%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP113^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n008.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 08:55:39 AM UTC 2026
% Result : Theorem 2.70s 1.59s
% Output : Refutation 2.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 8
% Syntax : Number of formulae : 36 ( 16 unt; 0 typ; 2 def)
% Number of atoms : 138 ( 42 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 283 ( 31 ~; 16 |; 9 &; 220 @)
% ( 2 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 230 ( 227 usr; 26 con; 0-3 aty)
% Number of variables : 35 ( 0 ^; 27 !; 8 ?; 35 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_Pr1928503567a_real: $tType ).
thf(type_def_7,type,
set_real: $tType ).
thf(type_def_8,type,
set_a: $tType ).
thf(type_def_9,type,
nat: $tType ).
thf(type_def_10,type,
real: $tType ).
thf(type_def_11,type,
product_prod_a_real: $tType ).
thf(type_def_12,type,
extended_ereal: $tType ).
thf(type_def_13,type,
set_Pr1158285382l_real: $tType ).
thf(type_def_14,type,
set_Pr147102617l_real: $tType ).
thf(type_def_15,type,
a: $tType ).
thf(type_def_16,type,
produc1691597095_set_a: $tType ).
thf(type_def_17,type,
product_prod_nat_nat: $tType ).
thf(type_def_18,type,
produc845587165t_real: $tType ).
thf(type_def_19,type,
produc581916509al_nat: $tType ).
thf(type_def_20,type,
produc957004601l_real: $tType ).
thf(type_def_21,type,
produc1312517202t_real: $tType ).
thf(type_def_22,type,
produc25772804_set_a: $tType ).
thf(type_def_23,type,
produc2021989178_set_a: $tType ).
thf(type_def_24,type,
produc211508562a_real: $tType ).
thf(type_def_25,type,
produc1931999653t_real: $tType ).
thf(type_def_26,type,
produc1350454001_set_a: $tType ).
thf(type_def_27,type,
produc304417648_set_a: $tType ).
thf(type_def_28,type,
produc509513372a_real: $tType ).
thf(type_def_29,type,
produc232195311t_real: $tType ).
thf(type_def_30,type,
produc792423270l_real: $tType ).
thf(func_def_0,type,
produc2021989178_set_a: $tType ).
thf(func_def_1,type,
produc211508562a_real: $tType ).
thf(func_def_2,type,
produc1312517202t_real: $tType ).
thf(func_def_3,type,
produc304417648_set_a: $tType ).
thf(func_def_4,type,
produc509513372a_real: $tType ).
thf(func_def_5,type,
produc25772804_set_a: $tType ).
thf(func_def_6,type,
set_Pr1158285382l_real: $tType ).
thf(func_def_7,type,
produc792423270l_real: $tType ).
thf(func_def_8,type,
produc1931999653t_real: $tType ).
thf(func_def_9,type,
produc232195311t_real: $tType ).
thf(func_def_10,type,
produc1350454001_set_a: $tType ).
thf(func_def_11,type,
set_Pr147102617l_real: $tType ).
thf(func_def_12,type,
produc1691597095_set_a: $tType ).
thf(func_def_13,type,
set_Pr1928503567a_real: $tType ).
thf(func_def_14,type,
produc957004601l_real: $tType ).
thf(func_def_15,type,
produc581916509al_nat: $tType ).
thf(func_def_16,type,
produc845587165t_real: $tType ).
thf(func_def_17,type,
product_prod_nat_nat: $tType ).
thf(func_def_18,type,
product_prod_a_real: $tType ).
thf(func_def_19,type,
set_real: $tType ).
thf(func_def_20,type,
extended_ereal: $tType ).
thf(func_def_21,type,
set_a: $tType ).
thf(func_def_22,type,
real: $tType ).
thf(func_def_23,type,
nat: $tType ).
thf(func_def_24,type,
a: $tType ).
thf(func_def_25,type,
elemen1695521870a_real: set_Pr1928503567a_real > set_Pr1928503567a_real ).
thf(func_def_26,type,
elemen1168706977e_real: set_real > set_real ).
thf(func_def_27,type,
elementary_closure_a: set_a > set_a ).
thf(func_def_28,type,
zero_zero_nat: nat ).
thf(func_def_29,type,
zero_zero_real: real ).
thf(func_def_30,type,
lower_870348689a_real: set_Pr1928503567a_real > ( product_prod_a_real > extended_ereal ) > set_Pr1158285382l_real ).
