%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP155^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:59 AM UTC 2026
% Result : Theorem 6.31s 2.12s
% Output : Refutation 6.31s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 11
% Syntax : Number of formulae : 74 ( 26 unt; 0 typ; 2 def)
% Number of atoms : 367 ( 105 equ; 0 cnn)
% Maximal formula atoms : 4 ( 4 avg)
% Number of connectives : 794 ( 53 ~; 50 |; 4 &; 632 @)
% ( 2 <=>; 37 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 255 ( 251 usr; 16 con; 0-3 aty)
% ( 16 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 100 ( 36 ^; 62 !; 2 ?; 100 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_nat: $tType ).
thf(type_def_6,type,
nat: $tType ).
thf(type_def_8,type,
set_Product_prod_a_a: $tType ).
thf(type_def_9,type,
set_a: $tType ).
thf(type_def_10,type,
set_Pr1986765409at_nat: $tType ).
thf(type_def_11,type,
set_Pr1948701895od_a_a: $tType ).
thf(type_def_12,type,
set_set_a: $tType ).
thf(type_def_13,type,
product_prod_a_a: $tType ).
thf(type_def_14,type,
a: $tType ).
thf(type_def_15,type,
produc1691597095_set_a: $tType ).
thf(type_def_16,type,
product_prod_nat_nat: $tType ).
thf(type_def_17,type,
produc1463940355_set_a: $tType ).
thf(type_def_18,type,
produc1404140509_a_nat: $tType ).
thf(type_def_19,type,
produc1572603623od_a_a: $tType ).
thf(func_def_0,type,
set_Pr1948701895od_a_a: $tType ).
thf(func_def_1,type,
produc1572603623od_a_a: $tType ).
thf(func_def_2,type,
produc1691597095_set_a: $tType ).
thf(func_def_3,type,
set_Pr1986765409at_nat: $tType ).
thf(func_def_4,type,
produc1404140509_a_nat: $tType ).
thf(func_def_5,type,
produc1463940355_set_a: $tType ).
thf(func_def_6,type,
set_Product_prod_a_a: $tType ).
thf(func_def_7,type,
product_prod_nat_nat: $tType ).
thf(func_def_8,type,
product_prod_a_a: $tType ).
thf(func_def_9,type,
set_set_a: $tType ).
thf(func_def_10,type,
set_nat: $tType ).
thf(func_def_11,type,
set_a: $tType ).
thf(func_def_12,type,
nat: $tType ).
thf(func_def_13,type,
a: $tType ).
thf(func_def_14,type,
finite_card_nat: set_nat > nat ).
thf(func_def_15,type,
finite1481642319od_a_a: set_Product_prod_a_a > nat ).
thf(func_def_16,type,
finite_card_a: set_a > nat ).
thf(func_def_17,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_18,type,
finite772653738at_nat: set_Pr1986765409at_nat > $o ).
thf(func_def_19,type,
finite1664988688od_a_a: set_Pr1948701895od_a_a > $o ).
thf(func_def_20,type,
finite179568208od_a_a: set_Product_prod_a_a > $o ).
thf(func_def_21,type,
finite_finite_set_a: set_set_a > $o ).
thf(func_def_22,type,
finite_finite_a: set_a > $o ).
thf(func_def_23,type,
zero_zero_nat: nat ).
thf(func_def_24,type,
order_1150839801on_nat: set_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_25,type,
order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_26,type,
order_preorder_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_27,type,
bot_bot_nat_o: nat > $o ).
thf(func_def_28,type,
bot_bo1293121322_a_a_o: product_prod_a_a > $o ).
thf(func_def_29,type,
bot_bot_a_o: a > $o ).
thf(func_def_30,type,
bot_bot_nat: nat ).
thf(func_def_31,type,
bot_bo1519632275_set_a: produc1691597095_set_a ).
thf(func_def_32,type,
bot_bot_set_nat: set_nat ).
thf(func_def_33,type,
bot_bo2131659635od_a_a: set_Product_prod_a_a ).
thf(func_def_34,type,
bot_bot_set_set_a: set_set_a ).
thf(func_def_35,type,
bot_bot_set_a: set_a ).
thf(func_def_36,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_37,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_38,type,
ord_le1302190241at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_39,type,
ord_le841160035_set_a: produc1463940355_set_a > produc1463940355_set_a > $o ).
thf(func_def_40,type,
ord_le781360189_a_nat: produc1404140509_a_nat > produc1404140509_a_nat > $o ).
thf(func_def_41,type,
ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
thf(func_def_42,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_43,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_44,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_45,type,
prefer1936906324as_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
thf(func_def_46,type,
prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_47,type,
prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_48,type,
prefer563798164od_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
thf(func_def_49,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_50,type,
prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_51,type,
prefer1672638789an_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
thf(func_def_52,type,
prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
thf(func_def_53,type,
prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a ).
thf(func_def_54,type,
prefer1293349710ce_nat: set_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_55,type,
prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_56,type,
prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_57,type,
prefer1147844220ce_nat: set_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_58,type,
prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_59,type,
prefer719835986ence_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_60,type,
prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_61,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_62,type,
produc1450889781_set_a: nat > set_a > produc1463940355_set_a ).
thf(func_def_63,type,
produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
thf(func_def_64,type,
produc515611863_a_nat: set_a > nat > produc1404140509_a_nat ).
thf(func_def_65,type,
produc1928581911_set_a: set_a > set_a > produc1691597095_set_a ).
thf(func_def_66,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_67,type,
refl_on_nat: set_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_68,type,
refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_69,type,
refl_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_70,type,
total_on_nat: set_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_71,type,
total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_72,type,
total_on_a: set_a > set_Product_prod_a_a > $o ).
thf(func_def_73,type,
trans_a: set_Product_prod_a_a > $o ).
thf(func_def_74,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_75,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_76,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_77,type,
is_empty_a: set_a > $o ).
thf(func_def_78,type,
member_nat: nat > set_nat > $o ).
thf(func_def_79,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_80,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_81,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_82,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_83,type,
member_a: a > set_a > $o ).
thf(func_def_84,type,
carrier: set_a ).
thf(func_def_85,type,
relation: set_Product_prod_a_a ).
thf(func_def_86,type,
x: a ).
thf(func_def_87,type,
y: a ).
thf(func_def_103,type,
sK0:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_104,type,
sK1:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_105,type,
sP2: ( product_prod_a_a > product_prod_a_a > $o ) > set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_106,type,
sP3: set_Product_prod_a_a > ( a > product_prod_a_a > $o ) > set_a > $o ).
thf(func_def_107,type,
sP4: set_nat > set_a > ( a > nat > $o ) > $o ).
thf(func_def_108,type,
sP5: ( product_prod_a_a > nat > $o ) > set_Product_prod_a_a > set_nat > $o ).
thf(func_def_109,type,
sP6: nat > nat > nat > $o ).
thf(func_def_110,type,
sP7: nat > nat > nat > $o ).
thf(func_def_111,type,
sP8: nat > nat > nat > $o ).
thf(func_def_112,type,
sP9: ( nat > product_prod_a_a > $o ) > set_nat > set_Product_prod_a_a > $o ).
thf(func_def_113,type,
sP10: ( product_prod_a_a > a > $o ) > set_a > set_Product_prod_a_a > $o ).
thf(func_def_114,type,
sP11: set_nat > set_nat > ( nat > nat > $o ) > $o ).
thf(func_def_115,type,
sP12: set_a > ( a > a > $o ) > set_a > $o ).
thf(func_def_116,type,
sP13: set_a > set_nat > ( nat > a > $o ) > $o ).
thf(func_def_117,type,
sK14: ( set_a > set_a ) > set_a ).
thf(func_def_118,type,
sK15: ( set_a > set_a ) > set_a ).
thf(func_def_119,type,
sK16: nat > set_a > set_a ).
thf(func_def_120,type,
sK17: product_prod_a_a > a ).
thf(func_def_121,type,
sK18: product_prod_a_a > a ).
thf(func_def_122,type,
sK19: ( product_prod_a_a > product_prod_a_a > $o ) > set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_123,type,
sK20: ( product_prod_a_a > product_prod_a_a > $o ) > set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_124,type,
sK21: ( product_prod_a_a > product_prod_a_a > $o ) > set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_125,type,
sK22: set_Product_prod_a_a > ( product_prod_a_a > product_prod_a_a > $o ) > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_126,type,
sK23: ( nat > nat > $o ) > nat ).
thf(func_def_127,type,
sK24: ( nat > nat > $o ) > nat ).
thf(func_def_128,type,
sK25: ( nat > nat > $o ) > nat ).
thf(func_def_129,type,
sK26: ( nat > nat > $o ) > nat ).
thf(func_def_130,type,
sK27: ( product_prod_a_a > product_prod_a_a > $o ) > product_prod_a_a ).
thf(func_def_131,type,
sK28: ( product_prod_a_a > product_prod_a_a > $o ) > product_prod_a_a ).
thf(func_def_132,type,
sK29: ( product_prod_a_a > product_prod_a_a > $o ) > product_prod_a_a ).
thf(func_def_133,type,
sK30: ( product_prod_a_a > product_prod_a_a > $o ) > product_prod_a_a ).
thf(func_def_134,type,
sK31: ( product_prod_a_a > product_prod_a_a > $o ) > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_135,type,
sK32: set_a > set_a > a ).
thf(func_def_136,type,
sK33: ( nat > $o ) > nat ).
thf(func_def_137,type,
sK34: ( nat > $o ) > nat > nat ).
thf(func_def_138,type,
sK35: ( a > nat > $o ) > set_a > a ).
thf(func_def_139,type,
sK36: ( a > nat > $o ) > set_a > nat ).
thf(func_def_140,type,
sK37: a > ( a > nat > $o ) > set_a > nat ).
thf(func_def_141,type,
sK38: set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_142,type,
sK39: ( set_a > nat ) > set_a ).
thf(func_def_143,type,
sK40: ( set_a > nat ) > set_a ).
thf(func_def_144,type,
sK41: nat > set_nat > set_nat ).
thf(func_def_145,type,
sK42: ( a > a > $o ) > a ).
thf(func_def_146,type,
sK43: ( a > a > $o ) > a ).
thf(func_def_147,type,
sK44: ( a > a > $o ) > a ).
thf(func_def_148,type,
sK45: ( a > a > $o ) > a ).
thf(func_def_149,type,
sK46: set_a > ( a > a > $o ) > a ).
thf(func_def_150,type,
sK47: nat > set_a > set_a ).
thf(func_def_151,type,
sK48: set_Product_prod_a_a > ( a > product_prod_a_a > $o ) > set_a > product_prod_a_a ).
thf(func_def_152,type,
sK49: set_Product_prod_a_a > ( a > product_prod_a_a > $o ) > set_a > a ).
thf(func_def_153,type,
sK50: set_Product_prod_a_a > ( a > product_prod_a_a > $o ) > set_a > a ).
thf(func_def_154,type,
sK51: set_a > ( a > product_prod_a_a > $o ) > set_Product_prod_a_a > a ).
thf(func_def_155,type,
sK52: set_Product_prod_a_a > nat > set_Product_prod_a_a ).
thf(func_def_156,type,
sK53: set_nat > nat > set_nat ).
thf(func_def_157,type,
sK54: ( nat > set_a ) > nat ).
thf(func_def_158,type,
sK55: ( nat > set_a ) > nat ).
thf(func_def_159,type,
sK56: ( set_a > nat ) > set_a ).
thf(func_def_160,type,
sK57: ( set_a > nat ) > set_a ).
thf(func_def_161,type,
sK58: nat > ( nat > $o ) > nat ).
thf(func_def_162,type,
sK59: ( nat > $o ) > nat ).
thf(func_def_163,type,
sK60: set_nat > set_a > ( a > nat > $o ) > a ).
thf(func_def_164,type,
sK61: set_nat > set_a > ( a > nat > $o ) > nat ).
thf(func_def_165,type,
sK62: set_nat > set_a > ( a > nat > $o ) > a ).
thf(func_def_166,type,
sK63: set_nat > set_a > ( a > nat > $o ) > a ).
thf(func_def_167,type,
sK64: ( nat > set_a ) > nat ).
thf(func_def_168,type,
sK65: ( nat > set_a ) > nat ).
thf(func_def_169,type,
sK66: set_Product_prod_a_a > a ).
thf(func_def_170,type,
sK67: set_Product_prod_a_a > a ).
thf(func_def_171,type,
sK68: set_Product_prod_a_a > a ).
thf(func_def_172,type,
sK69: set_set_a > set_a > set_a ).
thf(func_def_173,type,
sK70: ( set_a > set_a ) > set_a ).
thf(func_def_174,type,
sK71: ( set_a > set_a ) > set_a ).
thf(func_def_175,type,
sK72: set_Product_prod_a_a > set_Product_prod_a_a > a ).
thf(func_def_176,type,
sK73: set_Product_prod_a_a > set_Product_prod_a_a > a ).
thf(func_def_177,type,
sK74: nat > set_nat > nat ).
thf(func_def_178,type,
sK75: set_nat > set_nat > nat ).
thf(func_def_179,type,
sK76: ( product_prod_a_a > $o ) > a ).
thf(func_def_180,type,
sK77: ( product_prod_a_a > $o ) > a ).
thf(func_def_181,type,
sK78: ( product_prod_a_a > nat > $o ) > set_Product_prod_a_a > set_nat > product_prod_a_a ).
thf(func_def_182,type,
sK79: ( product_prod_a_a > nat > $o ) > set_Product_prod_a_a > set_nat > product_prod_a_a ).
thf(func_def_183,type,
sK80: ( product_prod_a_a > nat > $o ) > set_Product_prod_a_a > set_nat > nat ).
thf(func_def_184,type,
sK81: set_Product_prod_a_a > ( product_prod_a_a > nat > $o ) > set_nat > product_prod_a_a ).
thf(func_def_185,type,
sK82: set_set_a > set_a ).
thf(func_def_186,type,
sK83: ( nat > nat > $o ) > set_nat > nat ).
thf(func_def_187,type,
sK84: ( nat > nat > $o ) > set_nat > nat ).
thf(func_def_188,type,
sK85: nat > ( nat > nat > $o ) > set_nat > nat ).
thf(func_def_189,type,
sK86: set_a > nat > set_a ).
thf(func_def_190,type,
sK87: set_nat > nat > nat ).
thf(func_def_191,type,
sK88: ( nat > set_a ) > nat ).
thf(func_def_192,type,
sK89: ( nat > set_a ) > nat ).
thf(func_def_193,type,
sK90: set_Product_prod_a_a > set_a > a ).
thf(func_def_194,type,
sK91: set_Product_prod_a_a > set_a > a ).
thf(func_def_195,type,
sK92: set_Product_prod_a_a > set_a > a ).
thf(func_def_196,type,
sK93: set_Product_prod_a_a > set_a > a ).
thf(func_def_197,type,
sK94: ( nat > nat ) > nat ).
thf(func_def_198,type,
sK95: ( nat > nat ) > nat ).
thf(func_def_199,type,
sK96: set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_200,type,
sK97: set_a > a ).
thf(func_def_201,type,
sK98: product_prod_a_a > a ).
thf(func_def_202,type,
sK99: product_prod_a_a > a ).
thf(func_def_203,type,
sK100: set_nat > set_Pr1986765409at_nat > nat ).
thf(func_def_204,type,
sK101: set_nat > set_Pr1986765409at_nat > nat ).
thf(func_def_205,type,
sK102: set_nat > set_Pr1986765409at_nat > nat ).
thf(func_def_206,type,
sK103: set_nat > set_Pr1986765409at_nat > nat ).
thf(func_def_207,type,
sK104: set_Pr1948701895od_a_a > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_208,type,
sK105: set_Pr1948701895od_a_a > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_209,type,
sK106: ( nat > product_prod_a_a > $o ) > set_nat > set_Product_prod_a_a > nat ).
thf(func_def_210,type,
sK107: ( nat > product_prod_a_a > $o ) > set_nat > set_Product_prod_a_a > nat ).
thf(func_def_211,type,
sK108: ( nat > product_prod_a_a > $o ) > set_nat > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_212,type,
sK109: set_Product_prod_a_a > ( nat > product_prod_a_a > $o ) > set_nat > nat ).
thf(func_def_213,type,
sK110: ( nat > nat > $o ) > set_nat > nat ).
thf(func_def_214,type,
sK111: ( nat > nat > $o ) > nat ).
thf(func_def_215,type,
sK112: ( nat > nat > $o ) > nat ).
thf(func_def_216,type,
sK113: ( nat > nat > $o ) > nat ).
thf(func_def_217,type,
sK114: ( nat > nat > $o ) > nat ).
thf(func_def_218,type,
sK115: set_a > a ).
thf(func_def_219,type,
sK116: set_a > set_set_a > set_a ).
thf(func_def_220,type,
sK117: ( nat > nat ) > nat ).
thf(func_def_221,type,
sK118: ( nat > nat ) > nat ).
thf(func_def_222,type,
sK119: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_prod_a_a ).
thf(func_def_223,type,
sK120: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_prod_a_a ).
thf(func_def_224,type,
sK121: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_prod_a_a ).
thf(func_def_225,type,
sK122: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_prod_a_a ).
thf(func_def_226,type,
sK123: set_Pr1986765409at_nat > set_nat > nat ).
thf(func_def_227,type,
sK124: set_Pr1986765409at_nat > set_nat > nat ).
thf(func_def_228,type,
sK125: set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_229,type,
sK126: set_nat > nat ).
thf(func_def_230,type,
sK127: set_a > nat > set_a ).
thf(func_def_231,type,
sK128: ( nat > nat ) > nat ).
thf(func_def_232,type,
sK129: ( nat > nat ) > nat ).
thf(func_def_233,type,
sK130: ( set_a > nat ) > set_a ).
thf(func_def_234,type,
sK131: ( set_a > nat ) > set_a ).
thf(func_def_235,type,
sK132: set_set_a > set_a ).
thf(func_def_236,type,
sK133: ( product_prod_a_a > a > $o ) > set_a > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_237,type,
sK134: ( product_prod_a_a > a > $o ) > set_a > set_Product_prod_a_a > a ).
thf(func_def_238,type,
sK135: ( product_prod_a_a > a > $o ) > set_a > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_239,type,
sK136: ( product_prod_a_a > a > $o ) > set_Product_prod_a_a > set_a > product_prod_a_a ).
thf(func_def_240,type,
sK137: ( nat > $o ) > nat > nat ).
thf(func_def_241,type,
sK138: ( set_a > set_a ) > set_a ).
thf(func_def_242,type,
sK139: ( set_a > set_a ) > set_a ).
thf(func_def_243,type,
sK140: ( set_a > set_a ) > set_a ).
thf(func_def_244,type,
sK141: ( set_a > set_a ) > set_a ).
thf(func_def_245,type,
sK142: set_Product_prod_a_a > nat > set_Product_prod_a_a ).
thf(func_def_246,type,
sK143: ( nat > nat ) > nat ).
thf(func_def_247,type,
sK144: ( nat > nat ) > nat ).
thf(func_def_248,type,
sK145: ( set_a > nat ) > set_a ).
thf(func_def_249,type,
sK146: ( set_a > nat ) > set_a ).
thf(func_def_250,type,
sK147: ( product_prod_a_a > nat > $o ) > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_251,type,
sK148: ( product_prod_a_a > nat > $o ) > set_Product_prod_a_a > nat ).
thf(func_def_252,type,
sK149: product_prod_a_a > ( product_prod_a_a > nat > $o ) > set_Product_prod_a_a > nat ).
thf(func_def_253,type,
sK150: ( nat > set_a ) > nat ).
thf(func_def_254,type,
sK151: ( nat > set_a ) > nat ).
thf(func_def_255,type,
sK152: set_Product_prod_a_a > set_a > a ).
thf(func_def_256,type,
sK153: set_Product_prod_a_a > set_a > a ).
thf(func_def_257,type,
sK154: set_nat > nat ).
thf(func_def_258,type,
sK155: set_nat > set_nat > ( nat > nat > $o ) > nat ).
thf(func_def_259,type,
sK156: set_nat > set_nat > ( nat > nat > $o ) > nat ).
thf(func_def_260,type,
sK157: set_nat > set_nat > ( nat > nat > $o ) > nat ).
thf(func_def_261,type,
sK158: ( nat > nat > $o ) > set_nat > set_nat > nat ).
thf(func_def_262,type,
sK159: set_nat > nat ).
thf(func_def_263,type,
sK160: set_Product_prod_a_a > nat > set_Product_prod_a_a ).
thf(func_def_264,type,
sK161: set_nat > nat ).
thf(func_def_265,type,
sK162: set_a > ( a > a > $o ) > set_a > a ).
thf(func_def_266,type,
sK163: set_a > ( a > a > $o ) > set_a > a ).
thf(func_def_267,type,
sK164: set_a > ( a > a > $o ) > set_a > a ).
thf(func_def_268,type,
sK165: set_a > ( a > a > $o ) > set_a > a ).
thf(func_def_269,type,
sK166: ( product_prod_a_a > $o ) > a ).
thf(func_def_270,type,
sK167: ( product_prod_a_a > $o ) > a ).
thf(func_def_271,type,
sK168: set_a > set_nat > ( nat > a > $o ) > nat ).
thf(func_def_272,type,
sK169: set_a > set_nat > ( nat > a > $o ) > a ).
thf(func_def_273,type,
sK170: set_a > set_nat > ( nat > a > $o ) > nat ).
thf(func_def_274,type,
sK171: ( nat > a > $o ) > set_nat > set_a > nat ).
thf(func_def_275,type,
sK172: ( a > $o ) > ( a > $o ) > a ).
thf(func_def_276,type,
sK173: set_nat > nat > set_nat ).
thf(func_def_278,type,
ph175:
!>[X0: $tType] : X0 ).
thf(f3277,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3174,f3185,f3275]) ).
thf(f3275,plain,
~ spl174_15,
inference(avatar_contradiction_clause,[status(thm)],[f3274]) ).
thf(f3274,plain,
( $false
| ~ spl174_15 ),
inference(subsumption_resolution,[status(thm)],[f3242,f3243]) ).
thf(f3243,plain,
( ( $true
= ( member_a @ ( sK32 @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) )
| ~ spl174_15 ),
inference(superposition,[status(thm)],[f2933,f3173]) ).
thf(f3173,plain,
( ( ( prefer1676310729than_a @ x @ carrier @ relation )
= ( prefer1676310729than_a @ y @ carrier @ relation ) )
| ~ spl174_15 ),
inference(avatar_component_clause,[status(thm)],[f3171]) ).
thf(f3171,definition,
( spl174_15
<=> ( ( prefer1676310729than_a @ x @ carrier @ relation )
= ( prefer1676310729than_a @ y @ carrier @ relation ) ) ),
introduced(definition,[new_symbols(naming,[spl174_15])],[avatar_definition]) ).
thf(f2933,plain,
( ( member_a @ ( sK32 @ ( prefer1676310729than_a @ y @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2928]) ).
thf(f2928,plain,
( ( $true != $true )
| ( ( member_a @ ( sK32 @ ( prefer1676310729than_a @ y @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
= $true ) ),
inference(superposition,[status(thm)],[f2922,f2818]) ).
thf(f2818,plain,
! [X0: set_a,X1: set_a] :
( ( ( !! @ a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ X0 )
=> ( member_a @ Y0 @ X1 ) ) )
= $true )
| ( ( member_a @ ( sK32 @ X1 @ X0 ) @ X0 )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f2408]) ).
thf(f2408,plain,
! [X0: set_a,X1: set_a] :
( ( ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) )
@ X0
@ X1 )
= $true )
| ( ( member_a @ ( sK32 @ X1 @ X0 ) @ X0 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f1864,f2093]) ).
thf(f2093,plain,
( ord_less_eq_set_a
= ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f969]) ).
thf(f969,plain,
( ord_less_eq_set_a
= ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f968]) ).
thf(f968,plain,
( ( ^ [X0: set_a,X1: set_a] :
! [X2: a] :
( ( member_a @ X2 @ X0 )
=> ( member_a @ X2 @ X1 ) ) )
= ord_less_eq_set_a ),
inference(rectify,[status(thm)],[f149]) ).
thf(f149,axiom,
( ( ^ [X21: set_a,X22: set_a] :
! [X9: a] :
( ( member_a @ X9 @ X21 )
=> ( member_a @ X9 @ X22 ) ) )
= ord_less_eq_set_a ),
file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_148_subset__eq) ).
thf(f1864,plain,
! [X0: set_a,X1: set_a] :
( ( ( member_a @ ( sK32 @ X1 @ X0 ) @ X0 )
= $true )
| ( ( ord_less_eq_set_a @ X0 @ X1 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1619]) ).
thf(f1619,plain,
! [X0: set_a,X1: set_a] :
( ( ( $true
!= ( member_a @ ( sK32 @ X1 @ X0 ) @ X1 ) )
& ( ( member_a @ ( sK32 @ X1 @ X0 ) @ X0 )
= $true ) )
| ( ( ord_less_eq_set_a @ X0 @ X1 )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK32])],[f1395,f1618]) ).
thf(f1618,plain,
! [X0: set_a,X1: set_a] :
( ? [X2: a] :
( ( ( member_a @ X2 @ X1 )
!= $true )
& ( $true
= ( member_a @ X2 @ X0 ) ) )
=> ( ( $true
!= ( member_a @ ( sK32 @ X1 @ X0 ) @ X1 ) )
& ( ( member_a @ ( sK32 @ X1 @ X0 ) @ X0 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1395,plain,
! [X0: set_a,X1: set_a] :
( ? [X2: a] :
( ( ( member_a @ X2 @ X1 )
!= $true )
& ( $true
= ( member_a @ X2 @ X0 ) ) )
| ( ( ord_less_eq_set_a @ X0 @ X1 )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f729]) ).
thf(f729,plain,
! [X0: set_a,X1: set_a] :
( ! [X2: a] :
( ( $true
= ( member_a @ X2 @ X0 ) )
=> ( ( member_a @ X2 @ X1 )
= $true ) )
=> ( ( ord_less_eq_set_a @ X0 @ X1 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f728]) ).
thf(f728,plain,
! [X0: set_a,X1: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ X0 )
=> ( member_a @ X2 @ X1 ) )
=> ( ord_less_eq_set_a @ X0 @ X1 ) ),
inference(rectify,[status(thm)],[f120]) ).
thf(f120,axiom,
! [X15: set_a,X19: set_a] :
( ! [X11: a] :
( ( member_a @ X11 @ X15 )
=> ( member_a @ X11 @ X19 ) )
=> ( ord_less_eq_set_a @ X15 @ X19 ) ),
file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_119_subsetI) ).
thf(f2922,plain,
( ( !! @ a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
=> ( member_a @ Y0 @ ( prefer1676310729than_a @ y @ carrier @ relation ) ) ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f2432]) ).
thf(f2432,plain,
( $true
!= ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) )
@ ( prefer1676310729than_a @ x @ carrier @ relation )
@ ( prefer1676310729than_a @ y @ carrier @ relation ) ) ),
inference(definition_unfolding,[status(thm)],[f1913,f2093]) ).
thf(f1913,plain,
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1091]) ).
thf(f1091,plain,
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) )
!= $true ),
inference(flattening,[status(thm)],[f453]) ).
thf(f453,plain,
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f452]) ).
thf(f452,plain,
~ ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) ),
inference(rectify,[status(thm)],[f356]) ).
thf(f356,negated_conjecture,
~ ( ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) ),
inference(negated_conjecture,[status(cth)],[f355]) ).
thf(f355,conjecture,
ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ y @ carrier @ relation ),
file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',conj_0) ).
thf(f3242,plain,
( ( $true
!= ( member_a @ ( sK32 @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) )
| ~ spl174_15 ),
inference(superposition,[status(thm)],[f2939,f3173]) ).
thf(f2939,plain,
( ( member_a @ ( sK32 @ ( prefer1676310729than_a @ y @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2927]) ).
thf(f2927,plain,
( ( ( member_a @ ( sK32 @ ( prefer1676310729than_a @ y @ carrier @ relation ) @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2922,f2815]) ).
thf(f2815,plain,
! [X0: set_a,X1: set_a] :
( ( ( !! @ a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ X0 )
=> ( member_a @ Y0 @ X1 ) ) )
= $true )
| ( $true
!= ( member_a @ ( sK32 @ X1 @ X0 ) @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2407]) ).
thf(f2407,plain,
! [X0: set_a,X1: set_a] :
( ( $true
!= ( member_a @ ( sK32 @ X1 @ X0 ) @ X1 ) )
| ( ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) )
@ X0
@ X1 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f1865,f2093]) ).
thf(f1865,plain,
! [X0: set_a,X1: set_a] :
( ( $true
!= ( member_a @ ( sK32 @ X1 @ X0 ) @ X1 ) )
| ( ( ord_less_eq_set_a @ X0 @ X1 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1619]) ).
thf(f3185,plain,
spl174_12,
inference(avatar_contradiction_clause,[status(thm)],[f3184]) ).
thf(f3184,plain,
( $false
| spl174_12 ),
inference(subsumption_resolution,[status(thm)],[f3183,f2015]) ).
thf(f2015,plain,
( ( member_a @ x @ carrier )
= $true ),
inference(cnf_transformation,[status(thm)],[f1056]) ).
thf(f1056,plain,
( ( member_a @ x @ carrier )
= $true ),
inference(fool_elimination,[status(thm)],[f1055]) ).
thf(f1055,plain,
member_a @ x @ carrier,
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
member_a @ x @ carrier,
file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_1_assms_I1_J) ).
thf(f3183,plain,
( ( ( member_a @ x @ carrier )
!= $true )
| spl174_12 ),
inference(trivial_inequality_removal,[status(thm)],[f3182]) ).
thf(f3182,plain,
( ( ( member_a @ x @ carrier )
!= $true )
| ( $true != $true )
| spl174_12 ),
inference(superposition,[status(thm)],[f3158,f2017]) ).
thf(f2017,plain,
! [X0: a] :
( ( ( finite_finite_a @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) )
= $true )
| ( ( member_a @ X0 @ carrier )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1462]) ).
thf(f1462,plain,
! [X0: a] :
( ( ( finite_finite_a @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) )
= $true )
| ( ( member_a @ X0 @ carrier )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f632]) ).
thf(f632,plain,
! [X0: a] :
( ( ( member_a @ X0 @ carrier )
= $true )
=> ( ( finite_finite_a @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f631]) ).
thf(f631,plain,
! [X0: a] :
( ( member_a @ X0 @ carrier )
=> ( finite_finite_a @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) ) ),
inference(rectify,[status(thm)],[f16]) ).
thf(f16,axiom,
! [X2: a] :
( ( member_a @ X2 @ carrier )
=> ( finite_finite_a @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_15_fnt__carrier__fnt__nbt) ).
thf(f3158,plain,
( ( ( finite_finite_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
!= $true )
| spl174_12 ),
inference(avatar_component_clause,[status(thm)],[f3156]) ).
thf(f3156,definition,
( spl174_12
<=> ( ( finite_finite_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl174_12])],[avatar_definition]) ).
thf(f3174,plain,
( spl174_15
| ~ spl174_12 ),
inference(avatar_split_clause,[status(thm)],[f3169,f3156,f3171]) ).
thf(f3169,plain,
( ( ( prefer1676310729than_a @ x @ carrier @ relation )
= ( prefer1676310729than_a @ y @ carrier @ relation ) )
| ( ( finite_finite_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f3147,f2243]) ).
thf(f2243,plain,
( ( ord_less_eq_nat @ ( finite_card_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( finite_card_a @ ( prefer1676310729than_a @ y @ carrier @ relation ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f865]) ).
thf(f865,plain,
( ( ord_less_eq_nat @ ( finite_card_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( finite_card_a @ ( prefer1676310729than_a @ y @ carrier @ relation ) ) )
= $true ),
inference(fool_elimination,[status(thm)],[f864]) ).
thf(f864,plain,
ord_less_eq_nat @ ( finite_card_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( finite_card_a @ ( prefer1676310729than_a @ y @ carrier @ relation ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
ord_less_eq_nat @ ( finite_card_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( finite_card_a @ ( prefer1676310729than_a @ y @ carrier @ relation ) ),
file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062) ).
thf(f3147,plain,
( ( ( finite_finite_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
!= $true )
| ( ( prefer1676310729than_a @ x @ carrier @ relation )
= ( prefer1676310729than_a @ y @ carrier @ relation ) )
| ( ( ord_less_eq_nat @ ( finite_card_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( finite_card_a @ ( prefer1676310729than_a @ y @ carrier @ relation ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3119]) ).
thf(f3119,plain,
( ( $true != $true )
| ( ( finite_finite_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) )
!= $true )
| ( ( ord_less_eq_nat @ ( finite_card_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( finite_card_a @ ( prefer1676310729than_a @ y @ carrier @ relation ) ) )
!= $true )
| ( ( prefer1676310729than_a @ x @ carrier @ relation )
= ( prefer1676310729than_a @ y @ carrier @ relation ) ) ),
inference(superposition,[status(thm)],[f2826,f2937]) ).
thf(f2937,plain,
( $true
= ( !! @ a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ ( prefer1676310729than_a @ y @ carrier @ relation ) )
=> ( member_a @ Y0 @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2925]) ).
thf(f2925,plain,
( ( $true
= ( !! @ a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ ( prefer1676310729than_a @ y @ carrier @ relation ) )
=> ( member_a @ Y0 @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2922,f2731]) ).
thf(f2731,plain,
! [X0: a,X1: a] :
( ( $true
= ( !! @ a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) )
=> ( member_a @ Y0 @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) ) ) ) )
| ( ( !! @ a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) )
=> ( member_a @ Y0 @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) ) ) )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f2599]) ).
thf(f2599,plain,
! [X0: a,X1: a] :
( ( ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) )
@ ( prefer1676310729than_a @ X0 @ carrier @ relation )
@ ( prefer1676310729than_a @ X1 @ carrier @ relation ) )
= $true )
| ( ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) )
@ ( prefer1676310729than_a @ X1 @ carrier @ relation )
@ ( prefer1676310729than_a @ X0 @ carrier @ relation ) )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f2267,f2093,f2093]) ).
thf(f2267,plain,
! [X0: a,X1: a] :
( ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) )
= $true )
| ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f963]) ).
thf(f963,plain,
! [X0: a,X1: a] :
( ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) )
= $true )
| ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f962]) ).
thf(f962,plain,
! [X0: a,X1: a] :
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
! [X0: a,X1: a] :
( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) )
| ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X1 @ carrier @ relation ) @ ( prefer1676310729than_a @ X0 @ carrier @ relation ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_4_nbt__nest) ).
thf(f2826,plain,
! [X0: set_a,X1: set_a] :
( ( ( !! @ a
@ ^ [Y0: a] :
( ( member_a @ Y0 @ X0 )
=> ( member_a @ Y0 @ X1 ) ) )
!= $true )
| ( $true
!= ( ord_less_eq_nat @ ( finite_card_a @ X1 ) @ ( finite_card_a @ X0 ) ) )
| ( X0 = X1 )
| ( ( finite_finite_a @ X1 )
!= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f2648]) ).
thf(f2648,plain,
! [X0: set_a,X1: set_a] :
( ( ( finite_finite_a @ X1 )
!= $true )
| ( $true
!= ( ord_less_eq_nat @ ( finite_card_a @ X1 ) @ ( finite_card_a @ X0 ) ) )
| ( X0 = X1 )
| ( ( ^ [Y0: set_a,Y1: set_a] :
( !! @ a
@ ^ [Y2: a] :
( ( member_a @ Y2 @ Y0 )
=> ( member_a @ Y2 @ Y1 ) ) )
@ X0
@ X1 )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f2382,f2093]) ).
thf(f2382,plain,
! [X0: set_a,X1: set_a] :
( ( ( finite_finite_a @ X1 )
!= $true )
| ( $true
!= ( ord_less_eq_nat @ ( finite_card_a @ X1 ) @ ( finite_card_a @ X0 ) ) )
| ( ( ord_less_eq_set_a @ X0 @ X1 )
!= $true )
| ( X0 = X1 ) ),
inference(cnf_transformation,[status(thm)],[f1232]) ).
thf(f1232,plain,
! [X0: set_a,X1: set_a] :
( ( ( finite_finite_a @ X1 )
!= $true )
| ( $true
!= ( ord_less_eq_nat @ ( finite_card_a @ X1 ) @ ( finite_card_a @ X0 ) ) )
| ( ( ord_less_eq_set_a @ X0 @ X1 )
!= $true )
| ( X0 = X1 ) ),
inference(flattening,[status(thm)],[f1231]) ).
thf(f1231,plain,
! [X0: set_a,X1: set_a] :
( ( X0 = X1 )
| ( $true
!= ( ord_less_eq_nat @ ( finite_card_a @ X1 ) @ ( finite_card_a @ X0 ) ) )
| ( ( ord_less_eq_set_a @ X0 @ X1 )
!= $true )
| ( ( finite_finite_a @ X1 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f548]) ).
thf(f548,plain,
! [X0: set_a,X1: set_a] :
( ( ( finite_finite_a @ X1 )
= $true )
=> ( ( ( ord_less_eq_set_a @ X0 @ X1 )
= $true )
=> ( ( $true
= ( ord_less_eq_nat @ ( finite_card_a @ X1 ) @ ( finite_card_a @ X0 ) ) )
=> ( X0 = X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f547]) ).
thf(f547,plain,
! [X0: set_a,X1: set_a] :
( ( finite_finite_a @ X1 )
=> ( ( ord_less_eq_set_a @ X0 @ X1 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ X1 ) @ ( finite_card_a @ X0 ) )
=> ( X0 = X1 ) ) ) ),
inference(rectify,[status(thm)],[f212]) ).
thf(f212,axiom,
! [X15: set_a,X19: set_a] :
( ( finite_finite_a @ X19 )
=> ( ( ord_less_eq_set_a @ X15 @ X19 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ X19 ) @ ( finite_card_a @ X15 ) )
=> ( X15 = X19 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_211_card__seteq) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP155^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n006.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:28:00 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.47 ---- Original DTF file ---
% 0.29/0.47 thf(spec,logic,$$dhol).
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % File : ITP155^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.47 % Domain : Interactive Theorem Proving
% 0.29/0.47 % Problem : Sledgehammer Preferences problem prob_260__6251520_1
% 0.29/0.47 % Version : Especial.
% 0.29/0.47 % English :
% 0.29/0.47
% 0.29/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.47 % Source : [Des21]
% 0.29/0.47 % Names : Preferences/prob_260__6251520_1 [Des21]
% 0.29/0.47
% 0.29/0.47 % Status : Theorem
% 0.29/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.29/0.47 % Syntax : Number of formulae : 443 ( 98 unt; 88 typ; 0 def)
% 0.29/0.47 % Number of atoms : 1290 ( 214 equ; 0 cnn)
% 0.29/0.47 % Maximal formula atoms : 12 ( 3 avg)
% 0.29/0.47 % Number of connectives : 4010 ( 124 ~; 26 |; 135 &;3019 @)
% 0.29/0.47 % ( 0 <=>; 706 =>; 0 <=; 0 <~>)
% 0.29/0.47 % Maximal formula depth : 17 ( 8 avg)
% 0.29/0.47 % Number of types : 15 ( 14 usr)
% 0.29/0.47 % Number of type conns : 184 ( 184 >; 0 *; 0 +; 0 <<)
% 0.29/0.47 % Number of symbols : 75 ( 74 usr; 11 con; 0-3 aty)
% 0.29/0.47 % Number of variables : 1089 ( 61 ^; 986 !; 42 ?;1089 :)
% 0.29/0.47 % SPC : TH0_THM_EQU_NAR
% 0.29/0.47
% 0.29/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 15:45:04.162
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % Could-be-implicit typings (14)
% 0.29/0.47 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,
% 0.29/0.47 set_Pr1948701895od_a_a: $tType ).
% 0.29/0.47
% 0.29/0.47 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,
% 0.29/0.47 produc1572603623od_a_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 0.29/0.47 produc1691597095_set_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.29/0.47 set_Pr1986765409at_nat: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
% 0.29/0.47 produc1404140509_a_nat: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
% 0.29/0.47 produc1463940355_set_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.47 set_Product_prod_a_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.47 product_prod_nat_nat: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 product_prod_a_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.29/0.47 set_set_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.47 set_nat: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.29/0.47 set_a: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_t__Nat__Onat,type,
% 0.29/0.47 nat: $tType ).
% 0.29/0.47
% 0.29/0.47 thf(ty_n_tf__a,type,
% 0.29/0.47 a: $tType ).
% 0.29/0.47
% 0.29/0.47 % Explicit typings (74)
% 0.29/0.47 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 0.29/0.47 finite_card_nat: set_nat > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 finite1481642319od_a_a: set_Product_prod_a_a > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ocard_001tf__a,type,
% 0.29/0.47 finite_card_a: set_a > nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 0.29/0.47 finite_finite_nat: set_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.47 finite772653738at_nat: set_Pr1986765409at_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.47 finite1664988688od_a_a: set_Pr1948701895od_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 finite179568208od_a_a: set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.47 finite_finite_set_a: set_set_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
% 0.29/0.47 finite_finite_a: set_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.29/0.47 zero_zero_nat: nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Order__Relation_Opreorder__on_001t__Nat__Onat,type,
% 0.29/0.47 order_1150839801on_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Order__Relation_Opreorder__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Order__Relation_Opreorder__on_001tf__a,type,
% 0.29/0.47 order_preorder_on_a: set_a > set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 0.29/0.47 bot_bot_nat_o: nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
% 0.29/0.47 bot_bo1293121322_a_a_o: product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
% 0.29/0.47 bot_bot_a_o: a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 0.29/0.47 bot_bot_nat: nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 0.29/0.47 bot_bo1519632275_set_a: produc1691597095_set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.47 bot_bot_set_nat: set_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.47 bot_bo2131659635od_a_a: set_Product_prod_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.29/0.47 bot_bot_set_set_a: set_set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.47 bot_bot_set_a: set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.29/0.47 ord_less_nat: nat > nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.29/0.47 ord_less_eq_nat: nat > nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.47 ord_le1302190241at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
% 0.29/0.47 ord_le841160035_set_a: produc1463940355_set_a > produc1463940355_set_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
% 0.29/0.47 ord_le781360189_a_nat: produc1404140509_a_nat > produc1404140509_a_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 0.29/0.47 ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.47 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.29/0.47 ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.47 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oas__good__as_001t__Nat__Onat,type,
% 0.29/0.47 prefer1936906324as_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oas__good__as_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oas__good__as_001tf__a,type,
% 0.29/0.47 prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oat__least__as__good_001t__Nat__Onat,type,
% 0.29/0.47 prefer563798164od_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oat__least__as__good_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oat__least__as__good_001tf__a,type,
% 0.29/0.47 prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001t__Nat__Onat,type,
% 0.29/0.47 prefer1672638789an_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001tf__a,type,
% 0.29/0.47 prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Opreference_001t__Nat__Onat,type,
% 0.29/0.47 prefer1293349710ce_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Opreference_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Opreference_001tf__a,type,
% 0.29/0.47 prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001t__Nat__Onat,type,
% 0.29/0.47 prefer1147844220ce_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001tf__a,type,
% 0.29/0.47 prefer719835986ence_a: set_a > set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference__axioms_001tf__a,type,
% 0.29/0.47 prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.47 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.47 produc1450889781_set_a: nat > set_a > produc1463940355_set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
% 0.29/0.47 produc515611863_a_nat: set_a > nat > produc1404140509_a_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.47 produc1928581911_set_a: set_a > set_a > produc1691597095_set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 0.29/0.47 product_Pair_a_a: a > a > product_prod_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Orefl__on_001t__Nat__Onat,type,
% 0.29/0.47 refl_on_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Orefl__on_001tf__a,type,
% 0.29/0.47 refl_on_a: set_a > set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Ototal__on_001t__Nat__Onat,type,
% 0.29/0.47 total_on_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Ototal__on_001tf__a,type,
% 0.29/0.47 total_on_a: set_a > set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Relation_Otrans_001tf__a,type,
% 0.29/0.47 trans_a: set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.29/0.47 collect_nat: ( nat > $o ) > set_nat ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.29/0.47 collect_a: ( a > $o ) > set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_Set_Ois__empty_001tf__a,type,
% 0.29/0.47 is_empty_a: set_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member_001t__Nat__Onat,type,
% 0.29/0.47 member_nat: nat > set_nat > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.47 member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
% 0.29/0.47
% 0.29/0.47 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,
% 0.29/0.47 member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.29/0.47 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.29/0.47 member_set_a: set_a > set_set_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_c_member_001tf__a,type,
% 0.29/0.47 member_a: a > set_a > $o ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_carrier,type,
% 0.29/0.47 carrier: set_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_relation,type,
% 0.29/0.47 relation: set_Product_prod_a_a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_x,type,
% 0.29/0.47 x: a ).
