.62 = ( insert2058869377_a_nat @ A @ A4 ) )
% 3.27/1.62 = ( ( A = B )
% 3.27/1.62 & ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_298_singleton__insert__inj__eq_H,axiom,
% 3.27/1.62 ! [A: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
% 3.27/1.62 ( ( ( insert2058869377_a_nat @ A @ A4 )
% 3.27/1.62 = ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) )
% 3.27/1.62 = ( ( A = B )
% 3.27/1.62 & ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ B @ bot_bo2066531197_a_nat ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_299_Id__on__empty,axiom,
% 3.27/1.62 ( ( id_on_1401723275_a_nat @ bot_bo1119737266_a_nat )
% 3.27/1.62 = bot_bo2066531197_a_nat ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_300_Id__on__empty,axiom,
% 3.27/1.62 ( ( id_on_1649591758_a_nat @ bot_bo2066531197_a_nat )
% 3.27/1.62 = bot_bo1803599795_a_nat ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_301_subgraph__refl,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b] :
% 3.27/1.62 ( ( graph_1322245399_a_b_b @ G @ G @ ( id_on_b @ ( labeled_vertices_a_b @ G ) ) )
% 3.27/1.62 = ( G
% 3.27/1.62 = ( restrict_a_b @ G ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_302_subgraph__restrict,axiom,
% 3.27/1.62 ! [G: labeled_graph_a_b] :
% 3.27/1.62 ( ( graph_1322245399_a_b_b @ G @ ( restrict_a_b @ G ) @ ( id_on_b @ ( labeled_vertices_a_b @ G ) ) )
% 3.27/1.62 = ( G
% 3.27/1.62 = ( restrict_a_b @ G ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_303_graph__homomorphism__Id,axiom,
% 3.27/1.62 ! [A: labeled_graph_a_b] : ( graph_1322245399_a_b_b @ ( restrict_a_b @ A ) @ ( restrict_a_b @ A ) @ ( id_on_b @ ( labeled_vertices_a_b @ A ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_304_subgraph__subset_I1_J,axiom,
% 3.27/1.62 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 3.27/1.62 ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 3.27/1.62 => ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_305_Id__onE,axiom,
% 3.27/1.62 ! [C2: produc1168037607od_b_b,A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1652792080od_b_b @ C2 @ ( id_on_2019020932od_b_b @ A4 ) )
% 3.27/1.62 => ~ ! [X3: product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ X3 @ A4 )
% 3.27/1.62 => ( C2
% 3.27/1.62 != ( produc546497367od_b_b @ X3 @ X3 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_306_Id__onE,axiom,
% 3.27/1.62 ! [C2: product_prod_b_b,A4: set_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ C2 @ ( id_on_b @ A4 ) )
% 3.27/1.62 => ~ ! [X3: b] :
% 3.27/1.62 ( ( member_b @ X3 @ A4 )
% 3.27/1.62 => ( C2
% 3.27/1.62 != ( product_Pair_b_b @ X3 @ X3 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_307_Id__on__eqI,axiom,
% 3.27/1.62 ! [A: product_prod_b_b,B: product_prod_b_b,A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( A = B )
% 3.27/1.62 => ( ( member1285940496od_b_b @ A @ A4 )
% 3.27/1.62 => ( member1652792080od_b_b @ ( produc546497367od_b_b @ A @ B ) @ ( id_on_2019020932od_b_b @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_308_Id__on__eqI,axiom,
% 3.27/1.62 ! [A: b,B: b,A4: set_b] :
% 3.27/1.62 ( ( A = B )
% 3.27/1.62 => ( ( member_b @ A @ A4 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( id_on_b @ A4 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_309_Id__on__iff,axiom,
% 3.27/1.62 ! [X4: product_prod_b_b,Y4: product_prod_b_b,A4: set_Product_prod_b_b] :
% 3.27/1.62 ( ( member1652792080od_b_b @ ( produc546497367od_b_b @ X4 @ Y4 ) @ ( id_on_2019020932od_b_b @ A4 ) )
% 3.27/1.62 = ( ( X4 = Y4 )
% 3.27/1.62 & ( member1285940496od_b_b @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_310_Id__on__iff,axiom,
% 3.27/1.62 ! [X4: b,Y4: b,A4: set_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ ( id_on_b @ A4 ) )
% 3.27/1.62 = ( ( X4 = Y4 )
% 3.27/1.62 & ( member_b @ X4 @ A4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_311_bot_Oextremum__uniqueI,axiom,
% 3.27/1.62 ! [A: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( ord_le1374266033_a_nat @ A @ bot_bo2066531197_a_nat )
% 3.27/1.62 => ( A = bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_312_bot_Oextremum__unique,axiom,
% 3.27/1.62 ! [A: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ( ord_le1374266033_a_nat @ A @ bot_bo2066531197_a_nat )
% 3.27/1.62 = ( A = bot_bo2066531197_a_nat ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_313_bot_Oextremum,axiom,
% 3.27/1.62 ! [A: set_Pr1912306449_a_nat] : ( ord_le1374266033_a_nat @ bot_bo2066531197_a_nat @ A ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_314_subrelI,axiom,
% 3.27/1.62 ! [R2: set_Product_prod_b_b,S: set_Product_prod_b_b] :
% 3.27/1.62 ( ! [X3: b,Y3: b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ R2 )
% 3.27/1.62 => ( member1285940496od_b_b @ ( product_Pair_b_b @ X3 @ Y3 ) @ S ) )
% 3.27/1.62 => ( ord_le1036320359od_b_b @ R2 @ S ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_315_insert__mono,axiom,
% 3.27/1.62 ! [C4: set_Pr1912306449_a_nat,D2: set_Pr1912306449_a_nat,A: produc598811825_a_nat] :
% 3.27/1.62 ( ( ord_le1374266033_a_nat @ C4 @ D2 )
% 3.27/1.62 => ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ A @ C4 ) @ ( insert2058869377_a_nat @ A @ D2 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_316_subset__insert,axiom,
% 3.27/1.62 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
% 3.27/1.62 ( ~ ( member327031898_a_nat @ X4 @ A4 )
% 3.27/1.62 => ( ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ X4 @ B4 ) )
% 3.27/1.62 = ( ord_le1374266033_a_nat @ A4 @ B4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_317_subset__insert,axiom,
% 3.27/1.62 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b,B4: set_Product_prod_b_b] :
% 3.27/1.62 ( ~ ( member1285940496od_b_b @ X4 @ A4 )
% 3.27/1.62 => ( ( ord_le1036320359od_b_b @ A4 @ ( insert1952693431od_b_b @ X4 @ B4 ) )
% 3.27/1.62 = ( ord_le1036320359od_b_b @ A4 @ B4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_318_subset__insertI,axiom,
% 3.27/1.62 ! [B4: set_Pr1912306449_a_nat,A: produc598811825_a_nat] : ( ord_le1374266033_a_nat @ B4 @ ( insert2058869377_a_nat @ A @ B4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_319_subset__insertI2,axiom,
% 3.27/1.62 ! [A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat,B: produc598811825_a_nat] :
% 3.27/1.62 ( ( ord_le1374266033_a_nat @ A4 @ B4 )
% 3.27/1.62 => ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ B @ B4 ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_320_trans__Id__on,axiom,
% 3.27/1.62 ! [A4: set_b] : ( trans_b @ ( id_on_b @ A4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_321_refl__on__Id__on,axiom,
% 3.27/1.62 ! [A4: set_b] : ( refl_on_b @ A4 @ ( id_on_b @ A4 ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_322_subset__iff,axiom,
% 3.27/1.62 ( ord_le1036320359od_b_b
% 3.27/1.62 = ( ^ [A5: set_Product_prod_b_b,B6: set_Product_prod_b_b] :
% 3.27/1.62 ! [T: product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ T @ A5 )
% 3.27/1.62 => ( member1285940496od_b_b @ T @ B6 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_323_subset__eq,axiom,
% 3.27/1.62 ( ord_le1036320359od_b_b
% 3.27/1.62 = ( ^ [A5: set_Product_prod_b_b,B6: set_Product_prod_b_b] :
% 3.27/1.62 ! [X5: product_prod_b_b] :
% 3.27/1.62 ( ( member1285940496od_b_b @ X5 @ A5 )
% 3.27/1.62 => ( member1285940496od_b_b @ X5 @ B6 ) ) ) ) ).
