%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP152^1 : 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 : n028.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:58 AM UTC 2026
% Result : Theorem 3.09s 1.68s
% Output : Refutation 3.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 4
% Syntax : Number of formulae : 29 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 203 ( 62 equ; 0 cnn)
% Maximal formula atoms : 8 ( 7 avg)
% Number of connectives : 344 ( 42 ~; 27 |; 15 &; 252 @)
% ( 4 <=>; 4 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Number of types : 23 ( 22 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 165 ( 162 usr; 10 con; 0-3 aty)
% Number of variables : 18 ( 0 ^; 18 !; 0 ?; 18 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
multis599418605od_a_a: $tType ).
thf(type_def_7,type,
multiset_a: $tType ).
thf(type_def_8,type,
product_prod_a_a: $tType ).
thf(type_def_9,type,
a: $tType ).
thf(type_def_10,type,
multis1735859763od_a_a: $tType ).
thf(type_def_11,type,
multiset_multiset_a: $tType ).
thf(type_def_12,type,
set_Pr1948701895od_a_a: $tType ).
thf(type_def_13,type,
set_Pr1821581383od_a_a: $tType ).
thf(type_def_14,type,
set_Product_prod_a_a: $tType ).
thf(type_def_15,type,
set_Pr158363655iset_a: $tType ).
thf(type_def_16,type,
set_mu1281098573od_a_a: $tType ).
thf(type_def_17,type,
set_multiset_a: $tType ).
thf(type_def_18,type,
set_a: $tType ).
thf(type_def_19,type,
set_Pr1930443975od_a_a: $tType ).
thf(type_def_20,type,
set_Pr146455751iset_a: $tType ).
thf(type_def_21,type,
set_Pr1295299783od_a_a: $tType ).
thf(type_def_22,type,
produc2061001575od_a_a: $tType ).
thf(type_def_23,type,
produc1127127335iset_a: $tType ).
thf(type_def_24,type,
produc1572603623od_a_a: $tType ).
thf(type_def_25,type,
produc562496743od_a_a: $tType ).
thf(type_def_26,type,
produc558600423iset_a: $tType ).
thf(type_def_27,type,
produc1990712039od_a_a: $tType ).
thf(func_def_0,type,
set_Pr1930443975od_a_a: $tType ).
thf(func_def_1,type,
produc562496743od_a_a: $tType ).
thf(func_def_2,type,
set_Pr1295299783od_a_a: $tType ).
thf(func_def_3,type,
produc1990712039od_a_a: $tType ).
thf(func_def_4,type,
set_Pr146455751iset_a: $tType ).
thf(func_def_5,type,
produc558600423iset_a: $tType ).
thf(func_def_6,type,
set_Pr1821581383od_a_a: $tType ).
thf(func_def_7,type,
produc2061001575od_a_a: $tType ).
thf(func_def_8,type,
set_Pr1948701895od_a_a: $tType ).
thf(func_def_9,type,
produc1572603623od_a_a: $tType ).
thf(func_def_10,type,
set_Pr158363655iset_a: $tType ).
thf(func_def_11,type,
produc1127127335iset_a: $tType ).
thf(func_def_12,type,
multis1735859763od_a_a: $tType ).
thf(func_def_13,type,
set_mu1281098573od_a_a: $tType ).
thf(func_def_14,type,
multis599418605od_a_a: $tType ).
thf(func_def_15,type,
set_Product_prod_a_a: $tType ).
thf(func_def_16,type,
multiset_multiset_a: $tType ).
thf(func_def_17,type,
set_multiset_a: $tType ).
thf(func_def_18,type,
product_prod_a_a: $tType ).
thf(func_def_19,type,
multiset_a: $tType ).
thf(func_def_20,type,
set_a: $tType ).
thf(func_def_21,type,
a: $tType ).
thf(func_def_22,type,
minus_1257885652od_a_a: multis599418605od_a_a > multis599418605od_a_a > multis599418605od_a_a ).
thf(func_def_23,type,
minus_1649712171iset_a: multiset_a > multiset_a > multiset_a ).
thf(func_def_24,type,
plus_p404985124od_a_a: multis599418605od_a_a > multis599418605od_a_a > multis599418605od_a_a ).
thf(func_def_25,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(func_def_26,type,
zero_z160849828od_a_a: multis599418605od_a_a ).
thf(func_def_27,type,
zero_zero_multiset_a: multiset_a ).
thf(func_def_28,type,
add_ms1725977093od_a_a: product_prod_a_a > multis599418605od_a_a > multis599418605od_a_a ).
thf(func_def_29,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(func_def_30,type,
fold_m773816523od_a_a: ( multis599418605od_a_a > multis599418605od_a_a > multis599418605od_a_a ) > multis599418605od_a_a > multis1735859763od_a_a > multis599418605od_a_a ).
thf(func_def_31,type,
fold_m382157835iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
thf(func_def_32,type,
mult1_1719650714od_a_a: set_Pr1948701895od_a_a > set_Pr1821581383od_a_a ).
thf(func_def_33,type,
mult1_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
thf(func_def_34,type,
mult_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
thf(func_def_35,type,
multp_a: ( a > a > $o ) > multiset_a > multiset_a > $o ).
thf(func_def_36,type,
set_ms1869569194od_a_a: multis1735859763od_a_a > set_mu1281098573od_a_a ).
thf(func_def_37,type,
set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
thf(func_def_38,type,
set_ms1818337124od_a_a: multis599418605od_a_a > set_Product_prod_a_a ).
thf(func_def_39,type,
set_mset_a: multiset_a > set_a ).
thf(func_def_40,type,
order_1040740996od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o ).
thf(func_def_41,type,
order_750350043iset_a: set_multiset_a > set_Pr158363655iset_a > $o ).
thf(func_def_42,type,
order_1210379134od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o ).
thf(func_def_43,type,
order_761691966iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o ).
thf(func_def_44,type,
order_491428350od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_45,type,
order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_46,type,
order_preorder_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_47,type,
order_1505515173od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_48,type,
order_1558116988r_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_49,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_50,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_51,type,
top_to1483957574_a_a_o: product_prod_a_a > $o ).
thf(func_def_52,type,
top_top_a_o: a > $o ).
thf(func_def_53,type,
top_to398455383od_a_a: set_Product_prod_a_a ).
thf(func_def_54,type,
top_top_set_a: set_a ).
thf(func_def_55,type,
prefer1191731497od_a_a: multis599418605od_a_a > set_mu1281098573od_a_a > set_Pr1821581383od_a_a > set_mu1281098573od_a_a ).
thf(func_def_56,type,
prefer2130004608iset_a: multiset_a > set_multiset_a > set_Pr158363655iset_a > set_multiset_a ).
thf(func_def_57,type,
prefer737894307od_a_a: produc2061001575od_a_a > set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > set_Pr1821581383od_a_a ).
thf(func_def_58,type,
prefer619498851iset_a: produc1127127335iset_a > set_Pr158363655iset_a > set_Pr146455751iset_a > set_Pr158363655iset_a ).
thf(func_def_59,type,
prefer55414563od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a ).
thf(func_def_60,type,
prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_61,type,
prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_62,type,
prefer1340910575od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o ).
thf(func_def_63,type,
prefer1571048262iset_a: set_multiset_a > set_Pr158363655iset_a > $o ).
thf(func_def_64,type,
prefer273553257od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o ).
thf(func_def_65,type,
prefer1608839465iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o ).
thf(func_def_66,type,
prefer1409946857od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_67,type,
prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_68,type,
prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_69,type,
prefer1669415233od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o ).
thf(func_def_70,type,
prefer1779827352iset_a: set_multiset_a > set_Pr158363655iset_a > $o ).
thf(func_def_71,type,
prefer933983163od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o ).
thf(func_def_72,type,
prefer841781627iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o ).
thf(func_def_73,type,
prefer1828051771od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_74,type,
prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_75,type,
prefer719835986ence_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_76,type,
prefer1064779160od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_77,type,
prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_78,type,
produc1859144151od_a_a: multis599418605od_a_a > multis599418605od_a_a > produc2061001575od_a_a ).
thf(func_def_79,type,
produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a ).
thf(func_def_80,type,
produc978545495od_a_a: produc2061001575od_a_a > produc2061001575od_a_a > produc562496743od_a_a ).
thf(func_def_81,type,
produc1898392407iset_a: produc1127127335iset_a > produc1127127335iset_a > produc558600423iset_a ).
thf(func_def_82,type,
produc1935643479od_a_a: produc1572603623od_a_a > produc1572603623od_a_a > produc1990712039od_a_a ).
thf(func_def_83,type,
produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
thf(func_def_84,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_85,type,
produc1548936141_a_a_o: ( produc2061001575od_a_a > $o ) > multis599418605od_a_a > multis599418605od_a_a > $o ).
thf(func_def_86,type,
produc1530159757et_a_o: ( produc1127127335iset_a > $o ) > multiset_a > multiset_a > $o ).
thf(func_def_87,type,
produc94438221_a_a_o: ( produc1572603623od_a_a > $o ) > product_prod_a_a > product_prod_a_a > $o ).
thf(func_def_88,type,
product_curry_a_a_o: ( product_prod_a_a > $o ) > a > a > $o ).
thf(func_def_89,type,
produc143372156od_a_a: ( product_prod_a_a > set_Product_prod_a_a ) > a > a > set_Product_prod_a_a ).
thf(func_def_90,type,
produc580150227_set_a: ( product_prod_a_a > set_a ) > a > a > set_a ).
thf(func_def_91,type,
produc748978966_a_a_o: ( a > a > $o ) > product_prod_a_a > $o ).
thf(func_def_92,type,
produc1653225349od_a_a: ( a > a > set_Product_prod_a_a ) > product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_93,type,
produc694541532_set_a: ( a > a > set_a ) > product_prod_a_a > set_a ).
thf(func_def_94,type,
produc700022774_a_a_a: ( a > a ) > ( a > a ) > product_prod_a_a > product_prod_a_a ).
thf(func_def_95,type,
produc429414904iset_a: ( multiset_a > multiset_a > set_multiset_a ) > produc1127127335iset_a > set_multiset_a ).
thf(func_def_96,type,
produc2041367259od_a_a: ( multiset_a > multiset_a > set_Product_prod_a_a ) > produc1127127335iset_a > set_Product_prod_a_a ).
thf(func_def_97,type,
produc747255346_set_a: ( multiset_a > multiset_a > set_a ) > produc1127127335iset_a > set_a ).
thf(func_def_98,type,
produc734625720iset_a: ( product_prod_a_a > product_prod_a_a > set_multiset_a ) > produc1572603623od_a_a > set_multiset_a ).
thf(func_def_99,type,
produc869579762_set_a: ( product_prod_a_a > product_prod_a_a > set_a ) > produc1572603623od_a_a > set_a ).
thf(func_def_100,type,
produc1833107820_a_a_o: ( a > a > $o ) > product_prod_a_a > $o ).
thf(func_def_101,type,
produc1121147169od_a_a: ( a > a > set_mu1281098573od_a_a ) > product_prod_a_a > set_mu1281098573od_a_a ).
thf(func_def_102,type,
produc1828150264iset_a: ( a > a > set_multiset_a ) > product_prod_a_a > set_multiset_a ).
thf(func_def_103,type,
produc93594587iset_a: ( a > a > set_Pr158363655iset_a ) > product_prod_a_a > set_Pr158363655iset_a ).
thf(func_def_104,type,
produc1093061851od_a_a: ( a > a > set_Product_prod_a_a ) > product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_105,type,
produc961951794_set_a: ( a > a > set_a ) > product_prod_a_a > set_a ).
thf(func_def_106,type,
product_fst_a_a: product_prod_a_a > a ).
thf(func_def_107,type,
product_snd_a_a: product_prod_a_a > a ).
thf(func_def_108,type,
produc1913956215od_a_a: produc2061001575od_a_a > produc2061001575od_a_a ).
thf(func_def_109,type,
produc506430135iset_a: produc1127127335iset_a > produc1127127335iset_a ).
thf(func_def_110,type,
produc1641853175od_a_a: produc1572603623od_a_a > produc1572603623od_a_a ).
thf(func_def_111,type,
product_swap_a_a: product_prod_a_a > product_prod_a_a ).
thf(func_def_112,type,
asym_a: set_Product_prod_a_a > $o ).
thf(func_def_113,type,
inv_im1681759523_a_a_a: set_Pr1821581383od_a_a > ( a > multis599418605od_a_a ) > set_Product_prod_a_a ).
thf(func_def_114,type,
inv_im343774418iset_a: set_Pr158363655iset_a > ( multiset_a > multiset_a ) > set_Pr158363655iset_a ).
thf(func_def_115,type,
inv_im1716587189od_a_a: set_Pr158363655iset_a > ( product_prod_a_a > multiset_a ) > set_Pr1948701895od_a_a ).
thf(func_def_116,type,
inv_im975322764et_a_a: set_Pr158363655iset_a > ( a > multiset_a ) > set_Product_prod_a_a ).
thf(func_def_117,type,
inv_im184450735iset_a: set_Pr1948701895od_a_a > ( multiset_a > product_prod_a_a ) > set_Pr158363655iset_a ).
thf(func_def_118,type,
inv_im777352425_a_a_a: set_Pr1948701895od_a_a > ( a > product_prod_a_a ) > set_Product_prod_a_a ).
thf(func_def_119,type,
inv_im941243329od_a_a: set_Product_prod_a_a > ( multis599418605od_a_a > a ) > set_Pr1821581383od_a_a ).
thf(func_def_120,type,
inv_im479034648iset_a: set_Product_prod_a_a > ( multiset_a > a ) > set_Pr158363655iset_a ).
thf(func_def_121,type,
inv_im15016955od_a_a: set_Product_prod_a_a > ( product_prod_a_a > a ) > set_Pr1948701895od_a_a ).
thf(func_def_122,type,
inv_image_a_a: set_Product_prod_a_a > ( a > a ) > set_Product_prod_a_a ).
thf(func_def_123,type,
irrefl1672195874od_a_a: set_Pr1821581383od_a_a > $o ).
thf(func_def_124,type,
irrefl_multiset_a: set_Pr158363655iset_a > $o ).
thf(func_def_125,type,
irrefl76447196od_a_a: set_Pr1948701895od_a_a > $o ).
thf(func_def_126,type,
irrefl_a: set_Product_prod_a_a > $o ).
thf(func_def_127,type,
refl_o581572716od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o ).
thf(func_def_128,type,
refl_on_multiset_a: set_multiset_a > set_Pr158363655iset_a > $o ).
thf(func_def_129,type,
refl_o102522214od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o ).
thf(func_def_130,type,
refl_o1522155814iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o ).
thf(func_def_131,type,
refl_o198693862od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_132,type,
refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_133,type,
refl_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_134,type,
total_1024184433od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o ).
thf(func_def_135,type,
total_on_multiset_a: set_multiset_a > set_Pr158363655iset_a > $o ).
thf(func_def_136,type,
total_1170533099od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o ).
thf(func_def_137,type,
total_313747627iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o ).
thf(func_def_138,type,
total_1340260971od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_139,type,
total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_140,type,
total_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_141,type,
trans_1520871884od_a_a: set_Pr1821581383od_a_a > $o ).
thf(func_def_142,type,
trans_multiset_a: set_Pr158363655iset_a > $o ).
thf(func_def_143,type,
trans_972827014od_a_a: set_Pr1948701895od_a_a > $o ).
thf(func_def_144,type,
trans_a: set_Product_prod_a_a > $o ).
thf(func_def_145,type,
collec1948052440od_a_a: ( multis599418605od_a_a > $o ) > set_mu1281098573od_a_a ).
thf(func_def_146,type,
collect_multiset_a: ( multiset_a > $o ) > set_multiset_a ).
thf(func_def_147,type,
collec8264402od_a_a: ( produc2061001575od_a_a > $o ) > set_Pr1821581383od_a_a ).
thf(func_def_148,type,
collec1104713362iset_a: ( produc1127127335iset_a > $o ) > set_Pr158363655iset_a ).
thf(func_def_149,type,
collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
thf(func_def_150,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_151,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_152,type,
image_794773445od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_153,type,
image_340790172_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).
thf(func_def_154,type,
image_1725938350od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).
thf(func_def_155,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(func_def_156,type,
lex_prod_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a ).
thf(func_def_157,type,
same_fst_a_a: ( a > $o ) > ( a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
thf(func_def_158,type,
member1755255446od_a_a: multis599418605od_a_a > set_mu1281098573od_a_a > $o ).
thf(func_def_159,type,
member_multiset_a: multiset_a > set_multiset_a > $o ).
thf(func_def_160,type,
member1349055120od_a_a: produc2061001575od_a_a > set_Pr1821581383od_a_a > $o ).
thf(func_def_161,type,
member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o ).
thf(func_def_162,type,
member2007374608od_a_a: produc562496743od_a_a > set_Pr1930443975od_a_a > $o ).
thf(func_def_163,type,
member567864080iset_a: produc558600423iset_a > set_Pr146455751iset_a > $o ).
thf(func_def_164,type,
member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
thf(func_def_165,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_166,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_167,type,
member_a: a > set_a > $o ).
thf(func_def_168,type,
carrier: set_a ).
thf(func_def_169,type,
relation: set_Product_prod_a_a ).
thf(func_def_170,type,
x: a ).
thf(func_def_171,type,
y: a ).
thf(func_def_187,type,
sK0: ( product_prod_a_a > $o ) > a ).
thf(func_def_188,type,
sK1: ( product_prod_a_a > $o ) > a ).
thf(func_def_189,type,
sK2: product_prod_a_a > a ).
thf(func_def_190,type,
sK3: product_prod_a_a > a ).
thf(func_def_191,type,
sK4: ( product_prod_a_a > $o ) > a ).
thf(func_def_192,type,
sK5: ( product_prod_a_a > $o ) > a ).
thf(func_def_193,type,
sK6: product_prod_a_a > a ).
thf(func_def_194,type,
sK7: product_prod_a_a > a ).
thf(func_def_195,type,
sK8: set_Product_prod_a_a > a ).
thf(func_def_196,type,
sK9: set_Product_prod_a_a > a ).
thf(func_def_197,type,
sK10: set_Product_prod_a_a > a ).
thf(func_def_199,type,
ph12:
!>[X0: $tType] : X0 ).
thf(f1078,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1077,f1020]) ).
thf(f1020,plain,
( ( member_a @ y @ carrier )
= $true ),
inference(cnf_transformation,[status(thm)],[f367]) ).
thf(f367,plain,
( ( member_a @ y @ carrier )
= $true ),
inference(fool_elimination,[status(thm)],[f366]) ).
thf(f366,plain,
member_a @ y @ carrier,
inference(rectify,[status(thm)],[f357]) ).
thf(f357,axiom,
member_a @ y @ carrier,
file('/export/starexec/sandbox2/tmp/tmp.fZ0xgqz36i/DTF2THF_28173.p',conj_1) ).
thf(f1077,plain,
( ( member_a @ y @ carrier )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f1076,f1028]) ).
thf(f1028,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation )
!= $true ),
inference(cnf_transformation,[status(thm)],[f972]) ).
thf(f972,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation )
!= $true )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f891]) ).
thf(f891,plain,
~ ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation )
= $true ) ),
inference(fool_elimination,[status(thm)],[f890]) ).
thf(f890,plain,
~ ( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation ) ),
inference(rectify,[status(thm)],[f360]) ).
thf(f360,negated_conjecture,
~ ( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation ) ),
inference(negated_conjecture,[status(cth)],[f359]) ).
thf(f359,conjecture,
( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation )
| ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation ) ),
file('/export/starexec/sandbox2/tmp/tmp.fZ0xgqz36i/DTF2THF_28173.p',conj_3) ).
thf(f1076,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation )
= $true )
| ( ( member_a @ y @ carrier )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f1075,f1039]) ).
thf(f1039,plain,
( ( member_a @ x @ carrier )
= $true ),
inference(cnf_transformation,[status(thm)],[f409]) ).
thf(f409,plain,
( ( member_a @ x @ carrier )
= $true ),
inference(fool_elimination,[status(thm)],[f408]) ).
thf(f408,plain,
member_a @ x @ carrier,
inference(rectify,[status(thm)],[f356]) ).
thf(f356,axiom,
member_a @ x @ carrier,
file('/export/starexec/sandbox2/tmp/tmp.fZ0xgqz36i/DTF2THF_28173.p',conj_0) ).
thf(f1075,plain,
( ( ( member_a @ x @ carrier )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation )
= $true )
| ( ( member_a @ y @ carrier )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1061]) ).
thf(f1061,plain,
( ( ( member_a @ x @ carrier )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation )
= $true )
| ( ( member_a @ y @ carrier )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1027,f1018]) ).
thf(f1018,plain,
! [X0: a,X1: a] :
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
= $true )
| ( ( member_a @ X1 @ carrier )
!= $true )
| ( ( member_a @ X0 @ carrier )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f993]) ).
thf(f993,plain,
! [X0: a,X1: a] :
( ( ( member_a @ X1 @ carrier )
!= $true )
| ( ( member_a @ X0 @ carrier )
!= $true )
| ( ( ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
= $true ) )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
= $true ) )
& ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
!= $true ) ) ) ),
inference(flattening,[status(thm)],[f992]) ).
thf(f992,plain,
! [X0: a,X1: a] :
( ( ( member_a @ X1 @ carrier )
!= $true )
| ( ( member_a @ X0 @ carrier )
!= $true )
| ( ( ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
= $true ) )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
= $true ) )
& ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
= $true )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f962]) ).
thf(f962,plain,
! [X0: a,X1: a] :
( ( ( member_a @ X1 @ carrier )
!= $true )
| ( ( member_a @ X0 @ carrier )
!= $true )
| ( ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
= $true ) )
<=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true ) ) ),
inference(flattening,[status(thm)],[f961]) ).
thf(f961,plain,
! [X0: a,X1: a] :
( ( ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
= $true ) )
<=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true ) )
| ( ( member_a @ X1 @ carrier )
!= $true )
| ( ( member_a @ X0 @ carrier )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f954]) ).
thf(f954,plain,
! [X0: a,X1: a] :
( ( ( ( member_a @ X1 @ carrier )
= $true )
& ( ( member_a @ X0 @ carrier )
= $true ) )
=> ( ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
= $true ) )
<=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true ) ) ),
inference(flattening,[status(thm)],[f443]) ).
thf(f443,plain,
! [X0: a,X1: a] :
( ( ( ( member_a @ X1 @ carrier )
= $true )
& ( ( member_a @ X0 @ carrier )
= $true ) )
=> ( ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
= $true ) )
<=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
!= $true ) ) ),
inference(fool_elimination,[status(thm)],[f442]) ).
thf(f442,plain,
! [X0: a,X1: a] :
( ( ( member_a @ X1 @ carrier )
& ( member_a @ X0 @ carrier ) )
=> ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
= ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
! [X1: a,X0: a] :
( ( ( member_a @ X0 @ carrier )
& ( member_a @ X1 @ carrier ) )
=> ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation )
& ( ( member449909584od_a_a @ ( product_Pair_a_a @ X0 @ X1 ) @ relation )
= ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X1 @ X0 ) @ relation ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.fZ0xgqz36i/DTF2THF_28173.p',fact_5_strict__not__refl__weak) ).
thf(f1027,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
!= $true ),
inference(cnf_transformation,[status(thm)],[f972]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP152^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.32 % Computer : n028.cluster.edu
% 0.15/0.32 % Model : x86_64 x86_64
% 0.15/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.32 % Memory : 8042.1875MB
% 0.15/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.32 % CPULimit : 300
% 0.15/0.32 % WCLimit : 300
% 0.15/0.32 % DateTime : Tue Feb 24 08:28:12 EST 2026
% 0.15/0.32 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.44 ---- Original DTF file ---
% 0.18/0.44 thf(spec,logic,$$dhol).
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % File : ITP152^1 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.44 % Domain : Interactive Theorem Proving
% 0.18/0.44 % Problem : Sledgehammer Preferences problem prob_107__6248646_1
% 0.18/0.44 % Version : Especial.
% 0.18/0.44 % English :
% 0.18/0.44
% 0.18/0.44 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.44 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.44 % Source : [Des21]
% 0.18/0.44 % Names : Preferences/prob_107__6248646_1 [Des21]
% 0.18/0.44
% 0.18/0.44 % Status : Theorem
% 0.18/0.44 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.18/0.44 % Syntax : Number of formulae : 531 ( 129 unt; 172 typ; 0 def)
% 0.18/0.44 % Number of atoms : 1101 ( 269 equ; 0 cnn)
% 0.18/0.44 % Maximal formula atoms : 8 ( 3 avg)
% 0.18/0.44 % Number of connectives : 3857 ( 125 ~; 24 |; 99 &;3120 @)
% 0.18/0.44 % ( 0 <=>; 489 =>; 0 <=; 0 <~>)
% 0.18/0.44 % Maximal formula depth : 20 ( 8 avg)
% 0.18/0.44 % Number of types : 23 ( 22 usr)
% 0.18/0.44 % Number of type conns : 445 ( 445 >; 0 *; 0 +; 0 <<)
% 0.18/0.44 % Number of symbols : 151 ( 150 usr; 8 con; 0-3 aty)
% 0.18/0.44 % Number of variables : 1192 ( 57 ^;1111 !; 24 ?;1192 :)
% 0.18/0.44 % SPC : TH0_THM_EQU_NAR
% 0.18/0.44
% 0.18/0.44 % Comments : This file was generated by Sledgehammer 2021-02-23 15:43:28.915
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 % Could-be-implicit typings (22)
% 0.18/0.44 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J_J,type,
% 0.18/0.44 set_Pr1930443975od_a_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 0.18/0.44 produc562496743od_a_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 0.18/0.44 set_Pr1295299783od_a_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.18/0.44 produc1990712039od_a_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J,type,
% 0.18/0.44 set_Pr146455751iset_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.18/0.44 produc558600423iset_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 0.18/0.44 set_Pr1821581383od_a_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.18/0.44 produc2061001575od_a_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.18/0.44 set_Pr1948701895od_a_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.18/0.44 produc1572603623od_a_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.18/0.44 set_Pr158363655iset_a: $tType ).
% 0.18/0.44
% 0.18/0.44 thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
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% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.18/0.45 collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.18/0.45 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.18/0.45 collect_a: ( a > $o ) > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.18/0.45 image_794773445od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
% 0.18/0.45 image_340790172_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.18/0.45 image_1725938350od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
% 0.18/0.45 image_a_a: ( a > a ) > set_a > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Wellfounded_Olex__prod_001tf__a_001tf__a,type,
% 0.18/0.45 lex_prod_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Wfrec_Osame__fst_001tf__a_001tf__a,type,
% 0.18/0.45 same_fst_a_a: ( a > $o ) > ( a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.18/0.45 member1755255446od_a_a: multis599418605od_a_a > set_mu1281098573od_a_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
% 0.18/0.45 member_multiset_a: multiset_a > set_multiset_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.18/0.45 member1349055120od_a_a: produc2061001575od_a_a > set_Pr1821581383od_a_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 0.18/0.45 member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 0.18/0.45 member2007374608od_a_a: produc562496743od_a_a > set_Pr1930443975od_a_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 0.18/0.45 member567864080iset_a: produc558600423iset_a > set_Pr146455751iset_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.18/0.45 member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.18/0.45 member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.18/0.45 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001tf__a,type,
% 0.18/0.45 member_a: a > set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_carrier,type,
% 0.18/0.45 carrier: set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_relation,type,
% 0.18/0.45 relation: set_Product_prod_a_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_x,type,
% 0.18/0.45 x: a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_y,type,
% 0.18/0.45 y: a ).
% 0.18/0.45
% 0.18/0.45 % Relevant facts (355)
% 0.18/0.45 thf(fact_0_indiff__trans,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.18/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.18/0.45 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.18/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.18/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % indiff_trans
% 0.18/0.45 thf(fact_1_strict__trans,axiom,
% 0.18/0.45 ! [X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.18/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.18/0.45 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.18/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.18/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % strict_trans
% 0.18/0.45 thf(fact_2_rational__preference__axioms,axiom,
% 0.18/0.45 prefer719835986ence_a @ carrier @ relation ).
% 0.18/0.45
% 0.18/0.45 % rational_preference_axioms
% 0.18/0.45 thf(fact_3_compl,axiom,
% 0.18/0.45 ! [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ carrier )
% 0.18/0.45 => ! [Xa: a] :
% 0.18/0.45 ( ( member_a @ Xa @ carrier )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ relation )
% 0.18/0.45 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % compl
% 0.18/0.45 thf(fact_4_not__outside,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.18/0.45 => ( member_a @ X @ carrier ) ) ).
% 0.18/0.45
% 0.18/0.45 % not_outside
% 0.18/0.45 thf(fact_5_strict__not__refl__weak,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( ( member_a @ X @ carrier )
% 0.18/0.45 & ( member_a @ Y @ carrier ) )
% 0.18/0.45 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.18/0.45 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.18/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % strict_not_refl_weak
% 0.18/0.45 thf(fact_6_preference__axioms,axiom,
% 0.18/0.45 prefer1292084992ence_a @ carrier @ relation ).
% 0.18/0.45
% 0.18/0.45 % preference_axioms
% 0.18/0.45 thf(fact_7_trans__refl,axiom,
% 0.18/0.45 order_preorder_on_a @ carrier @ relation ).
% 0.18/0.45
% 0.18/0.45 % trans_refl
% 0.18/0.45 thf(fact_8_total,axiom,
% 0.18/0.45 total_on_a @ carrier @ relation ).
% 0.18/0.45
% 0.18/0.45 % total
% 0.18/0.45 thf(fact_9_reflexivity,axiom,
% 0.18/0.45 refl_on_a @ carrier @ relation ).
% 0.18/0.45
% 0.18/0.45 % reflexivity
% 0.18/0.45 thf(fact_10_prod_Oinject,axiom,
% 0.18/0.45 ! [X1: product_prod_a_a,X22: product_prod_a_a,Y1: product_prod_a_a,Y2: product_prod_a_a] :
% 0.18/0.45 ( ( ( produc1474507607od_a_a @ X1 @ X22 )
% 0.18/0.45 = ( produc1474507607od_a_a @ Y1 @ Y2 ) )
% 0.18/0.45 = ( ( X1 = Y1 )
% 0.18/0.45 & ( X22 = Y2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod.inject
% 0.18/0.45 thf(fact_11_prod_Oinject,axiom,
% 0.18/0.45 ! [X1: multiset_a,X22: multiset_a,Y1: multiset_a,Y2: multiset_a] :
% 0.18/0.45 ( ( ( produc2037245207iset_a @ X1 @ X22 )
% 0.18/0.45 = ( produc2037245207iset_a @ Y1 @ Y2 ) )
% 0.18/0.45 = ( ( X1 = Y1 )
% 0.18/0.45 & ( X22 = Y2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod.inject
% 0.18/0.45 thf(fact_12_prod_Oinject,axiom,
% 0.18/0.45 ! [X1: multis599418605od_a_a,X22: multis599418605od_a_a,Y1: multis599418605od_a_a,Y2: multis599418605od_a_a] :
% 0.18/0.45 ( ( ( produc1859144151od_a_a @ X1 @ X22 )
% 0.18/0.45 = ( produc1859144151od_a_a @ Y1 @ Y2 ) )
% 0.18/0.45 = ( ( X1 = Y1 )
% 0.18/0.45 & ( X22 = Y2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod.inject
% 0.18/0.45 thf(fact_13_prod_Oinject,axiom,
% 0.18/0.45 ! [X1: a,X22: a,Y1: a,Y2: a] :
% 0.18/0.45 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 0.18/0.45 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 0.18/0.45 = ( ( X1 = Y1 )
% 0.18/0.45 & ( X22 = Y2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod.inject
% 0.18/0.45 thf(fact_14_old_Oprod_Oinject,axiom,
% 0.18/0.45 ! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
% 0.18/0.45 ( ( ( produc1474507607od_a_a @ A @ B )
% 0.18/0.45 = ( produc1474507607od_a_a @ A2 @ B2 ) )
% 0.18/0.45 = ( ( A = A2 )
% 0.18/0.45 & ( B = B2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.inject
% 0.18/0.45 thf(fact_15_old_Oprod_Oinject,axiom,
% 0.18/0.45 ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
% 0.18/0.45 ( ( ( produc2037245207iset_a @ A @ B )
% 0.18/0.45 = ( produc2037245207iset_a @ A2 @ B2 ) )
% 0.18/0.45 = ( ( A = A2 )
% 0.18/0.45 & ( B = B2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.inject
% 0.18/0.45 thf(fact_16_old_Oprod_Oinject,axiom,
% 0.18/0.45 ! [A: multis599418605od_a_a,B: multis599418605od_a_a,A2: multis599418605od_a_a,B2: multis599418605od_a_a] :
% 0.18/0.45 ( ( ( produc1859144151od_a_a @ A @ B )
% 0.18/0.45 = ( produc1859144151od_a_a @ A2 @ B2 ) )
% 0.18/0.45 = ( ( A = A2 )
% 0.18/0.45 & ( B = B2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.inject
% 0.18/0.45 thf(fact_17_old_Oprod_Oinject,axiom,
% 0.18/0.45 ! [A: a,B: a,A2: a,B2: a] :
% 0.18/0.45 ( ( ( product_Pair_a_a @ A @ B )
% 0.18/0.45 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.18/0.45 = ( ( A = A2 )
% 0.18/0.45 & ( B = B2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.inject
% 0.18/0.45 thf(fact_18_transitivity,axiom,
% 0.18/0.45 trans_a @ relation ).