% 0.29/0.47
% 0.29/0.47 thf(sy_v_y,type,
% 0.29/0.47 y: a ).
% 0.29/0.47
% 0.29/0.47 % Relevant facts (354)
% 0.29/0.47 thf(fact_0_assms_I2_J,axiom,
% 0.29/0.47 member_a @ y @ carrier ).
% 0.29/0.47
% 0.29/0.47 % assms(2)
% 0.29/0.47 thf(fact_1_assms_I1_J,axiom,
% 0.29/0.47 member_a @ x @ carrier ).
% 0.29/0.47
% 0.29/0.47 % assms(1)
% 0.29/0.47 thf(fact_2_xy__in__eachothers__nbt,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( member_a @ X @ carrier )
% 0.29/0.47 => ( ( member_a @ Y @ carrier )
% 0.29/0.47 => ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ carrier @ relation ) )
% 0.29/0.47 | ( member_a @ Y @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % xy_in_eachothers_nbt
% 0.29/0.47 thf(fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062,axiom,
% 0.29/0.47 ord_less_eq_nat @ ( finite_card_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( finite_card_a @ ( prefer1676310729than_a @ y @ carrier @ relation ) ) ).
% 0.29/0.47
% 0.29/0.47 % \<open>card (no_better_than x carrier relation) \<le> card (no_better_than y carrier relation)\<close>
% 0.29/0.47 thf(fact_4_nbt__nest,axiom,
% 0.29/0.47 ! [Y: a,X: a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
% 0.29/0.47 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nbt_nest
% 0.29/0.47 thf(fact_5_nbt__subset__carrier,axiom,
% 0.29/0.47 ! [X: a] :
% 0.29/0.47 ( ( member_a @ X @ carrier )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ carrier ) ) ).
% 0.29/0.47
% 0.29/0.47 % nbt_subset_carrier
% 0.29/0.47 thf(fact_6_rational__preference__axioms,axiom,
% 0.29/0.47 prefer719835986ence_a @ carrier @ relation ).
% 0.29/0.47
% 0.29/0.47 % rational_preference_axioms
% 0.29/0.47 thf(fact_7__092_060open_062finite_Acarrier_092_060close_062,axiom,
% 0.29/0.47 finite_finite_a @ carrier ).
% 0.29/0.47
% 0.29/0.47 % \<open>finite carrier\<close>
% 0.29/0.47 thf(fact_8_preference__axioms,axiom,
% 0.29/0.47 prefer1292084992ence_a @ carrier @ relation ).
% 0.29/0.47
% 0.29/0.47 % preference_axioms
% 0.29/0.47 thf(fact_9_trans__refl,axiom,
% 0.29/0.47 order_preorder_on_a @ carrier @ relation ).
% 0.29/0.47
% 0.29/0.47 % trans_refl
% 0.29/0.47 thf(fact_10_nbt__subset,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( finite_finite_a @ carrier )
% 0.29/0.47 => ( ( member_a @ X @ carrier )
% 0.29/0.47 => ( ( member_a @ Y @ carrier )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
% 0.29/0.47 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % nbt_subset
% 0.29/0.47 thf(fact_11_no__better__subset__pref,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % no_better_subset_pref
% 0.29/0.47 thf(fact_12_no__better__thansubset__rel,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( member_a @ X @ carrier )
% 0.29/0.47 => ( ( member_a @ Y @ carrier )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % no_better_thansubset_rel
% 0.29/0.47 thf(fact_13_no__better__than__nonepty,axiom,
% 0.29/0.47 ! [X: a] :
% 0.29/0.47 ( ( carrier != bot_bot_set_a )
% 0.29/0.47 => ( ( member_a @ X @ carrier )
% 0.29/0.47 => ( ( prefer1676310729than_a @ X @ carrier @ relation )
% 0.29/0.47 != bot_bot_set_a ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % no_better_than_nonepty
% 0.29/0.47 thf(fact_14_total,axiom,
% 0.29/0.47 total_on_a @ carrier @ relation ).
% 0.29/0.47
% 0.29/0.47 % total
% 0.29/0.47 thf(fact_15_fnt__carrier__fnt__nbt,axiom,
% 0.29/0.47 ! [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ carrier )
% 0.29/0.47 => ( finite_finite_a @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % fnt_carrier_fnt_nbt
% 0.29/0.47 thf(fact_16_same__nbt__same__pref,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( member_a @ X @ carrier )
% 0.29/0.47 => ( ( member_a @ Y @ carrier )
% 0.29/0.47 => ( ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ carrier @ relation ) )
% 0.29/0.47 & ( member_a @ Y @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) )
% 0.29/0.47 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % same_nbt_same_pref
% 0.29/0.47 thf(fact_17_worse__in__no__better,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % worse_in_no_better
% 0.29/0.47 thf(fact_18_reflexivity,axiom,
% 0.29/0.47 refl_on_a @ carrier @ relation ).
% 0.29/0.47
% 0.29/0.47 % reflexivity
% 0.29/0.47 thf(fact_19_compl,axiom,
% 0.29/0.47 ! [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ carrier )
% 0.29/0.47 => ! [Xa: a] :
% 0.29/0.47 ( ( member_a @ Xa @ carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ relation )
% 0.29/0.47 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % compl
% 0.29/0.47 thf(fact_20_completeD,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( member_a @ X @ carrier )
% 0.29/0.47 => ( ( member_a @ Y @ carrier )
% 0.29/0.47 => ( ( X != Y )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % completeD
% 0.29/0.47 thf(fact_21_strict__trans,axiom,
% 0.29/0.47 ! [X: a,Y: a,Z: a] :
% 0.29/0.47 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % strict_trans
% 0.29/0.47 thf(fact_22_indiff__trans,axiom,
% 0.29/0.47 ! [X: a,Y: a,Z: a] :
% 0.29/0.47 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % indiff_trans
% 0.29/0.47 thf(fact_23_indifferent__imp__weak__pref_I1_J,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ).
% 0.29/0.47
% 0.29/0.47 % indifferent_imp_weak_pref(1)
% 0.29/0.47 thf(fact_24_indifferent__imp__weak__pref_I2_J,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ).
% 0.29/0.47
% 0.29/0.47 % indifferent_imp_weak_pref(2)
% 0.29/0.47 thf(fact_25_weak__is__transitive,axiom,
% 0.29/0.47 ! [X: a,Y: a,Z: a] :
% 0.29/0.47 ( ( ( member_a @ X @ carrier )
% 0.29/0.47 & ( member_a @ Y @ carrier )
% 0.29/0.47 & ( member_a @ Z @ carrier ) )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % weak_is_transitive
% 0.29/0.47 thf(fact_26_strict__is__neg__transitive,axiom,
% 0.29/0.47 ! [X: a,Y: a,Z: a] :
% 0.29/0.47 ( ( ( member_a @ X @ carrier )
% 0.29/0.47 & ( member_a @ Y @ carrier )
% 0.29/0.47 & ( member_a @ Z @ carrier ) )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) )
% 0.29/0.47 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % strict_is_neg_transitive
% 0.29/0.47 thf(fact_27_not__outside,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 => ( member_a @ X @ carrier ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_outside
% 0.29/0.47 thf(fact_28_strict__not__refl__weak,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( ( member_a @ X @ carrier )
% 0.29/0.47 & ( member_a @ Y @ carrier ) )
% 0.29/0.47 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.29/0.47 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % strict_not_refl_weak
% 0.29/0.47 thf(fact_29_rational__preference_Oaxioms_I1_J,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.axioms(1)
% 0.29/0.47 thf(fact_30_rational__preference_Oindifferent__imp__weak__pref_I2_J,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.indifferent_imp_weak_pref(2)
% 0.29/0.47 thf(fact_31_rational__preference_Oindifferent__imp__weak__pref_I1_J,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.indifferent_imp_weak_pref(1)
% 0.29/0.47 thf(fact_32_preference__def,axiom,
% 0.29/0.47 ( prefer1338892073od_a_a
% 0.29/0.47 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 0.29/0.47 ( ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 0.29/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.29/0.47 => ( member449909584od_a_a @ X3 @ Carrier2 ) )
% 0.29/0.47 & ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 0.29/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.29/0.47 => ( member449909584od_a_a @ Y2 @ Carrier2 ) )
% 0.29/0.47 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference_def
% 0.29/0.47 thf(fact_33_preference__def,axiom,
% 0.29/0.47 ( prefer1293349710ce_nat
% 0.29/0.47 = ( ^ [Carrier2: set_nat,Relation2: set_Pr1986765409at_nat] :
% 0.29/0.47 ( ! [X3: nat,Y2: nat] :
% 0.29/0.47 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ Relation2 )
% 0.29/0.47 => ( member_nat @ X3 @ Carrier2 ) )
% 0.29/0.47 & ! [X3: nat,Y2: nat] :
% 0.29/0.47 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ Relation2 )
% 0.29/0.47 => ( member_nat @ Y2 @ Carrier2 ) )
% 0.29/0.47 & ( order_1150839801on_nat @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference_def
% 0.29/0.47 thf(fact_34_preference__def,axiom,
% 0.29/0.47 ( prefer1292084992ence_a
% 0.29/0.47 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 0.29/0.47 ( ! [X3: a,Y2: a] :
% 0.29/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.29/0.47 => ( member_a @ X3 @ Carrier2 ) )
% 0.29/0.47 & ! [X3: a,Y2: a] :
% 0.29/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 0.29/0.47 => ( member_a @ Y2 @ Carrier2 ) )
% 0.29/0.47 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference_def
% 0.29/0.47 thf(fact_35_preference_Ointro,axiom,
% 0.29/0.47 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 0.29/0.47 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.29/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 0.29/0.47 => ( member449909584od_a_a @ X4 @ Carrier ) )
% 0.29/0.47 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.29/0.47 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 0.29/0.47 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 0.29/0.47 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference.intro
% 0.29/0.47 thf(fact_36_preference_Ointro,axiom,
% 0.29/0.47 ! [Relation: set_Pr1986765409at_nat,Carrier: set_nat] :
% 0.29/0.47 ( ! [X4: nat,Y3: nat] :
% 0.29/0.47 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ Relation )
% 0.29/0.47 => ( member_nat @ X4 @ Carrier ) )
% 0.29/0.47 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.47 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ Relation )
% 0.29/0.47 => ( member_nat @ Y3 @ Carrier ) )
% 0.29/0.47 => ( ( order_1150839801on_nat @ Carrier @ Relation )
% 0.29/0.47 => ( prefer1293349710ce_nat @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference.intro
% 0.29/0.47 thf(fact_37_preference_Ointro,axiom,
% 0.29/0.47 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 0.29/0.47 ( ! [X4: a,Y3: a] :
% 0.29/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 0.29/0.47 => ( member_a @ X4 @ Carrier ) )
% 0.29/0.47 => ( ! [X4: a,Y3: a] :
% 0.29/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 0.29/0.47 => ( member_a @ Y3 @ Carrier ) )
% 0.29/0.47 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 0.29/0.47 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference.intro
% 0.29/0.47 thf(fact_38_preference_Otrans__refl,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.47 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference.trans_refl
% 0.29/0.47 thf(fact_39_preference_Onot__outside,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 => ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference.not_outside
% 0.29/0.47 thf(fact_40_preference_Onot__outside,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1293349710ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 0.29/0.47 => ( member_nat @ X @ Carrier ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference.not_outside
% 0.29/0.47 thf(fact_41_preference_Onot__outside,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 => ( member_a @ X @ Carrier ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference.not_outside
% 0.29/0.47 thf(fact_42_preference_Oreflexivity,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.47 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( refl_on_a @ Carrier @ Relation ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference.reflexivity
% 0.29/0.47 thf(fact_43_preference_Oindiff__trans,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.29/0.47 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % preference.indiff_trans
% 0.29/0.47 thf(fact_44_rational__preference_Ocompl,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ! [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ Carrier )
% 0.29/0.47 => ! [Xa: a] :
% 0.29/0.47 ( ( member_a @ Xa @ Carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ Relation )
% 0.29/0.47 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.compl
% 0.29/0.47 thf(fact_45_rational__preference_Ototal,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( total_on_a @ Carrier @ Relation ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.total
% 0.29/0.47 thf(fact_46_rational__preference_OcompleteD,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.29/0.47 => ( ( X != Y )
% 0.29/0.47 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.completeD
% 0.29/0.47 thf(fact_47_rational__preference_OcompleteD,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_nat @ X @ Carrier )
% 0.29/0.47 => ( ( member_nat @ Y @ Carrier )
% 0.29/0.47 => ( ( X != Y )
% 0.29/0.47 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 0.29/0.47 | ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.completeD
% 0.29/0.47 thf(fact_48_rational__preference_OcompleteD,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_a @ X @ Carrier )
% 0.29/0.47 => ( ( member_a @ Y @ Carrier )
% 0.29/0.47 => ( ( X != Y )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.completeD
% 0.29/0.47 thf(fact_49_rational__preference_Ostrict__trans,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.strict_trans
% 0.29/0.47 thf(fact_50_rational__preference_Oweak__is__transitive,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 & ( member449909584od_a_a @ Y @ Carrier )
% 0.29/0.47 & ( member449909584od_a_a @ Z @ Carrier ) )
% 0.29/0.47 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 0.29/0.47 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.weak_is_transitive
% 0.29/0.47 thf(fact_51_rational__preference_Oweak__is__transitive,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat,Z: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member_nat @ X @ Carrier )
% 0.29/0.47 & ( member_nat @ Y @ Carrier )
% 0.29/0.47 & ( member_nat @ Z @ Carrier ) )
% 0.29/0.47 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 0.29/0.47 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ Relation )
% 0.29/0.47 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.weak_is_transitive
% 0.29/0.47 thf(fact_52_rational__preference_Oweak__is__transitive,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member_a @ X @ Carrier )
% 0.29/0.47 & ( member_a @ Y @ Carrier )
% 0.29/0.47 & ( member_a @ Z @ Carrier ) )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.weak_is_transitive
% 0.29/0.47 thf(fact_53_rational__preference_Ofnt__carrier__fnt__rel,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( finite179568208od_a_a @ Carrier )
% 0.29/0.47 => ( finite1664988688od_a_a @ Relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.fnt_carrier_fnt_rel
% 0.29/0.47 thf(fact_54_rational__preference_Ofnt__carrier__fnt__rel,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( finite_finite_nat @ Carrier )
% 0.29/0.47 => ( finite772653738at_nat @ Relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.fnt_carrier_fnt_rel
% 0.29/0.47 thf(fact_55_rational__preference_Ofnt__carrier__fnt__rel,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( finite_finite_a @ Carrier )
% 0.29/0.47 => ( finite179568208od_a_a @ Relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.fnt_carrier_fnt_rel
% 0.29/0.47 thf(fact_56_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 & ( member449909584od_a_a @ Y @ Carrier ) )
% 0.29/0.47 => ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.29/0.47 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.strict_not_refl_weak
% 0.29/0.47 thf(fact_57_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member_nat @ X @ Carrier )
% 0.29/0.47 & ( member_nat @ Y @ Carrier ) )
% 0.29/0.47 => ( ( ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) )
% 0.29/0.47 = ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.strict_not_refl_weak
% 0.29/0.47 thf(fact_58_rational__preference_Ostrict__not__refl__weak,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member_a @ X @ Carrier )
% 0.29/0.47 & ( member_a @ Y @ Carrier ) )
% 0.29/0.47 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.29/0.47 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.strict_not_refl_weak
% 0.29/0.47 thf(fact_59_mem__Collect__eq,axiom,
% 0.29/0.47 ! [A: a,P: a > $o] :
% 0.29/0.47 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.29/0.47 = ( P @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % mem_Collect_eq
% 0.29/0.47 thf(fact_60_mem__Collect__eq,axiom,
% 0.29/0.47 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 0.29/0.47 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 0.29/0.47 = ( P @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % mem_Collect_eq
% 0.29/0.47 thf(fact_61_mem__Collect__eq,axiom,
% 0.29/0.47 ! [A: nat,P: nat > $o] :
% 0.29/0.47 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.29/0.47 = ( P @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % mem_Collect_eq
% 0.29/0.47 thf(fact_62_Collect__mem__eq,axiom,
% 0.29/0.47 ! [A2: set_a] :
% 0.29/0.47 ( ( collect_a
% 0.29/0.47 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 0.29/0.47 = A2 ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_mem_eq
% 0.29/0.47 thf(fact_63_Collect__mem__eq,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a] :
% 0.29/0.47 ( ( collec645855634od_a_a
% 0.29/0.47 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A2 ) )
% 0.29/0.47 = A2 ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_mem_eq
% 0.29/0.47 thf(fact_64_Collect__mem__eq,axiom,
% 0.29/0.47 ! [A2: set_nat] :
% 0.29/0.47 ( ( collect_nat
% 0.29/0.47 @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 0.29/0.47 = A2 ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_mem_eq
% 0.29/0.47 thf(fact_65_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 & ( member449909584od_a_a @ Y @ Carrier )
% 0.29/0.47 & ( member449909584od_a_a @ Z @ Carrier ) )
% 0.29/0.47 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 0.29/0.47 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 0.29/0.47 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) )
% 0.29/0.47 | ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.strict_is_neg_transitive
% 0.29/0.47 thf(fact_66_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat,Z: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member_nat @ X @ Carrier )
% 0.29/0.47 & ( member_nat @ Y @ Carrier )
% 0.29/0.47 & ( member_nat @ Z @ Carrier ) )
% 0.29/0.47 => ( ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) )
% 0.29/0.47 => ( ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ Relation )
% 0.29/0.47 & ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ X ) @ Relation ) )
% 0.29/0.47 | ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.strict_is_neg_transitive
% 0.29/0.47 thf(fact_67_rational__preference_Ostrict__is__neg__transitive,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member_a @ X @ Carrier )
% 0.29/0.47 & ( member_a @ Y @ Carrier )
% 0.29/0.47 & ( member_a @ Z @ Carrier ) )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) )
% 0.29/0.47 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 0.29/0.47 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.strict_is_neg_transitive
% 0.29/0.47 thf(fact_68_rational__preference_Oworse__in__no__better,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 => ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.worse_in_no_better
% 0.29/0.47 thf(fact_69_rational__preference_Oworse__in__no__better,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 0.29/0.47 => ( member_nat @ Y @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.worse_in_no_better
% 0.29/0.47 thf(fact_70_rational__preference_Oworse__in__no__better,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.worse_in_no_better
% 0.29/0.47 thf(fact_71_rational__preference_Osame__nbt__same__pref,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ X @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 & ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) )
% 0.29/0.47 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.same_nbt_same_pref
% 0.29/0.47 thf(fact_72_rational__preference_Osame__nbt__same__pref,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_nat @ X @ Carrier )
% 0.29/0.47 => ( ( member_nat @ Y @ Carrier )
% 0.29/0.47 => ( ( ( member_nat @ X @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) )
% 0.29/0.47 & ( member_nat @ Y @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) ) )
% 0.29/0.47 = ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 0.29/0.47 & ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.same_nbt_same_pref
% 0.29/0.47 thf(fact_73_rational__preference_Osame__nbt__same__pref,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_a @ X @ Carrier )
% 0.29/0.47 => ( ( member_a @ Y @ Carrier )
% 0.29/0.47 => ( ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 & ( member_a @ Y @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) )
% 0.29/0.47 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.same_nbt_same_pref
% 0.29/0.47 thf(fact_74_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( finite179568208od_a_a @ Carrier )
% 0.29/0.47 => ! [X2: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X2 @ Carrier )
% 0.29/0.47 => ( finite179568208od_a_a @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.fnt_carrier_fnt_nbt
% 0.29/0.47 thf(fact_75_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( finite_finite_nat @ Carrier )
% 0.29/0.47 => ! [X2: nat] :
% 0.29/0.47 ( ( member_nat @ X2 @ Carrier )
% 0.29/0.47 => ( finite_finite_nat @ ( prefer1672638789an_nat @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.fnt_carrier_fnt_nbt
% 0.29/0.47 thf(fact_76_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( finite_finite_a @ Carrier )
% 0.29/0.47 => ! [X2: a] :
% 0.29/0.47 ( ( member_a @ X2 @ Carrier )
% 0.29/0.47 => ( finite_finite_a @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.fnt_carrier_fnt_nbt
% 0.29/0.47 thf(fact_77_rational__preference_Ono__better__than__nonepty,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( Carrier != bot_bo2131659635od_a_a )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 => ( ( prefer1498087410od_a_a @ X @ Carrier @ Relation )
% 0.29/0.47 != bot_bo2131659635od_a_a ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.no_better_than_nonepty
% 0.29/0.47 thf(fact_78_rational__preference_Ono__better__than__nonepty,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( Carrier != bot_bot_set_nat )
% 0.29/0.47 => ( ( member_nat @ X @ Carrier )
% 0.29/0.47 => ( ( prefer1672638789an_nat @ X @ Carrier @ Relation )
% 0.29/0.47 != bot_bot_set_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.no_better_than_nonepty
% 0.29/0.47 thf(fact_79_rational__preference_Ono__better__than__nonepty,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( Carrier != bot_bot_set_a )
% 0.29/0.47 => ( ( member_a @ X @ Carrier )
% 0.29/0.47 => ( ( prefer1676310729than_a @ X @ Carrier @ Relation )
% 0.29/0.47 != bot_bot_set_a ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.no_better_than_nonepty
% 0.29/0.47 thf(fact_80_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.29/0.47 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
% 0.29/0.47 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.no_better_thansubset_rel
% 0.29/0.47 thf(fact_81_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_nat @ X @ Carrier )
% 0.29/0.47 => ( ( member_nat @ Y @ Carrier )
% 0.29/0.47 => ( ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) )
% 0.29/0.47 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.no_better_thansubset_rel
% 0.29/0.47 thf(fact_82_rational__preference_Ono__better__thansubset__rel,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_a @ X @ Carrier )
% 0.29/0.47 => ( ( member_a @ Y @ Carrier )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.no_better_thansubset_rel
% 0.29/0.47 thf(fact_83_rational__preference_Ono__better__subset__pref,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.no_better_subset_pref
% 0.29/0.47 thf(fact_84_rational__preference_Onbt__subset,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( finite179568208od_a_a @ Carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.29/0.47 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
% 0.29/0.47 | ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.nbt_subset
% 0.29/0.47 thf(fact_85_rational__preference_Onbt__subset,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( finite_finite_nat @ Carrier )
% 0.29/0.47 => ( ( member_nat @ X @ Carrier )
% 0.29/0.47 => ( ( member_nat @ Y @ Carrier )
% 0.29/0.47 => ( ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) )
% 0.29/0.47 | ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.nbt_subset
% 0.29/0.47 thf(fact_86_rational__preference_Onbt__subset,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( finite_finite_a @ Carrier )
% 0.29/0.47 => ( ( member_a @ X @ Carrier )
% 0.29/0.47 => ( ( member_a @ Y @ Carrier )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 0.29/0.47 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.nbt_subset
% 0.29/0.47 thf(fact_87_rational__preference_Oxy__in__eachothers__nbt,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 | ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.xy_in_eachothers_nbt
% 0.29/0.47 thf(fact_88_rational__preference_Oxy__in__eachothers__nbt,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_nat @ X @ Carrier )
% 0.29/0.47 => ( ( member_nat @ Y @ Carrier )
% 0.29/0.47 => ( ( member_nat @ X @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) )
% 0.29/0.47 | ( member_nat @ Y @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.xy_in_eachothers_nbt
% 0.29/0.47 thf(fact_89_rational__preference_Oxy__in__eachothers__nbt,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_a @ X @ Carrier )
% 0.29/0.47 => ( ( member_a @ Y @ Carrier )
% 0.29/0.47 => ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 | ( member_a @ Y @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.xy_in_eachothers_nbt
% 0.29/0.47 thf(fact_90_rational__preference_Onbt__subset__carrier,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 => ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.nbt_subset_carrier
% 0.29/0.47 thf(fact_91_rational__preference_Onbt__subset__carrier,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_nat @ X @ Carrier )
% 0.29/0.47 => ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.nbt_subset_carrier
% 0.29/0.47 thf(fact_92_rational__preference_Onbt__subset__carrier,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_a @ X @ Carrier )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.nbt_subset_carrier
% 0.29/0.47 thf(fact_93_rational__preference_Onbt__nest,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Y: a,X: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 0.29/0.47 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.nbt_nest
% 0.29/0.47 thf(fact_94_at__lst__asgd__not__ge,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( carrier != bot_bot_set_a )
% 0.29/0.47 => ( ( member_a @ X @ carrier )
% 0.29/0.47 => ( ( member_a @ Y @ carrier )
% 0.29/0.47 => ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) )
% 0.29/0.47 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % at_lst_asgd_not_ge
% 0.29/0.47 thf(fact_95_same__at__least__as__equal,axiom,
% 0.29/0.47 ! [Z: a,Y: a] :
% 0.29/0.47 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) )
% 0.29/0.47 => ( ( prefer161218362good_a @ Z @ carrier @ relation )
% 0.29/0.47 = ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % same_at_least_as_equal
% 0.29/0.47 thf(fact_96_pref__in__at__least__as,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 => ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % pref_in_at_least_as
% 0.29/0.47 thf(fact_97_empty__subsetI,axiom,
% 0.29/0.47 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 0.29/0.47
% 0.29/0.47 % empty_subsetI
% 0.29/0.47 thf(fact_98_subset__empty,axiom,
% 0.29/0.47 ! [A2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 0.29/0.47 = ( A2 = bot_bot_set_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_empty
% 0.29/0.47 thf(fact_99_Pair__le,axiom,
% 0.29/0.47 ! [A: set_a,B: set_a,C: set_a,D: set_a] :
% 0.29/0.47 ( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A @ B ) @ ( produc1928581911_set_a @ C @ D ) )
% 0.29/0.47 = ( ( ord_less_eq_set_a @ A @ C )
% 0.29/0.47 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Pair_le
% 0.29/0.47 thf(fact_100_Pair__le,axiom,
% 0.29/0.47 ! [A: set_a,B: nat,C: set_a,D: nat] :
% 0.29/0.47 ( ( ord_le781360189_a_nat @ ( produc515611863_a_nat @ A @ B ) @ ( produc515611863_a_nat @ C @ D ) )
% 0.29/0.47 = ( ( ord_less_eq_set_a @ A @ C )
% 0.29/0.47 & ( ord_less_eq_nat @ B @ D ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Pair_le
% 0.29/0.47 thf(fact_101_Pair__le,axiom,
% 0.29/0.47 ! [A: nat,B: set_a,C: nat,D: set_a] :
% 0.29/0.47 ( ( ord_le841160035_set_a @ ( produc1450889781_set_a @ A @ B ) @ ( produc1450889781_set_a @ C @ D ) )
% 0.29/0.47 = ( ( ord_less_eq_nat @ A @ C )
% 0.29/0.47 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Pair_le
% 0.29/0.47 thf(fact_102_Pair__le,axiom,
% 0.29/0.47 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.47 ( ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
% 0.29/0.47 = ( ( ord_less_eq_nat @ A @ C )
% 0.29/0.47 & ( ord_less_eq_nat @ B @ D ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Pair_le
% 0.29/0.47 thf(fact_103_as__good__as__sameIff,axiom,
% 0.29/0.47 ! [Z: a,Y: a] :
% 0.29/0.47 ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 0.29/0.47 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % as_good_as_sameIff
% 0.29/0.47 thf(fact_104_as__good__asIff,axiom,
% 0.29/0.47 ! [X: a,Y: a] :
% 0.29/0.47 ( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 0.29/0.47 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % as_good_asIff
% 0.29/0.47 thf(fact_105_fnt__carrier__fnt__rel,axiom,
% 0.29/0.47 ( ( finite_finite_a @ carrier )
% 0.29/0.47 => ( finite179568208od_a_a @ relation ) ) ).
% 0.29/0.47
% 0.29/0.47 % fnt_carrier_fnt_rel
% 0.29/0.47 thf(fact_106_card__mono,axiom,
% 0.29/0.47 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 0.29/0.47 ( ( finite179568208od_a_a @ B2 )
% 0.29/0.47 => ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % card_mono
% 0.29/0.47 thf(fact_107_card__mono,axiom,
% 0.29/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.29/0.47 ( ( finite_finite_nat @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % card_mono
% 0.29/0.47 thf(fact_108_card__mono,axiom,
% 0.29/0.47 ! [B2: set_a,A2: set_a] :
% 0.29/0.47 ( ( finite_finite_a @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % card_mono
% 0.29/0.47 thf(fact_109_empty__Collect__eq,axiom,
% 0.29/0.47 ! [P: a > $o] :
% 0.29/0.47 ( ( bot_bot_set_a
% 0.29/0.47 = ( collect_a @ P ) )
% 0.29/0.47 = ( ! [X3: a] :
% 0.29/0.47 ~ ( P @ X3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % empty_Collect_eq
% 0.29/0.47 thf(fact_110_Collect__empty__eq,axiom,
% 0.29/0.47 ! [P: a > $o] :
% 0.29/0.47 ( ( ( collect_a @ P )
% 0.29/0.47 = bot_bot_set_a )
% 0.29/0.47 = ( ! [X3: a] :
% 0.29/0.47 ~ ( P @ X3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_empty_eq
% 0.29/0.47 thf(fact_111_all__not__in__conv,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ! [X3: product_prod_a_a] :
% 0.29/0.47 ~ ( member449909584od_a_a @ X3 @ A2 ) )
% 0.29/0.47 = ( A2 = bot_bo2131659635od_a_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % all_not_in_conv
% 0.29/0.47 thf(fact_112_all__not__in__conv,axiom,
% 0.29/0.47 ! [A2: set_nat] :
% 0.29/0.47 ( ( ! [X3: nat] :
% 0.29/0.47 ~ ( member_nat @ X3 @ A2 ) )
% 0.29/0.47 = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % all_not_in_conv
% 0.29/0.47 thf(fact_113_all__not__in__conv,axiom,
% 0.29/0.47 ! [A2: set_a] :
% 0.29/0.47 ( ( ! [X3: a] :
% 0.29/0.47 ~ ( member_a @ X3 @ A2 ) )
% 0.29/0.47 = ( A2 = bot_bot_set_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % all_not_in_conv
% 0.29/0.47 thf(fact_114_empty__iff,axiom,
% 0.29/0.47 ! [C: product_prod_a_a] :
% 0.29/0.47 ~ ( member449909584od_a_a @ C @ bot_bo2131659635od_a_a ) ).
% 0.29/0.47
% 0.29/0.47 % empty_iff
% 0.29/0.47 thf(fact_115_empty__iff,axiom,
% 0.29/0.47 ! [C: nat] :
% 0.29/0.47 ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 0.29/0.47
% 0.29/0.47 % empty_iff
% 0.29/0.47 thf(fact_116_empty__iff,axiom,
% 0.29/0.47 ! [C: a] :
% 0.29/0.47 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 0.29/0.47
% 0.29/0.47 % empty_iff
% 0.29/0.47 thf(fact_117_subsetI,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.29/0.47 ( ! [X4: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X4 @ A2 )
% 0.29/0.47 => ( member449909584od_a_a @ X4 @ B2 ) )
% 0.29/0.47 => ( ord_le1824328871od_a_a @ A2 @ B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % subsetI
% 0.29/0.47 thf(fact_118_subsetI,axiom,
% 0.29/0.47 ! [A2: set_nat,B2: set_nat] :
% 0.29/0.47 ( ! [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 => ( member_nat @ X4 @ B2 ) )
% 0.29/0.47 => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % subsetI
% 0.29/0.47 thf(fact_119_subsetI,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a] :
% 0.29/0.47 ( ! [X4: a] :
% 0.29/0.47 ( ( member_a @ X4 @ A2 )
% 0.29/0.47 => ( member_a @ X4 @ B2 ) )
% 0.29/0.47 => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % subsetI
% 0.29/0.47 thf(fact_120_subset__antisym,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ B2 @ A2 )
% 0.29/0.47 => ( A2 = B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_antisym
% 0.29/0.47 thf(fact_121_at__least__as__goodD,axiom,
% 0.29/0.47 ! [Z: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
% 0.29/0.47 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.29/0.47
% 0.29/0.47 % at_least_as_goodD
% 0.29/0.47 thf(fact_122_at__least__as__goodD,axiom,
% 0.29/0.47 ! [Z: nat,Y: nat,B2: set_nat,Pr: set_Pr1986765409at_nat] :
% 0.29/0.47 ( ( member_nat @ Z @ ( prefer563798164od_nat @ Y @ B2 @ Pr ) )
% 0.29/0.47 => ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ Pr ) ) ).
% 0.29/0.47
% 0.29/0.47 % at_least_as_goodD
% 0.29/0.47 thf(fact_123_at__least__as__goodD,axiom,
% 0.29/0.47 ! [Z: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
% 0.29/0.47 ( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% 0.29/0.47
% 0.29/0.47 % at_least_as_goodD
% 0.29/0.47 thf(fact_124_at__lst__asgd__ge,axiom,
% 0.29/0.47 ! [X: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
% 0.29/0.47 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% 0.29/0.47
% 0.29/0.47 % at_lst_asgd_ge
% 0.29/0.47 thf(fact_125_at__lst__asgd__ge,axiom,
% 0.29/0.47 ! [X: nat,Y: nat,B2: set_nat,Pr: set_Pr1986765409at_nat] :
% 0.29/0.47 ( ( member_nat @ X @ ( prefer563798164od_nat @ Y @ B2 @ Pr ) )
% 0.29/0.47 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Pr ) ) ).
% 0.29/0.47
% 0.29/0.47 % at_lst_asgd_ge
% 0.29/0.47 thf(fact_126_at__lst__asgd__ge,axiom,
% 0.29/0.47 ! [X: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
% 0.29/0.47 ( ( member_a @ X @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
% 0.29/0.47 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% 0.29/0.47
% 0.29/0.47 % at_lst_asgd_ge
% 0.29/0.47 thf(fact_127_ex__in__conv,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A2 ) )
% 0.29/0.47 = ( A2 != bot_bo2131659635od_a_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % ex_in_conv
% 0.29/0.47 thf(fact_128_ex__in__conv,axiom,
% 0.29/0.47 ! [A2: set_nat] :
% 0.29/0.47 ( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 0.29/0.47 = ( A2 != bot_bot_set_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % ex_in_conv
% 0.29/0.47 thf(fact_129_ex__in__conv,axiom,
% 0.29/0.47 ! [A2: set_a] :
% 0.29/0.47 ( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
% 0.29/0.47 = ( A2 != bot_bot_set_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % ex_in_conv
% 0.29/0.47 thf(fact_130_equals0I,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a] :
% 0.29/0.47 ( ! [Y3: product_prod_a_a] :
% 0.29/0.47 ~ ( member449909584od_a_a @ Y3 @ A2 )
% 0.29/0.47 => ( A2 = bot_bo2131659635od_a_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % equals0I
% 0.29/0.47 thf(fact_131_equals0I,axiom,
% 0.29/0.47 ! [A2: set_nat] :
% 0.29/0.47 ( ! [Y3: nat] :
% 0.29/0.47 ~ ( member_nat @ Y3 @ A2 )
% 0.29/0.47 => ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % equals0I
% 0.29/0.47 thf(fact_132_equals0I,axiom,
% 0.29/0.47 ! [A2: set_a] :
% 0.29/0.47 ( ! [Y3: a] :
% 0.29/0.47 ~ ( member_a @ Y3 @ A2 )
% 0.29/0.47 => ( A2 = bot_bot_set_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % equals0I
% 0.29/0.47 thf(fact_133_equals0D,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
% 0.29/0.47 ( ( A2 = bot_bo2131659635od_a_a )
% 0.29/0.47 => ~ ( member449909584od_a_a @ A @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % equals0D
% 0.29/0.47 thf(fact_134_equals0D,axiom,
% 0.29/0.47 ! [A2: set_nat,A: nat] :
% 0.29/0.47 ( ( A2 = bot_bot_set_nat )
% 0.29/0.47 => ~ ( member_nat @ A @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % equals0D
% 0.29/0.47 thf(fact_135_equals0D,axiom,
% 0.29/0.47 ! [A2: set_a,A: a] :
% 0.29/0.47 ( ( A2 = bot_bot_set_a )
% 0.29/0.47 => ~ ( member_a @ A @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % equals0D
% 0.29/0.47 thf(fact_136_emptyE,axiom,
% 0.29/0.47 ! [A: product_prod_a_a] :
% 0.29/0.47 ~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% 0.29/0.47
% 0.29/0.47 % emptyE
% 0.29/0.47 thf(fact_137_emptyE,axiom,
% 0.29/0.47 ! [A: nat] :
% 0.29/0.47 ~ ( member_nat @ A @ bot_bot_set_nat ) ).
% 0.29/0.47
% 0.29/0.47 % emptyE
% 0.29/0.47 thf(fact_138_emptyE,axiom,
% 0.29/0.47 ! [A: a] :
% 0.29/0.47 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 0.29/0.47
% 0.29/0.47 % emptyE
% 0.29/0.47 thf(fact_139_in__mono,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ A2 )
% 0.29/0.47 => ( member449909584od_a_a @ X @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % in_mono
% 0.29/0.47 thf(fact_140_in__mono,axiom,
% 0.29/0.47 ! [A2: set_nat,B2: set_nat,X: nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.47 => ( ( member_nat @ X @ A2 )
% 0.29/0.47 => ( member_nat @ X @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % in_mono
% 0.29/0.47 thf(fact_141_in__mono,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a,X: a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ( ( member_a @ X @ A2 )
% 0.29/0.47 => ( member_a @ X @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % in_mono
% 0.29/0.47 thf(fact_142_subsetD,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 0.29/0.47 => ( ( member449909584od_a_a @ C @ A2 )
% 0.29/0.47 => ( member449909584od_a_a @ C @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subsetD
% 0.29/0.47 thf(fact_143_subsetD,axiom,
% 0.29/0.47 ! [A2: set_nat,B2: set_nat,C: nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.47 => ( ( member_nat @ C @ A2 )
% 0.29/0.47 => ( member_nat @ C @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subsetD
% 0.29/0.47 thf(fact_144_subsetD,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a,C: a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ( ( member_a @ C @ A2 )
% 0.29/0.47 => ( member_a @ C @ B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subsetD
% 0.29/0.47 thf(fact_145_equalityE,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a] :
% 0.29/0.47 ( ( A2 = B2 )
% 0.29/0.47 => ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % equalityE
% 0.29/0.47 thf(fact_146_subset__eq,axiom,
% 0.29/0.47 ( ord_le1824328871od_a_a
% 0.29/0.47 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.29/0.47 ! [X3: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ X3 @ A3 )
% 0.29/0.47 => ( member449909584od_a_a @ X3 @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_eq
% 0.29/0.47 thf(fact_147_subset__eq,axiom,
% 0.29/0.47 ( ord_less_eq_set_nat
% 0.29/0.47 = ( ^ [A3: set_nat,B3: set_nat] :
% 0.29/0.47 ! [X3: nat] :
% 0.29/0.47 ( ( member_nat @ X3 @ A3 )
% 0.29/0.47 => ( member_nat @ X3 @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_eq
% 0.29/0.47 thf(fact_148_subset__eq,axiom,
% 0.29/0.47 ( ord_less_eq_set_a
% 0.29/0.47 = ( ^ [A3: set_a,B3: set_a] :
% 0.29/0.47 ! [X3: a] :
% 0.29/0.47 ( ( member_a @ X3 @ A3 )
% 0.29/0.47 => ( member_a @ X3 @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_eq
% 0.29/0.47 thf(fact_149_equalityD1,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a] :
% 0.29/0.47 ( ( A2 = B2 )
% 0.29/0.47 => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % equalityD1
% 0.29/0.47 thf(fact_150_equalityD2,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a] :
% 0.29/0.47 ( ( A2 = B2 )
% 0.29/0.47 => ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % equalityD2
% 0.29/0.47 thf(fact_151_subset__iff,axiom,
% 0.29/0.47 ( ord_le1824328871od_a_a
% 0.29/0.47 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 0.29/0.47 ! [T: product_prod_a_a] :
% 0.29/0.47 ( ( member449909584od_a_a @ T @ A3 )
% 0.29/0.47 => ( member449909584od_a_a @ T @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_iff
% 0.29/0.47 thf(fact_152_subset__iff,axiom,
% 0.29/0.47 ( ord_less_eq_set_nat
% 0.29/0.47 = ( ^ [A3: set_nat,B3: set_nat] :
% 0.29/0.47 ! [T: nat] :
% 0.29/0.47 ( ( member_nat @ T @ A3 )
% 0.29/0.47 => ( member_nat @ T @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_iff
% 0.29/0.47 thf(fact_153_subset__iff,axiom,
% 0.29/0.47 ( ord_less_eq_set_a
% 0.29/0.47 = ( ^ [A3: set_a,B3: set_a] :
% 0.29/0.47 ! [T: a] :
% 0.29/0.47 ( ( member_a @ T @ A3 )
% 0.29/0.47 => ( member_a @ T @ B3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_iff
% 0.29/0.47 thf(fact_154_subset__refl,axiom,
% 0.29/0.47 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 0.29/0.47
% 0.29/0.47 % subset_refl
% 0.29/0.47 thf(fact_155_Collect__mono,axiom,
% 0.29/0.47 ! [P: a > $o,Q: a > $o] :
% 0.29/0.47 ( ! [X4: a] :
% 0.29/0.47 ( ( P @ X4 )
% 0.29/0.47 => ( Q @ X4 ) )
% 0.29/0.47 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_mono
% 0.29/0.47 thf(fact_156_subset__trans,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a,C2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 0.29/0.47 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % subset_trans
% 0.29/0.47 thf(fact_157_set__eq__subset,axiom,
% 0.29/0.47 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.29/0.47 = ( ^ [A3: set_a,B3: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 0.29/0.47 & ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % set_eq_subset
% 0.29/0.47 thf(fact_158_Collect__mono__iff,axiom,
% 0.29/0.47 ! [P: a > $o,Q: a > $o] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.29/0.47 = ( ! [X3: a] :
% 0.29/0.47 ( ( P @ X3 )
% 0.29/0.47 => ( Q @ X3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Collect_mono_iff
% 0.29/0.47 thf(fact_159_rational__preference_Osame__at__least__as__equal,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) )
% 0.29/0.47 => ( ( prefer161218362good_a @ Z @ Carrier @ Relation )
% 0.29/0.47 = ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.same_at_least_as_equal
% 0.29/0.47 thf(fact_160_rational__preference_Opref__in__at__least__as,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 => ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.pref_in_at_least_as
% 0.29/0.47 thf(fact_161_rational__preference_Opref__in__at__least__as,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 0.29/0.47 => ( member_nat @ X @ ( prefer563798164od_nat @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.pref_in_at_least_as
% 0.29/0.47 thf(fact_162_rational__preference_Opref__in__at__least__as,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 => ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.pref_in_at_least_as
% 0.29/0.47 thf(fact_163_rational__preference_Oas__good__as__sameIff,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Z: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ Z @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 0.29/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.as_good_as_sameIff
% 0.29/0.47 thf(fact_164_rational__preference_Oas__good__as__sameIff,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,Z: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_nat @ Z @ ( prefer1936906324as_nat @ Y @ Carrier @ Relation ) )
% 0.29/0.47 = ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ Relation )
% 0.29/0.47 & ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.as_good_as_sameIff
% 0.29/0.47 thf(fact_165_rational__preference_Oas__good__as__sameIff,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.as_good_as_sameIff
% 0.29/0.47 thf(fact_166_rational__preference_Oas__good__asIff,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.as_good_asIff
% 0.29/0.47 thf(fact_167_rational__preference_Oas__good__asIff,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_nat @ X @ ( prefer1936906324as_nat @ Y @ Carrier @ Relation ) )
% 0.29/0.47 = ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 0.29/0.47 & ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.as_good_asIff
% 0.29/0.47 thf(fact_168_rational__preference_Oas__good__asIff,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 0.29/0.47 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.as_good_asIff
% 0.29/0.47 thf(fact_169_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.29/0.47 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.47 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( Carrier != bot_bo2131659635od_a_a )
% 0.29/0.47 => ( ( member449909584od_a_a @ X @ Carrier )
% 0.29/0.47 => ( ( member449909584od_a_a @ Y @ Carrier )
% 0.29/0.47 => ( ~ ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.at_lst_asgd_not_ge
% 0.29/0.47 thf(fact_170_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.29/0.47 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 0.29/0.47 => ( ( Carrier != bot_bot_set_nat )
% 0.29/0.47 => ( ( member_nat @ X @ Carrier )
% 0.29/0.47 => ( ( member_nat @ Y @ Carrier )
% 0.29/0.47 => ( ~ ( member_nat @ X @ ( prefer563798164od_nat @ Y @ Carrier @ Relation ) )
% 0.29/0.47 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.at_lst_asgd_not_ge
% 0.29/0.47 thf(fact_171_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 0.29/0.47 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.47 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.47 => ( ( Carrier != bot_bot_set_a )
% 0.29/0.47 => ( ( member_a @ X @ Carrier )
% 0.29/0.47 => ( ( member_a @ Y @ Carrier )
% 0.29/0.47 => ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) )
% 0.29/0.47 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rational_preference.at_lst_asgd_not_ge
% 0.29/0.47 thf(fact_172_Pair__mono,axiom,
% 0.29/0.47 ! [X: set_a,X5: set_a,Y: set_a,Y5: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ X @ X5 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 0.29/0.47 => ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X @ Y ) @ ( produc1928581911_set_a @ X5 @ Y5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Pair_mono
% 0.29/0.47 thf(fact_173_Pair__mono,axiom,
% 0.29/0.47 ! [X: set_a,X5: set_a,Y: nat,Y5: nat] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ X @ X5 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ Y @ Y5 )
% 0.29/0.47 => ( ord_le781360189_a_nat @ ( produc515611863_a_nat @ X @ Y ) @ ( produc515611863_a_nat @ X5 @ Y5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Pair_mono
% 0.29/0.47 thf(fact_174_Pair__mono,axiom,
% 0.29/0.47 ! [X: nat,X5: nat,Y: set_a,Y5: set_a] :
% 0.29/0.47 ( ( ord_less_eq_nat @ X @ X5 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 0.29/0.47 => ( ord_le841160035_set_a @ ( produc1450889781_set_a @ X @ Y ) @ ( produc1450889781_set_a @ X5 @ Y5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Pair_mono
% 0.29/0.47 thf(fact_175_Pair__mono,axiom,
% 0.29/0.47 ! [X: nat,X5: nat,Y: nat,Y5: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ X @ X5 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ Y @ Y5 )
% 0.29/0.47 => ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % Pair_mono
% 0.29/0.47 thf(fact_176_finite__has__maximal2,axiom,
% 0.29/0.47 ! [A2: set_set_a,A: set_a] :
% 0.29/0.47 ( ( finite_finite_set_a @ A2 )
% 0.29/0.47 => ( ( member_set_a @ A @ A2 )
% 0.29/0.47 => ? [X4: set_a] :
% 0.29/0.47 ( ( member_set_a @ X4 @ A2 )
% 0.29/0.47 & ( ord_less_eq_set_a @ A @ X4 )
% 0.29/0.47 & ! [Xa: set_a] :
% 0.29/0.47 ( ( member_set_a @ Xa @ A2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ X4 @ Xa )
% 0.29/0.47 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_has_maximal2
% 0.29/0.47 thf(fact_177_finite__has__maximal2,axiom,
% 0.29/0.47 ! [A2: set_nat,A: nat] :
% 0.29/0.47 ( ( finite_finite_nat @ A2 )
% 0.29/0.47 => ( ( member_nat @ A @ A2 )
% 0.29/0.47 => ? [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 & ( ord_less_eq_nat @ A @ X4 )
% 0.29/0.47 & ! [Xa: nat] :
% 0.29/0.47 ( ( member_nat @ Xa @ A2 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 0.29/0.47 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_has_maximal2
% 0.29/0.47 thf(fact_178_finite__has__minimal2,axiom,
% 0.29/0.47 ! [A2: set_set_a,A: set_a] :
% 0.29/0.47 ( ( finite_finite_set_a @ A2 )
% 0.29/0.47 => ( ( member_set_a @ A @ A2 )
% 0.29/0.47 => ? [X4: set_a] :
% 0.29/0.47 ( ( member_set_a @ X4 @ A2 )
% 0.29/0.47 & ( ord_less_eq_set_a @ X4 @ A )
% 0.29/0.47 & ! [Xa: set_a] :
% 0.29/0.47 ( ( member_set_a @ Xa @ A2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ Xa @ X4 )
% 0.29/0.47 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_has_minimal2
% 0.29/0.47 thf(fact_179_finite__has__minimal2,axiom,
% 0.29/0.47 ! [A2: set_nat,A: nat] :
% 0.29/0.47 ( ( finite_finite_nat @ A2 )
% 0.29/0.47 => ( ( member_nat @ A @ A2 )
% 0.29/0.47 => ? [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 & ( ord_less_eq_nat @ X4 @ A )
% 0.29/0.47 & ! [Xa: nat] :
% 0.29/0.47 ( ( member_nat @ Xa @ A2 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.29/0.47 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_has_minimal2
% 0.29/0.47 thf(fact_180_bot__prod__def,axiom,
% 0.29/0.47 ( bot_bo1519632275_set_a
% 0.29/0.47 = ( produc1928581911_set_a @ bot_bot_set_a @ bot_bot_set_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % bot_prod_def
% 0.29/0.47 thf(fact_181_infinite__imp__nonempty,axiom,
% 0.29/0.47 ! [S: set_Product_prod_a_a] :
% 0.29/0.47 ( ~ ( finite179568208od_a_a @ S )
% 0.29/0.47 => ( S != bot_bo2131659635od_a_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_imp_nonempty
% 0.29/0.47 thf(fact_182_infinite__imp__nonempty,axiom,
% 0.29/0.47 ! [S: set_nat] :
% 0.29/0.47 ( ~ ( finite_finite_nat @ S )
% 0.29/0.47 => ( S != bot_bot_set_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_imp_nonempty
% 0.29/0.47 thf(fact_183_infinite__imp__nonempty,axiom,
% 0.29/0.47 ! [S: set_a] :
% 0.29/0.47 ( ~ ( finite_finite_a @ S )
% 0.29/0.47 => ( S != bot_bot_set_a ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_imp_nonempty
% 0.29/0.47 thf(fact_184_finite_OemptyI,axiom,
% 0.29/0.47 finite179568208od_a_a @ bot_bo2131659635od_a_a ).