% 3.27/1.62
% 3.27/1.62 thf(fact_324_subsetD,axiom,
% 3.27/1.63 ! [A4: set_Product_prod_b_b,B4: set_Product_prod_b_b,C2: product_prod_b_b] :
% 3.27/1.63 ( ( ord_le1036320359od_b_b @ A4 @ B4 )
% 3.27/1.63 => ( ( member1285940496od_b_b @ C2 @ A4 )
% 3.27/1.63 => ( member1285940496od_b_b @ C2 @ B4 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_325_in__mono,axiom,
% 3.27/1.63 ! [A4: set_Product_prod_b_b,B4: set_Product_prod_b_b,X4: product_prod_b_b] :
% 3.27/1.63 ( ( ord_le1036320359od_b_b @ A4 @ B4 )
% 3.27/1.63 => ( ( member1285940496od_b_b @ X4 @ A4 )
% 3.27/1.63 => ( member1285940496od_b_b @ X4 @ B4 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_326_subset__singletonD,axiom,
% 3.27/1.63 ! [A4: set_Pr1912306449_a_nat,X4: produc598811825_a_nat] :
% 3.27/1.63 ( ( ord_le1374266033_a_nat @ A4 @ ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) )
% 3.27/1.63 => ( ( A4 = bot_bo2066531197_a_nat )
% 3.27/1.63 | ( A4
% 3.27/1.63 = ( insert2058869377_a_nat @ X4 @ bot_bo2066531197_a_nat ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_327_subset__singleton__iff,axiom,
% 3.27/1.63 ! [X6: set_Pr1912306449_a_nat,A: produc598811825_a_nat] :
% 3.27/1.63 ( ( ord_le1374266033_a_nat @ X6 @ ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) )
% 3.27/1.63 = ( ( X6 = bot_bo2066531197_a_nat )
% 3.27/1.63 | ( X6
% 3.27/1.63 = ( insert2058869377_a_nat @ A @ bot_bo2066531197_a_nat ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_328_subgraph__trans,axiom,
% 3.27/1.63 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b,G_3: labeled_graph_a_b] :
% 3.27/1.63 ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 3.27/1.63 => ( ( graph_1322245399_a_b_b @ G_2 @ G_3 @ ( id_on_b @ ( labeled_vertices_a_b @ G_2 ) ) )
% 3.27/1.63 => ( graph_1322245399_a_b_b @ G_1 @ G_3 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_329_in__quotient__imp__subset,axiom,
% 3.27/1.63 ! [A4: set_b,R2: set_Product_prod_b_b,X6: set_b] :
% 3.27/1.63 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.63 => ( ( member_set_b @ X6 @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.63 => ( ord_less_eq_set_b @ X6 @ A4 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_330_maintainedD2,axiom,
% 3.27/1.63 ! [A4: labeled_graph_a_nat,B4: labeled_graph_a_nat,G: labeled_graph_a_b,F: set_Pr2106913242_nat_b] :
% 3.27/1.63 ( ( maintained_a_nat_b @ ( produc2000589545_a_nat @ A4 @ B4 ) @ G )
% 3.27/1.63 => ( ( graph_914848022_nat_b @ A4 @ G @ F )
% 3.27/1.63 => ~ ! [G3: set_Pr2106913242_nat_b] :
% 3.27/1.63 ( ( graph_914848022_nat_b @ B4 @ G @ G3 )
% 3.27/1.63 => ~ ( ord_le1484132922_nat_b @ F @ G3 ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_331_subgraph__semantics,axiom,
% 3.27/1.63 ! [A4: labeled_graph_a_b,B4: labeled_graph_a_b,A: b,B: b,E: allegorical_term_a] :
% 3.27/1.63 ( ( graph_1322245399_a_b_b @ A4 @ B4 @ ( id_on_b @ ( labeled_vertices_a_b @ A4 ) ) )
% 3.27/1.63 => ( ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ A4 @ E ) )
% 3.27/1.63 => ( member1285940496od_b_b @ ( product_Pair_b_b @ A @ B ) @ ( semantics_a_b @ B4 @ E ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_332_proj__iff,axiom,
% 3.27/1.63 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b,Y4: b] :
% 3.27/1.63 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.63 => ( ( ord_less_eq_set_b @ ( insert_b @ X4 @ ( insert_b @ Y4 @ bot_bot_set_b ) ) @ A4 )
% 3.27/1.63 => ( ( ( equiv_proj_b_b @ R2 @ X4 )
% 3.27/1.63 = ( equiv_proj_b_b @ R2 @ Y4 ) )
% 3.27/1.63 = ( member1285940496od_b_b @ ( product_Pair_b_b @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_333_proj__iff,axiom,
% 3.27/1.63 ! [A4: set_Pr1912306449_a_nat,R2: set_Pr2074180935_a_nat,X4: produc598811825_a_nat,Y4: produc598811825_a_nat] :
% 3.27/1.63 ( ( equiv_281634663_a_nat @ A4 @ R2 )
% 3.27/1.63 => ( ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ ( insert2058869377_a_nat @ Y4 @ bot_bo2066531197_a_nat ) ) @ A4 )
% 3.27/1.63 => ( ( ( equiv_1873824970_a_nat @ R2 @ X4 )
% 3.27/1.63 = ( equiv_1873824970_a_nat @ R2 @ Y4 ) )
% 3.27/1.63 = ( member1542696848_a_nat @ ( produc410388695_a_nat @ X4 @ Y4 ) @ R2 ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_334_consequence__graphD_I1_J,axiom,
% 3.27/1.63 ! [Rs: set_Pr1912306449_a_nat,G: labeled_graph_a_b,R: produc598811825_a_nat] :
% 3.27/1.63 ( ( conseq1058228089_nat_b @ Rs @ G )
% 3.27/1.63 => ( ( member327031898_a_nat @ R @ Rs )
% 3.27/1.63 => ( maintained_a_nat_b @ R @ G ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_335_proj__in__iff,axiom,
% 3.27/1.63 ! [A4: set_Product_prod_b_b,R2: set_Pr2007082183od_b_b,X4: product_prod_b_b] :
% 3.27/1.63 ( ( equiv_1125628061od_b_b @ A4 @ R2 )
% 3.27/1.63 => ( ( member1050118384od_b_b @ ( equiv_270653002od_b_b @ R2 @ X4 ) @ ( equiv_1530858158od_b_b @ A4 @ R2 ) )
% 3.27/1.63 = ( member1285940496od_b_b @ X4 @ A4 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_336_proj__in__iff,axiom,
% 3.27/1.63 ! [A4: set_b,R2: set_Product_prod_b_b,X4: b] :
% 3.27/1.63 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.63 => ( ( member_set_b @ ( equiv_proj_b_b @ R2 @ X4 ) @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.63 = ( member_b @ X4 @ A4 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_337_less__by__empty,axiom,
% 3.27/1.63 ! [A4: set_Pr1912306449_a_nat,B4: set_Pr1912306449_a_nat] :
% 3.27/1.63 ( ( A4 = bot_bo2066531197_a_nat )
% 3.27/1.63 => ( ord_le1374266033_a_nat @ A4 @ B4 ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_338_insert__subsetI,axiom,
% 3.27/1.63 ! [X4: produc598811825_a_nat,A4: set_Pr1912306449_a_nat,X6: set_Pr1912306449_a_nat] :
% 3.27/1.63 ( ( member327031898_a_nat @ X4 @ A4 )
% 3.27/1.63 => ( ( ord_le1374266033_a_nat @ X6 @ A4 )
% 3.27/1.63 => ( ord_le1374266033_a_nat @ ( insert2058869377_a_nat @ X4 @ X6 ) @ A4 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_339_insert__subsetI,axiom,
% 3.27/1.63 ! [X4: product_prod_b_b,A4: set_Product_prod_b_b,X6: set_Product_prod_b_b] :
% 3.27/1.63 ( ( member1285940496od_b_b @ X4 @ A4 )
% 3.27/1.63 => ( ( ord_le1036320359od_b_b @ X6 @ A4 )
% 3.27/1.63 => ( ord_le1036320359od_b_b @ ( insert1952693431od_b_b @ X4 @ X6 ) @ A4 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_340_ssubst__Pair__rhs,axiom,
% 3.27/1.63 ! [R2: b,S: b,R: set_Product_prod_b_b,S2: b] :
% 3.27/1.63 ( ( member1285940496od_b_b @ ( product_Pair_b_b @ R2 @ S ) @ R )
% 3.27/1.63 => ( ( S2 = S )
% 3.27/1.63 => ( member1285940496od_b_b @ ( product_Pair_b_b @ R2 @ S2 ) @ R ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_341_subset__emptyI,axiom,
% 3.27/1.63 ! [A4: set_Product_prod_b_b] :
% 3.27/1.63 ( ! [X3: product_prod_b_b] :
% 3.27/1.63 ~ ( member1285940496od_b_b @ X3 @ A4 )
% 3.27/1.63 => ( ord_le1036320359od_b_b @ A4 @ bot_bo1343651123od_b_b ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_342_subset__emptyI,axiom,
% 3.27/1.63 ! [A4: set_Pr1912306449_a_nat] :
% 3.27/1.63 ( ! [X3: produc598811825_a_nat] :
% 3.27/1.63 ~ ( member327031898_a_nat @ X3 @ A4 )
% 3.27/1.63 => ( ord_le1374266033_a_nat @ A4 @ bot_bo2066531197_a_nat ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_343_subgraph__def,axiom,
% 3.27/1.63 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 3.27/1.63 ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 3.27/1.63 = ( ( G_1
% 3.27/1.63 = ( restrict_a_b @ G_1 ) )
% 3.27/1.63 & ( G_2
% 3.27/1.63 = ( restrict_a_b @ G_2 ) )
% 3.27/1.63 & ( ( graph_union_a_b @ G_1 @ G_2 )
% 3.27/1.63 = G_2 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_344_graph__union__preserves__restrict,axiom,
% 3.27/1.63 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 3.27/1.63 ( ( G_1
% 3.27/1.63 = ( restrict_a_b @ G_1 ) )
% 3.27/1.63 => ( ( G_2
% 3.27/1.63 = ( restrict_a_b @ G_2 ) )
% 3.27/1.63 => ( ( graph_union_a_b @ G_1 @ G_2 )
% 3.27/1.63 = ( restrict_a_b @ ( graph_union_a_b @ G_1 @ G_2 ) ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_345_univ__preserves,axiom,
% 3.27/1.63 ! [A4: set_b,R2: set_Product_prod_b_b,F: b > product_prod_b_b,B4: set_Product_prod_b_b] :
% 3.27/1.63 ( ( equiv_equiv_b @ A4 @ R2 )
% 3.27/1.63 => ( ( equiv_1778431934od_b_b @ R2 @ F )
% 3.27/1.63 => ( ! [X3: b] :
% 3.27/1.63 ( ( member_b @ X3 @ A4 )
% 3.27/1.63 => ( member1285940496od_b_b @ ( F @ X3 ) @ B4 ) )
% 3.27/1.63 => ! [X: set_b] :
% 3.27/1.63 ( ( member_set_b @ X @ ( equiv_quotient_b @ A4 @ R2 ) )
% 3.27/1.63 => ( member1285940496od_b_b @ ( bNF_Gr1693995344od_b_b @ F @ X ) @ B4 ) ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_346_relImage__proj,axiom,
% 3.27/1.63 ! [A4: set_b,R: set_Product_prod_b_b] :
% 3.27/1.63 ( ( equiv_equiv_b @ A4 @ R )
% 3.27/1.63 => ( ord_le2142668391_set_b @ ( bNF_Gr345950468_set_b @ R @ ( equiv_proj_b_b @ R ) ) @ ( id_on_set_b @ ( equiv_quotient_b @ A4 @ R ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_347_subgraph__def2,axiom,
% 3.27/1.63 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 3.27/1.63 ( ( G_1
% 3.27/1.63 = ( restrict_a_b @ G_1 ) )
% 3.27/1.63 => ( ( G_2
% 3.27/1.63 = ( restrict_a_b @ G_2 ) )
% 3.27/1.63 => ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 3.27/1.63 = ( ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) )