% 0.18/0.45
% 0.18/0.45 % transitivity
% 0.18/0.45 thf(fact_19_surj__pair,axiom,
% 0.18/0.45 ! [P: produc1572603623od_a_a] :
% 0.18/0.45 ? [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.18/0.45 ( P
% 0.18/0.45 = ( produc1474507607od_a_a @ X3 @ Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % surj_pair
% 0.18/0.45 thf(fact_20_surj__pair,axiom,
% 0.18/0.45 ! [P: produc1127127335iset_a] :
% 0.18/0.45 ? [X3: multiset_a,Y3: multiset_a] :
% 0.18/0.45 ( P
% 0.18/0.45 = ( produc2037245207iset_a @ X3 @ Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % surj_pair
% 0.18/0.45 thf(fact_21_surj__pair,axiom,
% 0.18/0.45 ! [P: produc2061001575od_a_a] :
% 0.18/0.45 ? [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
% 0.18/0.45 ( P
% 0.18/0.45 = ( produc1859144151od_a_a @ X3 @ Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % surj_pair
% 0.18/0.45 thf(fact_22_surj__pair,axiom,
% 0.18/0.45 ! [P: product_prod_a_a] :
% 0.18/0.45 ? [X3: a,Y3: a] :
% 0.18/0.45 ( P
% 0.18/0.45 = ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % surj_pair
% 0.18/0.45 thf(fact_23_prod__cases,axiom,
% 0.18/0.45 ! [P2: produc1572603623od_a_a > $o,P: produc1572603623od_a_a] :
% 0.18/0.45 ( ! [A3: product_prod_a_a,B3: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B3 ) )
% 0.18/0.45 => ( P2 @ P ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod_cases
% 0.18/0.45 thf(fact_24_prod__cases,axiom,
% 0.18/0.45 ! [P2: produc1127127335iset_a > $o,P: produc1127127335iset_a] :
% 0.18/0.45 ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
% 0.18/0.45 => ( P2 @ P ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod_cases
% 0.18/0.45 thf(fact_25_prod__cases,axiom,
% 0.18/0.45 ! [P2: produc2061001575od_a_a > $o,P: produc2061001575od_a_a] :
% 0.18/0.45 ( ! [A3: multis599418605od_a_a,B3: multis599418605od_a_a] : ( P2 @ ( produc1859144151od_a_a @ A3 @ B3 ) )
% 0.18/0.45 => ( P2 @ P ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod_cases
% 0.18/0.45 thf(fact_26_prod__cases,axiom,
% 0.18/0.45 ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
% 0.18/0.45 ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
% 0.18/0.45 => ( P2 @ P ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod_cases
% 0.18/0.45 thf(fact_27_Pair__inject,axiom,
% 0.18/0.45 ! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
% 0.18/0.45 ( ( ( produc1474507607od_a_a @ A @ B )
% 0.18/0.45 = ( produc1474507607od_a_a @ A2 @ B2 ) )
% 0.18/0.45 => ~ ( ( A = A2 )
% 0.18/0.45 => ( B != B2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Pair_inject
% 0.18/0.45 thf(fact_28_Pair__inject,axiom,
% 0.18/0.45 ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
% 0.18/0.45 ( ( ( produc2037245207iset_a @ A @ B )
% 0.18/0.45 = ( produc2037245207iset_a @ A2 @ B2 ) )
% 0.18/0.45 => ~ ( ( A = A2 )
% 0.18/0.45 => ( B != B2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Pair_inject
% 0.18/0.45 thf(fact_29_Pair__inject,axiom,
% 0.18/0.45 ! [A: multis599418605od_a_a,B: multis599418605od_a_a,A2: multis599418605od_a_a,B2: multis599418605od_a_a] :
% 0.18/0.45 ( ( ( produc1859144151od_a_a @ A @ B )
% 0.18/0.45 = ( produc1859144151od_a_a @ A2 @ B2 ) )
% 0.18/0.45 => ~ ( ( A = A2 )
% 0.18/0.45 => ( B != B2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Pair_inject
% 0.18/0.45 thf(fact_30_Pair__inject,axiom,
% 0.18/0.45 ! [A: a,B: a,A2: a,B2: a] :
% 0.18/0.45 ( ( ( product_Pair_a_a @ A @ B )
% 0.18/0.45 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.18/0.45 => ~ ( ( A = A2 )
% 0.18/0.45 => ( B != B2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Pair_inject
% 0.18/0.45 thf(fact_31_prod__cases3,axiom,
% 0.18/0.45 ! [Y: produc1572603623od_a_a] :
% 0.18/0.45 ~ ! [A3: product_prod_a_a,B3: a,C: a] :
% 0.18/0.45 ( Y
% 0.18/0.45 != ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B3 @ C ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod_cases3
% 0.18/0.45 thf(fact_32_rational__preference_Oaxioms_I1_J,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.axioms(1)
% 0.18/0.45 thf(fact_33_rational__preference_Oaxioms_I1_J,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.18/0.45 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.axioms(1)
% 0.18/0.45 thf(fact_34_preference_Otrans__refl,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( order_359341630od_a_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.trans_refl
% 0.18/0.45 thf(fact_35_preference_Otrans__refl,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.18/0.45 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.trans_refl
% 0.18/0.45 thf(fact_36_preference_Oreflexivity,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( refl_o1298442278od_a_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.reflexivity
% 0.18/0.45 thf(fact_37_preference_Oreflexivity,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.18/0.45 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( refl_on_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.reflexivity
% 0.18/0.45 thf(fact_38_preference_Otransitivity,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( trans_972827014od_a_a @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.transitivity
% 0.18/0.45 thf(fact_39_preference_Otransitivity,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.18/0.45 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( trans_a @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.transitivity
% 0.18/0.45 thf(fact_40_rational__preference_Ototal,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( total_1490170027od_a_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.total
% 0.18/0.45 thf(fact_41_rational__preference_Ototal,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.18/0.45 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( total_on_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.total
% 0.18/0.45 thf(fact_42_preference_Ointro,axiom,
% 0.18/0.45 ! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.18/0.45 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member2057358096od_a_a @ X3 @ Carrier ) )
% 0.18/0.45 => ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.18/0.45 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member2057358096od_a_a @ Y3 @ Carrier ) )
% 0.18/0.45 => ( ( order_491428350od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.intro
% 0.18/0.45 thf(fact_43_preference_Ointro,axiom,
% 0.18/0.45 ! [Relation: set_Pr146455751iset_a,Carrier: set_Pr158363655iset_a] :
% 0.18/0.45 ( ! [X3: produc1127127335iset_a,Y3: produc1127127335iset_a] :
% 0.18/0.45 ( ( member567864080iset_a @ ( produc1898392407iset_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member340150864iset_a @ X3 @ Carrier ) )
% 0.18/0.45 => ( ! [X3: produc1127127335iset_a,Y3: produc1127127335iset_a] :
% 0.18/0.45 ( ( member567864080iset_a @ ( produc1898392407iset_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member340150864iset_a @ Y3 @ Carrier ) )
% 0.18/0.45 => ( ( order_761691966iset_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1608839465iset_a @ Carrier @ Relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.intro
% 0.18/0.45 thf(fact_44_preference_Ointro,axiom,
% 0.18/0.45 ! [Relation: set_Pr1930443975od_a_a,Carrier: set_Pr1821581383od_a_a] :
% 0.18/0.45 ( ! [X3: produc2061001575od_a_a,Y3: produc2061001575od_a_a] :
% 0.18/0.45 ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member1349055120od_a_a @ X3 @ Carrier ) )
% 0.18/0.45 => ( ! [X3: produc2061001575od_a_a,Y3: produc2061001575od_a_a] :
% 0.18/0.45 ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member1349055120od_a_a @ Y3 @ Carrier ) )
% 0.18/0.45 => ( ( order_1210379134od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer273553257od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.intro
% 0.18/0.45 thf(fact_45_preference_Ointro,axiom,
% 0.18/0.45 ! [Relation: set_Pr158363655iset_a,Carrier: set_multiset_a] :
% 0.18/0.45 ( ! [X3: multiset_a,Y3: multiset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member_multiset_a @ X3 @ Carrier ) )
% 0.18/0.45 => ( ! [X3: multiset_a,Y3: multiset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member_multiset_a @ Y3 @ Carrier ) )
% 0.18/0.45 => ( ( order_750350043iset_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1571048262iset_a @ Carrier @ Relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.intro
% 0.18/0.45 thf(fact_46_preference_Ointro,axiom,
% 0.18/0.45 ! [Relation: set_Pr1821581383od_a_a,Carrier: set_mu1281098573od_a_a] :
% 0.18/0.45 ( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member1755255446od_a_a @ X3 @ Carrier ) )
% 0.18/0.45 => ( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member1755255446od_a_a @ Y3 @ Carrier ) )
% 0.18/0.45 => ( ( order_1040740996od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1340910575od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.intro
% 0.18/0.45 thf(fact_47_preference_Ointro,axiom,
% 0.18/0.45 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member449909584od_a_a @ X3 @ Carrier ) )
% 0.18/0.45 => ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 0.18/0.45 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.intro
% 0.18/0.45 thf(fact_48_preference_Ointro,axiom,
% 0.18/0.45 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 0.18/0.45 ( ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member_a @ X3 @ Carrier ) )
% 0.18/0.45 => ( ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
% 0.18/0.45 => ( member_a @ Y3 @ Carrier ) )
% 0.18/0.45 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.intro
% 0.18/0.45 thf(fact_49_preference__def,axiom,
% 0.18/0.45 ( prefer1409946857od_a_a
% 0.18/0.45 = ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
% 0.18/0.45 ( ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
% 0.18/0.45 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member2057358096od_a_a @ X4 @ Carrier2 ) )
% 0.18/0.45 & ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
% 0.18/0.45 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member2057358096od_a_a @ Y4 @ Carrier2 ) )
% 0.18/0.45 & ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference_def
% 0.18/0.45 thf(fact_50_preference__def,axiom,
% 0.18/0.45 ( prefer1608839465iset_a
% 0.18/0.45 = ( ^ [Carrier2: set_Pr158363655iset_a,Relation2: set_Pr146455751iset_a] :
% 0.18/0.45 ( ! [X4: produc1127127335iset_a,Y4: produc1127127335iset_a] :
% 0.18/0.45 ( ( member567864080iset_a @ ( produc1898392407iset_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member340150864iset_a @ X4 @ Carrier2 ) )
% 0.18/0.45 & ! [X4: produc1127127335iset_a,Y4: produc1127127335iset_a] :
% 0.18/0.45 ( ( member567864080iset_a @ ( produc1898392407iset_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member340150864iset_a @ Y4 @ Carrier2 ) )
% 0.18/0.45 & ( order_761691966iset_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference_def
% 0.18/0.45 thf(fact_51_preference__def,axiom,
% 0.18/0.45 ( prefer273553257od_a_a
% 0.18/0.45 = ( ^ [Carrier2: set_Pr1821581383od_a_a,Relation2: set_Pr1930443975od_a_a] :
% 0.18/0.45 ( ! [X4: produc2061001575od_a_a,Y4: produc2061001575od_a_a] :
% 0.18/0.45 ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member1349055120od_a_a @ X4 @ Carrier2 ) )
% 0.18/0.45 & ! [X4: produc2061001575od_a_a,Y4: produc2061001575od_a_a] :
% 0.18/0.45 ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member1349055120od_a_a @ Y4 @ Carrier2 ) )
% 0.18/0.45 & ( order_1210379134od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference_def
% 0.18/0.45 thf(fact_52_preference__def,axiom,
% 0.18/0.45 ( prefer1571048262iset_a
% 0.18/0.45 = ( ^ [Carrier2: set_multiset_a,Relation2: set_Pr158363655iset_a] :
% 0.18/0.45 ( ! [X4: multiset_a,Y4: multiset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member_multiset_a @ X4 @ Carrier2 ) )
% 0.18/0.45 & ! [X4: multiset_a,Y4: multiset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member_multiset_a @ Y4 @ Carrier2 ) )
% 0.18/0.45 & ( order_750350043iset_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference_def
% 0.18/0.45 thf(fact_53_preference__def,axiom,
% 0.18/0.45 ( prefer1340910575od_a_a
% 0.18/0.45 = ( ^ [Carrier2: set_mu1281098573od_a_a,Relation2: set_Pr1821581383od_a_a] :
% 0.18/0.45 ( ! [X4: multis599418605od_a_a,Y4: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member1755255446od_a_a @ X4 @ Carrier2 ) )
% 0.18/0.45 & ! [X4: multis599418605od_a_a,Y4: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member1755255446od_a_a @ Y4 @ Carrier2 ) )
% 0.18/0.45 & ( order_1040740996od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference_def
% 0.18/0.45 thf(fact_54_preference__def,axiom,
% 0.18/0.45 ( prefer1338892073od_a_a
% 0.18/0.45 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member449909584od_a_a @ X4 @ Carrier2 ) )
% 0.18/0.45 & ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member449909584od_a_a @ Y4 @ Carrier2 ) )
% 0.18/0.45 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference_def
% 0.18/0.45 thf(fact_55_preference__def,axiom,
% 0.18/0.45 ( prefer1292084992ence_a
% 0.18/0.45 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [X4: a,Y4: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member_a @ X4 @ Carrier2 ) )
% 0.18/0.45 & ! [X4: a,Y4: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
% 0.18/0.45 => ( member_a @ Y4 @ Carrier2 ) )
% 0.18/0.45 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference_def
% 0.18/0.45 thf(fact_56_preference_Oindiff__trans,axiom,
% 0.18/0.45 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 0.18/0.45 ( ( prefer1571048262iset_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 => ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ Relation )
% 0.18/0.45 & ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ Y ) @ Relation ) )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ Relation )
% 0.18/0.45 & ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.indiff_trans
% 0.18/0.45 thf(fact_57_preference_Oindiff__trans,axiom,
% 0.18/0.45 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
% 0.18/0.45 ( ( prefer1340910575od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 => ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ Relation )
% 0.18/0.45 & ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ Y ) @ Relation ) )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ Relation )
% 0.18/0.45 & ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.indiff_trans
% 0.18/0.45 thf(fact_58_preference_Oindiff__trans,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.18/0.45 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.18/0.45 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 0.18/0.45 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.indiff_trans
% 0.18/0.45 thf(fact_59_preference_Oindiff__trans,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.18/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 0.18/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.indiff_trans
% 0.18/0.45 thf(fact_60_preference_Onot__outside,axiom,
% 0.18/0.45 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.18/0.45 ( ( prefer1409946857od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 => ( member2057358096od_a_a @ X @ Carrier ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.not_outside
% 0.18/0.45 thf(fact_61_preference_Onot__outside,axiom,
% 0.18/0.45 ! [Carrier: set_Pr158363655iset_a,Relation: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
% 0.18/0.45 ( ( prefer1608839465iset_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ Relation )
% 0.18/0.45 => ( member340150864iset_a @ X @ Carrier ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.not_outside
% 0.18/0.45 thf(fact_62_preference_Onot__outside,axiom,
% 0.18/0.45 ! [Carrier: set_Pr1821581383od_a_a,Relation: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
% 0.18/0.45 ( ( prefer273553257od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 => ( member1349055120od_a_a @ X @ Carrier ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.not_outside
% 0.18/0.45 thf(fact_63_preference_Onot__outside,axiom,
% 0.18/0.45 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
% 0.18/0.45 ( ( prefer1571048262iset_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
% 0.18/0.45 => ( member_multiset_a @ X @ Carrier ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.not_outside
% 0.18/0.45 thf(fact_64_preference_Onot__outside,axiom,
% 0.18/0.45 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 0.18/0.45 ( ( prefer1340910575od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 => ( member1755255446od_a_a @ X @ Carrier ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.not_outside
% 0.18/0.45 thf(fact_65_preference_Onot__outside,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.18/0.45 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 => ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.not_outside
% 0.18/0.45 thf(fact_66_preference_Onot__outside,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.18/0.45 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 => ( member_a @ X @ Carrier ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preference.not_outside
% 0.18/0.45 thf(fact_67_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.18/0.45 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.18/0.45 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member2057358096od_a_a @ X @ Carrier )
% 0.18/0.45 & ( member2057358096od_a_a @ Y @ Carrier ) )
% 0.18/0.45 => ( ( ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_not_refl_weak
% 0.18/0.45 thf(fact_68_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.18/0.45 ! [Carrier: set_Pr158363655iset_a,Relation: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
% 0.18/0.45 ( ( prefer841781627iset_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member340150864iset_a @ X @ Carrier )
% 0.18/0.45 & ( member340150864iset_a @ Y @ Carrier ) )
% 0.18/0.45 => ( ( ~ ( member567864080iset_a @ ( produc1898392407iset_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 = ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member567864080iset_a @ ( produc1898392407iset_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_not_refl_weak
% 0.18/0.45 thf(fact_69_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.18/0.45 ! [Carrier: set_Pr1821581383od_a_a,Relation: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
% 0.18/0.45 ( ( prefer933983163od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member1349055120od_a_a @ X @ Carrier )
% 0.18/0.45 & ( member1349055120od_a_a @ Y @ Carrier ) )
% 0.18/0.45 => ( ( ~ ( member2007374608od_a_a @ ( produc978545495od_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 = ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member2007374608od_a_a @ ( produc978545495od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_not_refl_weak
% 0.18/0.45 thf(fact_70_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.18/0.45 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
% 0.18/0.45 ( ( prefer1779827352iset_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member_multiset_a @ X @ Carrier )
% 0.18/0.45 & ( member_multiset_a @ Y @ Carrier ) )
% 0.18/0.45 => ( ( ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 = ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_not_refl_weak
% 0.18/0.45 thf(fact_71_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.18/0.45 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 0.18/0.45 ( ( prefer1669415233od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member1755255446od_a_a @ X @ Carrier )
% 0.18/0.45 & ( member1755255446od_a_a @ Y @ Carrier ) )
% 0.18/0.45 => ( ( ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 = ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_not_refl_weak
% 0.18/0.45 thf(fact_72_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.18/0.45 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 0.18/0.45 & ( member449909584od_a_a @ Y @ Carrier ) )
% 0.18/0.45 => ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_not_refl_weak
% 0.18/0.45 thf(fact_73_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.18/0.45 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member_a @ X @ Carrier )
% 0.18/0.45 & ( member_a @ Y @ Carrier ) )
% 0.18/0.45 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_not_refl_weak
% 0.18/0.45 thf(fact_74_rational__preference_Ostrict__trans,axiom,
% 0.18/0.45 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 0.18/0.45 ( ( prefer1779827352iset_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 => ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ Relation )
% 0.18/0.45 & ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ Y ) @ Relation ) )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ Relation )
% 0.18/0.45 & ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_trans
% 0.18/0.45 thf(fact_75_rational__preference_Ostrict__trans,axiom,
% 0.18/0.45 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
% 0.18/0.45 ( ( prefer1669415233od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 => ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ Relation )
% 0.18/0.45 & ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ Y ) @ Relation ) )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ Relation )
% 0.18/0.45 & ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_trans
% 0.18/0.45 thf(fact_76_rational__preference_Ostrict__trans,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.18/0.45 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.18/0.45 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 0.18/0.45 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_trans
% 0.18/0.45 thf(fact_77_rational__preference_Ostrict__trans,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.18/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.18/0.45 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.18/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 0.18/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.strict_trans
% 0.18/0.45 thf(fact_78_rational__preference_Ocompl,axiom,
% 0.18/0.45 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a] :
% 0.18/0.45 ( ( prefer1779827352iset_a @ Carrier @ Relation )
% 0.18/0.45 => ! [X2: multiset_a] :
% 0.18/0.45 ( ( member_multiset_a @ X2 @ Carrier )
% 0.18/0.45 => ! [Xa: multiset_a] :
% 0.18/0.45 ( ( member_multiset_a @ Xa @ Carrier )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X2 @ Xa ) @ Relation )
% 0.18/0.45 | ( member340150864iset_a @ ( produc2037245207iset_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.compl
% 0.18/0.45 thf(fact_79_rational__preference_Ocompl,axiom,
% 0.18/0.45 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a] :
% 0.18/0.45 ( ( prefer1669415233od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ! [X2: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1755255446od_a_a @ X2 @ Carrier )
% 0.18/0.45 => ! [Xa: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1755255446od_a_a @ Xa @ Carrier )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X2 @ Xa ) @ Relation )
% 0.18/0.45 | ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.compl
% 0.18/0.45 thf(fact_80_rational__preference_Ocompl,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ! [X2: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.18/0.45 => ! [Xa: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ Xa @ Carrier )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Xa ) @ Relation )
% 0.18/0.45 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.compl
% 0.18/0.45 thf(fact_81_rational__preference_Ocompl,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.18/0.45 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.18/0.45 => ! [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ Carrier )
% 0.18/0.45 => ! [Xa: a] :
% 0.18/0.45 ( ( member_a @ Xa @ Carrier )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ Relation )
% 0.18/0.45 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.compl
% 0.18/0.45 thf(fact_82_old_Oprod_Oinducts,axiom,
% 0.18/0.45 ! [P2: produc1572603623od_a_a > $o,Prod: produc1572603623od_a_a] :
% 0.18/0.45 ( ! [A3: product_prod_a_a,B3: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B3 ) )
% 0.18/0.45 => ( P2 @ Prod ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.inducts
% 0.18/0.45 thf(fact_83_old_Oprod_Oinducts,axiom,
% 0.18/0.45 ! [P2: produc1127127335iset_a > $o,Prod: produc1127127335iset_a] :
% 0.18/0.45 ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
% 0.18/0.45 => ( P2 @ Prod ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.inducts
% 0.18/0.45 thf(fact_84_old_Oprod_Oinducts,axiom,
% 0.18/0.45 ! [P2: produc2061001575od_a_a > $o,Prod: produc2061001575od_a_a] :
% 0.18/0.45 ( ! [A3: multis599418605od_a_a,B3: multis599418605od_a_a] : ( P2 @ ( produc1859144151od_a_a @ A3 @ B3 ) )
% 0.18/0.45 => ( P2 @ Prod ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.inducts
% 0.18/0.45 thf(fact_85_old_Oprod_Oinducts,axiom,
% 0.18/0.45 ! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.18/0.45 ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
% 0.18/0.45 => ( P2 @ Prod ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.inducts
% 0.18/0.45 thf(fact_86_old_Oprod_Oexhaust,axiom,
% 0.18/0.45 ! [Y: produc1572603623od_a_a] :
% 0.18/0.45 ~ ! [A3: product_prod_a_a,B3: product_prod_a_a] :
% 0.18/0.45 ( Y
% 0.18/0.45 != ( produc1474507607od_a_a @ A3 @ B3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.exhaust
% 0.18/0.45 thf(fact_87_old_Oprod_Oexhaust,axiom,
% 0.18/0.45 ! [Y: produc1127127335iset_a] :
% 0.18/0.45 ~ ! [A3: multiset_a,B3: multiset_a] :
% 0.18/0.45 ( Y
% 0.18/0.45 != ( produc2037245207iset_a @ A3 @ B3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.exhaust
% 0.18/0.45 thf(fact_88_old_Oprod_Oexhaust,axiom,
% 0.18/0.45 ! [Y: produc2061001575od_a_a] :
% 0.18/0.45 ~ ! [A3: multis599418605od_a_a,B3: multis599418605od_a_a] :
% 0.18/0.45 ( Y
% 0.18/0.45 != ( produc1859144151od_a_a @ A3 @ B3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.exhaust
% 0.18/0.45 thf(fact_89_old_Oprod_Oexhaust,axiom,
% 0.18/0.45 ! [Y: product_prod_a_a] :
% 0.18/0.45 ~ ! [A3: a,B3: a] :
% 0.18/0.45 ( Y
% 0.18/0.45 != ( product_Pair_a_a @ A3 @ B3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.exhaust
% 0.18/0.45 thf(fact_90_prod__induct3,axiom,
% 0.18/0.45 ! [P2: produc1572603623od_a_a > $o,X: produc1572603623od_a_a] :
% 0.18/0.45 ( ! [A3: product_prod_a_a,B3: a,C: a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B3 @ C ) ) )
% 0.18/0.45 => ( P2 @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod_induct3
% 0.18/0.45 thf(fact_91_preorder__on__def,axiom,
% 0.18/0.45 ( order_359341630od_a_a
% 0.18/0.45 = ( ^ [A4: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( refl_o1298442278od_a_a @ A4 @ R )
% 0.18/0.45 & ( trans_972827014od_a_a @ R ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preorder_on_def
% 0.18/0.45 thf(fact_92_preorder__on__def,axiom,
% 0.18/0.45 ( order_preorder_on_a
% 0.18/0.45 = ( ^ [A4: set_a,R: set_Product_prod_a_a] :
% 0.18/0.45 ( ( refl_on_a @ A4 @ R )
% 0.18/0.45 & ( trans_a @ R ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % preorder_on_def
% 0.18/0.45 thf(fact_93_total__on__def,axiom,
% 0.18/0.45 ( total_on_multiset_a
% 0.18/0.45 = ( ^ [A4: set_multiset_a,R: set_Pr158363655iset_a] :
% 0.18/0.45 ! [X4: multiset_a] :
% 0.18/0.45 ( ( member_multiset_a @ X4 @ A4 )
% 0.18/0.45 => ! [Y4: multiset_a] :
% 0.18/0.45 ( ( member_multiset_a @ Y4 @ A4 )
% 0.18/0.45 => ( ( X4 != Y4 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ R )
% 0.18/0.45 | ( member340150864iset_a @ ( produc2037245207iset_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_on_def
% 0.18/0.45 thf(fact_94_total__on__def,axiom,
% 0.18/0.45 ( total_1024184433od_a_a
% 0.18/0.45 = ( ^ [A4: set_mu1281098573od_a_a,R: set_Pr1821581383od_a_a] :
% 0.18/0.45 ! [X4: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1755255446od_a_a @ X4 @ A4 )
% 0.18/0.45 => ! [Y4: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1755255446od_a_a @ Y4 @ A4 )
% 0.18/0.45 => ( ( X4 != Y4 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ R )
% 0.18/0.45 | ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_on_def
% 0.18/0.45 thf(fact_95_total__on__def,axiom,
% 0.18/0.45 ( total_1490170027od_a_a
% 0.18/0.45 = ( ^ [A4: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 0.18/0.45 ! [X4: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X4 @ A4 )
% 0.18/0.45 => ! [Y4: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ Y4 @ A4 )
% 0.18/0.45 => ( ( X4 != Y4 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R )
% 0.18/0.45 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_on_def
% 0.18/0.45 thf(fact_96_total__on__def,axiom,
% 0.18/0.45 ( total_on_a
% 0.18/0.45 = ( ^ [A4: set_a,R: set_Product_prod_a_a] :
% 0.18/0.45 ! [X4: a] :
% 0.18/0.45 ( ( member_a @ X4 @ A4 )
% 0.18/0.45 => ! [Y4: a] :
% 0.18/0.45 ( ( member_a @ Y4 @ A4 )
% 0.18/0.45 => ( ( X4 != Y4 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
% 0.18/0.45 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_on_def
% 0.18/0.45 thf(fact_97_total__onI,axiom,
% 0.18/0.45 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a] :
% 0.18/0.45 ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ X3 @ A5 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ Y3 @ A5 )
% 0.18/0.45 => ( ( X3 != Y3 )
% 0.18/0.45 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 | ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.18/0.45 => ( total_1340260971od_a_a @ A5 @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_onI
% 0.18/0.45 thf(fact_98_total__onI,axiom,
% 0.18/0.45 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a] :
% 0.18/0.45 ( ! [X3: produc1127127335iset_a,Y3: produc1127127335iset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ X3 @ A5 )
% 0.18/0.45 => ( ( member340150864iset_a @ Y3 @ A5 )
% 0.18/0.45 => ( ( X3 != Y3 )
% 0.18/0.45 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 | ( member567864080iset_a @ ( produc1898392407iset_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.18/0.45 => ( total_313747627iset_a @ A5 @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_onI
% 0.18/0.45 thf(fact_99_total__onI,axiom,
% 0.18/0.45 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a] :
% 0.18/0.45 ( ! [X3: produc2061001575od_a_a,Y3: produc2061001575od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ X3 @ A5 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ Y3 @ A5 )
% 0.18/0.45 => ( ( X3 != Y3 )
% 0.18/0.45 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 | ( member2007374608od_a_a @ ( produc978545495od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.18/0.45 => ( total_1170533099od_a_a @ A5 @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_onI
% 0.18/0.45 thf(fact_100_total__onI,axiom,
% 0.18/0.45 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a] :
% 0.18/0.45 ( ! [X3: multiset_a,Y3: multiset_a] :
% 0.18/0.45 ( ( member_multiset_a @ X3 @ A5 )
% 0.18/0.45 => ( ( member_multiset_a @ Y3 @ A5 )
% 0.18/0.45 => ( ( X3 != Y3 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 | ( member340150864iset_a @ ( produc2037245207iset_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.18/0.45 => ( total_on_multiset_a @ A5 @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_onI
% 0.18/0.45 thf(fact_101_total__onI,axiom,
% 0.18/0.45 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a] :
% 0.18/0.45 ( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1755255446od_a_a @ X3 @ A5 )
% 0.18/0.45 => ( ( member1755255446od_a_a @ Y3 @ A5 )
% 0.18/0.45 => ( ( X3 != Y3 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 | ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.18/0.45 => ( total_1024184433od_a_a @ A5 @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_onI
% 0.18/0.45 thf(fact_102_total__onI,axiom,
% 0.18/0.45 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X3 @ A5 )
% 0.18/0.45 => ( ( member449909584od_a_a @ Y3 @ A5 )
% 0.18/0.45 => ( ( X3 != Y3 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.18/0.45 => ( total_1490170027od_a_a @ A5 @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_onI
% 0.18/0.45 thf(fact_103_total__onI,axiom,
% 0.18/0.45 ! [A5: set_a,R2: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( member_a @ X3 @ A5 )
% 0.18/0.45 => ( ( member_a @ Y3 @ A5 )
% 0.18/0.45 => ( ( X3 != Y3 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 0.18/0.45 => ( total_on_a @ A5 @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_onI
% 0.18/0.45 thf(fact_104_refl__on__domain,axiom,
% 0.18/0.45 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,A: produc1572603623od_a_a,B: produc1572603623od_a_a] :
% 0.18/0.45 ( ( refl_o198693862od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B ) @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ A @ A5 )
% 0.18/0.45 & ( member2057358096od_a_a @ B @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_on_domain
% 0.18/0.45 thf(fact_105_refl__on__domain,axiom,
% 0.18/0.45 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,A: produc1127127335iset_a,B: produc1127127335iset_a] :
% 0.18/0.45 ( ( refl_o1522155814iset_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ A @ B ) @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ A @ A5 )
% 0.18/0.45 & ( member340150864iset_a @ B @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_on_domain
% 0.18/0.45 thf(fact_106_refl__on__domain,axiom,
% 0.18/0.45 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,A: produc2061001575od_a_a,B: produc2061001575od_a_a] :
% 0.18/0.45 ( ( refl_o102522214od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ A @ B ) @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ A @ A5 )
% 0.18/0.45 & ( member1349055120od_a_a @ B @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_on_domain
% 0.18/0.45 thf(fact_107_refl__on__domain,axiom,
% 0.18/0.45 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a,A: multiset_a,B: multiset_a] :
% 0.18/0.45 ( ( refl_on_multiset_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ A @ B ) @ R2 )
% 0.18/0.45 => ( ( member_multiset_a @ A @ A5 )
% 0.18/0.45 & ( member_multiset_a @ B @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_on_domain
% 0.18/0.45 thf(fact_108_refl__on__domain,axiom,
% 0.18/0.45 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,A: multis599418605od_a_a,B: multis599418605od_a_a] :
% 0.18/0.45 ( ( refl_o581572716od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A @ B ) @ R2 )
% 0.18/0.45 => ( ( member1755255446od_a_a @ A @ A5 )
% 0.18/0.45 & ( member1755255446od_a_a @ B @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_on_domain
% 0.18/0.45 thf(fact_109_refl__on__domain,axiom,
% 0.18/0.45 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
% 0.18/0.45 ( ( refl_o1298442278od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ A @ A5 )
% 0.18/0.45 & ( member449909584od_a_a @ B @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_on_domain
% 0.18/0.45 thf(fact_110_refl__on__domain,axiom,
% 0.18/0.45 ! [A5: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
% 0.18/0.45 ( ( refl_on_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
% 0.18/0.45 => ( ( member_a @ A @ A5 )
% 0.18/0.45 & ( member_a @ B @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_on_domain
% 0.18/0.45 thf(fact_111_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: produc1572603623od_a_a,P2: produc1572603623od_a_a > $o] :
% 0.18/0.45 ( ( member2057358096od_a_a @ A @ ( collec1635618130od_a_a @ P2 ) )
% 0.18/0.45 = ( P2 @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_112_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: produc1127127335iset_a,P2: produc1127127335iset_a > $o] :
% 0.18/0.45 ( ( member340150864iset_a @ A @ ( collec1104713362iset_a @ P2 ) )
% 0.18/0.45 = ( P2 @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_113_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: produc2061001575od_a_a,P2: produc2061001575od_a_a > $o] :
% 0.18/0.45 ( ( member1349055120od_a_a @ A @ ( collec8264402od_a_a @ P2 ) )
% 0.18/0.45 = ( P2 @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_114_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: multis599418605od_a_a,P2: multis599418605od_a_a > $o] :
% 0.18/0.45 ( ( member1755255446od_a_a @ A @ ( collec1948052440od_a_a @ P2 ) )
% 0.18/0.45 = ( P2 @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_115_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: multiset_a,P2: multiset_a > $o] :
% 0.18/0.45 ( ( member_multiset_a @ A @ ( collect_multiset_a @ P2 ) )
% 0.18/0.45 = ( P2 @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_116_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: a,P2: a > $o] :
% 0.18/0.45 ( ( member_a @ A @ ( collect_a @ P2 ) )
% 0.18/0.45 = ( P2 @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_117_mem__Collect__eq,axiom,
% 0.18/0.45 ! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
% 0.18/0.45 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
% 0.18/0.45 = ( P2 @ A ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_Collect_eq
% 0.18/0.45 thf(fact_118_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A5: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( collec1635618130od_a_a
% 0.18/0.45 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A5 ) )
% 0.18/0.45 = A5 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_119_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A5: set_Pr158363655iset_a] :
% 0.18/0.45 ( ( collec1104713362iset_a
% 0.18/0.45 @ ^ [X4: produc1127127335iset_a] : ( member340150864iset_a @ X4 @ A5 ) )
% 0.18/0.45 = A5 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_120_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A5: set_Pr1821581383od_a_a] :
% 0.18/0.45 ( ( collec8264402od_a_a
% 0.18/0.45 @ ^ [X4: produc2061001575od_a_a] : ( member1349055120od_a_a @ X4 @ A5 ) )
% 0.18/0.45 = A5 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_121_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A5: set_mu1281098573od_a_a] :
% 0.18/0.45 ( ( collec1948052440od_a_a
% 0.18/0.45 @ ^ [X4: multis599418605od_a_a] : ( member1755255446od_a_a @ X4 @ A5 ) )
% 0.18/0.45 = A5 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_122_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A5: set_multiset_a] :
% 0.18/0.45 ( ( collect_multiset_a
% 0.18/0.45 @ ^ [X4: multiset_a] : ( member_multiset_a @ X4 @ A5 ) )
% 0.18/0.45 = A5 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_123_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A5: set_a] :
% 0.18/0.45 ( ( collect_a
% 0.18/0.45 @ ^ [X4: a] : ( member_a @ X4 @ A5 ) )
% 0.18/0.45 = A5 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_124_Collect__mem__eq,axiom,
% 0.18/0.45 ! [A5: set_Product_prod_a_a] :
% 0.18/0.45 ( ( collec645855634od_a_a
% 0.18/0.45 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A5 ) )
% 0.18/0.45 = A5 ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_mem_eq
% 0.18/0.45 thf(fact_125_Collect__cong,axiom,
% 0.18/0.45 ! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.18/0.45 ( ! [X3: product_prod_a_a] :
% 0.18/0.45 ( ( P2 @ X3 )
% 0.18/0.45 = ( Q @ X3 ) )
% 0.18/0.45 => ( ( collec645855634od_a_a @ P2 )
% 0.18/0.45 = ( collec645855634od_a_a @ Q ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_cong
% 0.18/0.45 thf(fact_126_Collect__cong,axiom,
% 0.18/0.45 ! [P2: a > $o,Q: a > $o] :
% 0.18/0.45 ( ! [X3: a] :
% 0.18/0.45 ( ( P2 @ X3 )
% 0.18/0.45 = ( Q @ X3 ) )
% 0.18/0.45 => ( ( collect_a @ P2 )
% 0.18/0.45 = ( collect_a @ Q ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Collect_cong
% 0.18/0.45 thf(fact_127_refl__onD2,axiom,
% 0.18/0.45 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.18/0.45 ( ( refl_o198693862od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member2057358096od_a_a @ Y @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD2
% 0.18/0.45 thf(fact_128_refl__onD2,axiom,
% 0.18/0.45 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
% 0.18/0.45 ( ( refl_o1522155814iset_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member340150864iset_a @ Y @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD2
% 0.18/0.45 thf(fact_129_refl__onD2,axiom,
% 0.18/0.45 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
% 0.18/0.45 ( ( refl_o102522214od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member1349055120od_a_a @ Y @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD2
% 0.18/0.45 thf(fact_130_refl__onD2,axiom,
% 0.18/0.45 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
% 0.18/0.45 ( ( refl_on_multiset_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member_multiset_a @ Y @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD2
% 0.18/0.45 thf(fact_131_refl__onD2,axiom,
% 0.18/0.45 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 0.18/0.45 ( ( refl_o581572716od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member1755255446od_a_a @ Y @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD2
% 0.18/0.45 thf(fact_132_refl__onD2,axiom,
% 0.18/0.45 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.18/0.45 ( ( refl_o1298442278od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member449909584od_a_a @ Y @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD2
% 0.18/0.45 thf(fact_133_refl__onD2,axiom,
% 0.18/0.45 ! [A5: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
% 0.18/0.45 ( ( refl_on_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member_a @ Y @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD2
% 0.18/0.45 thf(fact_134_refl__onD1,axiom,
% 0.18/0.45 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 0.18/0.45 ( ( refl_o198693862od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member2057358096od_a_a @ X @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD1
% 0.18/0.45 thf(fact_135_refl__onD1,axiom,
% 0.18/0.45 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
% 0.18/0.45 ( ( refl_o1522155814iset_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member340150864iset_a @ X @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD1
% 0.18/0.45 thf(fact_136_refl__onD1,axiom,
% 0.18/0.45 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
% 0.18/0.45 ( ( refl_o102522214od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member1349055120od_a_a @ X @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD1
% 0.18/0.45 thf(fact_137_refl__onD1,axiom,
% 0.18/0.45 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
% 0.18/0.45 ( ( refl_on_multiset_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member_multiset_a @ X @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD1
% 0.18/0.45 thf(fact_138_refl__onD1,axiom,
% 0.18/0.45 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 0.18/0.45 ( ( refl_o581572716od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member1755255446od_a_a @ X @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD1
% 0.18/0.45 thf(fact_139_refl__onD1,axiom,
% 0.18/0.45 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.18/0.45 ( ( refl_o1298442278od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member449909584od_a_a @ X @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD1
% 0.18/0.45 thf(fact_140_refl__onD1,axiom,
% 0.18/0.45 ! [A5: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
% 0.18/0.45 ( ( refl_on_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( member_a @ X @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD1
% 0.18/0.45 thf(fact_141_refl__onD,axiom,
% 0.18/0.45 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,A: produc1572603623od_a_a] :
% 0.18/0.45 ( ( refl_o198693862od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ A @ A5 )
% 0.18/0.45 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ A ) @ R2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD
% 0.18/0.45 thf(fact_142_refl__onD,axiom,
% 0.18/0.45 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,A: produc1127127335iset_a] :
% 0.18/0.45 ( ( refl_o1522155814iset_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ A @ A5 )
% 0.18/0.45 => ( member567864080iset_a @ ( produc1898392407iset_a @ A @ A ) @ R2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD
% 0.18/0.45 thf(fact_143_refl__onD,axiom,
% 0.18/0.45 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,A: produc2061001575od_a_a] :
% 0.18/0.45 ( ( refl_o102522214od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ A @ A5 )
% 0.18/0.45 => ( member2007374608od_a_a @ ( produc978545495od_a_a @ A @ A ) @ R2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD
% 0.18/0.45 thf(fact_144_refl__onD,axiom,
% 0.18/0.45 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a,A: multiset_a] :
% 0.18/0.45 ( ( refl_on_multiset_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member_multiset_a @ A @ A5 )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ A @ A ) @ R2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD
% 0.18/0.45 thf(fact_145_refl__onD,axiom,
% 0.18/0.45 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,A: multis599418605od_a_a] :
% 0.18/0.45 ( ( refl_o581572716od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member1755255446od_a_a @ A @ A5 )
% 0.18/0.45 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A @ A ) @ R2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD
% 0.18/0.45 thf(fact_146_refl__onD,axiom,
% 0.18/0.45 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A: product_prod_a_a] :
% 0.18/0.45 ( ( refl_o1298442278od_a_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ A @ A5 )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ R2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD
% 0.18/0.45 thf(fact_147_refl__onD,axiom,
% 0.18/0.45 ! [A5: set_a,R2: set_Product_prod_a_a,A: a] :
% 0.18/0.45 ( ( refl_on_a @ A5 @ R2 )
% 0.18/0.45 => ( ( member_a @ A @ A5 )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % refl_onD
% 0.18/0.45 thf(fact_148_trans__def,axiom,
% 0.18/0.45 ( trans_972827014od_a_a
% 0.18/0.45 = ( ^ [R: set_Pr1948701895od_a_a] :
% 0.18/0.45 ! [X4: product_prod_a_a,Y4: product_prod_a_a,Z2: product_prod_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ Z2 ) @ R )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % trans_def
% 0.18/0.45 thf(fact_149_trans__def,axiom,
% 0.18/0.45 ( trans_multiset_a
% 0.18/0.45 = ( ^ [R: set_Pr158363655iset_a] :
% 0.18/0.45 ! [X4: multiset_a,Y4: multiset_a,Z2: multiset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ R )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y4 @ Z2 ) @ R )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % trans_def
% 0.18/0.45 thf(fact_150_trans__def,axiom,
% 0.18/0.45 ( trans_1520871884od_a_a
% 0.18/0.45 = ( ^ [R: set_Pr1821581383od_a_a] :
% 0.18/0.45 ! [X4: multis599418605od_a_a,Y4: multis599418605od_a_a,Z2: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ R )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y4 @ Z2 ) @ R )
% 0.18/0.45 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % trans_def
% 0.18/0.45 thf(fact_151_trans__def,axiom,
% 0.18/0.45 ( trans_a
% 0.18/0.45 = ( ^ [R: set_Product_prod_a_a] :
% 0.18/0.45 ! [X4: a,Y4: a,Z2: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ Z2 ) @ R )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % trans_def
% 0.18/0.45 thf(fact_152_transI,axiom,
% 0.18/0.45 ! [R2: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ Z3 ) @ R2 )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Z3 ) @ R2 ) ) )
% 0.18/0.45 => ( trans_972827014od_a_a @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % transI
% 0.18/0.45 thf(fact_153_transI,axiom,
% 0.18/0.45 ! [R2: set_Pr158363655iset_a] :
% 0.18/0.45 ( ! [X3: multiset_a,Y3: multiset_a,Z3: multiset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y3 @ Z3 ) @ R2 )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Z3 ) @ R2 ) ) )
% 0.18/0.45 => ( trans_multiset_a @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % transI
% 0.18/0.45 thf(fact_154_transI,axiom,
% 0.18/0.45 ! [R2: set_Pr1821581383od_a_a] :
% 0.18/0.45 ( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a,Z3: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y3 @ Z3 ) @ R2 )
% 0.18/0.45 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Z3 ) @ R2 ) ) )
% 0.18/0.45 => ( trans_1520871884od_a_a @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % transI
% 0.18/0.45 thf(fact_155_transI,axiom,
% 0.18/0.45 ! [R2: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [X3: a,Y3: a,Z3: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R2 )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z3 ) @ R2 ) ) )
% 0.18/0.45 => ( trans_a @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % transI
% 0.18/0.45 thf(fact_156_transD,axiom,
% 0.18/0.45 ! [R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.18/0.45 ( ( trans_972827014od_a_a @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ R2 )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % transD
% 0.18/0.45 thf(fact_157_transD,axiom,
% 0.18/0.45 ! [R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 0.18/0.45 ( ( trans_multiset_a @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R2 )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % transD
% 0.18/0.45 thf(fact_158_transD,axiom,
% 0.18/0.45 ! [R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
% 0.18/0.45 ( ( trans_1520871884od_a_a @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ R2 )
% 0.18/0.45 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % transD
% 0.18/0.45 thf(fact_159_transD,axiom,
% 0.18/0.45 ! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( trans_a @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % transD
% 0.18/0.45 thf(fact_160_transE,axiom,
% 0.18/0.45 ! [R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.18/0.45 ( ( trans_972827014od_a_a @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ R2 )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % transE
% 0.18/0.45 thf(fact_161_transE,axiom,
% 0.18/0.45 ! [R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 0.18/0.45 ( ( trans_multiset_a @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R2 )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % transE
% 0.18/0.45 thf(fact_162_transE,axiom,
% 0.18/0.45 ! [R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
% 0.18/0.45 ( ( trans_1520871884od_a_a @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ R2 )
% 0.18/0.45 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % transE
% 0.18/0.45 thf(fact_163_transE,axiom,
% 0.18/0.45 ! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.18/0.45 ( ( trans_a @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % transE
% 0.18/0.45 thf(fact_164_rational__preference_Ointro,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( prefer1064779160od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer697821563od_a_a @ Carrier @ Relation ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.intro
% 0.18/0.45 thf(fact_165_rational__preference_Ointro,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.18/0.45 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( ( prefer580340847ioms_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.intro
% 0.18/0.45 thf(fact_166_rational__preference__def,axiom,
% 0.18/0.45 ( prefer697821563od_a_a
% 0.18/0.45 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( prefer1338892073od_a_a @ Carrier2 @ Relation2 )
% 0.18/0.45 & ( prefer1064779160od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference_def
% 0.18/0.45 thf(fact_167_rational__preference__def,axiom,
% 0.18/0.45 ( prefer719835986ence_a
% 0.18/0.45 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 0.18/0.45 ( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
% 0.18/0.45 & ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference_def
% 0.18/0.45 thf(fact_168_rational__preference_Oaxioms_I2_J,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.axioms(2)
% 0.18/0.45 thf(fact_169_rational__preference_Oaxioms_I2_J,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.18/0.45 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference.axioms(2)
% 0.18/0.45 thf(fact_170_rational__preference__axioms_Ointro,axiom,
% 0.18/0.45 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( total_1490170027od_a_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference_axioms.intro
% 0.18/0.45 thf(fact_171_rational__preference__axioms_Ointro,axiom,
% 0.18/0.45 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.18/0.45 ( ( total_on_a @ Carrier @ Relation )
% 0.18/0.45 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 0.18/0.45
% 0.18/0.45 % rational_preference_axioms.intro
% 0.18/0.45 thf(fact_172_rational__preference__axioms__def,axiom,
% 0.18/0.45 prefer1064779160od_a_a = total_1490170027od_a_a ).