% 0.29/0.47
% 0.29/0.47 % finite.emptyI
% 0.29/0.47 thf(fact_185_finite_OemptyI,axiom,
% 0.29/0.47 finite_finite_nat @ bot_bot_set_nat ).
% 0.29/0.47
% 0.29/0.47 % finite.emptyI
% 0.29/0.47 thf(fact_186_finite_OemptyI,axiom,
% 0.29/0.47 finite_finite_a @ bot_bot_set_a ).
% 0.29/0.47
% 0.29/0.47 % finite.emptyI
% 0.29/0.47 thf(fact_187_finite__subset,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 0.29/0.47 => ( ( finite179568208od_a_a @ B2 )
% 0.29/0.47 => ( finite179568208od_a_a @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_subset
% 0.29/0.47 thf(fact_188_finite__subset,axiom,
% 0.29/0.47 ! [A2: set_nat,B2: set_nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.47 => ( ( finite_finite_nat @ B2 )
% 0.29/0.47 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_subset
% 0.29/0.47 thf(fact_189_finite__subset,axiom,
% 0.29/0.47 ! [A2: set_a,B2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ( ( finite_finite_a @ B2 )
% 0.29/0.47 => ( finite_finite_a @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_subset
% 0.29/0.47 thf(fact_190_infinite__super,axiom,
% 0.29/0.47 ! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ S @ T2 )
% 0.29/0.47 => ( ~ ( finite179568208od_a_a @ S )
% 0.29/0.47 => ~ ( finite179568208od_a_a @ T2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_super
% 0.29/0.47 thf(fact_191_infinite__super,axiom,
% 0.29/0.47 ! [S: set_nat,T2: set_nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.29/0.47 => ( ~ ( finite_finite_nat @ S )
% 0.29/0.47 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_super
% 0.29/0.47 thf(fact_192_infinite__super,axiom,
% 0.29/0.47 ! [S: set_a,T2: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ S @ T2 )
% 0.29/0.47 => ( ~ ( finite_finite_a @ S )
% 0.29/0.47 => ~ ( finite_finite_a @ T2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_super
% 0.29/0.47 thf(fact_193_rev__finite__subset,axiom,
% 0.29/0.47 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 0.29/0.47 ( ( finite179568208od_a_a @ B2 )
% 0.29/0.47 => ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 0.29/0.47 => ( finite179568208od_a_a @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rev_finite_subset
% 0.29/0.47 thf(fact_194_rev__finite__subset,axiom,
% 0.29/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.29/0.47 ( ( finite_finite_nat @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.47 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rev_finite_subset
% 0.29/0.47 thf(fact_195_rev__finite__subset,axiom,
% 0.29/0.47 ! [B2: set_a,A2: set_a] :
% 0.29/0.47 ( ( finite_finite_a @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ( finite_finite_a @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % rev_finite_subset
% 0.29/0.47 thf(fact_196_finite__has__minimal,axiom,
% 0.29/0.47 ! [A2: set_set_a] :
% 0.29/0.47 ( ( finite_finite_set_a @ A2 )
% 0.29/0.47 => ( ( A2 != bot_bot_set_set_a )
% 0.29/0.47 => ? [X4: set_a] :
% 0.29/0.47 ( ( member_set_a @ X4 @ A2 )
% 0.29/0.47 & ! [Xa: set_a] :
% 0.29/0.47 ( ( member_set_a @ Xa @ A2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ Xa @ X4 )
% 0.29/0.47 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_has_minimal
% 0.29/0.47 thf(fact_197_finite__has__minimal,axiom,
% 0.29/0.47 ! [A2: set_nat] :
% 0.29/0.47 ( ( finite_finite_nat @ A2 )
% 0.29/0.47 => ( ( A2 != bot_bot_set_nat )
% 0.29/0.47 => ? [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 & ! [Xa: nat] :
% 0.29/0.47 ( ( member_nat @ Xa @ A2 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.29/0.47 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_has_minimal
% 0.29/0.47 thf(fact_198_finite__has__maximal,axiom,
% 0.29/0.47 ! [A2: set_set_a] :
% 0.29/0.47 ( ( finite_finite_set_a @ A2 )
% 0.29/0.47 => ( ( A2 != bot_bot_set_set_a )
% 0.29/0.47 => ? [X4: set_a] :
% 0.29/0.47 ( ( member_set_a @ X4 @ A2 )
% 0.29/0.47 & ! [Xa: set_a] :
% 0.29/0.47 ( ( member_set_a @ Xa @ A2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ X4 @ Xa )
% 0.29/0.47 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_has_maximal
% 0.29/0.47 thf(fact_199_finite__has__maximal,axiom,
% 0.29/0.47 ! [A2: set_nat] :
% 0.29/0.47 ( ( finite_finite_nat @ A2 )
% 0.29/0.47 => ( ( A2 != bot_bot_set_nat )
% 0.29/0.47 => ? [X4: nat] :
% 0.29/0.47 ( ( member_nat @ X4 @ A2 )
% 0.29/0.47 & ! [Xa: nat] :
% 0.29/0.47 ( ( member_nat @ Xa @ A2 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 0.29/0.47 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_has_maximal
% 0.29/0.47 thf(fact_200_card__subset__eq,axiom,
% 0.29/0.47 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 0.29/0.47 ( ( finite179568208od_a_a @ B2 )
% 0.29/0.47 => ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 0.29/0.47 => ( ( ( finite1481642319od_a_a @ A2 )
% 0.29/0.47 = ( finite1481642319od_a_a @ B2 ) )
% 0.29/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % card_subset_eq
% 0.29/0.47 thf(fact_201_card__subset__eq,axiom,
% 0.29/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.29/0.47 ( ( finite_finite_nat @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.47 => ( ( ( finite_card_nat @ A2 )
% 0.29/0.47 = ( finite_card_nat @ B2 ) )
% 0.29/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % card_subset_eq
% 0.29/0.47 thf(fact_202_card__subset__eq,axiom,
% 0.29/0.47 ! [B2: set_a,A2: set_a] :
% 0.29/0.47 ( ( finite_finite_a @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ( ( ( finite_card_a @ A2 )
% 0.29/0.47 = ( finite_card_a @ B2 ) )
% 0.29/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % card_subset_eq
% 0.29/0.47 thf(fact_203_infinite__arbitrarily__large,axiom,
% 0.29/0.47 ! [A2: set_Product_prod_a_a,N: nat] :
% 0.29/0.47 ( ~ ( finite179568208od_a_a @ A2 )
% 0.29/0.47 => ? [B4: set_Product_prod_a_a] :
% 0.29/0.47 ( ( finite179568208od_a_a @ B4 )
% 0.29/0.47 & ( ( finite1481642319od_a_a @ B4 )
% 0.29/0.47 = N )
% 0.29/0.47 & ( ord_le1824328871od_a_a @ B4 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_arbitrarily_large
% 0.29/0.47 thf(fact_204_infinite__arbitrarily__large,axiom,
% 0.29/0.47 ! [A2: set_nat,N: nat] :
% 0.29/0.47 ( ~ ( finite_finite_nat @ A2 )
% 0.29/0.47 => ? [B4: set_nat] :
% 0.29/0.47 ( ( finite_finite_nat @ B4 )
% 0.29/0.47 & ( ( finite_card_nat @ B4 )
% 0.29/0.47 = N )
% 0.29/0.47 & ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_arbitrarily_large
% 0.29/0.47 thf(fact_205_infinite__arbitrarily__large,axiom,
% 0.29/0.47 ! [A2: set_a,N: nat] :
% 0.29/0.47 ( ~ ( finite_finite_a @ A2 )
% 0.29/0.47 => ? [B4: set_a] :
% 0.29/0.47 ( ( finite_finite_a @ B4 )
% 0.29/0.47 & ( ( finite_card_a @ B4 )
% 0.29/0.47 = N )
% 0.29/0.47 & ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % infinite_arbitrarily_large
% 0.29/0.47 thf(fact_206_finite__if__finite__subsets__card__bdd,axiom,
% 0.29/0.47 ! [F: set_Product_prod_a_a,C2: nat] :
% 0.29/0.47 ( ! [G: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ G @ F )
% 0.29/0.47 => ( ( finite179568208od_a_a @ G )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ G ) @ C2 ) ) )
% 0.29/0.47 => ( ( finite179568208od_a_a @ F )
% 0.29/0.47 & ( ord_less_eq_nat @ ( finite1481642319od_a_a @ F ) @ C2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_if_finite_subsets_card_bdd
% 0.29/0.47 thf(fact_207_finite__if__finite__subsets__card__bdd,axiom,
% 0.29/0.47 ! [F: set_nat,C2: nat] :
% 0.29/0.47 ( ! [G: set_nat] :
% 0.29/0.47 ( ( ord_less_eq_set_nat @ G @ F )
% 0.29/0.47 => ( ( finite_finite_nat @ G )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
% 0.29/0.47 => ( ( finite_finite_nat @ F )
% 0.29/0.47 & ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_if_finite_subsets_card_bdd
% 0.29/0.47 thf(fact_208_finite__if__finite__subsets__card__bdd,axiom,
% 0.29/0.47 ! [F: set_a,C2: nat] :
% 0.29/0.47 ( ! [G: set_a] :
% 0.29/0.47 ( ( ord_less_eq_set_a @ G @ F )
% 0.29/0.47 => ( ( finite_finite_a @ G )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C2 ) ) )
% 0.29/0.47 => ( ( finite_finite_a @ F )
% 0.29/0.47 & ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % finite_if_finite_subsets_card_bdd
% 0.29/0.47 thf(fact_209_card__seteq,axiom,
% 0.29/0.47 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 0.29/0.47 ( ( finite179568208od_a_a @ B2 )
% 0.29/0.47 => ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ ( finite1481642319od_a_a @ B2 ) @ ( finite1481642319od_a_a @ A2 ) )
% 0.29/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % card_seteq
% 0.29/0.47 thf(fact_210_card__seteq,axiom,
% 0.29/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.29/0.47 ( ( finite_finite_nat @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
% 0.29/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % card_seteq
% 0.29/0.47 thf(fact_211_card__seteq,axiom,
% 0.29/0.47 ! [B2: set_a,A2: set_a] :
% 0.29/0.47 ( ( finite_finite_a @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
% 0.29/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % card_seteq
% 0.29/0.47 thf(fact_212_obtain__subset__with__card__n,axiom,
% 0.29/0.47 ! [N: nat,S: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ord_less_eq_nat @ N @ ( finite1481642319od_a_a @ S ) )
% 0.29/0.47 => ~ ! [T3: set_Product_prod_a_a] :
% 0.29/0.47 ( ( ord_le1824328871od_a_a @ T3 @ S )
% 0.29/0.47 => ( ( ( finite1481642319od_a_a @ T3 )
% 0.29/0.47 = N )
% 0.29/0.47 => ~ ( finite179568208od_a_a @ T3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % obtain_subset_with_card_n
% 0.29/0.47 thf(fact_213_obtain__subset__with__card__n,axiom,
% 0.29/0.47 ! [N: nat,S: set_nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
% 0.29/0.48 => ~ ! [T3: set_nat] :
% 0.29/0.48 ( ( ord_less_eq_set_nat @ T3 @ S )
% 0.29/0.48 => ( ( ( finite_card_nat @ T3 )
% 0.29/0.48 = N )
% 0.29/0.48 => ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % obtain_subset_with_card_n
% 0.29/0.48 thf(fact_214_obtain__subset__with__card__n,axiom,
% 0.29/0.48 ! [N: nat,S: set_a] :
% 0.29/0.48 ( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
% 0.29/0.48 => ~ ! [T3: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ T3 @ S )
% 0.29/0.48 => ( ( ( finite_card_a @ T3 )
% 0.29/0.48 = N )
% 0.29/0.48 => ~ ( finite_finite_a @ T3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % obtain_subset_with_card_n
% 0.29/0.48 thf(fact_215_card__le__if__inj__on__rel,axiom,
% 0.29/0.48 ! [B2: set_a,A2: set_Product_prod_a_a,R: product_prod_a_a > a > $o] :
% 0.29/0.48 ( ( finite_finite_a @ B2 )
% 0.29/0.48 => ( ! [A4: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ A4 @ A2 )
% 0.29/0.48 => ? [B5: a] :
% 0.29/0.48 ( ( member_a @ B5 @ B2 )
% 0.29/0.48 & ( R @ A4 @ B5 ) ) )
% 0.29/0.48 => ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: a] :
% 0.29/0.48 ( ( member449909584od_a_a @ A1 @ A2 )
% 0.29/0.48 => ( ( member449909584od_a_a @ A22 @ A2 )
% 0.29/0.48 => ( ( member_a @ B6 @ B2 )
% 0.29/0.48 => ( ( R @ A1 @ B6 )
% 0.29/0.48 => ( ( R @ A22 @ B6 )
% 0.29/0.48 => ( A1 = A22 ) ) ) ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_le_if_inj_on_rel
% 0.29/0.48 thf(fact_216_card__le__if__inj__on__rel,axiom,
% 0.29/0.48 ! [B2: set_a,A2: set_nat,R: nat > a > $o] :
% 0.29/0.48 ( ( finite_finite_a @ B2 )
% 0.29/0.48 => ( ! [A4: nat] :
% 0.29/0.48 ( ( member_nat @ A4 @ A2 )
% 0.29/0.48 => ? [B5: a] :
% 0.29/0.48 ( ( member_a @ B5 @ B2 )
% 0.29/0.48 & ( R @ A4 @ B5 ) ) )
% 0.29/0.48 => ( ! [A1: nat,A22: nat,B6: a] :
% 0.29/0.48 ( ( member_nat @ A1 @ A2 )
% 0.29/0.48 => ( ( member_nat @ A22 @ A2 )
% 0.29/0.48 => ( ( member_a @ B6 @ B2 )
% 0.29/0.48 => ( ( R @ A1 @ B6 )
% 0.29/0.48 => ( ( R @ A22 @ B6 )
% 0.29/0.48 => ( A1 = A22 ) ) ) ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_le_if_inj_on_rel
% 0.29/0.48 thf(fact_217_card__le__if__inj__on__rel,axiom,
% 0.29/0.48 ! [B2: set_a,A2: set_a,R: a > a > $o] :
% 0.29/0.48 ( ( finite_finite_a @ B2 )
% 0.29/0.48 => ( ! [A4: a] :
% 0.29/0.48 ( ( member_a @ A4 @ A2 )
% 0.29/0.48 => ? [B5: a] :
% 0.29/0.48 ( ( member_a @ B5 @ B2 )
% 0.29/0.48 & ( R @ A4 @ B5 ) ) )
% 0.29/0.48 => ( ! [A1: a,A22: a,B6: a] :
% 0.29/0.48 ( ( member_a @ A1 @ A2 )
% 0.29/0.48 => ( ( member_a @ A22 @ A2 )
% 0.29/0.48 => ( ( member_a @ B6 @ B2 )
% 0.29/0.48 => ( ( R @ A1 @ B6 )
% 0.29/0.48 => ( ( R @ A22 @ B6 )
% 0.29/0.48 => ( A1 = A22 ) ) ) ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_le_if_inj_on_rel
% 0.29/0.48 thf(fact_218_card__le__if__inj__on__rel,axiom,
% 0.29/0.48 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a,R: product_prod_a_a > product_prod_a_a > $o] :
% 0.29/0.48 ( ( finite179568208od_a_a @ B2 )
% 0.29/0.48 => ( ! [A4: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ A4 @ A2 )
% 0.29/0.48 => ? [B5: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ B5 @ B2 )
% 0.29/0.48 & ( R @ A4 @ B5 ) ) )
% 0.29/0.48 => ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ A1 @ A2 )
% 0.29/0.48 => ( ( member449909584od_a_a @ A22 @ A2 )
% 0.29/0.48 => ( ( member449909584od_a_a @ B6 @ B2 )
% 0.29/0.48 => ( ( R @ A1 @ B6 )
% 0.29/0.48 => ( ( R @ A22 @ B6 )
% 0.29/0.48 => ( A1 = A22 ) ) ) ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_le_if_inj_on_rel
% 0.29/0.48 thf(fact_219_card__le__if__inj__on__rel,axiom,
% 0.29/0.48 ! [B2: set_Product_prod_a_a,A2: set_nat,R: nat > product_prod_a_a > $o] :
% 0.29/0.48 ( ( finite179568208od_a_a @ B2 )
% 0.29/0.48 => ( ! [A4: nat] :
% 0.29/0.48 ( ( member_nat @ A4 @ A2 )
% 0.29/0.48 => ? [B5: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ B5 @ B2 )
% 0.29/0.48 & ( R @ A4 @ B5 ) ) )
% 0.29/0.48 => ( ! [A1: nat,A22: nat,B6: product_prod_a_a] :
% 0.29/0.48 ( ( member_nat @ A1 @ A2 )
% 0.29/0.48 => ( ( member_nat @ A22 @ A2 )
% 0.29/0.48 => ( ( member449909584od_a_a @ B6 @ B2 )
% 0.29/0.48 => ( ( R @ A1 @ B6 )
% 0.29/0.48 => ( ( R @ A22 @ B6 )
% 0.29/0.48 => ( A1 = A22 ) ) ) ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_le_if_inj_on_rel
% 0.29/0.48 thf(fact_220_card__le__if__inj__on__rel,axiom,
% 0.29/0.48 ! [B2: set_Product_prod_a_a,A2: set_a,R: a > product_prod_a_a > $o] :
% 0.29/0.48 ( ( finite179568208od_a_a @ B2 )
% 0.29/0.48 => ( ! [A4: a] :
% 0.29/0.48 ( ( member_a @ A4 @ A2 )
% 0.29/0.48 => ? [B5: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ B5 @ B2 )
% 0.29/0.48 & ( R @ A4 @ B5 ) ) )
% 0.29/0.48 => ( ! [A1: a,A22: a,B6: product_prod_a_a] :
% 0.29/0.48 ( ( member_a @ A1 @ A2 )
% 0.29/0.48 => ( ( member_a @ A22 @ A2 )
% 0.29/0.48 => ( ( member449909584od_a_a @ B6 @ B2 )
% 0.29/0.48 => ( ( R @ A1 @ B6 )
% 0.29/0.48 => ( ( R @ A22 @ B6 )
% 0.29/0.48 => ( A1 = A22 ) ) ) ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_le_if_inj_on_rel
% 0.29/0.48 thf(fact_221_card__le__if__inj__on__rel,axiom,
% 0.29/0.48 ! [B2: set_nat,A2: set_Product_prod_a_a,R: product_prod_a_a > nat > $o] :
% 0.29/0.48 ( ( finite_finite_nat @ B2 )
% 0.29/0.48 => ( ! [A4: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ A4 @ A2 )
% 0.29/0.48 => ? [B5: nat] :
% 0.29/0.48 ( ( member_nat @ B5 @ B2 )
% 0.29/0.48 & ( R @ A4 @ B5 ) ) )
% 0.29/0.48 => ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: nat] :
% 0.29/0.48 ( ( member449909584od_a_a @ A1 @ A2 )
% 0.29/0.48 => ( ( member449909584od_a_a @ A22 @ A2 )
% 0.29/0.48 => ( ( member_nat @ B6 @ B2 )
% 0.29/0.48 => ( ( R @ A1 @ B6 )
% 0.29/0.48 => ( ( R @ A22 @ B6 )
% 0.29/0.48 => ( A1 = A22 ) ) ) ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_le_if_inj_on_rel
% 0.29/0.48 thf(fact_222_card__le__if__inj__on__rel,axiom,
% 0.29/0.48 ! [B2: set_nat,A2: set_nat,R: nat > nat > $o] :
% 0.29/0.48 ( ( finite_finite_nat @ B2 )
% 0.29/0.48 => ( ! [A4: nat] :
% 0.29/0.48 ( ( member_nat @ A4 @ A2 )
% 0.29/0.48 => ? [B5: nat] :
% 0.29/0.48 ( ( member_nat @ B5 @ B2 )
% 0.29/0.48 & ( R @ A4 @ B5 ) ) )
% 0.29/0.48 => ( ! [A1: nat,A22: nat,B6: nat] :
% 0.29/0.48 ( ( member_nat @ A1 @ A2 )
% 0.29/0.48 => ( ( member_nat @ A22 @ A2 )
% 0.29/0.48 => ( ( member_nat @ B6 @ B2 )
% 0.29/0.48 => ( ( R @ A1 @ B6 )
% 0.29/0.48 => ( ( R @ A22 @ B6 )
% 0.29/0.48 => ( A1 = A22 ) ) ) ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_le_if_inj_on_rel
% 0.29/0.48 thf(fact_223_card__le__if__inj__on__rel,axiom,
% 0.29/0.48 ! [B2: set_nat,A2: set_a,R: a > nat > $o] :
% 0.29/0.48 ( ( finite_finite_nat @ B2 )
% 0.29/0.48 => ( ! [A4: a] :
% 0.29/0.48 ( ( member_a @ A4 @ A2 )
% 0.29/0.48 => ? [B5: nat] :
% 0.29/0.48 ( ( member_nat @ B5 @ B2 )
% 0.29/0.48 & ( R @ A4 @ B5 ) ) )
% 0.29/0.48 => ( ! [A1: a,A22: a,B6: nat] :
% 0.29/0.48 ( ( member_a @ A1 @ A2 )
% 0.29/0.48 => ( ( member_a @ A22 @ A2 )
% 0.29/0.48 => ( ( member_nat @ B6 @ B2 )
% 0.29/0.48 => ( ( R @ A1 @ B6 )
% 0.29/0.48 => ( ( R @ A22 @ B6 )
% 0.29/0.48 => ( A1 = A22 ) ) ) ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_le_if_inj_on_rel
% 0.29/0.48 thf(fact_224_ex__card,axiom,
% 0.29/0.48 ! [N: nat,A2: set_a] :
% 0.29/0.48 ( ( ord_less_eq_nat @ N @ ( finite_card_a @ A2 ) )
% 0.29/0.48 => ? [S2: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ S2 @ A2 )
% 0.29/0.48 & ( ( finite_card_a @ S2 )
% 0.29/0.48 = N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ex_card
% 0.29/0.48 thf(fact_225_old_Oprod_Oinject,axiom,
% 0.29/0.48 ! [A: a,B: a,A5: a,B7: a] :
% 0.29/0.48 ( ( ( product_Pair_a_a @ A @ B )
% 0.29/0.48 = ( product_Pair_a_a @ A5 @ B7 ) )
% 0.29/0.48 = ( ( A = A5 )
% 0.29/0.48 & ( B = B7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % old.prod.inject
% 0.29/0.48 thf(fact_226_prod_Oinject,axiom,
% 0.29/0.48 ! [X1: a,X22: a,Y1: a,Y22: a] :
% 0.29/0.48 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 0.29/0.48 = ( product_Pair_a_a @ Y1 @ Y22 ) )
% 0.29/0.48 = ( ( X1 = Y1 )
% 0.29/0.48 & ( X22 = Y22 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % prod.inject
% 0.29/0.48 thf(fact_227_order__refl,axiom,
% 0.29/0.48 ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% 0.29/0.48
% 0.29/0.48 % order_refl
% 0.29/0.48 thf(fact_228_order__refl,axiom,
% 0.29/0.48 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.29/0.48
% 0.29/0.48 % order_refl
% 0.29/0.48 thf(fact_229_bot__set__def,axiom,
% 0.29/0.48 ( bot_bot_set_a
% 0.29/0.48 = ( collect_a @ bot_bot_a_o ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot_set_def
% 0.29/0.48 thf(fact_230_dual__order_Oantisym,axiom,
% 0.29/0.48 ! [B: set_a,A: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ B @ A )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.48 => ( A = B ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.antisym
% 0.29/0.48 thf(fact_231_dual__order_Oantisym,axiom,
% 0.29/0.48 ! [B: nat,A: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.48 => ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48 => ( A = B ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.antisym
% 0.29/0.48 thf(fact_232_dual__order_Oeq__iff,axiom,
% 0.29/0.48 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.29/0.48 = ( ^ [A6: set_a,B8: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ B8 @ A6 )
% 0.29/0.48 & ( ord_less_eq_set_a @ A6 @ B8 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.eq_iff
% 0.29/0.48 thf(fact_233_dual__order_Oeq__iff,axiom,
% 0.29/0.48 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 0.29/0.48 = ( ^ [A6: nat,B8: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ B8 @ A6 )
% 0.29/0.48 & ( ord_less_eq_nat @ A6 @ B8 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.eq_iff
% 0.29/0.48 thf(fact_234_dual__order_Otrans,axiom,
% 0.29/0.48 ! [B: set_a,A: set_a,C: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ B @ A )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ C @ B )
% 0.29/0.48 => ( ord_less_eq_set_a @ C @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.trans
% 0.29/0.48 thf(fact_235_dual__order_Otrans,axiom,
% 0.29/0.48 ! [B: nat,A: nat,C: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.48 => ( ( ord_less_eq_nat @ C @ B )
% 0.29/0.48 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.trans
% 0.29/0.48 thf(fact_236_linorder__wlog,axiom,
% 0.29/0.48 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.29/0.48 ( ! [A4: nat,B6: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A4 @ B6 )
% 0.29/0.48 => ( P @ A4 @ B6 ) )
% 0.29/0.48 => ( ! [A4: nat,B6: nat] :
% 0.29/0.48 ( ( P @ B6 @ A4 )
% 0.29/0.48 => ( P @ A4 @ B6 ) )
% 0.29/0.48 => ( P @ A @ B ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % linorder_wlog
% 0.29/0.48 thf(fact_237_dual__order_Orefl,axiom,
% 0.29/0.48 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.refl
% 0.29/0.48 thf(fact_238_dual__order_Orefl,axiom,
% 0.29/0.48 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.29/0.48
% 0.29/0.48 % dual_order.refl
% 0.29/0.48 thf(fact_239_order__trans,axiom,
% 0.29/0.48 ! [X: set_a,Y: set_a,Z: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X @ Y )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ Y @ Z )
% 0.29/0.48 => ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_trans
% 0.29/0.48 thf(fact_240_order__trans,axiom,
% 0.29/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X @ Y )
% 0.29/0.48 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.29/0.48 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_trans
% 0.29/0.48 thf(fact_241_order__class_Oorder_Oantisym,axiom,
% 0.29/0.48 ! [A: set_a,B: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ B @ A )
% 0.29/0.48 => ( A = B ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_class.order.antisym
% 0.29/0.48 thf(fact_242_order__class_Oorder_Oantisym,axiom,
% 0.29/0.48 ! [A: nat,B: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48 => ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.48 => ( A = B ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_class.order.antisym
% 0.29/0.48 thf(fact_243_ord__le__eq__trans,axiom,
% 0.29/0.48 ! [A: set_a,B: set_a,C: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.48 => ( ( B = C )
% 0.29/0.48 => ( ord_less_eq_set_a @ A @ C ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_le_eq_trans
% 0.29/0.48 thf(fact_244_ord__le__eq__trans,axiom,
% 0.29/0.48 ! [A: nat,B: nat,C: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48 => ( ( B = C )
% 0.29/0.48 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_le_eq_trans
% 0.29/0.48 thf(fact_245_ord__eq__le__trans,axiom,
% 0.29/0.48 ! [A: set_a,B: set_a,C: set_a] :
% 0.29/0.48 ( ( A = B )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ B @ C )
% 0.29/0.48 => ( ord_less_eq_set_a @ A @ C ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_eq_le_trans
% 0.29/0.48 thf(fact_246_ord__eq__le__trans,axiom,
% 0.29/0.48 ! [A: nat,B: nat,C: nat] :
% 0.29/0.48 ( ( A = B )
% 0.29/0.48 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.48 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_eq_le_trans
% 0.29/0.48 thf(fact_247_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.48 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.29/0.48 = ( ^ [A6: set_a,B8: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A6 @ B8 )
% 0.29/0.48 & ( ord_less_eq_set_a @ B8 @ A6 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_class.order.eq_iff
% 0.29/0.48 thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.48 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 0.29/0.48 = ( ^ [A6: nat,B8: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A6 @ B8 )
% 0.29/0.48 & ( ord_less_eq_nat @ B8 @ A6 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_class.order.eq_iff
% 0.29/0.48 thf(fact_249_antisym__conv,axiom,
% 0.29/0.48 ! [Y: set_a,X: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ Y @ X )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ X @ Y )
% 0.29/0.48 = ( X = Y ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % antisym_conv
% 0.29/0.48 thf(fact_250_antisym__conv,axiom,
% 0.29/0.48 ! [Y: nat,X: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ Y @ X )
% 0.29/0.48 => ( ( ord_less_eq_nat @ X @ Y )
% 0.29/0.48 = ( X = Y ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % antisym_conv
% 0.29/0.48 thf(fact_251_le__cases3,axiom,
% 0.29/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.29/0.48 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.29/0.48 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 0.29/0.48 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.29/0.48 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 0.29/0.48 => ( ( ( ord_less_eq_nat @ X @ Z )
% 0.29/0.48 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 0.29/0.48 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 0.29/0.48 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.29/0.48 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 0.29/0.48 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 0.29/0.48 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 0.29/0.48 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_cases3
% 0.29/0.48 thf(fact_252_order_Otrans,axiom,
% 0.29/0.48 ! [A: set_a,B: set_a,C: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ B @ C )
% 0.29/0.48 => ( ord_less_eq_set_a @ A @ C ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order.trans
% 0.29/0.48 thf(fact_253_order_Otrans,axiom,
% 0.29/0.48 ! [A: nat,B: nat,C: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.48 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order.trans
% 0.29/0.48 thf(fact_254_le__cases,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.29/0.48 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_cases
% 0.29/0.48 thf(fact_255_eq__refl,axiom,
% 0.29/0.48 ! [X: set_a,Y: set_a] :
% 0.29/0.48 ( ( X = Y )
% 0.29/0.48 => ( ord_less_eq_set_a @ X @ Y ) ) ).
% 0.29/0.48
% 0.29/0.48 % eq_refl
% 0.29/0.48 thf(fact_256_eq__refl,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( X = Y )
% 0.29/0.48 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.29/0.48
% 0.29/0.48 % eq_refl
% 0.29/0.48 thf(fact_257_linear,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X @ Y )
% 0.29/0.48 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.29/0.48
% 0.29/0.48 % linear
% 0.29/0.48 thf(fact_258_antisym,axiom,
% 0.29/0.48 ! [X: set_a,Y: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X @ Y )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ Y @ X )
% 0.29/0.48 => ( X = Y ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % antisym
% 0.29/0.48 thf(fact_259_antisym,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X @ Y )
% 0.29/0.48 => ( ( ord_less_eq_nat @ Y @ X )
% 0.29/0.48 => ( X = Y ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % antisym
% 0.29/0.48 thf(fact_260_eq__iff,axiom,
% 0.29/0.48 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 0.29/0.48 = ( ^ [X3: set_a,Y2: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 0.29/0.48 & ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % eq_iff
% 0.29/0.48 thf(fact_261_eq__iff,axiom,
% 0.29/0.48 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 0.29/0.48 = ( ^ [X3: nat,Y2: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.29/0.48 & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % eq_iff
% 0.29/0.48 thf(fact_262_ord__le__eq__subst,axiom,
% 0.29/0.48 ! [A: set_a,B: set_a,F2: set_a > set_a,C: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.48 => ( ( ( F2 @ B )
% 0.29/0.48 = C )
% 0.29/0.48 => ( ! [X4: set_a,Y3: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_le_eq_subst
% 0.29/0.48 thf(fact_263_ord__le__eq__subst,axiom,
% 0.29/0.48 ! [A: set_a,B: set_a,F2: set_a > nat,C: nat] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.48 => ( ( ( F2 @ B )
% 0.29/0.48 = C )
% 0.29/0.48 => ( ! [X4: set_a,Y3: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_le_eq_subst
% 0.29/0.48 thf(fact_264_ord__le__eq__subst,axiom,
% 0.29/0.48 ! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48 => ( ( ( F2 @ B )
% 0.29/0.48 = C )
% 0.29/0.48 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_le_eq_subst
% 0.29/0.48 thf(fact_265_ord__le__eq__subst,axiom,
% 0.29/0.48 ! [A: nat,B: nat,F2: nat > nat,C: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48 => ( ( ( F2 @ B )
% 0.29/0.48 = C )
% 0.29/0.48 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_le_eq_subst
% 0.29/0.48 thf(fact_266_ord__eq__le__subst,axiom,
% 0.29/0.48 ! [A: set_a,F2: set_a > set_a,B: set_a,C: set_a] :
% 0.29/0.48 ( ( A
% 0.29/0.48 = ( F2 @ B ) )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ B @ C )
% 0.29/0.48 => ( ! [X4: set_a,Y3: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_eq_le_subst
% 0.29/0.48 thf(fact_267_ord__eq__le__subst,axiom,
% 0.29/0.48 ! [A: nat,F2: set_a > nat,B: set_a,C: set_a] :
% 0.29/0.48 ( ( A
% 0.29/0.48 = ( F2 @ B ) )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ B @ C )
% 0.29/0.48 => ( ! [X4: set_a,Y3: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_eq_le_subst
% 0.29/0.48 thf(fact_268_ord__eq__le__subst,axiom,
% 0.29/0.48 ! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
% 0.29/0.48 ( ( A
% 0.29/0.48 = ( F2 @ B ) )
% 0.29/0.48 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.48 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_eq_le_subst
% 0.29/0.48 thf(fact_269_ord__eq__le__subst,axiom,
% 0.29/0.48 ! [A: nat,F2: nat > nat,B: nat,C: nat] :
% 0.29/0.48 ( ( A
% 0.29/0.48 = ( F2 @ B ) )
% 0.29/0.48 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.48 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ord_eq_le_subst
% 0.29/0.48 thf(fact_270_order__subst2,axiom,
% 0.29/0.48 ! [A: set_a,B: set_a,F2: set_a > set_a,C: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C )
% 0.29/0.48 => ( ! [X4: set_a,Y3: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst2
% 0.29/0.48 thf(fact_271_order__subst2,axiom,
% 0.29/0.48 ! [A: set_a,B: set_a,F2: set_a > nat,C: nat] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.48 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 0.29/0.48 => ( ! [X4: set_a,Y3: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst2
% 0.29/0.48 thf(fact_272_order__subst2,axiom,
% 0.29/0.48 ! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C )
% 0.29/0.48 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst2
% 0.29/0.48 thf(fact_273_order__subst2,axiom,
% 0.29/0.48 ! [A: nat,B: nat,F2: nat > nat,C: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 0.29/0.48 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst2
% 0.29/0.48 thf(fact_274_order__subst1,axiom,
% 0.29/0.48 ! [A: set_a,F2: set_a > set_a,B: set_a,C: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ B @ C )
% 0.29/0.48 => ( ! [X4: set_a,Y3: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst1
% 0.29/0.48 thf(fact_275_order__subst1,axiom,
% 0.29/0.48 ! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
% 0.29/0.48 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.48 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst1
% 0.29/0.48 thf(fact_276_order__subst1,axiom,
% 0.29/0.48 ! [A: nat,F2: set_a > nat,B: set_a,C: set_a] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 0.29/0.48 => ( ( ord_less_eq_set_a @ B @ C )
% 0.29/0.48 => ( ! [X4: set_a,Y3: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst1
% 0.29/0.48 thf(fact_277_order__subst1,axiom,
% 0.29/0.48 ! [A: nat,F2: nat > nat,B: nat,C: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 0.29/0.48 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.48 => ( ! [X4: nat,Y3: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 0.29/0.48 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.29/0.48 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % order_subst1
% 0.29/0.48 thf(fact_278_surj__pair,axiom,
% 0.29/0.48 ! [P2: product_prod_a_a] :
% 0.29/0.48 ? [X4: a,Y3: a] :
% 0.29/0.48 ( P2
% 0.29/0.48 = ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % surj_pair
% 0.29/0.48 thf(fact_279_prod__cases,axiom,
% 0.29/0.48 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 0.29/0.48 ( ! [A4: a,B6: a] : ( P @ ( product_Pair_a_a @ A4 @ B6 ) )
% 0.29/0.48 => ( P @ P2 ) ) ).
% 0.29/0.48
% 0.29/0.48 % prod_cases
% 0.29/0.48 thf(fact_280_Pair__inject,axiom,
% 0.29/0.48 ! [A: a,B: a,A5: a,B7: a] :
% 0.29/0.48 ( ( ( product_Pair_a_a @ A @ B )
% 0.29/0.48 = ( product_Pair_a_a @ A5 @ B7 ) )
% 0.29/0.48 => ~ ( ( A = A5 )
% 0.29/0.48 => ( B != B7 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Pair_inject
% 0.29/0.48 thf(fact_281_old_Oprod_Oexhaust,axiom,
% 0.29/0.48 ! [Y: product_prod_a_a] :
% 0.29/0.48 ~ ! [A4: a,B6: a] :
% 0.29/0.48 ( Y
% 0.29/0.48 != ( product_Pair_a_a @ A4 @ B6 ) ) ).
% 0.29/0.48
% 0.29/0.48 % old.prod.exhaust
% 0.29/0.48 thf(fact_282_old_Oprod_Oinducts,axiom,
% 0.29/0.48 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.29/0.48 ( ! [A4: a,B6: a] : ( P @ ( product_Pair_a_a @ A4 @ B6 ) )
% 0.29/0.48 => ( P @ Prod ) ) ).