thf(func_def_31,type,
lower_1607476196h_real: set_real > ( real > extended_ereal ) > set_Pr147102617l_real ).
thf(func_def_32,type,
lower_930854854raph_a: set_a > ( a > extended_ereal ) > set_Pr1928503567a_real ).
thf(func_def_33,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_34,type,
ord_le297355219_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
thf(func_def_35,type,
ord_less_real: real > real > $o ).
thf(func_def_36,type,
ord_le1379034467a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > $o ).
thf(func_def_37,type,
ord_less_set_real: set_real > set_real > $o ).
thf(func_def_38,type,
ord_less_set_a: set_a > set_a > $o ).
thf(func_def_39,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_40,type,
ord_le1302190241at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_41,type,
ord_le1950669437t_real: produc845587165t_real > produc845587165t_real > $o ).
thf(func_def_42,type,
ord_le1686998781al_nat: produc581916509al_nat > produc581916509al_nat > $o ).
thf(func_def_43,type,
ord_le1342644953l_real: produc957004601l_real > produc957004601l_real > $o ).
thf(func_def_44,type,
ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o ).
thf(func_def_45,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_46,type,
ord_le1586073967a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > $o ).
thf(func_def_47,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_48,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_49,type,
top_to1567080838real_o: product_prod_a_real > $o ).
thf(func_def_50,type,
top_top_real_o: real > $o ).
thf(func_def_51,type,
top_top_a_o: a > $o ).
thf(func_def_52,type,
top_to991334818t_real: produc1312517202t_real ).
thf(func_def_53,type,
top_to345582004_set_a: produc25772804_set_a ).
thf(func_def_54,type,
top_to522338794_set_a: produc2021989178_set_a ).
thf(func_def_55,type,
top_to2037809826a_real: produc211508562a_real ).
thf(func_def_56,type,
top_to95743733t_real: produc1931999653t_real ).
thf(func_def_57,type,
top_to1153982881_set_a: produc1350454001_set_a ).
thf(func_def_58,type,
top_to724687040_set_a: produc304417648_set_a ).
thf(func_def_59,type,
top_to829322572a_real: produc509513372a_real ).
thf(func_def_60,type,
top_to35724191t_real: produc232195311t_real ).
thf(func_def_61,type,
top_to1477940855_set_a: produc1691597095_set_a ).
thf(func_def_62,type,
top_to2138011583a_real: set_Pr1928503567a_real ).
thf(func_def_63,type,
top_top_set_real: set_real ).
thf(func_def_64,type,
top_top_set_a: set_a ).
thf(func_def_65,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_66,type,
produc49576085t_real: nat > real > produc845587165t_real ).
thf(func_def_67,type,
produc1214388894l_real: product_prod_a_real > real > produc792423270l_real ).
thf(func_def_68,type,
produc1541812501al_nat: real > nat > produc581916509al_nat ).
thf(func_def_69,type,
produc705216881l_real: real > real > produc957004601l_real ).
thf(func_def_70,type,
produc471223946t_real: set_Pr1928503567a_real > set_real > produc1312517202t_real ).
thf(func_def_71,type,
produc1220567734_set_a: set_Pr1928503567a_real > set_a > produc25772804_set_a ).
thf(func_def_72,type,
produc1162518380_set_a: set_real > produc1691597095_set_a > produc2021989178_set_a ).
thf(func_def_73,type,
produc150798858a_real: set_real > set_Pr1928503567a_real > produc211508562a_real ).
thf(func_def_74,type,
produc119681757t_real: set_real > set_real > produc1931999653t_real ).
thf(func_def_75,type,
produc190993315_set_a: set_real > set_a > produc1350454001_set_a ).
thf(func_def_76,type,
produc272677600_set_a: set_a > produc1691597095_set_a > produc304417648_set_a ).
thf(func_def_77,type,
produc383769494a_real: set_a > set_Pr1928503567a_real > produc509513372a_real ).
thf(func_def_78,type,
produc1266579561t_real: set_a > set_real > produc232195311t_real ).
thf(func_def_79,type,
produc1928581911_set_a: set_a > set_a > produc1691597095_set_a ).
thf(func_def_80,type,
product_Pair_a_real: a > real > product_prod_a_real ).
thf(func_def_81,type,
real_V404783528a_real: product_prod_a_real > product_prod_a_real > real ).