% 0.18/0.45
% 0.18/0.45 % rational_preference_axioms_def
% 0.18/0.45 thf(fact_173_rational__preference__axioms__def,axiom,
% 0.18/0.45 prefer580340847ioms_a = total_on_a ).
% 0.18/0.45
% 0.18/0.45 % rational_preference_axioms_def
% 0.18/0.45 thf(fact_174_ssubst__Pair__rhs,axiom,
% 0.18/0.45 ! [R2: product_prod_a_a,S: product_prod_a_a,R3: set_Pr1948701895od_a_a,S2: product_prod_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R2 @ S ) @ R3 )
% 0.18/0.45 => ( ( S2 = S )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R2 @ S2 ) @ R3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ssubst_Pair_rhs
% 0.18/0.45 thf(fact_175_ssubst__Pair__rhs,axiom,
% 0.18/0.45 ! [R2: multiset_a,S: multiset_a,R3: set_Pr158363655iset_a,S2: multiset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ R2 @ S ) @ R3 )
% 0.18/0.45 => ( ( S2 = S )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ R2 @ S2 ) @ R3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ssubst_Pair_rhs
% 0.18/0.45 thf(fact_176_ssubst__Pair__rhs,axiom,
% 0.18/0.45 ! [R2: multis599418605od_a_a,S: multis599418605od_a_a,R3: set_Pr1821581383od_a_a,S2: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ R2 @ S ) @ R3 )
% 0.18/0.45 => ( ( S2 = S )
% 0.18/0.45 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ R2 @ S2 ) @ R3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ssubst_Pair_rhs
% 0.18/0.45 thf(fact_177_ssubst__Pair__rhs,axiom,
% 0.18/0.45 ! [R2: a,S: a,R3: set_Product_prod_a_a,S2: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S ) @ R3 )
% 0.18/0.45 => ( ( S2 = S )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ R3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % ssubst_Pair_rhs
% 0.18/0.45 thf(fact_178_at__lst__asgd__ge,axiom,
% 0.18/0.45 ! [X: produc1572603623od_a_a,Y: produc1572603623od_a_a,B4: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_lst_asgd_ge
% 0.18/0.45 thf(fact_179_at__lst__asgd__ge,axiom,
% 0.18/0.45 ! [X: produc1127127335iset_a,Y: produc1127127335iset_a,B4: set_Pr158363655iset_a,Pr: set_Pr146455751iset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ X @ ( prefer619498851iset_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_lst_asgd_ge
% 0.18/0.45 thf(fact_180_at__lst__asgd__ge,axiom,
% 0.18/0.45 ! [X: produc2061001575od_a_a,Y: produc2061001575od_a_a,B4: set_Pr1821581383od_a_a,Pr: set_Pr1930443975od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ X @ ( prefer737894307od_a_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_lst_asgd_ge
% 0.18/0.45 thf(fact_181_at__lst__asgd__ge,axiom,
% 0.18/0.45 ! [X: multiset_a,Y: multiset_a,B4: set_multiset_a,Pr: set_Pr158363655iset_a] :
% 0.18/0.45 ( ( member_multiset_a @ X @ ( prefer2130004608iset_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_lst_asgd_ge
% 0.18/0.45 thf(fact_182_at__lst__asgd__ge,axiom,
% 0.18/0.45 ! [X: multis599418605od_a_a,Y: multis599418605od_a_a,B4: set_mu1281098573od_a_a,Pr: set_Pr1821581383od_a_a] :
% 0.18/0.45 ( ( member1755255446od_a_a @ X @ ( prefer1191731497od_a_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_lst_asgd_ge
% 0.18/0.45 thf(fact_183_at__lst__asgd__ge,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,Y: product_prod_a_a,B4: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_lst_asgd_ge
% 0.18/0.45 thf(fact_184_at__lst__asgd__ge,axiom,
% 0.18/0.45 ! [X: a,Y: a,B4: set_a,Pr: set_Product_prod_a_a] :
% 0.18/0.45 ( ( member_a @ X @ ( prefer161218362good_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_lst_asgd_ge
% 0.18/0.45 thf(fact_185_at__least__as__goodD,axiom,
% 0.18/0.45 ! [Z: produc1572603623od_a_a,Y: produc1572603623od_a_a,B4: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ Z @ ( prefer55414563od_a_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_least_as_goodD
% 0.18/0.45 thf(fact_186_at__least__as__goodD,axiom,
% 0.18/0.45 ! [Z: produc1127127335iset_a,Y: produc1127127335iset_a,B4: set_Pr158363655iset_a,Pr: set_Pr146455751iset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ Z @ ( prefer619498851iset_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member567864080iset_a @ ( produc1898392407iset_a @ Z @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_least_as_goodD
% 0.18/0.45 thf(fact_187_at__least__as__goodD,axiom,
% 0.18/0.45 ! [Z: produc2061001575od_a_a,Y: produc2061001575od_a_a,B4: set_Pr1821581383od_a_a,Pr: set_Pr1930443975od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ Z @ ( prefer737894307od_a_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member2007374608od_a_a @ ( produc978545495od_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_least_as_goodD
% 0.18/0.45 thf(fact_188_at__least__as__goodD,axiom,
% 0.18/0.45 ! [Z: multiset_a,Y: multiset_a,B4: set_multiset_a,Pr: set_Pr158363655iset_a] :
% 0.18/0.45 ( ( member_multiset_a @ Z @ ( prefer2130004608iset_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_least_as_goodD
% 0.18/0.45 thf(fact_189_at__least__as__goodD,axiom,
% 0.18/0.45 ! [Z: multis599418605od_a_a,Y: multis599418605od_a_a,B4: set_mu1281098573od_a_a,Pr: set_Pr1821581383od_a_a] :
% 0.18/0.45 ( ( member1755255446od_a_a @ Z @ ( prefer1191731497od_a_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_least_as_goodD
% 0.18/0.45 thf(fact_190_at__least__as__goodD,axiom,
% 0.18/0.45 ! [Z: product_prod_a_a,Y: product_prod_a_a,B4: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_least_as_goodD
% 0.18/0.45 thf(fact_191_at__least__as__goodD,axiom,
% 0.18/0.45 ! [Z: a,Y: a,B4: set_a,Pr: set_Product_prod_a_a] :
% 0.18/0.45 ( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B4 @ Pr ) )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.18/0.45
% 0.18/0.45 % at_least_as_goodD
% 0.18/0.45 thf(fact_192_curry__conv,axiom,
% 0.18/0.45 ( product_curry_a_a_o
% 0.18/0.45 = ( ^ [F: product_prod_a_a > $o,A6: a,B5: a] : ( F @ ( product_Pair_a_a @ A6 @ B5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % curry_conv
% 0.18/0.45 thf(fact_193_curryI,axiom,
% 0.18/0.45 ! [F2: produc1572603623od_a_a > $o,A: product_prod_a_a,B: product_prod_a_a] :
% 0.18/0.45 ( ( F2 @ ( produc1474507607od_a_a @ A @ B ) )
% 0.18/0.45 => ( produc94438221_a_a_o @ F2 @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % curryI
% 0.18/0.45 thf(fact_194_curryI,axiom,
% 0.18/0.45 ! [F2: produc1127127335iset_a > $o,A: multiset_a,B: multiset_a] :
% 0.18/0.45 ( ( F2 @ ( produc2037245207iset_a @ A @ B ) )
% 0.18/0.45 => ( produc1530159757et_a_o @ F2 @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % curryI
% 0.18/0.45 thf(fact_195_curryI,axiom,
% 0.18/0.45 ! [F2: produc2061001575od_a_a > $o,A: multis599418605od_a_a,B: multis599418605od_a_a] :
% 0.18/0.45 ( ( F2 @ ( produc1859144151od_a_a @ A @ B ) )
% 0.18/0.45 => ( produc1548936141_a_a_o @ F2 @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % curryI
% 0.18/0.45 thf(fact_196_curryI,axiom,
% 0.18/0.45 ! [F2: product_prod_a_a > $o,A: a,B: a] :
% 0.18/0.45 ( ( F2 @ ( product_Pair_a_a @ A @ B ) )
% 0.18/0.45 => ( product_curry_a_a_o @ F2 @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % curryI
% 0.18/0.45 thf(fact_197_in__inv__image,axiom,
% 0.18/0.45 ! [X: a,Y: a,R2: set_Product_prod_a_a,F2: a > a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_image_a_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_198_in__inv__image,axiom,
% 0.18/0.45 ! [X: a,Y: a,R2: set_Pr158363655iset_a,F2: a > multiset_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_im975322764et_a_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member340150864iset_a @ ( produc2037245207iset_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_199_in__inv__image,axiom,
% 0.18/0.45 ! [X: multiset_a,Y: multiset_a,R2: set_Product_prod_a_a,F2: multiset_a > a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ ( inv_im479034648iset_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_200_in__inv__image,axiom,
% 0.18/0.45 ! [X: a,Y: a,R2: set_Pr1948701895od_a_a,F2: a > product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_im777352425_a_a_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_201_in__inv__image,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,Y: product_prod_a_a,R2: set_Product_prod_a_a,F2: product_prod_a_a > a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( inv_im15016955od_a_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_202_in__inv__image,axiom,
% 0.18/0.45 ! [X: multiset_a,Y: multiset_a,R2: set_Pr158363655iset_a,F2: multiset_a > multiset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ ( inv_im343774418iset_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member340150864iset_a @ ( produc2037245207iset_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_203_in__inv__image,axiom,
% 0.18/0.45 ! [X: a,Y: a,R2: set_Pr1821581383od_a_a,F2: a > multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_im1681759523_a_a_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member1349055120od_a_a @ ( produc1859144151od_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_204_in__inv__image,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,Y: product_prod_a_a,R2: set_Pr158363655iset_a,F2: product_prod_a_a > multiset_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( inv_im1716587189od_a_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member340150864iset_a @ ( produc2037245207iset_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_205_in__inv__image,axiom,
% 0.18/0.45 ! [X: multiset_a,Y: multiset_a,R2: set_Pr1948701895od_a_a,F2: multiset_a > product_prod_a_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ ( inv_im184450735iset_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_206_in__inv__image,axiom,
% 0.18/0.45 ! [X: multis599418605od_a_a,Y: multis599418605od_a_a,R2: set_Product_prod_a_a,F2: multis599418605od_a_a > a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ ( inv_im941243329od_a_a @ R2 @ F2 ) )
% 0.18/0.45 = ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_inv_image
% 0.18/0.45 thf(fact_207_strict__linear__order__on__def,axiom,
% 0.18/0.45 ( order_1505515173od_a_a
% 0.18/0.45 = ( ^ [A4: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( trans_972827014od_a_a @ R )
% 0.18/0.45 & ( irrefl76447196od_a_a @ R )
% 0.18/0.45 & ( total_1490170027od_a_a @ A4 @ R ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % strict_linear_order_on_def
% 0.18/0.45 thf(fact_208_strict__linear__order__on__def,axiom,
% 0.18/0.45 ( order_1558116988r_on_a
% 0.18/0.45 = ( ^ [A4: set_a,R: set_Product_prod_a_a] :
% 0.18/0.45 ( ( trans_a @ R )
% 0.18/0.45 & ( irrefl_a @ R )
% 0.18/0.45 & ( total_on_a @ A4 @ R ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % strict_linear_order_on_def
% 0.18/0.45 thf(fact_209_swap__simp,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,Y: product_prod_a_a] :
% 0.18/0.45 ( ( produc1641853175od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
% 0.18/0.45 = ( produc1474507607od_a_a @ Y @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % swap_simp
% 0.18/0.45 thf(fact_210_swap__simp,axiom,
% 0.18/0.45 ! [X: multiset_a,Y: multiset_a] :
% 0.18/0.45 ( ( produc506430135iset_a @ ( produc2037245207iset_a @ X @ Y ) )
% 0.18/0.45 = ( produc2037245207iset_a @ Y @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % swap_simp
% 0.18/0.45 thf(fact_211_swap__simp,axiom,
% 0.18/0.45 ! [X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 0.18/0.45 ( ( produc1913956215od_a_a @ ( produc1859144151od_a_a @ X @ Y ) )
% 0.18/0.45 = ( produc1859144151od_a_a @ Y @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % swap_simp
% 0.18/0.45 thf(fact_212_swap__simp,axiom,
% 0.18/0.45 ! [X: a,Y: a] :
% 0.18/0.45 ( ( product_swap_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 0.18/0.45 = ( product_Pair_a_a @ Y @ X ) ) ).
% 0.18/0.45
% 0.18/0.45 % swap_simp
% 0.18/0.45 thf(fact_213_internal__case__prod__def,axiom,
% 0.18/0.45 produc1653225349od_a_a = produc1093061851od_a_a ).
% 0.18/0.45
% 0.18/0.45 % internal_case_prod_def
% 0.18/0.45 thf(fact_214_internal__case__prod__def,axiom,
% 0.18/0.45 produc694541532_set_a = produc961951794_set_a ).
% 0.18/0.45
% 0.18/0.45 % internal_case_prod_def
% 0.18/0.45 thf(fact_215_internal__case__prod__def,axiom,
% 0.18/0.45 produc748978966_a_a_o = produc1833107820_a_a_o ).
% 0.18/0.45
% 0.18/0.45 % internal_case_prod_def
% 0.18/0.45 thf(fact_216_swap__swap,axiom,
% 0.18/0.45 ! [P: product_prod_a_a] :
% 0.18/0.45 ( ( product_swap_a_a @ ( product_swap_a_a @ P ) )
% 0.18/0.45 = P ) ).
% 0.18/0.45
% 0.18/0.45 % swap_swap
% 0.18/0.45 thf(fact_217_case__prod__curry,axiom,
% 0.18/0.45 ! [F2: product_prod_a_a > set_Product_prod_a_a] :
% 0.18/0.45 ( ( produc1093061851od_a_a @ ( produc143372156od_a_a @ F2 ) )
% 0.18/0.45 = F2 ) ).
% 0.18/0.45
% 0.18/0.45 % case_prod_curry
% 0.18/0.45 thf(fact_218_case__prod__curry,axiom,
% 0.18/0.45 ! [F2: product_prod_a_a > set_a] :
% 0.18/0.45 ( ( produc961951794_set_a @ ( produc580150227_set_a @ F2 ) )
% 0.18/0.45 = F2 ) ).
% 0.18/0.45
% 0.18/0.45 % case_prod_curry
% 0.18/0.45 thf(fact_219_case__prod__curry,axiom,
% 0.18/0.45 ! [F2: product_prod_a_a > $o] :
% 0.18/0.45 ( ( produc1833107820_a_a_o @ ( product_curry_a_a_o @ F2 ) )
% 0.18/0.45 = F2 ) ).
% 0.18/0.45
% 0.18/0.45 % case_prod_curry
% 0.18/0.45 thf(fact_220_curry__case__prod,axiom,
% 0.18/0.45 ! [F2: a > a > set_Product_prod_a_a] :
% 0.18/0.45 ( ( produc143372156od_a_a @ ( produc1093061851od_a_a @ F2 ) )
% 0.18/0.45 = F2 ) ).
% 0.18/0.45
% 0.18/0.45 % curry_case_prod
% 0.18/0.45 thf(fact_221_curry__case__prod,axiom,
% 0.18/0.45 ! [F2: a > a > set_a] :
% 0.18/0.45 ( ( produc580150227_set_a @ ( produc961951794_set_a @ F2 ) )
% 0.18/0.45 = F2 ) ).
% 0.18/0.45
% 0.18/0.45 % curry_case_prod
% 0.18/0.45 thf(fact_222_curry__case__prod,axiom,
% 0.18/0.45 ! [F2: a > a > $o] :
% 0.18/0.45 ( ( product_curry_a_a_o @ ( produc1833107820_a_a_o @ F2 ) )
% 0.18/0.45 = F2 ) ).
% 0.18/0.45
% 0.18/0.45 % curry_case_prod
% 0.18/0.45 thf(fact_223_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: a,C2: a > a > set_a,P: product_prod_a_a] :
% 0.18/0.45 ( ( member_a @ Z @ ( produc961951794_set_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_224_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: product_prod_a_a,C2: a > a > set_Product_prod_a_a,P: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ Z @ ( produc1093061851od_a_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member449909584od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_225_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: multiset_a,C2: a > a > set_multiset_a,P: product_prod_a_a] :
% 0.18/0.45 ( ( member_multiset_a @ Z @ ( produc1828150264iset_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member_multiset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_226_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: a,C2: multiset_a > multiset_a > set_a,P: produc1127127335iset_a] :
% 0.18/0.45 ( ( member_a @ Z @ ( produc747255346_set_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: multiset_a,Y3: multiset_a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( produc2037245207iset_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_227_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: multis599418605od_a_a,C2: a > a > set_mu1281098573od_a_a,P: product_prod_a_a] :
% 0.18/0.45 ( ( member1755255446od_a_a @ Z @ ( produc1121147169od_a_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member1755255446od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_228_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: multiset_a,C2: multiset_a > multiset_a > set_multiset_a,P: produc1127127335iset_a] :
% 0.18/0.45 ( ( member_multiset_a @ Z @ ( produc429414904iset_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: multiset_a,Y3: multiset_a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( produc2037245207iset_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member_multiset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_229_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: produc1127127335iset_a,C2: a > a > set_Pr158363655iset_a,P: product_prod_a_a] :
% 0.18/0.45 ( ( member340150864iset_a @ Z @ ( produc93594587iset_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member340150864iset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_230_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: a,C2: product_prod_a_a > product_prod_a_a > set_a,P: produc1572603623od_a_a] :
% 0.18/0.45 ( ( member_a @ Z @ ( produc869579762_set_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( produc1474507607od_a_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_231_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: product_prod_a_a,C2: multiset_a > multiset_a > set_Product_prod_a_a,P: produc1127127335iset_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ Z @ ( produc2041367259od_a_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: multiset_a,Y3: multiset_a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( produc2037245207iset_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member449909584od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_232_mem__case__prodE,axiom,
% 0.18/0.45 ! [Z: multiset_a,C2: product_prod_a_a > product_prod_a_a > set_multiset_a,P: produc1572603623od_a_a] :
% 0.18/0.45 ( ( member_multiset_a @ Z @ ( produc734625720iset_a @ C2 @ P ) )
% 0.18/0.45 => ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 0.18/0.45 ( ( P
% 0.18/0.45 = ( produc1474507607od_a_a @ X3 @ Y3 ) )
% 0.18/0.45 => ~ ( member_multiset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mem_case_prodE
% 0.18/0.45 thf(fact_233_old_Oprod_Ocase,axiom,
% 0.18/0.45 ! [F2: a > a > set_Product_prod_a_a,X1: a,X22: a] :
% 0.18/0.45 ( ( produc1093061851od_a_a @ F2 @ ( product_Pair_a_a @ X1 @ X22 ) )
% 0.18/0.45 = ( F2 @ X1 @ X22 ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.case
% 0.18/0.45 thf(fact_234_old_Oprod_Ocase,axiom,
% 0.18/0.45 ! [F2: a > a > set_a,X1: a,X22: a] :
% 0.18/0.45 ( ( produc961951794_set_a @ F2 @ ( product_Pair_a_a @ X1 @ X22 ) )
% 0.18/0.45 = ( F2 @ X1 @ X22 ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.case
% 0.18/0.45 thf(fact_235_old_Oprod_Ocase,axiom,
% 0.18/0.45 ! [F2: a > a > $o,X1: a,X22: a] :
% 0.18/0.45 ( ( produc1833107820_a_a_o @ F2 @ ( product_Pair_a_a @ X1 @ X22 ) )
% 0.18/0.45 = ( F2 @ X1 @ X22 ) ) ).
% 0.18/0.45
% 0.18/0.45 % old.prod.case
% 0.18/0.45 thf(fact_236_irreflI,axiom,
% 0.18/0.45 ! [R3: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ! [A3: product_prod_a_a] :
% 0.18/0.45 ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A3 @ A3 ) @ R3 )
% 0.18/0.45 => ( irrefl76447196od_a_a @ R3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % irreflI
% 0.18/0.45 thf(fact_237_irreflI,axiom,
% 0.18/0.45 ! [R3: set_Pr158363655iset_a] :
% 0.18/0.45 ( ! [A3: multiset_a] :
% 0.18/0.45 ~ ( member340150864iset_a @ ( produc2037245207iset_a @ A3 @ A3 ) @ R3 )
% 0.18/0.45 => ( irrefl_multiset_a @ R3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % irreflI
% 0.18/0.45 thf(fact_238_irreflI,axiom,
% 0.18/0.45 ! [R3: set_Pr1821581383od_a_a] :
% 0.18/0.45 ( ! [A3: multis599418605od_a_a] :
% 0.18/0.45 ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A3 @ A3 ) @ R3 )
% 0.18/0.45 => ( irrefl1672195874od_a_a @ R3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % irreflI
% 0.18/0.45 thf(fact_239_irreflI,axiom,
% 0.18/0.45 ! [R3: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [A3: a] :
% 0.18/0.45 ~ ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R3 )
% 0.18/0.45 => ( irrefl_a @ R3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % irreflI
% 0.18/0.45 thf(fact_240_irrefl__def,axiom,
% 0.18/0.45 ( irrefl76447196od_a_a
% 0.18/0.45 = ( ^ [R: set_Pr1948701895od_a_a] :
% 0.18/0.45 ! [A6: product_prod_a_a] :
% 0.18/0.45 ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A6 @ A6 ) @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % irrefl_def
% 0.18/0.45 thf(fact_241_irrefl__def,axiom,
% 0.18/0.45 ( irrefl_multiset_a
% 0.18/0.45 = ( ^ [R: set_Pr158363655iset_a] :
% 0.18/0.45 ! [A6: multiset_a] :
% 0.18/0.45 ~ ( member340150864iset_a @ ( produc2037245207iset_a @ A6 @ A6 ) @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % irrefl_def
% 0.18/0.45 thf(fact_242_irrefl__def,axiom,
% 0.18/0.45 ( irrefl1672195874od_a_a
% 0.18/0.45 = ( ^ [R: set_Pr1821581383od_a_a] :
% 0.18/0.45 ! [A6: multis599418605od_a_a] :
% 0.18/0.45 ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A6 @ A6 ) @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % irrefl_def
% 0.18/0.45 thf(fact_243_irrefl__def,axiom,
% 0.18/0.45 ( irrefl_a
% 0.18/0.45 = ( ^ [R: set_Product_prod_a_a] :
% 0.18/0.45 ! [A6: a] :
% 0.18/0.45 ~ ( member449909584od_a_a @ ( product_Pair_a_a @ A6 @ A6 ) @ R ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % irrefl_def
% 0.18/0.45 thf(fact_244_curryD,axiom,
% 0.18/0.45 ! [F2: produc1127127335iset_a > $o,A: multiset_a,B: multiset_a] :
% 0.18/0.45 ( ( produc1530159757et_a_o @ F2 @ A @ B )
% 0.18/0.45 => ( F2 @ ( produc2037245207iset_a @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % curryD
% 0.18/0.45 thf(fact_245_curryD,axiom,
% 0.18/0.45 ! [F2: produc2061001575od_a_a > $o,A: multis599418605od_a_a,B: multis599418605od_a_a] :
% 0.18/0.45 ( ( produc1548936141_a_a_o @ F2 @ A @ B )
% 0.18/0.45 => ( F2 @ ( produc1859144151od_a_a @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % curryD
% 0.18/0.45 thf(fact_246_curryD,axiom,
% 0.18/0.45 ! [F2: product_prod_a_a > $o,A: a,B: a] :
% 0.18/0.45 ( ( product_curry_a_a_o @ F2 @ A @ B )
% 0.18/0.45 => ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % curryD
% 0.18/0.45 thf(fact_247_curryE,axiom,
% 0.18/0.45 ! [F2: product_prod_a_a > $o,A: a,B: a] :
% 0.18/0.45 ( ( product_curry_a_a_o @ F2 @ A @ B )
% 0.18/0.45 => ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % curryE
% 0.18/0.45 thf(fact_248_trans__inv__image,axiom,
% 0.18/0.45 ! [R2: set_Product_prod_a_a,F2: a > a] :
% 0.18/0.45 ( ( trans_a @ R2 )
% 0.18/0.45 => ( trans_a @ ( inv_image_a_a @ R2 @ F2 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % trans_inv_image
% 0.18/0.45 thf(fact_249_pair__in__swap__image,axiom,
% 0.18/0.45 ! [Y: a,X: a,A5: set_Product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( image_794773445od_a_a @ product_swap_a_a @ A5 ) )
% 0.18/0.45 = ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ A5 ) ) ).
% 0.18/0.45
% 0.18/0.45 % pair_in_swap_image
% 0.18/0.45 thf(fact_250_multp__iff,axiom,
% 0.18/0.45 ! [R3: set_Product_prod_a_a,P2: a > a > $o,N: multiset_a,M: multiset_a] :
% 0.18/0.45 ( ( irrefl_a @ R3 )
% 0.18/0.45 => ( ( trans_a @ R3 )
% 0.18/0.45 => ( ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( P2 @ X3 @ Y3 )
% 0.18/0.45 = ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R3 ) )
% 0.18/0.45 => ( ( multp_a @ P2 @ N @ M )
% 0.18/0.45 = ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult_a @ R3 ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multp_iff
% 0.18/0.45 thf(fact_251_asym_Oinducts,axiom,
% 0.18/0.45 ! [X: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
% 0.18/0.45 ( ( asym_a @ X )
% 0.18/0.45 => ( ! [R4: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [A7: a,B6: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A7 @ B6 ) @ R4 )
% 0.18/0.45 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A7 ) @ R4 ) )
% 0.18/0.45 => ( P2 @ R4 ) )
% 0.18/0.45 => ( P2 @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % asym.inducts
% 0.18/0.45 thf(fact_252_asym_Ointros,axiom,
% 0.18/0.45 ! [R3: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [A3: a,B3: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 )
% 0.18/0.45 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A3 ) @ R3 ) )
% 0.18/0.45 => ( asym_a @ R3 ) ) ).
% 0.18/0.45
% 0.18/0.45 % asym.intros
% 0.18/0.45 thf(fact_253_asym_Osimps,axiom,
% 0.18/0.45 ( asym_a
% 0.18/0.45 = ( ^ [A6: set_Product_prod_a_a] :
% 0.18/0.45 ? [R5: set_Product_prod_a_a] :
% 0.18/0.45 ( ( A6 = R5 )
% 0.18/0.45 & ! [X4: a,Y4: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R5 )
% 0.18/0.45 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R5 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % asym.simps
% 0.18/0.45 thf(fact_254_asym_Ocases,axiom,
% 0.18/0.45 ! [A: set_Product_prod_a_a] :
% 0.18/0.45 ( ( asym_a @ A )
% 0.18/0.45 => ! [A7: a,B6: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A7 @ B6 ) @ A )
% 0.18/0.45 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A7 ) @ A ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % asym.cases
% 0.18/0.45 thf(fact_255_asym__iff,axiom,
% 0.18/0.45 ( asym_a
% 0.18/0.45 = ( ^ [R5: set_Product_prod_a_a] :
% 0.18/0.45 ! [X4: a,Y4: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R5 )
% 0.18/0.45 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % asym_iff
% 0.18/0.45 thf(fact_256_asymD,axiom,
% 0.18/0.45 ! [R3: set_Product_prod_a_a,X: a,Y: a] :
% 0.18/0.45 ( ( asym_a @ R3 )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R3 )
% 0.18/0.45 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % asymD
% 0.18/0.45 thf(fact_257_image__eqI,axiom,
% 0.18/0.45 ! [B: a,F2: a > a,X: a,A5: set_a] :
% 0.18/0.45 ( ( B
% 0.18/0.45 = ( F2 @ X ) )
% 0.18/0.45 => ( ( member_a @ X @ A5 )
% 0.18/0.45 => ( member_a @ B @ ( image_a_a @ F2 @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % image_eqI
% 0.18/0.45 thf(fact_258_image__eqI,axiom,
% 0.18/0.45 ! [B: product_prod_a_a,F2: a > product_prod_a_a,X: a,A5: set_a] :
% 0.18/0.45 ( ( B
% 0.18/0.45 = ( F2 @ X ) )
% 0.18/0.45 => ( ( member_a @ X @ A5 )
% 0.18/0.45 => ( member449909584od_a_a @ B @ ( image_1725938350od_a_a @ F2 @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % image_eqI
% 0.18/0.45 thf(fact_259_image__eqI,axiom,
% 0.18/0.45 ! [B: a,F2: product_prod_a_a > a,X: product_prod_a_a,A5: set_Product_prod_a_a] :
% 0.18/0.45 ( ( B
% 0.18/0.45 = ( F2 @ X ) )
% 0.18/0.45 => ( ( member449909584od_a_a @ X @ A5 )
% 0.18/0.45 => ( member_a @ B @ ( image_340790172_a_a_a @ F2 @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % image_eqI
% 0.18/0.45 thf(fact_260_image__eqI,axiom,
% 0.18/0.45 ! [B: product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,A5: set_Product_prod_a_a] :
% 0.18/0.45 ( ( B
% 0.18/0.45 = ( F2 @ X ) )
% 0.18/0.45 => ( ( member449909584od_a_a @ X @ A5 )
% 0.18/0.45 => ( member449909584od_a_a @ B @ ( image_794773445od_a_a @ F2 @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % image_eqI
% 0.18/0.45 thf(fact_261_mult__cancel,axiom,
% 0.18/0.45 ! [S: set_Product_prod_a_a,X5: multiset_a,Z4: multiset_a,Y5: multiset_a] :
% 0.18/0.45 ( ( trans_a @ S )
% 0.18/0.45 => ( ( irrefl_a @ S )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ X5 @ Z4 ) @ ( plus_plus_multiset_a @ Y5 @ Z4 ) ) @ ( mult_a @ S ) )
% 0.18/0.45 = ( member340150864iset_a @ ( produc2037245207iset_a @ X5 @ Y5 ) @ ( mult_a @ S ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mult_cancel
% 0.18/0.45 thf(fact_262_mult__cancel__add__mset,axiom,
% 0.18/0.45 ! [S: set_Product_prod_a_a,Uu: a,X5: multiset_a,Y5: multiset_a] :
% 0.18/0.45 ( ( trans_a @ S )
% 0.18/0.45 => ( ( irrefl_a @ S )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X5 ) @ ( add_mset_a @ Uu @ Y5 ) ) @ ( mult_a @ S ) )
% 0.18/0.45 = ( member340150864iset_a @ ( produc2037245207iset_a @ X5 @ Y5 ) @ ( mult_a @ S ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mult_cancel_add_mset
% 0.18/0.45 thf(fact_263_imageI,axiom,
% 0.18/0.45 ! [X: a,A5: set_a,F2: a > a] :
% 0.18/0.45 ( ( member_a @ X @ A5 )
% 0.18/0.45 => ( member_a @ ( F2 @ X ) @ ( image_a_a @ F2 @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % imageI
% 0.18/0.45 thf(fact_264_imageI,axiom,
% 0.18/0.45 ! [X: a,A5: set_a,F2: a > product_prod_a_a] :
% 0.18/0.45 ( ( member_a @ X @ A5 )
% 0.18/0.45 => ( member449909584od_a_a @ ( F2 @ X ) @ ( image_1725938350od_a_a @ F2 @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % imageI
% 0.18/0.45 thf(fact_265_imageI,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ A5 )
% 0.18/0.45 => ( member_a @ ( F2 @ X ) @ ( image_340790172_a_a_a @ F2 @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % imageI
% 0.18/0.45 thf(fact_266_imageI,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ A5 )
% 0.18/0.45 => ( member449909584od_a_a @ ( F2 @ X ) @ ( image_794773445od_a_a @ F2 @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % imageI
% 0.18/0.45 thf(fact_267_rev__image__eqI,axiom,
% 0.18/0.45 ! [X: a,A5: set_a,B: a,F2: a > a] :
% 0.18/0.45 ( ( member_a @ X @ A5 )
% 0.18/0.45 => ( ( B
% 0.18/0.45 = ( F2 @ X ) )
% 0.18/0.45 => ( member_a @ B @ ( image_a_a @ F2 @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rev_image_eqI
% 0.18/0.45 thf(fact_268_rev__image__eqI,axiom,
% 0.18/0.45 ! [X: a,A5: set_a,B: product_prod_a_a,F2: a > product_prod_a_a] :
% 0.18/0.45 ( ( member_a @ X @ A5 )
% 0.18/0.45 => ( ( B
% 0.18/0.45 = ( F2 @ X ) )
% 0.18/0.45 => ( member449909584od_a_a @ B @ ( image_1725938350od_a_a @ F2 @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rev_image_eqI
% 0.18/0.45 thf(fact_269_rev__image__eqI,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,A5: set_Product_prod_a_a,B: a,F2: product_prod_a_a > a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ A5 )
% 0.18/0.45 => ( ( B
% 0.18/0.45 = ( F2 @ X ) )
% 0.18/0.45 => ( member_a @ B @ ( image_340790172_a_a_a @ F2 @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rev_image_eqI
% 0.18/0.45 thf(fact_270_rev__image__eqI,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,A5: set_Product_prod_a_a,B: product_prod_a_a,F2: product_prod_a_a > product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ A5 )
% 0.18/0.45 => ( ( B
% 0.18/0.45 = ( F2 @ X ) )
% 0.18/0.45 => ( member449909584od_a_a @ B @ ( image_794773445od_a_a @ F2 @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % rev_image_eqI
% 0.18/0.45 thf(fact_271_top__empty__eq,axiom,
% 0.18/0.45 ( top_top_a_o
% 0.18/0.45 = ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % top_empty_eq
% 0.18/0.45 thf(fact_272_top__empty__eq,axiom,
% 0.18/0.45 ( top_to1483957574_a_a_o
% 0.18/0.45 = ( ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ top_to398455383od_a_a ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % top_empty_eq
% 0.18/0.45 thf(fact_273_total__lex__prod,axiom,
% 0.18/0.45 ! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 0.18/0.45 ( ( total_on_a @ top_top_set_a @ R2 )
% 0.18/0.45 => ( ( total_on_a @ top_top_set_a @ S )
% 0.18/0.45 => ( total_1490170027od_a_a @ top_to398455383od_a_a @ ( lex_prod_a_a @ R2 @ S ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % total_lex_prod
% 0.18/0.45 thf(fact_274_map__prod__simp,axiom,
% 0.18/0.45 ! [F2: a > a,G: a > a,A: a,B: a] :
% 0.18/0.45 ( ( produc700022774_a_a_a @ F2 @ G @ ( product_Pair_a_a @ A @ B ) )
% 0.18/0.45 = ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % map_prod_simp
% 0.18/0.45 thf(fact_275_in__lex__prod,axiom,
% 0.18/0.45 ! [A: a,B: a,A2: a,B2: a,R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 0.18/0.45 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Pair_a_a @ A2 @ B2 ) ) @ ( lex_prod_a_a @ R2 @ S ) )
% 0.18/0.45 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R2 )
% 0.18/0.45 | ( ( A = A2 )
% 0.18/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ B @ B2 ) @ S ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_lex_prod
% 0.18/0.45 thf(fact_276_trans__lex__prod,axiom,
% 0.18/0.45 ! [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a] :
% 0.18/0.45 ( ( trans_a @ R1 )
% 0.18/0.45 => ( ( trans_a @ R22 )
% 0.18/0.45 => ( trans_972827014od_a_a @ ( lex_prod_a_a @ R1 @ R22 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % trans_lex_prod
% 0.18/0.45 thf(fact_277_prod_Ocollapse,axiom,
% 0.18/0.45 ! [Prod: product_prod_a_a] :
% 0.18/0.45 ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
% 0.18/0.45 = Prod ) ).