% 0.29/0.48
% 0.29/0.48 % old.prod.inducts
% 0.29/0.48 thf(fact_283_bounded__Max__nat,axiom,
% 0.29/0.48 ! [P: nat > $o,X: nat,M: nat] :
% 0.29/0.48 ( ( P @ X )
% 0.29/0.48 => ( ! [X4: nat] :
% 0.29/0.48 ( ( P @ X4 )
% 0.29/0.48 => ( ord_less_eq_nat @ X4 @ M ) )
% 0.29/0.48 => ~ ! [M2: nat] :
% 0.29/0.48 ( ( P @ M2 )
% 0.29/0.48 => ~ ! [X2: nat] :
% 0.29/0.48 ( ( P @ X2 )
% 0.29/0.48 => ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % bounded_Max_nat
% 0.29/0.48 thf(fact_284_finite__nat__set__iff__bounded__le,axiom,
% 0.29/0.48 ( finite_finite_nat
% 0.29/0.48 = ( ^ [N2: set_nat] :
% 0.29/0.48 ? [M3: nat] :
% 0.29/0.48 ! [X3: nat] :
% 0.29/0.48 ( ( member_nat @ X3 @ N2 )
% 0.29/0.48 => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_nat_set_iff_bounded_le
% 0.29/0.48 thf(fact_285_bot_Oextremum__uniqueI,axiom,
% 0.29/0.48 ! [A: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 0.29/0.48 => ( A = bot_bot_set_a ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot.extremum_uniqueI
% 0.29/0.48 thf(fact_286_bot_Oextremum__uniqueI,axiom,
% 0.29/0.48 ! [A: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.29/0.48 => ( A = bot_bot_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot.extremum_uniqueI
% 0.29/0.48 thf(fact_287_bot_Oextremum__unique,axiom,
% 0.29/0.48 ! [A: set_a] :
% 0.29/0.48 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 0.29/0.48 = ( A = bot_bot_set_a ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot.extremum_unique
% 0.29/0.48 thf(fact_288_bot_Oextremum__unique,axiom,
% 0.29/0.48 ! [A: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.29/0.48 = ( A = bot_bot_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot.extremum_unique
% 0.29/0.48 thf(fact_289_bot_Oextremum,axiom,
% 0.29/0.48 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 0.29/0.48
% 0.29/0.48 % bot.extremum
% 0.29/0.48 thf(fact_290_bot_Oextremum,axiom,
% 0.29/0.48 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 0.29/0.48
% 0.29/0.48 % bot.extremum
% 0.29/0.48 thf(fact_291_preorder__on__empty,axiom,
% 0.29/0.48 order_preorder_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 0.29/0.48
% 0.29/0.48 % preorder_on_empty
% 0.29/0.48 thf(fact_292_total__onI,axiom,
% 0.29/0.48 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 0.29/0.48 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ X4 @ A2 )
% 0.29/0.48 => ( ( member449909584od_a_a @ Y3 @ A2 )
% 0.29/0.48 => ( ( X4 != Y3 )
% 0.29/0.48 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R )
% 0.29/0.48 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.29/0.48 => ( total_1490170027od_a_a @ A2 @ R ) ) ).
% 0.29/0.48
% 0.29/0.48 % total_onI
% 0.29/0.48 thf(fact_293_total__onI,axiom,
% 0.29/0.48 ! [A2: set_nat,R: set_Pr1986765409at_nat] :
% 0.29/0.48 ( ! [X4: nat,Y3: nat] :
% 0.29/0.48 ( ( member_nat @ X4 @ A2 )
% 0.29/0.48 => ( ( member_nat @ Y3 @ A2 )
% 0.29/0.48 => ( ( X4 != Y3 )
% 0.29/0.48 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
% 0.29/0.48 | ( member701585322at_nat @ ( product_Pair_nat_nat @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.29/0.48 => ( total_on_nat @ A2 @ R ) ) ).
% 0.29/0.48
% 0.29/0.48 % total_onI
% 0.29/0.48 thf(fact_294_total__onI,axiom,
% 0.29/0.48 ! [A2: set_a,R: set_Product_prod_a_a] :
% 0.29/0.48 ( ! [X4: a,Y3: a] :
% 0.29/0.48 ( ( member_a @ X4 @ A2 )
% 0.29/0.48 => ( ( member_a @ Y3 @ A2 )
% 0.29/0.48 => ( ( X4 != Y3 )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 0.29/0.48 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 0.29/0.48 => ( total_on_a @ A2 @ R ) ) ).
% 0.29/0.48
% 0.29/0.48 % total_onI
% 0.29/0.48 thf(fact_295_total__on__def,axiom,
% 0.29/0.48 ( total_on_a
% 0.29/0.48 = ( ^ [A3: set_a,R2: set_Product_prod_a_a] :
% 0.29/0.48 ! [X3: a] :
% 0.29/0.48 ( ( member_a @ X3 @ A3 )
% 0.29/0.48 => ! [Y2: a] :
% 0.29/0.48 ( ( member_a @ Y2 @ A3 )
% 0.29/0.48 => ( ( X3 != Y2 )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
% 0.29/0.48 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % total_on_def
% 0.29/0.48 thf(fact_296_bot__empty__eq,axiom,
% 0.29/0.48 ( bot_bo1293121322_a_a_o
% 0.29/0.48 = ( ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ bot_bo2131659635od_a_a ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot_empty_eq
% 0.29/0.48 thf(fact_297_bot__empty__eq,axiom,
% 0.29/0.48 ( bot_bot_nat_o
% 0.29/0.48 = ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot_empty_eq
% 0.29/0.48 thf(fact_298_bot__empty__eq,axiom,
% 0.29/0.48 ( bot_bot_a_o
% 0.29/0.48 = ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot_empty_eq
% 0.29/0.48 thf(fact_299_subrelI,axiom,
% 0.29/0.48 ! [R: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 0.29/0.48 ( ! [X4: a,Y3: a] :
% 0.29/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 0.29/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S3 ) )
% 0.29/0.48 => ( ord_le1824328871od_a_a @ R @ S3 ) ) ).
% 0.29/0.48
% 0.29/0.48 % subrelI
% 0.29/0.48 thf(fact_300_refl__on__empty,axiom,
% 0.29/0.48 refl_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 0.29/0.48
% 0.29/0.48 % refl_on_empty
% 0.29/0.48 thf(fact_301_refl__on__domain,axiom,
% 0.29/0.48 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
% 0.29/0.48 ( ( refl_o1298442278od_a_a @ A2 @ R )
% 0.29/0.48 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ A @ A2 )
% 0.29/0.48 & ( member449909584od_a_a @ B @ A2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_on_domain
% 0.29/0.48 thf(fact_302_refl__on__domain,axiom,
% 0.29/0.48 ! [A2: set_nat,R: set_Pr1986765409at_nat,A: nat,B: nat] :
% 0.29/0.48 ( ( refl_on_nat @ A2 @ R )
% 0.29/0.48 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
% 0.29/0.48 => ( ( member_nat @ A @ A2 )
% 0.29/0.48 & ( member_nat @ B @ A2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_on_domain
% 0.29/0.48 thf(fact_303_refl__on__domain,axiom,
% 0.29/0.48 ! [A2: set_a,R: set_Product_prod_a_a,A: a,B: a] :
% 0.29/0.48 ( ( refl_on_a @ A2 @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
% 0.29/0.48 => ( ( member_a @ A @ A2 )
% 0.29/0.48 & ( member_a @ B @ A2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_on_domain
% 0.29/0.48 thf(fact_304_refl__onD2,axiom,
% 0.29/0.48 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.48 ( ( refl_o1298442278od_a_a @ A2 @ R )
% 0.29/0.48 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R )
% 0.29/0.48 => ( member449909584od_a_a @ Y @ A2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_onD2
% 0.29/0.48 thf(fact_305_refl__onD2,axiom,
% 0.29/0.48 ! [A2: set_nat,R: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.48 ( ( refl_on_nat @ A2 @ R )
% 0.29/0.48 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 0.29/0.48 => ( member_nat @ Y @ A2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_onD2
% 0.29/0.48 thf(fact_306_refl__onD2,axiom,
% 0.29/0.48 ! [A2: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.48 ( ( refl_on_a @ A2 @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 0.29/0.48 => ( member_a @ Y @ A2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_onD2
% 0.29/0.48 thf(fact_307_refl__onD1,axiom,
% 0.29/0.48 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 0.29/0.48 ( ( refl_o1298442278od_a_a @ A2 @ R )
% 0.29/0.48 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R )
% 0.29/0.48 => ( member449909584od_a_a @ X @ A2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_onD1
% 0.29/0.48 thf(fact_308_refl__onD1,axiom,
% 0.29/0.48 ! [A2: set_nat,R: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 0.29/0.48 ( ( refl_on_nat @ A2 @ R )
% 0.29/0.48 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 0.29/0.48 => ( member_nat @ X @ A2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_onD1
% 0.29/0.48 thf(fact_309_refl__onD1,axiom,
% 0.29/0.48 ! [A2: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
% 0.29/0.48 ( ( refl_on_a @ A2 @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 0.29/0.48 => ( member_a @ X @ A2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_onD1
% 0.29/0.48 thf(fact_310_refl__onD,axiom,
% 0.29/0.48 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a] :
% 0.29/0.48 ( ( refl_o1298442278od_a_a @ A2 @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ A @ A2 )
% 0.29/0.48 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ R ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_onD
% 0.29/0.48 thf(fact_311_refl__onD,axiom,
% 0.29/0.48 ! [A2: set_nat,R: set_Pr1986765409at_nat,A: nat] :
% 0.29/0.48 ( ( refl_on_nat @ A2 @ R )
% 0.29/0.48 => ( ( member_nat @ A @ A2 )
% 0.29/0.48 => ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_onD
% 0.29/0.48 thf(fact_312_refl__onD,axiom,
% 0.29/0.48 ! [A2: set_a,R: set_Product_prod_a_a,A: a] :
% 0.29/0.48 ( ( refl_on_a @ A2 @ R )
% 0.29/0.48 => ( ( member_a @ A @ A2 )
% 0.29/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % refl_onD
% 0.29/0.48 thf(fact_313_total__on__empty,axiom,
% 0.29/0.48 ! [R: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R ) ).
% 0.29/0.48
% 0.29/0.48 % total_on_empty
% 0.29/0.48 thf(fact_314_Collect__empty__eq__bot,axiom,
% 0.29/0.48 ! [P: a > $o] :
% 0.29/0.48 ( ( ( collect_a @ P )
% 0.29/0.48 = bot_bot_set_a )
% 0.29/0.48 = ( P = bot_bot_a_o ) ) ).
% 0.29/0.48
% 0.29/0.48 % Collect_empty_eq_bot
% 0.29/0.48 thf(fact_315_infinite__nat__iff__unbounded__le,axiom,
% 0.29/0.48 ! [S: set_nat] :
% 0.29/0.48 ( ( ~ ( finite_finite_nat @ S ) )
% 0.29/0.48 = ( ! [M3: nat] :
% 0.29/0.48 ? [N3: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.29/0.48 & ( member_nat @ N3 @ S ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % infinite_nat_iff_unbounded_le
% 0.29/0.48 thf(fact_316_finite__transitivity__chain,axiom,
% 0.29/0.48 ! [A2: set_Product_prod_a_a,R3: product_prod_a_a > product_prod_a_a > $o] :
% 0.29/0.48 ( ( finite179568208od_a_a @ A2 )
% 0.29/0.48 => ( ! [X4: product_prod_a_a] :
% 0.29/0.48 ~ ( R3 @ X4 @ X4 )
% 0.29/0.48 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
% 0.29/0.48 ( ( R3 @ X4 @ Y3 )
% 0.29/0.48 => ( ( R3 @ Y3 @ Z3 )
% 0.29/0.48 => ( R3 @ X4 @ Z3 ) ) )
% 0.29/0.48 => ( ! [X4: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ X4 @ A2 )
% 0.29/0.48 => ? [Y6: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ Y6 @ A2 )
% 0.29/0.48 & ( R3 @ X4 @ Y6 ) ) )
% 0.29/0.48 => ( A2 = bot_bo2131659635od_a_a ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_transitivity_chain
% 0.29/0.48 thf(fact_317_finite__transitivity__chain,axiom,
% 0.29/0.48 ! [A2: set_nat,R3: nat > nat > $o] :
% 0.29/0.48 ( ( finite_finite_nat @ A2 )
% 0.29/0.48 => ( ! [X4: nat] :
% 0.29/0.48 ~ ( R3 @ X4 @ X4 )
% 0.29/0.48 => ( ! [X4: nat,Y3: nat,Z3: nat] :
% 0.29/0.48 ( ( R3 @ X4 @ Y3 )
% 0.29/0.48 => ( ( R3 @ Y3 @ Z3 )
% 0.29/0.48 => ( R3 @ X4 @ Z3 ) ) )
% 0.29/0.48 => ( ! [X4: nat] :
% 0.29/0.48 ( ( member_nat @ X4 @ A2 )
% 0.29/0.48 => ? [Y6: nat] :
% 0.29/0.48 ( ( member_nat @ Y6 @ A2 )
% 0.29/0.48 & ( R3 @ X4 @ Y6 ) ) )
% 0.29/0.48 => ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_transitivity_chain
% 0.29/0.48 thf(fact_318_finite__transitivity__chain,axiom,
% 0.29/0.48 ! [A2: set_a,R3: a > a > $o] :
% 0.29/0.48 ( ( finite_finite_a @ A2 )
% 0.29/0.48 => ( ! [X4: a] :
% 0.29/0.48 ~ ( R3 @ X4 @ X4 )
% 0.29/0.48 => ( ! [X4: a,Y3: a,Z3: a] :
% 0.29/0.48 ( ( R3 @ X4 @ Y3 )
% 0.29/0.48 => ( ( R3 @ Y3 @ Z3 )
% 0.29/0.48 => ( R3 @ X4 @ Z3 ) ) )
% 0.29/0.48 => ( ! [X4: a] :
% 0.29/0.48 ( ( member_a @ X4 @ A2 )
% 0.29/0.48 => ? [Y6: a] :
% 0.29/0.48 ( ( member_a @ Y6 @ A2 )
% 0.29/0.48 & ( R3 @ X4 @ Y6 ) ) )
% 0.29/0.48 => ( A2 = bot_bot_set_a ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_transitivity_chain
% 0.29/0.48 thf(fact_319_subset__emptyI,axiom,
% 0.29/0.48 ! [A2: set_Product_prod_a_a] :
% 0.29/0.48 ( ! [X4: product_prod_a_a] :
% 0.29/0.48 ~ ( member449909584od_a_a @ X4 @ A2 )
% 0.29/0.48 => ( ord_le1824328871od_a_a @ A2 @ bot_bo2131659635od_a_a ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_emptyI
% 0.29/0.48 thf(fact_320_subset__emptyI,axiom,
% 0.29/0.48 ! [A2: set_nat] :
% 0.29/0.48 ( ! [X4: nat] :
% 0.29/0.48 ~ ( member_nat @ X4 @ A2 )
% 0.29/0.48 => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_emptyI
% 0.29/0.48 thf(fact_321_subset__emptyI,axiom,
% 0.29/0.48 ! [A2: set_a] :
% 0.29/0.48 ( ! [X4: a] :
% 0.29/0.48 ~ ( member_a @ X4 @ A2 )
% 0.29/0.48 => ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% 0.29/0.48
% 0.29/0.48 % subset_emptyI
% 0.29/0.48 thf(fact_322_rational__preference__def,axiom,
% 0.29/0.48 ( prefer719835986ence_a
% 0.29/0.48 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 0.29/0.48 ( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
% 0.29/0.48 & ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % rational_preference_def
% 0.29/0.48 thf(fact_323_ssubst__Pair__rhs,axiom,
% 0.29/0.48 ! [R: a,S3: a,R3: set_Product_prod_a_a,S4: a] :
% 0.29/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S3 ) @ R3 )
% 0.29/0.48 => ( ( S4 = S3 )
% 0.29/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S4 ) @ R3 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ssubst_Pair_rhs
% 0.29/0.48 thf(fact_324_rational__preference__axioms_Ointro,axiom,
% 0.29/0.48 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.48 ( ( total_on_a @ Carrier @ Relation )
% 0.29/0.48 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 0.29/0.48
% 0.29/0.48 % rational_preference_axioms.intro
% 0.29/0.48 thf(fact_325_rational__preference__axioms__def,axiom,
% 0.29/0.48 prefer580340847ioms_a = total_on_a ).
% 0.29/0.48
% 0.29/0.48 % rational_preference_axioms_def
% 0.29/0.48 thf(fact_326_rational__preference_Oaxioms_I2_J,axiom,
% 0.29/0.48 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.48 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 0.29/0.48 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 0.29/0.48
% 0.29/0.48 % rational_preference.axioms(2)
% 0.29/0.48 thf(fact_327_rational__preference_Ointro,axiom,
% 0.29/0.48 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.48 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.29/0.48 => ( ( prefer580340847ioms_a @ Carrier @ Relation )
% 0.29/0.48 => ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % rational_preference.intro
% 0.29/0.48 thf(fact_328_finite__indexed__bound,axiom,
% 0.29/0.48 ! [A2: set_a,P: a > nat > $o] :
% 0.29/0.48 ( ( finite_finite_a @ A2 )
% 0.29/0.48 => ( ! [X4: a] :
% 0.29/0.48 ( ( member_a @ X4 @ A2 )
% 0.29/0.48 => ? [X_1: nat] : ( P @ X4 @ X_1 ) )
% 0.29/0.48 => ? [M2: nat] :
% 0.29/0.48 ! [X2: a] :
% 0.29/0.48 ( ( member_a @ X2 @ A2 )
% 0.29/0.48 => ? [K: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ K @ M2 )
% 0.29/0.48 & ( P @ X2 @ K ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_indexed_bound
% 0.29/0.48 thf(fact_329_finite__indexed__bound,axiom,
% 0.29/0.48 ! [A2: set_Product_prod_a_a,P: product_prod_a_a > nat > $o] :
% 0.29/0.48 ( ( finite179568208od_a_a @ A2 )
% 0.29/0.48 => ( ! [X4: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ X4 @ A2 )
% 0.29/0.48 => ? [X_1: nat] : ( P @ X4 @ X_1 ) )
% 0.29/0.48 => ? [M2: nat] :
% 0.29/0.48 ! [X2: product_prod_a_a] :
% 0.29/0.48 ( ( member449909584od_a_a @ X2 @ A2 )
% 0.29/0.48 => ? [K: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ K @ M2 )
% 0.29/0.48 & ( P @ X2 @ K ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_indexed_bound
% 0.29/0.48 thf(fact_330_finite__indexed__bound,axiom,
% 0.29/0.48 ! [A2: set_nat,P: nat > nat > $o] :
% 0.29/0.48 ( ( finite_finite_nat @ A2 )
% 0.29/0.48 => ( ! [X4: nat] :
% 0.29/0.48 ( ( member_nat @ X4 @ A2 )
% 0.29/0.48 => ? [X_1: nat] : ( P @ X4 @ X_1 ) )
% 0.29/0.48 => ? [M2: nat] :
% 0.29/0.48 ! [X2: nat] :
% 0.29/0.48 ( ( member_nat @ X2 @ A2 )
% 0.29/0.48 => ? [K: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ K @ M2 )
% 0.29/0.48 & ( P @ X2 @ K ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % finite_indexed_bound
% 0.29/0.48 thf(fact_331_transitivity,axiom,
% 0.29/0.48 trans_a @ relation ).
% 0.29/0.48
% 0.29/0.48 % transitivity
% 0.29/0.48 thf(fact_332_trans__empty,axiom,
% 0.29/0.48 trans_a @ bot_bo2131659635od_a_a ).
% 0.29/0.48
% 0.29/0.48 % trans_empty
% 0.29/0.48 thf(fact_333_transD,axiom,
% 0.29/0.48 ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.29/0.48 ( ( trans_a @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 0.29/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % transD
% 0.29/0.48 thf(fact_334_transE,axiom,
% 0.29/0.48 ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 0.29/0.48 ( ( trans_a @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 0.29/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % transE
% 0.29/0.48 thf(fact_335_transI,axiom,
% 0.29/0.48 ! [R: set_Product_prod_a_a] :
% 0.29/0.48 ( ! [X4: a,Y3: a,Z3: a] :
% 0.29/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
% 0.29/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R ) ) )
% 0.29/0.48 => ( trans_a @ R ) ) ).
% 0.29/0.48
% 0.29/0.48 % transI
% 0.29/0.48 thf(fact_336_trans__def,axiom,
% 0.29/0.48 ( trans_a
% 0.29/0.48 = ( ^ [R2: set_Product_prod_a_a] :
% 0.29/0.48 ! [X3: a,Y2: a,Z4: a] :
% 0.29/0.48 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
% 0.29/0.48 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R2 )
% 0.29/0.48 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % trans_def
% 0.29/0.48 thf(fact_337_preference_Otransitivity,axiom,
% 0.29/0.48 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 0.29/0.48 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 0.29/0.48 => ( trans_a @ Relation ) ) ).
% 0.29/0.48
% 0.29/0.48 % preference.transitivity
% 0.29/0.48 thf(fact_338_preorder__on__def,axiom,
% 0.29/0.48 ( order_preorder_on_a
% 0.29/0.48 = ( ^ [A3: set_a,R2: set_Product_prod_a_a] :
% 0.29/0.48 ( ( refl_on_a @ A3 @ R2 )
% 0.29/0.48 & ( trans_a @ R2 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % preorder_on_def
% 0.29/0.48 thf(fact_339_Set_Ois__empty__def,axiom,
% 0.29/0.48 ( is_empty_a
% 0.29/0.48 = ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Set.is_empty_def
% 0.29/0.48 thf(fact_340_card__0__eq,axiom,
% 0.29/0.48 ! [A2: set_a] :
% 0.29/0.48 ( ( finite_finite_a @ A2 )
% 0.29/0.48 => ( ( ( finite_card_a @ A2 )
% 0.29/0.48 = zero_zero_nat )
% 0.29/0.48 = ( A2 = bot_bot_set_a ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % card_0_eq
% 0.29/0.48 thf(fact_341_bot__nat__0_Oextremum,axiom,
% 0.29/0.48 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.29/0.48
% 0.29/0.48 % bot_nat_0.extremum
% 0.29/0.48 thf(fact_342_le0,axiom,
% 0.29/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.29/0.48
% 0.29/0.48 % le0
% 0.29/0.48 thf(fact_343_Nat_Oex__has__greatest__nat,axiom,
% 0.29/0.48 ! [P: nat > $o,K2: nat,B: nat] :
% 0.29/0.48 ( ( P @ K2 )
% 0.29/0.48 => ( ! [Y3: nat] :
% 0.29/0.48 ( ( P @ Y3 )
% 0.29/0.48 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.29/0.48 => ? [X4: nat] :
% 0.29/0.48 ( ( P @ X4 )
% 0.29/0.48 & ! [Y6: nat] :
% 0.29/0.48 ( ( P @ Y6 )
% 0.29/0.48 => ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % Nat.ex_has_greatest_nat
% 0.29/0.48 thf(fact_344_nat__le__linear,axiom,
% 0.29/0.48 ! [M4: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ M4 @ N )
% 0.29/0.48 | ( ord_less_eq_nat @ N @ M4 ) ) ).
% 0.29/0.48
% 0.29/0.48 % nat_le_linear
% 0.29/0.48 thf(fact_345_le__antisym,axiom,
% 0.29/0.48 ! [M4: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ M4 @ N )
% 0.29/0.48 => ( ( ord_less_eq_nat @ N @ M4 )
% 0.29/0.48 => ( M4 = N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_antisym
% 0.29/0.48 thf(fact_346_eq__imp__le,axiom,
% 0.29/0.48 ! [M4: nat,N: nat] :
% 0.29/0.48 ( ( M4 = N )
% 0.29/0.48 => ( ord_less_eq_nat @ M4 @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 % eq_imp_le
% 0.29/0.48 thf(fact_347_le__trans,axiom,
% 0.29/0.48 ! [I: nat,J: nat,K2: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.48 => ( ( ord_less_eq_nat @ J @ K2 )
% 0.29/0.48 => ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_trans
% 0.29/0.48 thf(fact_348_le__refl,axiom,
% 0.29/0.48 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.29/0.48
% 0.29/0.48 % le_refl
% 0.29/0.48 thf(fact_349_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.48 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.29/0.48
% 0.29/0.48 % less_eq_nat.simps(1)
% 0.29/0.48 thf(fact_350_le__0__eq,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.29/0.48 = ( N = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % le_0_eq
% 0.29/0.48 thf(fact_351_bot__nat__0_Oextremum__unique,axiom,
% 0.29/0.48 ! [A: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.48 = ( A = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot_nat_0.extremum_unique
% 0.29/0.48 thf(fact_352_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.29/0.48 ! [A: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.48 => ( A = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 % bot_nat_0.extremum_uniqueI
% 0.29/0.48 thf(fact_353_ex__least__nat__le,axiom,
% 0.29/0.48 ! [P: nat > $o,N: nat] :
% 0.29/0.48 ( ( P @ N )
% 0.29/0.48 => ( ~ ( P @ zero_zero_nat )
% 0.29/0.48 => ? [K: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.48 & ! [I2: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48 => ~ ( P @ I2 ) )
% 0.29/0.48 & ( P @ K ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 % ex_least_nat_le
% 0.29/0.48
% 0.29/0.48 % Conjectures (1)
% 0.29/0.48 thf(conj_0,conjecture,
% 0.29/0.48 ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) ).
% 0.29/0.48
% 0.29/0.48 %------------------------------------------------------------------------------
% 0.29/0.48 ------------------------
% 2.99/1.58 ---- Embedded in THF ---
% 2.99/1.58 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.99/1.58 %%% Generated on Tue Feb 24 08:28:01 EST 2026
% 2.99/1.58 %%% using '$$dhol' embedding, version 1.3.0.
% 2.99/1.58 %%% Logic specification used:
% 2.99/1.58 %%% thf(spec, logic, $$dhol).
% 2.99/1.58
% 2.99/1.58 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2DTF13001.p
% See solution above
% 2.99/1.58 ------------------------
% 3.21/1.61 ---- Cleaned THF ---
% 3.21/1.61 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,
% 3.21/1.61 set_Pr1948701895od_a_a: $tType ).
% 3.21/1.61
% 3.21/1.61 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,
% 3.21/1.61 produc1572603623od_a_a: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 3.21/1.61 produc1691597095_set_a: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 3.21/1.61 set_Pr1986765409at_nat: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
% 3.21/1.61 produc1404140509_a_nat: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
% 3.21/1.61 produc1463940355_set_a: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.21/1.61 set_Product_prod_a_a: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.21/1.61 product_prod_nat_nat: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 product_prod_a_a: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 3.21/1.61 set_set_a: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 3.21/1.61 set_nat: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 3.21/1.61 set_a: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_t__Nat__Onat,type,
% 3.21/1.61 nat: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(ty_n_tf__a,type,
% 3.21/1.61 a: $tType ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 3.21/1.61 finite_card_nat: set_nat > nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 finite1481642319od_a_a: set_Product_prod_a_a > nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Finite__Set_Ocard_001tf__a,type,
% 3.21/1.61 finite_card_a: set_a > nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 3.21/1.61 finite_finite_nat: set_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.21/1.61 finite772653738at_nat: set_Pr1986765409at_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.21/1.61 finite1664988688od_a_a: set_Pr1948701895od_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 finite179568208od_a_a: set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
% 3.21/1.61 finite_finite_set_a: set_set_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
% 3.21/1.61 finite_finite_a: set_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 3.21/1.61 zero_zero_nat: nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Order__Relation_Opreorder__on_001t__Nat__Onat,type,
% 3.21/1.61 order_1150839801on_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Order__Relation_Opreorder__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Order__Relation_Opreorder__on_001tf__a,type,
% 3.21/1.61 order_preorder_on_a: set_a > set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 3.21/1.61 bot_bot_nat_o: nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
% 3.21/1.61 bot_bo1293121322_a_a_o: product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
% 3.21/1.61 bot_bot_a_o: a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 3.21/1.61 bot_bot_nat: nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 3.21/1.61 bot_bo1519632275_set_a: produc1691597095_set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.21/1.61 bot_bot_set_nat: set_nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.21/1.61 bot_bo2131659635od_a_a: set_Product_prod_a_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 3.21/1.61 bot_bot_set_set_a: set_set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 3.21/1.61 bot_bot_set_a: set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 3.21/1.61 ord_less_nat: nat > nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 3.21/1.61 ord_less_eq_nat: nat > nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.21/1.61 ord_le1302190241at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_Itf__a_J_J,type,
% 3.21/1.61 ord_le841160035_set_a: produc1463940355_set_a > produc1463940355_set_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Nat__Onat_J,type,
% 3.21/1.61 ord_le781360189_a_nat: produc1404140509_a_nat > produc1404140509_a_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 3.21/1.61 ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 3.21/1.61 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.21/1.61 ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 3.21/1.61 ord_less_eq_set_a: set_a > set_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oas__good__as_001t__Nat__Onat,type,
% 3.21/1.61 prefer1936906324as_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oas__good__as_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oas__good__as_001tf__a,type,
% 3.21/1.61 prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oat__least__as__good_001t__Nat__Onat,type,
% 3.21/1.61 prefer563798164od_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oat__least__as__good_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Oat__least__as__good_001tf__a,type,
% 3.21/1.61 prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001t__Nat__Onat,type,
% 3.21/1.61 prefer1672638789an_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Ono__better__than_001tf__a,type,
% 3.21/1.61 prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Opreference_001t__Nat__Onat,type,
% 3.21/1.61 prefer1293349710ce_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Opreference_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Opreference_001tf__a,type,
% 3.21/1.61 prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001t__Nat__Onat,type,
% 3.21/1.61 prefer1147844220ce_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference_001tf__a,type,
% 3.21/1.61 prefer719835986ence_a: set_a > set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Preferences__Mirabelle__wwlsriwuiu_Orational__preference__axioms_001tf__a,type,
% 3.21/1.61 prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 3.21/1.61 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
% 3.21/1.61 produc1450889781_set_a: nat > set_a > produc1463940355_set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
% 3.21/1.61 produc515611863_a_nat: set_a > nat > produc1404140509_a_nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
% 3.21/1.61 produc1928581911_set_a: set_a > set_a > produc1691597095_set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 3.21/1.61 product_Pair_a_a: a > a > product_prod_a_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Relation_Orefl__on_001t__Nat__Onat,type,
% 3.21/1.61 refl_on_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Relation_Orefl__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Relation_Orefl__on_001tf__a,type,
% 3.21/1.61 refl_on_a: set_a > set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Relation_Ototal__on_001t__Nat__Onat,type,
% 3.21/1.61 total_on_nat: set_nat > set_Pr1986765409at_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Relation_Ototal__on_001tf__a,type,
% 3.21/1.61 total_on_a: set_a > set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Relation_Otrans_001tf__a,type,
% 3.21/1.61 trans_a: set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 3.21/1.61 collect_nat: ( nat > $o ) > set_nat ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Set_OCollect_001tf__a,type,
% 3.21/1.61 collect_a: ( a > $o ) > set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_Set_Ois__empty_001tf__a,type,
% 3.21/1.61 is_empty_a: set_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_member_001t__Nat__Onat,type,
% 3.21/1.61 member_nat: nat > set_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 3.21/1.61 member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.21/1.61 member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.21/1.61 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 3.21/1.61 member_set_a: set_a > set_set_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_c_member_001tf__a,type,
% 3.21/1.61 member_a: a > set_a > $o ).
% 3.21/1.61
% 3.21/1.61 thf(sy_v_carrier,type,
% 3.21/1.61 carrier: set_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_v_relation,type,
% 3.21/1.61 relation: set_Product_prod_a_a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_v_x,type,
% 3.21/1.61 x: a ).
% 3.21/1.61
% 3.21/1.61 thf(sy_v_y,type,
% 3.21/1.61 y: a ).
% 3.21/1.61
% 3.21/1.61 thf(fact_0_assms_I2_J,axiom,
% 3.21/1.61 member_a @ y @ carrier ).
% 3.21/1.61
% 3.21/1.61 thf(fact_1_assms_I1_J,axiom,
% 3.21/1.61 member_a @ x @ carrier ).
% 3.21/1.61
% 3.21/1.61 thf(fact_2_xy__in__eachothers__nbt,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( member_a @ X @ carrier )
% 3.21/1.61 => ( ( member_a @ Y @ carrier )
% 3.21/1.61 => ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ carrier @ relation ) )
% 3.21/1.61 | ( member_a @ Y @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062,axiom,
% 3.21/1.61 ord_less_eq_nat @ ( finite_card_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) ) @ ( finite_card_a @ ( prefer1676310729than_a @ y @ carrier @ relation ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_4_nbt__nest,axiom,
% 3.21/1.61 ! [Y: a,X: a] :
% 3.21/1.61 ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
% 3.21/1.61 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_5_nbt__subset__carrier,axiom,
% 3.21/1.61 ! [X: a] :
% 3.21/1.61 ( ( member_a @ X @ carrier )
% 3.21/1.61 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ carrier ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_6_rational__preference__axioms,axiom,
% 3.21/1.61 prefer719835986ence_a @ carrier @ relation ).
% 3.21/1.61
% 3.21/1.61 thf(fact_7__092_060open_062finite_Acarrier_092_060close_062,axiom,
% 3.21/1.61 finite_finite_a @ carrier ).
% 3.21/1.61
% 3.21/1.61 thf(fact_8_preference__axioms,axiom,
% 3.21/1.61 prefer1292084992ence_a @ carrier @ relation ).
% 3.21/1.61
% 3.21/1.61 thf(fact_9_trans__refl,axiom,
% 3.21/1.61 order_preorder_on_a @ carrier @ relation ).
% 3.21/1.61
% 3.21/1.61 thf(fact_10_nbt__subset,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( finite_finite_a @ carrier )
% 3.21/1.61 => ( ( member_a @ X @ carrier )
% 3.21/1.61 => ( ( member_a @ Y @ carrier )
% 3.21/1.61 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
% 3.21/1.61 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_11_no__better__subset__pref,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_12_no__better__thansubset__rel,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( member_a @ X @ carrier )
% 3.21/1.61 => ( ( member_a @ Y @ carrier )
% 3.21/1.61 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ carrier @ relation ) @ ( prefer1676310729than_a @ X @ carrier @ relation ) )
% 3.21/1.61 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_13_no__better__than__nonepty,axiom,
% 3.21/1.61 ! [X: a] :
% 3.21/1.61 ( ( carrier != bot_bot_set_a )
% 3.21/1.61 => ( ( member_a @ X @ carrier )
% 3.21/1.61 => ( ( prefer1676310729than_a @ X @ carrier @ relation )
% 3.21/1.61 != bot_bot_set_a ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_14_total,axiom,
% 3.21/1.61 total_on_a @ carrier @ relation ).
% 3.21/1.61
% 3.21/1.61 thf(fact_15_fnt__carrier__fnt__nbt,axiom,
% 3.21/1.61 ! [X2: a] :
% 3.21/1.61 ( ( member_a @ X2 @ carrier )
% 3.21/1.61 => ( finite_finite_a @ ( prefer1676310729than_a @ X2 @ carrier @ relation ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_16_same__nbt__same__pref,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( member_a @ X @ carrier )
% 3.21/1.61 => ( ( member_a @ Y @ carrier )
% 3.21/1.61 => ( ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ carrier @ relation ) )
% 3.21/1.61 & ( member_a @ Y @ ( prefer1676310729than_a @ X @ carrier @ relation ) ) )
% 3.21/1.61 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_17_worse__in__no__better,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ carrier @ relation ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_18_reflexivity,axiom,
% 3.21/1.61 refl_on_a @ carrier @ relation ).