thf(func_def_82,type,
real_V1934908667t_real: real > real > real ).
thf(func_def_83,type,
real_V1514887919dist_a: a > a > real ).
thf(func_def_84,type,
collec1714955950a_real: ( product_prod_a_real > $o ) > set_Pr1928503567a_real ).
thf(func_def_85,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_86,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_87,type,
member565909519l_real: produc792423270l_real > set_Pr1158285382l_real > $o ).
thf(func_def_88,type,
member1068169442l_real: produc957004601l_real > set_Pr147102617l_real > $o ).
thf(func_def_89,type,
member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).
thf(func_def_90,type,
member_real: real > set_real > $o ).
thf(func_def_91,type,
member_a: a > set_a > $o ).
thf(func_def_92,type,
e: real ).
thf(func_def_93,type,
f: a > extended_ereal ).
thf(func_def_95,type,
x: a ).
thf(func_def_96,type,
y: real ).
thf(func_def_97,type,
z: real ).
thf(func_def_112,type,
sP0: nat > nat > nat > $o ).
thf(func_def_113,type,
sP1: nat > nat > nat > $o ).
thf(func_def_114,type,
sP2: nat > nat > nat > $o ).
thf(func_def_115,type,
sP3: real > real > real > $o ).
thf(func_def_116,type,
sP4: real > real > real > $o ).
thf(func_def_117,type,
sP5: real > real > real > $o ).
thf(func_def_118,type,
sK6: ( real > real ) > real ).
thf(func_def_119,type,
sK7: ( real > real ) > real ).
thf(func_def_120,type,
sK8: ( real > nat ) > real ).
thf(func_def_121,type,
sK9: ( real > nat ) > real ).
thf(func_def_122,type,
sK10: ( real > real ) > real ).
thf(func_def_123,type,
sK11: ( real > real ) > real ).
thf(func_def_124,type,
sK12: product_prod_a_real > a ).
thf(func_def_125,type,
sK13: product_prod_a_real > real ).
thf(func_def_126,type,
sK14: ( real > nat ) > real ).
thf(func_def_127,type,
sK15: ( real > nat ) > real ).
thf(func_def_128,type,
sK16: product_prod_a_real ).
thf(func_def_129,type,
sK17: ( nat > real ) > nat ).
thf(func_def_130,type,
sK18: ( nat > real ) > nat ).
thf(func_def_131,type,
sK19: ( real > real ) > real ).
thf(func_def_132,type,
sK20: ( real > real ) > real ).
thf(func_def_133,type,
sK21: real > real > real ).
thf(func_def_134,type,
sK22: product_prod_a_real ).
thf(func_def_135,type,
sK23: ( nat > nat ) > nat ).
thf(func_def_136,type,
sK24: ( nat > nat ) > nat ).
thf(func_def_137,type,
sK25: ( nat > nat ) > nat ).
thf(func_def_138,type,
sK26: ( nat > nat ) > nat ).
thf(func_def_139,type,
sK27: real > real ).
thf(func_def_140,type,
sK28: ( nat > $o ) > nat ).
thf(func_def_141,type,
sK29: real > real > real ).
thf(func_def_142,type,
sK30: ( nat > nat ) > nat ).
thf(func_def_143,type,
sK31: ( nat > nat ) > nat ).
thf(func_def_144,type,
sK32: ( real > nat ) > real ).
thf(func_def_145,type,
sK33: ( real > nat ) > real ).
thf(func_def_146,type,
sK34: ( real > nat ) > real ).
thf(func_def_147,type,
sK35: ( real > nat ) > real ).
thf(func_def_148,type,
sK36: ( nat > real ) > nat ).
thf(func_def_149,type,
sK37: ( nat > real ) > nat ).
thf(func_def_150,type,
sK38: ( nat > nat ) > nat ).
thf(func_def_151,type,
sK39: ( nat > nat ) > nat ).
thf(func_def_152,type,
sK40: ( real > nat ) > real ).
thf(func_def_153,type,
sK41: ( real > nat ) > real ).
thf(func_def_154,type,
sK42: ( nat > real ) > nat ).
thf(func_def_155,type,
sK43: ( nat > real ) > nat ).
thf(func_def_156,type,
sK44: real > real > real ).
thf(func_def_157,type,
sK45: ( nat > nat ) > nat ).
thf(func_def_158,type,
sK46: ( nat > nat ) > nat ).