% 0.18/0.45
% 0.18/0.45 % prod.collapse
% 0.18/0.45 thf(fact_278_map__prod__imageI,axiom,
% 0.18/0.45 ! [A: a,B: a,R3: set_Product_prod_a_a,F2: a > a,G: a > a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R3 )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F2 @ G ) @ R3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % map_prod_imageI
% 0.18/0.45 thf(fact_279_Product__Type_OCollect__case__prodD,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,A5: a > a > $o] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ ( collec645855634od_a_a @ ( produc1833107820_a_a_o @ A5 ) ) )
% 0.18/0.45 => ( A5 @ ( product_fst_a_a @ X ) @ ( product_snd_a_a @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % Product_Type.Collect_case_prodD
% 0.18/0.45 thf(fact_280_prod_Oexhaust__sel,axiom,
% 0.18/0.45 ! [Prod: product_prod_a_a] :
% 0.18/0.45 ( Prod
% 0.18/0.45 = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod.exhaust_sel
% 0.18/0.45 thf(fact_281_surjective__pairing,axiom,
% 0.18/0.45 ! [T: product_prod_a_a] :
% 0.18/0.45 ( T
% 0.18/0.45 = ( product_Pair_a_a @ ( product_fst_a_a @ T ) @ ( product_snd_a_a @ T ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % surjective_pairing
% 0.18/0.45 thf(fact_282_fst__conv,axiom,
% 0.18/0.45 ! [X1: a,X22: a] :
% 0.18/0.45 ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
% 0.18/0.45 = X1 ) ).
% 0.18/0.45
% 0.18/0.45 % fst_conv
% 0.18/0.45 thf(fact_283_fst__eqD,axiom,
% 0.18/0.45 ! [X: a,Y: a,A: a] :
% 0.18/0.45 ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 0.18/0.45 = A )
% 0.18/0.45 => ( X = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % fst_eqD
% 0.18/0.45 thf(fact_284_snd__conv,axiom,
% 0.18/0.45 ! [X1: a,X22: a] :
% 0.18/0.45 ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
% 0.18/0.45 = X22 ) ).
% 0.18/0.45
% 0.18/0.45 % snd_conv
% 0.18/0.45 thf(fact_285_snd__eqD,axiom,
% 0.18/0.45 ! [X: a,Y: a,A: a] :
% 0.18/0.45 ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 0.18/0.45 = A )
% 0.18/0.45 => ( Y = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % snd_eqD
% 0.18/0.45 thf(fact_286_prod_Oswap__def,axiom,
% 0.18/0.45 ( product_swap_a_a
% 0.18/0.45 = ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod.swap_def
% 0.18/0.45 thf(fact_287_prod__fun__imageE,axiom,
% 0.18/0.45 ! [C2: product_prod_a_a,F2: a > a,G: a > a,R3: set_Product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ C2 @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F2 @ G ) @ R3 ) )
% 0.18/0.45 => ~ ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( C2
% 0.18/0.45 = ( product_Pair_a_a @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
% 0.18/0.45 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R3 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % prod_fun_imageE
% 0.18/0.45 thf(fact_288_exI__realizer,axiom,
% 0.18/0.45 ! [P2: a > a > $o,Y: a,X: a] :
% 0.18/0.45 ( ( P2 @ Y @ X )
% 0.18/0.45 => ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % exI_realizer
% 0.18/0.45 thf(fact_289_conjI__realizer,axiom,
% 0.18/0.45 ! [P2: a > $o,P: a,Q: a > $o,Q2: a] :
% 0.18/0.45 ( ( P2 @ P )
% 0.18/0.45 => ( ( Q @ Q2 )
% 0.18/0.45 => ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) )
% 0.18/0.45 & ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % conjI_realizer
% 0.18/0.45 thf(fact_290_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.18/0.45 ! [P2: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
% 0.18/0.45 ( ( P2 @ X @ Y )
% 0.18/0.45 => ( ( A
% 0.18/0.45 = ( product_Pair_a_a @ X @ Y ) )
% 0.18/0.45 => ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % BNF_Greatest_Fixpoint.subst_Pair
% 0.18/0.45 thf(fact_291_sndI,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,Y: a,Z: a] :
% 0.18/0.45 ( ( X
% 0.18/0.45 = ( product_Pair_a_a @ Y @ Z ) )
% 0.18/0.45 => ( ( product_snd_a_a @ X )
% 0.18/0.45 = Z ) ) ).
% 0.18/0.45
% 0.18/0.45 % sndI
% 0.18/0.45 thf(fact_292_fstI,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,Y: a,Z: a] :
% 0.18/0.45 ( ( X
% 0.18/0.45 = ( product_Pair_a_a @ Y @ Z ) )
% 0.18/0.45 => ( ( product_fst_a_a @ X )
% 0.18/0.45 = Y ) ) ).
% 0.18/0.45
% 0.18/0.45 % fstI
% 0.18/0.45 thf(fact_293_eq__snd__iff,axiom,
% 0.18/0.45 ! [B: a,P: product_prod_a_a] :
% 0.18/0.45 ( ( B
% 0.18/0.45 = ( product_snd_a_a @ P ) )
% 0.18/0.45 = ( ? [A6: a] :
% 0.18/0.45 ( P
% 0.18/0.45 = ( product_Pair_a_a @ A6 @ B ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % eq_snd_iff
% 0.18/0.45 thf(fact_294_eq__fst__iff,axiom,
% 0.18/0.45 ! [A: a,P: product_prod_a_a] :
% 0.18/0.45 ( ( A
% 0.18/0.45 = ( product_fst_a_a @ P ) )
% 0.18/0.45 = ( ? [B5: a] :
% 0.18/0.45 ( P
% 0.18/0.45 = ( product_Pair_a_a @ A @ B5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % eq_fst_iff
% 0.18/0.45 thf(fact_295_subrelI,axiom,
% 0.18/0.45 ! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [X3: a,Y3: a] :
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S ) )
% 0.18/0.45 => ( ord_le1824328871od_a_a @ R2 @ S ) ) ).
% 0.18/0.45
% 0.18/0.45 % subrelI
% 0.18/0.45 thf(fact_296_image__subsetI,axiom,
% 0.18/0.45 ! [A5: set_a,F2: a > a,B4: set_a] :
% 0.18/0.45 ( ! [X3: a] :
% 0.18/0.45 ( ( member_a @ X3 @ A5 )
% 0.18/0.45 => ( member_a @ ( F2 @ X3 ) @ B4 ) )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A5 ) @ B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % image_subsetI
% 0.18/0.45 thf(fact_297_image__subsetI,axiom,
% 0.18/0.45 ! [A5: set_a,F2: a > product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [X3: a] :
% 0.18/0.45 ( ( member_a @ X3 @ A5 )
% 0.18/0.45 => ( member449909584od_a_a @ ( F2 @ X3 ) @ B4 ) )
% 0.18/0.45 => ( ord_le1824328871od_a_a @ ( image_1725938350od_a_a @ F2 @ A5 ) @ B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % image_subsetI
% 0.18/0.45 thf(fact_298_image__subsetI,axiom,
% 0.18/0.45 ! [A5: set_Product_prod_a_a,F2: product_prod_a_a > a,B4: set_a] :
% 0.18/0.45 ( ! [X3: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X3 @ A5 )
% 0.18/0.45 => ( member_a @ ( F2 @ X3 ) @ B4 ) )
% 0.18/0.45 => ( ord_less_eq_set_a @ ( image_340790172_a_a_a @ F2 @ A5 ) @ B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % image_subsetI
% 0.18/0.45 thf(fact_299_image__subsetI,axiom,
% 0.18/0.45 ! [A5: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.18/0.45 ( ! [X3: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X3 @ A5 )
% 0.18/0.45 => ( member449909584od_a_a @ ( F2 @ X3 ) @ B4 ) )
% 0.18/0.45 => ( ord_le1824328871od_a_a @ ( image_794773445od_a_a @ F2 @ A5 ) @ B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % image_subsetI
% 0.18/0.45 thf(fact_300_same__fstI,axiom,
% 0.18/0.45 ! [P2: a > $o,X: a,Y6: a,Y: a,R3: a > set_Product_prod_a_a] :
% 0.18/0.45 ( ( P2 @ X )
% 0.18/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y6 @ Y ) @ ( R3 @ X ) )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ X @ Y6 ) @ ( product_Pair_a_a @ X @ Y ) ) @ ( same_fst_a_a @ P2 @ R3 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % same_fstI
% 0.18/0.45 thf(fact_301_mult1E,axiom,
% 0.18/0.45 ! [N: multis599418605od_a_a,M: multis599418605od_a_a,R2: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R2 ) )
% 0.18/0.45 => ~ ! [A3: product_prod_a_a,M0: multis599418605od_a_a] :
% 0.18/0.45 ( ( M
% 0.18/0.45 = ( add_ms1725977093od_a_a @ A3 @ M0 ) )
% 0.18/0.45 => ! [K: multis599418605od_a_a] :
% 0.18/0.45 ( ( N
% 0.18/0.45 = ( plus_p404985124od_a_a @ M0 @ K ) )
% 0.18/0.45 => ~ ! [B6: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ B6 @ ( set_ms1818337124od_a_a @ K ) )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B6 @ A3 ) @ R2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mult1E
% 0.18/0.45 thf(fact_302_mult1E,axiom,
% 0.18/0.45 ! [N: multiset_a,M: multiset_a,R2: set_Product_prod_a_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R2 ) )
% 0.18/0.45 => ~ ! [A3: a,M0: multiset_a] :
% 0.18/0.45 ( ( M
% 0.18/0.45 = ( add_mset_a @ A3 @ M0 ) )
% 0.18/0.45 => ! [K: multiset_a] :
% 0.18/0.45 ( ( N
% 0.18/0.45 = ( plus_plus_multiset_a @ M0 @ K ) )
% 0.18/0.45 => ~ ! [B6: a] :
% 0.18/0.45 ( ( member_a @ B6 @ ( set_mset_a @ K ) )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A3 ) @ R2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mult1E
% 0.18/0.45 thf(fact_303_mult1I,axiom,
% 0.18/0.45 ! [M: multis599418605od_a_a,A: product_prod_a_a,M02: multis599418605od_a_a,N: multis599418605od_a_a,K2: multis599418605od_a_a,R2: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( M
% 0.18/0.45 = ( add_ms1725977093od_a_a @ A @ M02 ) )
% 0.18/0.45 => ( ( N
% 0.18/0.45 = ( plus_p404985124od_a_a @ M02 @ K2 ) )
% 0.18/0.45 => ( ! [B3: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ B3 @ ( set_ms1818337124od_a_a @ K2 ) )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B3 @ A ) @ R2 ) )
% 0.18/0.45 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mult1I
% 0.18/0.45 thf(fact_304_mult1I,axiom,
% 0.18/0.45 ! [M: multiset_a,A: a,M02: multiset_a,N: multiset_a,K2: multiset_a,R2: set_Product_prod_a_a] :
% 0.18/0.45 ( ( M
% 0.18/0.45 = ( add_mset_a @ A @ M02 ) )
% 0.18/0.45 => ( ( N
% 0.18/0.45 = ( plus_plus_multiset_a @ M02 @ K2 ) )
% 0.18/0.45 => ( ! [B3: a] :
% 0.18/0.45 ( ( member_a @ B3 @ ( set_mset_a @ K2 ) )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R2 ) )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mult1I
% 0.18/0.45 thf(fact_305_less__add,axiom,
% 0.18/0.45 ! [N: multis599418605od_a_a,A: product_prod_a_a,M02: multis599418605od_a_a,R2: set_Pr1948701895od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ M02 ) ) @ ( mult1_1719650714od_a_a @ R2 ) )
% 0.18/0.45 => ( ? [M2: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ M2 @ M02 ) @ ( mult1_1719650714od_a_a @ R2 ) )
% 0.18/0.45 & ( N
% 0.18/0.45 = ( add_ms1725977093od_a_a @ A @ M2 ) ) )
% 0.18/0.45 | ? [K: multis599418605od_a_a] :
% 0.18/0.45 ( ! [B6: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ B6 @ ( set_ms1818337124od_a_a @ K ) )
% 0.18/0.45 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B6 @ A ) @ R2 ) )
% 0.18/0.45 & ( N
% 0.18/0.45 = ( plus_p404985124od_a_a @ M02 @ K ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_add
% 0.18/0.45 thf(fact_306_less__add,axiom,
% 0.18/0.45 ! [N: multiset_a,A: a,M02: multiset_a,R2: set_Product_prod_a_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ ( add_mset_a @ A @ M02 ) ) @ ( mult1_a @ R2 ) )
% 0.18/0.45 => ( ? [M2: multiset_a] :
% 0.18/0.45 ( ( member340150864iset_a @ ( produc2037245207iset_a @ M2 @ M02 ) @ ( mult1_a @ R2 ) )
% 0.18/0.45 & ( N
% 0.18/0.45 = ( add_mset_a @ A @ M2 ) ) )
% 0.18/0.45 | ? [K: multiset_a] :
% 0.18/0.45 ( ! [B6: a] :
% 0.18/0.45 ( ( member_a @ B6 @ ( set_mset_a @ K ) )
% 0.18/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A ) @ R2 ) )
% 0.18/0.45 & ( N
% 0.18/0.45 = ( plus_plus_multiset_a @ M02 @ K ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % less_add
% 0.18/0.45 thf(fact_307_mset__add,axiom,
% 0.18/0.45 ! [A: a,A5: multiset_a] :
% 0.18/0.45 ( ( member_a @ A @ ( set_mset_a @ A5 ) )
% 0.18/0.45 => ~ ! [B7: multiset_a] :
% 0.18/0.45 ( A5
% 0.18/0.45 != ( add_mset_a @ A @ B7 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mset_add
% 0.18/0.45 thf(fact_308_mset__add,axiom,
% 0.18/0.45 ! [A: product_prod_a_a,A5: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A5 ) )
% 0.18/0.45 => ~ ! [B7: multis599418605od_a_a] :
% 0.18/0.45 ( A5
% 0.18/0.45 != ( add_ms1725977093od_a_a @ A @ B7 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mset_add
% 0.18/0.45 thf(fact_309_multi__member__split,axiom,
% 0.18/0.45 ! [X: a,M: multiset_a] :
% 0.18/0.45 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.18/0.45 => ? [A8: multiset_a] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( add_mset_a @ X @ A8 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_member_split
% 0.18/0.45 thf(fact_310_multi__member__split,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 0.18/0.45 => ? [A8: multis599418605od_a_a] :
% 0.18/0.45 ( M
% 0.18/0.45 = ( add_ms1725977093od_a_a @ X @ A8 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_member_split
% 0.18/0.45 thf(fact_311_insert__noteq__member,axiom,
% 0.18/0.45 ! [B: a,B4: multiset_a,C2: a,C3: multiset_a] :
% 0.18/0.45 ( ( ( add_mset_a @ B @ B4 )
% 0.18/0.45 = ( add_mset_a @ C2 @ C3 ) )
% 0.18/0.45 => ( ( B != C2 )
% 0.18/0.45 => ( member_a @ C2 @ ( set_mset_a @ B4 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % insert_noteq_member
% 0.18/0.45 thf(fact_312_insert__noteq__member,axiom,
% 0.18/0.45 ! [B: product_prod_a_a,B4: multis599418605od_a_a,C2: product_prod_a_a,C3: multis599418605od_a_a] :
% 0.18/0.45 ( ( ( add_ms1725977093od_a_a @ B @ B4 )
% 0.18/0.45 = ( add_ms1725977093od_a_a @ C2 @ C3 ) )
% 0.18/0.45 => ( ( B != C2 )
% 0.18/0.45 => ( member449909584od_a_a @ C2 @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % insert_noteq_member
% 0.18/0.45 thf(fact_313_union__single__eq__member,axiom,
% 0.18/0.45 ! [X: a,M: multiset_a,N: multiset_a] :
% 0.18/0.45 ( ( ( add_mset_a @ X @ M )
% 0.18/0.45 = N )
% 0.18/0.45 => ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % union_single_eq_member
% 0.18/0.45 thf(fact_314_union__single__eq__member,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
% 0.18/0.45 ( ( ( add_ms1725977093od_a_a @ X @ M )
% 0.18/0.45 = N )
% 0.18/0.45 => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % union_single_eq_member
% 0.18/0.45 thf(fact_315_union__iff,axiom,
% 0.18/0.45 ! [A: a,A5: multiset_a,B4: multiset_a] :
% 0.18/0.45 ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A5 @ B4 ) ) )
% 0.18/0.45 = ( ( member_a @ A @ ( set_mset_a @ A5 ) )
% 0.18/0.45 | ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % union_iff
% 0.18/0.45 thf(fact_316_union__iff,axiom,
% 0.18/0.45 ! [A: product_prod_a_a,A5: multis599418605od_a_a,B4: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ A5 @ B4 ) ) )
% 0.18/0.45 = ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A5 ) )
% 0.18/0.45 | ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % union_iff
% 0.18/0.45 thf(fact_317_mult__implies__one__step,axiom,
% 0.18/0.45 ! [R2: set_Product_prod_a_a,M: multiset_a,N: multiset_a] :
% 0.18/0.45 ( ( trans_a @ R2 )
% 0.18/0.45 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ M @ N ) @ ( mult_a @ R2 ) )
% 0.18/0.45 => ? [I: multiset_a,J: multiset_a] :
% 0.18/0.45 ( ( N
% 0.18/0.45 = ( plus_plus_multiset_a @ I @ J ) )
% 0.18/0.45 & ? [K: multiset_a] :
% 0.18/0.45 ( ( M
% 0.18/0.45 = ( plus_plus_multiset_a @ I @ K ) )
% 0.18/0.45 & ( J != zero_zero_multiset_a )
% 0.18/0.45 & ! [X2: a] :
% 0.18/0.45 ( ( member_a @ X2 @ ( set_mset_a @ K ) )
% 0.18/0.45 => ? [Xa2: a] :
% 0.18/0.45 ( ( member_a @ Xa2 @ ( set_mset_a @ J ) )
% 0.18/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa2 ) @ R2 ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % mult_implies_one_step
% 0.18/0.45 thf(fact_318_one__step__implies__mult,axiom,
% 0.18/0.45 ! [J2: multiset_a,K2: multiset_a,R2: set_Product_prod_a_a,I2: multiset_a] :
% 0.18/0.45 ( ( J2 != zero_zero_multiset_a )
% 0.18/0.45 => ( ! [X3: a] :
% 0.18/0.45 ( ( member_a @ X3 @ ( set_mset_a @ K2 ) )
% 0.18/0.45 => ? [Xa: a] :
% 0.18/0.45 ( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
% 0.18/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Xa ) @ R2 ) ) )
% 0.18/0.45 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I2 @ K2 ) @ ( plus_plus_multiset_a @ I2 @ J2 ) ) @ ( mult_a @ R2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % one_step_implies_mult
% 0.18/0.45 thf(fact_319_multiset__nonemptyE,axiom,
% 0.18/0.45 ! [A5: multiset_a] :
% 0.18/0.45 ( ( A5 != zero_zero_multiset_a )
% 0.18/0.45 => ~ ! [X3: a] :
% 0.18/0.45 ~ ( member_a @ X3 @ ( set_mset_a @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multiset_nonemptyE
% 0.18/0.45 thf(fact_320_multiset__nonemptyE,axiom,
% 0.18/0.45 ! [A5: multis599418605od_a_a] :
% 0.18/0.45 ( ( A5 != zero_z160849828od_a_a )
% 0.18/0.45 => ~ ! [X3: product_prod_a_a] :
% 0.18/0.45 ~ ( member449909584od_a_a @ X3 @ ( set_ms1818337124od_a_a @ A5 ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multiset_nonemptyE
% 0.18/0.45 thf(fact_321_multi__member__last,axiom,
% 0.18/0.45 ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_member_last
% 0.18/0.45 thf(fact_322_multi__member__last,axiom,
% 0.18/0.45 ! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_member_last
% 0.18/0.45 thf(fact_323_multi__member__skip,axiom,
% 0.18/0.45 ! [X: a,XS: multiset_a,Y: a] :
% 0.18/0.45 ( ( member_a @ X @ ( set_mset_a @ XS ) )
% 0.18/0.45 => ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_member_skip
% 0.18/0.45 thf(fact_324_multi__member__skip,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,XS: multis599418605od_a_a,Y: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ XS ) )
% 0.18/0.45 => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) @ XS ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_member_skip
% 0.18/0.45 thf(fact_325_multi__member__this,axiom,
% 0.18/0.45 ! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_member_this
% 0.18/0.45 thf(fact_326_multi__member__this,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,XS: multis599418605od_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) @ XS ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_member_this
% 0.18/0.45 thf(fact_327_in__mset__fold__plus__iff,axiom,
% 0.18/0.45 ! [X: a,M: multiset_a,NN: multiset_multiset_a] :
% 0.18/0.45 ( ( member_a @ X @ ( set_mset_a @ ( fold_m382157835iset_a @ plus_plus_multiset_a @ M @ NN ) ) )
% 0.18/0.45 = ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.18/0.45 | ? [N2: multiset_a] :
% 0.18/0.45 ( ( member_multiset_a @ N2 @ ( set_mset_multiset_a @ NN ) )
% 0.18/0.45 & ( member_a @ X @ ( set_mset_a @ N2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_mset_fold_plus_iff
% 0.18/0.45 thf(fact_328_in__mset__fold__plus__iff,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,M: multis599418605od_a_a,NN: multis1735859763od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( fold_m773816523od_a_a @ plus_p404985124od_a_a @ M @ NN ) ) )
% 0.18/0.45 = ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 0.18/0.45 | ? [N2: multis599418605od_a_a] :
% 0.18/0.45 ( ( member1755255446od_a_a @ N2 @ ( set_ms1869569194od_a_a @ NN ) )
% 0.18/0.45 & ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ N2 ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_mset_fold_plus_iff
% 0.18/0.45 thf(fact_329_insert__DiffM2,axiom,
% 0.18/0.45 ! [X: a,M: multiset_a] :
% 0.18/0.45 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.18/0.45 => ( ( plus_plus_multiset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 0.18/0.45 = M ) ) ).
% 0.18/0.45
% 0.18/0.45 % insert_DiffM2
% 0.18/0.45 thf(fact_330_insert__DiffM2,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 0.18/0.45 => ( ( plus_p404985124od_a_a @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) )
% 0.18/0.45 = M ) ) ).
% 0.18/0.45
% 0.18/0.45 % insert_DiffM2
% 0.18/0.45 thf(fact_331_diff__add__mset__swap,axiom,
% 0.18/0.45 ! [B: a,A5: multiset_a,M: multiset_a] :
% 0.18/0.45 ( ~ ( member_a @ B @ ( set_mset_a @ A5 ) )
% 0.18/0.45 => ( ( minus_1649712171iset_a @ ( add_mset_a @ B @ M ) @ A5 )
% 0.18/0.45 = ( add_mset_a @ B @ ( minus_1649712171iset_a @ M @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_add_mset_swap
% 0.18/0.45 thf(fact_332_diff__add__mset__swap,axiom,
% 0.18/0.45 ! [B: product_prod_a_a,A5: multis599418605od_a_a,M: multis599418605od_a_a] :
% 0.18/0.45 ( ~ ( member449909584od_a_a @ B @ ( set_ms1818337124od_a_a @ A5 ) )
% 0.18/0.45 => ( ( minus_1257885652od_a_a @ ( add_ms1725977093od_a_a @ B @ M ) @ A5 )
% 0.18/0.45 = ( add_ms1725977093od_a_a @ B @ ( minus_1257885652od_a_a @ M @ A5 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_add_mset_swap
% 0.18/0.45 thf(fact_333_insert__DiffM,axiom,
% 0.18/0.45 ! [X: a,M: multiset_a] :
% 0.18/0.45 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.18/0.45 => ( ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
% 0.18/0.45 = M ) ) ).
% 0.18/0.45
% 0.18/0.45 % insert_DiffM
% 0.18/0.45 thf(fact_334_insert__DiffM,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 0.18/0.45 => ( ( add_ms1725977093od_a_a @ X @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) )
% 0.18/0.45 = M ) ) ).
% 0.18/0.45
% 0.18/0.45 % insert_DiffM
% 0.18/0.45 thf(fact_335_diff__union__swap2,axiom,
% 0.18/0.45 ! [Y: a,M: multiset_a,X: a] :
% 0.18/0.45 ( ( member_a @ Y @ ( set_mset_a @ M ) )
% 0.18/0.45 => ( ( minus_1649712171iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 0.18/0.45 = ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_union_swap2
% 0.18/0.45 thf(fact_336_diff__union__swap2,axiom,
% 0.18/0.45 ! [Y: product_prod_a_a,M: multis599418605od_a_a,X: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ Y @ ( set_ms1818337124od_a_a @ M ) )
% 0.18/0.45 => ( ( minus_1257885652od_a_a @ ( add_ms1725977093od_a_a @ X @ M ) @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) )
% 0.18/0.45 = ( add_ms1725977093od_a_a @ X @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_union_swap2
% 0.18/0.45 thf(fact_337_in__diffD,axiom,
% 0.18/0.45 ! [A: a,M: multiset_a,N: multiset_a] :
% 0.18/0.45 ( ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ N ) ) )
% 0.18/0.45 => ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_diffD
% 0.18/0.45 thf(fact_338_in__diffD,axiom,
% 0.18/0.45 ! [A: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( minus_1257885652od_a_a @ M @ N ) ) )
% 0.18/0.45 => ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ M ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_diffD
% 0.18/0.45 thf(fact_339_diff__single__trivial,axiom,
% 0.18/0.45 ! [X: a,M: multiset_a] :
% 0.18/0.45 ( ~ ( member_a @ X @ ( set_mset_a @ M ) )
% 0.18/0.45 => ( ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 0.18/0.45 = M ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_single_trivial
% 0.18/0.45 thf(fact_340_diff__single__trivial,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 0.18/0.45 ( ~ ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 0.18/0.45 => ( ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) )
% 0.18/0.45 = M ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_single_trivial
% 0.18/0.45 thf(fact_341_diff__single__eq__union,axiom,
% 0.18/0.45 ! [X: a,M: multiset_a,N: multiset_a] :
% 0.18/0.45 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 0.18/0.45 => ( ( ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 0.18/0.45 = N )
% 0.18/0.45 = ( M
% 0.18/0.45 = ( add_mset_a @ X @ N ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_single_eq_union
% 0.18/0.45 thf(fact_342_diff__single__eq__union,axiom,
% 0.18/0.45 ! [X: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 0.18/0.45 => ( ( ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) )
% 0.18/0.45 = N )
% 0.18/0.45 = ( M
% 0.18/0.45 = ( add_ms1725977093od_a_a @ X @ N ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_single_eq_union
% 0.18/0.45 thf(fact_343_multi__drop__mem__not__eq,axiom,
% 0.18/0.45 ! [C2: a,B4: multiset_a] :
% 0.18/0.45 ( ( member_a @ C2 @ ( set_mset_a @ B4 ) )
% 0.18/0.45 => ( ( minus_1649712171iset_a @ B4 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) )
% 0.18/0.45 != B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_drop_mem_not_eq
% 0.18/0.45 thf(fact_344_multi__drop__mem__not__eq,axiom,
% 0.18/0.45 ! [C2: product_prod_a_a,B4: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ C2 @ ( set_ms1818337124od_a_a @ B4 ) )
% 0.18/0.45 => ( ( minus_1257885652od_a_a @ B4 @ ( add_ms1725977093od_a_a @ C2 @ zero_z160849828od_a_a ) )
% 0.18/0.45 != B4 ) ) ).
% 0.18/0.45
% 0.18/0.45 % multi_drop_mem_not_eq
% 0.18/0.45 thf(fact_345_add__mset__remove__trivial__If,axiom,
% 0.18/0.45 ! [A: a,N: multiset_a] :
% 0.18/0.45 ( ( ( member_a @ A @ ( set_mset_a @ N ) )
% 0.18/0.45 => ( ( add_mset_a @ A @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
% 0.18/0.45 = N ) )
% 0.18/0.45 & ( ~ ( member_a @ A @ ( set_mset_a @ N ) )
% 0.18/0.45 => ( ( add_mset_a @ A @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
% 0.18/0.45 = ( add_mset_a @ A @ N ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % add_mset_remove_trivial_If
% 0.18/0.45 thf(fact_346_add__mset__remove__trivial__If,axiom,
% 0.18/0.45 ! [A: product_prod_a_a,N: multis599418605od_a_a] :
% 0.18/0.45 ( ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ N ) )
% 0.18/0.45 => ( ( add_ms1725977093od_a_a @ A @ ( minus_1257885652od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) )
% 0.18/0.45 = N ) )
% 0.18/0.45 & ( ~ ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ N ) )
% 0.18/0.45 => ( ( add_ms1725977093od_a_a @ A @ ( minus_1257885652od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) )
% 0.18/0.45 = ( add_ms1725977093od_a_a @ A @ N ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % add_mset_remove_trivial_If
% 0.18/0.45 thf(fact_347_add__mset__remove__trivial__eq,axiom,
% 0.18/0.45 ! [N: multiset_a,A: a] :
% 0.18/0.45 ( ( N
% 0.18/0.45 = ( add_mset_a @ A @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
% 0.18/0.45 = ( member_a @ A @ ( set_mset_a @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % add_mset_remove_trivial_eq
% 0.18/0.45 thf(fact_348_add__mset__remove__trivial__eq,axiom,
% 0.18/0.45 ! [N: multis599418605od_a_a,A: product_prod_a_a] :
% 0.18/0.45 ( ( N
% 0.18/0.45 = ( add_ms1725977093od_a_a @ A @ ( minus_1257885652od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) ) )
% 0.18/0.45 = ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % add_mset_remove_trivial_eq
% 0.18/0.45 thf(fact_349_multiset__add__sub__el__shuffle,axiom,
% 0.18/0.45 ! [C2: a,B4: multiset_a,B: a] :
% 0.18/0.45 ( ( member_a @ C2 @ ( set_mset_a @ B4 ) )
% 0.18/0.45 => ( ( B != C2 )
% 0.18/0.45 => ( ( add_mset_a @ B @ ( minus_1649712171iset_a @ B4 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) )
% 0.18/0.45 = ( minus_1649712171iset_a @ ( add_mset_a @ B @ B4 ) @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multiset_add_sub_el_shuffle
% 0.18/0.45 thf(fact_350_multiset__add__sub__el__shuffle,axiom,
% 0.18/0.45 ! [C2: product_prod_a_a,B4: multis599418605od_a_a,B: product_prod_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ C2 @ ( set_ms1818337124od_a_a @ B4 ) )
% 0.18/0.45 => ( ( B != C2 )
% 0.18/0.45 => ( ( add_ms1725977093od_a_a @ B @ ( minus_1257885652od_a_a @ B4 @ ( add_ms1725977093od_a_a @ C2 @ zero_z160849828od_a_a ) ) )
% 0.18/0.45 = ( minus_1257885652od_a_a @ ( add_ms1725977093od_a_a @ B @ B4 ) @ ( add_ms1725977093od_a_a @ C2 @ zero_z160849828od_a_a ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % multiset_add_sub_el_shuffle
% 0.18/0.45 thf(fact_351_more__than__one__mset__mset__diff,axiom,
% 0.18/0.45 ! [A: a,M: multiset_a] :
% 0.18/0.45 ( ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
% 0.18/0.45 => ( ( set_mset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
% 0.18/0.45 = ( set_mset_a @ M ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % more_than_one_mset_mset_diff
% 0.18/0.45 thf(fact_352_more__than__one__mset__mset__diff,axiom,
% 0.18/0.45 ! [A: product_prod_a_a,M: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) ) )
% 0.18/0.45 => ( ( set_ms1818337124od_a_a @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) )
% 0.18/0.45 = ( set_ms1818337124od_a_a @ M ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % more_than_one_mset_mset_diff
% 0.18/0.45 thf(fact_353_diff__union__single__conv,axiom,
% 0.18/0.45 ! [A: a,J2: multiset_a,I2: multiset_a] :
% 0.18/0.45 ( ( member_a @ A @ ( set_mset_a @ J2 ) )
% 0.18/0.45 => ( ( minus_1649712171iset_a @ ( plus_plus_multiset_a @ I2 @ J2 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 0.18/0.45 = ( plus_plus_multiset_a @ I2 @ ( minus_1649712171iset_a @ J2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_union_single_conv
% 0.18/0.45 thf(fact_354_diff__union__single__conv,axiom,
% 0.18/0.45 ! [A: product_prod_a_a,J2: multis599418605od_a_a,I2: multis599418605od_a_a] :
% 0.18/0.45 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ J2 ) )
% 0.18/0.45 => ( ( minus_1257885652od_a_a @ ( plus_p404985124od_a_a @ I2 @ J2 ) @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) )
% 0.18/0.45 = ( plus_p404985124od_a_a @ I2 @ ( minus_1257885652od_a_a @ J2 @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % diff_union_single_conv
% 0.18/0.45
% 0.18/0.45 % Conjectures (4)
% 0.18/0.45 thf(conj_0,hypothesis,
% 0.18/0.45 member_a @ x @ carrier ).
% 0.18/0.45
% 0.18/0.45 thf(conj_1,hypothesis,
% 0.18/0.45 member_a @ y @ carrier ).
% 0.18/0.45
% 0.18/0.45 thf(conj_2,hypothesis,
% 0.18/0.45 x != y ).
% 0.18/0.45
% 0.18/0.45 thf(conj_3,conjecture,
% 0.18/0.45 ( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
% 0.18/0.45 | ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation ) ) ).
% 0.18/0.45
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 ------------------------
% 2.97/1.52 ---- Embedded in THF ---
% 2.97/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.97/1.52 %%% Generated on Tue Feb 24 08:28:13 EST 2026
% 2.97/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.97/1.52 %%% Logic specification used:
% 2.97/1.52 %%% thf(spec, logic, $$dhol).
% 2.97/1.52
% 2.97/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.fZ0xgqz36i/DTF2DTF28173.p
% See solution above
% 2.97/1.52 ------------------------
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% 2.97/1.59 refl_o198693862od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.97/1.59 refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Orefl__on_001tf__a,type,
% 2.97/1.59 refl_on_a: set_a > set_Product_prod_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Ototal__on_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.97/1.59 total_1024184433od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Ototal__on_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.97/1.59 total_on_multiset_a: set_multiset_a > set_Pr158363655iset_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.97/1.59 total_1170533099od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 2.97/1.59 total_313747627iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.97/1.59 total_1340260971od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.97/1.59 total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Ototal__on_001tf__a,type,
% 2.97/1.59 total_on_a: set_a > set_Product_prod_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Otrans_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.97/1.59 trans_1520871884od_a_a: set_Pr1821581383od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Otrans_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.97/1.59 trans_multiset_a: set_Pr158363655iset_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Otrans_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.97/1.59 trans_972827014od_a_a: set_Pr1948701895od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Relation_Otrans_001tf__a,type,
% 2.97/1.59 trans_a: set_Product_prod_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_OCollect_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.97/1.59 collec1948052440od_a_a: ( multis599418605od_a_a > $o ) > set_mu1281098573od_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_OCollect_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.97/1.59 collect_multiset_a: ( multiset_a > $o ) > set_multiset_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.97/1.59 collec8264402od_a_a: ( produc2061001575od_a_a > $o ) > set_Pr1821581383od_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 2.97/1.59 collec1104713362iset_a: ( produc1127127335iset_a > $o ) > set_Pr158363655iset_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.97/1.59 collec1635618130od_a_a: ( produc1572603623od_a_a > $o ) > set_Pr1948701895od_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.97/1.59 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.97/1.59 collect_a: ( a > $o ) > set_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.97/1.59 image_794773445od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
% 2.97/1.59 image_340790172_a_a_a: ( product_prod_a_a > a ) > set_Product_prod_a_a > set_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_Oimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.97/1.59 image_1725938350od_a_a: ( a > product_prod_a_a ) > set_a > set_Product_prod_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
% 2.97/1.59 image_a_a: ( a > a ) > set_a > set_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Wellfounded_Olex__prod_001tf__a_001tf__a,type,
% 2.97/1.59 lex_prod_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_Wfrec_Osame__fst_001tf__a_001tf__a,type,
% 2.97/1.59 same_fst_a_a: ( a > $o ) > ( a > set_Product_prod_a_a ) > set_Pr1948701895od_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.97/1.59 member1755255446od_a_a: multis599418605od_a_a > set_mu1281098573od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
% 2.97/1.59 member_multiset_a: multiset_a > set_multiset_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.97/1.59 member1349055120od_a_a: produc2061001575od_a_a > set_Pr1821581383od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
% 2.97/1.59 member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
% 2.97/1.59 member2007374608od_a_a: produc562496743od_a_a > set_Pr1930443975od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
% 2.97/1.59 member567864080iset_a: produc558600423iset_a > set_Pr146455751iset_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.97/1.59 member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.97/1.59 member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.97/1.59 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_c_member_001tf__a,type,
% 2.97/1.59 member_a: a > set_a > $o ).