% 3.21/1.61
% 3.21/1.61 thf(fact_19_compl,axiom,
% 3.21/1.61 ! [X2: a] :
% 3.21/1.61 ( ( member_a @ X2 @ carrier )
% 3.21/1.61 => ! [Xa: a] :
% 3.21/1.61 ( ( member_a @ Xa @ carrier )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ relation )
% 3.21/1.61 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_20_completeD,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( member_a @ X @ carrier )
% 3.21/1.61 => ( ( member_a @ Y @ carrier )
% 3.21/1.61 => ( ( X != Y )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_21_strict__trans,axiom,
% 3.21/1.61 ! [X: a,Y: a,Z: a] :
% 3.21/1.61 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_22_indiff__trans,axiom,
% 3.21/1.61 ! [X: a,Y: a,Z: a] :
% 3.21/1.61 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_23_indifferent__imp__weak__pref_I1_J,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 3.21/1.61 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_24_indifferent__imp__weak__pref_I2_J,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 3.21/1.61 => ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_25_weak__is__transitive,axiom,
% 3.21/1.61 ! [X: a,Y: a,Z: a] :
% 3.21/1.61 ( ( ( member_a @ X @ carrier )
% 3.21/1.61 & ( member_a @ Y @ carrier )
% 3.21/1.61 & ( member_a @ Z @ carrier ) )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 3.21/1.61 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_26_strict__is__neg__transitive,axiom,
% 3.21/1.61 ! [X: a,Y: a,Z: a] :
% 3.21/1.61 ( ( ( member_a @ X @ carrier )
% 3.21/1.61 & ( member_a @ Y @ carrier )
% 3.21/1.61 & ( member_a @ Z @ carrier ) )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) )
% 3.21/1.61 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_27_not__outside,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 => ( member_a @ X @ carrier ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_28_strict__not__refl__weak,axiom,
% 3.21/1.61 ! [X: a,Y: a] :
% 3.21/1.61 ( ( ( member_a @ X @ carrier )
% 3.21/1.61 & ( member_a @ Y @ carrier ) )
% 3.21/1.61 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 3.21/1.61 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_29_rational__preference_Oaxioms_I1_J,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_30_rational__preference_Oindifferent__imp__weak__pref_I2_J,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 3.21/1.61 => ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_31_rational__preference_Oindifferent__imp__weak__pref_I1_J,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 3.21/1.61 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_32_preference__def,axiom,
% 3.21/1.61 ( prefer1338892073od_a_a
% 3.21/1.61 = ( ^ [Carrier2: set_Product_prod_a_a,Relation2: set_Pr1948701895od_a_a] :
% 3.21/1.61 ( ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 3.21/1.61 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 3.21/1.61 => ( member449909584od_a_a @ X3 @ Carrier2 ) )
% 3.21/1.61 & ! [X3: product_prod_a_a,Y2: product_prod_a_a] :
% 3.21/1.61 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X3 @ Y2 ) @ Relation2 )
% 3.21/1.61 => ( member449909584od_a_a @ Y2 @ Carrier2 ) )
% 3.21/1.61 & ( order_359341630od_a_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_33_preference__def,axiom,
% 3.21/1.61 ( prefer1293349710ce_nat
% 3.21/1.61 = ( ^ [Carrier2: set_nat,Relation2: set_Pr1986765409at_nat] :
% 3.21/1.61 ( ! [X3: nat,Y2: nat] :
% 3.21/1.61 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ Relation2 )
% 3.21/1.61 => ( member_nat @ X3 @ Carrier2 ) )
% 3.21/1.61 & ! [X3: nat,Y2: nat] :
% 3.21/1.61 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ Relation2 )
% 3.21/1.61 => ( member_nat @ Y2 @ Carrier2 ) )
% 3.21/1.61 & ( order_1150839801on_nat @ Carrier2 @ Relation2 ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_34_preference__def,axiom,
% 3.21/1.61 ( prefer1292084992ence_a
% 3.21/1.61 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 3.21/1.61 ( ! [X3: a,Y2: a] :
% 3.21/1.61 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 3.21/1.61 => ( member_a @ X3 @ Carrier2 ) )
% 3.21/1.61 & ! [X3: a,Y2: a] :
% 3.21/1.61 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ Relation2 )
% 3.21/1.61 => ( member_a @ Y2 @ Carrier2 ) )
% 3.21/1.61 & ( order_preorder_on_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_35_preference_Ointro,axiom,
% 3.21/1.61 ! [Relation: set_Pr1948701895od_a_a,Carrier: set_Product_prod_a_a] :
% 3.21/1.61 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 3.21/1.61 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 3.21/1.61 => ( member449909584od_a_a @ X4 @ Carrier ) )
% 3.21/1.61 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 3.21/1.61 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Relation )
% 3.21/1.61 => ( member449909584od_a_a @ Y3 @ Carrier ) )
% 3.21/1.61 => ( ( order_359341630od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( prefer1338892073od_a_a @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_36_preference_Ointro,axiom,
% 3.21/1.61 ! [Relation: set_Pr1986765409at_nat,Carrier: set_nat] :
% 3.21/1.61 ( ! [X4: nat,Y3: nat] :
% 3.21/1.61 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ Relation )
% 3.21/1.61 => ( member_nat @ X4 @ Carrier ) )
% 3.21/1.61 => ( ! [X4: nat,Y3: nat] :
% 3.21/1.61 ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ Relation )
% 3.21/1.61 => ( member_nat @ Y3 @ Carrier ) )
% 3.21/1.61 => ( ( order_1150839801on_nat @ Carrier @ Relation )
% 3.21/1.61 => ( prefer1293349710ce_nat @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_37_preference_Ointro,axiom,
% 3.21/1.61 ! [Relation: set_Product_prod_a_a,Carrier: set_a] :
% 3.21/1.61 ( ! [X4: a,Y3: a] :
% 3.21/1.61 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 3.21/1.61 => ( member_a @ X4 @ Carrier ) )
% 3.21/1.61 => ( ! [X4: a,Y3: a] :
% 3.21/1.61 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Relation )
% 3.21/1.61 => ( member_a @ Y3 @ Carrier ) )
% 3.21/1.61 => ( ( order_preorder_on_a @ Carrier @ Relation )
% 3.21/1.61 => ( prefer1292084992ence_a @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_38_preference_Otrans__refl,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.61 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( order_preorder_on_a @ Carrier @ Relation ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_39_preference_Onot__outside,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.61 ( ( prefer1338892073od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 => ( member449909584od_a_a @ X @ Carrier ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_40_preference_Onot__outside,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.61 ( ( prefer1293349710ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 3.21/1.61 => ( member_nat @ X @ Carrier ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_41_preference_Onot__outside,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.61 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 => ( member_a @ X @ Carrier ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_42_preference_Oreflexivity,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.61 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( refl_on_a @ Carrier @ Relation ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_43_preference_Oindiff__trans,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 3.21/1.61 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_44_rational__preference_Ocompl,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ! [X2: a] :
% 3.21/1.61 ( ( member_a @ X2 @ Carrier )
% 3.21/1.61 => ! [Xa: a] :
% 3.21/1.61 ( ( member_a @ Xa @ Carrier )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ Relation )
% 3.21/1.61 | ( member449909584od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_45_rational__preference_Ototal,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( total_on_a @ Carrier @ Relation ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_46_rational__preference_OcompleteD,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.61 => ( ( member449909584od_a_a @ Y @ Carrier )
% 3.21/1.61 => ( ( X != Y )
% 3.21/1.61 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_47_rational__preference_OcompleteD,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( member_nat @ X @ Carrier )
% 3.21/1.61 => ( ( member_nat @ Y @ Carrier )
% 3.21/1.61 => ( ( X != Y )
% 3.21/1.61 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 3.21/1.61 | ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_48_rational__preference_OcompleteD,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member_a @ X @ Carrier )
% 3.21/1.61 => ( ( member_a @ Y @ Carrier )
% 3.21/1.61 => ( ( X != Y )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_49_rational__preference_Ostrict__trans,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation ) )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_50_rational__preference_Oweak__is__transitive,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.61 & ( member449909584od_a_a @ Y @ Carrier )
% 3.21/1.61 & ( member449909584od_a_a @ Z @ Carrier ) )
% 3.21/1.61 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation )
% 3.21/1.61 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_51_rational__preference_Oweak__is__transitive,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat,Z: nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member_nat @ X @ Carrier )
% 3.21/1.61 & ( member_nat @ Y @ Carrier )
% 3.21/1.61 & ( member_nat @ Z @ Carrier ) )
% 3.21/1.61 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 3.21/1.61 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ Relation )
% 3.21/1.61 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_52_rational__preference_Oweak__is__transitive,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member_a @ X @ Carrier )
% 3.21/1.61 & ( member_a @ Y @ Carrier )
% 3.21/1.61 & ( member_a @ Z @ Carrier ) )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation )
% 3.21/1.61 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_53_rational__preference_Ofnt__carrier__fnt__rel,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( finite179568208od_a_a @ Carrier )
% 3.21/1.61 => ( finite1664988688od_a_a @ Relation ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_54_rational__preference_Ofnt__carrier__fnt__rel,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( finite_finite_nat @ Carrier )
% 3.21/1.61 => ( finite772653738at_nat @ Relation ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_55_rational__preference_Ofnt__carrier__fnt__rel,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( finite_finite_a @ Carrier )
% 3.21/1.61 => ( finite179568208od_a_a @ Relation ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_56_rational__preference_Ostrict__not__refl__weak,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.61 & ( member449909584od_a_a @ Y @ Carrier ) )
% 3.21/1.61 => ( ( ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 3.21/1.61 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_57_rational__preference_Ostrict__not__refl__weak,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member_nat @ X @ Carrier )
% 3.21/1.61 & ( member_nat @ Y @ Carrier ) )
% 3.21/1.61 => ( ( ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) )
% 3.21/1.61 = ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_58_rational__preference_Ostrict__not__refl__weak,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member_a @ X @ Carrier )
% 3.21/1.61 & ( member_a @ Y @ Carrier ) )
% 3.21/1.61 => ( ( ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 3.21/1.61 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_59_mem__Collect__eq,axiom,
% 3.21/1.61 ! [A: a,P: a > $o] :
% 3.21/1.61 ( ( member_a @ A @ ( collect_a @ P ) )
% 3.21/1.61 = ( P @ A ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_60_mem__Collect__eq,axiom,
% 3.21/1.61 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 3.21/1.61 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 3.21/1.61 = ( P @ A ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_61_mem__Collect__eq,axiom,
% 3.21/1.61 ! [A: nat,P: nat > $o] :
% 3.21/1.61 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 3.21/1.61 = ( P @ A ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_62_Collect__mem__eq,axiom,
% 3.21/1.61 ! [A2: set_a] :
% 3.21/1.61 ( ( collect_a
% 3.21/1.61 @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
% 3.21/1.61 = A2 ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_63_Collect__mem__eq,axiom,
% 3.21/1.61 ! [A2: set_Product_prod_a_a] :
% 3.21/1.61 ( ( collec645855634od_a_a
% 3.21/1.61 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A2 ) )
% 3.21/1.61 = A2 ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_64_Collect__mem__eq,axiom,
% 3.21/1.61 ! [A2: set_nat] :
% 3.21/1.61 ( ( collect_nat
% 3.21/1.61 @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 3.21/1.61 = A2 ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_65_rational__preference_Ostrict__is__neg__transitive,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a,Z: product_prod_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.61 & ( member449909584od_a_a @ Y @ Carrier )
% 3.21/1.61 & ( member449909584od_a_a @ Z @ Carrier ) )
% 3.21/1.61 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) )
% 3.21/1.61 => ( ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Z ) @ Relation )
% 3.21/1.61 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ X ) @ Relation ) )
% 3.21/1.61 | ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_66_rational__preference_Ostrict__is__neg__transitive,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat,Z: nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member_nat @ X @ Carrier )
% 3.21/1.61 & ( member_nat @ Y @ Carrier )
% 3.21/1.61 & ( member_nat @ Z @ Carrier ) )
% 3.21/1.61 => ( ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) )
% 3.21/1.61 => ( ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ Relation )
% 3.21/1.61 & ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ X ) @ Relation ) )
% 3.21/1.61 | ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_67_rational__preference_Ostrict__is__neg__transitive,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( ( member_a @ X @ Carrier )
% 3.21/1.61 & ( member_a @ Y @ Carrier )
% 3.21/1.61 & ( member_a @ Z @ Carrier ) )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ Relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ Relation ) )
% 3.21/1.61 | ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 3.21/1.61 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_68_rational__preference_Oworse__in__no__better,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 => ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_69_rational__preference_Oworse__in__no__better,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 3.21/1.61 => ( member_nat @ Y @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_70_rational__preference_Oworse__in__no__better,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 => ( member_a @ Y @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_71_rational__preference_Osame__nbt__same__pref,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.61 => ( ( member449909584od_a_a @ Y @ Carrier )
% 3.21/1.61 => ( ( ( member449909584od_a_a @ X @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) )
% 3.21/1.61 & ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) )
% 3.21/1.61 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_72_rational__preference_Osame__nbt__same__pref,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( member_nat @ X @ Carrier )
% 3.21/1.61 => ( ( member_nat @ Y @ Carrier )
% 3.21/1.61 => ( ( ( member_nat @ X @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) )
% 3.21/1.61 & ( member_nat @ Y @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) ) )
% 3.21/1.61 = ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 3.21/1.61 & ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_73_rational__preference_Osame__nbt__same__pref,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member_a @ X @ Carrier )
% 3.21/1.61 => ( ( member_a @ Y @ Carrier )
% 3.21/1.61 => ( ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) )
% 3.21/1.61 & ( member_a @ Y @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) )
% 3.21/1.61 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_74_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( finite179568208od_a_a @ Carrier )
% 3.21/1.61 => ! [X2: product_prod_a_a] :
% 3.21/1.61 ( ( member449909584od_a_a @ X2 @ Carrier )
% 3.21/1.61 => ( finite179568208od_a_a @ ( prefer1498087410od_a_a @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_75_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( finite_finite_nat @ Carrier )
% 3.21/1.61 => ! [X2: nat] :
% 3.21/1.61 ( ( member_nat @ X2 @ Carrier )
% 3.21/1.61 => ( finite_finite_nat @ ( prefer1672638789an_nat @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_76_rational__preference_Ofnt__carrier__fnt__nbt,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( finite_finite_a @ Carrier )
% 3.21/1.61 => ! [X2: a] :
% 3.21/1.61 ( ( member_a @ X2 @ Carrier )
% 3.21/1.61 => ( finite_finite_a @ ( prefer1676310729than_a @ X2 @ Carrier @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_77_rational__preference_Ono__better__than__nonepty,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( Carrier != bot_bo2131659635od_a_a )
% 3.21/1.61 => ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.61 => ( ( prefer1498087410od_a_a @ X @ Carrier @ Relation )
% 3.21/1.61 != bot_bo2131659635od_a_a ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_78_rational__preference_Ono__better__than__nonepty,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( Carrier != bot_bot_set_nat )
% 3.21/1.61 => ( ( member_nat @ X @ Carrier )
% 3.21/1.61 => ( ( prefer1672638789an_nat @ X @ Carrier @ Relation )
% 3.21/1.61 != bot_bot_set_nat ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_79_rational__preference_Ono__better__than__nonepty,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( Carrier != bot_bot_set_a )
% 3.21/1.61 => ( ( member_a @ X @ Carrier )
% 3.21/1.61 => ( ( prefer1676310729than_a @ X @ Carrier @ Relation )
% 3.21/1.61 != bot_bot_set_a ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_80_rational__preference_Ono__better__thansubset__rel,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.61 => ( ( member449909584od_a_a @ Y @ Carrier )
% 3.21/1.61 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
% 3.21/1.61 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_81_rational__preference_Ono__better__thansubset__rel,axiom,
% 3.21/1.61 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.61 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.61 => ( ( member_nat @ X @ Carrier )
% 3.21/1.61 => ( ( member_nat @ Y @ Carrier )
% 3.21/1.61 => ( ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) )
% 3.21/1.61 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_82_rational__preference_Ono__better__thansubset__rel,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member_a @ X @ Carrier )
% 3.21/1.61 => ( ( member_a @ Y @ Carrier )
% 3.21/1.61 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 3.21/1.61 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_83_rational__preference_Ono__better__subset__pref,axiom,
% 3.21/1.61 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.61 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.61 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.61
% 3.21/1.61 thf(fact_84_rational__preference_Onbt__subset,axiom,
% 3.21/1.61 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.61 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.61 => ( ( finite179568208od_a_a @ Carrier )
% 3.21/1.61 => ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.61 => ( ( member449909584od_a_a @ Y @ Carrier )
% 3.21/1.61 => ( ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) )
% 3.21/1.61 | ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_85_rational__preference_Onbt__subset,axiom,
% 3.21/1.62 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.62 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.62 => ( ( finite_finite_nat @ Carrier )
% 3.21/1.62 => ( ( member_nat @ X @ Carrier )
% 3.21/1.62 => ( ( member_nat @ Y @ Carrier )
% 3.21/1.62 => ( ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) )
% 3.21/1.62 | ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_86_rational__preference_Onbt__subset,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( finite_finite_a @ Carrier )
% 3.21/1.62 => ( ( member_a @ X @ Carrier )
% 3.21/1.62 => ( ( member_a @ Y @ Carrier )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 3.21/1.62 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_87_rational__preference_Oxy__in__eachothers__nbt,axiom,
% 3.21/1.62 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.62 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.62 => ( ( member449909584od_a_a @ Y @ Carrier )
% 3.21/1.62 => ( ( member449909584od_a_a @ X @ ( prefer1498087410od_a_a @ Y @ Carrier @ Relation ) )
% 3.21/1.62 | ( member449909584od_a_a @ Y @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_88_rational__preference_Oxy__in__eachothers__nbt,axiom,
% 3.21/1.62 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.62 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.62 => ( ( member_nat @ X @ Carrier )
% 3.21/1.62 => ( ( member_nat @ Y @ Carrier )
% 3.21/1.62 => ( ( member_nat @ X @ ( prefer1672638789an_nat @ Y @ Carrier @ Relation ) )
% 3.21/1.62 | ( member_nat @ Y @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_89_rational__preference_Oxy__in__eachothers__nbt,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member_a @ X @ Carrier )
% 3.21/1.62 => ( ( member_a @ Y @ Carrier )
% 3.21/1.62 => ( ( member_a @ X @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) )
% 3.21/1.62 | ( member_a @ Y @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_90_rational__preference_Onbt__subset__carrier,axiom,
% 3.21/1.62 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a] :
% 3.21/1.62 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.62 => ( ord_le1824328871od_a_a @ ( prefer1498087410od_a_a @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_91_rational__preference_Onbt__subset__carrier,axiom,
% 3.21/1.62 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat] :
% 3.21/1.62 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.62 => ( ( member_nat @ X @ Carrier )
% 3.21/1.62 => ( ord_less_eq_set_nat @ ( prefer1672638789an_nat @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_92_rational__preference_Onbt__subset__carrier,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member_a @ X @ Carrier )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ Carrier ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_93_rational__preference_Onbt__nest,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Y: a,X: a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) )
% 3.21/1.62 | ( ord_less_eq_set_a @ ( prefer1676310729than_a @ X @ Carrier @ Relation ) @ ( prefer1676310729than_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_94_at__lst__asgd__not__ge,axiom,
% 3.21/1.62 ! [X: a,Y: a] :
% 3.21/1.62 ( ( carrier != bot_bot_set_a )
% 3.21/1.62 => ( ( member_a @ X @ carrier )
% 3.21/1.62 => ( ( member_a @ Y @ carrier )
% 3.21/1.62 => ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) )
% 3.21/1.62 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_95_same__at__least__as__equal,axiom,
% 3.21/1.62 ! [Z: a,Y: a] :
% 3.21/1.62 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 3.21/1.62 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) )
% 3.21/1.62 => ( ( prefer161218362good_a @ Z @ carrier @ relation )
% 3.21/1.62 = ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_96_pref__in__at__least__as,axiom,
% 3.21/1.62 ! [X: a,Y: a] :
% 3.21/1.62 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.62 => ( member_a @ X @ ( prefer161218362good_a @ Y @ carrier @ relation ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_97_empty__subsetI,axiom,
% 3.21/1.62 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_98_subset__empty,axiom,
% 3.21/1.62 ! [A2: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 3.21/1.62 = ( A2 = bot_bot_set_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_99_Pair__le,axiom,
% 3.21/1.62 ! [A: set_a,B: set_a,C: set_a,D: set_a] :
% 3.21/1.62 ( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A @ B ) @ ( produc1928581911_set_a @ C @ D ) )
% 3.21/1.62 = ( ( ord_less_eq_set_a @ A @ C )
% 3.21/1.62 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_100_Pair__le,axiom,
% 3.21/1.62 ! [A: set_a,B: nat,C: set_a,D: nat] :
% 3.21/1.62 ( ( ord_le781360189_a_nat @ ( produc515611863_a_nat @ A @ B ) @ ( produc515611863_a_nat @ C @ D ) )
% 3.21/1.62 = ( ( ord_less_eq_set_a @ A @ C )
% 3.21/1.62 & ( ord_less_eq_nat @ B @ D ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_101_Pair__le,axiom,
% 3.21/1.62 ! [A: nat,B: set_a,C: nat,D: set_a] :
% 3.21/1.62 ( ( ord_le841160035_set_a @ ( produc1450889781_set_a @ A @ B ) @ ( produc1450889781_set_a @ C @ D ) )
% 3.21/1.62 = ( ( ord_less_eq_nat @ A @ C )
% 3.21/1.62 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_102_Pair__le,axiom,
% 3.21/1.62 ! [A: nat,B: nat,C: nat,D: nat] :
% 3.21/1.62 ( ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
% 3.21/1.62 = ( ( ord_less_eq_nat @ A @ C )
% 3.21/1.62 & ( ord_less_eq_nat @ B @ D ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_103_as__good__as__sameIff,axiom,
% 3.21/1.62 ! [Z: a,Y: a] :
% 3.21/1.62 ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 3.21/1.62 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation )
% 3.21/1.62 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_104_as__good__asIff,axiom,
% 3.21/1.62 ! [X: a,Y: a] :
% 3.21/1.62 ( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ carrier @ relation ) )
% 3.21/1.62 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 3.21/1.62 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_105_fnt__carrier__fnt__rel,axiom,
% 3.21/1.62 ( ( finite_finite_a @ carrier )
% 3.21/1.62 => ( finite179568208od_a_a @ relation ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_106_card__mono,axiom,
% 3.21/1.62 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( finite179568208od_a_a @ B2 )
% 3.21/1.62 => ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_107_card__mono,axiom,
% 3.21/1.62 ! [B2: set_nat,A2: set_nat] :
% 3.21/1.62 ( ( finite_finite_nat @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_108_card__mono,axiom,
% 3.21/1.62 ! [B2: set_a,A2: set_a] :
% 3.21/1.62 ( ( finite_finite_a @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_109_empty__Collect__eq,axiom,
% 3.21/1.62 ! [P: a > $o] :
% 3.21/1.62 ( ( bot_bot_set_a
% 3.21/1.62 = ( collect_a @ P ) )
% 3.21/1.62 = ( ! [X3: a] :
% 3.21/1.62 ~ ( P @ X3 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_110_Collect__empty__eq,axiom,
% 3.21/1.62 ! [P: a > $o] :
% 3.21/1.62 ( ( ( collect_a @ P )
% 3.21/1.62 = bot_bot_set_a )
% 3.21/1.62 = ( ! [X3: a] :
% 3.21/1.62 ~ ( P @ X3 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_111_all__not__in__conv,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( ! [X3: product_prod_a_a] :
% 3.21/1.62 ~ ( member449909584od_a_a @ X3 @ A2 ) )
% 3.21/1.62 = ( A2 = bot_bo2131659635od_a_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_112_all__not__in__conv,axiom,
% 3.21/1.62 ! [A2: set_nat] :
% 3.21/1.62 ( ( ! [X3: nat] :
% 3.21/1.62 ~ ( member_nat @ X3 @ A2 ) )
% 3.21/1.62 = ( A2 = bot_bot_set_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_113_all__not__in__conv,axiom,
% 3.21/1.62 ! [A2: set_a] :
% 3.21/1.62 ( ( ! [X3: a] :
% 3.21/1.62 ~ ( member_a @ X3 @ A2 ) )
% 3.21/1.62 = ( A2 = bot_bot_set_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_114_empty__iff,axiom,
% 3.21/1.62 ! [C: product_prod_a_a] :
% 3.21/1.62 ~ ( member449909584od_a_a @ C @ bot_bo2131659635od_a_a ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_115_empty__iff,axiom,
% 3.21/1.62 ! [C: nat] :
% 3.21/1.62 ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_116_empty__iff,axiom,
% 3.21/1.62 ! [C: a] :
% 3.21/1.62 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_117_subsetI,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 3.21/1.62 ( ! [X4: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ X4 @ A2 )
% 3.21/1.62 => ( member449909584od_a_a @ X4 @ B2 ) )
% 3.21/1.62 => ( ord_le1824328871od_a_a @ A2 @ B2 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_118_subsetI,axiom,
% 3.21/1.62 ! [A2: set_nat,B2: set_nat] :
% 3.21/1.62 ( ! [X4: nat] :
% 3.21/1.62 ( ( member_nat @ X4 @ A2 )
% 3.21/1.62 => ( member_nat @ X4 @ B2 ) )
% 3.21/1.62 => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_119_subsetI,axiom,
% 3.21/1.62 ! [A2: set_a,B2: set_a] :
% 3.21/1.62 ( ! [X4: a] :
% 3.21/1.62 ( ( member_a @ X4 @ A2 )
% 3.21/1.62 => ( member_a @ X4 @ B2 ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_120_subset__antisym,axiom,
% 3.21/1.62 ! [A2: set_a,B2: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ B2 @ A2 )
% 3.21/1.62 => ( A2 = B2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_121_at__least__as__goodD,axiom,
% 3.21/1.62 ! [Z: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ Z @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
% 3.21/1.62 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Pr ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_122_at__least__as__goodD,axiom,
% 3.21/1.62 ! [Z: nat,Y: nat,B2: set_nat,Pr: set_Pr1986765409at_nat] :
% 3.21/1.62 ( ( member_nat @ Z @ ( prefer563798164od_nat @ Y @ B2 @ Pr ) )
% 3.21/1.62 => ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ Pr ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_123_at__least__as__goodD,axiom,
% 3.21/1.62 ! [Z: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
% 3.21/1.62 ( ( member_a @ Z @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
% 3.21/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Pr ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_124_at__lst__asgd__ge,axiom,
% 3.21/1.62 ! [X: product_prod_a_a,Y: product_prod_a_a,B2: set_Product_prod_a_a,Pr: set_Pr1948701895od_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ B2 @ Pr ) )
% 3.21/1.62 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Pr ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_125_at__lst__asgd__ge,axiom,
% 3.21/1.62 ! [X: nat,Y: nat,B2: set_nat,Pr: set_Pr1986765409at_nat] :
% 3.21/1.62 ( ( member_nat @ X @ ( prefer563798164od_nat @ Y @ B2 @ Pr ) )
% 3.21/1.62 => ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Pr ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_126_at__lst__asgd__ge,axiom,
% 3.21/1.62 ! [X: a,Y: a,B2: set_a,Pr: set_Product_prod_a_a] :
% 3.21/1.62 ( ( member_a @ X @ ( prefer161218362good_a @ Y @ B2 @ Pr ) )
% 3.21/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_127_ex__in__conv,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A2 ) )
% 3.21/1.62 = ( A2 != bot_bo2131659635od_a_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_128_ex__in__conv,axiom,
% 3.21/1.62 ! [A2: set_nat] :
% 3.21/1.62 ( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
% 3.21/1.62 = ( A2 != bot_bot_set_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_129_ex__in__conv,axiom,
% 3.21/1.62 ! [A2: set_a] :
% 3.21/1.62 ( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
% 3.21/1.62 = ( A2 != bot_bot_set_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_130_equals0I,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a] :
% 3.21/1.62 ( ! [Y3: product_prod_a_a] :
% 3.21/1.62 ~ ( member449909584od_a_a @ Y3 @ A2 )
% 3.21/1.62 => ( A2 = bot_bo2131659635od_a_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_131_equals0I,axiom,
% 3.21/1.62 ! [A2: set_nat] :
% 3.21/1.62 ( ! [Y3: nat] :
% 3.21/1.62 ~ ( member_nat @ Y3 @ A2 )
% 3.21/1.62 => ( A2 = bot_bot_set_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_132_equals0I,axiom,
% 3.21/1.62 ! [A2: set_a] :
% 3.21/1.62 ( ! [Y3: a] :
% 3.21/1.62 ~ ( member_a @ Y3 @ A2 )
% 3.21/1.62 => ( A2 = bot_bot_set_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_133_equals0D,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
% 3.21/1.62 ( ( A2 = bot_bo2131659635od_a_a )
% 3.21/1.62 => ~ ( member449909584od_a_a @ A @ A2 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_134_equals0D,axiom,
% 3.21/1.62 ! [A2: set_nat,A: nat] :
% 3.21/1.62 ( ( A2 = bot_bot_set_nat )
% 3.21/1.62 => ~ ( member_nat @ A @ A2 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_135_equals0D,axiom,
% 3.21/1.62 ! [A2: set_a,A: a] :
% 3.21/1.62 ( ( A2 = bot_bot_set_a )
% 3.21/1.62 => ~ ( member_a @ A @ A2 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_136_emptyE,axiom,
% 3.21/1.62 ! [A: product_prod_a_a] :
% 3.21/1.62 ~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_137_emptyE,axiom,
% 3.21/1.62 ! [A: nat] :
% 3.21/1.62 ~ ( member_nat @ A @ bot_bot_set_nat ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_138_emptyE,axiom,
% 3.21/1.62 ! [A: a] :
% 3.21/1.62 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_139_in__mono,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,X: product_prod_a_a] :
% 3.21/1.62 ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ X @ A2 )
% 3.21/1.62 => ( member449909584od_a_a @ X @ B2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_140_in__mono,axiom,
% 3.21/1.62 ! [A2: set_nat,B2: set_nat,X: nat] :
% 3.21/1.62 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 3.21/1.62 => ( ( member_nat @ X @ A2 )
% 3.21/1.62 => ( member_nat @ X @ B2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_141_in__mono,axiom,
% 3.21/1.62 ! [A2: set_a,B2: set_a,X: a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ( ( member_a @ X @ A2 )
% 3.21/1.62 => ( member_a @ X @ B2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_142_subsetD,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a,C: product_prod_a_a] :
% 3.21/1.62 ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ C @ A2 )
% 3.21/1.62 => ( member449909584od_a_a @ C @ B2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_143_subsetD,axiom,
% 3.21/1.62 ! [A2: set_nat,B2: set_nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 3.21/1.62 => ( ( member_nat @ C @ A2 )
% 3.21/1.62 => ( member_nat @ C @ B2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_144_subsetD,axiom,
% 3.21/1.62 ! [A2: set_a,B2: set_a,C: a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ( ( member_a @ C @ A2 )
% 3.21/1.62 => ( member_a @ C @ B2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_145_equalityE,axiom,
% 3.21/1.62 ! [A2: set_a,B2: set_a] :
% 3.21/1.62 ( ( A2 = B2 )
% 3.21/1.62 => ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_146_subset__eq,axiom,
% 3.21/1.62 ( ord_le1824328871od_a_a
% 3.21/1.62 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 3.21/1.62 ! [X3: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ X3 @ A3 )
% 3.21/1.62 => ( member449909584od_a_a @ X3 @ B3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_147_subset__eq,axiom,
% 3.21/1.62 ( ord_less_eq_set_nat
% 3.21/1.62 = ( ^ [A3: set_nat,B3: set_nat] :
% 3.21/1.62 ! [X3: nat] :
% 3.21/1.62 ( ( member_nat @ X3 @ A3 )
% 3.21/1.62 => ( member_nat @ X3 @ B3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_148_subset__eq,axiom,
% 3.21/1.62 ( ord_less_eq_set_a
% 3.21/1.62 = ( ^ [A3: set_a,B3: set_a] :
% 3.21/1.62 ! [X3: a] :
% 3.21/1.62 ( ( member_a @ X3 @ A3 )
% 3.21/1.62 => ( member_a @ X3 @ B3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_149_equalityD1,axiom,
% 3.21/1.62 ! [A2: set_a,B2: set_a] :
% 3.21/1.62 ( ( A2 = B2 )
% 3.21/1.62 => ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_150_equalityD2,axiom,
% 3.21/1.62 ! [A2: set_a,B2: set_a] :
% 3.21/1.62 ( ( A2 = B2 )
% 3.21/1.62 => ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_151_subset__iff,axiom,
% 3.21/1.62 ( ord_le1824328871od_a_a
% 3.21/1.62 = ( ^ [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
% 3.21/1.62 ! [T: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ T @ A3 )
% 3.21/1.62 => ( member449909584od_a_a @ T @ B3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_152_subset__iff,axiom,
% 3.21/1.62 ( ord_less_eq_set_nat
% 3.21/1.62 = ( ^ [A3: set_nat,B3: set_nat] :
% 3.21/1.62 ! [T: nat] :
% 3.21/1.62 ( ( member_nat @ T @ A3 )
% 3.21/1.62 => ( member_nat @ T @ B3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_153_subset__iff,axiom,
% 3.21/1.62 ( ord_less_eq_set_a
% 3.21/1.62 = ( ^ [A3: set_a,B3: set_a] :
% 3.21/1.62 ! [T: a] :
% 3.21/1.62 ( ( member_a @ T @ A3 )
% 3.21/1.62 => ( member_a @ T @ B3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_154_subset__refl,axiom,
% 3.21/1.62 ! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_155_Collect__mono,axiom,
% 3.21/1.62 ! [P: a > $o,Q: a > $o] :
% 3.21/1.62 ( ! [X4: a] :
% 3.21/1.62 ( ( P @ X4 )
% 3.21/1.62 => ( Q @ X4 ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_156_subset__trans,axiom,
% 3.21/1.62 ! [A2: set_a,B2: set_a,C2: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ B2 @ C2 )
% 3.21/1.62 => ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_157_set__eq__subset,axiom,
% 3.21/1.62 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 3.21/1.62 = ( ^ [A3: set_a,B3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A3 @ B3 )
% 3.21/1.62 & ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_158_Collect__mono__iff,axiom,
% 3.21/1.62 ! [P: a > $o,Q: a > $o] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 3.21/1.62 = ( ! [X3: a] :
% 3.21/1.62 ( ( P @ X3 )
% 3.21/1.62 => ( Q @ X3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_159_rational__preference_Osame__at__least__as__equal,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 3.21/1.62 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) )
% 3.21/1.62 => ( ( prefer161218362good_a @ Z @ Carrier @ Relation )
% 3.21/1.62 = ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_160_rational__preference_Opref__in__at__least__as,axiom,
% 3.21/1.62 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.62 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.21/1.62 => ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_161_rational__preference_Opref__in__at__least__as,axiom,
% 3.21/1.62 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.62 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.62 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 3.21/1.62 => ( member_nat @ X @ ( prefer563798164od_nat @ Y @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_162_rational__preference_Opref__in__at__least__as,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.62 => ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_163_rational__preference_Oas__good__as__sameIff,axiom,
% 3.21/1.62 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,Z: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.62 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member449909584od_a_a @ Z @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 3.21/1.62 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Z @ Y ) @ Relation )
% 3.21/1.62 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_164_rational__preference_Oas__good__as__sameIff,axiom,
% 3.21/1.62 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,Z: nat,Y: nat] :
% 3.21/1.62 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.62 => ( ( member_nat @ Z @ ( prefer1936906324as_nat @ Y @ Carrier @ Relation ) )
% 3.21/1.62 = ( ( member701585322at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ Relation )
% 3.21/1.62 & ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_165_rational__preference_Oas__good__as__sameIff,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a,Z: a,Y: a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member_a @ Z @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 3.21/1.62 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ Relation )
% 3.21/1.62 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_166_rational__preference_Oas__good__asIff,axiom,
% 3.21/1.62 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.62 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member449909584od_a_a @ X @ ( prefer74720675od_a_a @ Y @ Carrier @ Relation ) )
% 3.21/1.62 = ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation )
% 3.21/1.62 & ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_167_rational__preference_Oas__good__asIff,axiom,
% 3.21/1.62 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.62 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.62 => ( ( member_nat @ X @ ( prefer1936906324as_nat @ Y @ Carrier @ Relation ) )
% 3.21/1.62 = ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation )
% 3.21/1.62 & ( member701585322at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_168_rational__preference_Oas__good__asIff,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( member_a @ X @ ( prefer436369274d_as_a @ Y @ Carrier @ Relation ) )
% 3.21/1.62 = ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation )
% 3.21/1.62 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ Relation ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_169_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 3.21/1.62 ! [Carrier: set_Product_prod_a_a,Relation: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.62 ( ( prefer697821563od_a_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( Carrier != bot_bo2131659635od_a_a )
% 3.21/1.62 => ( ( member449909584od_a_a @ X @ Carrier )
% 3.21/1.62 => ( ( member449909584od_a_a @ Y @ Carrier )
% 3.21/1.62 => ( ~ ( member449909584od_a_a @ X @ ( prefer477097315od_a_a @ Y @ Carrier @ Relation ) )
% 3.21/1.62 => ~ ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_170_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 3.21/1.62 ! [Carrier: set_nat,Relation: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.62 ( ( prefer1147844220ce_nat @ Carrier @ Relation )
% 3.21/1.62 => ( ( Carrier != bot_bot_set_nat )
% 3.21/1.62 => ( ( member_nat @ X @ Carrier )
% 3.21/1.62 => ( ( member_nat @ Y @ Carrier )
% 3.21/1.62 => ( ~ ( member_nat @ X @ ( prefer563798164od_nat @ Y @ Carrier @ Relation ) )
% 3.21/1.62 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_171_rational__preference_Oat__lst__asgd__not__ge,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( Carrier != bot_bot_set_a )
% 3.21/1.62 => ( ( member_a @ X @ Carrier )
% 3.21/1.62 => ( ( member_a @ Y @ Carrier )
% 3.21/1.62 => ( ~ ( member_a @ X @ ( prefer161218362good_a @ Y @ Carrier @ Relation ) )
% 3.21/1.62 => ~ ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Relation ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_172_Pair__mono,axiom,
% 3.21/1.62 ! [X: set_a,X5: set_a,Y: set_a,Y5: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X @ X5 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 3.21/1.62 => ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X @ Y ) @ ( produc1928581911_set_a @ X5 @ Y5 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_173_Pair__mono,axiom,
% 3.21/1.62 ! [X: set_a,X5: set_a,Y: nat,Y5: nat] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X @ X5 )
% 3.21/1.62 => ( ( ord_less_eq_nat @ Y @ Y5 )
% 3.21/1.62 => ( ord_le781360189_a_nat @ ( produc515611863_a_nat @ X @ Y ) @ ( produc515611863_a_nat @ X5 @ Y5 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_174_Pair__mono,axiom,
% 3.21/1.62 ! [X: nat,X5: nat,Y: set_a,Y5: set_a] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X @ X5 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ Y @ Y5 )
% 3.21/1.62 => ( ord_le841160035_set_a @ ( produc1450889781_set_a @ X @ Y ) @ ( produc1450889781_set_a @ X5 @ Y5 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_175_Pair__mono,axiom,
% 3.21/1.62 ! [X: nat,X5: nat,Y: nat,Y5: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X @ X5 )
% 3.21/1.62 => ( ( ord_less_eq_nat @ Y @ Y5 )
% 3.21/1.62 => ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_176_finite__has__maximal2,axiom,
% 3.21/1.62 ! [A2: set_set_a,A: set_a] :
% 3.21/1.62 ( ( finite_finite_set_a @ A2 )
% 3.21/1.62 => ( ( member_set_a @ A @ A2 )
% 3.21/1.62 => ? [X4: set_a] :
% 3.21/1.62 ( ( member_set_a @ X4 @ A2 )
% 3.21/1.62 & ( ord_less_eq_set_a @ A @ X4 )
% 3.21/1.62 & ! [Xa: set_a] :
% 3.21/1.62 ( ( member_set_a @ Xa @ A2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ X4 @ Xa )
% 3.21/1.62 => ( X4 = Xa ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_177_finite__has__maximal2,axiom,
% 3.21/1.62 ! [A2: set_nat,A: nat] :
% 3.21/1.62 ( ( finite_finite_nat @ A2 )
% 3.21/1.62 => ( ( member_nat @ A @ A2 )
% 3.21/1.62 => ? [X4: nat] :
% 3.21/1.62 ( ( member_nat @ X4 @ A2 )
% 3.21/1.62 & ( ord_less_eq_nat @ A @ X4 )
% 3.21/1.62 & ! [Xa: nat] :
% 3.21/1.62 ( ( member_nat @ Xa @ A2 )
% 3.21/1.62 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 3.21/1.62 => ( X4 = Xa ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_178_finite__has__minimal2,axiom,
% 3.21/1.62 ! [A2: set_set_a,A: set_a] :
% 3.21/1.62 ( ( finite_finite_set_a @ A2 )
% 3.21/1.62 => ( ( member_set_a @ A @ A2 )
% 3.21/1.62 => ? [X4: set_a] :
% 3.21/1.62 ( ( member_set_a @ X4 @ A2 )
% 3.21/1.62 & ( ord_less_eq_set_a @ X4 @ A )
% 3.21/1.62 & ! [Xa: set_a] :
% 3.21/1.62 ( ( member_set_a @ Xa @ A2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ Xa @ X4 )
% 3.21/1.62 => ( X4 = Xa ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_179_finite__has__minimal2,axiom,
% 3.21/1.62 ! [A2: set_nat,A: nat] :
% 3.21/1.62 ( ( finite_finite_nat @ A2 )
% 3.21/1.62 => ( ( member_nat @ A @ A2 )
% 3.21/1.62 => ? [X4: nat] :
% 3.21/1.62 ( ( member_nat @ X4 @ A2 )
% 3.21/1.62 & ( ord_less_eq_nat @ X4 @ A )
% 3.21/1.62 & ! [Xa: nat] :
% 3.21/1.62 ( ( member_nat @ Xa @ A2 )
% 3.21/1.62 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 3.21/1.62 => ( X4 = Xa ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_180_bot__prod__def,axiom,
% 3.21/1.62 ( bot_bo1519632275_set_a
% 3.21/1.62 = ( produc1928581911_set_a @ bot_bot_set_a @ bot_bot_set_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_181_infinite__imp__nonempty,axiom,
% 3.21/1.62 ! [S: set_Product_prod_a_a] :
% 3.21/1.62 ( ~ ( finite179568208od_a_a @ S )
% 3.21/1.62 => ( S != bot_bo2131659635od_a_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_182_infinite__imp__nonempty,axiom,
% 3.21/1.62 ! [S: set_nat] :
% 3.21/1.62 ( ~ ( finite_finite_nat @ S )
% 3.21/1.62 => ( S != bot_bot_set_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_183_infinite__imp__nonempty,axiom,
% 3.21/1.62 ! [S: set_a] :
% 3.21/1.62 ( ~ ( finite_finite_a @ S )
% 3.21/1.62 => ( S != bot_bot_set_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_184_finite_OemptyI,axiom,
% 3.21/1.62 finite179568208od_a_a @ bot_bo2131659635od_a_a ).
% 3.21/1.62
% 3.21/1.62 thf(fact_185_finite_OemptyI,axiom,
% 3.21/1.62 finite_finite_nat @ bot_bot_set_nat ).
% 3.21/1.62
% 3.21/1.62 thf(fact_186_finite_OemptyI,axiom,
% 3.21/1.62 finite_finite_a @ bot_bot_set_a ).