thf(func_def_159,type,
sK47: ( real > nat ) > real ).
thf(func_def_160,type,
sK48: ( real > nat ) > real ).
thf(func_def_161,type,
sK49: ( nat > nat > $o ) > nat ).
thf(func_def_162,type,
sK50: ( nat > nat > $o ) > nat ).
thf(func_def_163,type,
sK51: ( nat > nat > $o ) > nat ).
thf(func_def_164,type,
sK52: ( nat > nat > $o ) > nat ).
thf(func_def_165,type,
sK53: ( nat > nat > $o ) > nat ).
thf(func_def_166,type,
sK54: real > real ).
thf(func_def_167,type,
sK55: ( nat > nat ) > nat ).
thf(func_def_168,type,
sK56: ( nat > nat ) > nat ).
thf(func_def_169,type,
sK57: ( product_prod_a_real > $o ) > real ).
thf(func_def_170,type,
sK58: ( product_prod_a_real > $o ) > a ).
thf(func_def_171,type,
sK59: ( real > real ) > real ).
thf(func_def_172,type,
sK60: ( real > real ) > real ).
thf(func_def_173,type,
sK61: ( real > nat ) > real ).
thf(func_def_174,type,
sK62: ( real > nat ) > real ).
thf(func_def_175,type,
sK63: ( nat > nat > $o ) > nat ).
thf(func_def_176,type,
sK64: ( nat > nat > $o ) > nat ).
thf(func_def_177,type,
sK65: ( nat > nat > $o ) > nat ).
thf(func_def_178,type,
sK66: ( nat > nat > $o ) > nat ).
thf(func_def_179,type,
sK67: ( nat > real ) > nat ).
thf(func_def_180,type,
sK68: ( nat > real ) > nat ).
thf(func_def_181,type,
sK69: ( nat > $o ) > nat > nat ).
thf(func_def_182,type,
sK70: ( nat > $o ) > nat ).
thf(func_def_183,type,
sK71: ( nat > nat ) > nat ).
thf(func_def_184,type,
sK72: ( nat > nat ) > nat ).
thf(func_def_185,type,
sK73: set_Pr1928503567a_real > product_prod_a_real ).
thf(func_def_186,type,
sK74: ( nat > real ) > nat ).
thf(func_def_187,type,
sK75: ( nat > real ) > nat ).
thf(func_def_188,type,
sK76: real > real > real > real ).
thf(func_def_189,type,
sK77: nat > ( nat > real ) > nat ).
thf(func_def_190,type,
sK78: ( nat > real ) > nat > ( nat > real ) > nat ).
thf(func_def_191,type,
sK79: ( real > nat ) > real ).
thf(func_def_192,type,
sK80: ( real > nat ) > real ).
thf(func_def_193,type,
sK81: ( nat > real ) > nat ).
thf(func_def_194,type,
sK82: ( nat > real ) > nat ).
thf(func_def_195,type,
sK83: ( nat > real ) > nat ).
thf(func_def_196,type,
sK84: ( nat > real ) > nat ).
thf(func_def_197,type,
sK85: ( nat > $o ) > nat ).
thf(func_def_198,type,
sK86: ( real > real ) > real ).
thf(func_def_199,type,
sK87: ( real > real ) > real ).
thf(func_def_200,type,
sK88: ( nat > real ) > nat ).
thf(func_def_201,type,
sK89: ( nat > real ) > nat ).
thf(func_def_202,type,
sK90: ( product_prod_a_real > $o ) > real ).
thf(func_def_203,type,
sK91: ( product_prod_a_real > $o ) > a ).
thf(func_def_204,type,
sK92: ( nat > nat ) > nat ).
thf(func_def_205,type,
sK93: ( nat > nat ) > nat ).
thf(func_def_206,type,
sK94: ( real > real > $o ) > real ).
thf(func_def_207,type,
sK95: ( real > real > $o ) > real ).
thf(func_def_208,type,
sK96: ( real > real > $o ) > real ).
thf(func_def_209,type,
sK97: ( real > real > $o ) > real ).
thf(func_def_210,type,
sK98: ( real > real > $o ) > real ).
thf(func_def_211,type,
sK99: ( nat > nat ) > nat ).
thf(func_def_212,type,
sK100: ( nat > nat ) > nat ).
thf(func_def_213,type,
sK101: ( real > real > $o ) > real ).
thf(func_def_214,type,
sK102: ( real > real > $o ) > real ).