% 2.97/1.59
% 2.97/1.59 thf(sy_v_carrier,type,
% 2.97/1.59 carrier: set_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_v_relation,type,
% 2.97/1.59 relation: set_Product_prod_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_v_x,type,
% 2.97/1.59 x: a ).
% 2.97/1.59
% 2.97/1.59 thf(sy_v_y,type,
% 2.97/1.59 y: a ).
% 2.97/1.59
% 2.97/1.59 thf(fact_0_indiff__trans,axiom,
% 2.97/1.59 ! [X: a,Y: a,Z: a] :
% 2.97/1.59 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.97/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.97/1.59 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.97/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 2.97/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_1_strict__trans,axiom,
% 2.97/1.59 ! [X: a,Y: a,Z: a] :
% 2.97/1.59 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.97/1.59 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.97/1.59 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.97/1.59 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 2.97/1.59 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_2_rational__preference__axioms,axiom,
% 2.97/1.59 prefer719835986ence_a @ carrier @ relation ).
% 2.97/1.59
% 2.97/1.59 thf(fact_3_compl,axiom,
% 2.97/1.59 ! [X2: a] :
% 2.97/1.59 ( ( member_a @ X2 @ carrier )
% 2.97/1.59 => ! [Xa: a] :
% 2.97/1.59 ( ( member_a @ Xa @ carrier )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ relation )
% 2.97/1.59 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_4_not__outside,axiom,
% 2.97/1.59 ! [X: a,Y: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.97/1.59 => ( member_a @ X @ carrier ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_5_strict__not__refl__weak,axiom,
% 2.97/1.59 ! [X: a,Y: a] :
% 2.97/1.59 ( ( ( member_a @ X @ carrier )
% 2.97/1.59 & ( member_a @ Y @ carrier ) )
% 2.97/1.59 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.97/1.59 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.97/1.59 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_6_preference__axioms,axiom,
% 2.97/1.59 prefer1292084992ence_a @ carrier @ relation ).
% 2.97/1.59
% 2.97/1.59 thf(fact_7_trans__refl,axiom,
% 2.97/1.59 order_preorder_on_a @ carrier @ relation ).
% 2.97/1.59
% 2.97/1.59 thf(fact_8_total,axiom,
% 2.97/1.59 total_on_a @ carrier @ relation ).
% 2.97/1.59
% 2.97/1.59 thf(fact_9_reflexivity,axiom,
% 2.97/1.59 refl_on_a @ carrier @ relation ).
% 2.97/1.59
% 2.97/1.59 thf(fact_10_prod_Oinject,axiom,
% 2.97/1.59 ! [X1: product_prod_a_a,X22: product_prod_a_a,Y1: product_prod_a_a,Y2: product_prod_a_a] :
% 2.97/1.59 ( ( ( produc1474507607od_a_a @ X1 @ X22 )
% 2.97/1.59 = ( produc1474507607od_a_a @ Y1 @ Y2 ) )
% 2.97/1.59 = ( ( X1 = Y1 )
% 2.97/1.59 & ( X22 = Y2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_11_prod_Oinject,axiom,
% 2.97/1.59 ! [X1: multiset_a,X22: multiset_a,Y1: multiset_a,Y2: multiset_a] :
% 2.97/1.59 ( ( ( produc2037245207iset_a @ X1 @ X22 )
% 2.97/1.59 = ( produc2037245207iset_a @ Y1 @ Y2 ) )
% 2.97/1.59 = ( ( X1 = Y1 )
% 2.97/1.59 & ( X22 = Y2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_12_prod_Oinject,axiom,
% 2.97/1.59 ! [X1: multis599418605od_a_a,X22: multis599418605od_a_a,Y1: multis599418605od_a_a,Y2: multis599418605od_a_a] :
% 2.97/1.59 ( ( ( produc1859144151od_a_a @ X1 @ X22 )
% 2.97/1.59 = ( produc1859144151od_a_a @ Y1 @ Y2 ) )
% 2.97/1.59 = ( ( X1 = Y1 )
% 2.97/1.59 & ( X22 = Y2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_13_prod_Oinject,axiom,
% 2.97/1.59 ! [X1: a,X22: a,Y1: a,Y2: a] :
% 2.97/1.59 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 2.97/1.59 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 2.97/1.59 = ( ( X1 = Y1 )
% 2.97/1.59 & ( X22 = Y2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_14_old_Oprod_Oinject,axiom,
% 2.97/1.59 ! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
% 2.97/1.59 ( ( ( produc1474507607od_a_a @ A @ B )
% 2.97/1.59 = ( produc1474507607od_a_a @ A2 @ B2 ) )
% 2.97/1.59 = ( ( A = A2 )
% 2.97/1.59 & ( B = B2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_15_old_Oprod_Oinject,axiom,
% 2.97/1.59 ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
% 2.97/1.59 ( ( ( produc2037245207iset_a @ A @ B )
% 2.97/1.59 = ( produc2037245207iset_a @ A2 @ B2 ) )
% 2.97/1.59 = ( ( A = A2 )
% 2.97/1.59 & ( B = B2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_16_old_Oprod_Oinject,axiom,
% 2.97/1.59 ! [A: multis599418605od_a_a,B: multis599418605od_a_a,A2: multis599418605od_a_a,B2: multis599418605od_a_a] :
% 2.97/1.59 ( ( ( produc1859144151od_a_a @ A @ B )
% 2.97/1.59 = ( produc1859144151od_a_a @ A2 @ B2 ) )
% 2.97/1.59 = ( ( A = A2 )
% 2.97/1.59 & ( B = B2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_17_old_Oprod_Oinject,axiom,
% 2.97/1.59 ! [A: a,B: a,A2: a,B2: a] :
% 2.97/1.59 ( ( ( product_Pair_a_a @ A @ B )
% 2.97/1.59 = ( product_Pair_a_a @ A2 @ B2 ) )
% 2.97/1.59 = ( ( A = A2 )
% 2.97/1.59 & ( B = B2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_18_transitivity,axiom,
% 2.97/1.59 trans_a @ relation ).
% 2.97/1.59
% 2.97/1.59 thf(fact_19_surj__pair,axiom,
% 2.97/1.59 ! [P: produc1572603623od_a_a] :
% 2.97/1.59 ? [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 2.97/1.59 ( P
% 2.97/1.59 = ( produc1474507607od_a_a @ X3 @ Y3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_20_surj__pair,axiom,
% 2.97/1.59 ! [P: produc1127127335iset_a] :
% 2.97/1.59 ? [X3: multiset_a,Y3: multiset_a] :
% 2.97/1.59 ( P
% 2.97/1.59 = ( produc2037245207iset_a @ X3 @ Y3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_21_surj__pair,axiom,
% 2.97/1.59 ! [P: produc2061001575od_a_a] :
% 2.97/1.59 ? [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
% 2.97/1.59 ( P
% 2.97/1.59 = ( produc1859144151od_a_a @ X3 @ Y3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_22_surj__pair,axiom,
% 2.97/1.59 ! [P: product_prod_a_a] :
% 2.97/1.59 ? [X3: a,Y3: a] :
% 2.97/1.59 ( P
% 2.97/1.59 = ( product_Pair_a_a @ X3 @ Y3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_23_prod__cases,axiom,
% 2.97/1.59 ! [P2: produc1572603623od_a_a > $o,P: produc1572603623od_a_a] :
% 2.97/1.59 ( ! [A3: product_prod_a_a,B3: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B3 ) )
% 2.97/1.59 => ( P2 @ P ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_24_prod__cases,axiom,
% 2.97/1.59 ! [P2: produc1127127335iset_a > $o,P: produc1127127335iset_a] :
% 2.97/1.59 ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
% 2.97/1.59 => ( P2 @ P ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_25_prod__cases,axiom,
% 2.97/1.59 ! [P2: produc2061001575od_a_a > $o,P: produc2061001575od_a_a] :
% 2.97/1.59 ( ! [A3: multis599418605od_a_a,B3: multis599418605od_a_a] : ( P2 @ ( produc1859144151od_a_a @ A3 @ B3 ) )
% 2.97/1.59 => ( P2 @ P ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_26_prod__cases,axiom,
% 2.97/1.59 ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
% 2.97/1.59 ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
% 2.97/1.59 => ( P2 @ P ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_27_Pair__inject,axiom,
% 2.97/1.59 ! [A: product_prod_a_a,B: product_prod_a_a,A2: product_prod_a_a,B2: product_prod_a_a] :
% 2.97/1.59 ( ( ( produc1474507607od_a_a @ A @ B )
% 2.97/1.59 = ( produc1474507607od_a_a @ A2 @ B2 ) )
% 2.97/1.59 => ~ ( ( A = A2 )
% 2.97/1.59 => ( B != B2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_28_Pair__inject,axiom,
% 2.97/1.59 ! [A: multiset_a,B: multiset_a,A2: multiset_a,B2: multiset_a] :
% 2.97/1.59 ( ( ( produc2037245207iset_a @ A @ B )
% 2.97/1.59 = ( produc2037245207iset_a @ A2 @ B2 ) )
% 2.97/1.59 => ~ ( ( A = A2 )
% 2.97/1.59 => ( B != B2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_29_Pair__inject,axiom,
% 2.97/1.59 ! [A: multis599418605od_a_a,B: multis599418605od_a_a,A2: multis599418605od_a_a,B2: multis599418605od_a_a] :
% 2.97/1.59 ( ( ( produc1859144151od_a_a @ A @ B )
% 2.97/1.59 = ( produc1859144151od_a_a @ A2 @ B2 ) )
% 2.97/1.59 => ~ ( ( A = A2 )
% 2.97/1.59 => ( B != B2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_30_Pair__inject,axiom,
% 2.97/1.59 ! [A: a,B: a,A2: a,B2: a] :
% 2.97/1.59 ( ( ( product_Pair_a_a @ A @ B )
% 2.97/1.59 = ( product_Pair_a_a @ A2 @ B2 ) )
% 2.97/1.59 => ~ ( ( A = A2 )
% 2.97/1.59 => ( B != B2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_31_prod__cases3,axiom,
% 2.97/1.59 ! [Y: produc1572603623od_a_a] :
% 2.97/1.59 ~ ! [A3: product_prod_a_a,B3: a,C: a] :
% 2.97/1.59 ( Y
% 2.97/1.59 != ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B3 @ C ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_32_rational__preference_Oaxioms_I1_J,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_33_rational__preference_Oaxioms_I1_J,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.97/1.59 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_34_preference_Otrans__refl,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( order_359341630od_a_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_35_preference_Otrans__refl,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.97/1.59 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_36_preference_Oreflexivity,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( refl_o1298442278od_a_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_37_preference_Oreflexivity,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.97/1.59 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( refl_on_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_38_preference_Otransitivity,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( trans_972827014od_a_a @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_39_preference_Otransitivity,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.97/1.59 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( trans_a @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_40_rational__preference_Ototal,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( total_1490170027od_a_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_41_rational__preference_Ototal,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.97/1.59 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( total_on_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_42_preference_Ointro,axiom,
% 2.97/1.59 ! [Relation: set_Pr1295299783od_a_a,Carrier: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 2.97/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member2057358096od_a_a @ X3 @ Carrier ) )
% 2.97/1.59 => ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 2.97/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member2057358096od_a_a @ Y3 @ Carrier ) )
% 2.97/1.59 => ( ( order_491428350od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1409946857od_a_a @ Carrier @ Relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_43_preference_Ointro,axiom,
% 2.97/1.59 ! [Relation: set_Pr146455751iset_a,Carrier: set_Pr158363655iset_a] :
% 2.97/1.59 ( ! [X3: produc1127127335iset_a,Y3: produc1127127335iset_a] :
% 2.97/1.59 ( ( member567864080iset_a @ ( produc1898392407iset_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member340150864iset_a @ X3 @ Carrier ) )
% 2.97/1.59 => ( ! [X3: produc1127127335iset_a,Y3: produc1127127335iset_a] :
% 2.97/1.59 ( ( member567864080iset_a @ ( produc1898392407iset_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member340150864iset_a @ Y3 @ Carrier ) )
% 2.97/1.59 => ( ( order_761691966iset_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1608839465iset_a @ Carrier @ Relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_44_preference_Ointro,axiom,
% 2.97/1.59 ! [Relation: set_Pr1930443975od_a_a,Carrier: set_Pr1821581383od_a_a] :
% 2.97/1.59 ( ! [X3: produc2061001575od_a_a,Y3: produc2061001575od_a_a] :
% 2.97/1.59 ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member1349055120od_a_a @ X3 @ Carrier ) )
% 2.97/1.59 => ( ! [X3: produc2061001575od_a_a,Y3: produc2061001575od_a_a] :
% 2.97/1.59 ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member1349055120od_a_a @ Y3 @ Carrier ) )
% 2.97/1.59 => ( ( order_1210379134od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer273553257od_a_a @ Carrier @ Relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_45_preference_Ointro,axiom,
% 2.97/1.59 ! [Relation: set_Pr158363655iset_a,Carrier: set_multiset_a] :
% 2.97/1.59 ( ! [X3: multiset_a,Y3: multiset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member_multiset_a @ X3 @ Carrier ) )
% 2.97/1.59 => ( ! [X3: multiset_a,Y3: multiset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member_multiset_a @ Y3 @ Carrier ) )
% 2.97/1.59 => ( ( order_750350043iset_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1571048262iset_a @ Carrier @ Relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_46_preference_Ointro,axiom,
% 2.97/1.59 ! [Relation: set_Pr1821581383od_a_a,Carrier: set_mu1281098573od_a_a] :
% 2.97/1.59 ( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member1755255446od_a_a @ X3 @ Carrier ) )
% 2.97/1.59 => ( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member1755255446od_a_a @ Y3 @ Carrier ) )
% 2.97/1.59 => ( ( order_1040740996od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1340910575od_a_a @ Carrier @ Relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_47_preference_Ointro,axiom,
% 2.97/1.59 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member449909584od_a_a @ X3 @ Carrier ) )
% 2.97/1.59 => ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 2.97/1.59 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_48_preference_Ointro,axiom,
% 2.97/1.59 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 2.97/1.59 ( ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member_a @ X3 @ Carrier ) )
% 2.97/1.59 => ( ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ Relation )
% 2.97/1.59 => ( member_a @ Y3 @ Carrier ) )
% 2.97/1.59 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_49_preference__def,axiom,
% 2.97/1.59 ( prefer1409946857od_a_a
% 2.97/1.59 = ( ^ [Carrier2: set_Pr1948701895od_a_a,Relation2: set_Pr1295299783od_a_a] :
% 2.97/1.59 ( ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
% 2.97/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member2057358096od_a_a @ X4 @ Carrier2 ) )
% 2.97/1.59 & ! [X4: produc1572603623od_a_a,Y4: produc1572603623od_a_a] :
% 2.97/1.59 ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member2057358096od_a_a @ Y4 @ Carrier2 ) )
% 2.97/1.59 & ( order_491428350od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_50_preference__def,axiom,
% 2.97/1.59 ( prefer1608839465iset_a
% 2.97/1.59 = ( ^ [Carrier2: set_Pr158363655iset_a,Relation2: set_Pr146455751iset_a] :
% 2.97/1.59 ( ! [X4: produc1127127335iset_a,Y4: produc1127127335iset_a] :
% 2.97/1.59 ( ( member567864080iset_a @ ( produc1898392407iset_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member340150864iset_a @ X4 @ Carrier2 ) )
% 2.97/1.59 & ! [X4: produc1127127335iset_a,Y4: produc1127127335iset_a] :
% 2.97/1.59 ( ( member567864080iset_a @ ( produc1898392407iset_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member340150864iset_a @ Y4 @ Carrier2 ) )
% 2.97/1.59 & ( order_761691966iset_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_51_preference__def,axiom,
% 2.97/1.59 ( prefer273553257od_a_a
% 2.97/1.59 = ( ^ [Carrier2: set_Pr1821581383od_a_a,Relation2: set_Pr1930443975od_a_a] :
% 2.97/1.59 ( ! [X4: produc2061001575od_a_a,Y4: produc2061001575od_a_a] :
% 2.97/1.59 ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member1349055120od_a_a @ X4 @ Carrier2 ) )
% 2.97/1.59 & ! [X4: produc2061001575od_a_a,Y4: produc2061001575od_a_a] :
% 2.97/1.59 ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member1349055120od_a_a @ Y4 @ Carrier2 ) )
% 2.97/1.59 & ( order_1210379134od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_52_preference__def,axiom,
% 2.97/1.59 ( prefer1571048262iset_a
% 2.97/1.59 = ( ^ [Carrier2: set_multiset_a,Relation2: set_Pr158363655iset_a] :
% 2.97/1.59 ( ! [X4: multiset_a,Y4: multiset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member_multiset_a @ X4 @ Carrier2 ) )
% 2.97/1.59 & ! [X4: multiset_a,Y4: multiset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member_multiset_a @ Y4 @ Carrier2 ) )
% 2.97/1.59 & ( order_750350043iset_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_53_preference__def,axiom,
% 2.97/1.59 ( prefer1340910575od_a_a
% 2.97/1.59 = ( ^ [Carrier2: set_mu1281098573od_a_a,Relation2: set_Pr1821581383od_a_a] :
% 2.97/1.59 ( ! [X4: multis599418605od_a_a,Y4: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member1755255446od_a_a @ X4 @ Carrier2 ) )
% 2.97/1.59 & ! [X4: multis599418605od_a_a,Y4: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member1755255446od_a_a @ Y4 @ Carrier2 ) )
% 2.97/1.59 & ( order_1040740996od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_54_preference__def,axiom,
% 2.97/1.59 ( prefer1338892073od_a_a
% 2.97/1.59 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member449909584od_a_a @ X4 @ Carrier2 ) )
% 2.97/1.59 & ! [X4: product_prod_a_a,Y4: product_prod_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member449909584od_a_a @ Y4 @ Carrier2 ) )
% 2.97/1.59 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_55_preference__def,axiom,
% 2.97/1.59 ( prefer1292084992ence_a
% 2.97/1.59 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [X4: a,Y4: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member_a @ X4 @ Carrier2 ) )
% 2.97/1.59 & ! [X4: a,Y4: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ Relation2 )
% 2.97/1.59 => ( member_a @ Y4 @ Carrier2 ) )
% 2.97/1.59 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_56_preference_Oindiff__trans,axiom,
% 2.97/1.59 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 2.97/1.59 ( ( prefer1571048262iset_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 => ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ Relation )
% 2.97/1.59 & ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ Y ) @ Relation ) )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ Relation )
% 2.97/1.59 & ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_57_preference_Oindiff__trans,axiom,
% 2.97/1.59 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
% 2.97/1.59 ( ( prefer1340910575od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 => ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ Relation )
% 2.97/1.59 & ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ Y ) @ Relation ) )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ Relation )
% 2.97/1.59 & ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_58_preference_Oindiff__trans,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.97/1.59 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 2.97/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 2.97/1.59 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_59_preference_Oindiff__trans,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.97/1.59 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 2.97/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 2.97/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_60_preference_Onot__outside,axiom,
% 2.97/1.59 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.97/1.59 ( ( prefer1409946857od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 => ( member2057358096od_a_a @ X @ Carrier ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_61_preference_Onot__outside,axiom,
% 2.97/1.59 ! [Carrier: set_Pr158363655iset_a,Relation: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
% 2.97/1.59 ( ( prefer1608839465iset_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ Relation )
% 2.97/1.59 => ( member340150864iset_a @ X @ Carrier ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_62_preference_Onot__outside,axiom,
% 2.97/1.59 ! [Carrier: set_Pr1821581383od_a_a,Relation: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
% 2.97/1.59 ( ( prefer273553257od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 => ( member1349055120od_a_a @ X @ Carrier ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_63_preference_Onot__outside,axiom,
% 2.97/1.59 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
% 2.97/1.59 ( ( prefer1571048262iset_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
% 2.97/1.59 => ( member_multiset_a @ X @ Carrier ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_64_preference_Onot__outside,axiom,
% 2.97/1.59 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 2.97/1.59 ( ( prefer1340910575od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 => ( member1755255446od_a_a @ X @ Carrier ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_65_preference_Onot__outside,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.97/1.59 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 => ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_66_preference_Onot__outside,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.97/1.59 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 => ( member_a @ X @ Carrier ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_67_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.97/1.59 ! [Carrier: set_Pr1948701895od_a_a,Relation: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.97/1.59 ( ( prefer1828051771od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member2057358096od_a_a @ X @ Carrier )
% 2.97/1.59 & ( member2057358096od_a_a @ Y @ Carrier ) )
% 2.97/1.59 => ( ( ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 = ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_68_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.97/1.59 ! [Carrier: set_Pr158363655iset_a,Relation: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
% 2.97/1.59 ( ( prefer841781627iset_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member340150864iset_a @ X @ Carrier )
% 2.97/1.59 & ( member340150864iset_a @ Y @ Carrier ) )
% 2.97/1.59 => ( ( ~ ( member567864080iset_a @ ( produc1898392407iset_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 = ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member567864080iset_a @ ( produc1898392407iset_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_69_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.97/1.59 ! [Carrier: set_Pr1821581383od_a_a,Relation: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
% 2.97/1.59 ( ( prefer933983163od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member1349055120od_a_a @ X @ Carrier )
% 2.97/1.59 & ( member1349055120od_a_a @ Y @ Carrier ) )
% 2.97/1.59 => ( ( ~ ( member2007374608od_a_a @ ( produc978545495od_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 = ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member2007374608od_a_a @ ( produc978545495od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_70_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.97/1.59 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
% 2.97/1.59 ( ( prefer1779827352iset_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member_multiset_a @ X @ Carrier )
% 2.97/1.59 & ( member_multiset_a @ Y @ Carrier ) )
% 2.97/1.59 => ( ( ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 = ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_71_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.97/1.59 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 2.97/1.59 ( ( prefer1669415233od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member1755255446od_a_a @ X @ Carrier )
% 2.97/1.59 & ( member1755255446od_a_a @ Y @ Carrier ) )
% 2.97/1.59 => ( ( ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 = ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_72_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.97/1.59 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 2.97/1.59 & ( member449909584od_a_a @ Y @ Carrier ) )
% 2.97/1.59 => ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_73_rational__preference_Ostrict__not__refl__weak,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 2.97/1.59 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member_a @ X @ Carrier )
% 2.97/1.59 & ( member_a @ Y @ Carrier ) )
% 2.97/1.59 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_74_rational__preference_Ostrict__trans,axiom,
% 2.97/1.59 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 2.97/1.59 ( ( prefer1779827352iset_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 => ( ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ Relation )
% 2.97/1.59 & ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ Y ) @ Relation ) )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ Relation )
% 2.97/1.59 & ~ ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_75_rational__preference_Ostrict__trans,axiom,
% 2.97/1.59 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
% 2.97/1.59 ( ( prefer1669415233od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 => ( ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ Relation )
% 2.97/1.59 & ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ Y ) @ Relation ) )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ Relation )
% 2.97/1.59 & ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_76_rational__preference_Ostrict__trans,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.97/1.59 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 2.97/1.59 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation ) )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 2.97/1.59 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_77_rational__preference_Ostrict__trans,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.97/1.59 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 2.97/1.59 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 2.97/1.59 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 2.97/1.59 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 2.97/1.59 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_78_rational__preference_Ocompl,axiom,
% 2.97/1.59 ! [Carrier: set_multiset_a,Relation: set_Pr158363655iset_a] :
% 2.97/1.59 ( ( prefer1779827352iset_a @ Carrier @ Relation )
% 2.97/1.59 => ! [X2: multiset_a] :
% 2.97/1.59 ( ( member_multiset_a @ X2 @ Carrier )
% 2.97/1.59 => ! [Xa: multiset_a] :
% 2.97/1.59 ( ( member_multiset_a @ Xa @ Carrier )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X2 @ Xa ) @ Relation )
% 2.97/1.59 | ( member340150864iset_a @ ( produc2037245207iset_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_79_rational__preference_Ocompl,axiom,
% 2.97/1.59 ! [Carrier: set_mu1281098573od_a_a,Relation: set_Pr1821581383od_a_a] :
% 2.97/1.59 ( ( prefer1669415233od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ! [X2: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1755255446od_a_a @ X2 @ Carrier )
% 2.97/1.59 => ! [Xa: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1755255446od_a_a @ Xa @ Carrier )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X2 @ Xa ) @ Relation )
% 2.97/1.59 | ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_80_rational__preference_Ocompl,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ! [X2: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X2 @ Carrier )
% 2.97/1.59 => ! [Xa: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ Xa @ Carrier )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X2 @ Xa ) @ Relation )
% 2.97/1.59 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_81_rational__preference_Ocompl,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.97/1.59 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.97/1.59 => ! [X2: a] :
% 2.97/1.59 ( ( member_a @ X2 @ Carrier )
% 2.97/1.59 => ! [Xa: a] :
% 2.97/1.59 ( ( member_a @ Xa @ Carrier )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ Relation )
% 2.97/1.59 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_82_old_Oprod_Oinducts,axiom,
% 2.97/1.59 ! [P2: produc1572603623od_a_a > $o,Prod: produc1572603623od_a_a] :
% 2.97/1.59 ( ! [A3: product_prod_a_a,B3: product_prod_a_a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ B3 ) )
% 2.97/1.59 => ( P2 @ Prod ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_83_old_Oprod_Oinducts,axiom,
% 2.97/1.59 ! [P2: produc1127127335iset_a > $o,Prod: produc1127127335iset_a] :
% 2.97/1.59 ( ! [A3: multiset_a,B3: multiset_a] : ( P2 @ ( produc2037245207iset_a @ A3 @ B3 ) )
% 2.97/1.59 => ( P2 @ Prod ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_84_old_Oprod_Oinducts,axiom,
% 2.97/1.59 ! [P2: produc2061001575od_a_a > $o,Prod: produc2061001575od_a_a] :
% 2.97/1.59 ( ! [A3: multis599418605od_a_a,B3: multis599418605od_a_a] : ( P2 @ ( produc1859144151od_a_a @ A3 @ B3 ) )
% 2.97/1.59 => ( P2 @ Prod ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_85_old_Oprod_Oinducts,axiom,
% 2.97/1.59 ! [P2: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 2.97/1.59 ( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
% 2.97/1.59 => ( P2 @ Prod ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_86_old_Oprod_Oexhaust,axiom,
% 2.97/1.59 ! [Y: produc1572603623od_a_a] :
% 2.97/1.59 ~ ! [A3: product_prod_a_a,B3: product_prod_a_a] :
% 2.97/1.59 ( Y
% 2.97/1.59 != ( produc1474507607od_a_a @ A3 @ B3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_87_old_Oprod_Oexhaust,axiom,
% 2.97/1.59 ! [Y: produc1127127335iset_a] :
% 2.97/1.59 ~ ! [A3: multiset_a,B3: multiset_a] :
% 2.97/1.59 ( Y
% 2.97/1.59 != ( produc2037245207iset_a @ A3 @ B3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_88_old_Oprod_Oexhaust,axiom,
% 2.97/1.59 ! [Y: produc2061001575od_a_a] :
% 2.97/1.59 ~ ! [A3: multis599418605od_a_a,B3: multis599418605od_a_a] :
% 2.97/1.59 ( Y
% 2.97/1.59 != ( produc1859144151od_a_a @ A3 @ B3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_89_old_Oprod_Oexhaust,axiom,
% 2.97/1.59 ! [Y: product_prod_a_a] :
% 2.97/1.59 ~ ! [A3: a,B3: a] :
% 2.97/1.59 ( Y
% 2.97/1.59 != ( product_Pair_a_a @ A3 @ B3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_90_prod__induct3,axiom,
% 2.97/1.59 ! [P2: produc1572603623od_a_a > $o,X: produc1572603623od_a_a] :
% 2.97/1.59 ( ! [A3: product_prod_a_a,B3: a,C: a] : ( P2 @ ( produc1474507607od_a_a @ A3 @ ( product_Pair_a_a @ B3 @ C ) ) )
% 2.97/1.59 => ( P2 @ X ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_91_preorder__on__def,axiom,
% 2.97/1.59 ( order_359341630od_a_a
% 2.97/1.59 = ( ^ [A4: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( refl_o1298442278od_a_a @ A4 @ R )
% 2.97/1.59 & ( trans_972827014od_a_a @ R ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_92_preorder__on__def,axiom,
% 2.97/1.59 ( order_preorder_on_a
% 2.97/1.59 = ( ^ [A4: set_a,R: set_Product_prod_a_a] :
% 2.97/1.59 ( ( refl_on_a @ A4 @ R )
% 2.97/1.59 & ( trans_a @ R ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_93_total__on__def,axiom,
% 2.97/1.59 ( total_on_multiset_a
% 2.97/1.59 = ( ^ [A4: set_multiset_a,R: set_Pr158363655iset_a] :
% 2.97/1.59 ! [X4: multiset_a] :
% 2.97/1.59 ( ( member_multiset_a @ X4 @ A4 )
% 2.97/1.59 => ! [Y4: multiset_a] :
% 2.97/1.59 ( ( member_multiset_a @ Y4 @ A4 )
% 2.97/1.59 => ( ( X4 != Y4 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ R )
% 2.97/1.59 | ( member340150864iset_a @ ( produc2037245207iset_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_94_total__on__def,axiom,
% 2.97/1.59 ( total_1024184433od_a_a
% 2.97/1.59 = ( ^ [A4: set_mu1281098573od_a_a,R: set_Pr1821581383od_a_a] :
% 2.97/1.59 ! [X4: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1755255446od_a_a @ X4 @ A4 )
% 2.97/1.59 => ! [Y4: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1755255446od_a_a @ Y4 @ A4 )
% 2.97/1.59 => ( ( X4 != Y4 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ R )
% 2.97/1.59 | ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_95_total__on__def,axiom,
% 2.97/1.59 ( total_1490170027od_a_a
% 2.97/1.59 = ( ^ [A4: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 2.97/1.59 ! [X4: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X4 @ A4 )
% 2.97/1.59 => ! [Y4: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ Y4 @ A4 )
% 2.97/1.59 => ( ( X4 != Y4 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R )
% 2.97/1.59 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_96_total__on__def,axiom,
% 2.97/1.59 ( total_on_a
% 2.97/1.59 = ( ^ [A4: set_a,R: set_Product_prod_a_a] :
% 2.97/1.59 ! [X4: a] :
% 2.97/1.59 ( ( member_a @ X4 @ A4 )
% 2.97/1.59 => ! [Y4: a] :
% 2.97/1.59 ( ( member_a @ Y4 @ A4 )
% 2.97/1.59 => ( ( X4 != Y4 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
% 2.97/1.59 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R ) ) ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_97_total__onI,axiom,
% 2.97/1.59 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a] :
% 2.97/1.59 ( ! [X3: produc1572603623od_a_a,Y3: produc1572603623od_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ X3 @ A5 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ Y3 @ A5 )
% 2.97/1.59 => ( ( X3 != Y3 )
% 2.97/1.59 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 | ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 2.97/1.59 => ( total_1340260971od_a_a @ A5 @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_98_total__onI,axiom,
% 2.97/1.59 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a] :
% 2.97/1.59 ( ! [X3: produc1127127335iset_a,Y3: produc1127127335iset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ X3 @ A5 )
% 2.97/1.59 => ( ( member340150864iset_a @ Y3 @ A5 )
% 2.97/1.59 => ( ( X3 != Y3 )
% 2.97/1.59 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 | ( member567864080iset_a @ ( produc1898392407iset_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 2.97/1.59 => ( total_313747627iset_a @ A5 @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_99_total__onI,axiom,
% 2.97/1.59 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a] :
% 2.97/1.59 ( ! [X3: produc2061001575od_a_a,Y3: produc2061001575od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ X3 @ A5 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ Y3 @ A5 )
% 2.97/1.59 => ( ( X3 != Y3 )
% 2.97/1.59 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 | ( member2007374608od_a_a @ ( produc978545495od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 2.97/1.59 => ( total_1170533099od_a_a @ A5 @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_100_total__onI,axiom,
% 2.97/1.59 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a] :
% 2.97/1.59 ( ! [X3: multiset_a,Y3: multiset_a] :
% 2.97/1.59 ( ( member_multiset_a @ X3 @ A5 )
% 2.97/1.59 => ( ( member_multiset_a @ Y3 @ A5 )
% 2.97/1.59 => ( ( X3 != Y3 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 | ( member340150864iset_a @ ( produc2037245207iset_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 2.97/1.59 => ( total_on_multiset_a @ A5 @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_101_total__onI,axiom,
% 2.97/1.59 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a] :
% 2.97/1.59 ( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1755255446od_a_a @ X3 @ A5 )
% 2.97/1.59 => ( ( member1755255446od_a_a @ Y3 @ A5 )
% 2.97/1.59 => ( ( X3 != Y3 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 | ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 2.97/1.59 => ( total_1024184433od_a_a @ A5 @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_102_total__onI,axiom,
% 2.97/1.59 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X3 @ A5 )
% 2.97/1.59 => ( ( member449909584od_a_a @ Y3 @ A5 )
% 2.97/1.59 => ( ( X3 != Y3 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 2.97/1.59 => ( total_1490170027od_a_a @ A5 @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_103_total__onI,axiom,
% 2.97/1.59 ! [A5: set_a,R2: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( member_a @ X3 @ A5 )
% 2.97/1.59 => ( ( member_a @ Y3 @ A5 )
% 2.97/1.59 => ( ( X3 != Y3 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X3 ) @ R2 ) ) ) ) )
% 2.97/1.59 => ( total_on_a @ A5 @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_104_refl__on__domain,axiom,
% 2.97/1.59 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,A: produc1572603623od_a_a,B: produc1572603623od_a_a] :
% 2.97/1.59 ( ( refl_o198693862od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ B ) @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ A @ A5 )
% 2.97/1.59 & ( member2057358096od_a_a @ B @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_105_refl__on__domain,axiom,
% 2.97/1.59 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,A: produc1127127335iset_a,B: produc1127127335iset_a] :
% 2.97/1.59 ( ( refl_o1522155814iset_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ A @ B ) @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ A @ A5 )
% 2.97/1.59 & ( member340150864iset_a @ B @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_106_refl__on__domain,axiom,
% 2.97/1.59 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,A: produc2061001575od_a_a,B: produc2061001575od_a_a] :
% 2.97/1.59 ( ( refl_o102522214od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ A @ B ) @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ A @ A5 )
% 2.97/1.59 & ( member1349055120od_a_a @ B @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_107_refl__on__domain,axiom,
% 2.97/1.59 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a,A: multiset_a,B: multiset_a] :
% 2.97/1.59 ( ( refl_on_multiset_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ A @ B ) @ R2 )
% 2.97/1.59 => ( ( member_multiset_a @ A @ A5 )
% 2.97/1.59 & ( member_multiset_a @ B @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_108_refl__on__domain,axiom,
% 2.97/1.59 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,A: multis599418605od_a_a,B: multis599418605od_a_a] :