% 3.21/1.62
% 3.21/1.62 thf(fact_187_finite__subset,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 3.21/1.62 => ( ( finite179568208od_a_a @ B2 )
% 3.21/1.62 => ( finite179568208od_a_a @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_188_finite__subset,axiom,
% 3.21/1.62 ! [A2: set_nat,B2: set_nat] :
% 3.21/1.62 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 3.21/1.62 => ( ( finite_finite_nat @ B2 )
% 3.21/1.62 => ( finite_finite_nat @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_189_finite__subset,axiom,
% 3.21/1.62 ! [A2: set_a,B2: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ( ( finite_finite_a @ B2 )
% 3.21/1.62 => ( finite_finite_a @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_190_infinite__super,axiom,
% 3.21/1.62 ! [S: set_Product_prod_a_a,T2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( ord_le1824328871od_a_a @ S @ T2 )
% 3.21/1.62 => ( ~ ( finite179568208od_a_a @ S )
% 3.21/1.62 => ~ ( finite179568208od_a_a @ T2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_191_infinite__super,axiom,
% 3.21/1.62 ! [S: set_nat,T2: set_nat] :
% 3.21/1.62 ( ( ord_less_eq_set_nat @ S @ T2 )
% 3.21/1.62 => ( ~ ( finite_finite_nat @ S )
% 3.21/1.62 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_192_infinite__super,axiom,
% 3.21/1.62 ! [S: set_a,T2: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ S @ T2 )
% 3.21/1.62 => ( ~ ( finite_finite_a @ S )
% 3.21/1.62 => ~ ( finite_finite_a @ T2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_193_rev__finite__subset,axiom,
% 3.21/1.62 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( finite179568208od_a_a @ B2 )
% 3.21/1.62 => ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 3.21/1.62 => ( finite179568208od_a_a @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_194_rev__finite__subset,axiom,
% 3.21/1.62 ! [B2: set_nat,A2: set_nat] :
% 3.21/1.62 ( ( finite_finite_nat @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 3.21/1.62 => ( finite_finite_nat @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_195_rev__finite__subset,axiom,
% 3.21/1.62 ! [B2: set_a,A2: set_a] :
% 3.21/1.62 ( ( finite_finite_a @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ( finite_finite_a @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_196_finite__has__minimal,axiom,
% 3.21/1.62 ! [A2: set_set_a] :
% 3.21/1.62 ( ( finite_finite_set_a @ A2 )
% 3.21/1.62 => ( ( A2 != bot_bot_set_set_a )
% 3.21/1.62 => ? [X4: set_a] :
% 3.21/1.62 ( ( member_set_a @ X4 @ A2 )
% 3.21/1.62 & ! [Xa: set_a] :
% 3.21/1.62 ( ( member_set_a @ Xa @ A2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ Xa @ X4 )
% 3.21/1.62 => ( X4 = Xa ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_197_finite__has__minimal,axiom,
% 3.21/1.62 ! [A2: set_nat] :
% 3.21/1.62 ( ( finite_finite_nat @ A2 )
% 3.21/1.62 => ( ( A2 != bot_bot_set_nat )
% 3.21/1.62 => ? [X4: nat] :
% 3.21/1.62 ( ( member_nat @ X4 @ A2 )
% 3.21/1.62 & ! [Xa: nat] :
% 3.21/1.62 ( ( member_nat @ Xa @ A2 )
% 3.21/1.62 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 3.21/1.62 => ( X4 = Xa ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_198_finite__has__maximal,axiom,
% 3.21/1.62 ! [A2: set_set_a] :
% 3.21/1.62 ( ( finite_finite_set_a @ A2 )
% 3.21/1.62 => ( ( A2 != bot_bot_set_set_a )
% 3.21/1.62 => ? [X4: set_a] :
% 3.21/1.62 ( ( member_set_a @ X4 @ A2 )
% 3.21/1.62 & ! [Xa: set_a] :
% 3.21/1.62 ( ( member_set_a @ Xa @ A2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ X4 @ Xa )
% 3.21/1.62 => ( X4 = Xa ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_199_finite__has__maximal,axiom,
% 3.21/1.62 ! [A2: set_nat] :
% 3.21/1.62 ( ( finite_finite_nat @ A2 )
% 3.21/1.62 => ( ( A2 != bot_bot_set_nat )
% 3.21/1.62 => ? [X4: nat] :
% 3.21/1.62 ( ( member_nat @ X4 @ A2 )
% 3.21/1.62 & ! [Xa: nat] :
% 3.21/1.62 ( ( member_nat @ Xa @ A2 )
% 3.21/1.62 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 3.21/1.62 => ( X4 = Xa ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_200_card__subset__eq,axiom,
% 3.21/1.62 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( finite179568208od_a_a @ B2 )
% 3.21/1.62 => ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 3.21/1.62 => ( ( ( finite1481642319od_a_a @ A2 )
% 3.21/1.62 = ( finite1481642319od_a_a @ B2 ) )
% 3.21/1.62 => ( A2 = B2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_201_card__subset__eq,axiom,
% 3.21/1.62 ! [B2: set_nat,A2: set_nat] :
% 3.21/1.62 ( ( finite_finite_nat @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 3.21/1.62 => ( ( ( finite_card_nat @ A2 )
% 3.21/1.62 = ( finite_card_nat @ B2 ) )
% 3.21/1.62 => ( A2 = B2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_202_card__subset__eq,axiom,
% 3.21/1.62 ! [B2: set_a,A2: set_a] :
% 3.21/1.62 ( ( finite_finite_a @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ( ( ( finite_card_a @ A2 )
% 3.21/1.62 = ( finite_card_a @ B2 ) )
% 3.21/1.62 => ( A2 = B2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_203_infinite__arbitrarily__large,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,N: nat] :
% 3.21/1.62 ( ~ ( finite179568208od_a_a @ A2 )
% 3.21/1.62 => ? [B4: set_Product_prod_a_a] :
% 3.21/1.62 ( ( finite179568208od_a_a @ B4 )
% 3.21/1.62 & ( ( finite1481642319od_a_a @ B4 )
% 3.21/1.62 = N )
% 3.21/1.62 & ( ord_le1824328871od_a_a @ B4 @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_204_infinite__arbitrarily__large,axiom,
% 3.21/1.62 ! [A2: set_nat,N: nat] :
% 3.21/1.62 ( ~ ( finite_finite_nat @ A2 )
% 3.21/1.62 => ? [B4: set_nat] :
% 3.21/1.62 ( ( finite_finite_nat @ B4 )
% 3.21/1.62 & ( ( finite_card_nat @ B4 )
% 3.21/1.62 = N )
% 3.21/1.62 & ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_205_infinite__arbitrarily__large,axiom,
% 3.21/1.62 ! [A2: set_a,N: nat] :
% 3.21/1.62 ( ~ ( finite_finite_a @ A2 )
% 3.21/1.62 => ? [B4: set_a] :
% 3.21/1.62 ( ( finite_finite_a @ B4 )
% 3.21/1.62 & ( ( finite_card_a @ B4 )
% 3.21/1.62 = N )
% 3.21/1.62 & ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_206_finite__if__finite__subsets__card__bdd,axiom,
% 3.21/1.62 ! [F: set_Product_prod_a_a,C2: nat] :
% 3.21/1.62 ( ! [G: set_Product_prod_a_a] :
% 3.21/1.62 ( ( ord_le1824328871od_a_a @ G @ F )
% 3.21/1.62 => ( ( finite179568208od_a_a @ G )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ G ) @ C2 ) ) )
% 3.21/1.62 => ( ( finite179568208od_a_a @ F )
% 3.21/1.62 & ( ord_less_eq_nat @ ( finite1481642319od_a_a @ F ) @ C2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_207_finite__if__finite__subsets__card__bdd,axiom,
% 3.21/1.62 ! [F: set_nat,C2: nat] :
% 3.21/1.62 ( ! [G: set_nat] :
% 3.21/1.62 ( ( ord_less_eq_set_nat @ G @ F )
% 3.21/1.62 => ( ( finite_finite_nat @ G )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
% 3.21/1.62 => ( ( finite_finite_nat @ F )
% 3.21/1.62 & ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_208_finite__if__finite__subsets__card__bdd,axiom,
% 3.21/1.62 ! [F: set_a,C2: nat] :
% 3.21/1.62 ( ! [G: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ G @ F )
% 3.21/1.62 => ( ( finite_finite_a @ G )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C2 ) ) )
% 3.21/1.62 => ( ( finite_finite_a @ F )
% 3.21/1.62 & ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_209_card__seteq,axiom,
% 3.21/1.62 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( finite179568208od_a_a @ B2 )
% 3.21/1.62 => ( ( ord_le1824328871od_a_a @ A2 @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_nat @ ( finite1481642319od_a_a @ B2 ) @ ( finite1481642319od_a_a @ A2 ) )
% 3.21/1.62 => ( A2 = B2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_210_card__seteq,axiom,
% 3.21/1.62 ! [B2: set_nat,A2: set_nat] :
% 3.21/1.62 ( ( finite_finite_nat @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
% 3.21/1.62 => ( A2 = B2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_211_card__seteq,axiom,
% 3.21/1.62 ! [B2: set_a,A2: set_a] :
% 3.21/1.62 ( ( finite_finite_a @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ A2 @ B2 )
% 3.21/1.62 => ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
% 3.21/1.62 => ( A2 = B2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_212_obtain__subset__with__card__n,axiom,
% 3.21/1.62 ! [N: nat,S: set_Product_prod_a_a] :
% 3.21/1.62 ( ( ord_less_eq_nat @ N @ ( finite1481642319od_a_a @ S ) )
% 3.21/1.62 => ~ ! [T3: set_Product_prod_a_a] :
% 3.21/1.62 ( ( ord_le1824328871od_a_a @ T3 @ S )
% 3.21/1.62 => ( ( ( finite1481642319od_a_a @ T3 )
% 3.21/1.62 = N )
% 3.21/1.62 => ~ ( finite179568208od_a_a @ T3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_213_obtain__subset__with__card__n,axiom,
% 3.21/1.62 ! [N: nat,S: set_nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
% 3.21/1.62 => ~ ! [T3: set_nat] :
% 3.21/1.62 ( ( ord_less_eq_set_nat @ T3 @ S )
% 3.21/1.62 => ( ( ( finite_card_nat @ T3 )
% 3.21/1.62 = N )
% 3.21/1.62 => ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_214_obtain__subset__with__card__n,axiom,
% 3.21/1.62 ! [N: nat,S: set_a] :
% 3.21/1.62 ( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
% 3.21/1.62 => ~ ! [T3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ T3 @ S )
% 3.21/1.62 => ( ( ( finite_card_a @ T3 )
% 3.21/1.62 = N )
% 3.21/1.62 => ~ ( finite_finite_a @ T3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_215_card__le__if__inj__on__rel,axiom,
% 3.21/1.62 ! [B2: set_a,A2: set_Product_prod_a_a,R: product_prod_a_a > a > $o] :
% 3.21/1.62 ( ( finite_finite_a @ B2 )
% 3.21/1.62 => ( ! [A4: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ A4 @ A2 )
% 3.21/1.62 => ? [B5: a] :
% 3.21/1.62 ( ( member_a @ B5 @ B2 )
% 3.21/1.62 & ( R @ A4 @ B5 ) ) )
% 3.21/1.62 => ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: a] :
% 3.21/1.62 ( ( member449909584od_a_a @ A1 @ A2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ A22 @ A2 )
% 3.21/1.62 => ( ( member_a @ B6 @ B2 )
% 3.21/1.62 => ( ( R @ A1 @ B6 )
% 3.21/1.62 => ( ( R @ A22 @ B6 )
% 3.21/1.62 => ( A1 = A22 ) ) ) ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_216_card__le__if__inj__on__rel,axiom,
% 3.21/1.62 ! [B2: set_a,A2: set_nat,R: nat > a > $o] :
% 3.21/1.62 ( ( finite_finite_a @ B2 )
% 3.21/1.62 => ( ! [A4: nat] :
% 3.21/1.62 ( ( member_nat @ A4 @ A2 )
% 3.21/1.62 => ? [B5: a] :
% 3.21/1.62 ( ( member_a @ B5 @ B2 )
% 3.21/1.62 & ( R @ A4 @ B5 ) ) )
% 3.21/1.62 => ( ! [A1: nat,A22: nat,B6: a] :
% 3.21/1.62 ( ( member_nat @ A1 @ A2 )
% 3.21/1.62 => ( ( member_nat @ A22 @ A2 )
% 3.21/1.62 => ( ( member_a @ B6 @ B2 )
% 3.21/1.62 => ( ( R @ A1 @ B6 )
% 3.21/1.62 => ( ( R @ A22 @ B6 )
% 3.21/1.62 => ( A1 = A22 ) ) ) ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_217_card__le__if__inj__on__rel,axiom,
% 3.21/1.62 ! [B2: set_a,A2: set_a,R: a > a > $o] :
% 3.21/1.62 ( ( finite_finite_a @ B2 )
% 3.21/1.62 => ( ! [A4: a] :
% 3.21/1.62 ( ( member_a @ A4 @ A2 )
% 3.21/1.62 => ? [B5: a] :
% 3.21/1.62 ( ( member_a @ B5 @ B2 )
% 3.21/1.62 & ( R @ A4 @ B5 ) ) )
% 3.21/1.62 => ( ! [A1: a,A22: a,B6: a] :
% 3.21/1.62 ( ( member_a @ A1 @ A2 )
% 3.21/1.62 => ( ( member_a @ A22 @ A2 )
% 3.21/1.62 => ( ( member_a @ B6 @ B2 )
% 3.21/1.62 => ( ( R @ A1 @ B6 )
% 3.21/1.62 => ( ( R @ A22 @ B6 )
% 3.21/1.62 => ( A1 = A22 ) ) ) ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_218_card__le__if__inj__on__rel,axiom,
% 3.21/1.62 ! [B2: set_Product_prod_a_a,A2: set_Product_prod_a_a,R: product_prod_a_a > product_prod_a_a > $o] :
% 3.21/1.62 ( ( finite179568208od_a_a @ B2 )
% 3.21/1.62 => ( ! [A4: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ A4 @ A2 )
% 3.21/1.62 => ? [B5: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ B5 @ B2 )
% 3.21/1.62 & ( R @ A4 @ B5 ) ) )
% 3.21/1.62 => ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ A1 @ A2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ A22 @ A2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ B6 @ B2 )
% 3.21/1.62 => ( ( R @ A1 @ B6 )
% 3.21/1.62 => ( ( R @ A22 @ B6 )
% 3.21/1.62 => ( A1 = A22 ) ) ) ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_219_card__le__if__inj__on__rel,axiom,
% 3.21/1.62 ! [B2: set_Product_prod_a_a,A2: set_nat,R: nat > product_prod_a_a > $o] :
% 3.21/1.62 ( ( finite179568208od_a_a @ B2 )
% 3.21/1.62 => ( ! [A4: nat] :
% 3.21/1.62 ( ( member_nat @ A4 @ A2 )
% 3.21/1.62 => ? [B5: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ B5 @ B2 )
% 3.21/1.62 & ( R @ A4 @ B5 ) ) )
% 3.21/1.62 => ( ! [A1: nat,A22: nat,B6: product_prod_a_a] :
% 3.21/1.62 ( ( member_nat @ A1 @ A2 )
% 3.21/1.62 => ( ( member_nat @ A22 @ A2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ B6 @ B2 )
% 3.21/1.62 => ( ( R @ A1 @ B6 )
% 3.21/1.62 => ( ( R @ A22 @ B6 )
% 3.21/1.62 => ( A1 = A22 ) ) ) ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_220_card__le__if__inj__on__rel,axiom,
% 3.21/1.62 ! [B2: set_Product_prod_a_a,A2: set_a,R: a > product_prod_a_a > $o] :
% 3.21/1.62 ( ( finite179568208od_a_a @ B2 )
% 3.21/1.62 => ( ! [A4: a] :
% 3.21/1.62 ( ( member_a @ A4 @ A2 )
% 3.21/1.62 => ? [B5: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ B5 @ B2 )
% 3.21/1.62 & ( R @ A4 @ B5 ) ) )
% 3.21/1.62 => ( ! [A1: a,A22: a,B6: product_prod_a_a] :
% 3.21/1.62 ( ( member_a @ A1 @ A2 )
% 3.21/1.62 => ( ( member_a @ A22 @ A2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ B6 @ B2 )
% 3.21/1.62 => ( ( R @ A1 @ B6 )
% 3.21/1.62 => ( ( R @ A22 @ B6 )
% 3.21/1.62 => ( A1 = A22 ) ) ) ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite1481642319od_a_a @ B2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_221_card__le__if__inj__on__rel,axiom,
% 3.21/1.62 ! [B2: set_nat,A2: set_Product_prod_a_a,R: product_prod_a_a > nat > $o] :
% 3.21/1.62 ( ( finite_finite_nat @ B2 )
% 3.21/1.62 => ( ! [A4: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ A4 @ A2 )
% 3.21/1.62 => ? [B5: nat] :
% 3.21/1.62 ( ( member_nat @ B5 @ B2 )
% 3.21/1.62 & ( R @ A4 @ B5 ) ) )
% 3.21/1.62 => ( ! [A1: product_prod_a_a,A22: product_prod_a_a,B6: nat] :
% 3.21/1.62 ( ( member449909584od_a_a @ A1 @ A2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ A22 @ A2 )
% 3.21/1.62 => ( ( member_nat @ B6 @ B2 )
% 3.21/1.62 => ( ( R @ A1 @ B6 )
% 3.21/1.62 => ( ( R @ A22 @ B6 )
% 3.21/1.62 => ( A1 = A22 ) ) ) ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite1481642319od_a_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_222_card__le__if__inj__on__rel,axiom,
% 3.21/1.62 ! [B2: set_nat,A2: set_nat,R: nat > nat > $o] :
% 3.21/1.62 ( ( finite_finite_nat @ B2 )
% 3.21/1.62 => ( ! [A4: nat] :
% 3.21/1.62 ( ( member_nat @ A4 @ A2 )
% 3.21/1.62 => ? [B5: nat] :
% 3.21/1.62 ( ( member_nat @ B5 @ B2 )
% 3.21/1.62 & ( R @ A4 @ B5 ) ) )
% 3.21/1.62 => ( ! [A1: nat,A22: nat,B6: nat] :
% 3.21/1.62 ( ( member_nat @ A1 @ A2 )
% 3.21/1.62 => ( ( member_nat @ A22 @ A2 )
% 3.21/1.62 => ( ( member_nat @ B6 @ B2 )
% 3.21/1.62 => ( ( R @ A1 @ B6 )
% 3.21/1.62 => ( ( R @ A22 @ B6 )
% 3.21/1.62 => ( A1 = A22 ) ) ) ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_223_card__le__if__inj__on__rel,axiom,
% 3.21/1.62 ! [B2: set_nat,A2: set_a,R: a > nat > $o] :
% 3.21/1.62 ( ( finite_finite_nat @ B2 )
% 3.21/1.62 => ( ! [A4: a] :
% 3.21/1.62 ( ( member_a @ A4 @ A2 )
% 3.21/1.62 => ? [B5: nat] :
% 3.21/1.62 ( ( member_nat @ B5 @ B2 )
% 3.21/1.62 & ( R @ A4 @ B5 ) ) )
% 3.21/1.62 => ( ! [A1: a,A22: a,B6: nat] :
% 3.21/1.62 ( ( member_a @ A1 @ A2 )
% 3.21/1.62 => ( ( member_a @ A22 @ A2 )
% 3.21/1.62 => ( ( member_nat @ B6 @ B2 )
% 3.21/1.62 => ( ( R @ A1 @ B6 )
% 3.21/1.62 => ( ( R @ A22 @ B6 )
% 3.21/1.62 => ( A1 = A22 ) ) ) ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_224_ex__card,axiom,
% 3.21/1.62 ! [N: nat,A2: set_a] :
% 3.21/1.62 ( ( ord_less_eq_nat @ N @ ( finite_card_a @ A2 ) )
% 3.21/1.62 => ? [S2: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ S2 @ A2 )
% 3.21/1.62 & ( ( finite_card_a @ S2 )
% 3.21/1.62 = N ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_225_old_Oprod_Oinject,axiom,
% 3.21/1.62 ! [A: a,B: a,A5: a,B7: a] :
% 3.21/1.62 ( ( ( product_Pair_a_a @ A @ B )
% 3.21/1.62 = ( product_Pair_a_a @ A5 @ B7 ) )
% 3.21/1.62 = ( ( A = A5 )
% 3.21/1.62 & ( B = B7 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_226_prod_Oinject,axiom,
% 3.21/1.62 ! [X1: a,X22: a,Y1: a,Y22: a] :
% 3.21/1.62 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 3.21/1.62 = ( product_Pair_a_a @ Y1 @ Y22 ) )
% 3.21/1.62 = ( ( X1 = Y1 )
% 3.21/1.62 & ( X22 = Y22 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_227_order__refl,axiom,
% 3.21/1.62 ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_228_order__refl,axiom,
% 3.21/1.62 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_229_bot__set__def,axiom,
% 3.21/1.62 ( bot_bot_set_a
% 3.21/1.62 = ( collect_a @ bot_bot_a_o ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_230_dual__order_Oantisym,axiom,
% 3.21/1.62 ! [B: set_a,A: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ B @ A )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ A @ B )
% 3.21/1.62 => ( A = B ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_231_dual__order_Oantisym,axiom,
% 3.21/1.62 ! [B: nat,A: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ B @ A )
% 3.21/1.62 => ( ( ord_less_eq_nat @ A @ B )
% 3.21/1.62 => ( A = B ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_232_dual__order_Oeq__iff,axiom,
% 3.21/1.62 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 3.21/1.62 = ( ^ [A6: set_a,B8: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ B8 @ A6 )
% 3.21/1.62 & ( ord_less_eq_set_a @ A6 @ B8 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_233_dual__order_Oeq__iff,axiom,
% 3.21/1.62 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 3.21/1.62 = ( ^ [A6: nat,B8: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ B8 @ A6 )
% 3.21/1.62 & ( ord_less_eq_nat @ A6 @ B8 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_234_dual__order_Otrans,axiom,
% 3.21/1.62 ! [B: set_a,A: set_a,C: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ B @ A )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ C @ B )
% 3.21/1.62 => ( ord_less_eq_set_a @ C @ A ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_235_dual__order_Otrans,axiom,
% 3.21/1.62 ! [B: nat,A: nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ B @ A )
% 3.21/1.62 => ( ( ord_less_eq_nat @ C @ B )
% 3.21/1.62 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_236_linorder__wlog,axiom,
% 3.21/1.62 ! [P: nat > nat > $o,A: nat,B: nat] :
% 3.21/1.62 ( ! [A4: nat,B6: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A4 @ B6 )
% 3.21/1.62 => ( P @ A4 @ B6 ) )
% 3.21/1.62 => ( ! [A4: nat,B6: nat] :
% 3.21/1.62 ( ( P @ B6 @ A4 )
% 3.21/1.62 => ( P @ A4 @ B6 ) )
% 3.21/1.62 => ( P @ A @ B ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_237_dual__order_Orefl,axiom,
% 3.21/1.62 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_238_dual__order_Orefl,axiom,
% 3.21/1.62 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_239_order__trans,axiom,
% 3.21/1.62 ! [X: set_a,Y: set_a,Z: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X @ Y )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ Y @ Z )
% 3.21/1.62 => ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_240_order__trans,axiom,
% 3.21/1.62 ! [X: nat,Y: nat,Z: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X @ Y )
% 3.21/1.62 => ( ( ord_less_eq_nat @ Y @ Z )
% 3.21/1.62 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_241_order__class_Oorder_Oantisym,axiom,
% 3.21/1.62 ! [A: set_a,B: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ B )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ B @ A )
% 3.21/1.62 => ( A = B ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_242_order__class_Oorder_Oantisym,axiom,
% 3.21/1.62 ! [A: nat,B: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ B )
% 3.21/1.62 => ( ( ord_less_eq_nat @ B @ A )
% 3.21/1.62 => ( A = B ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_243_ord__le__eq__trans,axiom,
% 3.21/1.62 ! [A: set_a,B: set_a,C: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ B )
% 3.21/1.62 => ( ( B = C )
% 3.21/1.62 => ( ord_less_eq_set_a @ A @ C ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_244_ord__le__eq__trans,axiom,
% 3.21/1.62 ! [A: nat,B: nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ B )
% 3.21/1.62 => ( ( B = C )
% 3.21/1.62 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_245_ord__eq__le__trans,axiom,
% 3.21/1.62 ! [A: set_a,B: set_a,C: set_a] :
% 3.21/1.62 ( ( A = B )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ B @ C )
% 3.21/1.62 => ( ord_less_eq_set_a @ A @ C ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_246_ord__eq__le__trans,axiom,
% 3.21/1.62 ! [A: nat,B: nat,C: nat] :
% 3.21/1.62 ( ( A = B )
% 3.21/1.62 => ( ( ord_less_eq_nat @ B @ C )
% 3.21/1.62 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_247_order__class_Oorder_Oeq__iff,axiom,
% 3.21/1.62 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 3.21/1.62 = ( ^ [A6: set_a,B8: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A6 @ B8 )
% 3.21/1.62 & ( ord_less_eq_set_a @ B8 @ A6 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_248_order__class_Oorder_Oeq__iff,axiom,
% 3.21/1.62 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 3.21/1.62 = ( ^ [A6: nat,B8: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A6 @ B8 )
% 3.21/1.62 & ( ord_less_eq_nat @ B8 @ A6 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_249_antisym__conv,axiom,
% 3.21/1.62 ! [Y: set_a,X: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ Y @ X )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ X @ Y )
% 3.21/1.62 = ( X = Y ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_250_antisym__conv,axiom,
% 3.21/1.62 ! [Y: nat,X: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ Y @ X )
% 3.21/1.62 => ( ( ord_less_eq_nat @ X @ Y )
% 3.21/1.62 = ( X = Y ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_251_le__cases3,axiom,
% 3.21/1.62 ! [X: nat,Y: nat,Z: nat] :
% 3.21/1.62 ( ( ( ord_less_eq_nat @ X @ Y )
% 3.21/1.62 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 3.21/1.62 => ( ( ( ord_less_eq_nat @ Y @ X )
% 3.21/1.62 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 3.21/1.62 => ( ( ( ord_less_eq_nat @ X @ Z )
% 3.21/1.62 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 3.21/1.62 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 3.21/1.62 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 3.21/1.62 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 3.21/1.62 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 3.21/1.62 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 3.21/1.62 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_252_order_Otrans,axiom,
% 3.21/1.62 ! [A: set_a,B: set_a,C: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ B )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ B @ C )
% 3.21/1.62 => ( ord_less_eq_set_a @ A @ C ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_253_order_Otrans,axiom,
% 3.21/1.62 ! [A: nat,B: nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ B )
% 3.21/1.62 => ( ( ord_less_eq_nat @ B @ C )
% 3.21/1.62 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_254_le__cases,axiom,
% 3.21/1.62 ! [X: nat,Y: nat] :
% 3.21/1.62 ( ~ ( ord_less_eq_nat @ X @ Y )
% 3.21/1.62 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_255_eq__refl,axiom,
% 3.21/1.62 ! [X: set_a,Y: set_a] :
% 3.21/1.62 ( ( X = Y )
% 3.21/1.62 => ( ord_less_eq_set_a @ X @ Y ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_256_eq__refl,axiom,
% 3.21/1.62 ! [X: nat,Y: nat] :
% 3.21/1.62 ( ( X = Y )
% 3.21/1.62 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_257_linear,axiom,
% 3.21/1.62 ! [X: nat,Y: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X @ Y )
% 3.21/1.62 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_258_antisym,axiom,
% 3.21/1.62 ! [X: set_a,Y: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X @ Y )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ Y @ X )
% 3.21/1.62 => ( X = Y ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_259_antisym,axiom,
% 3.21/1.62 ! [X: nat,Y: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X @ Y )
% 3.21/1.62 => ( ( ord_less_eq_nat @ Y @ X )
% 3.21/1.62 => ( X = Y ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_260_eq__iff,axiom,
% 3.21/1.62 ( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
% 3.21/1.62 = ( ^ [X3: set_a,Y2: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X3 @ Y2 )
% 3.21/1.62 & ( ord_less_eq_set_a @ Y2 @ X3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_261_eq__iff,axiom,
% 3.21/1.62 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 3.21/1.62 = ( ^ [X3: nat,Y2: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 3.21/1.62 & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_262_ord__le__eq__subst,axiom,
% 3.21/1.62 ! [A: set_a,B: set_a,F2: set_a > set_a,C: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ B )
% 3.21/1.62 => ( ( ( F2 @ B )
% 3.21/1.62 = C )
% 3.21/1.62 => ( ! [X4: set_a,Y3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_263_ord__le__eq__subst,axiom,
% 3.21/1.62 ! [A: set_a,B: set_a,F2: set_a > nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ B )
% 3.21/1.62 => ( ( ( F2 @ B )
% 3.21/1.62 = C )
% 3.21/1.62 => ( ! [X4: set_a,Y3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_264_ord__le__eq__subst,axiom,
% 3.21/1.62 ! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ B )
% 3.21/1.62 => ( ( ( F2 @ B )
% 3.21/1.62 = C )
% 3.21/1.62 => ( ! [X4: nat,Y3: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_265_ord__le__eq__subst,axiom,
% 3.21/1.62 ! [A: nat,B: nat,F2: nat > nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ B )
% 3.21/1.62 => ( ( ( F2 @ B )
% 3.21/1.62 = C )
% 3.21/1.62 => ( ! [X4: nat,Y3: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_266_ord__eq__le__subst,axiom,
% 3.21/1.62 ! [A: set_a,F2: set_a > set_a,B: set_a,C: set_a] :
% 3.21/1.62 ( ( A
% 3.21/1.62 = ( F2 @ B ) )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ B @ C )
% 3.21/1.62 => ( ! [X4: set_a,Y3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_267_ord__eq__le__subst,axiom,
% 3.21/1.62 ! [A: nat,F2: set_a > nat,B: set_a,C: set_a] :
% 3.21/1.62 ( ( A
% 3.21/1.62 = ( F2 @ B ) )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ B @ C )
% 3.21/1.62 => ( ! [X4: set_a,Y3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_268_ord__eq__le__subst,axiom,
% 3.21/1.62 ! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
% 3.21/1.62 ( ( A
% 3.21/1.62 = ( F2 @ B ) )
% 3.21/1.62 => ( ( ord_less_eq_nat @ B @ C )
% 3.21/1.62 => ( ! [X4: nat,Y3: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_269_ord__eq__le__subst,axiom,
% 3.21/1.62 ! [A: nat,F2: nat > nat,B: nat,C: nat] :
% 3.21/1.62 ( ( A
% 3.21/1.62 = ( F2 @ B ) )
% 3.21/1.62 => ( ( ord_less_eq_nat @ B @ C )
% 3.21/1.62 => ( ! [X4: nat,Y3: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_270_order__subst2,axiom,
% 3.21/1.62 ! [A: set_a,B: set_a,F2: set_a > set_a,C: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ B )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C )
% 3.21/1.62 => ( ! [X4: set_a,Y3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_271_order__subst2,axiom,
% 3.21/1.62 ! [A: set_a,B: set_a,F2: set_a > nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ B )
% 3.21/1.62 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 3.21/1.62 => ( ! [X4: set_a,Y3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_272_order__subst2,axiom,
% 3.21/1.62 ! [A: nat,B: nat,F2: nat > set_a,C: set_a] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ B )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ ( F2 @ B ) @ C )
% 3.21/1.62 => ( ! [X4: nat,Y3: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_273_order__subst2,axiom,
% 3.21/1.62 ! [A: nat,B: nat,F2: nat > nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ B )
% 3.21/1.62 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 3.21/1.62 => ( ! [X4: nat,Y3: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_274_order__subst1,axiom,
% 3.21/1.62 ! [A: set_a,F2: set_a > set_a,B: set_a,C: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ B @ C )
% 3.21/1.62 => ( ! [X4: set_a,Y3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_275_order__subst1,axiom,
% 3.21/1.62 ! [A: set_a,F2: nat > set_a,B: nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
% 3.21/1.62 => ( ( ord_less_eq_nat @ B @ C )
% 3.21/1.62 => ( ! [X4: nat,Y3: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_set_a @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_276_order__subst1,axiom,
% 3.21/1.62 ! [A: nat,F2: set_a > nat,B: set_a,C: set_a] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 3.21/1.62 => ( ( ord_less_eq_set_a @ B @ C )
% 3.21/1.62 => ( ! [X4: set_a,Y3: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_277_order__subst1,axiom,
% 3.21/1.62 ! [A: nat,F2: nat > nat,B: nat,C: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 3.21/1.62 => ( ( ord_less_eq_nat @ B @ C )
% 3.21/1.62 => ( ! [X4: nat,Y3: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ X4 @ Y3 )
% 3.21/1.62 => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 3.21/1.62 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_278_surj__pair,axiom,
% 3.21/1.62 ! [P2: product_prod_a_a] :
% 3.21/1.62 ? [X4: a,Y3: a] :
% 3.21/1.62 ( P2
% 3.21/1.62 = ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_279_prod__cases,axiom,
% 3.21/1.62 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 3.21/1.62 ( ! [A4: a,B6: a] : ( P @ ( product_Pair_a_a @ A4 @ B6 ) )
% 3.21/1.62 => ( P @ P2 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_280_Pair__inject,axiom,
% 3.21/1.62 ! [A: a,B: a,A5: a,B7: a] :
% 3.21/1.62 ( ( ( product_Pair_a_a @ A @ B )
% 3.21/1.62 = ( product_Pair_a_a @ A5 @ B7 ) )
% 3.21/1.62 => ~ ( ( A = A5 )
% 3.21/1.62 => ( B != B7 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_281_old_Oprod_Oexhaust,axiom,
% 3.21/1.62 ! [Y: product_prod_a_a] :
% 3.21/1.62 ~ ! [A4: a,B6: a] :
% 3.21/1.62 ( Y
% 3.21/1.62 != ( product_Pair_a_a @ A4 @ B6 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_282_old_Oprod_Oinducts,axiom,
% 3.21/1.62 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 3.21/1.62 ( ! [A4: a,B6: a] : ( P @ ( product_Pair_a_a @ A4 @ B6 ) )
% 3.21/1.62 => ( P @ Prod ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_283_bounded__Max__nat,axiom,
% 3.21/1.62 ! [P: nat > $o,X: nat,M: nat] :
% 3.21/1.62 ( ( P @ X )
% 3.21/1.62 => ( ! [X4: nat] :
% 3.21/1.62 ( ( P @ X4 )
% 3.21/1.62 => ( ord_less_eq_nat @ X4 @ M ) )
% 3.21/1.62 => ~ ! [M2: nat] :
% 3.21/1.62 ( ( P @ M2 )
% 3.21/1.62 => ~ ! [X2: nat] :
% 3.21/1.62 ( ( P @ X2 )
% 3.21/1.62 => ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_284_finite__nat__set__iff__bounded__le,axiom,
% 3.21/1.62 ( finite_finite_nat
% 3.21/1.62 = ( ^ [N2: set_nat] :
% 3.21/1.62 ? [M3: nat] :
% 3.21/1.62 ! [X3: nat] :
% 3.21/1.62 ( ( member_nat @ X3 @ N2 )
% 3.21/1.62 => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_285_bot_Oextremum__uniqueI,axiom,
% 3.21/1.62 ! [A: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 3.21/1.62 => ( A = bot_bot_set_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_286_bot_Oextremum__uniqueI,axiom,
% 3.21/1.62 ! [A: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 3.21/1.62 => ( A = bot_bot_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_287_bot_Oextremum__unique,axiom,
% 3.21/1.62 ! [A: set_a] :
% 3.21/1.62 ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
% 3.21/1.62 = ( A = bot_bot_set_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_288_bot_Oextremum__unique,axiom,
% 3.21/1.62 ! [A: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 3.21/1.62 = ( A = bot_bot_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_289_bot_Oextremum,axiom,
% 3.21/1.62 ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_290_bot_Oextremum,axiom,
% 3.21/1.62 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_291_preorder__on__empty,axiom,
% 3.21/1.62 order_preorder_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 3.21/1.62
% 3.21/1.62 thf(fact_292_total__onI,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a] :
% 3.21/1.62 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ X4 @ A2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ Y3 @ A2 )
% 3.21/1.62 => ( ( X4 != Y3 )
% 3.21/1.62 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ R )
% 3.21/1.62 | ( member2057358096od_a_a @ ( produc1474507607od_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 3.21/1.62 => ( total_1490170027od_a_a @ A2 @ R ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_293_total__onI,axiom,
% 3.21/1.62 ! [A2: set_nat,R: set_Pr1986765409at_nat] :
% 3.21/1.62 ( ! [X4: nat,Y3: nat] :
% 3.21/1.62 ( ( member_nat @ X4 @ A2 )
% 3.21/1.62 => ( ( member_nat @ Y3 @ A2 )
% 3.21/1.62 => ( ( X4 != Y3 )
% 3.21/1.62 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
% 3.21/1.62 | ( member701585322at_nat @ ( product_Pair_nat_nat @ Y3 @ X4 ) @ R ) ) ) ) )
% 3.21/1.62 => ( total_on_nat @ A2 @ R ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_294_total__onI,axiom,
% 3.21/1.62 ! [A2: set_a,R: set_Product_prod_a_a] :
% 3.21/1.62 ( ! [X4: a,Y3: a] :
% 3.21/1.62 ( ( member_a @ X4 @ A2 )
% 3.21/1.62 => ( ( member_a @ Y3 @ A2 )
% 3.21/1.62 => ( ( X4 != Y3 )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 3.21/1.62 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R ) ) ) ) )
% 3.21/1.62 => ( total_on_a @ A2 @ R ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_295_total__on__def,axiom,
% 3.21/1.62 ( total_on_a
% 3.21/1.62 = ( ^ [A3: set_a,R2: set_Product_prod_a_a] :
% 3.21/1.62 ! [X3: a] :
% 3.21/1.62 ( ( member_a @ X3 @ A3 )
% 3.21/1.62 => ! [Y2: a] :
% 3.21/1.62 ( ( member_a @ Y2 @ A3 )
% 3.21/1.62 => ( ( X3 != Y2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
% 3.21/1.62 | ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R2 ) ) ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_296_bot__empty__eq,axiom,
% 3.21/1.62 ( bot_bo1293121322_a_a_o
% 3.21/1.62 = ( ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ bot_bo2131659635od_a_a ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_297_bot__empty__eq,axiom,
% 3.21/1.62 ( bot_bot_nat_o
% 3.21/1.62 = ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_298_bot__empty__eq,axiom,
% 3.21/1.62 ( bot_bot_a_o
% 3.21/1.62 = ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_299_subrelI,axiom,
% 3.21/1.62 ! [R: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
% 3.21/1.62 ( ! [X4: a,Y3: a] :
% 3.21/1.62 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 3.21/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S3 ) )
% 3.21/1.62 => ( ord_le1824328871od_a_a @ R @ S3 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_300_refl__on__empty,axiom,
% 3.21/1.62 refl_on_a @ bot_bot_set_a @ bot_bo2131659635od_a_a ).
% 3.21/1.62
% 3.21/1.62 thf(fact_301_refl__on__domain,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a,B: product_prod_a_a] :
% 3.21/1.62 ( ( refl_o1298442278od_a_a @ A2 @ R )
% 3.21/1.62 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ B ) @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ A @ A2 )
% 3.21/1.62 & ( member449909584od_a_a @ B @ A2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_302_refl__on__domain,axiom,
% 3.21/1.62 ! [A2: set_nat,R: set_Pr1986765409at_nat,A: nat,B: nat] :
% 3.21/1.62 ( ( refl_on_nat @ A2 @ R )
% 3.21/1.62 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
% 3.21/1.62 => ( ( member_nat @ A @ A2 )
% 3.21/1.62 & ( member_nat @ B @ A2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_303_refl__on__domain,axiom,
% 3.21/1.62 ! [A2: set_a,R: set_Product_prod_a_a,A: a,B: a] :
% 3.21/1.62 ( ( refl_on_a @ A2 @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
% 3.21/1.62 => ( ( member_a @ A @ A2 )
% 3.21/1.62 & ( member_a @ B @ A2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_304_refl__onD2,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.62 ( ( refl_o1298442278od_a_a @ A2 @ R )
% 3.21/1.62 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R )
% 3.21/1.62 => ( member449909584od_a_a @ Y @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_305_refl__onD2,axiom,
% 3.21/1.62 ! [A2: set_nat,R: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.62 ( ( refl_on_nat @ A2 @ R )
% 3.21/1.62 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 3.21/1.62 => ( member_nat @ Y @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_306_refl__onD2,axiom,
% 3.21/1.62 ! [A2: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.62 ( ( refl_on_a @ A2 @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 3.21/1.62 => ( member_a @ Y @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_307_refl__onD1,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,X: product_prod_a_a,Y: product_prod_a_a] :
% 3.21/1.62 ( ( refl_o1298442278od_a_a @ A2 @ R )
% 3.21/1.62 => ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X @ Y ) @ R )
% 3.21/1.62 => ( member449909584od_a_a @ X @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_308_refl__onD1,axiom,
% 3.21/1.62 ! [A2: set_nat,R: set_Pr1986765409at_nat,X: nat,Y: nat] :
% 3.21/1.62 ( ( refl_on_nat @ A2 @ R )
% 3.21/1.62 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 3.21/1.62 => ( member_nat @ X @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_309_refl__onD1,axiom,
% 3.21/1.62 ! [A2: set_a,R: set_Product_prod_a_a,X: a,Y: a] :
% 3.21/1.62 ( ( refl_on_a @ A2 @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 3.21/1.62 => ( member_a @ X @ A2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_310_refl__onD,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,R: set_Pr1948701895od_a_a,A: product_prod_a_a] :
% 3.21/1.62 ( ( refl_o1298442278od_a_a @ A2 @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ A @ A2 )
% 3.21/1.62 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ A @ A ) @ R ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_311_refl__onD,axiom,
% 3.21/1.62 ! [A2: set_nat,R: set_Pr1986765409at_nat,A: nat] :
% 3.21/1.62 ( ( refl_on_nat @ A2 @ R )
% 3.21/1.62 => ( ( member_nat @ A @ A2 )
% 3.21/1.62 => ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_312_refl__onD,axiom,
% 3.21/1.62 ! [A2: set_a,R: set_Product_prod_a_a,A: a] :
% 3.21/1.62 ( ( refl_on_a @ A2 @ R )
% 3.21/1.62 => ( ( member_a @ A @ A2 )
% 3.21/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_313_total__on__empty,axiom,
% 3.21/1.62 ! [R: set_Product_prod_a_a] : ( total_on_a @ bot_bot_set_a @ R ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_314_Collect__empty__eq__bot,axiom,
% 3.21/1.62 ! [P: a > $o] :
% 3.21/1.62 ( ( ( collect_a @ P )
% 3.21/1.62 = bot_bot_set_a )
% 3.21/1.62 = ( P = bot_bot_a_o ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_315_infinite__nat__iff__unbounded__le,axiom,
% 3.21/1.62 ! [S: set_nat] :
% 3.21/1.62 ( ( ~ ( finite_finite_nat @ S ) )
% 3.21/1.62 = ( ! [M3: nat] :
% 3.21/1.62 ? [N3: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ M3 @ N3 )
% 3.21/1.62 & ( member_nat @ N3 @ S ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_316_finite__transitivity__chain,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,R3: product_prod_a_a > product_prod_a_a > $o] :
% 3.21/1.62 ( ( finite179568208od_a_a @ A2 )
% 3.21/1.62 => ( ! [X4: product_prod_a_a] :
% 3.21/1.62 ~ ( R3 @ X4 @ X4 )
% 3.21/1.62 => ( ! [X4: product_prod_a_a,Y3: product_prod_a_a,Z3: product_prod_a_a] :
% 3.21/1.62 ( ( R3 @ X4 @ Y3 )
% 3.21/1.62 => ( ( R3 @ Y3 @ Z3 )
% 3.21/1.62 => ( R3 @ X4 @ Z3 ) ) )
% 3.21/1.62 => ( ! [X4: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ X4 @ A2 )
% 3.21/1.62 => ? [Y6: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ Y6 @ A2 )
% 3.21/1.62 & ( R3 @ X4 @ Y6 ) ) )
% 3.21/1.62 => ( A2 = bot_bo2131659635od_a_a ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_317_finite__transitivity__chain,axiom,
% 3.21/1.62 ! [A2: set_nat,R3: nat > nat > $o] :
% 3.21/1.62 ( ( finite_finite_nat @ A2 )
% 3.21/1.62 => ( ! [X4: nat] :
% 3.21/1.62 ~ ( R3 @ X4 @ X4 )
% 3.21/1.62 => ( ! [X4: nat,Y3: nat,Z3: nat] :
% 3.21/1.62 ( ( R3 @ X4 @ Y3 )
% 3.21/1.62 => ( ( R3 @ Y3 @ Z3 )
% 3.21/1.62 => ( R3 @ X4 @ Z3 ) ) )
% 3.21/1.62 => ( ! [X4: nat] :
% 3.21/1.62 ( ( member_nat @ X4 @ A2 )
% 3.21/1.62 => ? [Y6: nat] :
% 3.21/1.62 ( ( member_nat @ Y6 @ A2 )
% 3.21/1.62 & ( R3 @ X4 @ Y6 ) ) )
% 3.21/1.62 => ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_318_finite__transitivity__chain,axiom,
% 3.21/1.62 ! [A2: set_a,R3: a > a > $o] :
% 3.21/1.62 ( ( finite_finite_a @ A2 )
% 3.21/1.62 => ( ! [X4: a] :
% 3.21/1.62 ~ ( R3 @ X4 @ X4 )
% 3.21/1.62 => ( ! [X4: a,Y3: a,Z3: a] :
% 3.21/1.62 ( ( R3 @ X4 @ Y3 )
% 3.21/1.62 => ( ( R3 @ Y3 @ Z3 )
% 3.21/1.62 => ( R3 @ X4 @ Z3 ) ) )
% 3.21/1.62 => ( ! [X4: a] :
% 3.21/1.62 ( ( member_a @ X4 @ A2 )
% 3.21/1.62 => ? [Y6: a] :
% 3.21/1.62 ( ( member_a @ Y6 @ A2 )
% 3.21/1.62 & ( R3 @ X4 @ Y6 ) ) )
% 3.21/1.62 => ( A2 = bot_bot_set_a ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_319_subset__emptyI,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a] :
% 3.21/1.62 ( ! [X4: product_prod_a_a] :
% 3.21/1.62 ~ ( member449909584od_a_a @ X4 @ A2 )
% 3.21/1.62 => ( ord_le1824328871od_a_a @ A2 @ bot_bo2131659635od_a_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_320_subset__emptyI,axiom,
% 3.21/1.62 ! [A2: set_nat] :
% 3.21/1.62 ( ! [X4: nat] :
% 3.21/1.62 ~ ( member_nat @ X4 @ A2 )
% 3.21/1.62 => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_321_subset__emptyI,axiom,
% 3.21/1.62 ! [A2: set_a] :
% 3.21/1.62 ( ! [X4: a] :
% 3.21/1.62 ~ ( member_a @ X4 @ A2 )
% 3.21/1.62 => ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_322_rational__preference__def,axiom,
% 3.21/1.62 ( prefer719835986ence_a
% 3.21/1.62 = ( ^ [Carrier2: set_a,Relation2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( prefer1292084992ence_a @ Carrier2 @ Relation2 )
% 3.21/1.62 & ( prefer580340847ioms_a @ Carrier2 @ Relation2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_323_ssubst__Pair__rhs,axiom,
% 3.21/1.62 ! [R: a,S3: a,R3: set_Product_prod_a_a,S4: a] :
% 3.21/1.62 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S3 ) @ R3 )
% 3.21/1.62 => ( ( S4 = S3 )
% 3.21/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S4 ) @ R3 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_324_rational__preference__axioms_Ointro,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.62 ( ( total_on_a @ Carrier @ Relation )
% 3.21/1.62 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_325_rational__preference__axioms__def,axiom,
% 3.21/1.62 prefer580340847ioms_a = total_on_a ).
% 3.21/1.62
% 3.21/1.62 thf(fact_326_rational__preference_Oaxioms_I2_J,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.62 ( ( prefer719835986ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( prefer580340847ioms_a @ Carrier @ Relation ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_327_rational__preference_Ointro,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.62 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( ( prefer580340847ioms_a @ Carrier @ Relation )
% 3.21/1.62 => ( prefer719835986ence_a @ Carrier @ Relation ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_328_finite__indexed__bound,axiom,
% 3.21/1.62 ! [A2: set_a,P: a > nat > $o] :
% 3.21/1.62 ( ( finite_finite_a @ A2 )
% 3.21/1.62 => ( ! [X4: a] :
% 3.21/1.62 ( ( member_a @ X4 @ A2 )
% 3.21/1.62 => ? [X_1: nat] : ( P @ X4 @ X_1 ) )
% 3.21/1.62 => ? [M2: nat] :
% 3.21/1.62 ! [X2: a] :
% 3.21/1.62 ( ( member_a @ X2 @ A2 )
% 3.21/1.62 => ? [K: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ K @ M2 )
% 3.21/1.62 & ( P @ X2 @ K ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_329_finite__indexed__bound,axiom,
% 3.21/1.62 ! [A2: set_Product_prod_a_a,P: product_prod_a_a > nat > $o] :
% 3.21/1.62 ( ( finite179568208od_a_a @ A2 )
% 3.21/1.62 => ( ! [X4: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ X4 @ A2 )
% 3.21/1.62 => ? [X_1: nat] : ( P @ X4 @ X_1 ) )
% 3.21/1.62 => ? [M2: nat] :
% 3.21/1.62 ! [X2: product_prod_a_a] :
% 3.21/1.62 ( ( member449909584od_a_a @ X2 @ A2 )
% 3.21/1.62 => ? [K: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ K @ M2 )
% 3.21/1.62 & ( P @ X2 @ K ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_330_finite__indexed__bound,axiom,
% 3.21/1.62 ! [A2: set_nat,P: nat > nat > $o] :
% 3.21/1.62 ( ( finite_finite_nat @ A2 )
% 3.21/1.62 => ( ! [X4: nat] :
% 3.21/1.62 ( ( member_nat @ X4 @ A2 )
% 3.21/1.62 => ? [X_1: nat] : ( P @ X4 @ X_1 ) )
% 3.21/1.62 => ? [M2: nat] :
% 3.21/1.62 ! [X2: nat] :
% 3.21/1.62 ( ( member_nat @ X2 @ A2 )
% 3.21/1.62 => ? [K: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ K @ M2 )
% 3.21/1.62 & ( P @ X2 @ K ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_331_transitivity,axiom,
% 3.21/1.62 trans_a @ relation ).
% 3.21/1.62
% 3.21/1.62 thf(fact_332_trans__empty,axiom,
% 3.21/1.62 trans_a @ bot_bo2131659635od_a_a ).
% 3.21/1.62
% 3.21/1.62 thf(fact_333_transD,axiom,
% 3.21/1.62 ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 3.21/1.62 ( ( trans_a @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 3.21/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_334_transE,axiom,
% 3.21/1.62 ! [R: set_Product_prod_a_a,X: a,Y: a,Z: a] :
% 3.21/1.62 ( ( trans_a @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R )
% 3.21/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_335_transI,axiom,
% 3.21/1.62 ! [R: set_Product_prod_a_a] :
% 3.21/1.62 ( ! [X4: a,Y3: a,Z3: a] :
% 3.21/1.62 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y3 @ Z3 ) @ R )
% 3.21/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Z3 ) @ R ) ) )
% 3.21/1.62 => ( trans_a @ R ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_336_trans__def,axiom,
% 3.21/1.62 ( trans_a
% 3.21/1.62 = ( ^ [R2: set_Product_prod_a_a] :
% 3.21/1.62 ! [X3: a,Y2: a,Z4: a] :
% 3.21/1.62 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
% 3.21/1.62 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y2 @ Z4 ) @ R2 )
% 3.21/1.62 => ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R2 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_337_preference_Otransitivity,axiom,
% 3.21/1.62 ! [Carrier: set_a,Relation: set_Product_prod_a_a] :
% 3.21/1.62 ( ( prefer1292084992ence_a @ Carrier @ Relation )
% 3.21/1.62 => ( trans_a @ Relation ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_338_preorder__on__def,axiom,
% 3.21/1.62 ( order_preorder_on_a
% 3.21/1.62 = ( ^ [A3: set_a,R2: set_Product_prod_a_a] :
% 3.21/1.62 ( ( refl_on_a @ A3 @ R2 )
% 3.21/1.62 & ( trans_a @ R2 ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_339_Set_Ois__empty__def,axiom,
% 3.21/1.62 ( is_empty_a
% 3.21/1.62 = ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_340_card__0__eq,axiom,
% 3.21/1.62 ! [A2: set_a] :
% 3.21/1.62 ( ( finite_finite_a @ A2 )
% 3.21/1.62 => ( ( ( finite_card_a @ A2 )
% 3.21/1.62 = zero_zero_nat )
% 3.21/1.62 = ( A2 = bot_bot_set_a ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_341_bot__nat__0_Oextremum,axiom,
% 3.21/1.62 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_342_le0,axiom,
% 3.21/1.62 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_343_Nat_Oex__has__greatest__nat,axiom,
% 3.21/1.62 ! [P: nat > $o,K2: nat,B: nat] :
% 3.21/1.62 ( ( P @ K2 )
% 3.21/1.62 => ( ! [Y3: nat] :
% 3.21/1.62 ( ( P @ Y3 )
% 3.21/1.62 => ( ord_less_eq_nat @ Y3 @ B ) )
% 3.21/1.62 => ? [X4: nat] :
% 3.21/1.62 ( ( P @ X4 )
% 3.21/1.62 & ! [Y6: nat] :
% 3.21/1.62 ( ( P @ Y6 )
% 3.21/1.62 => ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_344_nat__le__linear,axiom,
% 3.21/1.62 ! [M4: nat,N: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ M4 @ N )
% 3.21/1.62 | ( ord_less_eq_nat @ N @ M4 ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_345_le__antisym,axiom,
% 3.21/1.62 ! [M4: nat,N: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ M4 @ N )
% 3.21/1.62 => ( ( ord_less_eq_nat @ N @ M4 )
% 3.21/1.62 => ( M4 = N ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_346_eq__imp__le,axiom,
% 3.21/1.62 ! [M4: nat,N: nat] :
% 3.21/1.62 ( ( M4 = N )
% 3.21/1.62 => ( ord_less_eq_nat @ M4 @ N ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_347_le__trans,axiom,
% 3.21/1.62 ! [I: nat,J: nat,K2: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ I @ J )
% 3.21/1.62 => ( ( ord_less_eq_nat @ J @ K2 )
% 3.21/1.62 => ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_348_le__refl,axiom,
% 3.21/1.62 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_349_less__eq__nat_Osimps_I1_J,axiom,
% 3.21/1.62 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_350_le__0__eq,axiom,
% 3.21/1.62 ! [N: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 3.21/1.62 = ( N = zero_zero_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_351_bot__nat__0_Oextremum__unique,axiom,
% 3.21/1.62 ! [A: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 3.21/1.62 = ( A = zero_zero_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_352_bot__nat__0_Oextremum__uniqueI,axiom,
% 3.21/1.62 ! [A: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 3.21/1.62 => ( A = zero_zero_nat ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(fact_353_ex__least__nat__le,axiom,
% 3.21/1.62 ! [P: nat > $o,N: nat] :
% 3.21/1.62 ( ( P @ N )
% 3.21/1.62 => ( ~ ( P @ zero_zero_nat )
% 3.21/1.62 => ? [K: nat] :
% 3.21/1.62 ( ( ord_less_eq_nat @ K @ N )
% 3.21/1.62 & ! [I2: nat] :
% 3.21/1.62 ( ( ord_less_nat @ I2 @ K )
% 3.21/1.62 => ~ ( P @ I2 ) )
% 3.21/1.62 & ( P @ K ) ) ) ) ).