% 3.27/1.63 & ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G_1 ) @ ( labeled_edges_a_b @ G_2 ) ) ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_348_labeled__graph_Ocollapse,axiom,
% 3.27/1.63 ! [Labeled_graph: labeled_graph_a_b] :
% 3.27/1.63 ( ( labeled_LG_a_b @ ( labeled_edges_a_b @ Labeled_graph ) @ ( labeled_vertices_a_b @ Labeled_graph ) )
% 3.27/1.63 = Labeled_graph ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_349_graph__unionI,axiom,
% 3.27/1.63 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 3.27/1.63 ( ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G_1 ) @ ( labeled_edges_a_b @ G_2 ) )
% 3.27/1.63 => ( ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) )
% 3.27/1.63 => ( ( graph_union_a_b @ G_1 @ G_2 )
% 3.27/1.63 = G_2 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_350_restrict__subsD,axiom,
% 3.27/1.63 ! [G: labeled_graph_a_b] :
% 3.27/1.63 ( ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G ) @ ( labeled_edges_a_b @ ( restrict_a_b @ G ) ) )
% 3.27/1.63 => ( G
% 3.27/1.63 = ( restrict_a_b @ G ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_351_labeled__graph_Oexhaust__sel,axiom,
% 3.27/1.63 ! [Labeled_graph: labeled_graph_a_b] :
% 3.27/1.63 ( Labeled_graph
% 3.27/1.63 = ( labeled_LG_a_b @ ( labeled_edges_a_b @ Labeled_graph ) @ ( labeled_vertices_a_b @ Labeled_graph ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_352_labeled__graph_Oexpand,axiom,
% 3.27/1.63 ! [Labeled_graph: labeled_graph_a_b,Labeled_graph2: labeled_graph_a_b] :
% 3.27/1.63 ( ( ( ( labeled_edges_a_b @ Labeled_graph )
% 3.27/1.63 = ( labeled_edges_a_b @ Labeled_graph2 ) )
% 3.27/1.63 & ( ( labeled_vertices_a_b @ Labeled_graph )
% 3.27/1.63 = ( labeled_vertices_a_b @ Labeled_graph2 ) ) )
% 3.27/1.63 => ( Labeled_graph = Labeled_graph2 ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_353_graph__union__iff,axiom,
% 3.27/1.63 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 3.27/1.63 ( ( ( graph_union_a_b @ G_1 @ G_2 )
% 3.27/1.63 = G_2 )
% 3.27/1.63 = ( ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ G_1 ) @ ( labeled_edges_a_b @ G_2 ) )
% 3.27/1.63 & ( ord_less_eq_set_b @ ( labeled_vertices_a_b @ G_1 ) @ ( labeled_vertices_a_b @ G_2 ) ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(fact_354_subgraph__subset_I2_J,axiom,
% 3.27/1.63 ! [G_1: labeled_graph_a_b,G_2: labeled_graph_a_b] :
% 3.27/1.63 ( ( graph_1322245399_a_b_b @ G_1 @ G_2 @ ( id_on_b @ ( labeled_vertices_a_b @ G_1 ) ) )
% 3.27/1.63 => ( ord_le887919760od_b_b @ ( labeled_edges_a_b @ ( restrict_a_b @ G_1 ) ) @ ( labeled_edges_a_b @ G_2 ) ) ) ).
% 3.27/1.63
% 3.27/1.63 thf(conj_0,conjecture,
% 3.27/1.63 member1285940496od_b_b @ ( product_Pair_b_b @ z @ y ) @ ( getRel_a_b @ i @ g ) ).
% 3.27/1.63 ------------------------
% 3.27/1.65 % (16571)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_16406 for (2999ds/2Mi)
% 3.27/1.65 % (16567)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16406 for (2999ds/183Mi)
% 3.27/1.65 % (16569)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_16406 for (2999ds/27Mi)
% 3.27/1.65 % (16568)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_16406 for (2999ds/4Mi)
% 3.27/1.65 % (16570)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16406 for (2999ds/2Mi)
% 3.27/1.65 % (16572)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_16406 for (2999ds/275Mi)
% 3.27/1.65 % (16573)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16406 for (2999ds/18Mi)
% 3.27/1.65 % (16570)Instruction limit reached!
% 3.27/1.65 % (16570)------------------------------
% 3.27/1.65 % (16570)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.65 % (16570)Termination reason: Unknown
% 3.27/1.65 % (16570)Termination phase: shuffling
% 3.27/1.65
% 3.27/1.65 % (16570)Memory used [KB]: 1535
% 3.27/1.65 % (16570)Time elapsed: 0.002 s
% 3.27/1.65 % (16570)Instructions burned: 2 (million)
% 3.27/1.65 % (16570)------------------------------
% 3.27/1.65 % (16570)------------------------------
% 3.27/1.65 % (16571)Instruction limit reached!
% 3.27/1.65 % (16571)------------------------------
% 3.27/1.65 % (16571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.65 % (16571)Termination reason: Unknown
% 3.27/1.65 % (16571)Termination phase: shuffling
% 3.27/1.65
% 3.27/1.65 % (16571)Memory used [KB]: 1535
% 3.27/1.65 % (16571)Time elapsed: 0.002 s
% 3.27/1.65 % (16571)Instructions burned: 2 (million)
% 3.27/1.65 % (16571)------------------------------
% 3.27/1.65 % (16571)------------------------------
% 3.27/1.65 % (16568)Instruction limit reached!
% 3.27/1.65 % (16568)------------------------------
% 3.27/1.65 % (16568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.65 % (16568)Termination reason: Unknown
% 3.27/1.65 % (16568)Termination phase: shuffling
% 3.27/1.65
% 3.27/1.65 % (16568)Memory used [KB]: 1663
% 3.27/1.65 % (16568)Time elapsed: 0.003 s
% 3.27/1.65 % (16568)Instructions burned: 5 (million)
% 3.27/1.65 % (16568)------------------------------
% 3.27/1.65 % (16568)------------------------------
% 3.27/1.66 % (16573)Instruction limit reached!
% 3.27/1.66 % (16573)------------------------------
% 3.27/1.66 % (16573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.66 % (16573)Termination reason: Unknown
% 3.27/1.66 % (16573)Termination phase: shuffling
% 3.27/1.66
% 3.27/1.66 % (16573)Memory used [KB]: 1918
% 3.27/1.66 % (16573)Time elapsed: 0.007 s
% 3.27/1.66 % (16573)Instructions burned: 18 (million)
% 3.27/1.66 % (16573)------------------------------
% 3.27/1.66 % (16573)------------------------------
% 3.27/1.66 % (16569)Instruction limit reached!
% 3.27/1.66 % (16569)------------------------------
% 3.27/1.66 % (16569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.66 % (16569)Termination reason: Unknown
% 3.27/1.66 % (16569)Termination phase: shuffling
% 3.27/1.66
% 3.27/1.66 % (16569)Memory used [KB]: 2046
% 3.27/1.66 % (16569)Time elapsed: 0.010 s
% 3.27/1.66 % (16569)Instructions burned: 27 (million)
% 3.27/1.66 % (16569)------------------------------
% 3.27/1.66 % (16569)------------------------------
% 3.27/1.66 % (16574)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16406 for (2999ds/3Mi)
% 3.27/1.66 % (16575)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_16406 for (2999ds/37Mi)
% 3.27/1.66 % (16576)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_16406 for (2999ds/15Mi)
% 3.27/1.66 % (16574)Instruction limit reached!
% 3.27/1.66 % (16574)------------------------------
% 3.27/1.66 % (16574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.66 % (16574)Termination reason: Unknown
% 3.27/1.66 % (16574)Termination phase: shuffling
% 3.27/1.66
% 3.27/1.66 % (16574)Memory used [KB]: 1663
% 3.27/1.66 % (16574)Time elapsed: 0.003 s
% 3.27/1.66 % (16574)Instructions burned: 5 (million)
% 3.27/1.66 % (16574)------------------------------
% 3.27/1.66 % (16574)------------------------------
% 3.27/1.67 % (16577)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_16406 for (2999ds/3Mi)
% 3.27/1.67 % (16576)Instruction limit reached!
% 3.27/1.67 % (16576)------------------------------
% 3.27/1.67 % (16576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.67 % (16576)Termination reason: Unknown
% 3.27/1.67 % (16576)Termination phase: shuffling
% 3.27/1.67
% 3.27/1.67 % (16576)Memory used [KB]: 1791
% 3.27/1.67 % (16576)Time elapsed: 0.006 s
% 3.27/1.67 % (16576)Instructions burned: 15 (million)
% 3.27/1.67 % (16576)------------------------------
% 3.27/1.67 % (16576)------------------------------
% 3.27/1.67 % (16577)Instruction limit reached!
% 3.27/1.67 % (16577)------------------------------
% 3.27/1.67 % (16577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.67 % (16577)Termination reason: Unknown
% 3.27/1.67 % (16577)Termination phase: shuffling
% 3.27/1.67
% 3.27/1.67 % (16577)Memory used [KB]: 1663
% 3.27/1.67 % (16577)Time elapsed: 0.003 s
% 3.27/1.67 % (16577)Instructions burned: 5 (million)
% 3.27/1.67 % (16577)------------------------------
% 3.27/1.67 % (16577)------------------------------
% 3.27/1.67 % (16578)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_16406 for (2999ds/1041Mi)
% 3.27/1.67 % (16572)First to succeed.
% 3.27/1.67 % (16575)Instruction limit reached!
% 3.27/1.67 % (16575)------------------------------
% 3.27/1.67 % (16575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.67 % (16575)Termination reason: Unknown
% 3.27/1.67 % (16575)Termination phase: shuffling
% 3.27/1.67
% 3.27/1.67 % (16575)Memory used [KB]: 2430
% 3.27/1.67 % (16575)Time elapsed: 0.012 s
% 3.27/1.67 % (16575)Instructions burned: 40 (million)
% 3.27/1.67 % (16575)------------------------------
% 3.27/1.67 % (16575)------------------------------
% 3.27/1.67 % (16579)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16406 for (2999ds/7Mi)
% 3.27/1.67 % (16572)Refutation found. Thanks to Tanya!
% 3.27/1.67 % SZS status Theorem for DTF2THF_16406
% 3.27/1.67 % SZS output start Proof for DTF2THF_16406
% 3.27/1.67 thf(type_def_5, type, set_Product_prod_b_b: $tType).