% 2.97/1.59 ( ( refl_o581572716od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A @ B ) @ R2 )
% 2.97/1.59 => ( ( member1755255446od_a_a @ A @ A5 )
% 2.97/1.59 & ( member1755255446od_a_a @ B @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_109_refl__on__domain,axiom,
% 2.97/1.59 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
% 2.97/1.59 ( ( refl_o1298442278od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ A @ A5 )
% 2.97/1.59 & ( member449909584od_a_a @ B @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_110_refl__on__domain,axiom,
% 2.97/1.59 ! [A5: set_a,R2: set_Product_prod_a_a,A: a,B: a] :
% 2.97/1.59 ( ( refl_on_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
% 2.97/1.59 => ( ( member_a @ A @ A5 )
% 2.97/1.59 & ( member_a @ B @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_111_mem__Collect__eq,axiom,
% 2.97/1.59 ! [A: produc1572603623od_a_a,P2: produc1572603623od_a_a > $o] :
% 2.97/1.59 ( ( member2057358096od_a_a @ A @ ( collec1635618130od_a_a @ P2 ) )
% 2.97/1.59 = ( P2 @ A ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_112_mem__Collect__eq,axiom,
% 2.97/1.59 ! [A: produc1127127335iset_a,P2: produc1127127335iset_a > $o] :
% 2.97/1.59 ( ( member340150864iset_a @ A @ ( collec1104713362iset_a @ P2 ) )
% 2.97/1.59 = ( P2 @ A ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_113_mem__Collect__eq,axiom,
% 2.97/1.59 ! [A: produc2061001575od_a_a,P2: produc2061001575od_a_a > $o] :
% 2.97/1.59 ( ( member1349055120od_a_a @ A @ ( collec8264402od_a_a @ P2 ) )
% 2.97/1.59 = ( P2 @ A ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_114_mem__Collect__eq,axiom,
% 2.97/1.59 ! [A: multis599418605od_a_a,P2: multis599418605od_a_a > $o] :
% 2.97/1.59 ( ( member1755255446od_a_a @ A @ ( collec1948052440od_a_a @ P2 ) )
% 2.97/1.59 = ( P2 @ A ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_115_mem__Collect__eq,axiom,
% 2.97/1.59 ! [A: multiset_a,P2: multiset_a > $o] :
% 2.97/1.59 ( ( member_multiset_a @ A @ ( collect_multiset_a @ P2 ) )
% 2.97/1.59 = ( P2 @ A ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_116_mem__Collect__eq,axiom,
% 2.97/1.59 ! [A: a,P2: a > $o] :
% 2.97/1.59 ( ( member_a @ A @ ( collect_a @ P2 ) )
% 2.97/1.59 = ( P2 @ A ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_117_mem__Collect__eq,axiom,
% 2.97/1.59 ! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
% 2.97/1.59 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P2 ) )
% 2.97/1.59 = ( P2 @ A ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_118_Collect__mem__eq,axiom,
% 2.97/1.59 ! [A5: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( collec1635618130od_a_a
% 2.97/1.59 @ ^ [X4: produc1572603623od_a_a] : ( member2057358096od_a_a @ X4 @ A5 ) )
% 2.97/1.59 = A5 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_119_Collect__mem__eq,axiom,
% 2.97/1.59 ! [A5: set_Pr158363655iset_a] :
% 2.97/1.59 ( ( collec1104713362iset_a
% 2.97/1.59 @ ^ [X4: produc1127127335iset_a] : ( member340150864iset_a @ X4 @ A5 ) )
% 2.97/1.59 = A5 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_120_Collect__mem__eq,axiom,
% 2.97/1.59 ! [A5: set_Pr1821581383od_a_a] :
% 2.97/1.59 ( ( collec8264402od_a_a
% 2.97/1.59 @ ^ [X4: produc2061001575od_a_a] : ( member1349055120od_a_a @ X4 @ A5 ) )
% 2.97/1.59 = A5 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_121_Collect__mem__eq,axiom,
% 2.97/1.59 ! [A5: set_mu1281098573od_a_a] :
% 2.97/1.59 ( ( collec1948052440od_a_a
% 2.97/1.59 @ ^ [X4: multis599418605od_a_a] : ( member1755255446od_a_a @ X4 @ A5 ) )
% 2.97/1.59 = A5 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_122_Collect__mem__eq,axiom,
% 2.97/1.59 ! [A5: set_multiset_a] :
% 2.97/1.59 ( ( collect_multiset_a
% 2.97/1.59 @ ^ [X4: multiset_a] : ( member_multiset_a @ X4 @ A5 ) )
% 2.97/1.59 = A5 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_123_Collect__mem__eq,axiom,
% 2.97/1.59 ! [A5: set_a] :
% 2.97/1.59 ( ( collect_a
% 2.97/1.59 @ ^ [X4: a] : ( member_a @ X4 @ A5 ) )
% 2.97/1.59 = A5 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_124_Collect__mem__eq,axiom,
% 2.97/1.59 ! [A5: set_Product_prod_a_a] :
% 2.97/1.59 ( ( collec645855634od_a_a
% 2.97/1.59 @ ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ A5 ) )
% 2.97/1.59 = A5 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_125_Collect__cong,axiom,
% 2.97/1.59 ! [P2: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 2.97/1.59 ( ! [X3: product_prod_a_a] :
% 2.97/1.59 ( ( P2 @ X3 )
% 2.97/1.59 = ( Q @ X3 ) )
% 2.97/1.59 => ( ( collec645855634od_a_a @ P2 )
% 2.97/1.59 = ( collec645855634od_a_a @ Q ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_126_Collect__cong,axiom,
% 2.97/1.59 ! [P2: a > $o,Q: a > $o] :
% 2.97/1.59 ( ! [X3: a] :
% 2.97/1.59 ( ( P2 @ X3 )
% 2.97/1.59 = ( Q @ X3 ) )
% 2.97/1.59 => ( ( collect_a @ P2 )
% 2.97/1.59 = ( collect_a @ Q ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_127_refl__onD2,axiom,
% 2.97/1.59 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.97/1.59 ( ( refl_o198693862od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member2057358096od_a_a @ Y @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_128_refl__onD2,axiom,
% 2.97/1.59 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
% 2.97/1.59 ( ( refl_o1522155814iset_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member340150864iset_a @ Y @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_129_refl__onD2,axiom,
% 2.97/1.59 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
% 2.97/1.59 ( ( refl_o102522214od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member1349055120od_a_a @ Y @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_130_refl__onD2,axiom,
% 2.97/1.59 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
% 2.97/1.59 ( ( refl_on_multiset_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member_multiset_a @ Y @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_131_refl__onD2,axiom,
% 2.97/1.59 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 2.97/1.59 ( ( refl_o581572716od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member1755255446od_a_a @ Y @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_132_refl__onD2,axiom,
% 2.97/1.59 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.97/1.59 ( ( refl_o1298442278od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member449909584od_a_a @ Y @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_133_refl__onD2,axiom,
% 2.97/1.59 ! [A5: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
% 2.97/1.59 ( ( refl_on_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member_a @ Y @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_134_refl__onD1,axiom,
% 2.97/1.59 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,X: produc1572603623od_a_a,Y: produc1572603623od_a_a] :
% 2.97/1.59 ( ( refl_o198693862od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member2057358096od_a_a @ X @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_135_refl__onD1,axiom,
% 2.97/1.59 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,X: produc1127127335iset_a,Y: produc1127127335iset_a] :
% 2.97/1.59 ( ( refl_o1522155814iset_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member340150864iset_a @ X @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_136_refl__onD1,axiom,
% 2.97/1.59 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,X: produc2061001575od_a_a,Y: produc2061001575od_a_a] :
% 2.97/1.59 ( ( refl_o102522214od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member1349055120od_a_a @ X @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_137_refl__onD1,axiom,
% 2.97/1.59 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a] :
% 2.97/1.59 ( ( refl_on_multiset_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member_multiset_a @ X @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_138_refl__onD1,axiom,
% 2.97/1.59 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 2.97/1.59 ( ( refl_o581572716od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member1755255446od_a_a @ X @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_139_refl__onD1,axiom,
% 2.97/1.59 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 2.97/1.59 ( ( refl_o1298442278od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member449909584od_a_a @ X @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_140_refl__onD1,axiom,
% 2.97/1.59 ! [A5: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
% 2.97/1.59 ( ( refl_on_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( member_a @ X @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_141_refl__onD,axiom,
% 2.97/1.59 ! [A5: set_Pr1948701895od_a_a,R2: set_Pr1295299783od_a_a,A: produc1572603623od_a_a] :
% 2.97/1.59 ( ( refl_o198693862od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ A @ A5 )
% 2.97/1.59 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ A @ A ) @ R2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_142_refl__onD,axiom,
% 2.97/1.59 ! [A5: set_Pr158363655iset_a,R2: set_Pr146455751iset_a,A: produc1127127335iset_a] :
% 2.97/1.59 ( ( refl_o1522155814iset_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ A @ A5 )
% 2.97/1.59 => ( member567864080iset_a @ ( produc1898392407iset_a @ A @ A ) @ R2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_143_refl__onD,axiom,
% 2.97/1.59 ! [A5: set_Pr1821581383od_a_a,R2: set_Pr1930443975od_a_a,A: produc2061001575od_a_a] :
% 2.97/1.59 ( ( refl_o102522214od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ A @ A5 )
% 2.97/1.59 => ( member2007374608od_a_a @ ( produc978545495od_a_a @ A @ A ) @ R2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_144_refl__onD,axiom,
% 2.97/1.59 ! [A5: set_multiset_a,R2: set_Pr158363655iset_a,A: multiset_a] :
% 2.97/1.59 ( ( refl_on_multiset_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member_multiset_a @ A @ A5 )
% 2.97/1.59 => ( member340150864iset_a @ ( produc2037245207iset_a @ A @ A ) @ R2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_145_refl__onD,axiom,
% 2.97/1.59 ! [A5: set_mu1281098573od_a_a,R2: set_Pr1821581383od_a_a,A: multis599418605od_a_a] :
% 2.97/1.59 ( ( refl_o581572716od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member1755255446od_a_a @ A @ A5 )
% 2.97/1.59 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A @ A ) @ R2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_146_refl__onD,axiom,
% 2.97/1.59 ! [A5: set_Product_prod_a_a,R2: set_Pr1948701895od_a_a,A: product_prod_a_a] :
% 2.97/1.59 ( ( refl_o1298442278od_a_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ A @ A5 )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ R2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_147_refl__onD,axiom,
% 2.97/1.59 ! [A5: set_a,R2: set_Product_prod_a_a,A: a] :
% 2.97/1.59 ( ( refl_on_a @ A5 @ R2 )
% 2.97/1.59 => ( ( member_a @ A @ A5 )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ R2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_148_trans__def,axiom,
% 2.97/1.59 ( trans_972827014od_a_a
% 2.97/1.59 = ( ^ [R: set_Pr1948701895od_a_a] :
% 2.97/1.59 ! [X4: product_prod_a_a,Y4: product_prod_a_a,Z2: product_prod_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y4 ) @ R )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y4 @ Z2 ) @ R )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_149_trans__def,axiom,
% 2.97/1.59 ( trans_multiset_a
% 2.97/1.59 = ( ^ [R: set_Pr158363655iset_a] :
% 2.97/1.59 ! [X4: multiset_a,Y4: multiset_a,Z2: multiset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Y4 ) @ R )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y4 @ Z2 ) @ R )
% 2.97/1.59 => ( member340150864iset_a @ ( produc2037245207iset_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_150_trans__def,axiom,
% 2.97/1.59 ( trans_1520871884od_a_a
% 2.97/1.59 = ( ^ [R: set_Pr1821581383od_a_a] :
% 2.97/1.59 ! [X4: multis599418605od_a_a,Y4: multis599418605od_a_a,Z2: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Y4 ) @ R )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y4 @ Z2 ) @ R )
% 2.97/1.59 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_151_trans__def,axiom,
% 2.97/1.59 ( trans_a
% 2.97/1.59 = ( ^ [R: set_Product_prod_a_a] :
% 2.97/1.59 ! [X4: a,Y4: a,Z2: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ Z2 ) @ R )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z2 ) @ R ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_152_transI,axiom,
% 2.97/1.59 ! [R2: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ! [X3: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ Z3 ) @ R2 )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Z3 ) @ R2 ) ) )
% 2.97/1.59 => ( trans_972827014od_a_a @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_153_transI,axiom,
% 2.97/1.59 ! [R2: set_Pr158363655iset_a] :
% 2.97/1.59 ( ! [X3: multiset_a,Y3: multiset_a,Z3: multiset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y3 @ Z3 ) @ R2 )
% 2.97/1.59 => ( member340150864iset_a @ ( produc2037245207iset_a @ X3 @ Z3 ) @ R2 ) ) )
% 2.97/1.59 => ( trans_multiset_a @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_154_transI,axiom,
% 2.97/1.59 ! [R2: set_Pr1821581383od_a_a] :
% 2.97/1.59 ( ! [X3: multis599418605od_a_a,Y3: multis599418605od_a_a,Z3: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y3 @ Z3 ) @ R2 )
% 2.97/1.59 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X3 @ Z3 ) @ R2 ) ) )
% 2.97/1.59 => ( trans_1520871884od_a_a @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_155_transI,axiom,
% 2.97/1.59 ! [R2: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [X3: a,Y3: a,Z3: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R2 )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z3 ) @ R2 ) ) )
% 2.97/1.59 => ( trans_a @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_156_transD,axiom,
% 2.97/1.59 ! [R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.97/1.59 ( ( trans_972827014od_a_a @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ R2 )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_157_transD,axiom,
% 2.97/1.59 ! [R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 2.97/1.59 ( ( trans_multiset_a @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R2 )
% 2.97/1.59 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_158_transD,axiom,
% 2.97/1.59 ! [R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
% 2.97/1.59 ( ( trans_1520871884od_a_a @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ R2 )
% 2.97/1.59 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_159_transD,axiom,
% 2.97/1.59 ! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.97/1.59 ( ( trans_a @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_160_transE,axiom,
% 2.97/1.59 ! [R2: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 2.97/1.59 ( ( trans_972827014od_a_a @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ R2 )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_161_transE,axiom,
% 2.97/1.59 ! [R2: set_Pr158363655iset_a,X: multiset_a,Y: multiset_a,Z: multiset_a] :
% 2.97/1.59 ( ( trans_multiset_a @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ Y @ Z ) @ R2 )
% 2.97/1.59 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_162_transE,axiom,
% 2.97/1.59 ! [R2: set_Pr1821581383od_a_a,X: multis599418605od_a_a,Y: multis599418605od_a_a,Z: multis599418605od_a_a] :
% 2.97/1.59 ( ( trans_1520871884od_a_a @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Y @ Z ) @ R2 )
% 2.97/1.59 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_163_transE,axiom,
% 2.97/1.59 ! [R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 2.97/1.59 ( ( trans_a @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_164_rational__preference_Ointro,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( prefer1064779160od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer697821563od_a_a @ Carrier @ Relation ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_165_rational__preference_Ointro,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.97/1.59 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( ( prefer580340847ioms_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_166_rational__preference__def,axiom,
% 2.97/1.59 ( prefer697821563od_a_a
% 2.97/1.59 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( prefer1338892073od_a_a @ Carrier2 @ Relation2 )
% 2.97/1.59 & ( prefer1064779160od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_167_rational__preference__def,axiom,
% 2.97/1.59 ( prefer719835986ence_a
% 2.97/1.59 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 2.97/1.59 ( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
% 2.97/1.59 & ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_168_rational__preference_Oaxioms_I2_J,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_169_rational__preference_Oaxioms_I2_J,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.97/1.59 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_170_rational__preference__axioms_Ointro,axiom,
% 2.97/1.59 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( total_1490170027od_a_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer1064779160od_a_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_171_rational__preference__axioms_Ointro,axiom,
% 2.97/1.59 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 2.97/1.59 ( ( total_on_a @ Carrier @ Relation )
% 2.97/1.59 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_172_rational__preference__axioms__def,axiom,
% 2.97/1.59 prefer1064779160od_a_a = total_1490170027od_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(fact_173_rational__preference__axioms__def,axiom,
% 2.97/1.59 prefer580340847ioms_a = total_on_a ).
% 2.97/1.59
% 2.97/1.59 thf(fact_174_ssubst__Pair__rhs,axiom,
% 2.97/1.59 ! [R2: product_prod_a_a,S: product_prod_a_a,R3: set_Pr1948701895od_a_a,S2: product_prod_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R2 @ S ) @ R3 )
% 2.97/1.59 => ( ( S2 = S )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ R2 @ S2 ) @ R3 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_175_ssubst__Pair__rhs,axiom,
% 2.97/1.59 ! [R2: multiset_a,S: multiset_a,R3: set_Pr158363655iset_a,S2: multiset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ R2 @ S ) @ R3 )
% 2.97/1.59 => ( ( S2 = S )
% 2.97/1.59 => ( member340150864iset_a @ ( produc2037245207iset_a @ R2 @ S2 ) @ R3 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_176_ssubst__Pair__rhs,axiom,
% 2.97/1.59 ! [R2: multis599418605od_a_a,S: multis599418605od_a_a,R3: set_Pr1821581383od_a_a,S2: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ R2 @ S ) @ R3 )
% 2.97/1.59 => ( ( S2 = S )
% 2.97/1.59 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ R2 @ S2 ) @ R3 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_177_ssubst__Pair__rhs,axiom,
% 2.97/1.59 ! [R2: a,S: a,R3: set_Product_prod_a_a,S2: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S ) @ R3 )
% 2.97/1.59 => ( ( S2 = S )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ R3 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_178_at__lst__asgd__ge,axiom,
% 2.97/1.59 ! [X: produc1572603623od_a_a,Y: produc1572603623od_a_a,B4: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ X @ ( prefer55414563od_a_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ X @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_179_at__lst__asgd__ge,axiom,
% 2.97/1.59 ! [X: produc1127127335iset_a,Y: produc1127127335iset_a,B4: set_Pr158363655iset_a,Pr: set_Pr146455751iset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ X @ ( prefer619498851iset_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member567864080iset_a @ ( produc1898392407iset_a @ X @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_180_at__lst__asgd__ge,axiom,
% 2.97/1.59 ! [X: produc2061001575od_a_a,Y: produc2061001575od_a_a,B4: set_Pr1821581383od_a_a,Pr: set_Pr1930443975od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ X @ ( prefer737894307od_a_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member2007374608od_a_a @ ( produc978545495od_a_a @ X @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_181_at__lst__asgd__ge,axiom,
% 2.97/1.59 ! [X: multiset_a,Y: multiset_a,B4: set_multiset_a,Pr: set_Pr158363655iset_a] :
% 2.97/1.59 ( ( member_multiset_a @ X @ ( prefer2130004608iset_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_182_at__lst__asgd__ge,axiom,
% 2.97/1.59 ! [X: multis599418605od_a_a,Y: multis599418605od_a_a,B4: set_mu1281098573od_a_a,Pr: set_Pr1821581383od_a_a] :
% 2.97/1.59 ( ( member1755255446od_a_a @ X @ ( prefer1191731497od_a_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_183_at__lst__asgd__ge,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,Y: product_prod_a_a,B4: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_184_at__lst__asgd__ge,axiom,
% 2.97/1.59 ! [X: a,Y: a,B4: set_a,Pr: set_Product_prod_a_a] :
% 2.97/1.59 ( ( member_a @ X @ ( prefer161218362good_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_185_at__least__as__goodD,axiom,
% 2.97/1.59 ! [Z: produc1572603623od_a_a,Y: produc1572603623od_a_a,B4: set_Pr1948701895od_a_a,Pr: set_Pr1295299783od_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ Z @ ( prefer55414563od_a_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member1899387664od_a_a @ ( produc1935643479od_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_186_at__least__as__goodD,axiom,
% 2.97/1.59 ! [Z: produc1127127335iset_a,Y: produc1127127335iset_a,B4: set_Pr158363655iset_a,Pr: set_Pr146455751iset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ Z @ ( prefer619498851iset_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member567864080iset_a @ ( produc1898392407iset_a @ Z @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_187_at__least__as__goodD,axiom,
% 2.97/1.59 ! [Z: produc2061001575od_a_a,Y: produc2061001575od_a_a,B4: set_Pr1821581383od_a_a,Pr: set_Pr1930443975od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ Z @ ( prefer737894307od_a_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member2007374608od_a_a @ ( produc978545495od_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_188_at__least__as__goodD,axiom,
% 2.97/1.59 ! [Z: multiset_a,Y: multiset_a,B4: set_multiset_a,Pr: set_Pr158363655iset_a] :
% 2.97/1.59 ( ( member_multiset_a @ Z @ ( prefer2130004608iset_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member340150864iset_a @ ( produc2037245207iset_a @ Z @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_189_at__least__as__goodD,axiom,
% 2.97/1.59 ! [Z: multis599418605od_a_a,Y: multis599418605od_a_a,B4: set_mu1281098573od_a_a,Pr: set_Pr1821581383od_a_a] :
% 2.97/1.59 ( ( member1755255446od_a_a @ Z @ ( prefer1191731497od_a_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_190_at__least__as__goodD,axiom,
% 2.97/1.59 ! [Z: product_prod_a_a,Y: product_prod_a_a,B4: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_191_at__least__as__goodD,axiom,
% 2.97/1.59 ! [Z: a,Y: a,B4: set_a,Pr: set_Product_prod_a_a] :
% 2.97/1.59 ( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B4 @ Pr ) )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_192_curry__conv,axiom,
% 2.97/1.59 ( product_curry_a_a_o
% 2.97/1.59 = ( ^ [F: product_prod_a_a > $o,A6: a,B5: a] : ( F @ ( product_Pair_a_a @ A6 @ B5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_193_curryI,axiom,
% 2.97/1.59 ! [F2: produc1572603623od_a_a > $o,A: product_prod_a_a,B: product_prod_a_a] :
% 2.97/1.59 ( ( F2 @ ( produc1474507607od_a_a @ A @ B ) )
% 2.97/1.59 => ( produc94438221_a_a_o @ F2 @ A @ B ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_194_curryI,axiom,
% 2.97/1.59 ! [F2: produc1127127335iset_a > $o,A: multiset_a,B: multiset_a] :
% 2.97/1.59 ( ( F2 @ ( produc2037245207iset_a @ A @ B ) )
% 2.97/1.59 => ( produc1530159757et_a_o @ F2 @ A @ B ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_195_curryI,axiom,
% 2.97/1.59 ! [F2: produc2061001575od_a_a > $o,A: multis599418605od_a_a,B: multis599418605od_a_a] :
% 2.97/1.59 ( ( F2 @ ( produc1859144151od_a_a @ A @ B ) )
% 2.97/1.59 => ( produc1548936141_a_a_o @ F2 @ A @ B ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_196_curryI,axiom,
% 2.97/1.59 ! [F2: product_prod_a_a > $o,A: a,B: a] :
% 2.97/1.59 ( ( F2 @ ( product_Pair_a_a @ A @ B ) )
% 2.97/1.59 => ( product_curry_a_a_o @ F2 @ A @ B ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_197_in__inv__image,axiom,
% 2.97/1.59 ! [X: a,Y: a,R2: set_Product_prod_a_a,F2: a > a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_image_a_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_198_in__inv__image,axiom,
% 2.97/1.59 ! [X: a,Y: a,R2: set_Pr158363655iset_a,F2: a > multiset_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_im975322764et_a_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member340150864iset_a @ ( produc2037245207iset_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_199_in__inv__image,axiom,
% 2.97/1.59 ! [X: multiset_a,Y: multiset_a,R2: set_Product_prod_a_a,F2: multiset_a > a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ ( inv_im479034648iset_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_200_in__inv__image,axiom,
% 2.97/1.59 ! [X: a,Y: a,R2: set_Pr1948701895od_a_a,F2: a > product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_im777352425_a_a_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_201_in__inv__image,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,Y: product_prod_a_a,R2: set_Product_prod_a_a,F2: product_prod_a_a > a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( inv_im15016955od_a_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_202_in__inv__image,axiom,
% 2.97/1.59 ! [X: multiset_a,Y: multiset_a,R2: set_Pr158363655iset_a,F2: multiset_a > multiset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ ( inv_im343774418iset_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member340150864iset_a @ ( produc2037245207iset_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_203_in__inv__image,axiom,
% 2.97/1.59 ! [X: a,Y: a,R2: set_Pr1821581383od_a_a,F2: a > multis599418605od_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( inv_im1681759523_a_a_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member1349055120od_a_a @ ( produc1859144151od_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_204_in__inv__image,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,Y: product_prod_a_a,R2: set_Pr158363655iset_a,F2: product_prod_a_a > multiset_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ ( inv_im1716587189od_a_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member340150864iset_a @ ( produc2037245207iset_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_205_in__inv__image,axiom,
% 2.97/1.59 ! [X: multiset_a,Y: multiset_a,R2: set_Pr1948701895od_a_a,F2: multiset_a > product_prod_a_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ X @ Y ) @ ( inv_im184450735iset_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_206_in__inv__image,axiom,
% 2.97/1.59 ! [X: multis599418605od_a_a,Y: multis599418605od_a_a,R2: set_Product_prod_a_a,F2: multis599418605od_a_a > a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ X @ Y ) @ ( inv_im941243329od_a_a @ R2 @ F2 ) )
% 2.97/1.59 = ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R2 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_207_strict__linear__order__on__def,axiom,
% 2.97/1.59 ( order_1505515173od_a_a
% 2.97/1.59 = ( ^ [A4: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( trans_972827014od_a_a @ R )
% 2.97/1.59 & ( irrefl76447196od_a_a @ R )
% 2.97/1.59 & ( total_1490170027od_a_a @ A4 @ R ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_208_strict__linear__order__on__def,axiom,
% 2.97/1.59 ( order_1558116988r_on_a
% 2.97/1.59 = ( ^ [A4: set_a,R: set_Product_prod_a_a] :
% 2.97/1.59 ( ( trans_a @ R )
% 2.97/1.59 & ( irrefl_a @ R )
% 2.97/1.59 & ( total_on_a @ A4 @ R ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_209_swap__simp,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,Y: product_prod_a_a] :
% 2.97/1.59 ( ( produc1641853175od_a_a @ ( produc1474507607od_a_a @ X @ Y ) )
% 2.97/1.59 = ( produc1474507607od_a_a @ Y @ X ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_210_swap__simp,axiom,
% 2.97/1.59 ! [X: multiset_a,Y: multiset_a] :
% 2.97/1.59 ( ( produc506430135iset_a @ ( produc2037245207iset_a @ X @ Y ) )
% 2.97/1.59 = ( produc2037245207iset_a @ Y @ X ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_211_swap__simp,axiom,
% 2.97/1.59 ! [X: multis599418605od_a_a,Y: multis599418605od_a_a] :
% 2.97/1.59 ( ( produc1913956215od_a_a @ ( produc1859144151od_a_a @ X @ Y ) )
% 2.97/1.59 = ( produc1859144151od_a_a @ Y @ X ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_212_swap__simp,axiom,
% 2.97/1.59 ! [X: a,Y: a] :
% 2.97/1.59 ( ( product_swap_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 2.97/1.59 = ( product_Pair_a_a @ Y @ X ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_213_internal__case__prod__def,axiom,
% 2.97/1.59 produc1653225349od_a_a = produc1093061851od_a_a ).
% 2.97/1.59
% 2.97/1.59 thf(fact_214_internal__case__prod__def,axiom,
% 2.97/1.59 produc694541532_set_a = produc961951794_set_a ).
% 2.97/1.59
% 2.97/1.59 thf(fact_215_internal__case__prod__def,axiom,
% 2.97/1.59 produc748978966_a_a_o = produc1833107820_a_a_o ).
% 2.97/1.59
% 2.97/1.59 thf(fact_216_swap__swap,axiom,
% 2.97/1.59 ! [P: product_prod_a_a] :
% 2.97/1.59 ( ( product_swap_a_a @ ( product_swap_a_a @ P ) )
% 2.97/1.59 = P ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_217_case__prod__curry,axiom,
% 2.97/1.59 ! [F2: product_prod_a_a > set_Product_prod_a_a] :
% 2.97/1.59 ( ( produc1093061851od_a_a @ ( produc143372156od_a_a @ F2 ) )
% 2.97/1.59 = F2 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_218_case__prod__curry,axiom,
% 2.97/1.59 ! [F2: product_prod_a_a > set_a] :
% 2.97/1.59 ( ( produc961951794_set_a @ ( produc580150227_set_a @ F2 ) )
% 2.97/1.59 = F2 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_219_case__prod__curry,axiom,
% 2.97/1.59 ! [F2: product_prod_a_a > $o] :
% 2.97/1.59 ( ( produc1833107820_a_a_o @ ( product_curry_a_a_o @ F2 ) )
% 2.97/1.59 = F2 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_220_curry__case__prod,axiom,
% 2.97/1.59 ! [F2: a > a > set_Product_prod_a_a] :
% 2.97/1.59 ( ( produc143372156od_a_a @ ( produc1093061851od_a_a @ F2 ) )
% 2.97/1.59 = F2 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_221_curry__case__prod,axiom,
% 2.97/1.59 ! [F2: a > a > set_a] :
% 2.97/1.59 ( ( produc580150227_set_a @ ( produc961951794_set_a @ F2 ) )
% 2.97/1.59 = F2 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_222_curry__case__prod,axiom,
% 2.97/1.59 ! [F2: a > a > $o] :
% 2.97/1.59 ( ( product_curry_a_a_o @ ( produc1833107820_a_a_o @ F2 ) )
% 2.97/1.59 = F2 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_223_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: a,C2: a > a > set_a,P: product_prod_a_a] :
% 2.97/1.59 ( ( member_a @ Z @ ( produc961951794_set_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_224_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: product_prod_a_a,C2: a > a > set_Product_prod_a_a,P: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ Z @ ( produc1093061851od_a_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member449909584od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_225_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: multiset_a,C2: a > a > set_multiset_a,P: product_prod_a_a] :
% 2.97/1.59 ( ( member_multiset_a @ Z @ ( produc1828150264iset_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member_multiset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_226_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: a,C2: multiset_a > multiset_a > set_a,P: produc1127127335iset_a] :
% 2.97/1.59 ( ( member_a @ Z @ ( produc747255346_set_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: multiset_a,Y3: multiset_a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( produc2037245207iset_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_227_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: multis599418605od_a_a,C2: a > a > set_mu1281098573od_a_a,P: product_prod_a_a] :
% 2.97/1.59 ( ( member1755255446od_a_a @ Z @ ( produc1121147169od_a_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member1755255446od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_228_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: multiset_a,C2: multiset_a > multiset_a > set_multiset_a,P: produc1127127335iset_a] :
% 2.97/1.59 ( ( member_multiset_a @ Z @ ( produc429414904iset_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: multiset_a,Y3: multiset_a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( produc2037245207iset_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member_multiset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_229_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: produc1127127335iset_a,C2: a > a > set_Pr158363655iset_a,P: product_prod_a_a] :
% 2.97/1.59 ( ( member340150864iset_a @ Z @ ( produc93594587iset_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( product_Pair_a_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member340150864iset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_230_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: a,C2: product_prod_a_a > product_prod_a_a > set_a,P: produc1572603623od_a_a] :
% 2.97/1.59 ( ( member_a @ Z @ ( produc869579762_set_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( produc1474507607od_a_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_231_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: product_prod_a_a,C2: multiset_a > multiset_a > set_Product_prod_a_a,P: produc1127127335iset_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ Z @ ( produc2041367259od_a_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: multiset_a,Y3: multiset_a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( produc2037245207iset_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member449909584od_a_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_232_mem__case__prodE,axiom,
% 2.97/1.59 ! [Z: multiset_a,C2: product_prod_a_a > product_prod_a_a > set_multiset_a,P: produc1572603623od_a_a] :
% 2.97/1.59 ( ( member_multiset_a @ Z @ ( produc734625720iset_a @ C2 @ P ) )
% 2.97/1.59 => ~ ! [X3: product_prod_a_a,Y3: product_prod_a_a] :
% 2.97/1.59 ( ( P
% 2.97/1.59 = ( produc1474507607od_a_a @ X3 @ Y3 ) )
% 2.97/1.59 => ~ ( member_multiset_a @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_233_old_Oprod_Ocase,axiom,
% 2.97/1.59 ! [F2: a > a > set_Product_prod_a_a,X1: a,X22: a] :
% 2.97/1.59 ( ( produc1093061851od_a_a @ F2 @ ( product_Pair_a_a @ X1 @ X22 ) )
% 2.97/1.59 = ( F2 @ X1 @ X22 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_234_old_Oprod_Ocase,axiom,
% 2.97/1.59 ! [F2: a > a > set_a,X1: a,X22: a] :
% 2.97/1.59 ( ( produc961951794_set_a @ F2 @ ( product_Pair_a_a @ X1 @ X22 ) )
% 2.97/1.59 = ( F2 @ X1 @ X22 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_235_old_Oprod_Ocase,axiom,
% 2.97/1.59 ! [F2: a > a > $o,X1: a,X22: a] :
% 2.97/1.59 ( ( produc1833107820_a_a_o @ F2 @ ( product_Pair_a_a @ X1 @ X22 ) )
% 2.97/1.59 = ( F2 @ X1 @ X22 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_236_irreflI,axiom,
% 2.97/1.59 ! [R3: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ! [A3: product_prod_a_a] :
% 2.97/1.59 ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A3 @ A3 ) @ R3 )
% 2.97/1.59 => ( irrefl76447196od_a_a @ R3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_237_irreflI,axiom,
% 2.97/1.59 ! [R3: set_Pr158363655iset_a] :
% 2.97/1.59 ( ! [A3: multiset_a] :
% 2.97/1.59 ~ ( member340150864iset_a @ ( produc2037245207iset_a @ A3 @ A3 ) @ R3 )
% 2.97/1.59 => ( irrefl_multiset_a @ R3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_238_irreflI,axiom,
% 2.97/1.59 ! [R3: set_Pr1821581383od_a_a] :
% 2.97/1.59 ( ! [A3: multis599418605od_a_a] :
% 2.97/1.59 ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A3 @ A3 ) @ R3 )
% 2.97/1.59 => ( irrefl1672195874od_a_a @ R3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_239_irreflI,axiom,
% 2.97/1.59 ! [R3: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [A3: a] :
% 2.97/1.59 ~ ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R3 )
% 2.97/1.59 => ( irrefl_a @ R3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_240_irrefl__def,axiom,
% 2.97/1.59 ( irrefl76447196od_a_a
% 2.97/1.59 = ( ^ [R: set_Pr1948701895od_a_a] :
% 2.97/1.59 ! [A6: product_prod_a_a] :
% 2.97/1.59 ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A6 @ A6 ) @ R ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_241_irrefl__def,axiom,
% 2.97/1.59 ( irrefl_multiset_a
% 2.97/1.59 = ( ^ [R: set_Pr158363655iset_a] :
% 2.97/1.59 ! [A6: multiset_a] :
% 2.97/1.59 ~ ( member340150864iset_a @ ( produc2037245207iset_a @ A6 @ A6 ) @ R ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_242_irrefl__def,axiom,
% 2.97/1.59 ( irrefl1672195874od_a_a
% 2.97/1.59 = ( ^ [R: set_Pr1821581383od_a_a] :
% 2.97/1.59 ! [A6: multis599418605od_a_a] :
% 2.97/1.59 ~ ( member1349055120od_a_a @ ( produc1859144151od_a_a @ A6 @ A6 ) @ R ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_243_irrefl__def,axiom,
% 2.97/1.59 ( irrefl_a
% 2.97/1.59 = ( ^ [R: set_Product_prod_a_a] :
% 2.97/1.59 ! [A6: a] :
% 2.97/1.59 ~ ( member449909584od_a_a @ ( product_Pair_a_a @ A6 @ A6 ) @ R ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_244_curryD,axiom,
% 2.97/1.59 ! [F2: produc1127127335iset_a > $o,A: multiset_a,B: multiset_a] :
% 2.97/1.59 ( ( produc1530159757et_a_o @ F2 @ A @ B )
% 2.97/1.59 => ( F2 @ ( produc2037245207iset_a @ A @ B ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_245_curryD,axiom,