% 3.21/1.62
% 3.21/1.62 thf(conj_0,conjecture,
% 3.21/1.62 ord_less_eq_set_a @ ( prefer1676310729than_a @ x @ carrier @ relation ) @ ( prefer1676310729than_a @ y @ carrier @ relation ) ).
% 3.21/1.62 ------------------------
% 3.21/1.64 % (13199)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/183Mi)
% 3.21/1.64 % (13201)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_13001 for (2999ds/27Mi)
% 3.21/1.64 % (13202)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/2Mi)
% 3.21/1.64 % (13202)Instruction limit reached!
% 3.21/1.64 % (13202)------------------------------
% 3.21/1.64 % (13202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.64 % (13202)Termination reason: Unknown
% 3.21/1.64 % (13202)Termination phase: shuffling
% 3.21/1.64
% 3.21/1.64 % (13202)Memory used [KB]: 1407
% 3.21/1.64 % (13202)Time elapsed: 0.002 s
% 3.21/1.64 % (13202)Instructions burned: 2 (million)
% 3.21/1.64 % (13202)------------------------------
% 3.21/1.64 % (13202)------------------------------
% 3.21/1.64 % (13204)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_13001 for (2999ds/275Mi)
% 3.21/1.64 % (13200)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_13001 for (2999ds/4Mi)
% 3.21/1.65 % (13205)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/18Mi)
% 3.21/1.65 % (13200)Instruction limit reached!
% 3.21/1.65 % (13200)------------------------------
% 3.21/1.65 % (13200)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.65 % (13200)Termination reason: Unknown
% 3.21/1.65 % (13200)Termination phase: shuffling
% 3.21/1.65
% 3.21/1.65 % (13200)Memory used [KB]: 1535
% 3.21/1.65 % (13200)Time elapsed: 0.003 s
% 3.21/1.65 % (13200)Instructions burned: 5 (million)
% 3.21/1.65 % (13200)------------------------------
% 3.21/1.65 % (13200)------------------------------
% 3.21/1.65 % (13203)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_13001 for (2999ds/2Mi)
% 3.21/1.65 % (13203)Instruction limit reached!
% 3.21/1.65 % (13203)------------------------------
% 3.21/1.65 % (13203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.65 % (13203)Termination reason: Unknown
% 3.21/1.65 % (13203)Termination phase: shuffling
% 3.21/1.65
% 3.21/1.65 % (13203)Memory used [KB]: 1407
% 3.21/1.65 % (13203)Time elapsed: 0.003 s
% 3.21/1.65 % (13203)Instructions burned: 2 (million)
% 3.21/1.65 % (13203)------------------------------
% 3.21/1.65 % (13203)------------------------------
% 3.21/1.65 % (13201)Instruction limit reached!
% 3.21/1.65 % (13201)------------------------------
% 3.21/1.65 % (13201)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.65 % (13201)Termination reason: Unknown
% 3.21/1.65 % (13201)Termination phase: shuffling
% 3.21/1.65
% 3.21/1.65 % (13201)Memory used [KB]: 2046
% 3.21/1.65 % (13201)Time elapsed: 0.010 s
% 3.21/1.65 % (13201)Instructions burned: 28 (million)
% 3.21/1.65 % (13201)------------------------------
% 3.21/1.65 % (13201)------------------------------
% 3.21/1.65 % (13205)Instruction limit reached!
% 3.21/1.65 % (13205)------------------------------
% 3.21/1.65 % (13205)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.65 % (13205)Termination reason: Unknown
% 3.21/1.65 % (13205)Termination phase: shuffling
% 3.21/1.65
% 3.21/1.65 % (13205)Memory used [KB]: 1791
% 3.21/1.65 % (13205)Time elapsed: 0.008 s
% 3.21/1.65 % (13205)Instructions burned: 20 (million)
% 3.21/1.65 % (13205)------------------------------
% 3.21/1.65 % (13205)------------------------------
% 3.21/1.65 % (13206)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/3Mi)
% 3.21/1.65 % (13206)Instruction limit reached!
% 3.21/1.65 % (13206)------------------------------
% 3.21/1.65 % (13206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.65 % (13206)Termination reason: Unknown
% 3.21/1.65 % (13206)Termination phase: shuffling
% 3.21/1.65
% 3.21/1.65 % (13206)Memory used [KB]: 1535
% 3.21/1.65 % (13206)Time elapsed: 0.002 s
% 3.21/1.65 % (13206)Instructions burned: 3 (million)
% 3.21/1.65 % (13206)------------------------------
% 3.21/1.65 % (13206)------------------------------
% 3.21/1.66 % (13207)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_13001 for (2999ds/37Mi)
% 3.21/1.66 % (13209)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_13001 for (2999ds/3Mi)
% 3.21/1.66 % (13209)Instruction limit reached!
% 3.21/1.66 % (13209)------------------------------
% 3.21/1.66 % (13209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.66 % (13209)Termination reason: Unknown
% 3.21/1.66 % (13209)Termination phase: shuffling
% 3.21/1.66
% 3.21/1.66 % (13209)Memory used [KB]: 1535
% 3.21/1.66 % (13209)Time elapsed: 0.002 s
% 3.21/1.66 % (13209)Instructions burned: 3 (million)
% 3.21/1.66 % (13209)------------------------------
% 3.21/1.66 % (13209)------------------------------
% 3.21/1.66 % (13210)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_13001 for (2999ds/1041Mi)
% 3.21/1.66 % (13211)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/7Mi)
% 3.21/1.67 % (13211)Instruction limit reached!
% 3.21/1.67 % (13211)------------------------------
% 3.21/1.67 % (13211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.67 % (13211)Termination reason: Unknown
% 3.21/1.67 % (13211)Termination phase: shuffling
% 3.21/1.67
% 3.21/1.67 % (13211)Memory used [KB]: 1663
% 3.21/1.67 % (13211)Time elapsed: 0.004 s
% 3.21/1.67 % (13211)Instructions burned: 10 (million)
% 3.21/1.67 % (13211)------------------------------
% 3.21/1.67 % (13211)------------------------------
% 3.21/1.67 % (13208)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_13001 for (2999ds/15Mi)
% 3.21/1.67 % (13207)Instruction limit reached!
% 3.21/1.67 % (13207)------------------------------
% 3.21/1.67 % (13207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.67 % (13207)Termination reason: Unknown
% 3.21/1.67 % (13207)Termination phase: shuffling
% 3.21/1.67
% 3.21/1.67 % (13207)Memory used [KB]: 2174
% 3.21/1.67 % (13207)Time elapsed: 0.013 s
% 3.21/1.67 % (13207)Instructions burned: 39 (million)
% 3.21/1.67 % (13207)------------------------------
% 3.21/1.67 % (13207)------------------------------
% 3.21/1.67 % (13213)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/16Mi)
% 3.21/1.68 % (13208)Instruction limit reached!
% 3.21/1.68 % (13208)------------------------------
% 3.21/1.68 % (13208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.68 % (13208)Termination reason: Unknown
% 3.21/1.68 % (13208)Termination phase: shuffling
% 3.21/1.68
% 3.21/1.68 % (13208)Memory used [KB]: 1791
% 3.21/1.68 % (13208)Time elapsed: 0.011 s
% 3.21/1.68 % (13208)Instructions burned: 15 (million)
% 3.21/1.68 % (13208)------------------------------
% 3.21/1.68 % (13208)------------------------------
% 3.21/1.68 % (13215)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/3Mi)
% 3.21/1.68 % (13213)Instruction limit reached!
% 3.21/1.68 % (13213)------------------------------
% 3.21/1.68 % (13213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.68 % (13213)Termination reason: Unknown
% 3.21/1.68 % (13213)Termination phase: shuffling
% 3.21/1.68
% 3.21/1.68 % (13213)Memory used [KB]: 1791
% 3.21/1.68 % (13213)Time elapsed: 0.006 s
% 3.21/1.68 % (13213)Instructions burned: 16 (million)
% 3.21/1.68 % (13213)------------------------------
% 3.21/1.68 % (13213)------------------------------
% 3.21/1.68 % (13215)Instruction limit reached!
% 3.21/1.68 % (13215)------------------------------
% 3.21/1.68 % (13215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.68 % (13215)Termination reason: Unknown
% 3.21/1.68 % (13215)Termination phase: shuffling
% 3.21/1.68
% 3.21/1.68 % (13215)Memory used [KB]: 1535
% 3.21/1.68 % (13215)Time elapsed: 0.002 s
% 3.21/1.68 % (13215)Instructions burned: 3 (million)
% 3.21/1.68 % (13215)------------------------------
% 3.21/1.68 % (13215)------------------------------
% 3.21/1.68 % (13216)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/3Mi)
% 3.21/1.68 % (13216)Instruction limit reached!
% 3.21/1.68 % (13216)------------------------------
% 3.21/1.68 % (13216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.68 % (13216)Termination reason: Unknown
% 3.21/1.68 % (13216)Termination phase: shuffling
% 3.21/1.68
% 3.21/1.68 % (13216)Memory used [KB]: 1535
% 3.21/1.68 % (13216)Time elapsed: 0.003 s
% 3.21/1.68 % (13216)Instructions burned: 4 (million)
% 3.21/1.68 % (13216)------------------------------
% 3.21/1.68 % (13216)------------------------------
% 3.21/1.69 % (13218)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/3Mi)
% 3.21/1.69 % (13219)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/4Mi)
% 3.21/1.69 % (13220)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/18Mi)
% 3.21/1.69 % (13218)Instruction limit reached!
% 3.21/1.69 % (13218)------------------------------
% 3.21/1.69 % (13218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.69 % (13218)Termination reason: Unknown
% 3.21/1.69 % (13218)Termination phase: shuffling
% 3.21/1.69
% 3.21/1.69 % (13218)Memory used [KB]: 1407
% 3.21/1.69 % (13218)Time elapsed: 0.003 s
% 3.21/1.69 % (13218)Instructions burned: 3 (million)
% 3.21/1.69 % (13218)------------------------------
% 3.21/1.69 % (13218)------------------------------
% 3.21/1.69 % (13217)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/7Mi)
% 3.21/1.69 % (13219)Instruction limit reached!
% 3.21/1.69 % (13219)------------------------------
% 3.21/1.69 % (13219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.69 % (13219)Termination reason: Unknown
% 3.21/1.69 % (13219)Termination phase: shuffling
% 3.21/1.69
% 3.21/1.69 % (13219)Memory used [KB]: 1535
% 3.21/1.69 % (13219)Time elapsed: 0.004 s
% 3.21/1.69 % (13219)Instructions burned: 4 (million)
% 3.21/1.69 % (13219)------------------------------
% 3.21/1.69 % (13219)------------------------------
% 3.21/1.69 % (13217)Instruction limit reached!
% 3.21/1.69 % (13217)------------------------------
% 3.21/1.69 % (13217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.69 % (13217)Termination reason: Unknown
% 3.21/1.69 % (13217)Termination phase: shuffling
% 3.21/1.69
% 3.21/1.69 % (13217)Memory used [KB]: 1535
% 3.21/1.69 % (13217)Time elapsed: 0.005 s
% 3.21/1.69 % (13217)Instructions burned: 7 (million)
% 3.21/1.69 % (13217)------------------------------
% 3.21/1.69 % (13217)------------------------------
% 3.21/1.70 % (13220)Instruction limit reached!
% 3.21/1.70 % (13220)------------------------------
% 3.21/1.70 % (13220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.70 % (13220)Termination reason: Unknown
% 3.21/1.70 % (13220)Termination phase: shuffling
% 3.21/1.70
% 3.21/1.70 % (13220)Memory used [KB]: 1791
% 3.21/1.70 % (13220)Time elapsed: 0.008 s
% 3.21/1.70 % (13220)Instructions burned: 18 (million)
% 3.21/1.70 % (13220)------------------------------
% 3.21/1.70 % (13220)------------------------------
% 3.23/1.70 % (13199)Instruction limit reached!
% 3.23/1.70 % (13199)------------------------------
% 3.23/1.70 % (13199)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.70 % (13199)Termination reason: Unknown
% 3.23/1.70 % (13199)Termination phase: Saturation
% 3.23/1.70
% 3.23/1.70 % (13199)Memory used [KB]: 7419
% 3.23/1.70 % (13199)Time elapsed: 0.061 s
% 3.23/1.70 % (13199)Instructions burned: 183 (million)
% 3.23/1.70 % (13199)------------------------------
% 3.23/1.70 % (13199)------------------------------
% 3.23/1.70 % (13228)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/710Mi)
% 3.23/1.71 % (13229)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/6Mi)
% 3.23/1.71 % (13231)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/21Mi)
% 3.23/1.71 % (13229)Instruction limit reached!
% 3.23/1.71 % (13229)------------------------------
% 3.23/1.71 % (13229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.71 % (13229)Termination reason: Unknown
% 3.23/1.71 % (13229)Termination phase: shuffling
% 3.23/1.71
% 3.23/1.71 % (13229)Memory used [KB]: 1535
% 3.23/1.71 % (13229)Time elapsed: 0.005 s
% 3.23/1.71 % (13229)Instructions burned: 7 (million)
% 3.23/1.71 % (13229)------------------------------
% 3.23/1.71 % (13229)------------------------------
% 3.23/1.71 % (13230)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/902Mi)
% 3.23/1.72 % (13231)Instruction limit reached!
% 3.23/1.72 % (13231)------------------------------
% 3.23/1.72 % (13231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.72 % (13231)Termination reason: Unknown
% 3.23/1.72 % (13231)Termination phase: shuffling
% 3.23/1.72
% 3.23/1.72 % (13231)Memory used [KB]: 1791
% 3.23/1.72 % (13231)Time elapsed: 0.013 s
% 3.23/1.72 % (13231)Instructions burned: 21 (million)
% 3.23/1.72 % (13231)------------------------------
% 3.23/1.72 % (13231)------------------------------
% 3.23/1.72 % (13235)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/5Mi)
% 3.23/1.72 % (13235)Instruction limit reached!
% 3.23/1.72 % (13235)------------------------------
% 3.23/1.72 % (13235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.72 % (13235)Termination reason: Unknown
% 3.23/1.72 % (13235)Termination phase: shuffling
% 3.23/1.72
% 3.23/1.72 % (13235)Memory used [KB]: 1535
% 3.23/1.72 % (13235)Time elapsed: 0.004 s
% 3.23/1.72 % (13235)Instructions burned: 5 (million)
% 3.23/1.72 % (13235)------------------------------
% 3.23/1.72 % (13235)------------------------------
% 3.23/1.73 % (13237)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13001 for (2999ds/6Mi)
% 3.23/1.73 % (13237)Instruction limit reached!
% 3.23/1.73 % (13237)------------------------------
% 3.23/1.73 % (13237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.73 % (13237)Termination reason: Unknown
% 3.23/1.73 % (13237)Termination phase: shuffling
% 3.23/1.73
% 3.23/1.73 % (13237)Memory used [KB]: 1663
% 3.23/1.73 % (13237)Time elapsed: 0.005 s
% 3.23/1.73 % (13237)Instructions burned: 8 (million)
% 3.23/1.73 % (13237)------------------------------
% 3.23/1.73 % (13237)------------------------------
% 3.23/1.74 % (13239)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13001 for (2998ds/377Mi)
% 3.23/1.74 % (13242)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_13001 for (2998ds/779Mi)
% 3.23/1.74 % (13204)Instruction limit reached!
% 3.23/1.74 % (13204)------------------------------
% 3.23/1.74 % (13204)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.74 % (13204)Termination reason: Unknown
% 3.23/1.74 % (13204)Termination phase: Saturation
% 3.23/1.74
% 3.23/1.74 % (13204)Memory used [KB]: 7419
% 3.23/1.74 % (13204)Time elapsed: 0.101 s
% 3.23/1.74 % (13204)Instructions burned: 275 (million)
% 3.23/1.74 % (13204)------------------------------
% 3.23/1.74 % (13204)------------------------------
% 3.23/1.75 % (13244)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_13001 for (2998ds/19Mi)
% 3.23/1.76 % (13244)Instruction limit reached!
% 3.23/1.76 % (13244)------------------------------
% 3.23/1.76 % (13244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.76 % (13244)Termination reason: Unknown
% 3.23/1.76 % (13244)Termination phase: shuffling
% 3.23/1.76
% 3.23/1.76 % (13244)Memory used [KB]: 1791
% 3.23/1.76 % (13244)Time elapsed: 0.010 s
% 3.23/1.76 % (13244)Instructions burned: 19 (million)
% 3.23/1.76 % (13244)------------------------------
% 3.23/1.76 % (13244)------------------------------
% 3.23/1.76 % (13249)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13001 for (2998ds/879Mi)
% 3.23/1.77 % (13255)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_13001 for (2998ds/17Mi)
% 3.23/1.78 % (13255)Instruction limit reached!
% 3.23/1.78 % (13255)------------------------------
% 3.23/1.78 % (13255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.78 % (13255)Termination reason: Unknown
% 3.23/1.78 % (13255)Termination phase: shuffling
% 3.23/1.78
% 3.23/1.78 % (13255)Memory used [KB]: 1791
% 3.23/1.78 % (13255)Time elapsed: 0.007 s
% 3.23/1.78 % (13255)Instructions burned: 18 (million)
% 3.23/1.78 % (13255)------------------------------
% 3.23/1.78 % (13255)------------------------------
% 3.23/1.79 % (13259)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13001 for (2998ds/3Mi)
% 3.23/1.79 % (13259)Instruction limit reached!
% 3.23/1.79 % (13259)------------------------------
% 3.23/1.79 % (13259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.79 % (13259)Termination reason: Unknown
% 3.23/1.79 % (13259)Termination phase: shuffling
% 3.23/1.79
% 3.23/1.79 % (13259)Memory used [KB]: 1535
% 3.23/1.79 % (13259)Time elapsed: 0.003 s
% 3.23/1.79 % (13259)Instructions burned: 5 (million)
% 3.23/1.79 % (13259)------------------------------
% 3.23/1.79 % (13259)------------------------------
% 3.23/1.80 % (13261)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_13001 for (2998ds/30Mi)
% 3.23/1.82 % (13261)Instruction limit reached!
% 3.23/1.82 % (13261)------------------------------
% 3.23/1.82 % (13261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.82 % (13261)Termination reason: Unknown
% 3.23/1.82 % (13261)Termination phase: shuffling
% 3.23/1.82
% 3.23/1.82 % (13261)Memory used [KB]: 2046
% 3.23/1.82 % (13261)Time elapsed: 0.017 s
% 3.23/1.82 % (13261)Instructions burned: 30 (million)
% 3.23/1.82 % (13261)------------------------------
% 3.23/1.82 % (13261)------------------------------
% 3.23/1.83 % (13264)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_13001 for (2998ds/127Mi)
% 3.23/1.86 % (13264)Instruction limit reached!
% 3.23/1.86 % (13264)------------------------------
% 3.23/1.86 % (13264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.86 % (13264)Termination reason: Unknown
% 3.23/1.86 % (13264)Termination phase: Property scanning
% 3.23/1.86
% 3.23/1.86 % (13264)Memory used [KB]: 2942
% 3.23/1.86 % (13264)Time elapsed: 0.038 s
% 3.23/1.86 % (13264)Instructions burned: 129 (million)
% 3.23/1.86 % (13264)------------------------------
% 3.23/1.86 % (13264)------------------------------
% 3.23/1.87 % (13267)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_13001 for (2997ds/100Mi)
% 3.23/1.88 % (13239)Instruction limit reached!
% 3.23/1.88 % (13239)------------------------------
% 3.23/1.88 % (13239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.88 % (13239)Termination reason: Unknown
% 3.23/1.88 % (13239)Termination phase: Saturation
% 3.23/1.88
% 3.23/1.88 % (13239)Memory used [KB]: 8443
% 3.23/1.88 % (13239)Time elapsed: 0.147 s
% 3.23/1.88 % (13239)Instructions burned: 378 (million)
% 3.23/1.88 % (13239)------------------------------
% 3.23/1.88 % (13239)------------------------------
% 3.23/1.90 % (13269)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_13001 for (2997ds/3Mi)
% 3.23/1.90 % (13267)Instruction limit reached!
% 3.23/1.90 % (13267)------------------------------
% 3.23/1.90 % (13267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.90 % (13267)Termination reason: Unknown
% 3.23/1.90 % (13267)Termination phase: Property scanning
% 3.23/1.90
% 3.23/1.90 % (13267)Memory used [KB]: 2814
% 3.23/1.90 % (13267)Time elapsed: 0.027 s
% 3.23/1.90 % (13267)Instructions burned: 100 (million)
% 3.23/1.90 % (13267)------------------------------
% 3.23/1.90 % (13267)------------------------------
% 3.23/1.90 % (13269)Instruction limit reached!
% 3.23/1.90 % (13269)------------------------------
% 3.23/1.90 % (13269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.90 % (13269)Termination reason: Unknown
% 3.23/1.90 % (13269)Termination phase: shuffling
% 3.23/1.90
% 3.23/1.90 % (13269)Memory used [KB]: 1535
% 3.23/1.90 % (13269)Time elapsed: 0.004 s
% 3.23/1.90 % (13269)Instructions burned: 4 (million)
% 3.23/1.90 % (13269)------------------------------
% 3.23/1.90 % (13269)------------------------------
% 3.23/1.91 % (13271)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_13001 for (2997ds/20Mi)
% 3.23/1.91 % (13272)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_13001 for (2997ds/86Mi)
% 3.23/1.91 % (13271)Instruction limit reached!
% 3.23/1.91 % (13271)------------------------------
% 3.23/1.91 % (13271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.91 % (13271)Termination reason: Unknown
% 3.23/1.91 % (13271)Termination phase: shuffling
% 3.23/1.91
% 3.23/1.91 % (13271)Memory used [KB]: 1918
% 3.23/1.91 % (13271)Time elapsed: 0.008 s
% 3.23/1.91 % (13271)Instructions burned: 21 (million)
% 3.23/1.91 % (13271)------------------------------
% 3.23/1.91 % (13271)------------------------------
% 3.23/1.92 % (13274)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_13001 for (2997ds/107Mi)
% 3.23/1.94 % (13272)Instruction limit reached!
% 3.23/1.94 % (13272)------------------------------
% 3.23/1.94 % (13272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.94 % (13272)Termination reason: Unknown
% 3.23/1.94 % (13272)Termination phase: Property scanning
% 3.23/1.94
% 3.23/1.94 % (13272)Memory used [KB]: 2814
% 3.23/1.94 % (13272)Time elapsed: 0.028 s
% 3.23/1.94 % (13272)Instructions burned: 86 (million)
% 3.23/1.94 % (13272)------------------------------
% 3.23/1.94 % (13272)------------------------------
% 3.23/1.95 % (13275)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_13001 for (2996ds/39Mi)
% 3.23/1.96 % (13274)Instruction limit reached!
% 3.23/1.96 % (13274)------------------------------
% 3.23/1.96 % (13274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.96 % (13274)Termination reason: Unknown
% 3.23/1.96 % (13274)Termination phase: Property scanning
% 3.23/1.96
% 3.23/1.96 % (13274)Memory used [KB]: 2814
% 3.23/1.96 % (13274)Time elapsed: 0.034 s
% 3.23/1.96 % (13274)Instructions burned: 109 (million)
% 3.23/1.96 % (13274)------------------------------
% 3.23/1.96 % (13274)------------------------------
% 3.23/1.96 % (13275)Instruction limit reached!
% 3.23/1.96 % (13275)------------------------------
% 3.23/1.96 % (13275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.96 % (13275)Termination reason: Unknown
% 3.23/1.96 % (13275)Termination phase: shuffling
% 3.23/1.96
% 3.23/1.96 % (13275)Memory used [KB]: 2302
% 3.23/1.96 % (13275)Time elapsed: 0.013 s
% 3.23/1.96 % (13275)Instructions burned: 42 (million)
% 3.23/1.96 % (13275)------------------------------
% 3.23/1.96 % (13275)------------------------------
% 3.23/1.97 % (13276)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_13001 for (2996ds/448Mi)
% 3.23/1.97 % (13277)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_13001 for (2996ds/46Mi)
% 3.23/1.99 % (13277)Instruction limit reached!
% 3.23/1.99 % (13277)------------------------------
% 3.23/1.99 % (13277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.99 % (13277)Termination reason: Unknown
% 3.23/1.99 % (13277)Termination phase: Property scanning
% 3.23/1.99
% 3.23/1.99 % (13277)Memory used [KB]: 2046
% 3.23/1.99 % (13277)Time elapsed: 0.015 s
% 3.23/1.99 % (13277)Instructions burned: 47 (million)
% 3.23/1.99 % (13277)------------------------------
% 3.23/1.99 % (13277)------------------------------
% 3.23/1.99 % (13228)Instruction limit reached!
% 3.23/1.99 % (13228)------------------------------
% 3.23/1.99 % (13228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.99 % (13228)Termination reason: Unknown
% 3.23/1.99 % (13228)Termination phase: Saturation
% 3.23/1.99
% 3.23/1.99 % (13228)Memory used [KB]: 8443
% 3.23/1.99 % (13228)Time elapsed: 0.287 s
% 3.23/1.99 % (13228)Instructions burned: 710 (million)
% 3.23/1.99 % (13228)------------------------------
% 3.23/1.99 % (13228)------------------------------
% 3.23/1.99 % (13278)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_13001 for (2996ds/98Mi)
% 3.23/2.00 % (13279)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_13001 for (2996ds/507Mi)
% 5.79/2.02 % (13278)Instruction limit reached!
% 5.79/2.02 % (13278)------------------------------
% 5.79/2.02 % (13278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.02 % (13278)Termination reason: Unknown
% 5.79/2.02 % (13278)Termination phase: Naming
% 5.79/2.02
% 5.79/2.02 % (13278)Memory used [KB]: 2686
% 5.79/2.02 % (13278)Time elapsed: 0.027 s
% 5.79/2.02 % (13278)Instructions burned: 101 (million)
% 5.79/2.02 % (13278)------------------------------
% 5.79/2.02 % (13278)------------------------------
% 5.79/2.03 % (13280)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_13001 for (2996ds/149Mi)
% 5.79/2.03 % (13242)Instruction limit reached!
% 5.79/2.03 % (13242)------------------------------
% 5.79/2.03 % (13242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.03 % (13242)Termination reason: Unknown
% 5.79/2.03 % (13242)Termination phase: Saturation
% 5.79/2.03
% 5.79/2.03 % (13242)Memory used [KB]: 9978
% 5.79/2.03 % (13242)Time elapsed: 0.294 s
% 5.79/2.03 % (13242)Instructions burned: 779 (million)
% 5.79/2.03 % (13242)------------------------------
% 5.79/2.03 % (13242)------------------------------
% 5.79/2.04 % (13283)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/39Mi)
% 5.79/2.05 % (13283)Instruction limit reached!
% 5.79/2.05 % (13283)------------------------------
% 5.79/2.05 % (13283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.05 % (13283)Termination reason: Unknown
% 5.79/2.05 % (13283)Termination phase: Property scanning
% 5.79/2.05
% 5.79/2.05 % (13283)Memory used [KB]: 2046
% 5.79/2.05 % (13283)Time elapsed: 0.014 s
% 5.79/2.05 % (13283)Instructions burned: 42 (million)
% 5.79/2.05 % (13283)------------------------------
% 5.79/2.05 % (13283)------------------------------
% 5.79/2.06 % (13210)Instruction limit reached!
% 5.79/2.06 % (13210)------------------------------
% 5.79/2.06 % (13210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.06 % (13210)Termination reason: Unknown
% 5.79/2.06 % (13210)Termination phase: Saturation
% 5.79/2.06
% 5.79/2.06 % (13210)Memory used [KB]: 10106
% 5.79/2.06 % (13210)Time elapsed: 0.424 s
% 5.79/2.06 % (13210)Instructions burned: 1043 (million)
% 5.79/2.06 % (13210)------------------------------
% 5.79/2.06 % (13210)------------------------------
% 5.79/2.06 % (13293)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/6Mi)
% 5.79/2.07 % (13293)Instruction limit reached!
% 5.79/2.07 % (13293)------------------------------
% 5.79/2.07 % (13293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.07 % (13293)Termination reason: Unknown
% 5.79/2.07 % (13293)Termination phase: shuffling
% 5.79/2.07
% 5.79/2.07 % (13293)Memory used [KB]: 1663
% 5.79/2.07 % (13293)Time elapsed: 0.004 s
% 5.79/2.07 % (13293)Instructions burned: 9 (million)
% 5.79/2.07 % (13293)------------------------------
% 5.79/2.07 % (13293)------------------------------
% 5.79/2.07 % (13230)Instruction limit reached!
% 5.79/2.07 % (13230)------------------------------
% 5.79/2.07 % (13230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.07 % (13230)Termination reason: Unknown
% 5.79/2.07 % (13230)Termination phase: Saturation
% 5.79/2.07
% 5.79/2.07 % (13230)Memory used [KB]: 9083
% 5.79/2.07 % (13230)Time elapsed: 0.361 s
% 5.79/2.07 % (13230)Instructions burned: 902 (million)
% 5.79/2.07 % (13230)------------------------------
% 5.79/2.07 % (13230)------------------------------
% 5.79/2.07 % (13280)Instruction limit reached!
% 5.79/2.07 % (13280)------------------------------
% 5.79/2.07 % (13280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.07 % (13280)Termination reason: Unknown
% 5.79/2.07 % (13280)Termination phase: Property scanning
% 5.79/2.07
% 5.79/2.07 % (13280)Memory used [KB]: 2942
% 5.79/2.07 % (13280)Time elapsed: 0.042 s
% 5.79/2.07 % (13280)Instructions burned: 149 (million)
% 5.79/2.07 % (13280)------------------------------
% 5.79/2.07 % (13280)------------------------------
% 5.79/2.07 % (13298)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/1087Mi)
% 5.79/2.08 % (13300)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/47Mi)
% 5.79/2.08 % (13303)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/44Mi)
% 5.79/2.08 % (13304)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/432Mi)
% 5.79/2.09 % (13276)Instruction limit reached!
% 5.79/2.09 % (13276)------------------------------
% 5.79/2.09 % (13276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.09 % (13276)Termination reason: Unknown
% 5.79/2.09 % (13276)Termination phase: Saturation
% 5.79/2.09
% 5.79/2.09 % (13276)Memory used [KB]: 8955
% 5.79/2.09 % (13276)Time elapsed: 0.122 s
% 5.79/2.09 % (13276)Instructions burned: 450 (million)
% 5.79/2.09 % (13276)------------------------------
% 5.79/2.09 % (13276)------------------------------
% 5.79/2.09 % (13300)Instruction limit reached!
% 5.79/2.09 % (13300)------------------------------
% 5.79/2.09 % (13300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.09 % (13300)Termination reason: Unknown
% 5.79/2.09 % (13300)Termination phase: Property scanning
% 5.79/2.09
% 5.79/2.09 % (13300)Memory used [KB]: 2046
% 5.79/2.09 % (13300)Time elapsed: 0.016 s
% 5.79/2.09 % (13300)Instructions burned: 49 (million)
% 5.79/2.09 % (13300)------------------------------
% 5.79/2.09 % (13300)------------------------------
% 5.79/2.09 % (13249)Instruction limit reached!
% 5.79/2.09 % (13249)------------------------------
% 5.79/2.09 % (13249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.09 % (13249)Termination reason: Unknown
% 5.79/2.09 % (13249)Termination phase: Saturation
% 5.79/2.09
% 5.79/2.09 % (13249)Memory used [KB]: 8187
% 5.79/2.09 % (13249)Time elapsed: 0.333 s
% 5.79/2.09 % (13249)Instructions burned: 879 (million)
% 5.79/2.09 % (13249)------------------------------
% 5.79/2.09 % (13249)------------------------------
% 5.79/2.09 % (13303)Instruction limit reached!
% 5.79/2.09 % (13303)------------------------------
% 5.79/2.09 % (13303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.09 % (13303)Termination reason: Unknown
% 5.79/2.09 % (13303)Termination phase: shuffling
% 5.79/2.09
% 5.79/2.09 % (13303)Memory used [KB]: 2302
% 5.79/2.09 % (13303)Time elapsed: 0.014 s
% 5.79/2.09 % (13303)Instructions burned: 47 (million)
% 5.79/2.09 % (13303)------------------------------
% 5.79/2.09 % (13303)------------------------------
% 5.79/2.10 % (13311)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/62Mi)
% 5.79/2.10 % (13313)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/31Mi)
% 5.79/2.10 % (13315)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/96Mi)
% 5.79/2.10 % (13316)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_13001 for (2995ds/453Mi)
% 5.79/2.11 % (13313)Instruction limit reached!
% 5.79/2.11 % (13313)------------------------------
% 5.79/2.11 % (13313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.11 % (13313)Termination reason: Unknown
% 5.79/2.11 % (13313)Termination phase: shuffling
% 5.79/2.11
% 5.79/2.11 % (13313)Memory used [KB]: 2046
% 5.79/2.11 % (13313)Time elapsed: 0.011 s
% 5.79/2.11 % (13313)Instructions burned: 31 (million)
% 5.79/2.11 % (13313)------------------------------
% 5.79/2.11 % (13313)------------------------------
% 5.79/2.11 % (13279)First to succeed.
% 5.79/2.11 % (13311)Instruction limit reached!
% 5.79/2.11 % (13311)------------------------------
% 5.79/2.11 % (13311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.11 % (13311)Termination reason: Unknown
% 5.79/2.11 % (13311)Termination phase: Property scanning
% 5.79/2.11
% 5.79/2.11 % (13311)Memory used [KB]: 2814
% 5.79/2.11 % (13311)Time elapsed: 0.018 s
% 5.79/2.11 % (13311)Instructions burned: 62 (million)
% 5.79/2.11 % (13311)------------------------------
% 5.79/2.11 % (13311)------------------------------
% 6.31/2.12 % (13279)Refutation found. Thanks to Tanya!
% 6.31/2.12 % SZS status Theorem for DTF2THF_13001
% 6.31/2.12 % SZS output start Proof for DTF2THF_13001
% 6.31/2.12 thf(type_def_5, type, set_nat: $tType).
% 6.31/2.12 thf(type_def_6, type, nat: $tType).
% 6.31/2.12 thf(type_def_8, type, set_Product_prod_a_a: $tType).
% 6.31/2.12 thf(type_def_9, type, set_a: $tType).
% 6.31/2.12 thf(type_def_10, type, set_Pr1986765409at_nat: $tType).
% 6.31/2.12 thf(type_def_11, type, set_Pr1948701895od_a_a: $tType).
% 6.31/2.12 thf(type_def_12, type, set_set_a: $tType).
% 6.31/2.12 thf(type_def_13, type, product_prod_a_a: $tType).
% 6.31/2.12 thf(type_def_14, type, a: $tType).
% 6.31/2.12 thf(type_def_15, type, produc1691597095_set_a: $tType).
% 6.31/2.12 thf(type_def_16, type, product_prod_nat_nat: $tType).
% 6.31/2.12 thf(type_def_17, type, produc1463940355_set_a: $tType).
% 6.31/2.12 thf(type_def_18, type, produc1404140509_a_nat: $tType).
% 6.31/2.12 thf(type_def_19, type, produc1572603623od_a_a: $tType).
% 6.31/2.12 thf(func_def_0, type, set_Pr1948701895od_a_a: $tType).
% 6.31/2.12 thf(func_def_1, type, produc1572603623od_a_a: $tType).
% 6.31/2.12 thf(func_def_2, type, produc1691597095_set_a: $tType).
% 6.31/2.12 thf(func_def_3, type, set_Pr1986765409at_nat: $tType).
% 6.31/2.12 thf(func_def_4, type, produc1404140509_a_nat: $tType).
% 6.31/2.12 thf(func_def_5, type, produc1463940355_set_a: $tType).
% 6.31/2.12 thf(func_def_6, type, set_Product_prod_a_a: $tType).
% 6.31/2.12 thf(func_def_7, type, product_prod_nat_nat: $tType).
% 6.31/2.12 thf(func_def_8, type, product_prod_a_a: $tType).
% 6.31/2.12 thf(func_def_9, type, set_set_a: $tType).
% 6.31/2.12 thf(func_def_10, type, set_nat: $tType).
% 6.31/2.12 thf(func_def_11, type, set_a: $tType).
% 6.31/2.12 thf(func_def_12, type, nat: $tType).
% 6.31/2.12 thf(func_def_13, type, a: $tType).
% 6.31/2.12 thf(func_def_14, type, finite_card_nat: set_nat > nat).
% 6.31/2.12 thf(func_def_15, type, finite1481642319od_a_a: set_Product_prod_a_a > nat).
% 6.31/2.12 thf(func_def_16, type, finite_card_a: set_a > nat).
% 6.31/2.12 thf(func_def_17, type, finite_finite_nat: set_nat > $o).
% 6.31/2.12 thf(func_def_18, type, finite772653738at_nat: set_Pr1986765409at_nat > $o).
% 6.31/2.12 thf(func_def_19, type, finite1664988688od_a_a: set_Pr1948701895od_a_a > $o).
% 6.31/2.12 thf(func_def_20, type, finite179568208od_a_a: set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_21, type, finite_finite_set_a: set_set_a > $o).
% 6.31/2.12 thf(func_def_22, type, finite_finite_a: set_a > $o).
% 6.31/2.12 thf(func_def_23, type, zero_zero_nat: nat).
% 6.31/2.12 thf(func_def_24, type, order_1150839801on_nat: set_nat > set_Pr1986765409at_nat > $o).
% 6.31/2.12 thf(func_def_25, type, order_359341630od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 6.31/2.12 thf(func_def_26, type, order_preorder_on_a: set_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_27, type, bot_bot_nat_o: nat > $o).
% 6.31/2.12 thf(func_def_28, type, bot_bo1293121322_a_a_o: product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_29, type, bot_bot_a_o: a > $o).
% 6.31/2.12 thf(func_def_30, type, bot_bot_nat: nat).
% 6.31/2.12 thf(func_def_31, type, bot_bo1519632275_set_a: produc1691597095_set_a).
% 6.31/2.12 thf(func_def_32, type, bot_bot_set_nat: set_nat).
% 6.31/2.12 thf(func_def_33, type, bot_bo2131659635od_a_a: set_Product_prod_a_a).
% 6.31/2.12 thf(func_def_34, type, bot_bot_set_set_a: set_set_a).
% 6.31/2.12 thf(func_def_35, type, bot_bot_set_a: set_a).
% 6.31/2.12 thf(func_def_36, type, ord_less_nat: nat > nat > $o).
% 6.31/2.12 thf(func_def_37, type, ord_less_eq_nat: nat > nat > $o).
% 6.31/2.12 thf(func_def_38, type, ord_le1302190241at_nat: product_prod_nat_nat > product_prod_nat_nat > $o).
% 6.31/2.12 thf(func_def_39, type, ord_le841160035_set_a: produc1463940355_set_a > produc1463940355_set_a > $o).
% 6.31/2.12 thf(func_def_40, type, ord_le781360189_a_nat: produc1404140509_a_nat > produc1404140509_a_nat > $o).
% 6.31/2.12 thf(func_def_41, type, ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o).
% 6.31/2.12 thf(func_def_42, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 6.31/2.12 thf(func_def_43, type, ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_44, type, ord_less_eq_set_a: set_a > set_a > $o).
% 6.31/2.12 thf(func_def_45, type, prefer1936906324as_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat).
% 6.31/2.12 thf(func_def_46, type, prefer74720675od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 6.31/2.12 thf(func_def_47, type, prefer436369274d_as_a: a > set_a > set_Product_prod_a_a > set_a).
% 6.31/2.12 thf(func_def_48, type, prefer563798164od_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat).
% 6.31/2.12 thf(func_def_49, type, prefer477097315od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 6.31/2.12 thf(func_def_50, type, prefer161218362good_a: a > set_a > set_Product_prod_a_a > set_a).
% 6.31/2.12 thf(func_def_51, type, prefer1672638789an_nat: nat > set_nat > set_Pr1986765409at_nat > set_nat).
% 6.31/2.12 thf(func_def_52, type, prefer1498087410od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Pr1948701895od_a_a > set_Product_prod_a_a).
% 6.31/2.12 thf(func_def_53, type, prefer1676310729than_a: a > set_a > set_Product_prod_a_a > set_a).
% 6.31/2.12 thf(func_def_54, type, prefer1293349710ce_nat: set_nat > set_Pr1986765409at_nat > $o).
% 6.31/2.12 thf(func_def_55, type, prefer1338892073od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 6.31/2.12 thf(func_def_56, type, prefer1292084992ence_a: set_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_57, type, prefer1147844220ce_nat: set_nat > set_Pr1986765409at_nat > $o).
% 6.31/2.12 thf(func_def_58, type, prefer697821563od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 6.31/2.12 thf(func_def_59, type, prefer719835986ence_a: set_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_60, type, prefer580340847ioms_a: set_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_61, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 6.31/2.12 thf(func_def_62, type, produc1450889781_set_a: nat > set_a > produc1463940355_set_a).
% 6.31/2.12 thf(func_def_63, type, produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a).
% 6.31/2.12 thf(func_def_64, type, produc515611863_a_nat: set_a > nat > produc1404140509_a_nat).
% 6.31/2.12 thf(func_def_65, type, produc1928581911_set_a: set_a > set_a > produc1691597095_set_a).
% 6.31/2.12 thf(func_def_66, type, product_Pair_a_a: a > a > product_prod_a_a).
% 6.31/2.12 thf(func_def_67, type, refl_on_nat: set_nat > set_Pr1986765409at_nat > $o).
% 6.31/2.12 thf(func_def_68, type, refl_o1298442278od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 6.31/2.12 thf(func_def_69, type, refl_on_a: set_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_70, type, total_on_nat: set_nat > set_Pr1986765409at_nat > $o).
% 6.31/2.12 thf(func_def_71, type, total_1490170027od_a_a: set_Product_prod_a_a > set_Pr1948701895od_a_a > $o).
% 6.31/2.12 thf(func_def_72, type, total_on_a: set_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_73, type, trans_a: set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_74, type, collect_nat: (nat > $o) > set_nat).
% 6.31/2.12 thf(func_def_75, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 6.31/2.12 thf(func_def_76, type, collect_a: (a > $o) > set_a).
% 6.31/2.12 thf(func_def_77, type, is_empty_a: set_a > $o).
% 6.31/2.12 thf(func_def_78, type, member_nat: nat > set_nat > $o).
% 6.31/2.12 thf(func_def_79, type, member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o).
% 6.31/2.12 thf(func_def_80, type, member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o).
% 6.31/2.12 thf(func_def_81, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_82, type, member_set_a: set_a > set_set_a > $o).