% 3.27/1.67 thf(type_def_6, type, b: $tType).
% 3.27/1.67 thf(type_def_7, type, set_b: $tType).
% 3.27/1.67 thf(type_def_9, type, set_Pr1463111367_set_b: $tType).
% 3.27/1.67 thf(type_def_10, type, product_prod_b_b: $tType).
% 3.27/1.67 thf(type_def_11, type, set_Pr1912306449_a_nat: $tType).
% 3.27/1.67 thf(type_def_12, type, set_Pr2074180935_a_nat: $tType).
% 3.27/1.67 thf(type_def_13, type, set_Pr2007082183od_b_b: $tType).
% 3.27/1.67 thf(type_def_14, type, produc598811825_a_nat: $tType).
% 3.27/1.67 thf(type_def_15, type, set_se480333169_a_nat: $tType).
% 3.27/1.67 thf(type_def_16, type, set_se668355239od_b_b: $tType).
% 3.27/1.67 thf(type_def_17, type, set_set_b: $tType).
% 3.27/1.67 thf(type_def_18, type, a: $tType).
% 3.27/1.67 thf(type_def_19, type, labele553847424_a_nat: $tType).
% 3.27/1.67 thf(type_def_20, type, labele1586099759_a_nat: $tType).
% 3.27/1.67 thf(type_def_21, type, labele488176869od_b_b: $tType).
% 3.27/1.67 thf(type_def_22, type, labeled_graph_a_b: $tType).
% 3.27/1.67 thf(type_def_23, type, allegorical_term_a: $tType).
% 3.27/1.67 thf(type_def_24, type, labeled_graph_a_nat: $tType).
% 3.27/1.67 thf(type_def_25, type, set_Pr2106913242_nat_b: $tType).
% 3.27/1.67 thf(type_def_26, type, set_Pr1914572181_nat_b: $tType).
% 3.27/1.67 thf(type_def_27, type, set_Pr21610864_a_nat: $tType).
% 3.27/1.67 thf(type_def_28, type, set_Pr1900477680od_b_b: $tType).
% 3.27/1.67 thf(type_def_29, type, set_la1108361502_a_nat: $tType).
% 3.27/1.67 thf(type_def_30, type, set_Pr1370504391_a_nat: $tType).
% 3.27/1.67 thf(type_def_31, type, set_Pr1870121716_a_nat: $tType).
% 3.27/1.67 thf(type_def_32, type, set_Pr1223907581_a_nat: $tType).
% 3.27/1.67 thf(type_def_33, type, set_Pr471017666_nat_b: $tType).
% 3.27/1.67 thf(type_def_34, type, set_Pr2028709629od_b_b: $tType).
% 3.27/1.67 thf(type_def_35, type, set_Pr1549208893od_b_b: $tType).
% 3.27/1.67 thf(type_def_36, type, set_Pr357419743_b_b_b: $tType).
% 3.27/1.67 thf(type_def_37, type, set_Pr1391123143od_b_b: $tType).
% 3.27/1.67 thf(type_def_38, type, set_Pr1271557767od_b_b: $tType).
% 3.27/1.67 thf(type_def_39, type, set_Pr2010644300_a_nat: $tType).
% 3.27/1.67 thf(type_def_40, type, set_Pr2054759953_a_nat: $tType).
% 3.27/1.67 thf(type_def_41, type, set_Pr1626978257_a_nat: $tType).
% 3.27/1.67 thf(type_def_42, type, set_Pr352593785_a_nat: $tType).
% 3.27/1.67 thf(type_def_43, type, set_Pr2072721159_a_nat: $tType).
% 3.27/1.67 thf(type_def_44, type, set_Pr621705391od_b_b: $tType).
% 3.27/1.67 thf(type_def_45, type, set_Pr1229594311od_b_b: $tType).
% 3.27/1.67 thf(type_def_46, type, set_Pr406163591od_b_b: $tType).
% 3.27/1.67 thf(type_def_47, type, set_Pr454079687od_b_b: $tType).
% 3.27/1.67 thf(type_def_48, type, set_Pr963356295od_b_b: $tType).
% 3.27/1.67 thf(type_def_49, type, set_Pr1045675335_a_nat: $tType).
% 3.27/1.67 thf(type_def_50, type, produc297983550_a_nat: $tType).
% 3.27/1.67 thf(type_def_51, type, produc1248251020_nat_b: $tType).
% 3.27/1.67 thf(type_def_52, type, produc1099860071_a_nat: $tType).
% 3.27/1.67 thf(type_def_53, type, produc1428649885_a_nat: $tType).
% 3.27/1.67 thf(type_def_54, type, produc1734972647_a_nat: $tType).
% 3.27/1.67 thf(type_def_55, type, produc588599965_a_nat: $tType).
% 3.27/1.67 thf(type_def_56, type, produc759315249_a_nat: $tType).
% 3.27/1.67 thf(type_def_57, type, produc1168037607od_b_b: $tType).
% 3.27/1.67 thf(type_def_58, type, produc130774173od_b_b: $tType).
% 3.27/1.67 thf(type_def_59, type, produc1529836775od_b_b: $tType).
% 3.27/1.67 thf(type_def_60, type, produc300546791od_b_b: $tType).
% 3.27/1.67 thf(type_def_61, type, produc1230257585_a_nat: $tType).
% 3.27/1.67 thf(type_def_62, type, produc1554726631od_b_b: $tType).
% 3.27/1.67 thf(type_def_63, type, produc1921243879od_b_b: $tType).
% 3.27/1.67 thf(type_def_64, type, produc1063300186_a_nat: $tType).
% 3.27/1.67 thf(type_def_65, type, produc1133639312od_b_b: $tType).
% 3.27/1.67 thf(type_def_66, type, produc401211613od_b_b: $tType).
% 3.27/1.67 thf(type_def_67, type, produc1614181008od_b_b: $tType).
% 3.27/1.67 thf(type_def_68, type, produc1114535079od_b_b: $tType).
% 3.27/1.67 thf(type_def_69, type, produc776320369_a_nat: $tType).
% 3.27/1.67 thf(type_def_70, type, produc1942261264_a_nat: $tType).
% 3.27/1.67 thf(type_def_71, type, produc2106791463_a_nat: $tType).
% 3.27/1.67 thf(type_def_72, type, produc1053464231od_b_b: $tType).
% 3.27/1.67 thf(type_def_73, type, produc197046951od_b_b: $tType).
% 3.27/1.67 thf(type_def_74, type, produc2112523263_b_b_b: $tType).
% 3.27/1.67 thf(type_def_75, type, produc1869574247_a_nat: $tType).
% 3.27/1.67 thf(type_def_76, type, produc51693287od_b_b: $tType).
% 3.27/1.67 thf(type_def_77, type, produc150548199_set_b: $tType).
% 3.27/1.67 thf(type_def_78, type, produc126142486_a_nat: $tType).
% 3.27/1.67 thf(type_def_79, type, produc207618585_a_nat: $tType).
% 3.27/1.67 thf(type_def_80, type, produc255402447od_b_b: $tType).
% 3.27/1.67 thf(type_def_81, type, set_Pr773647559od_b_b: $tType).
% 3.27/1.67 thf(func_def_0, type, set_Pr1223907581_a_nat: $tType).
% 3.27/1.67 thf(func_def_1, type, produc588599965_a_nat: $tType).
% 3.27/1.67 thf(func_def_2, type, set_Pr1370504391_a_nat: $tType).
% 3.27/1.67 thf(func_def_3, type, produc1734972647_a_nat: $tType).
% 3.27/1.67 thf(func_def_4, type, produc1428649885_a_nat: $tType).
% 3.27/1.67 thf(func_def_5, type, set_Pr2072721159_a_nat: $tType).
% 3.27/1.67 thf(func_def_6, type, produc2106791463_a_nat: $tType).
% 3.27/1.67 thf(func_def_7, type, set_Pr2054759953_a_nat: $tType).
% 3.27/1.67 thf(func_def_8, type, set_Pr2028709629od_b_b: $tType).
% 3.27/1.67 thf(func_def_9, type, set_Pr21610864_a_nat: $tType).
% 3.27/1.67 thf(func_def_10, type, set_Pr1045675335_a_nat: $tType).
% 3.27/1.67 thf(func_def_11, type, produc1230257585_a_nat: $tType).
% 3.27/1.67 thf(func_def_12, type, produc130774173od_b_b: $tType).
% 3.27/1.67 thf(func_def_13, type, produc1942261264_a_nat: $tType).
% 3.27/1.67 thf(func_def_14, type, produc1869574247_a_nat: $tType).
% 3.27/1.67 thf(func_def_15, type, set_Pr1229594311od_b_b: $tType).
% 3.27/1.67 thf(func_def_16, type, set_Pr1391123143od_b_b: $tType).
% 3.27/1.67 thf(func_def_17, type, set_Pr2074180935_a_nat: $tType).
% 3.27/1.67 thf(func_def_18, type, produc1554726631od_b_b: $tType).
% 3.27/1.67 thf(func_def_19, type, produc300546791od_b_b: $tType).
% 3.27/1.67 thf(func_def_20, type, produc1099860071_a_nat: $tType).
% 3.27/1.67 thf(func_def_21, type, set_Pr1626978257_a_nat: $tType).
% 3.27/1.67 thf(func_def_22, type, set_Pr1549208893od_b_b: $tType).
% 3.27/1.67 thf(func_def_23, type, produc776320369_a_nat: $tType).
% 3.27/1.67 thf(func_def_24, type, produc401211613od_b_b: $tType).
% 3.27/1.67 thf(func_def_25, type, set_Pr406163591od_b_b: $tType).
% 3.27/1.67 thf(func_def_26, type, set_Pr1271557767od_b_b: $tType).
% 3.27/1.67 thf(func_def_27, type, set_Pr454079687od_b_b: $tType).
% 3.27/1.67 thf(func_def_28, type, produc1053464231od_b_b: $tType).
% 3.27/1.67 thf(func_def_29, type, produc1114535079od_b_b: $tType).
% 3.27/1.67 thf(func_def_30, type, produc1921243879od_b_b: $tType).
% 3.27/1.67 thf(func_def_31, type, set_Pr1870121716_a_nat: $tType).
% 3.27/1.67 thf(func_def_32, type, produc297983550_a_nat: $tType).