% 2.97/1.59 ! [F2: produc2061001575od_a_a > $o,A: multis599418605od_a_a,B: multis599418605od_a_a] :
% 2.97/1.59 ( ( produc1548936141_a_a_o @ F2 @ A @ B )
% 2.97/1.59 => ( F2 @ ( produc1859144151od_a_a @ A @ B ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_246_curryD,axiom,
% 2.97/1.59 ! [F2: product_prod_a_a > $o,A: a,B: a] :
% 2.97/1.59 ( ( product_curry_a_a_o @ F2 @ A @ B )
% 2.97/1.59 => ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_247_curryE,axiom,
% 2.97/1.59 ! [F2: product_prod_a_a > $o,A: a,B: a] :
% 2.97/1.59 ( ( product_curry_a_a_o @ F2 @ A @ B )
% 2.97/1.59 => ( F2 @ ( product_Pair_a_a @ A @ B ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_248_trans__inv__image,axiom,
% 2.97/1.59 ! [R2: set_Product_prod_a_a,F2: a > a] :
% 2.97/1.59 ( ( trans_a @ R2 )
% 2.97/1.59 => ( trans_a @ ( inv_image_a_a @ R2 @ F2 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_249_pair__in__swap__image,axiom,
% 2.97/1.59 ! [Y: a,X: a,A5: set_Product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( image_794773445od_a_a @ product_swap_a_a @ A5 ) )
% 2.97/1.59 = ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ A5 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_250_multp__iff,axiom,
% 2.97/1.59 ! [R3: set_Product_prod_a_a,P2: a > a > $o,N: multiset_a,M: multiset_a] :
% 2.97/1.59 ( ( irrefl_a @ R3 )
% 2.97/1.59 => ( ( trans_a @ R3 )
% 2.97/1.59 => ( ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( P2 @ X3 @ Y3 )
% 2.97/1.59 = ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R3 ) )
% 2.97/1.59 => ( ( multp_a @ P2 @ N @ M )
% 2.97/1.59 = ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult_a @ R3 ) ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_251_asym_Oinducts,axiom,
% 2.97/1.59 ! [X: set_Product_prod_a_a,P2: set_Product_prod_a_a > $o] :
% 2.97/1.59 ( ( asym_a @ X )
% 2.97/1.59 => ( ! [R4: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [A7: a,B6: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A7 @ B6 ) @ R4 )
% 2.97/1.59 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A7 ) @ R4 ) )
% 2.97/1.59 => ( P2 @ R4 ) )
% 2.97/1.59 => ( P2 @ X ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_252_asym_Ointros,axiom,
% 2.97/1.59 ! [R3: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [A3: a,B3: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 )
% 2.97/1.59 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A3 ) @ R3 ) )
% 2.97/1.59 => ( asym_a @ R3 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_253_asym_Osimps,axiom,
% 2.97/1.59 ( asym_a
% 2.97/1.59 = ( ^ [A6: set_Product_prod_a_a] :
% 2.97/1.59 ? [R5: set_Product_prod_a_a] :
% 2.97/1.59 ( ( A6 = R5 )
% 2.97/1.59 & ! [X4: a,Y4: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R5 )
% 2.97/1.59 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R5 ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_254_asym_Ocases,axiom,
% 2.97/1.59 ! [A: set_Product_prod_a_a] :
% 2.97/1.59 ( ( asym_a @ A )
% 2.97/1.59 => ! [A7: a,B6: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A7 @ B6 ) @ A )
% 2.97/1.59 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A7 ) @ A ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_255_asym__iff,axiom,
% 2.97/1.59 ( asym_a
% 2.97/1.59 = ( ^ [R5: set_Product_prod_a_a] :
% 2.97/1.59 ! [X4: a,Y4: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y4 ) @ R5 )
% 2.97/1.59 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ R5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_256_asymD,axiom,
% 2.97/1.59 ! [R3: set_Product_prod_a_a,X: a,Y: a] :
% 2.97/1.59 ( ( asym_a @ R3 )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R3 )
% 2.97/1.59 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R3 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_257_image__eqI,axiom,
% 2.97/1.59 ! [B: a,F2: a > a,X: a,A5: set_a] :
% 2.97/1.59 ( ( B
% 2.97/1.59 = ( F2 @ X ) )
% 2.97/1.59 => ( ( member_a @ X @ A5 )
% 2.97/1.59 => ( member_a @ B @ ( image_a_a @ F2 @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_258_image__eqI,axiom,
% 2.97/1.59 ! [B: product_prod_a_a,F2: a > product_prod_a_a,X: a,A5: set_a] :
% 2.97/1.59 ( ( B
% 2.97/1.59 = ( F2 @ X ) )
% 2.97/1.59 => ( ( member_a @ X @ A5 )
% 2.97/1.59 => ( member449909584od_a_a @ B @ ( image_1725938350od_a_a @ F2 @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_259_image__eqI,axiom,
% 2.97/1.59 ! [B: a,F2: product_prod_a_a > a,X: product_prod_a_a,A5: set_Product_prod_a_a] :
% 2.97/1.59 ( ( B
% 2.97/1.59 = ( F2 @ X ) )
% 2.97/1.59 => ( ( member449909584od_a_a @ X @ A5 )
% 2.97/1.59 => ( member_a @ B @ ( image_340790172_a_a_a @ F2 @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_260_image__eqI,axiom,
% 2.97/1.59 ! [B: product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,X: product_prod_a_a,A5: set_Product_prod_a_a] :
% 2.97/1.59 ( ( B
% 2.97/1.59 = ( F2 @ X ) )
% 2.97/1.59 => ( ( member449909584od_a_a @ X @ A5 )
% 2.97/1.59 => ( member449909584od_a_a @ B @ ( image_794773445od_a_a @ F2 @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_261_mult__cancel,axiom,
% 2.97/1.59 ! [S: set_Product_prod_a_a,X5: multiset_a,Z4: multiset_a,Y5: multiset_a] :
% 2.97/1.59 ( ( trans_a @ S )
% 2.97/1.59 => ( ( irrefl_a @ S )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ X5 @ Z4 ) @ ( plus_plus_multiset_a @ Y5 @ Z4 ) ) @ ( mult_a @ S ) )
% 2.97/1.59 = ( member340150864iset_a @ ( produc2037245207iset_a @ X5 @ Y5 ) @ ( mult_a @ S ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_262_mult__cancel__add__mset,axiom,
% 2.97/1.59 ! [S: set_Product_prod_a_a,Uu: a,X5: multiset_a,Y5: multiset_a] :
% 2.97/1.59 ( ( trans_a @ S )
% 2.97/1.59 => ( ( irrefl_a @ S )
% 2.97/1.59 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X5 ) @ ( add_mset_a @ Uu @ Y5 ) ) @ ( mult_a @ S ) )
% 2.97/1.59 = ( member340150864iset_a @ ( produc2037245207iset_a @ X5 @ Y5 ) @ ( mult_a @ S ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_263_imageI,axiom,
% 2.97/1.59 ! [X: a,A5: set_a,F2: a > a] :
% 2.97/1.59 ( ( member_a @ X @ A5 )
% 2.97/1.59 => ( member_a @ ( F2 @ X ) @ ( image_a_a @ F2 @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_264_imageI,axiom,
% 2.97/1.59 ! [X: a,A5: set_a,F2: a > product_prod_a_a] :
% 2.97/1.59 ( ( member_a @ X @ A5 )
% 2.97/1.59 => ( member449909584od_a_a @ ( F2 @ X ) @ ( image_1725938350od_a_a @ F2 @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_265_imageI,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X @ A5 )
% 2.97/1.59 => ( member_a @ ( F2 @ X ) @ ( image_340790172_a_a_a @ F2 @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_266_imageI,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,A5: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X @ A5 )
% 2.97/1.59 => ( member449909584od_a_a @ ( F2 @ X ) @ ( image_794773445od_a_a @ F2 @ A5 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_267_rev__image__eqI,axiom,
% 2.97/1.59 ! [X: a,A5: set_a,B: a,F2: a > a] :
% 2.97/1.59 ( ( member_a @ X @ A5 )
% 2.97/1.59 => ( ( B
% 2.97/1.59 = ( F2 @ X ) )
% 2.97/1.59 => ( member_a @ B @ ( image_a_a @ F2 @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_268_rev__image__eqI,axiom,
% 2.97/1.59 ! [X: a,A5: set_a,B: product_prod_a_a,F2: a > product_prod_a_a] :
% 2.97/1.59 ( ( member_a @ X @ A5 )
% 2.97/1.59 => ( ( B
% 2.97/1.59 = ( F2 @ X ) )
% 2.97/1.59 => ( member449909584od_a_a @ B @ ( image_1725938350od_a_a @ F2 @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_269_rev__image__eqI,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,A5: set_Product_prod_a_a,B: a,F2: product_prod_a_a > a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X @ A5 )
% 2.97/1.59 => ( ( B
% 2.97/1.59 = ( F2 @ X ) )
% 2.97/1.59 => ( member_a @ B @ ( image_340790172_a_a_a @ F2 @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_270_rev__image__eqI,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,A5: set_Product_prod_a_a,B: product_prod_a_a,F2: product_prod_a_a > product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X @ A5 )
% 2.97/1.59 => ( ( B
% 2.97/1.59 = ( F2 @ X ) )
% 2.97/1.59 => ( member449909584od_a_a @ B @ ( image_794773445od_a_a @ F2 @ A5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_271_top__empty__eq,axiom,
% 2.97/1.59 ( top_top_a_o
% 2.97/1.59 = ( ^ [X4: a] : ( member_a @ X4 @ top_top_set_a ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_272_top__empty__eq,axiom,
% 2.97/1.59 ( top_to1483957574_a_a_o
% 2.97/1.59 = ( ^ [X4: product_prod_a_a] : ( member449909584od_a_a @ X4 @ top_to398455383od_a_a ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_273_total__lex__prod,axiom,
% 2.97/1.59 ! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 2.97/1.59 ( ( total_on_a @ top_top_set_a @ R2 )
% 2.97/1.59 => ( ( total_on_a @ top_top_set_a @ S )
% 2.97/1.59 => ( total_1490170027od_a_a @ top_to398455383od_a_a @ ( lex_prod_a_a @ R2 @ S ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_274_map__prod__simp,axiom,
% 2.97/1.59 ! [F2: a > a,G: a > a,A: a,B: a] :
% 2.97/1.59 ( ( produc700022774_a_a_a @ F2 @ G @ ( product_Pair_a_a @ A @ B ) )
% 2.97/1.59 = ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_275_in__lex__prod,axiom,
% 2.97/1.59 ! [A: a,B: a,A2: a,B2: a,R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 2.97/1.59 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Pair_a_a @ A2 @ B2 ) ) @ ( lex_prod_a_a @ R2 @ S ) )
% 2.97/1.59 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A2 ) @ R2 )
% 2.97/1.59 | ( ( A = A2 )
% 2.97/1.59 & ( member449909584od_a_a @ ( product_Pair_a_a @ B @ B2 ) @ S ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_276_trans__lex__prod,axiom,
% 2.97/1.59 ! [R1: set_Product_prod_a_a,R22: set_Product_prod_a_a] :
% 2.97/1.59 ( ( trans_a @ R1 )
% 2.97/1.59 => ( ( trans_a @ R22 )
% 2.97/1.59 => ( trans_972827014od_a_a @ ( lex_prod_a_a @ R1 @ R22 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_277_prod_Ocollapse,axiom,
% 2.97/1.59 ! [Prod: product_prod_a_a] :
% 2.97/1.59 ( ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) )
% 2.97/1.59 = Prod ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_278_map__prod__imageI,axiom,
% 2.97/1.59 ! [A: a,B: a,R3: set_Product_prod_a_a,F2: a > a,G: a > a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R3 )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F2 @ G ) @ R3 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_279_Product__Type_OCollect__case__prodD,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,A5: a > a > $o] :
% 2.97/1.59 ( ( member449909584od_a_a @ X @ ( collec645855634od_a_a @ ( produc1833107820_a_a_o @ A5 ) ) )
% 2.97/1.59 => ( A5 @ ( product_fst_a_a @ X ) @ ( product_snd_a_a @ X ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_280_prod_Oexhaust__sel,axiom,
% 2.97/1.59 ! [Prod: product_prod_a_a] :
% 2.97/1.59 ( Prod
% 2.97/1.59 = ( product_Pair_a_a @ ( product_fst_a_a @ Prod ) @ ( product_snd_a_a @ Prod ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_281_surjective__pairing,axiom,
% 2.97/1.59 ! [T: product_prod_a_a] :
% 2.97/1.59 ( T
% 2.97/1.59 = ( product_Pair_a_a @ ( product_fst_a_a @ T ) @ ( product_snd_a_a @ T ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_282_fst__conv,axiom,
% 2.97/1.59 ! [X1: a,X22: a] :
% 2.97/1.59 ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
% 2.97/1.59 = X1 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_283_fst__eqD,axiom,
% 2.97/1.59 ! [X: a,Y: a,A: a] :
% 2.97/1.59 ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 2.97/1.59 = A )
% 2.97/1.59 => ( X = A ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_284_snd__conv,axiom,
% 2.97/1.59 ! [X1: a,X22: a] :
% 2.97/1.59 ( ( product_snd_a_a @ ( product_Pair_a_a @ X1 @ X22 ) )
% 2.97/1.59 = X22 ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_285_snd__eqD,axiom,
% 2.97/1.59 ! [X: a,Y: a,A: a] :
% 2.97/1.59 ( ( ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) )
% 2.97/1.59 = A )
% 2.97/1.59 => ( Y = A ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_286_prod_Oswap__def,axiom,
% 2.97/1.59 ( product_swap_a_a
% 2.97/1.59 = ( ^ [P3: product_prod_a_a] : ( product_Pair_a_a @ ( product_snd_a_a @ P3 ) @ ( product_fst_a_a @ P3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_287_prod__fun__imageE,axiom,
% 2.97/1.59 ! [C2: product_prod_a_a,F2: a > a,G: a > a,R3: set_Product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ C2 @ ( image_794773445od_a_a @ ( produc700022774_a_a_a @ F2 @ G ) @ R3 ) )
% 2.97/1.59 => ~ ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( C2
% 2.97/1.59 = ( product_Pair_a_a @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
% 2.97/1.59 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R3 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_288_exI__realizer,axiom,
% 2.97/1.59 ! [P2: a > a > $o,Y: a,X: a] :
% 2.97/1.59 ( ( P2 @ Y @ X )
% 2.97/1.59 => ( P2 @ ( product_snd_a_a @ ( product_Pair_a_a @ X @ Y ) ) @ ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_289_conjI__realizer,axiom,
% 2.97/1.59 ! [P2: a > $o,P: a,Q: a > $o,Q2: a] :
% 2.97/1.59 ( ( P2 @ P )
% 2.97/1.59 => ( ( Q @ Q2 )
% 2.97/1.59 => ( ( P2 @ ( product_fst_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) )
% 2.97/1.59 & ( Q @ ( product_snd_a_a @ ( product_Pair_a_a @ P @ Q2 ) ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_290_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 2.97/1.59 ! [P2: a > a > $o,X: a,Y: a,A: product_prod_a_a] :
% 2.97/1.59 ( ( P2 @ X @ Y )
% 2.97/1.59 => ( ( A
% 2.97/1.59 = ( product_Pair_a_a @ X @ Y ) )
% 2.97/1.59 => ( P2 @ ( product_fst_a_a @ A ) @ ( product_snd_a_a @ A ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_291_sndI,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,Y: a,Z: a] :
% 2.97/1.59 ( ( X
% 2.97/1.59 = ( product_Pair_a_a @ Y @ Z ) )
% 2.97/1.59 => ( ( product_snd_a_a @ X )
% 2.97/1.59 = Z ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_292_fstI,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,Y: a,Z: a] :
% 2.97/1.59 ( ( X
% 2.97/1.59 = ( product_Pair_a_a @ Y @ Z ) )
% 2.97/1.59 => ( ( product_fst_a_a @ X )
% 2.97/1.59 = Y ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_293_eq__snd__iff,axiom,
% 2.97/1.59 ! [B: a,P: product_prod_a_a] :
% 2.97/1.59 ( ( B
% 2.97/1.59 = ( product_snd_a_a @ P ) )
% 2.97/1.59 = ( ? [A6: a] :
% 2.97/1.59 ( P
% 2.97/1.59 = ( product_Pair_a_a @ A6 @ B ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_294_eq__fst__iff,axiom,
% 2.97/1.59 ! [A: a,P: product_prod_a_a] :
% 2.97/1.59 ( ( A
% 2.97/1.59 = ( product_fst_a_a @ P ) )
% 2.97/1.59 = ( ? [B5: a] :
% 2.97/1.59 ( P
% 2.97/1.59 = ( product_Pair_a_a @ A @ B5 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_295_subrelI,axiom,
% 2.97/1.59 ! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [X3: a,Y3: a] :
% 2.97/1.59 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R2 )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ S ) )
% 2.97/1.59 => ( ord_le1824328871od_a_a @ R2 @ S ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_296_image__subsetI,axiom,
% 2.97/1.59 ! [A5: set_a,F2: a > a,B4: set_a] :
% 2.97/1.59 ( ! [X3: a] :
% 2.97/1.59 ( ( member_a @ X3 @ A5 )
% 2.97/1.59 => ( member_a @ ( F2 @ X3 ) @ B4 ) )
% 2.97/1.59 => ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A5 ) @ B4 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_297_image__subsetI,axiom,
% 2.97/1.59 ! [A5: set_a,F2: a > product_prod_a_a,B4: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [X3: a] :
% 2.97/1.59 ( ( member_a @ X3 @ A5 )
% 2.97/1.59 => ( member449909584od_a_a @ ( F2 @ X3 ) @ B4 ) )
% 2.97/1.59 => ( ord_le1824328871od_a_a @ ( image_1725938350od_a_a @ F2 @ A5 ) @ B4 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_298_image__subsetI,axiom,
% 2.97/1.59 ! [A5: set_Product_prod_a_a,F2: product_prod_a_a > a,B4: set_a] :
% 2.97/1.59 ( ! [X3: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X3 @ A5 )
% 2.97/1.59 => ( member_a @ ( F2 @ X3 ) @ B4 ) )
% 2.97/1.59 => ( ord_less_eq_set_a @ ( image_340790172_a_a_a @ F2 @ A5 ) @ B4 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_299_image__subsetI,axiom,
% 2.97/1.59 ! [A5: set_Product_prod_a_a,F2: product_prod_a_a > product_prod_a_a,B4: set_Product_prod_a_a] :
% 2.97/1.59 ( ! [X3: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X3 @ A5 )
% 2.97/1.59 => ( member449909584od_a_a @ ( F2 @ X3 ) @ B4 ) )
% 2.97/1.59 => ( ord_le1824328871od_a_a @ ( image_794773445od_a_a @ F2 @ A5 ) @ B4 ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_300_same__fstI,axiom,
% 2.97/1.59 ! [P2: a > $o,X: a,Y6: a,Y: a,R3: a > set_Product_prod_a_a] :
% 2.97/1.59 ( ( P2 @ X )
% 2.97/1.59 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y6 @ Y ) @ ( R3 @ X ) )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( product_Pair_a_a @ X @ Y6 ) @ ( product_Pair_a_a @ X @ Y ) ) @ ( same_fst_a_a @ P2 @ R3 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_301_mult1E,axiom,
% 2.97/1.59 ! [N: multis599418605od_a_a,M: multis599418605od_a_a,R2: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R2 ) )
% 2.97/1.59 => ~ ! [A3: product_prod_a_a,M0: multis599418605od_a_a] :
% 2.97/1.59 ( ( M
% 2.97/1.59 = ( add_ms1725977093od_a_a @ A3 @ M0 ) )
% 2.97/1.59 => ! [K: multis599418605od_a_a] :
% 2.97/1.59 ( ( N
% 2.97/1.59 = ( plus_p404985124od_a_a @ M0 @ K ) )
% 2.97/1.59 => ~ ! [B6: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ B6 @ ( set_ms1818337124od_a_a @ K ) )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B6 @ A3 ) @ R2 ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_302_mult1E,axiom,
% 2.97/1.59 ! [N: multiset_a,M: multiset_a,R2: set_Product_prod_a_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R2 ) )
% 2.97/1.59 => ~ ! [A3: a,M0: multiset_a] :
% 2.97/1.59 ( ( M
% 2.97/1.59 = ( add_mset_a @ A3 @ M0 ) )
% 2.97/1.59 => ! [K: multiset_a] :
% 2.97/1.59 ( ( N
% 2.97/1.59 = ( plus_plus_multiset_a @ M0 @ K ) )
% 2.97/1.59 => ~ ! [B6: a] :
% 2.97/1.59 ( ( member_a @ B6 @ ( set_mset_a @ K ) )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A3 ) @ R2 ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_303_mult1I,axiom,
% 2.97/1.59 ! [M: multis599418605od_a_a,A: product_prod_a_a,M02: multis599418605od_a_a,N: multis599418605od_a_a,K2: multis599418605od_a_a,R2: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( M
% 2.97/1.59 = ( add_ms1725977093od_a_a @ A @ M02 ) )
% 2.97/1.59 => ( ( N
% 2.97/1.59 = ( plus_p404985124od_a_a @ M02 @ K2 ) )
% 2.97/1.59 => ( ! [B3: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ B3 @ ( set_ms1818337124od_a_a @ K2 ) )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B3 @ A ) @ R2 ) )
% 2.97/1.59 => ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ M ) @ ( mult1_1719650714od_a_a @ R2 ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_304_mult1I,axiom,
% 2.97/1.59 ! [M: multiset_a,A: a,M02: multiset_a,N: multiset_a,K2: multiset_a,R2: set_Product_prod_a_a] :
% 2.97/1.59 ( ( M
% 2.97/1.59 = ( add_mset_a @ A @ M02 ) )
% 2.97/1.59 => ( ( N
% 2.97/1.59 = ( plus_plus_multiset_a @ M02 @ K2 ) )
% 2.97/1.59 => ( ! [B3: a] :
% 2.97/1.59 ( ( member_a @ B3 @ ( set_mset_a @ K2 ) )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R2 ) )
% 2.97/1.59 => ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R2 ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_305_less__add,axiom,
% 2.97/1.59 ! [N: multis599418605od_a_a,A: product_prod_a_a,M02: multis599418605od_a_a,R2: set_Pr1948701895od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ M02 ) ) @ ( mult1_1719650714od_a_a @ R2 ) )
% 2.97/1.59 => ( ? [M2: multis599418605od_a_a] :
% 2.97/1.59 ( ( member1349055120od_a_a @ ( produc1859144151od_a_a @ M2 @ M02 ) @ ( mult1_1719650714od_a_a @ R2 ) )
% 2.97/1.59 & ( N
% 2.97/1.59 = ( add_ms1725977093od_a_a @ A @ M2 ) ) )
% 2.97/1.59 | ? [K: multis599418605od_a_a] :
% 2.97/1.59 ( ! [B6: product_prod_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ B6 @ ( set_ms1818337124od_a_a @ K ) )
% 2.97/1.59 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ B6 @ A ) @ R2 ) )
% 2.97/1.59 & ( N
% 2.97/1.59 = ( plus_p404985124od_a_a @ M02 @ K ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_306_less__add,axiom,
% 2.97/1.59 ! [N: multiset_a,A: a,M02: multiset_a,R2: set_Product_prod_a_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ ( add_mset_a @ A @ M02 ) ) @ ( mult1_a @ R2 ) )
% 2.97/1.59 => ( ? [M2: multiset_a] :
% 2.97/1.59 ( ( member340150864iset_a @ ( produc2037245207iset_a @ M2 @ M02 ) @ ( mult1_a @ R2 ) )
% 2.97/1.59 & ( N
% 2.97/1.59 = ( add_mset_a @ A @ M2 ) ) )
% 2.97/1.59 | ? [K: multiset_a] :
% 2.97/1.59 ( ! [B6: a] :
% 2.97/1.59 ( ( member_a @ B6 @ ( set_mset_a @ K ) )
% 2.97/1.59 => ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A ) @ R2 ) )
% 2.97/1.59 & ( N
% 2.97/1.59 = ( plus_plus_multiset_a @ M02 @ K ) ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_307_mset__add,axiom,
% 2.97/1.59 ! [A: a,A5: multiset_a] :
% 2.97/1.59 ( ( member_a @ A @ ( set_mset_a @ A5 ) )
% 2.97/1.59 => ~ ! [B7: multiset_a] :
% 2.97/1.59 ( A5
% 2.97/1.59 != ( add_mset_a @ A @ B7 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_308_mset__add,axiom,
% 2.97/1.59 ! [A: product_prod_a_a,A5: multis599418605od_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A5 ) )
% 2.97/1.59 => ~ ! [B7: multis599418605od_a_a] :
% 2.97/1.59 ( A5
% 2.97/1.59 != ( add_ms1725977093od_a_a @ A @ B7 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_309_multi__member__split,axiom,
% 2.97/1.59 ! [X: a,M: multiset_a] :
% 2.97/1.59 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.97/1.59 => ? [A8: multiset_a] :
% 2.97/1.59 ( M
% 2.97/1.59 = ( add_mset_a @ X @ A8 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_310_multi__member__split,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 2.97/1.59 => ? [A8: multis599418605od_a_a] :
% 2.97/1.59 ( M
% 2.97/1.59 = ( add_ms1725977093od_a_a @ X @ A8 ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_311_insert__noteq__member,axiom,
% 2.97/1.59 ! [B: a,B4: multiset_a,C2: a,C3: multiset_a] :
% 2.97/1.59 ( ( ( add_mset_a @ B @ B4 )
% 2.97/1.59 = ( add_mset_a @ C2 @ C3 ) )
% 2.97/1.59 => ( ( B != C2 )
% 2.97/1.59 => ( member_a @ C2 @ ( set_mset_a @ B4 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_312_insert__noteq__member,axiom,
% 2.97/1.59 ! [B: product_prod_a_a,B4: multis599418605od_a_a,C2: product_prod_a_a,C3: multis599418605od_a_a] :
% 2.97/1.59 ( ( ( add_ms1725977093od_a_a @ B @ B4 )
% 2.97/1.59 = ( add_ms1725977093od_a_a @ C2 @ C3 ) )
% 2.97/1.59 => ( ( B != C2 )
% 2.97/1.59 => ( member449909584od_a_a @ C2 @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_313_union__single__eq__member,axiom,
% 2.97/1.59 ! [X: a,M: multiset_a,N: multiset_a] :
% 2.97/1.59 ( ( ( add_mset_a @ X @ M )
% 2.97/1.59 = N )
% 2.97/1.59 => ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_314_union__single__eq__member,axiom,
% 2.97/1.59 ! [X: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
% 2.97/1.59 ( ( ( add_ms1725977093od_a_a @ X @ M )
% 2.97/1.59 = N )
% 2.97/1.59 => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ N ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_315_union__iff,axiom,
% 2.97/1.59 ! [A: a,A5: multiset_a,B4: multiset_a] :
% 2.97/1.59 ( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A5 @ B4 ) ) )
% 2.97/1.59 = ( ( member_a @ A @ ( set_mset_a @ A5 ) )
% 2.97/1.59 | ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_316_union__iff,axiom,
% 2.97/1.59 ! [A: product_prod_a_a,A5: multis599418605od_a_a,B4: multis599418605od_a_a] :
% 2.97/1.59 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ A5 @ B4 ) ) )
% 2.97/1.59 = ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ A5 ) )
% 2.97/1.59 | ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ B4 ) ) ) ) ).
% 2.97/1.59
% 2.97/1.59 thf(fact_317_mult__implies__one__step,axiom,
% 2.97/1.59 ! [R2: set_Product_prod_a_a,M: multiset_a,N: multiset_a] :
% 2.97/1.60 ( ( trans_a @ R2 )
% 2.97/1.60 => ( ( member340150864iset_a @ ( produc2037245207iset_a @ M @ N ) @ ( mult_a @ R2 ) )
% 2.97/1.60 => ? [I: multiset_a,J: multiset_a] :
% 2.97/1.60 ( ( N
% 2.97/1.60 = ( plus_plus_multiset_a @ I @ J ) )
% 2.97/1.60 & ? [K: multiset_a] :
% 2.97/1.60 ( ( M
% 2.97/1.60 = ( plus_plus_multiset_a @ I @ K ) )
% 2.97/1.60 & ( J != zero_zero_multiset_a )
% 2.97/1.60 & ! [X2: a] :
% 2.97/1.60 ( ( member_a @ X2 @ ( set_mset_a @ K ) )
% 2.97/1.60 => ? [Xa2: a] :
% 2.97/1.60 ( ( member_a @ Xa2 @ ( set_mset_a @ J ) )
% 2.97/1.60 & ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa2 ) @ R2 ) ) ) ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_318_one__step__implies__mult,axiom,
% 2.97/1.60 ! [J2: multiset_a,K2: multiset_a,R2: set_Product_prod_a_a,I2: multiset_a] :
% 2.97/1.60 ( ( J2 != zero_zero_multiset_a )
% 2.97/1.60 => ( ! [X3: a] :
% 2.97/1.60 ( ( member_a @ X3 @ ( set_mset_a @ K2 ) )
% 2.97/1.60 => ? [Xa: a] :
% 2.97/1.60 ( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
% 2.97/1.60 & ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Xa ) @ R2 ) ) )
% 2.97/1.60 => ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I2 @ K2 ) @ ( plus_plus_multiset_a @ I2 @ J2 ) ) @ ( mult_a @ R2 ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_319_multiset__nonemptyE,axiom,
% 2.97/1.60 ! [A5: multiset_a] :
% 2.97/1.60 ( ( A5 != zero_zero_multiset_a )
% 2.97/1.60 => ~ ! [X3: a] :
% 2.97/1.60 ~ ( member_a @ X3 @ ( set_mset_a @ A5 ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_320_multiset__nonemptyE,axiom,
% 2.97/1.60 ! [A5: multis599418605od_a_a] :
% 2.97/1.60 ( ( A5 != zero_z160849828od_a_a )
% 2.97/1.60 => ~ ! [X3: product_prod_a_a] :
% 2.97/1.60 ~ ( member449909584od_a_a @ X3 @ ( set_ms1818337124od_a_a @ A5 ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_321_multi__member__last,axiom,
% 2.97/1.60 ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_322_multi__member__last,axiom,
% 2.97/1.60 ! [X: product_prod_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_323_multi__member__skip,axiom,
% 2.97/1.60 ! [X: a,XS: multiset_a,Y: a] :
% 2.97/1.60 ( ( member_a @ X @ ( set_mset_a @ XS ) )
% 2.97/1.60 => ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_324_multi__member__skip,axiom,
% 2.97/1.60 ! [X: product_prod_a_a,XS: multis599418605od_a_a,Y: product_prod_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ XS ) )
% 2.97/1.60 => ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) @ XS ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_325_multi__member__this,axiom,
% 2.97/1.60 ! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_326_multi__member__this,axiom,
% 2.97/1.60 ! [X: product_prod_a_a,XS: multis599418605od_a_a] : ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( plus_p404985124od_a_a @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) @ XS ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_327_in__mset__fold__plus__iff,axiom,
% 2.97/1.60 ! [X: a,M: multiset_a,NN: multiset_multiset_a] :
% 2.97/1.60 ( ( member_a @ X @ ( set_mset_a @ ( fold_m382157835iset_a @ plus_plus_multiset_a @ M @ NN ) ) )
% 2.97/1.60 = ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.97/1.60 | ? [N2: multiset_a] :
% 2.97/1.60 ( ( member_multiset_a @ N2 @ ( set_mset_multiset_a @ NN ) )
% 2.97/1.60 & ( member_a @ X @ ( set_mset_a @ N2 ) ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_328_in__mset__fold__plus__iff,axiom,
% 2.97/1.60 ! [X: product_prod_a_a,M: multis599418605od_a_a,NN: multis1735859763od_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ ( fold_m773816523od_a_a @ plus_p404985124od_a_a @ M @ NN ) ) )
% 2.97/1.60 = ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 2.97/1.60 | ? [N2: multis599418605od_a_a] :
% 2.97/1.60 ( ( member1755255446od_a_a @ N2 @ ( set_ms1869569194od_a_a @ NN ) )
% 2.97/1.60 & ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ N2 ) ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_329_insert__DiffM2,axiom,
% 2.97/1.60 ! [X: a,M: multiset_a] :
% 2.97/1.60 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.97/1.60 => ( ( plus_plus_multiset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 2.97/1.60 = M ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_330_insert__DiffM2,axiom,
% 2.97/1.60 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 2.97/1.60 => ( ( plus_p404985124od_a_a @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) )
% 2.97/1.60 = M ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_331_diff__add__mset__swap,axiom,
% 2.97/1.60 ! [B: a,A5: multiset_a,M: multiset_a] :
% 2.97/1.60 ( ~ ( member_a @ B @ ( set_mset_a @ A5 ) )
% 2.97/1.60 => ( ( minus_1649712171iset_a @ ( add_mset_a @ B @ M ) @ A5 )
% 2.97/1.60 = ( add_mset_a @ B @ ( minus_1649712171iset_a @ M @ A5 ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_332_diff__add__mset__swap,axiom,
% 2.97/1.60 ! [B: product_prod_a_a,A5: multis599418605od_a_a,M: multis599418605od_a_a] :
% 2.97/1.60 ( ~ ( member449909584od_a_a @ B @ ( set_ms1818337124od_a_a @ A5 ) )
% 2.97/1.60 => ( ( minus_1257885652od_a_a @ ( add_ms1725977093od_a_a @ B @ M ) @ A5 )
% 2.97/1.60 = ( add_ms1725977093od_a_a @ B @ ( minus_1257885652od_a_a @ M @ A5 ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_333_insert__DiffM,axiom,
% 2.97/1.60 ! [X: a,M: multiset_a] :
% 2.97/1.60 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.97/1.60 => ( ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
% 2.97/1.60 = M ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_334_insert__DiffM,axiom,
% 2.97/1.60 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 2.97/1.60 => ( ( add_ms1725977093od_a_a @ X @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) ) )
% 2.97/1.60 = M ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_335_diff__union__swap2,axiom,
% 2.97/1.60 ! [Y: a,M: multiset_a,X: a] :
% 2.97/1.60 ( ( member_a @ Y @ ( set_mset_a @ M ) )
% 2.97/1.60 => ( ( minus_1649712171iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
% 2.97/1.60 = ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_336_diff__union__swap2,axiom,
% 2.97/1.60 ! [Y: product_prod_a_a,M: multis599418605od_a_a,X: product_prod_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ Y @ ( set_ms1818337124od_a_a @ M ) )
% 2.97/1.60 => ( ( minus_1257885652od_a_a @ ( add_ms1725977093od_a_a @ X @ M ) @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) )
% 2.97/1.60 = ( add_ms1725977093od_a_a @ X @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ Y @ zero_z160849828od_a_a ) ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_337_in__diffD,axiom,
% 2.97/1.60 ! [A: a,M: multiset_a,N: multiset_a] :
% 2.97/1.60 ( ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ N ) ) )
% 2.97/1.60 => ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_338_in__diffD,axiom,
% 2.97/1.60 ! [A: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( minus_1257885652od_a_a @ M @ N ) ) )
% 2.97/1.60 => ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ M ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_339_diff__single__trivial,axiom,
% 2.97/1.60 ! [X: a,M: multiset_a] :
% 2.97/1.60 ( ~ ( member_a @ X @ ( set_mset_a @ M ) )
% 2.97/1.60 => ( ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 2.97/1.60 = M ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_340_diff__single__trivial,axiom,
% 2.97/1.60 ! [X: product_prod_a_a,M: multis599418605od_a_a] :
% 2.97/1.60 ( ~ ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 2.97/1.60 => ( ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) )
% 2.97/1.60 = M ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_341_diff__single__eq__union,axiom,
% 2.97/1.60 ! [X: a,M: multiset_a,N: multiset_a] :
% 2.97/1.60 ( ( member_a @ X @ ( set_mset_a @ M ) )
% 2.97/1.60 => ( ( ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
% 2.97/1.60 = N )
% 2.97/1.60 = ( M
% 2.97/1.60 = ( add_mset_a @ X @ N ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_342_diff__single__eq__union,axiom,
% 2.97/1.60 ! [X: product_prod_a_a,M: multis599418605od_a_a,N: multis599418605od_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ X @ ( set_ms1818337124od_a_a @ M ) )
% 2.97/1.60 => ( ( ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ X @ zero_z160849828od_a_a ) )
% 2.97/1.60 = N )
% 2.97/1.60 = ( M
% 2.97/1.60 = ( add_ms1725977093od_a_a @ X @ N ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_343_multi__drop__mem__not__eq,axiom,
% 2.97/1.60 ! [C2: a,B4: multiset_a] :
% 2.97/1.60 ( ( member_a @ C2 @ ( set_mset_a @ B4 ) )
% 2.97/1.60 => ( ( minus_1649712171iset_a @ B4 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) )
% 2.97/1.60 != B4 ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_344_multi__drop__mem__not__eq,axiom,
% 2.97/1.60 ! [C2: product_prod_a_a,B4: multis599418605od_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ C2 @ ( set_ms1818337124od_a_a @ B4 ) )
% 2.97/1.60 => ( ( minus_1257885652od_a_a @ B4 @ ( add_ms1725977093od_a_a @ C2 @ zero_z160849828od_a_a ) )
% 2.97/1.60 != B4 ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_345_add__mset__remove__trivial__If,axiom,
% 2.97/1.60 ! [A: a,N: multiset_a] :
% 2.97/1.60 ( ( ( member_a @ A @ ( set_mset_a @ N ) )
% 2.97/1.60 => ( ( add_mset_a @ A @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
% 2.97/1.60 = N ) )
% 2.97/1.60 & ( ~ ( member_a @ A @ ( set_mset_a @ N ) )
% 2.97/1.60 => ( ( add_mset_a @ A @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
% 2.97/1.60 = ( add_mset_a @ A @ N ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_346_add__mset__remove__trivial__If,axiom,
% 2.97/1.60 ! [A: product_prod_a_a,N: multis599418605od_a_a] :
% 2.97/1.60 ( ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ N ) )
% 2.97/1.60 => ( ( add_ms1725977093od_a_a @ A @ ( minus_1257885652od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) )
% 2.97/1.60 = N ) )
% 2.97/1.60 & ( ~ ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ N ) )
% 2.97/1.60 => ( ( add_ms1725977093od_a_a @ A @ ( minus_1257885652od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) )
% 2.97/1.60 = ( add_ms1725977093od_a_a @ A @ N ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_347_add__mset__remove__trivial__eq,axiom,
% 2.97/1.60 ! [N: multiset_a,A: a] :
% 2.97/1.60 ( ( N
% 2.97/1.60 = ( add_mset_a @ A @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
% 2.97/1.60 = ( member_a @ A @ ( set_mset_a @ N ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_348_add__mset__remove__trivial__eq,axiom,
% 2.97/1.60 ! [N: multis599418605od_a_a,A: product_prod_a_a] :
% 2.97/1.60 ( ( N
% 2.97/1.60 = ( add_ms1725977093od_a_a @ A @ ( minus_1257885652od_a_a @ N @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) ) )
% 2.97/1.60 = ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ N ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_349_multiset__add__sub__el__shuffle,axiom,
% 2.97/1.60 ! [C2: a,B4: multiset_a,B: a] :
% 2.97/1.60 ( ( member_a @ C2 @ ( set_mset_a @ B4 ) )
% 2.97/1.60 => ( ( B != C2 )
% 2.97/1.60 => ( ( add_mset_a @ B @ ( minus_1649712171iset_a @ B4 @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) )
% 2.97/1.60 = ( minus_1649712171iset_a @ ( add_mset_a @ B @ B4 ) @ ( add_mset_a @ C2 @ zero_zero_multiset_a ) ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_350_multiset__add__sub__el__shuffle,axiom,
% 2.97/1.60 ! [C2: product_prod_a_a,B4: multis599418605od_a_a,B: product_prod_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ C2 @ ( set_ms1818337124od_a_a @ B4 ) )
% 2.97/1.60 => ( ( B != C2 )
% 2.97/1.60 => ( ( add_ms1725977093od_a_a @ B @ ( minus_1257885652od_a_a @ B4 @ ( add_ms1725977093od_a_a @ C2 @ zero_z160849828od_a_a ) ) )
% 2.97/1.60 = ( minus_1257885652od_a_a @ ( add_ms1725977093od_a_a @ B @ B4 ) @ ( add_ms1725977093od_a_a @ C2 @ zero_z160849828od_a_a ) ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_351_more__than__one__mset__mset__diff,axiom,
% 2.97/1.60 ! [A: a,M: multiset_a] :
% 2.97/1.60 ( ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
% 2.97/1.60 => ( ( set_mset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
% 2.97/1.60 = ( set_mset_a @ M ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_352_more__than__one__mset__mset__diff,axiom,
% 2.97/1.60 ! [A: product_prod_a_a,M: multis599418605od_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) ) )
% 2.97/1.60 => ( ( set_ms1818337124od_a_a @ ( minus_1257885652od_a_a @ M @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) )
% 2.97/1.60 = ( set_ms1818337124od_a_a @ M ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_353_diff__union__single__conv,axiom,
% 2.97/1.60 ! [A: a,J2: multiset_a,I2: multiset_a] :
% 2.97/1.60 ( ( member_a @ A @ ( set_mset_a @ J2 ) )
% 2.97/1.60 => ( ( minus_1649712171iset_a @ ( plus_plus_multiset_a @ I2 @ J2 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) )
% 2.97/1.60 = ( plus_plus_multiset_a @ I2 @ ( minus_1649712171iset_a @ J2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(fact_354_diff__union__single__conv,axiom,
% 2.97/1.60 ! [A: product_prod_a_a,J2: multis599418605od_a_a,I2: multis599418605od_a_a] :
% 2.97/1.60 ( ( member449909584od_a_a @ A @ ( set_ms1818337124od_a_a @ J2 ) )
% 2.97/1.60 => ( ( minus_1257885652od_a_a @ ( plus_p404985124od_a_a @ I2 @ J2 ) @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) )
% 2.97/1.60 = ( plus_p404985124od_a_a @ I2 @ ( minus_1257885652od_a_a @ J2 @ ( add_ms1725977093od_a_a @ A @ zero_z160849828od_a_a ) ) ) ) ) ).
% 2.97/1.60
% 2.97/1.60 thf(conj_0,hypothesis,
% 2.97/1.60 member_a @ x @ carrier ).
% 2.97/1.60
% 2.97/1.60 thf(conj_1,hypothesis,
% 2.97/1.60 member_a @ y @ carrier ).
% 2.97/1.60
% 2.97/1.60 thf(conj_2,hypothesis,
% 2.97/1.60 x != y ).
% 2.97/1.60
% 2.97/1.60 thf(conj_3,conjecture,
% 2.97/1.60 ( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
% 2.97/1.60 | ( member449909584od_a_a @ ( product_Pair_a_a @ y @ x ) @ relation ) ) ).