thf(func_def_215,type,
sK103: ( real > real > $o ) > real ).
thf(func_def_216,type,
sK104: ( real > real > $o ) > real ).
thf(func_def_217,type,
sK105: ( real > real ) > real ).
thf(func_def_218,type,
sK106: ( real > real ) > real ).
thf(func_def_219,type,
sK107: ( nat > real ) > nat ).
thf(func_def_220,type,
sK108: ( nat > real ) > nat ).
thf(func_def_221,type,
sK109: ( real > real ) > real ).
thf(func_def_222,type,
sK110: ( real > real ) > real ).
thf(func_def_223,type,
sK111: product_prod_a_real > product_prod_a_real > set_Pr1928503567a_real > product_prod_a_real ).
thf(func_def_224,type,
sK112: ( real > real ) > real ).
thf(func_def_225,type,
sK113: ( real > real ) > real ).
thf(func_def_226,type,
sK114: ( real > nat ) > real ).
thf(func_def_227,type,
sK115: ( real > nat ) > real ).
thf(func_def_228,type,
sK116: ( nat > nat ) > nat ).
thf(func_def_229,type,
sK117: ( nat > nat ) > nat ).
thf(func_def_230,type,
sK118: nat > nat ).
thf(func_def_231,type,
sK119: ( nat > nat ) > nat ).
thf(func_def_232,type,
sK120: ( nat > nat ) > nat ).
thf(func_def_233,type,
sK121: ( nat > nat ) > nat ).
thf(func_def_234,type,
sK122: ( nat > nat ) > nat ).
thf(func_def_235,type,
sK123: ( nat > real ) > nat ).
thf(func_def_236,type,
sK124: ( nat > real ) > nat ).
thf(func_def_237,type,
sK125: ( nat > real ) > nat ).
thf(func_def_238,type,
sK126: ( nat > real ) > nat ).
thf(func_def_239,type,
sK127: ( nat > real ) > nat ).
thf(func_def_240,type,
sK128: ( nat > real ) > nat ).
thf(func_def_241,type,
sK129: real > real > real > real ).
thf(func_def_242,type,
sK130: ( real > real ) > real ).
thf(func_def_243,type,
sK131: ( real > real ) > real ).
thf(func_def_244,type,
sK132: ( nat > nat ) > nat ).
thf(func_def_245,type,
sK133: ( nat > nat ) > nat ).
thf(func_def_246,type,
sK134: ( real > nat ) > real ).
thf(func_def_247,type,
sK135: ( real > nat ) > real ).
thf(func_def_248,type,
sK136: ( nat > $o ) > nat > nat ).
thf(func_def_249,type,
sK137: ( nat > $o ) > nat > nat ).
thf(func_def_250,type,
sK138: ( real > nat ) > real ).
thf(func_def_251,type,
sK139: ( real > nat ) > real ).
thf(func_def_252,type,
sK140: ( real > real ) > real ).
thf(func_def_253,type,
sK141: ( real > real ) > real ).
thf(func_def_254,type,
sK142: ( nat > $o ) > nat ).
thf(func_def_255,type,
sK143: ( real > real ) > real ).
thf(func_def_256,type,
sK144: ( real > real ) > real ).
thf(func_def_257,type,
sK145: ( real > nat ) > real ).
thf(func_def_258,type,
sK146: ( real > nat ) > real ).
thf(func_def_259,type,
sK147: product_prod_a_real > a ).
thf(func_def_260,type,
sK148: product_prod_a_real > real ).
thf(func_def_261,type,
sK149: ( real > real ) > real ).
thf(func_def_262,type,
sK150: ( real > real ) > real ).
thf(func_def_264,type,
ph152:
!>[X0: $tType] : X0 ).
thf(f2113,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f2014,f2015,f2112]) ).
thf(f2112,plain,
~ spl151_1,
inference(avatar_contradiction_clause,[status(thm)],[f2111]) ).
thf(f2111,plain,
( $false
| ~ spl151_1 ),
inference(subsumption_resolution,[status(thm)],[f2097,f1598]) ).
thf(f1598,plain,
( ( member1103263856a_real @ sK16 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1367]) ).
thf(f1367,plain,
( ( ( ord_less_real @ ( real_V404783528a_real @ sK16 @ ( product_Pair_a_real @ x @ y ) ) @ e )
= $true )
& ( ( member1103263856a_real @ sK16 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK16])],[f534,f1366]) ).