% 3.27/1.67 thf(func_def_33, type, produc759315249_a_nat: $tType).
% 3.27/1.67 thf(func_def_34, type, set_Pr963356295od_b_b: $tType).
% 3.27/1.67 thf(func_def_35, type, set_Pr352593785_a_nat: $tType).
% 3.27/1.67 thf(func_def_36, type, set_Pr1914572181_nat_b: $tType).
% 3.27/1.67 thf(func_def_37, type, labele1586099759_a_nat: $tType).
% 3.27/1.67 thf(func_def_38, type, produc1529836775od_b_b: $tType).
% 3.27/1.67 thf(func_def_39, type, produc197046951od_b_b: $tType).
% 3.27/1.67 thf(func_def_40, type, produc207618585_a_nat: $tType).
% 3.27/1.67 thf(func_def_41, type, produc1063300186_a_nat: $tType).
% 3.27/1.67 thf(func_def_42, type, set_se480333169_a_nat: $tType).
% 3.27/1.67 thf(func_def_43, type, set_Pr1912306449_a_nat: $tType).
% 3.27/1.67 thf(func_def_44, type, set_Pr773647559od_b_b: $tType).
% 3.27/1.67 thf(func_def_45, type, produc1614181008od_b_b: $tType).
% 3.27/1.67 thf(func_def_46, type, produc598811825_a_nat: $tType).
% 3.27/1.67 thf(func_def_47, type, produc51693287od_b_b: $tType).
% 3.27/1.67 thf(func_def_48, type, set_Pr2007082183od_b_b: $tType).
% 3.27/1.67 thf(func_def_49, type, produc1168037607od_b_b: $tType).
% 3.27/1.67 thf(func_def_50, type, set_Pr2010644300_a_nat: $tType).
% 3.27/1.67 thf(func_def_51, type, set_Pr471017666_nat_b: $tType).
% 3.27/1.67 thf(func_def_52, type, labele553847424_a_nat: $tType).
% 3.27/1.67 thf(func_def_53, type, produc126142486_a_nat: $tType).
% 3.27/1.67 thf(func_def_54, type, produc1248251020_nat_b: $tType).
% 3.27/1.67 thf(func_def_55, type, set_Pr621705391od_b_b: $tType).
% 3.27/1.67 thf(func_def_56, type, set_Pr1900477680od_b_b: $tType).
% 3.27/1.67 thf(func_def_57, type, set_Pr357419743_b_b_b: $tType).
% 3.27/1.67 thf(func_def_58, type, set_Pr1463111367_set_b: $tType).
% 3.27/1.67 thf(func_def_59, type, labele488176869od_b_b: $tType).
% 3.27/1.67 thf(func_def_60, type, produc255402447od_b_b: $tType).
% 3.27/1.67 thf(func_def_61, type, produc1133639312od_b_b: $tType).
% 3.27/1.67 thf(func_def_62, type, produc2112523263_b_b_b: $tType).
% 3.27/1.67 thf(func_def_63, type, set_la1108361502_a_nat: $tType).
% 3.27/1.67 thf(func_def_64, type, set_se668355239od_b_b: $tType).
% 3.27/1.67 thf(func_def_65, type, produc150548199_set_b: $tType).
% 3.27/1.67 thf(func_def_66, type, set_Pr2106913242_nat_b: $tType).
% 3.27/1.67 thf(func_def_67, type, labeled_graph_a_nat: $tType).
% 3.27/1.67 thf(func_def_68, type, set_Product_prod_b_b: $tType).
% 3.27/1.67 thf(func_def_69, type, allegorical_term_a: $tType).
% 3.27/1.67 thf(func_def_70, type, labeled_graph_a_b: $tType).
% 3.27/1.67 thf(func_def_71, type, product_prod_b_b: $tType).
% 3.27/1.67 thf(func_def_72, type, set_set_b: $tType).
% 3.27/1.67 thf(func_def_73, type, set_b: $tType).
% 3.27/1.67 thf(func_def_74, type, b: $tType).
% 3.27/1.67 thf(func_def_75, type, a: $tType).
% 3.27/1.67 thf(func_def_76, type, bNF_Gr345950468_set_b: set_Product_prod_b_b > (b > set_b) > set_Pr1463111367_set_b).
% 3.27/1.67 thf(func_def_77, type, bNF_Gr1693995344od_b_b: (b > product_prod_b_b) > set_b > product_prod_b_b).
% 3.27/1.67 thf(func_def_78, type, equiv_1778431934od_b_b: set_Product_prod_b_b > (b > product_prod_b_b) > $o).
% 3.27/1.67 thf(func_def_79, type, equiv_281634663_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o).
% 3.27/1.67 thf(func_def_80, type, equiv_1125628061od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o).
% 3.27/1.67 thf(func_def_81, type, equiv_equiv_b: set_b > set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_82, type, equiv_1873824970_a_nat: set_Pr2074180935_a_nat > produc598811825_a_nat > set_Pr1912306449_a_nat).
% 3.27/1.67 thf(func_def_83, type, equiv_270653002od_b_b: set_Pr2007082183od_b_b > product_prod_b_b > set_Product_prod_b_b).
% 3.27/1.67 thf(func_def_84, type, equiv_proj_b_b: set_Product_prod_b_b > b > set_b).
% 3.27/1.67 thf(func_def_85, type, equiv_1338033912_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > set_se480333169_a_nat).
% 3.27/1.67 thf(func_def_86, type, equiv_1530858158od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > set_se668355239od_b_b).
% 3.27/1.67 thf(func_def_87, type, equiv_quotient_b: set_b > set_Product_prod_b_b > set_set_b).
% 3.27/1.67 thf(func_def_88, type, getRel1657740669_a_nat: a > labele553847424_a_nat > set_Pr1912306449_a_nat).
% 3.27/1.67 thf(func_def_89, type, getRel1248627292_a_nat: a > labele1586099759_a_nat > set_Pr2074180935_a_nat).
% 3.27/1.67 thf(func_def_90, type, getRel1183868946od_b_b: a > labele488176869od_b_b > set_Pr2007082183od_b_b).
% 3.27/1.67 thf(func_def_91, type, getRel_a_b: a > labeled_graph_a_b > set_Product_prod_b_b).
% 3.27/1.67 thf(func_def_92, type, semantics_a_b: labeled_graph_a_b > allegorical_term_a > set_Product_prod_b_b).
% 3.27/1.67 thf(func_def_93, type, graph_914848022_nat_b: labeled_graph_a_nat > labeled_graph_a_b > set_Pr2106913242_nat_b > $o).
% 3.27/1.67 thf(func_def_94, type, graph_23275493_nat_b: labele1586099759_a_nat > labeled_graph_a_b > set_Pr1914572181_nat_b > $o).
% 3.27/1.67 thf(func_def_95, type, graph_1322245399_a_b_b: labeled_graph_a_b > labeled_graph_a_b > set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_96, type, graph_union_a_b: labeled_graph_a_b > labeled_graph_a_b > labeled_graph_a_b).
% 3.27/1.67 thf(func_def_97, type, labele1171788421_a_nat: set_Pr21610864_a_nat > set_Pr1912306449_a_nat > labele1586099759_a_nat).
% 3.27/1.67 thf(func_def_98, type, labeled_LG_a_b: set_Pr1900477680od_b_b > set_b > labeled_graph_a_b).
% 3.27/1.67 thf(func_def_99, type, labeled_edges_a_b: labeled_graph_a_b > set_Pr1900477680od_b_b).
% 3.27/1.67 thf(func_def_100, type, labele953802518_a_nat: labele553847424_a_nat > set_la1108361502_a_nat).
% 3.27/1.67 thf(func_def_101, type, labele198153475_a_nat: labele1586099759_a_nat > set_Pr1912306449_a_nat).
% 3.27/1.67 thf(func_def_102, type, labele240831545od_b_b: labele488176869od_b_b > set_Product_prod_b_b).
% 3.27/1.67 thf(func_def_103, type, labeled_vertices_a_b: labeled_graph_a_b > set_b).
% 3.27/1.67 thf(func_def_104, type, on_tri232494965_a_nat: set_Pr1912306449_a_nat > set_Pr1370504391_a_nat).
% 3.27/1.67 thf(func_def_105, type, on_tri1766439058_a_nat: set_Pr1870121716_a_nat > set_Pr1223907581_a_nat).
% 3.27/1.67 thf(func_def_106, type, on_tri1706508026od_b_b: set_Pr471017666_nat_b > set_Pr2028709629od_b_b).
% 3.27/1.67 thf(func_def_107, type, on_tri1993240081at_b_a: set_Pr471017666_nat_b > set_Pr1549208893od_b_b).
% 3.27/1.67 thf(func_def_108, type, on_tri97791909od_b_b: set_Pr357419743_b_b_b > set_Pr1391123143od_b_b).
% 3.27/1.67 thf(func_def_109, type, on_tri1764442556_b_b_a: set_Pr357419743_b_b_b > set_Pr1271557767od_b_b).
% 3.27/1.67 thf(func_def_110, type, on_tri1837931688od_b_b: set_Pr2010644300_a_nat > set_Pr2054759953_a_nat).
% 3.27/1.67 thf(func_def_111, type, on_tri2117183935_nat_a: set_Pr2010644300_a_nat > set_Pr1626978257_a_nat).
% 3.27/1.67 thf(func_def_112, type, on_tri1794757026_nat_a: set_Pr352593785_a_nat > set_Pr2072721159_a_nat).
% 3.27/1.67 thf(func_def_113, type, on_tri927008213od_b_b: set_Pr621705391od_b_b > set_Pr1229594311od_b_b).
% 3.27/1.67 thf(func_def_114, type, on_tri2003683308_b_b_a: set_Pr621705391od_b_b > set_Pr406163591od_b_b).
% 3.27/1.67 thf(func_def_115, type, on_tri111564221od_b_b: set_Product_prod_b_b > set_Pr454079687od_b_b).
% 3.27/1.67 thf(func_def_116, type, on_triple_b_b_a: set_Product_prod_b_b > set_Pr963356295od_b_b).
% 3.27/1.67 thf(func_def_117, type, restri1808752350_a_nat: labele553847424_a_nat > labele553847424_a_nat).