% 2.97/1.60 ------------------------
% 3.09/1.64 % (28394)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_28173 for (2999ds/27Mi)
% 3.09/1.65 % (28392)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_28173 for (2999ds/183Mi)
% 3.09/1.65 % (28393)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_28173 for (2999ds/4Mi)
% 3.09/1.65 % (28395)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_28173 for (2999ds/2Mi)
% 3.09/1.65 % (28396)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_28173 for (2999ds/2Mi)
% 3.09/1.65 % (28397)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_28173 for (2999ds/275Mi)
% 3.09/1.65 % (28398)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28173 for (2999ds/18Mi)
% 3.09/1.65 % (28395)Instruction limit reached!
% 3.09/1.65 % (28395)------------------------------
% 3.09/1.65 % (28395)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.65 % (28395)Termination reason: Unknown
% 3.09/1.65 % (28395)Termination phase: shuffling
% 3.09/1.65
% 3.09/1.65 % (28395)Memory used [KB]: 1663
% 3.09/1.65 % (28395)Time elapsed: 0.004 s
% 3.09/1.65 % (28395)Instructions burned: 2 (million)
% 3.09/1.65 % (28395)------------------------------
% 3.09/1.65 % (28395)------------------------------
% 3.09/1.65 % (28396)Instruction limit reached!
% 3.09/1.65 % (28396)------------------------------
% 3.09/1.65 % (28396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.65 % (28396)Termination reason: Unknown
% 3.09/1.65 % (28396)Termination phase: shuffling
% 3.09/1.65
% 3.09/1.65 % (28396)Memory used [KB]: 1663
% 3.09/1.65 % (28396)Time elapsed: 0.004 s
% 3.09/1.65 % (28396)Instructions burned: 2 (million)
% 3.09/1.65 % (28396)------------------------------
% 3.09/1.65 % (28396)------------------------------
% 3.09/1.65 % (28393)Instruction limit reached!
% 3.09/1.65 % (28393)------------------------------
% 3.09/1.65 % (28393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.65 % (28393)Termination reason: Unknown
% 3.09/1.65 % (28393)Termination phase: shuffling
% 3.09/1.65
% 3.09/1.65 % (28393)Memory used [KB]: 1663
% 3.09/1.65 % (28393)Time elapsed: 0.005 s
% 3.09/1.65 % (28393)Instructions burned: 4 (million)
% 3.09/1.65 % (28393)------------------------------
% 3.09/1.65 % (28393)------------------------------
% 3.09/1.65 % (28394)Instruction limit reached!
% 3.09/1.65 % (28394)------------------------------
% 3.09/1.65 % (28394)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.65 % (28394)Termination reason: Unknown
% 3.09/1.65 % (28394)Termination phase: shuffling
% 3.09/1.65
% 3.09/1.65 % (28394)Memory used [KB]: 2174
% 3.09/1.65 % (28394)Time elapsed: 0.013 s
% 3.09/1.65 % (28394)Instructions burned: 27 (million)
% 3.09/1.65 % (28394)------------------------------
% 3.09/1.65 % (28394)------------------------------
% 3.09/1.66 % (28398)Instruction limit reached!
% 3.09/1.66 % (28398)------------------------------
% 3.09/1.66 % (28398)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.66 % (28398)Termination reason: Unknown
% 3.09/1.66 % (28398)Termination phase: shuffling
% 3.09/1.66
% 3.09/1.66 % (28398)Memory used [KB]: 1918
% 3.09/1.66 % (28398)Time elapsed: 0.014 s
% 3.09/1.66 % (28398)Instructions burned: 19 (million)
% 3.09/1.66 % (28398)------------------------------
% 3.09/1.66 % (28398)------------------------------
% 3.09/1.66 % (28402)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_28173 for (2999ds/3Mi)
% 3.09/1.66 % (28402)Instruction limit reached!
% 3.09/1.66 % (28402)------------------------------
% 3.09/1.66 % (28402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.66 % (28402)Termination reason: Unknown
% 3.09/1.66 % (28402)Termination phase: shuffling
% 3.09/1.66
% 3.09/1.66 % (28402)Memory used [KB]: 1663
% 3.09/1.66 % (28402)Time elapsed: 0.002 s
% 3.09/1.66 % (28402)Instructions burned: 3 (million)
% 3.09/1.66 % (28402)------------------------------
% 3.09/1.66 % (28402)------------------------------
% 3.09/1.67 % (28399)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28173 for (2999ds/3Mi)
% 3.09/1.67 % (28400)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_28173 for (2999ds/37Mi)
% 3.09/1.67 % (28401)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_28173 for (2999ds/15Mi)
% 3.09/1.67 % (28399)Instruction limit reached!
% 3.09/1.67 % (28399)------------------------------
% 3.09/1.67 % (28399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.67 % (28399)Termination reason: Unknown
% 3.09/1.67 % (28399)Termination phase: shuffling
% 3.09/1.67
% 3.09/1.67 % (28399)Memory used [KB]: 1663
% 3.09/1.67 % (28399)Time elapsed: 0.004 s
% 3.09/1.67 % (28399)Instructions burned: 4 (million)
% 3.09/1.67 % (28399)------------------------------
% 3.09/1.67 % (28399)------------------------------
% 3.09/1.67 % (28404)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_28173 for (2999ds/7Mi)
% 3.09/1.68 % (28404)Instruction limit reached!
% 3.09/1.68 % (28404)------------------------------
% 3.09/1.68 % (28404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.68 % (28404)Termination reason: Unknown
% 3.09/1.68 % (28404)Termination phase: shuffling
% 3.09/1.68
% 3.09/1.68 % (28404)Memory used [KB]: 1791
% 3.09/1.68 % (28404)Time elapsed: 0.004 s
% 3.09/1.68 % (28404)Instructions burned: 8 (million)
% 3.09/1.68 % (28404)------------------------------
% 3.09/1.68 % (28404)------------------------------
% 3.09/1.68 % (28403)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_28173 for (2999ds/1041Mi)
% 3.09/1.68 % (28401)Instruction limit reached!
% 3.09/1.68 % (28401)------------------------------
% 3.09/1.68 % (28401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.68 % (28401)Termination reason: Unknown
% 3.09/1.68 % (28401)Termination phase: shuffling
% 3.09/1.68
% 3.09/1.68 % (28401)Memory used [KB]: 1918
% 3.09/1.68 % (28401)Time elapsed: 0.012 s
% 3.09/1.68 % (28401)Instructions burned: 16 (million)
% 3.09/1.68 % (28401)------------------------------
% 3.09/1.68 % (28401)------------------------------
% 3.09/1.68 % (28405)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_28173 for (2999ds/16Mi)
% 3.09/1.68 % (28397)First to succeed.
% 3.09/1.68 % (28405)Instruction limit reached!
% 3.09/1.68 % (28405)------------------------------
% 3.09/1.68 % (28405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.68 % (28405)Termination reason: Unknown
% 3.09/1.68 % (28405)Termination phase: shuffling
% 3.09/1.68
% 3.09/1.68 % (28405)Memory used [KB]: 1918
% 3.09/1.68 % (28405)Time elapsed: 0.007 s
% 3.09/1.68 % (28405)Instructions burned: 18 (million)
% 3.09/1.68 % (28405)------------------------------
% 3.09/1.68 % (28405)------------------------------
% 3.09/1.68 % (28397)Refutation found. Thanks to Tanya!
% 3.09/1.68 % SZS status Theorem for DTF2THF_28173
% 3.09/1.68 % SZS output start Proof for DTF2THF_28173
% 3.09/1.68 thf(type_def_5, type, multis599418605od_a_a: $tType).
% 3.09/1.68 thf(type_def_7, type, multiset_a: $tType).
% 3.09/1.68 thf(type_def_8, type, product_prod_a_a: $tType).
% 3.09/1.68 thf(type_def_9, type, a: $tType).
% 3.09/1.68 thf(type_def_10, type, multis1735859763od_a_a: $tType).
% 3.09/1.68 thf(type_def_11, type, multiset_multiset_a: $tType).
% 3.09/1.68 thf(type_def_12, type, set_Pr1948701895od_a_a: $tType).
% 3.09/1.68 thf(type_def_13, type, set_Pr1821581383od_a_a: $tType).
% 3.09/1.68 thf(type_def_14, type, set_Product_prod_a_a: $tType).
% 3.09/1.68 thf(type_def_15, type, set_Pr158363655iset_a: $tType).
% 3.09/1.68 thf(type_def_16, type, set_mu1281098573od_a_a: $tType).
% 3.09/1.68 thf(type_def_17, type, set_multiset_a: $tType).
% 3.09/1.68 thf(type_def_18, type, set_a: $tType).
% 3.09/1.68 thf(type_def_19, type, set_Pr1930443975od_a_a: $tType).
% 3.09/1.68 thf(type_def_20, type, set_Pr146455751iset_a: $tType).
% 3.09/1.68 thf(type_def_21, type, set_Pr1295299783od_a_a: $tType).
% 3.09/1.68 thf(type_def_22, type, produc2061001575od_a_a: $tType).
% 3.09/1.68 thf(type_def_23, type, produc1127127335iset_a: $tType).
% 3.09/1.68 thf(type_def_24, type, produc1572603623od_a_a: $tType).
% 3.09/1.68 thf(type_def_25, type, produc562496743od_a_a: $tType).
% 3.09/1.68 thf(type_def_26, type, produc558600423iset_a: $tType).
% 3.09/1.68 thf(type_def_27, type, produc1990712039od_a_a: $tType).
% 3.09/1.68 thf(func_def_0, type, set_Pr1930443975od_a_a: $tType).
% 3.09/1.68 thf(func_def_1, type, produc562496743od_a_a: $tType).
% 3.09/1.68 thf(func_def_2, type, set_Pr1295299783od_a_a: $tType).
% 3.09/1.68 thf(func_def_3, type, produc1990712039od_a_a: $tType).
% 3.09/1.68 thf(func_def_4, type, set_Pr146455751iset_a: $tType).
% 3.09/1.68 thf(func_def_5, type, produc558600423iset_a: $tType).
% 3.09/1.68 thf(func_def_6, type, set_Pr1821581383od_a_a: $tType).
% 3.09/1.68 thf(func_def_7, type, produc2061001575od_a_a: $tType).
% 3.09/1.68 thf(func_def_8, type, set_Pr1948701895od_a_a: $tType).
% 3.09/1.68 thf(func_def_9, type, produc1572603623od_a_a: $tType).
% 3.09/1.68 thf(func_def_10, type, set_Pr158363655iset_a: $tType).
% 3.09/1.68 thf(func_def_11, type, produc1127127335iset_a: $tType).
% 3.09/1.68 thf(func_def_12, type, multis1735859763od_a_a: $tType).
% 3.09/1.68 thf(func_def_13, type, set_mu1281098573od_a_a: $tType).
% 3.09/1.68 thf(func_def_14, type, multis599418605od_a_a: $tType).
% 3.09/1.68 thf(func_def_15, type, set_Product_prod_a_a: $tType).
% 3.09/1.68 thf(func_def_16, type, multiset_multiset_a: $tType).
% 3.09/1.68 thf(func_def_17, type, set_multiset_a: $tType).
% 3.09/1.68 thf(func_def_18, type, product_prod_a_a: $tType).
% 3.09/1.68 thf(func_def_19, type, multiset_a: $tType).
% 3.09/1.68 thf(func_def_20, type, set_a: $tType).
% 3.09/1.68 thf(func_def_21, type, a: $tType).
% 3.09/1.68 thf(func_def_22, type, minus_1257885652od_a_a: multis599418605od_a_a > multis599418605od_a_a > multis599418605od_a_a).
% 3.09/1.68 thf(func_def_23, type, minus_1649712171iset_a: multiset_a > multiset_a > multiset_a).
% 3.09/1.68 thf(func_def_24, type, plus_p404985124od_a_a: multis599418605od_a_a > multis599418605od_a_a > multis599418605od_a_a).
% 3.09/1.68 thf(func_def_25, type, plus_plus_multiset_a: multiset_a > multiset_a > multiset_a).
% 3.09/1.68 thf(func_def_26, type, zero_z160849828od_a_a: multis599418605od_a_a).
% 3.09/1.68 thf(func_def_27, type, zero_zero_multiset_a: multiset_a).
% 3.09/1.68 thf(func_def_28, type, add_ms1725977093od_a_a: product_prod_a_a > multis599418605od_a_a > multis599418605od_a_a).
% 3.09/1.68 thf(func_def_29, type, add_mset_a: a > multiset_a > multiset_a).
% 3.09/1.68 thf(func_def_30, type, fold_m773816523od_a_a: (multis599418605od_a_a > multis599418605od_a_a > multis599418605od_a_a) > multis599418605od_a_a > multis1735859763od_a_a > multis599418605od_a_a).
% 3.09/1.68 thf(func_def_31, type, fold_m382157835iset_a: (multiset_a > multiset_a > multiset_a) > multiset_a > multiset_multiset_a > multiset_a).
% 3.09/1.68 thf(func_def_32, type, mult1_1719650714od_a_a: set_Pr1948701895od_a_a > set_Pr1821581383od_a_a).
% 3.09/1.68 thf(func_def_33, type, mult1_a: set_Product_prod_a_a > set_Pr158363655iset_a).
% 3.09/1.68 thf(func_def_34, type, mult_a: set_Product_prod_a_a > set_Pr158363655iset_a).
% 3.09/1.68 thf(func_def_35, type, multp_a: (a > a > $o) > multiset_a > multiset_a > $o).
% 3.09/1.68 thf(func_def_36, type, set_ms1869569194od_a_a: multis1735859763od_a_a > set_mu1281098573od_a_a).
% 3.09/1.68 thf(func_def_37, type, set_mset_multiset_a: multiset_multiset_a > set_multiset_a).
% 3.09/1.68 thf(func_def_38, type, set_ms1818337124od_a_a: multis599418605od_a_a > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_39, type, set_mset_a: multiset_a > set_a).
% 3.09/1.68 thf(func_def_40, type, order_1040740996od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o).
% 3.09/1.68 thf(func_def_41, type, order_750350043iset_a: set_multiset_a > set_Pr158363655iset_a > $o).
% 3.09/1.68 thf(func_def_42, type, order_1210379134od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o).
% 3.09/1.68 thf(func_def_43, type, order_761691966iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o).
% 3.09/1.68 thf(func_def_44, type, order_491428350od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 3.09/1.68 thf(func_def_45, type, order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_46, type, order_preorder_on_a: set_a > set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_47, type, order_1505515173od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_48, type, order_1558116988r_on_a: set_a > set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_49, type, ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_50, type, ord_less_eq_set_a: set_a > set_a > $o).
% 3.09/1.68 thf(func_def_51, type, top_to1483957574_a_a_o: product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_52, type, top_top_a_o: a > $o).
% 3.09/1.68 thf(func_def_53, type, top_to398455383od_a_a: set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_54, type, top_top_set_a: set_a).
% 3.09/1.68 thf(func_def_55, type, prefer1191731497od_a_a: multis599418605od_a_a > set_mu1281098573od_a_a > set_Pr1821581383od_a_a > set_mu1281098573od_a_a).
% 3.09/1.68 thf(func_def_56, type, prefer2130004608iset_a: multiset_a > set_multiset_a > set_Pr158363655iset_a > set_multiset_a).
% 3.09/1.68 thf(func_def_57, type, prefer737894307od_a_a: produc2061001575od_a_a > set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > set_Pr1821581383od_a_a).
% 3.09/1.68 thf(func_def_58, type, prefer619498851iset_a: produc1127127335iset_a > set_Pr158363655iset_a > set_Pr146455751iset_a > set_Pr158363655iset_a).
% 3.09/1.68 thf(func_def_59, type, prefer55414563od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > set_Pr1948701895od_a_a).
% 3.09/1.68 thf(func_def_60, type, prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_61, type, prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a).
% 3.09/1.68 thf(func_def_62, type, prefer1340910575od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o).
% 3.09/1.68 thf(func_def_63, type, prefer1571048262iset_a: set_multiset_a > set_Pr158363655iset_a > $o).
% 3.09/1.68 thf(func_def_64, type, prefer273553257od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o).
% 3.09/1.68 thf(func_def_65, type, prefer1608839465iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o).
% 3.09/1.68 thf(func_def_66, type, prefer1409946857od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 3.09/1.68 thf(func_def_67, type, prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_68, type, prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_69, type, prefer1669415233od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o).
% 3.09/1.68 thf(func_def_70, type, prefer1779827352iset_a: set_multiset_a > set_Pr158363655iset_a > $o).
% 3.09/1.68 thf(func_def_71, type, prefer933983163od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o).
% 3.09/1.68 thf(func_def_72, type, prefer841781627iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o).
% 3.09/1.68 thf(func_def_73, type, prefer1828051771od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 3.09/1.68 thf(func_def_74, type, prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_75, type, prefer719835986ence_a: set_a > set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_76, type, prefer1064779160od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_77, type, prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_78, type, produc1859144151od_a_a: multis599418605od_a_a > multis599418605od_a_a > produc2061001575od_a_a).
% 3.09/1.68 thf(func_def_79, type, produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a).
% 3.09/1.68 thf(func_def_80, type, produc978545495od_a_a: produc2061001575od_a_a > produc2061001575od_a_a > produc562496743od_a_a).
% 3.09/1.68 thf(func_def_81, type, produc1898392407iset_a: produc1127127335iset_a > produc1127127335iset_a > produc558600423iset_a).
% 3.09/1.68 thf(func_def_82, type, produc1935643479od_a_a: produc1572603623od_a_a > produc1572603623od_a_a > produc1990712039od_a_a).
% 3.09/1.68 thf(func_def_83, type, produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a).
% 3.09/1.68 thf(func_def_84, type, product_Pair_a_a: a > a > product_prod_a_a).
% 3.09/1.68 thf(func_def_85, type, produc1548936141_a_a_o: (produc2061001575od_a_a > $o) > multis599418605od_a_a > multis599418605od_a_a > $o).
% 3.09/1.68 thf(func_def_86, type, produc1530159757et_a_o: (produc1127127335iset_a > $o) > multiset_a > multiset_a > $o).
% 3.09/1.68 thf(func_def_87, type, produc94438221_a_a_o: (produc1572603623od_a_a > $o) > product_prod_a_a > product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_88, type, product_curry_a_a_o: (product_prod_a_a > $o) > a > a > $o).
% 3.09/1.68 thf(func_def_89, type, produc143372156od_a_a: (product_prod_a_a > set_Product_prod_a_a) > a > a > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_90, type, produc580150227_set_a: (product_prod_a_a > set_a) > a > a > set_a).
% 3.09/1.68 thf(func_def_91, type, produc748978966_a_a_o: (a > a > $o) > product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_92, type, produc1653225349od_a_a: (a > a > set_Product_prod_a_a) > product_prod_a_a > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_93, type, produc694541532_set_a: (a > a > set_a) > product_prod_a_a > set_a).
% 3.09/1.68 thf(func_def_94, type, produc700022774_a_a_a: (a > a) > (a > a) > product_prod_a_a > product_prod_a_a).
% 3.09/1.68 thf(func_def_95, type, produc429414904iset_a: (multiset_a > multiset_a > set_multiset_a) > produc1127127335iset_a > set_multiset_a).
% 3.09/1.68 thf(func_def_96, type, produc2041367259od_a_a: (multiset_a > multiset_a > set_Product_prod_a_a) > produc1127127335iset_a > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_97, type, produc747255346_set_a: (multiset_a > multiset_a > set_a) > produc1127127335iset_a > set_a).
% 3.09/1.68 thf(func_def_98, type, produc734625720iset_a: (product_prod_a_a > product_prod_a_a > set_multiset_a) > produc1572603623od_a_a > set_multiset_a).
% 3.09/1.68 thf(func_def_99, type, produc869579762_set_a: (product_prod_a_a > product_prod_a_a > set_a) > produc1572603623od_a_a > set_a).
% 3.09/1.68 thf(func_def_100, type, produc1833107820_a_a_o: (a > a > $o) > product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_101, type, produc1121147169od_a_a: (a > a > set_mu1281098573od_a_a) > product_prod_a_a > set_mu1281098573od_a_a).
% 3.09/1.68 thf(func_def_102, type, produc1828150264iset_a: (a > a > set_multiset_a) > product_prod_a_a > set_multiset_a).
% 3.09/1.68 thf(func_def_103, type, produc93594587iset_a: (a > a > set_Pr158363655iset_a) > product_prod_a_a > set_Pr158363655iset_a).
% 3.09/1.68 thf(func_def_104, type, produc1093061851od_a_a: (a > a > set_Product_prod_a_a) > product_prod_a_a > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_105, type, produc961951794_set_a: (a > a > set_a) > product_prod_a_a > set_a).
% 3.09/1.68 thf(func_def_106, type, product_fst_a_a: product_prod_a_a > a).
% 3.09/1.68 thf(func_def_107, type, product_snd_a_a: product_prod_a_a > a).
% 3.09/1.68 thf(func_def_108, type, produc1913956215od_a_a: produc2061001575od_a_a > produc2061001575od_a_a).
% 3.09/1.68 thf(func_def_109, type, produc506430135iset_a: produc1127127335iset_a > produc1127127335iset_a).
% 3.09/1.68 thf(func_def_110, type, produc1641853175od_a_a: produc1572603623od_a_a > produc1572603623od_a_a).
% 3.09/1.68 thf(func_def_111, type, product_swap_a_a: product_prod_a_a > product_prod_a_a).
% 3.09/1.68 thf(func_def_112, type, asym_a: set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_113, type, inv_im1681759523_a_a_a: set_Pr1821581383od_a_a > (a > multis599418605od_a_a) > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_114, type, inv_im343774418iset_a: set_Pr158363655iset_a > (multiset_a > multiset_a) > set_Pr158363655iset_a).
% 3.09/1.68 thf(func_def_115, type, inv_im1716587189od_a_a: set_Pr158363655iset_a > (product_prod_a_a > multiset_a) > set_Pr1948701895od_a_a).
% 3.09/1.68 thf(func_def_116, type, inv_im975322764et_a_a: set_Pr158363655iset_a > (a > multiset_a) > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_117, type, inv_im184450735iset_a: set_Pr1948701895od_a_a > (multiset_a > product_prod_a_a) > set_Pr158363655iset_a).
% 3.09/1.68 thf(func_def_118, type, inv_im777352425_a_a_a: set_Pr1948701895od_a_a > (a > product_prod_a_a) > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_119, type, inv_im941243329od_a_a: set_Product_prod_a_a > (multis599418605od_a_a > a) > set_Pr1821581383od_a_a).
% 3.09/1.68 thf(func_def_120, type, inv_im479034648iset_a: set_Product_prod_a_a > (multiset_a > a) > set_Pr158363655iset_a).
% 3.09/1.68 thf(func_def_121, type, inv_im15016955od_a_a: set_Product_prod_a_a > (product_prod_a_a > a) > set_Pr1948701895od_a_a).
% 3.09/1.68 thf(func_def_122, type, inv_image_a_a: set_Product_prod_a_a > (a > a) > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_123, type, irrefl1672195874od_a_a: set_Pr1821581383od_a_a > $o).
% 3.09/1.68 thf(func_def_124, type, irrefl_multiset_a: set_Pr158363655iset_a > $o).
% 3.09/1.68 thf(func_def_125, type, irrefl76447196od_a_a: set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_126, type, irrefl_a: set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_127, type, refl_o581572716od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o).
% 3.09/1.68 thf(func_def_128, type, refl_on_multiset_a: set_multiset_a > set_Pr158363655iset_a > $o).
% 3.09/1.68 thf(func_def_129, type, refl_o102522214od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o).
% 3.09/1.68 thf(func_def_130, type, refl_o1522155814iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o).
% 3.09/1.68 thf(func_def_131, type, refl_o198693862od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 3.09/1.68 thf(func_def_132, type, refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_133, type, refl_on_a: set_a > set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_134, type, total_1024184433od_a_a: set_mu1281098573od_a_a > set_Pr1821581383od_a_a > $o).
% 3.09/1.68 thf(func_def_135, type, total_on_multiset_a: set_multiset_a > set_Pr158363655iset_a > $o).
% 3.09/1.68 thf(func_def_136, type, total_1170533099od_a_a: set_Pr1821581383od_a_a > set_Pr1930443975od_a_a > $o).
% 3.09/1.68 thf(func_def_137, type, total_313747627iset_a: set_Pr158363655iset_a > set_Pr146455751iset_a > $o).
% 3.09/1.68 thf(func_def_138, type, total_1340260971od_a_a: set_Pr1948701895od_a_a > set_Pr1295299783od_a_a > $o).
% 3.09/1.68 thf(func_def_139, type, total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_140, type, total_on_a: set_a > set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_141, type, trans_1520871884od_a_a: set_Pr1821581383od_a_a > $o).
% 3.09/1.68 thf(func_def_142, type, trans_multiset_a: set_Pr158363655iset_a > $o).
% 3.09/1.68 thf(func_def_143, type, trans_972827014od_a_a: set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_144, type, trans_a: set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_145, type, collec1948052440od_a_a: (multis599418605od_a_a > $o) > set_mu1281098573od_a_a).
% 3.09/1.68 thf(func_def_146, type, collect_multiset_a: (multiset_a > $o) > set_multiset_a).
% 3.09/1.68 thf(func_def_147, type, collec8264402od_a_a: (produc2061001575od_a_a > $o) > set_Pr1821581383od_a_a).
% 3.09/1.68 thf(func_def_148, type, collec1104713362iset_a: (produc1127127335iset_a > $o) > set_Pr158363655iset_a).
% 3.09/1.68 thf(func_def_149, type, collec1635618130od_a_a: (produc1572603623od_a_a > $o) > set_Pr1948701895od_a_a).
% 3.09/1.68 thf(func_def_150, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_151, type, collect_a: (a > $o) > set_a).
% 3.09/1.68 thf(func_def_152, type, image_794773445od_a_a: (product_prod_a_a > product_prod_a_a) > set_Product_prod_a_a > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_153, type, image_340790172_a_a_a: (product_prod_a_a > a) > set_Product_prod_a_a > set_a).
% 3.09/1.68 thf(func_def_154, type, image_1725938350od_a_a: (a > product_prod_a_a) > set_a > set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_155, type, image_a_a: (a > a) > set_a > set_a).
% 3.09/1.68 thf(func_def_156, type, lex_prod_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a).
% 3.09/1.68 thf(func_def_157, type, same_fst_a_a: (a > $o) > (a > set_Product_prod_a_a) > set_Pr1948701895od_a_a).
% 3.09/1.68 thf(func_def_158, type, member1755255446od_a_a: multis599418605od_a_a > set_mu1281098573od_a_a > $o).
% 3.09/1.68 thf(func_def_159, type, member_multiset_a: multiset_a > set_multiset_a > $o).
% 3.09/1.68 thf(func_def_160, type, member1349055120od_a_a: produc2061001575od_a_a > set_Pr1821581383od_a_a > $o).
% 3.09/1.68 thf(func_def_161, type, member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o).
% 3.09/1.68 thf(func_def_162, type, member2007374608od_a_a: produc562496743od_a_a > set_Pr1930443975od_a_a > $o).
% 3.09/1.68 thf(func_def_163, type, member567864080iset_a: produc558600423iset_a > set_Pr146455751iset_a > $o).
% 3.09/1.68 thf(func_def_164, type, member1899387664od_a_a: produc1990712039od_a_a > set_Pr1295299783od_a_a > $o).
% 3.09/1.68 thf(func_def_165, type, member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o).
% 3.09/1.68 thf(func_def_166, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 3.09/1.68 thf(func_def_167, type, member_a: a > set_a > $o).
% 3.09/1.68 thf(func_def_168, type, carrier: set_a).
% 3.09/1.68 thf(func_def_169, type, relation: set_Product_prod_a_a).
% 3.09/1.68 thf(func_def_170, type, x: a).
% 3.09/1.68 thf(func_def_171, type, y: a).
% 3.09/1.68 thf(func_def_187, type, sK0: (product_prod_a_a > $o) > a).
% 3.09/1.68 thf(func_def_188, type, sK1: (product_prod_a_a > $o) > a).
% 3.09/1.68 thf(func_def_189, type, sK2: product_prod_a_a > a).
% 3.09/1.68 thf(func_def_190, type, sK3: product_prod_a_a > a).
% 3.09/1.68 thf(func_def_191, type, sK4: (product_prod_a_a > $o) > a).
% 3.09/1.68 thf(func_def_192, type, sK5: (product_prod_a_a > $o) > a).
% 3.09/1.68 thf(func_def_193, type, sK6: product_prod_a_a > a).
% 3.09/1.68 thf(func_def_194, type, sK7: product_prod_a_a > a).
% 3.09/1.68 thf(func_def_195, type, sK8: set_Product_prod_a_a > a).
% 3.09/1.68 thf(func_def_196, type, sK9: set_Product_prod_a_a > a).
% 3.09/1.68 thf(func_def_197, type, sK10: set_Product_prod_a_a > a).
% 3.09/1.68 thf(func_def_199, type, ph12: !>[X0: $tType]:(X0)).
% 3.09/1.68 thf(f1078,plain,(
% 3.09/1.68 $false),
% 3.09/1.68 inference(subsumption_resolution,[status(thm)],[f1077,f1020])).
% 3.09/1.68 thf(f1020,plain,(
% 3.09/1.68 ((member_a @ y @ carrier) = $true)),
% 3.09/1.68 inference(cnf_transformation,[status(thm)],[f367])).
% 3.09/1.68 thf(f367,plain,(
% 3.09/1.68 ((member_a @ y @ carrier) = $true)),
% 3.09/1.68 inference(fool_elimination,[status(thm)],[f366])).
% 3.09/1.68 thf(f366,plain,(
% 3.09/1.68 (member_a @ y @ carrier)),
% 3.09/1.68 inference(rectify,[status(thm)],[f357])).
% 3.09/1.68 thf(f357,axiom,(
% 3.09/1.68 (member_a @ y @ carrier)),
% 3.09/1.68 file('/export/starexec/sandbox2/tmp/tmp.fZ0xgqz36i/DTF2THF_28173.p',conj_1)).
% 3.09/1.68 thf(f1077,plain,(
% 3.09/1.68 ((member_a @ y @ carrier) != $true)),
% 3.09/1.68 inference(subsumption_resolution,[status(thm)],[f1076,f1028])).
% 3.09/1.68 thf(f1028,plain,(
% 3.09/1.68 ((member449909584od_a_a @ (product_Pair_a_a @ y @ x) @ relation) != $true)),
% 3.09/1.68 inference(cnf_transformation,[status(thm)],[f972])).
% 3.09/1.68 thf(f972,plain,(
% 3.09/1.68 ((member449909584od_a_a @ (product_Pair_a_a @ y @ x) @ relation) != $true) & ((member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation) != $true)),
% 3.09/1.68 inference(ennf_transformation,[status(thm)],[f891])).
% 3.09/1.68 thf(f891,plain,(
% 3.09/1.68 ~(((member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation) = $true) | ((member449909584od_a_a @ (product_Pair_a_a @ y @ x) @ relation) = $true))),
% 3.09/1.68 inference(fool_elimination,[status(thm)],[f890])).
% 3.09/1.68 thf(f890,plain,(
% 3.09/1.68 ~((member449909584od_a_a @ (product_Pair_a_a @ y @ x) @ relation) | (member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation))),
% 3.09/1.68 inference(rectify,[status(thm)],[f360])).
% 3.09/1.68 thf(f360,negated_conjecture,(
% 3.09/1.68 ~((member449909584od_a_a @ (product_Pair_a_a @ y @ x) @ relation) | (member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation))),
% 3.09/1.68 inference(negated_conjecture,[status(cth)],[f359])).
% 3.09/1.68 thf(f359,conjecture,(
% 3.09/1.68 (member449909584od_a_a @ (product_Pair_a_a @ y @ x) @ relation) | (member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation)),
% 3.09/1.68 file('/export/starexec/sandbox2/tmp/tmp.fZ0xgqz36i/DTF2THF_28173.p',conj_3)).
% 3.09/1.68 thf(f1076,plain,(
% 3.09/1.68 ((member449909584od_a_a @ (product_Pair_a_a @ y @ x) @ relation) = $true) | ((member_a @ y @ carrier) != $true)),
% 3.09/1.68 inference(subsumption_resolution,[status(thm)],[f1075,f1039])).
% 3.09/1.68 thf(f1039,plain,(
% 3.09/1.68 ((member_a @ x @ carrier) = $true)),
% 3.09/1.68 inference(cnf_transformation,[status(thm)],[f409])).
% 3.09/1.68 thf(f409,plain,(
% 3.09/1.68 ((member_a @ x @ carrier) = $true)),
% 3.09/1.68 inference(fool_elimination,[status(thm)],[f408])).
% 3.09/1.68 thf(f408,plain,(
% 3.09/1.68 (member_a @ x @ carrier)),
% 3.09/1.68 inference(rectify,[status(thm)],[f356])).
% 3.09/1.68 thf(f356,axiom,(
% 3.09/1.68 (member_a @ x @ carrier)),
% 3.09/1.68 file('/export/starexec/sandbox2/tmp/tmp.fZ0xgqz36i/DTF2THF_28173.p',conj_0)).
% 3.09/1.68 thf(f1075,plain,(
% 3.09/1.68 ((member_a @ x @ carrier) != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ y @ x) @ relation) = $true) | ((member_a @ y @ carrier) != $true)),
% 3.09/1.68 inference(trivial_inequality_removal,[status(thm)],[f1061])).
% 3.09/1.68 thf(f1061,plain,(
% 3.09/1.68 ((member_a @ x @ carrier) != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ y @ x) @ relation) = $true) | ((member_a @ y @ carrier) != $true) | ($true != $true)),
% 3.09/1.68 inference(superposition,[status(thm)],[f1027,f1018])).
% 3.09/1.68 thf(f1018,plain,(
% 3.09/1.68 ( ! [X0 : a,X1 : a] : (((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) = $true) | ((member_a @ X1 @ carrier) != $true) | ((member_a @ X0 @ carrier) != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) = $true)) )),
% 3.09/1.68 inference(cnf_transformation,[status(thm)],[f993])).
% 3.09/1.68 thf(f993,plain,(
% 3.09/1.68 ! [X0 : a,X1 : a] : (((member_a @ X1 @ carrier) != $true) | ((member_a @ X0 @ carrier) != $true) | (((((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true) & ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) = $true)) | ((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) = $true)) & (((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true) | ((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) = $true) | ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) != $true))))),
% 3.09/1.68 inference(flattening,[status(thm)],[f992])).
% 3.09/1.68 thf(f992,plain,(
% 3.09/1.68 ! [X0 : a,X1 : a] : (((member_a @ X1 @ carrier) != $true) | ((member_a @ X0 @ carrier) != $true) | (((((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true) & ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) = $true)) | ((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) = $true)) & (((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true) | (((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) = $true) | ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) != $true)))))),
% 3.09/1.68 inference(nnf_transformation,[status(thm)],[f962])).
% 3.09/1.68 thf(f962,plain,(
% 3.09/1.68 ! [X0 : a,X1 : a] : (((member_a @ X1 @ carrier) != $true) | ((member_a @ X0 @ carrier) != $true) | ((((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true) & ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) = $true)) <=> ((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true)))),
% 3.09/1.68 inference(flattening,[status(thm)],[f961])).
% 3.09/1.68 thf(f961,plain,(
% 3.09/1.68 ! [X0 : a,X1 : a] : (((((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true) & ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) = $true)) <=> ((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true)) | (((member_a @ X1 @ carrier) != $true) | ((member_a @ X0 @ carrier) != $true)))),
% 3.09/1.68 inference(ennf_transformation,[status(thm)],[f954])).
% 3.09/1.68 thf(f954,plain,(
% 3.09/1.68 ! [X0 : a,X1 : a] : ((((member_a @ X1 @ carrier) = $true) & ((member_a @ X0 @ carrier) = $true)) => ((((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true) & ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) = $true)) <=> ((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) != $true)))),
% 3.09/1.68 inference(flattening,[status(thm)],[f443])).
% 3.09/1.68 thf(f443,plain,(
% 3.09/1.68 ! [X0 : a,X1 : a] : ((((member_a @ X1 @ carrier) = $true) & ((member_a @ X0 @ carrier) = $true)) => ((~((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) = $true) & ((member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) = $true)) <=> ~((member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) = $true)))),
% 3.09/1.68 inference(fool_elimination,[status(thm)],[f442])).
% 3.09/1.68 thf(f442,plain,(
% 3.09/1.68 ! [X0 : a,X1 : a] : (((member_a @ X1 @ carrier) & (member_a @ X0 @ carrier)) => (~(member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) & (member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) = ~(member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation)))),
% 3.09/1.68 inference(rectify,[status(thm)],[f6])).
% 3.09/1.68 thf(f6,axiom,(
% 3.09/1.68 ! [X1 : a,X0 : a] : (((member_a @ X0 @ carrier) & (member_a @ X1 @ carrier)) => (~(member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation) & (member449909584od_a_a @ (product_Pair_a_a @ X0 @ X1) @ relation) = ~(member449909584od_a_a @ (product_Pair_a_a @ X1 @ X0) @ relation)))),
% 3.09/1.68 file('/export/starexec/sandbox2/tmp/tmp.fZ0xgqz36i/DTF2THF_28173.p',fact_5_strict__not__refl__weak)).
% 3.09/1.68 thf(f1027,plain,(
% 3.09/1.68 ((member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation) != $true)),
% 3.09/1.68 inference(cnf_transformation,[status(thm)],[f972])).
% 3.09/1.68 % SZS output end Proof for DTF2THF_28173
% 3.09/1.68 % (28397)------------------------------
% 3.09/1.68 % (28397)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.09/1.68 % (28397)Termination reason: Refutation
% 3.09/1.68
% 3.09/1.68 % (28397)Memory used [KB]: 6908
% 3.09/1.68 % (28397)Time elapsed: 0.039 s
% 3.09/1.68 % (28397)Instructions burned: 73 (million)
% 3.09/1.68 % (28397)------------------------------
% 3.09/1.68 % (28397)------------------------------
% 3.09/1.68 % (28391)Success in time 0.081 s
%------------------------------------------------------------------------------