% 6.31/2.12 thf(func_def_83, type, member_a: a > set_a > $o).
% 6.31/2.12 thf(func_def_84, type, carrier: set_a).
% 6.31/2.12 thf(func_def_85, type, relation: set_Product_prod_a_a).
% 6.31/2.12 thf(func_def_86, type, x: a).
% 6.31/2.12 thf(func_def_87, type, y: a).
% 6.31/2.12 thf(func_def_103, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 6.31/2.12 thf(func_def_104, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 6.31/2.12 thf(func_def_105, type, sP2: (product_prod_a_a > product_prod_a_a > $o) > set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_106, type, sP3: set_Product_prod_a_a > (a > product_prod_a_a > $o) > set_a > $o).
% 6.31/2.12 thf(func_def_107, type, sP4: set_nat > set_a > (a > nat > $o) > $o).
% 6.31/2.12 thf(func_def_108, type, sP5: (product_prod_a_a > nat > $o) > set_Product_prod_a_a > set_nat > $o).
% 6.31/2.12 thf(func_def_109, type, sP6: nat > nat > nat > $o).
% 6.31/2.12 thf(func_def_110, type, sP7: nat > nat > nat > $o).
% 6.31/2.12 thf(func_def_111, type, sP8: nat > nat > nat > $o).
% 6.31/2.12 thf(func_def_112, type, sP9: (nat > product_prod_a_a > $o) > set_nat > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_113, type, sP10: (product_prod_a_a > a > $o) > set_a > set_Product_prod_a_a > $o).
% 6.31/2.12 thf(func_def_114, type, sP11: set_nat > set_nat > (nat > nat > $o) > $o).
% 6.31/2.12 thf(func_def_115, type, sP12: set_a > (a > a > $o) > set_a > $o).
% 6.31/2.12 thf(func_def_116, type, sP13: set_a > set_nat > (nat > a > $o) > $o).
% 6.31/2.12 thf(func_def_117, type, sK14: (set_a > set_a) > set_a).
% 6.31/2.12 thf(func_def_118, type, sK15: (set_a > set_a) > set_a).
% 6.31/2.12 thf(func_def_119, type, sK16: nat > set_a > set_a).
% 6.31/2.12 thf(func_def_120, type, sK17: product_prod_a_a > a).
% 6.31/2.12 thf(func_def_121, type, sK18: product_prod_a_a > a).
% 6.31/2.12 thf(func_def_122, type, sK19: (product_prod_a_a > product_prod_a_a > $o) > set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_123, type, sK20: (product_prod_a_a > product_prod_a_a > $o) > set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_124, type, sK21: (product_prod_a_a > product_prod_a_a > $o) > set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_125, type, sK22: set_Product_prod_a_a > (product_prod_a_a > product_prod_a_a > $o) > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_126, type, sK23: (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_127, type, sK24: (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_128, type, sK25: (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_129, type, sK26: (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_130, type, sK27: (product_prod_a_a > product_prod_a_a > $o) > product_prod_a_a).
% 6.31/2.12 thf(func_def_131, type, sK28: (product_prod_a_a > product_prod_a_a > $o) > product_prod_a_a).
% 6.31/2.12 thf(func_def_132, type, sK29: (product_prod_a_a > product_prod_a_a > $o) > product_prod_a_a).
% 6.31/2.12 thf(func_def_133, type, sK30: (product_prod_a_a > product_prod_a_a > $o) > product_prod_a_a).
% 6.31/2.12 thf(func_def_134, type, sK31: (product_prod_a_a > product_prod_a_a > $o) > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_135, type, sK32: set_a > set_a > a).
% 6.31/2.12 thf(func_def_136, type, sK33: (nat > $o) > nat).
% 6.31/2.12 thf(func_def_137, type, sK34: (nat > $o) > nat > nat).
% 6.31/2.12 thf(func_def_138, type, sK35: (a > nat > $o) > set_a > a).
% 6.31/2.12 thf(func_def_139, type, sK36: (a > nat > $o) > set_a > nat).
% 6.31/2.12 thf(func_def_140, type, sK37: a > (a > nat > $o) > set_a > nat).
% 6.31/2.12 thf(func_def_141, type, sK38: set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_142, type, sK39: (set_a > nat) > set_a).
% 6.31/2.12 thf(func_def_143, type, sK40: (set_a > nat) > set_a).
% 6.31/2.12 thf(func_def_144, type, sK41: nat > set_nat > set_nat).
% 6.31/2.12 thf(func_def_145, type, sK42: (a > a > $o) > a).
% 6.31/2.12 thf(func_def_146, type, sK43: (a > a > $o) > a).
% 6.31/2.12 thf(func_def_147, type, sK44: (a > a > $o) > a).
% 6.31/2.12 thf(func_def_148, type, sK45: (a > a > $o) > a).
% 6.31/2.12 thf(func_def_149, type, sK46: set_a > (a > a > $o) > a).
% 6.31/2.12 thf(func_def_150, type, sK47: nat > set_a > set_a).
% 6.31/2.12 thf(func_def_151, type, sK48: set_Product_prod_a_a > (a > product_prod_a_a > $o) > set_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_152, type, sK49: set_Product_prod_a_a > (a > product_prod_a_a > $o) > set_a > a).
% 6.31/2.12 thf(func_def_153, type, sK50: set_Product_prod_a_a > (a > product_prod_a_a > $o) > set_a > a).
% 6.31/2.12 thf(func_def_154, type, sK51: set_a > (a > product_prod_a_a > $o) > set_Product_prod_a_a > a).
% 6.31/2.12 thf(func_def_155, type, sK52: set_Product_prod_a_a > nat > set_Product_prod_a_a).
% 6.31/2.12 thf(func_def_156, type, sK53: set_nat > nat > set_nat).
% 6.31/2.12 thf(func_def_157, type, sK54: (nat > set_a) > nat).
% 6.31/2.12 thf(func_def_158, type, sK55: (nat > set_a) > nat).
% 6.31/2.12 thf(func_def_159, type, sK56: (set_a > nat) > set_a).
% 6.31/2.12 thf(func_def_160, type, sK57: (set_a > nat) > set_a).
% 6.31/2.12 thf(func_def_161, type, sK58: nat > (nat > $o) > nat).
% 6.31/2.12 thf(func_def_162, type, sK59: (nat > $o) > nat).
% 6.31/2.12 thf(func_def_163, type, sK60: set_nat > set_a > (a > nat > $o) > a).
% 6.31/2.12 thf(func_def_164, type, sK61: set_nat > set_a > (a > nat > $o) > nat).
% 6.31/2.12 thf(func_def_165, type, sK62: set_nat > set_a > (a > nat > $o) > a).
% 6.31/2.12 thf(func_def_166, type, sK63: set_nat > set_a > (a > nat > $o) > a).
% 6.31/2.12 thf(func_def_167, type, sK64: (nat > set_a) > nat).
% 6.31/2.12 thf(func_def_168, type, sK65: (nat > set_a) > nat).
% 6.31/2.12 thf(func_def_169, type, sK66: set_Product_prod_a_a > a).
% 6.31/2.12 thf(func_def_170, type, sK67: set_Product_prod_a_a > a).
% 6.31/2.12 thf(func_def_171, type, sK68: set_Product_prod_a_a > a).
% 6.31/2.12 thf(func_def_172, type, sK69: set_set_a > set_a > set_a).
% 6.31/2.12 thf(func_def_173, type, sK70: (set_a > set_a) > set_a).
% 6.31/2.12 thf(func_def_174, type, sK71: (set_a > set_a) > set_a).
% 6.31/2.12 thf(func_def_175, type, sK72: set_Product_prod_a_a > set_Product_prod_a_a > a).
% 6.31/2.12 thf(func_def_176, type, sK73: set_Product_prod_a_a > set_Product_prod_a_a > a).
% 6.31/2.12 thf(func_def_177, type, sK74: nat > set_nat > nat).
% 6.31/2.12 thf(func_def_178, type, sK75: set_nat > set_nat > nat).
% 6.31/2.12 thf(func_def_179, type, sK76: (product_prod_a_a > $o) > a).
% 6.31/2.12 thf(func_def_180, type, sK77: (product_prod_a_a > $o) > a).
% 6.31/2.12 thf(func_def_181, type, sK78: (product_prod_a_a > nat > $o) > set_Product_prod_a_a > set_nat > product_prod_a_a).
% 6.31/2.12 thf(func_def_182, type, sK79: (product_prod_a_a > nat > $o) > set_Product_prod_a_a > set_nat > product_prod_a_a).
% 6.31/2.12 thf(func_def_183, type, sK80: (product_prod_a_a > nat > $o) > set_Product_prod_a_a > set_nat > nat).
% 6.31/2.12 thf(func_def_184, type, sK81: set_Product_prod_a_a > (product_prod_a_a > nat > $o) > set_nat > product_prod_a_a).
% 6.31/2.12 thf(func_def_185, type, sK82: set_set_a > set_a).
% 6.31/2.12 thf(func_def_186, type, sK83: (nat > nat > $o) > set_nat > nat).
% 6.31/2.12 thf(func_def_187, type, sK84: (nat > nat > $o) > set_nat > nat).
% 6.31/2.12 thf(func_def_188, type, sK85: nat > (nat > nat > $o) > set_nat > nat).
% 6.31/2.12 thf(func_def_189, type, sK86: set_a > nat > set_a).
% 6.31/2.12 thf(func_def_190, type, sK87: set_nat > nat > nat).
% 6.31/2.12 thf(func_def_191, type, sK88: (nat > set_a) > nat).
% 6.31/2.12 thf(func_def_192, type, sK89: (nat > set_a) > nat).
% 6.31/2.12 thf(func_def_193, type, sK90: set_Product_prod_a_a > set_a > a).
% 6.31/2.12 thf(func_def_194, type, sK91: set_Product_prod_a_a > set_a > a).
% 6.31/2.12 thf(func_def_195, type, sK92: set_Product_prod_a_a > set_a > a).
% 6.31/2.12 thf(func_def_196, type, sK93: set_Product_prod_a_a > set_a > a).
% 6.31/2.12 thf(func_def_197, type, sK94: (nat > nat) > nat).
% 6.31/2.12 thf(func_def_198, type, sK95: (nat > nat) > nat).
% 6.31/2.12 thf(func_def_199, type, sK96: set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_200, type, sK97: set_a > a).
% 6.31/2.12 thf(func_def_201, type, sK98: product_prod_a_a > a).
% 6.31/2.12 thf(func_def_202, type, sK99: product_prod_a_a > a).
% 6.31/2.12 thf(func_def_203, type, sK100: set_nat > set_Pr1986765409at_nat > nat).
% 6.31/2.12 thf(func_def_204, type, sK101: set_nat > set_Pr1986765409at_nat > nat).
% 6.31/2.12 thf(func_def_205, type, sK102: set_nat > set_Pr1986765409at_nat > nat).
% 6.31/2.12 thf(func_def_206, type, sK103: set_nat > set_Pr1986765409at_nat > nat).
% 6.31/2.12 thf(func_def_207, type, sK104: set_Pr1948701895od_a_a > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_208, type, sK105: set_Pr1948701895od_a_a > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_209, type, sK106: (nat > product_prod_a_a > $o) > set_nat > set_Product_prod_a_a > nat).
% 6.31/2.12 thf(func_def_210, type, sK107: (nat > product_prod_a_a > $o) > set_nat > set_Product_prod_a_a > nat).
% 6.31/2.12 thf(func_def_211, type, sK108: (nat > product_prod_a_a > $o) > set_nat > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_212, type, sK109: set_Product_prod_a_a > (nat > product_prod_a_a > $o) > set_nat > nat).
% 6.31/2.12 thf(func_def_213, type, sK110: (nat > nat > $o) > set_nat > nat).
% 6.31/2.12 thf(func_def_214, type, sK111: (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_215, type, sK112: (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_216, type, sK113: (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_217, type, sK114: (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_218, type, sK115: set_a > a).
% 6.31/2.12 thf(func_def_219, type, sK116: set_a > set_set_a > set_a).
% 6.31/2.12 thf(func_def_220, type, sK117: (nat > nat) > nat).
% 6.31/2.12 thf(func_def_221, type, sK118: (nat > nat) > nat).
% 6.31/2.12 thf(func_def_222, type, sK119: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_223, type, sK120: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_224, type, sK121: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_225, type, sK122: set_Product_prod_a_a > set_Pr1948701895od_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_226, type, sK123: set_Pr1986765409at_nat > set_nat > nat).
% 6.31/2.12 thf(func_def_227, type, sK124: set_Pr1986765409at_nat > set_nat > nat).
% 6.31/2.12 thf(func_def_228, type, sK125: set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_229, type, sK126: set_nat > nat).
% 6.31/2.12 thf(func_def_230, type, sK127: set_a > nat > set_a).
% 6.31/2.12 thf(func_def_231, type, sK128: (nat > nat) > nat).
% 6.31/2.12 thf(func_def_232, type, sK129: (nat > nat) > nat).
% 6.31/2.12 thf(func_def_233, type, sK130: (set_a > nat) > set_a).
% 6.31/2.12 thf(func_def_234, type, sK131: (set_a > nat) > set_a).
% 6.31/2.12 thf(func_def_235, type, sK132: set_set_a > set_a).
% 6.31/2.12 thf(func_def_236, type, sK133: (product_prod_a_a > a > $o) > set_a > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_237, type, sK134: (product_prod_a_a > a > $o) > set_a > set_Product_prod_a_a > a).
% 6.31/2.12 thf(func_def_238, type, sK135: (product_prod_a_a > a > $o) > set_a > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_239, type, sK136: (product_prod_a_a > a > $o) > set_Product_prod_a_a > set_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_240, type, sK137: (nat > $o) > nat > nat).
% 6.31/2.12 thf(func_def_241, type, sK138: (set_a > set_a) > set_a).
% 6.31/2.12 thf(func_def_242, type, sK139: (set_a > set_a) > set_a).
% 6.31/2.12 thf(func_def_243, type, sK140: (set_a > set_a) > set_a).
% 6.31/2.12 thf(func_def_244, type, sK141: (set_a > set_a) > set_a).
% 6.31/2.12 thf(func_def_245, type, sK142: set_Product_prod_a_a > nat > set_Product_prod_a_a).
% 6.31/2.12 thf(func_def_246, type, sK143: (nat > nat) > nat).
% 6.31/2.12 thf(func_def_247, type, sK144: (nat > nat) > nat).
% 6.31/2.12 thf(func_def_248, type, sK145: (set_a > nat) > set_a).
% 6.31/2.12 thf(func_def_249, type, sK146: (set_a > nat) > set_a).
% 6.31/2.12 thf(func_def_250, type, sK147: (product_prod_a_a > nat > $o) > set_Product_prod_a_a > product_prod_a_a).
% 6.31/2.12 thf(func_def_251, type, sK148: (product_prod_a_a > nat > $o) > set_Product_prod_a_a > nat).
% 6.31/2.12 thf(func_def_252, type, sK149: product_prod_a_a > (product_prod_a_a > nat > $o) > set_Product_prod_a_a > nat).
% 6.31/2.12 thf(func_def_253, type, sK150: (nat > set_a) > nat).
% 6.31/2.12 thf(func_def_254, type, sK151: (nat > set_a) > nat).
% 6.31/2.12 thf(func_def_255, type, sK152: set_Product_prod_a_a > set_a > a).
% 6.31/2.12 thf(func_def_256, type, sK153: set_Product_prod_a_a > set_a > a).
% 6.31/2.12 thf(func_def_257, type, sK154: set_nat > nat).
% 6.31/2.12 thf(func_def_258, type, sK155: set_nat > set_nat > (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_259, type, sK156: set_nat > set_nat > (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_260, type, sK157: set_nat > set_nat > (nat > nat > $o) > nat).
% 6.31/2.12 thf(func_def_261, type, sK158: (nat > nat > $o) > set_nat > set_nat > nat).
% 6.31/2.12 thf(func_def_262, type, sK159: set_nat > nat).
% 6.31/2.12 thf(func_def_263, type, sK160: set_Product_prod_a_a > nat > set_Product_prod_a_a).
% 6.31/2.12 thf(func_def_264, type, sK161: set_nat > nat).
% 6.31/2.12 thf(func_def_265, type, sK162: set_a > (a > a > $o) > set_a > a).
% 6.31/2.12 thf(func_def_266, type, sK163: set_a > (a > a > $o) > set_a > a).
% 6.31/2.12 thf(func_def_267, type, sK164: set_a > (a > a > $o) > set_a > a).
% 6.31/2.12 thf(func_def_268, type, sK165: set_a > (a > a > $o) > set_a > a).
% 6.31/2.12 thf(func_def_269, type, sK166: (product_prod_a_a > $o) > a).
% 6.31/2.12 thf(func_def_270, type, sK167: (product_prod_a_a > $o) > a).
% 6.31/2.12 thf(func_def_271, type, sK168: set_a > set_nat > (nat > a > $o) > nat).
% 6.31/2.12 thf(func_def_272, type, sK169: set_a > set_nat > (nat > a > $o) > a).
% 6.31/2.12 thf(func_def_273, type, sK170: set_a > set_nat > (nat > a > $o) > nat).
% 6.31/2.12 thf(func_def_274, type, sK171: (nat > a > $o) > set_nat > set_a > nat).
% 6.31/2.12 thf(func_def_275, type, sK172: (a > $o) > (a > $o) > a).
% 6.31/2.12 thf(func_def_276, type, sK173: set_nat > nat > set_nat).
% 6.31/2.12 thf(func_def_278, type, ph175: !>[X0: $tType]:(X0)).
% 6.31/2.12 thf(f3277,plain,(
% 6.31/2.12 $false),
% 6.31/2.12 inference(avatar_sat_refutation,[status(thm)],[f3174,f3185,f3275])).
% 6.31/2.12 thf(f3275,plain,(
% 6.31/2.12 ~spl174_15),
% 6.31/2.12 inference(avatar_contradiction_clause,[status(thm)],[f3274])).
% 6.31/2.12 thf(f3274,plain,(
% 6.31/2.12 $false | ~spl174_15),
% 6.31/2.12 inference(subsumption_resolution,[status(thm)],[f3242,f3243])).
% 6.31/2.12 thf(f3243,plain,(
% 6.31/2.12 ($true = (member_a @ (sK32 @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (prefer1676310729than_a @ x @ carrier @ relation))) | ~spl174_15),
% 6.31/2.12 inference(superposition,[status(thm)],[f2933,f3173])).
% 6.31/2.12 thf(f3173,plain,(
% 6.31/2.12 ((prefer1676310729than_a @ x @ carrier @ relation) = (prefer1676310729than_a @ y @ carrier @ relation)) | ~spl174_15),
% 6.31/2.12 inference(avatar_component_clause,[status(thm)],[f3171])).
% 6.31/2.12 thf(f3171,definition,(
% 6.31/2.12 spl174_15 <=> ((prefer1676310729than_a @ x @ carrier @ relation) = (prefer1676310729than_a @ y @ carrier @ relation))),
% 6.31/2.12 introduced(definition,[new_symbols(naming,[spl174_15])],[avatar_definition])).
% 6.31/2.12 thf(f2933,plain,(
% 6.31/2.12 ((member_a @ (sK32 @ (prefer1676310729than_a @ y @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (prefer1676310729than_a @ x @ carrier @ relation)) = $true)),
% 6.31/2.12 inference(trivial_inequality_removal,[status(thm)],[f2928])).
% 6.31/2.12 thf(f2928,plain,(
% 6.31/2.12 ($true != $true) | ((member_a @ (sK32 @ (prefer1676310729than_a @ y @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (prefer1676310729than_a @ x @ carrier @ relation)) = $true)),
% 6.31/2.12 inference(superposition,[status(thm)],[f2922,f2818])).
% 6.31/2.12 thf(f2818,plain,(
% 6.31/2.12 ( ! [X0 : set_a,X1 : set_a] : (((!! @ a @ (^[Y0 : a]: ((member_a @ Y0 @ X0) => (member_a @ Y0 @ X1)))) = $true) | ((member_a @ (sK32 @ X1 @ X0) @ X0) = $true)) )),
% 6.31/2.12 inference(beta_eta_normalization,[status(thm)],[f2408])).
% 6.31/2.12 thf(f2408,plain,(
% 6.31/2.12 ( ! [X0 : set_a,X1 : set_a] : ((((^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))) @ X0 @ X1) = $true) | ((member_a @ (sK32 @ X1 @ X0) @ X0) = $true)) )),
% 6.31/2.12 inference(definition_unfolding,[status(thm)],[f1864,f2093])).
% 6.31/2.12 thf(f2093,plain,(
% 6.31/2.12 (ord_less_eq_set_a = (^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))))),
% 6.31/2.12 inference(cnf_transformation,[status(thm)],[f969])).
% 6.31/2.12 thf(f969,plain,(
% 6.31/2.12 (ord_less_eq_set_a = (^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))))),
% 6.31/2.12 inference(fool_elimination,[status(thm)],[f968])).
% 6.31/2.12 thf(f968,plain,(
% 6.31/2.12 ((^[X0 : set_a, X1 : set_a] : (! [X2 : a] : ((member_a @ X2 @ X0) => (member_a @ X2 @ X1)))) = ord_less_eq_set_a)),
% 6.31/2.12 inference(rectify,[status(thm)],[f149])).
% 6.31/2.12 thf(f149,axiom,(
% 6.31/2.12 ((^[X21 : set_a, X22 : set_a] : (! [X9 : a] : ((member_a @ X9 @ X21) => (member_a @ X9 @ X22)))) = ord_less_eq_set_a)),
% 6.31/2.12 file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_148_subset__eq)).
% 6.31/2.12 thf(f1864,plain,(
% 6.31/2.12 ( ! [X0 : set_a,X1 : set_a] : (((member_a @ (sK32 @ X1 @ X0) @ X0) = $true) | ((ord_less_eq_set_a @ X0 @ X1) = $true)) )),
% 6.31/2.12 inference(cnf_transformation,[status(thm)],[f1619])).
% 6.31/2.12 thf(f1619,plain,(
% 6.31/2.12 ! [X0 : set_a,X1 : set_a] : ((($true != (member_a @ (sK32 @ X1 @ X0) @ X1)) & ((member_a @ (sK32 @ X1 @ X0) @ X0) = $true)) | ((ord_less_eq_set_a @ X0 @ X1) = $true))),
% 6.31/2.12 inference(skolemisation,[status(esa),new_symbols(skolem,[sK32])],[f1395,f1618])).
% 6.31/2.12 thf(f1618,plain,(
% 6.31/2.12 ! [X0 : set_a,X1 : set_a] : (? [X2 : a] : (((member_a @ X2 @ X1) != $true) & ($true = (member_a @ X2 @ X0))) => (($true != (member_a @ (sK32 @ X1 @ X0) @ X1)) & ((member_a @ (sK32 @ X1 @ X0) @ X0) = $true)))),
% 6.31/2.12 introduced(definition,[],[choice_axiom])).
% 6.31/2.12 thf(f1395,plain,(
% 6.31/2.12 ! [X0 : set_a,X1 : set_a] : (? [X2 : a] : (((member_a @ X2 @ X1) != $true) & ($true = (member_a @ X2 @ X0))) | ((ord_less_eq_set_a @ X0 @ X1) = $true))),
% 6.31/2.12 inference(ennf_transformation,[status(thm)],[f729])).
% 6.31/2.12 thf(f729,plain,(
% 6.31/2.12 ! [X0 : set_a,X1 : set_a] : (! [X2 : a] : (($true = (member_a @ X2 @ X0)) => ((member_a @ X2 @ X1) = $true)) => ((ord_less_eq_set_a @ X0 @ X1) = $true))),
% 6.31/2.12 inference(fool_elimination,[status(thm)],[f728])).
% 6.31/2.12 thf(f728,plain,(
% 6.31/2.12 ! [X0 : set_a,X1 : set_a] : (! [X2 : a] : ((member_a @ X2 @ X0) => (member_a @ X2 @ X1)) => (ord_less_eq_set_a @ X0 @ X1))),
% 6.31/2.12 inference(rectify,[status(thm)],[f120])).
% 6.31/2.12 thf(f120,axiom,(
% 6.31/2.12 ! [X15 : set_a,X19 : set_a] : (! [X11 : a] : ((member_a @ X11 @ X15) => (member_a @ X11 @ X19)) => (ord_less_eq_set_a @ X15 @ X19))),
% 6.31/2.12 file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_119_subsetI)).
% 6.31/2.12 thf(f2922,plain,(
% 6.31/2.12 ((!! @ a @ (^[Y0 : a]: ((member_a @ Y0 @ (prefer1676310729than_a @ x @ carrier @ relation)) => (member_a @ Y0 @ (prefer1676310729than_a @ y @ carrier @ relation))))) != $true)),
% 6.31/2.12 inference(beta_eta_normalization,[status(thm)],[f2432])).
% 6.31/2.12 thf(f2432,plain,(
% 6.31/2.12 ($true != ((^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))) @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ y @ carrier @ relation)))),
% 6.31/2.12 inference(definition_unfolding,[status(thm)],[f1913,f2093])).
% 6.31/2.12 thf(f1913,plain,(
% 6.31/2.12 ((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ y @ carrier @ relation)) != $true)),
% 6.31/2.12 inference(cnf_transformation,[status(thm)],[f1091])).
% 6.31/2.12 thf(f1091,plain,(
% 6.31/2.12 ((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ y @ carrier @ relation)) != $true)),
% 6.31/2.12 inference(flattening,[status(thm)],[f453])).
% 6.31/2.12 thf(f453,plain,(
% 6.31/2.12 ~((ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ y @ carrier @ relation)) = $true)),
% 6.31/2.12 inference(fool_elimination,[status(thm)],[f452])).
% 6.31/2.12 thf(f452,plain,(
% 6.31/2.12 ~(ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ y @ carrier @ relation))),
% 6.31/2.12 inference(rectify,[status(thm)],[f356])).
% 6.31/2.12 thf(f356,negated_conjecture,(
% 6.31/2.12 ~(ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ y @ carrier @ relation))),
% 6.31/2.12 inference(negated_conjecture,[status(cth)],[f355])).
% 6.31/2.12 thf(f355,conjecture,(
% 6.31/2.12 (ord_less_eq_set_a @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ y @ carrier @ relation))),
% 6.31/2.12 file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',conj_0)).
% 6.31/2.12 thf(f3242,plain,(
% 6.31/2.12 ($true != (member_a @ (sK32 @ (prefer1676310729than_a @ x @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (prefer1676310729than_a @ x @ carrier @ relation))) | ~spl174_15),
% 6.31/2.12 inference(superposition,[status(thm)],[f2939,f3173])).
% 6.31/2.12 thf(f2939,plain,(
% 6.31/2.12 ((member_a @ (sK32 @ (prefer1676310729than_a @ y @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (prefer1676310729than_a @ y @ carrier @ relation)) != $true)),
% 6.31/2.12 inference(trivial_inequality_removal,[status(thm)],[f2927])).
% 6.31/2.12 thf(f2927,plain,(
% 6.31/2.12 ((member_a @ (sK32 @ (prefer1676310729than_a @ y @ carrier @ relation) @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (prefer1676310729than_a @ y @ carrier @ relation)) != $true) | ($true != $true)),
% 6.31/2.12 inference(superposition,[status(thm)],[f2922,f2815])).
% 6.31/2.12 thf(f2815,plain,(
% 6.31/2.12 ( ! [X0 : set_a,X1 : set_a] : (((!! @ a @ (^[Y0 : a]: ((member_a @ Y0 @ X0) => (member_a @ Y0 @ X1)))) = $true) | ($true != (member_a @ (sK32 @ X1 @ X0) @ X1))) )),
% 6.31/2.12 inference(beta_eta_normalization,[status(thm)],[f2407])).
% 6.31/2.12 thf(f2407,plain,(
% 6.31/2.12 ( ! [X0 : set_a,X1 : set_a] : (($true != (member_a @ (sK32 @ X1 @ X0) @ X1)) | (((^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))) @ X0 @ X1) = $true)) )),
% 6.31/2.12 inference(definition_unfolding,[status(thm)],[f1865,f2093])).
% 6.31/2.12 thf(f1865,plain,(
% 6.31/2.12 ( ! [X0 : set_a,X1 : set_a] : (($true != (member_a @ (sK32 @ X1 @ X0) @ X1)) | ((ord_less_eq_set_a @ X0 @ X1) = $true)) )),
% 6.31/2.12 inference(cnf_transformation,[status(thm)],[f1619])).
% 6.31/2.12 thf(f3185,plain,(
% 6.31/2.12 spl174_12),
% 6.31/2.12 inference(avatar_contradiction_clause,[status(thm)],[f3184])).
% 6.31/2.12 thf(f3184,plain,(
% 6.31/2.12 $false | spl174_12),
% 6.31/2.12 inference(subsumption_resolution,[status(thm)],[f3183,f2015])).
% 6.31/2.12 thf(f2015,plain,(
% 6.31/2.12 ((member_a @ x @ carrier) = $true)),
% 6.31/2.12 inference(cnf_transformation,[status(thm)],[f1056])).
% 6.31/2.12 thf(f1056,plain,(
% 6.31/2.12 ((member_a @ x @ carrier) = $true)),
% 6.31/2.12 inference(fool_elimination,[status(thm)],[f1055])).
% 6.31/2.12 thf(f1055,plain,(
% 6.31/2.12 (member_a @ x @ carrier)),
% 6.31/2.12 inference(rectify,[status(thm)],[f2])).
% 6.31/2.12 thf(f2,axiom,(
% 6.31/2.12 (member_a @ x @ carrier)),
% 6.31/2.12 file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_1_assms_I1_J)).
% 6.31/2.12 thf(f3183,plain,(
% 6.31/2.12 ((member_a @ x @ carrier) != $true) | spl174_12),
% 6.31/2.12 inference(trivial_inequality_removal,[status(thm)],[f3182])).
% 6.31/2.12 thf(f3182,plain,(
% 6.31/2.12 ((member_a @ x @ carrier) != $true) | ($true != $true) | spl174_12),
% 6.31/2.12 inference(superposition,[status(thm)],[f3158,f2017])).
% 6.31/2.12 thf(f2017,plain,(
% 6.31/2.12 ( ! [X0 : a] : (((finite_finite_a @ (prefer1676310729than_a @ X0 @ carrier @ relation)) = $true) | ((member_a @ X0 @ carrier) != $true)) )),
% 6.31/2.12 inference(cnf_transformation,[status(thm)],[f1462])).
% 6.31/2.12 thf(f1462,plain,(
% 6.31/2.12 ! [X0 : a] : (((finite_finite_a @ (prefer1676310729than_a @ X0 @ carrier @ relation)) = $true) | ((member_a @ X0 @ carrier) != $true))),
% 6.31/2.12 inference(ennf_transformation,[status(thm)],[f632])).
% 6.31/2.12 thf(f632,plain,(
% 6.31/2.12 ! [X0 : a] : (((member_a @ X0 @ carrier) = $true) => ((finite_finite_a @ (prefer1676310729than_a @ X0 @ carrier @ relation)) = $true))),
% 6.31/2.12 inference(fool_elimination,[status(thm)],[f631])).
% 6.31/2.12 thf(f631,plain,(
% 6.31/2.12 ! [X0 : a] : ((member_a @ X0 @ carrier) => (finite_finite_a @ (prefer1676310729than_a @ X0 @ carrier @ relation)))),
% 6.31/2.12 inference(rectify,[status(thm)],[f16])).
% 6.31/2.12 thf(f16,axiom,(
% 6.31/2.12 ! [X2 : a] : ((member_a @ X2 @ carrier) => (finite_finite_a @ (prefer1676310729than_a @ X2 @ carrier @ relation)))),
% 6.31/2.12 file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_15_fnt__carrier__fnt__nbt)).
% 6.31/2.12 thf(f3158,plain,(
% 6.31/2.12 ((finite_finite_a @ (prefer1676310729than_a @ x @ carrier @ relation)) != $true) | spl174_12),
% 6.31/2.12 inference(avatar_component_clause,[status(thm)],[f3156])).
% 6.31/2.12 thf(f3156,definition,(
% 6.31/2.12 spl174_12 <=> ((finite_finite_a @ (prefer1676310729than_a @ x @ carrier @ relation)) = $true)),
% 6.31/2.12 introduced(definition,[new_symbols(naming,[spl174_12])],[avatar_definition])).
% 6.31/2.12 thf(f3174,plain,(
% 6.31/2.12 spl174_15 | ~spl174_12),
% 6.31/2.12 inference(avatar_split_clause,[status(thm)],[f3169,f3156,f3171])).
% 6.31/2.12 thf(f3169,plain,(
% 6.31/2.12 ((prefer1676310729than_a @ x @ carrier @ relation) = (prefer1676310729than_a @ y @ carrier @ relation)) | ((finite_finite_a @ (prefer1676310729than_a @ x @ carrier @ relation)) != $true)),
% 6.31/2.12 inference(subsumption_resolution,[status(thm)],[f3147,f2243])).
% 6.31/2.12 thf(f2243,plain,(
% 6.31/2.12 ((ord_less_eq_nat @ (finite_card_a @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (finite_card_a @ (prefer1676310729than_a @ y @ carrier @ relation))) = $true)),
% 6.31/2.12 inference(cnf_transformation,[status(thm)],[f865])).
% 6.31/2.12 thf(f865,plain,(
% 6.31/2.12 ((ord_less_eq_nat @ (finite_card_a @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (finite_card_a @ (prefer1676310729than_a @ y @ carrier @ relation))) = $true)),
% 6.31/2.12 inference(fool_elimination,[status(thm)],[f864])).
% 6.31/2.12 thf(f864,plain,(
% 6.31/2.12 (ord_less_eq_nat @ (finite_card_a @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (finite_card_a @ (prefer1676310729than_a @ y @ carrier @ relation)))),
% 6.31/2.12 inference(rectify,[status(thm)],[f4])).
% 6.31/2.12 thf(f4,axiom,(
% 6.31/2.12 (ord_less_eq_nat @ (finite_card_a @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (finite_card_a @ (prefer1676310729than_a @ y @ carrier @ relation)))),
% 6.31/2.12 file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_3__092_060open_062card_A_Ino__better__than_Ax_Acarrier_Arelation_J_A_092_060le_062_Acard_A_Ino__better__than_Ay_Acarrier_Arelation_J_092_060close_062)).
% 6.31/2.12 thf(f3147,plain,(
% 6.31/2.12 ((finite_finite_a @ (prefer1676310729than_a @ x @ carrier @ relation)) != $true) | ((prefer1676310729than_a @ x @ carrier @ relation) = (prefer1676310729than_a @ y @ carrier @ relation)) | ((ord_less_eq_nat @ (finite_card_a @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (finite_card_a @ (prefer1676310729than_a @ y @ carrier @ relation))) != $true)),
% 6.31/2.12 inference(trivial_inequality_removal,[status(thm)],[f3119])).
% 6.31/2.12 thf(f3119,plain,(
% 6.31/2.12 ($true != $true) | ((finite_finite_a @ (prefer1676310729than_a @ x @ carrier @ relation)) != $true) | ((ord_less_eq_nat @ (finite_card_a @ (prefer1676310729than_a @ x @ carrier @ relation)) @ (finite_card_a @ (prefer1676310729than_a @ y @ carrier @ relation))) != $true) | ((prefer1676310729than_a @ x @ carrier @ relation) = (prefer1676310729than_a @ y @ carrier @ relation))),
% 6.31/2.12 inference(superposition,[status(thm)],[f2826,f2937])).
% 6.31/2.12 thf(f2937,plain,(
% 6.31/2.12 ($true = (!! @ a @ (^[Y0 : a]: ((member_a @ Y0 @ (prefer1676310729than_a @ y @ carrier @ relation)) => (member_a @ Y0 @ (prefer1676310729than_a @ x @ carrier @ relation))))))),
% 6.31/2.12 inference(trivial_inequality_removal,[status(thm)],[f2925])).
% 6.31/2.12 thf(f2925,plain,(
% 6.31/2.12 ($true = (!! @ a @ (^[Y0 : a]: ((member_a @ Y0 @ (prefer1676310729than_a @ y @ carrier @ relation)) => (member_a @ Y0 @ (prefer1676310729than_a @ x @ carrier @ relation)))))) | ($true != $true)),
% 6.31/2.12 inference(superposition,[status(thm)],[f2922,f2731])).
% 6.31/2.12 thf(f2731,plain,(
% 6.31/2.12 ( ! [X0 : a,X1 : a] : (($true = (!! @ a @ (^[Y0 : a]: ((member_a @ Y0 @ (prefer1676310729than_a @ X1 @ carrier @ relation)) => (member_a @ Y0 @ (prefer1676310729than_a @ X0 @ carrier @ relation)))))) | ((!! @ a @ (^[Y0 : a]: ((member_a @ Y0 @ (prefer1676310729than_a @ X0 @ carrier @ relation)) => (member_a @ Y0 @ (prefer1676310729than_a @ X1 @ carrier @ relation))))) = $true)) )),
% 6.31/2.12 inference(beta_eta_normalization,[status(thm)],[f2599])).
% 6.31/2.12 thf(f2599,plain,(
% 6.31/2.12 ( ! [X0 : a,X1 : a] : ((((^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))) @ (prefer1676310729than_a @ X0 @ carrier @ relation) @ (prefer1676310729than_a @ X1 @ carrier @ relation)) = $true) | (((^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))) @ (prefer1676310729than_a @ X1 @ carrier @ relation) @ (prefer1676310729than_a @ X0 @ carrier @ relation)) = $true)) )),
% 6.31/2.12 inference(definition_unfolding,[status(thm)],[f2267,f2093,f2093])).
% 6.31/2.12 thf(f2267,plain,(
% 6.31/2.12 ( ! [X0 : a,X1 : a] : (((ord_less_eq_set_a @ (prefer1676310729than_a @ X1 @ carrier @ relation) @ (prefer1676310729than_a @ X0 @ carrier @ relation)) = $true) | ((ord_less_eq_set_a @ (prefer1676310729than_a @ X0 @ carrier @ relation) @ (prefer1676310729than_a @ X1 @ carrier @ relation)) = $true)) )),
% 6.31/2.12 inference(cnf_transformation,[status(thm)],[f963])).
% 6.31/2.12 thf(f963,plain,(
% 6.31/2.12 ! [X0 : a,X1 : a] : (((ord_less_eq_set_a @ (prefer1676310729than_a @ X1 @ carrier @ relation) @ (prefer1676310729than_a @ X0 @ carrier @ relation)) = $true) | ((ord_less_eq_set_a @ (prefer1676310729than_a @ X0 @ carrier @ relation) @ (prefer1676310729than_a @ X1 @ carrier @ relation)) = $true))),
% 6.31/2.12 inference(fool_elimination,[status(thm)],[f962])).
% 6.31/2.12 thf(f962,plain,(
% 6.31/2.12 ! [X0 : a,X1 : a] : ((ord_less_eq_set_a @ (prefer1676310729than_a @ X0 @ carrier @ relation) @ (prefer1676310729than_a @ X1 @ carrier @ relation)) | (ord_less_eq_set_a @ (prefer1676310729than_a @ X1 @ carrier @ relation) @ (prefer1676310729than_a @ X0 @ carrier @ relation)))),
% 6.31/2.12 inference(rectify,[status(thm)],[f5])).
% 6.31/2.12 thf(f5,axiom,(
% 6.31/2.12 ! [X0 : a,X1 : a] : ((ord_less_eq_set_a @ (prefer1676310729than_a @ X0 @ carrier @ relation) @ (prefer1676310729than_a @ X1 @ carrier @ relation)) | (ord_less_eq_set_a @ (prefer1676310729than_a @ X1 @ carrier @ relation) @ (prefer1676310729than_a @ X0 @ carrier @ relation)))),
% 6.31/2.12 file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_4_nbt__nest)).
% 6.31/2.12 thf(f2826,plain,(
% 6.31/2.12 ( ! [X0 : set_a,X1 : set_a] : (((!! @ a @ (^[Y0 : a]: ((member_a @ Y0 @ X0) => (member_a @ Y0 @ X1)))) != $true) | ($true != (ord_less_eq_nat @ (finite_card_a @ X1) @ (finite_card_a @ X0))) | (X0 = X1) | ((finite_finite_a @ X1) != $true)) )),
% 6.31/2.12 inference(beta_eta_normalization,[status(thm)],[f2648])).
% 6.31/2.12 thf(f2648,plain,(
% 6.31/2.12 ( ! [X0 : set_a,X1 : set_a] : (((finite_finite_a @ X1) != $true) | ($true != (ord_less_eq_nat @ (finite_card_a @ X1) @ (finite_card_a @ X0))) | (X0 = X1) | (((^[Y0 : set_a]: ((^[Y1 : set_a]: (!! @ a @ (^[Y2 : a]: ((member_a @ Y2 @ Y0) => (member_a @ Y2 @ Y1))))))) @ X0 @ X1) != $true)) )),
% 6.31/2.12 inference(definition_unfolding,[status(thm)],[f2382,f2093])).
% 6.31/2.12 thf(f2382,plain,(
% 6.31/2.12 ( ! [X0 : set_a,X1 : set_a] : (((finite_finite_a @ X1) != $true) | ($true != (ord_less_eq_nat @ (finite_card_a @ X1) @ (finite_card_a @ X0))) | ((ord_less_eq_set_a @ X0 @ X1) != $true) | (X0 = X1)) )),
% 6.31/2.12 inference(cnf_transformation,[status(thm)],[f1232])).
% 6.31/2.12 thf(f1232,plain,(
% 6.31/2.12 ! [X0 : set_a,X1 : set_a] : (((finite_finite_a @ X1) != $true) | ($true != (ord_less_eq_nat @ (finite_card_a @ X1) @ (finite_card_a @ X0))) | ((ord_less_eq_set_a @ X0 @ X1) != $true) | (X0 = X1))),
% 6.31/2.12 inference(flattening,[status(thm)],[f1231])).
% 6.31/2.12 thf(f1231,plain,(
% 6.31/2.12 ! [X0 : set_a,X1 : set_a] : ((((X0 = X1) | ($true != (ord_less_eq_nat @ (finite_card_a @ X1) @ (finite_card_a @ X0)))) | ((ord_less_eq_set_a @ X0 @ X1) != $true)) | ((finite_finite_a @ X1) != $true))),
% 6.31/2.12 inference(ennf_transformation,[status(thm)],[f548])).
% 6.31/2.12 thf(f548,plain,(
% 6.31/2.12 ! [X0 : set_a,X1 : set_a] : (((finite_finite_a @ X1) = $true) => (((ord_less_eq_set_a @ X0 @ X1) = $true) => (($true = (ord_less_eq_nat @ (finite_card_a @ X1) @ (finite_card_a @ X0))) => (X0 = X1))))),
% 6.31/2.12 inference(fool_elimination,[status(thm)],[f547])).
% 6.31/2.12 thf(f547,plain,(
% 6.31/2.12 ! [X0 : set_a,X1 : set_a] : ((finite_finite_a @ X1) => ((ord_less_eq_set_a @ X0 @ X1) => ((ord_less_eq_nat @ (finite_card_a @ X1) @ (finite_card_a @ X0)) => (X0 = X1))))),
% 6.31/2.12 inference(rectify,[status(thm)],[f212])).
% 6.31/2.12 thf(f212,axiom,(
% 6.31/2.12 ! [X15 : set_a,X19 : set_a] : ((finite_finite_a @ X19) => ((ord_less_eq_set_a @ X15 @ X19) => ((ord_less_eq_nat @ (finite_card_a @ X19) @ (finite_card_a @ X15)) => (X15 = X19))))),
% 6.31/2.12 file('/export/starexec/sandbox/tmp/tmp.ntfdFrZiAq/DTF2THF_13001.p',fact_211_card__seteq)).
% 6.31/2.12 % SZS output end Proof for DTF2THF_13001
% 6.31/2.12 % (13279)------------------------------
% 6.31/2.12 % (13279)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.31/2.12 % (13279)Termination reason: Refutation
% 6.31/2.12
% 6.31/2.12 % (13279)Memory used [KB]: 8699
% 6.31/2.12 % (13279)Time elapsed: 0.115 s
% 6.31/2.12 % (13279)Instructions burned: 386 (million)
% 6.31/2.12 % (13279)------------------------------
% 6.31/2.12 % (13279)------------------------------
% 6.31/2.12 % (13193)Success in time 0.504 s
%------------------------------------------------------------------------------