% 3.27/1.67 thf(func_def_118, type, restri147273531_a_nat: labele1586099759_a_nat > labele1586099759_a_nat).
% 3.27/1.67 thf(func_def_119, type, restri431340145od_b_b: labele488176869od_b_b > labele488176869od_b_b).
% 3.27/1.67 thf(func_def_120, type, restrict_a_b: labeled_graph_a_b > labeled_graph_a_b).
% 3.27/1.67 thf(func_def_121, type, idempo848975736_a_nat: set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_122, type, idempo365306145_a_nat: set_Pr2074180935_a_nat > $o).
% 3.27/1.67 thf(func_def_123, type, idempo371692375od_b_b: set_Pr2007082183od_b_b > $o).
% 3.27/1.67 thf(func_def_124, type, idempotent_b: set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_125, type, order_1770871027_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_126, type, order_1458491494_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o).
% 3.27/1.67 thf(func_def_127, type, order_653178932r_on_b: set_b > set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_128, type, order_989282449_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_129, type, order_145089608_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o).
% 3.27/1.67 thf(func_def_130, type, order_265664679od_b_b: set_Pr1900477680od_b_b > set_Pr963356295od_b_b > $o).
% 3.27/1.67 thf(func_def_131, type, order_1195372542od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o).
% 3.27/1.67 thf(func_def_132, type, order_640573096_a_nat: set_se480333169_a_nat > set_Pr1045675335_a_nat > $o).
% 3.27/1.67 thf(func_def_133, type, order_preorder_on_b: set_b > set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_134, type, bot_bo186023328_nat_o: produc598811825_a_nat > $o).
% 3.27/1.67 thf(func_def_135, type, bot_bo1302443370_b_b_o: product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_136, type, bot_bo1119737266_a_nat: set_la1108361502_a_nat).
% 3.27/1.67 thf(func_def_137, type, bot_bo2066531197_a_nat: set_Pr1912306449_a_nat).
% 3.27/1.67 thf(func_def_138, type, bot_bo1803599795_a_nat: set_Pr2074180935_a_nat).
% 3.27/1.67 thf(func_def_139, type, bot_bo1936722433_nat_b: set_Pr1914572181_nat_b).
% 3.27/1.67 thf(func_def_140, type, bot_bo896299827_a_nat: set_Pr1370504391_a_nat).
% 3.27/1.67 thf(func_def_141, type, bot_bo1856908531od_b_b: set_Pr963356295od_b_b).
% 3.27/1.67 thf(func_def_142, type, bot_bo1174586163od_b_b: set_Pr2007082183od_b_b).
% 3.27/1.67 thf(func_def_143, type, bot_bo2142478259_a_nat: set_Pr1045675335_a_nat).
% 3.27/1.67 thf(func_def_144, type, bot_bo1463793628_a_nat: set_Pr21610864_a_nat).
% 3.27/1.67 thf(func_def_145, type, bot_bo519661404od_b_b: set_Pr1900477680od_b_b).
% 3.27/1.67 thf(func_def_146, type, bot_bo1343651123od_b_b: set_Product_prod_b_b).
% 3.27/1.67 thf(func_def_147, type, bot_bo1746593245_a_nat: set_se480333169_a_nat).
% 3.27/1.67 thf(func_def_148, type, bot_bot_set_b: set_b).
% 3.27/1.67 thf(func_def_149, type, ord_le1374266033_a_nat: set_Pr1912306449_a_nat > set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_150, type, ord_le1484132922_nat_b: set_Pr2106913242_nat_b > set_Pr2106913242_nat_b > $o).
% 3.27/1.67 thf(func_def_151, type, ord_le2142668391_set_b: set_Pr1463111367_set_b > set_Pr1463111367_set_b > $o).
% 3.27/1.67 thf(func_def_152, type, ord_le887919760od_b_b: set_Pr1900477680od_b_b > set_Pr1900477680od_b_b > $o).
% 3.27/1.67 thf(func_def_153, type, ord_le1036320359od_b_b: set_Product_prod_b_b > set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_154, type, ord_less_eq_set_b: set_b > set_b > $o).
% 3.27/1.67 thf(func_def_155, type, produc2000589545_a_nat: labeled_graph_a_nat > labeled_graph_a_nat > produc598811825_a_nat).
% 3.27/1.67 thf(func_def_156, type, produc2092742640_a_nat: labeled_graph_a_nat > produc598811825_a_nat > produc297983550_a_nat).
% 3.27/1.67 thf(func_def_157, type, produc1600512702_nat_b: labeled_graph_a_nat > b > produc1248251020_nat_b).
% 3.27/1.67 thf(func_def_158, type, produc410388695_a_nat: produc598811825_a_nat > produc598811825_a_nat > produc1099860071_a_nat).
% 3.27/1.67 thf(func_def_159, type, produc639906189_a_nat: produc598811825_a_nat > produc1099860071_a_nat > produc1428649885_a_nat).
% 3.27/1.67 thf(func_def_160, type, produc404908887_a_nat: produc1099860071_a_nat > produc1099860071_a_nat > produc1734972647_a_nat).
% 3.27/1.67 thf(func_def_161, type, produc1645191821_a_nat: produc1099860071_a_nat > produc1428649885_a_nat > produc588599965_a_nat).
% 3.27/1.67 thf(func_def_162, type, produc926922637od_b_b: produc759315249_a_nat > produc1168037607od_b_b > produc130774173od_b_b).
% 3.27/1.67 thf(func_def_163, type, produc885982551od_b_b: produc1529836775od_b_b > produc1168037607od_b_b > produc300546791od_b_b).
% 3.27/1.67 thf(func_def_164, type, produc1783652257_a_nat: produc1168037607od_b_b > produc759315249_a_nat > produc1230257585_a_nat).
% 3.27/1.67 thf(func_def_165, type, produc962089303od_b_b: produc1168037607od_b_b > produc1529836775od_b_b > produc1554726631od_b_b).
% 3.27/1.67 thf(func_def_166, type, produc1403229015od_b_b: produc1168037607od_b_b > produc1168037607od_b_b > produc1921243879od_b_b).
% 3.27/1.67 thf(func_def_167, type, produc179745357od_b_b: produc1063300186_a_nat > produc1133639312od_b_b > produc401211613od_b_b).
% 3.27/1.67 thf(func_def_168, type, produc1116925335od_b_b: produc1614181008od_b_b > produc1133639312od_b_b > produc1114535079od_b_b).
% 3.27/1.67 thf(func_def_169, type, produc966067681_a_nat: produc1133639312od_b_b > produc1063300186_a_nat > produc776320369_a_nat).
% 3.27/1.67 thf(func_def_170, type, produc797892631_a_nat: produc1133639312od_b_b > produc1942261264_a_nat > produc2106791463_a_nat).
% 3.27/1.67 thf(func_def_171, type, produc789452695od_b_b: produc1133639312od_b_b > produc1614181008od_b_b > produc1053464231od_b_b).
% 3.27/1.67 thf(func_def_172, type, produc1037687191od_b_b: produc1133639312od_b_b > produc1133639312od_b_b > produc197046951od_b_b).
% 3.27/1.67 thf(func_def_173, type, produc90796321_a_nat: product_prod_b_b > produc598811825_a_nat > produc759315249_a_nat).
% 3.27/1.67 thf(func_def_174, type, produc1850071895od_b_b: product_prod_b_b > produc1168037607od_b_b > produc1529836775od_b_b).
% 3.27/1.67 thf(func_def_175, type, produc546497367od_b_b: product_prod_b_b > product_prod_b_b > produc1168037607od_b_b).
% 3.27/1.67 thf(func_def_176, type, produc1001682799_b_b_b: product_prod_b_b > b > produc2112523263_b_b_b).
% 3.27/1.67 thf(func_def_177, type, produc1767291095_a_nat: set_Pr1912306449_a_nat > set_Pr1912306449_a_nat > produc1869574247_a_nat).
% 3.27/1.67 thf(func_def_178, type, produc1027169111od_b_b: set_Product_prod_b_b > set_Product_prod_b_b > produc51693287od_b_b).
% 3.27/1.67 thf(func_def_179, type, produc1403713879_set_b: set_b > set_b > produc150548199_set_b).
% 3.27/1.67 thf(func_def_180, type, produc1110849866_a_nat: a > produc598811825_a_nat > produc1063300186_a_nat).
% 3.27/1.67 thf(func_def_181, type, produc1904940160_a_nat: a > produc1099860071_a_nat > produc1942261264_a_nat).
% 3.27/1.67 thf(func_def_182, type, produc690925568od_b_b: a > produc1168037607od_b_b > produc1614181008od_b_b).
% 3.27/1.67 thf(func_def_183, type, produc2135284224od_b_b: a > product_prod_b_b > produc1133639312od_b_b).
% 3.27/1.67 thf(func_def_184, type, produc1311571856_a_nat: b > labeled_graph_a_nat > produc126142486_a_nat).
% 3.27/1.67 thf(func_def_185, type, produc255168265_a_nat: b > produc598811825_a_nat > produc207618585_a_nat).
% 3.27/1.67 thf(func_def_186, type, produc1257047359od_b_b: b > product_prod_b_b > produc255402447od_b_b).
% 3.27/1.67 thf(func_def_187, type, product_Pair_b_b: b > b > product_prod_b_b).
% 3.27/1.67 thf(func_def_188, type, id_on_1401723275_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat).
% 3.27/1.67 thf(func_def_189, type, id_on_1649591758_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat).
% 3.27/1.67 thf(func_def_190, type, id_on_2019020932od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b).
% 3.27/1.67 thf(func_def_191, type, id_on_set_b: set_set_b > set_Pr1463111367_set_b).
% 3.27/1.67 thf(func_def_192, type, id_on_b: set_b > set_Product_prod_b_b).
% 3.27/1.67 thf(func_def_193, type, refl_o1018941609_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_194, type, refl_o206400048_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o).
% 3.27/1.67 thf(func_def_195, type, refl_o2093460070_a_nat: set_Pr2074180935_a_nat > set_Pr1370504391_a_nat > $o).
% 3.27/1.67 thf(func_def_196, type, refl_o1941611494od_b_b: set_Pr2007082183od_b_b > set_Pr454079687od_b_b > $o).
% 3.27/1.67 thf(func_def_197, type, refl_o686103183od_b_b: set_Pr1900477680od_b_b > set_Pr963356295od_b_b > $o).
% 3.27/1.67 thf(func_def_198, type, refl_o2134473190od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o).
% 3.27/1.67 thf(func_def_199, type, refl_o730481808_a_nat: set_se480333169_a_nat > set_Pr1045675335_a_nat > $o).
% 3.27/1.67 thf(func_def_200, type, refl_o1899808198od_b_b: set_se668355239od_b_b > set_Pr773647559od_b_b > $o).
% 3.27/1.67 thf(func_def_201, type, refl_on_set_b: set_set_b > set_Pr1463111367_set_b > $o).
% 3.27/1.67 thf(func_def_202, type, refl_on_b: set_b > set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_203, type, sym_b: set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_204, type, total_6094180_a_nat: set_la1108361502_a_nat > set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_205, type, total_2145009205_a_nat: set_Pr1912306449_a_nat > set_Pr2074180935_a_nat > $o).
% 3.27/1.67 thf(func_def_206, type, total_178717291od_b_b: set_Product_prod_b_b > set_Pr2007082183od_b_b > $o).
% 3.27/1.67 thf(func_def_207, type, total_on_b: set_b > set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_208, type, trans_1002055497_a_nat: set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_209, type, trans_444496784_a_nat: set_Pr2074180935_a_nat > $o).
% 3.27/1.67 thf(func_def_210, type, trans_1808857926od_b_b: set_Pr2007082183od_b_b > $o).
% 3.27/1.67 thf(func_def_211, type, trans_b: set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_212, type, conseq1058228089_nat_b: set_Pr1912306449_a_nat > labeled_graph_a_b > $o).
% 3.27/1.67 thf(func_def_213, type, extensible_a_nat_b: produc598811825_a_nat > labeled_graph_a_b > set_Pr2106913242_nat_b > $o).
% 3.27/1.67 thf(func_def_214, type, fin_ma603645735_nat_b: produc598811825_a_nat > labeled_graph_a_b > $o).
% 3.27/1.67 thf(func_def_215, type, mainta535344367_a_nat: produc598811825_a_nat > labele553847424_a_nat > $o).
% 3.27/1.67 thf(func_def_216, type, mainta1627462506_a_nat: produc598811825_a_nat > labele1586099759_a_nat > $o).
% 3.27/1.67 thf(func_def_217, type, mainta552012832od_b_b: produc598811825_a_nat > labele488176869od_b_b > $o).
% 3.27/1.67 thf(func_def_218, type, maintained_a_nat_b: produc598811825_a_nat > labeled_graph_a_b > $o).
% 3.27/1.67 thf(func_def_219, type, collec978177436_a_nat: (produc598811825_a_nat > $o) > set_Pr1912306449_a_nat).
% 3.27/1.67 thf(func_def_220, type, collec1231052114od_b_b: (produc1168037607od_b_b > $o) > set_Pr2007082183od_b_b).
% 3.27/1.67 thf(func_def_221, type, collec1481886546od_b_b: (product_prod_b_b > $o) > set_Product_prod_b_b).
% 3.27/1.67 thf(func_def_222, type, collec971438076_a_nat: (set_Pr1912306449_a_nat > $o) > set_se480333169_a_nat).
% 3.27/1.67 thf(func_def_223, type, collec1543202610od_b_b: (set_Product_prod_b_b > $o) > set_se668355239od_b_b).
% 3.27/1.67 thf(func_def_224, type, collect_set_b: (set_b > $o) > set_set_b).
% 3.27/1.67 thf(func_def_225, type, collect_b: (b > $o) > set_b).
% 3.27/1.67 thf(func_def_226, type, insert1337853464_a_nat: labeled_graph_a_nat > set_la1108361502_a_nat > set_la1108361502_a_nat).
% 3.27/1.67 thf(func_def_227, type, insert2058869377_a_nat: produc598811825_a_nat > set_Pr1912306449_a_nat > set_Pr1912306449_a_nat).
% 3.27/1.67 thf(func_def_228, type, insert354391607_a_nat: produc1099860071_a_nat > set_Pr2074180935_a_nat > set_Pr2074180935_a_nat).
% 3.27/1.67 thf(func_def_229, type, insert1772835511_a_nat: produc1734972647_a_nat > set_Pr1370504391_a_nat > set_Pr1370504391_a_nat).
% 3.27/1.67 thf(func_def_230, type, insert428421239od_b_b: produc197046951od_b_b > set_Pr963356295od_b_b > set_Pr963356295od_b_b).
% 3.27/1.67 thf(func_def_231, type, insert1098249399od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > set_Pr2007082183od_b_b).
% 3.27/1.67 thf(func_def_232, type, insert885128759_a_nat: produc1869574247_a_nat > set_Pr1045675335_a_nat > set_Pr1045675335_a_nat).
% 3.27/1.67 thf(func_def_233, type, insert1655137376od_b_b: produc1133639312od_b_b > set_Pr1900477680od_b_b > set_Pr1900477680od_b_b).
% 3.27/1.67 thf(func_def_234, type, insert1952693431od_b_b: product_prod_b_b > set_Product_prod_b_b > set_Product_prod_b_b).
% 3.27/1.67 thf(func_def_235, type, insert1220735201_a_nat: set_Pr1912306449_a_nat > set_se480333169_a_nat > set_se480333169_a_nat).
% 3.27/1.67 thf(func_def_236, type, insert_b: b > set_b > set_b).
% 3.27/1.67 thf(func_def_237, type, is_emp1506632323_a_nat: set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_238, type, is_sin1787449765_a_nat: set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_239, type, is_sin1713057243od_b_b: set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_240, type, the_el537460006_a_nat: set_Pr1912306449_a_nat > produc598811825_a_nat).
% 3.27/1.67 thf(func_def_241, type, standa414753934rule_a: a > a > produc598811825_a_nat).
% 3.27/1.67 thf(func_def_242, type, standa879863265rule_a: produc598811825_a_nat).
% 3.27/1.67 thf(func_def_243, type, standa1329480012rule_a: a > produc598811825_a_nat).
% 3.27/1.67 thf(func_def_244, type, standa709637829rule_a: a > produc598811825_a_nat).
% 3.27/1.67 thf(func_def_245, type, standa390703484rule_a: a > produc598811825_a_nat).
% 3.27/1.67 thf(func_def_246, type, same_fst_b_b: (b > $o) > (b > set_Product_prod_b_b) > set_Pr2007082183od_b_b).
% 3.27/1.67 thf(func_def_247, type, member154916863_a_nat: labeled_graph_a_nat > set_la1108361502_a_nat > $o).
% 3.27/1.67 thf(func_def_248, type, member327031898_a_nat: produc598811825_a_nat > set_Pr1912306449_a_nat > $o).
% 3.27/1.67 thf(func_def_249, type, member2103088853_a_nat: produc297983550_a_nat > set_Pr1870121716_a_nat > $o).
% 3.27/1.67 thf(func_def_250, type, member1885627811_nat_b: produc1248251020_nat_b > set_Pr471017666_nat_b > $o).
% 3.27/1.67 thf(func_def_251, type, member1542696848_a_nat: produc1099860071_a_nat > set_Pr2074180935_a_nat > $o).
% 3.27/1.67 thf(func_def_252, type, member121756432_a_nat: produc1734972647_a_nat > set_Pr1370504391_a_nat > $o).
% 3.27/1.67 thf(func_def_253, type, member428378182_a_nat: produc588599965_a_nat > set_Pr1223907581_a_nat > $o).
% 3.27/1.67 thf(func_def_254, type, member670041670od_b_b: produc130774173od_b_b > set_Pr2028709629od_b_b > $o).
% 3.27/1.67 thf(func_def_255, type, member178477328od_b_b: produc300546791od_b_b > set_Pr1391123143od_b_b > $o).
% 3.27/1.67 thf(func_def_256, type, member1769525082_a_nat: produc1230257585_a_nat > set_Pr2054759953_a_nat > $o).
% 3.27/1.67 thf(func_def_257, type, member1432657168od_b_b: produc1554726631od_b_b > set_Pr1229594311od_b_b > $o).
% 3.27/1.67 thf(func_def_258, type, member1829919504od_b_b: produc1921243879od_b_b > set_Pr454079687od_b_b > $o).
% 3.27/1.67 thf(func_def_259, type, member442676358od_b_b: produc401211613od_b_b > set_Pr1549208893od_b_b > $o).
% 3.27/1.67 thf(func_def_260, type, member1023210704od_b_b: produc1114535079od_b_b > set_Pr1271557767od_b_b > $o).
% 3.27/1.67 thf(func_def_261, type, member817785114_a_nat: produc776320369_a_nat > set_Pr1626978257_a_nat > $o).
% 3.27/1.67 thf(func_def_262, type, member1379268944_a_nat: produc2106791463_a_nat > set_Pr2072721159_a_nat > $o).
% 3.27/1.67 thf(func_def_263, type, member962139856od_b_b: produc1053464231od_b_b > set_Pr406163591od_b_b > $o).
% 3.27/1.67 thf(func_def_264, type, member1694913744od_b_b: produc197046951od_b_b > set_Pr963356295od_b_b > $o).
% 3.27/1.67 thf(func_def_265, type, member1652792080od_b_b: produc1168037607od_b_b > set_Pr2007082183od_b_b > $o).
% 3.27/1.67 thf(func_def_266, type, member351494440_b_b_b: produc2112523263_b_b_b > set_Pr357419743_b_b_b > $o).
% 3.27/1.67 thf(func_def_267, type, member1072844688_a_nat: produc1869574247_a_nat > set_Pr1045675335_a_nat > $o).
% 3.27/1.67 thf(func_def_268, type, member1009730320od_b_b: produc51693287od_b_b > set_Pr773647559od_b_b > $o).
% 3.27/1.67 thf(func_def_269, type, member1199885584_set_b: produc150548199_set_b > set_Pr1463111367_set_b > $o).
% 3.27/1.67 thf(func_def_270, type, member763519277_a_nat: produc126142486_a_nat > set_Pr2010644300_a_nat > $o).
% 3.27/1.67 thf(func_def_271, type, member692218050_a_nat: produc207618585_a_nat > set_Pr352593785_a_nat > $o).
% 3.27/1.67 thf(func_def_272, type, member641857272od_b_b: produc255402447od_b_b > set_Pr621705391od_b_b > $o).
% 3.27/1.67 thf(func_def_273, type, member1285940496od_b_b: product_prod_b_b > set_Product_prod_b_b > $o).
% 3.27/1.67 thf(func_def_274, type, member101535418_a_nat: set_Pr1912306449_a_nat > set_se480333169_a_nat > $o).
% 3.27/1.67 thf(func_def_275, type, member1050118384od_b_b: set_Product_prod_b_b > set_se668355239od_b_b > $o).
% 3.27/1.67 thf(func_def_276, type, member_set_b: set_b > set_set_b > $o).
% 3.27/1.67 thf(func_def_277, type, member_b: b > set_b > $o).
% 3.27/1.67 thf(func_def_278, type, g: labeled_graph_a_b).
% 3.27/1.67 thf(func_def_279, type, i: a).
% 3.27/1.67 thf(func_def_280, type, l: a).
% 3.27/1.67 thf(func_def_281, type, y: b).
% 3.27/1.67 thf(func_def_282, type, z: b).
% 3.27/1.67 thf(func_def_298, type, sK0: (product_prod_b_b > $o) > b).
% 3.27/1.67 thf(func_def_299, type, sK1: (product_prod_b_b > $o) > b).
% 3.27/1.67 thf(func_def_300, type, sK2: product_prod_b_b > b).
% 3.27/1.67 thf(func_def_301, type, sK3: product_prod_b_b > b).
% 3.27/1.67 thf(func_def_302, type, sK4: (product_prod_b_b > $o) > b).
% 3.27/1.67 thf(func_def_303, type, sK5: (product_prod_b_b > $o) > b).
% 3.27/1.67 thf(func_def_304, type, sK6: product_prod_b_b > b).
% 3.27/1.67 thf(func_def_305, type, sK7: product_prod_b_b > b).
% 3.27/1.67 thf(func_def_306, type, sK8: set_Product_prod_b_b > b).
% 3.27/1.67 thf(func_def_307, type, sK9: set_Product_prod_b_b > b).
% 3.27/1.67 thf(func_def_308, type, sK10: set_Product_prod_b_b > b).
% 3.27/1.67 thf(func_def_310, type, ph12: !>[X0: $tType]:(X0)).
% 3.27/1.67 thf(f1074,plain,(
% 3.27/1.67 $false),
% 3.27/1.67 inference(subsumption_resolution,[status(thm)],[f1070,f1056])).
% 3.27/1.67 thf(f1056,plain,(
% 3.27/1.67 ((member1285940496od_b_b @ (product_Pair_b_b @ y @ z) @ (getRel_a_b @ i @ g)) = $true)),
% 3.27/1.67 inference(cnf_transformation,[status(thm)],[f638])).
% 3.27/1.67 thf(f638,plain,(
% 3.27/1.67 ((member1285940496od_b_b @ (product_Pair_b_b @ y @ z) @ (getRel_a_b @ i @ g)) = $true)),
% 3.27/1.67 inference(fool_elimination,[status(thm)],[f637])).
% 3.27/1.67 thf(f637,plain,(
% 3.27/1.67 (member1285940496od_b_b @ (product_Pair_b_b @ y @ z) @ (getRel_a_b @ i @ g))),
% 3.27/1.67 inference(rectify,[status(thm)],[f2])).
% 3.27/1.67 thf(f2,axiom,(
% 3.27/1.67 (member1285940496od_b_b @ (product_Pair_b_b @ y @ z) @ (getRel_a_b @ i @ g))),
% 3.27/1.67 file('/export/starexec/sandbox2/tmp/tmp.2AfYOHPK5t/DTF2THF_16406.p',fact_1_aI)).
% 3.27/1.67 thf(f1070,plain,(
% 3.27/1.67 ((member1285940496od_b_b @ (product_Pair_b_b @ y @ z) @ (getRel_a_b @ i @ g)) != $true)),
% 3.27/1.67 inference(trivial_inequality_removal,[status(thm)],[f1067])).
% 3.27/1.67 thf(f1067,plain,(
% 3.27/1.67 ((member1285940496od_b_b @ (product_Pair_b_b @ y @ z) @ (getRel_a_b @ i @ g)) != $true) | ($true != $true)),
% 3.27/1.67 inference(superposition,[status(thm)],[f1024,f1050])).
% 3.27/1.67 thf(f1050,plain,(
% 3.27/1.67 ( ! [X0 : b,X1 : b] : (((member1285940496od_b_b @ (product_Pair_b_b @ X0 @ X1) @ (getRel_a_b @ i @ g)) = $true) | ((member1285940496od_b_b @ (product_Pair_b_b @ X1 @ X0) @ (getRel_a_b @ i @ g)) != $true)) )),
% 3.27/1.67 inference(cnf_transformation,[status(thm)],[f994])).
% 3.27/1.67 thf(f994,plain,(
% 3.27/1.67 ((trans_b @ (getRel_a_b @ i @ g)) = $true) & ((refl_on_b @ (labeled_vertices_a_b @ g) @ (getRel_a_b @ i @ g)) = $true) & ! [X0 : b,X1 : b] : (((member1285940496od_b_b @ (product_Pair_b_b @ X1 @ X0) @ (getRel_a_b @ i @ g)) != $true) | ((member1285940496od_b_b @ (product_Pair_b_b @ X0 @ X1) @ (getRel_a_b @ i @ g)) = $true))),
% 3.27/1.67 inference(ennf_transformation,[status(thm)],[f928])).
% 3.27/1.67 thf(f928,plain,(
% 3.27/1.67 ((refl_on_b @ (labeled_vertices_a_b @ g) @ (getRel_a_b @ i @ g)) = $true) & ! [X0 : b,X1 : b] : (((member1285940496od_b_b @ (product_Pair_b_b @ X1 @ X0) @ (getRel_a_b @ i @ g)) = $true) => ((member1285940496od_b_b @ (product_Pair_b_b @ X0 @ X1) @ (getRel_a_b @ i @ g)) = $true)) & ((trans_b @ (getRel_a_b @ i @ g)) = $true)),
% 3.27/1.67 inference(fool_elimination,[status(thm)],[f927])).
% 3.27/1.67 thf(f927,plain,(
% 3.27/1.67 (trans_b @ (getRel_a_b @ i @ g)) & ! [X0 : b,X1 : b] : ((member1285940496od_b_b @ (product_Pair_b_b @ X1 @ X0) @ (getRel_a_b @ i @ g)) => (member1285940496od_b_b @ (product_Pair_b_b @ X0 @ X1) @ (getRel_a_b @ i @ g))) & (refl_on_b @ (labeled_vertices_a_b @ g) @ (getRel_a_b @ i @ g))),
% 3.27/1.67 inference(rectify,[status(thm)],[f1])).
% 3.27/1.67 thf(f1,axiom,(
% 3.27/1.67 (trans_b @ (getRel_a_b @ i @ g)) & ! [X1 : b,X0 : b] : ((member1285940496od_b_b @ (product_Pair_b_b @ X0 @ X1) @ (getRel_a_b @ i @ g)) => (member1285940496od_b_b @ (product_Pair_b_b @ X1 @ X0) @ (getRel_a_b @ i @ g))) & (refl_on_b @ (labeled_vertices_a_b @ g) @ (getRel_a_b @ i @ g))),
% 3.27/1.67 file('/export/starexec/sandbox2/tmp/tmp.2AfYOHPK5t/DTF2THF_16406.p',fact_0__092_060open_062refl__on_A_Ivertices_AG_J_A_IgetRel_AI_AG_J_A_092_060and_062_A_I_092_060forall_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_AgetRel_AI_AG_A_092_060longrightarrow_062_A_Iy_M_Ax_J_A_092_060in_062_AgetRel_AI_AG_J_A_092_060and_062_Atrans_A_IgetRel_AI_AG_J_092_060close_062)).
% 3.27/1.67 thf(f1024,plain,(
% 3.27/1.67 ((member1285940496od_b_b @ (product_Pair_b_b @ z @ y) @ (getRel_a_b @ i @ g)) != $true)),
% 3.27/1.67 inference(cnf_transformation,[status(thm)],[f979])).
% 3.27/1.67 thf(f979,plain,(
% 3.27/1.67 ((member1285940496od_b_b @ (product_Pair_b_b @ z @ y) @ (getRel_a_b @ i @ g)) != $true)),
% 3.27/1.67 inference(flattening,[status(thm)],[f398])).
% 3.27/1.67 thf(f398,plain,(
% 3.27/1.67 ~((member1285940496od_b_b @ (product_Pair_b_b @ z @ y) @ (getRel_a_b @ i @ g)) = $true)),
% 3.27/1.67 inference(fool_elimination,[status(thm)],[f397])).
% 3.27/1.67 thf(f397,plain,(
% 3.27/1.67 ~(member1285940496od_b_b @ (product_Pair_b_b @ z @ y) @ (getRel_a_b @ i @ g))),
% 3.27/1.67 inference(rectify,[status(thm)],[f357])).
% 3.27/1.67 thf(f357,negated_conjecture,(
% 3.27/1.67 ~(member1285940496od_b_b @ (product_Pair_b_b @ z @ y) @ (getRel_a_b @ i @ g))),
% 3.27/1.67 inference(negated_conjecture,[status(cth)],[f356])).
% 3.27/1.67 thf(f356,conjecture,(
% 3.27/1.67 (member1285940496od_b_b @ (product_Pair_b_b @ z @ y) @ (getRel_a_b @ i @ g))),
% 3.27/1.67 file('/export/starexec/sandbox2/tmp/tmp.2AfYOHPK5t/DTF2THF_16406.p',conj_0)).
% 3.27/1.67 % SZS output end Proof for DTF2THF_16406
% 3.27/1.67 % (16572)------------------------------
% 3.27/1.67 % (16572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.27/1.67 % (16572)Termination reason: Refutation
% 3.27/1.67
% 3.27/1.67 % (16572)Memory used [KB]: 6652
% 3.27/1.67 % (16572)Time elapsed: 0.022 s
% 3.27/1.67 % (16572)Instructions burned: 60 (million)
% 3.27/1.67 % (16572)------------------------------
% 3.27/1.67 % (16572)------------------------------
% 3.27/1.67 % (16565)Success in time 0.044 s
%------------------------------------------------------------------------------