thf(f1366,plain,
( ? [X0: product_prod_a_real] :
( ( ( ord_less_real @ ( real_V404783528a_real @ X0 @ ( product_Pair_a_real @ x @ y ) ) @ e )
= $true )
& ( ( member1103263856a_real @ X0 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
= $true ) )
=> ( ( ( ord_less_real @ ( real_V404783528a_real @ sK16 @ ( product_Pair_a_real @ x @ y ) ) @ e )
= $true )
& ( ( member1103263856a_real @ sK16 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f534,plain,
? [X0: product_prod_a_real] :
( ( ( ord_less_real @ ( real_V404783528a_real @ X0 @ ( product_Pair_a_real @ x @ y ) ) @ e )
= $true )
& ( ( member1103263856a_real @ X0 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f533]) ).
thf(f533,plain,
? [X0: product_prod_a_real] :
( ( member1103263856a_real @ X0 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
& ( ord_less_real @ ( real_V404783528a_real @ X0 @ ( product_Pair_a_real @ x @ y ) ) @ e ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
? [X0: product_prod_a_real] :
( ( member1103263856a_real @ X0 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
& ( ord_less_real @ ( real_V404783528a_real @ X0 @ ( product_Pair_a_real @ x @ y ) ) @ e ) ),
file('/export/starexec/sandbox2/tmp/tmp.5gsfs41n1a/DTF2THF_2567.p',fact_0__092_060open_062_092_060exists_062ya_092_060in_062Epigraph_AUNIV_Af_O_Adist_Aya_A_Ix_M_Ay_J_A_060_Ae_092_060close_062) ).
thf(f2097,plain,
( ( ( member1103263856a_real @ sK16 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
!= $true )
| ~ spl151_1 ),
inference(trivial_inequality_removal,[status(thm)],[f2067]) ).
thf(f2067,plain,
( ( ( member1103263856a_real @ sK16 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
!= $true )
| ( $true != $true )
| ~ spl151_1 ),
inference(superposition,[status(thm)],[f2017,f1599]) ).
thf(f1599,plain,
( ( ord_less_real @ ( real_V404783528a_real @ sK16 @ ( product_Pair_a_real @ x @ y ) ) @ e )
= $true ),
inference(cnf_transformation,[status(thm)],[f1367]) ).
thf(f2017,plain,
( ! [X0: product_prod_a_real] :
( ( ( ord_less_real @ ( real_V404783528a_real @ X0 @ ( product_Pair_a_real @ x @ y ) ) @ e )
!= $true )
| ( ( member1103263856a_real @ X0 @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
!= $true ) )
| ~ spl151_1 ),
inference(superposition,[status(thm)],[f2009,f1588]) ).
thf(f1588,plain,
! [X0: product_prod_a_real] :
( ( product_Pair_a_real @ ( sK12 @ X0 ) @ ( sK13 @ X0 ) )
= X0 ),
inference(cnf_transformation,[status(thm)],[f1362]) ).
thf(f1362,plain,
! [X0: product_prod_a_real] :
( ( product_Pair_a_real @ ( sK12 @ X0 ) @ ( sK13 @ X0 ) )
= X0 ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK12,sK13])],[f1283,f1361]) ).
thf(f1361,plain,
! [X0: product_prod_a_real] :
( ? [X1: a,X2: real] :
( ( product_Pair_a_real @ X1 @ X2 )
= X0 )
=> ( ( product_Pair_a_real @ ( sK12 @ X0 ) @ ( sK13 @ X0 ) )
= X0 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1283,plain,
! [X0: product_prod_a_real] :
? [X1: a,X2: real] :
( ( product_Pair_a_real @ X1 @ X2 )
= X0 ),
inference(ennf_transformation,[status(thm)],[f1004]) ).
thf(f1004,plain,
! [X0: product_prod_a_real] :
~ ! [X1: a,X2: real] :
( ( product_Pair_a_real @ X1 @ X2 )
!= X0 ),
inference(rectify,[status(thm)],[f281]) ).
thf(f281,axiom,
! [X10: product_prod_a_real] :
~ ! [X27: a,X28: real] :
( ( product_Pair_a_real @ X27 @ X28 )
!= X10 ),
file('/export/starexec/sandbox2/tmp/tmp.5gsfs41n1a/DTF2THF_2567.p',fact_280_old_Oprod_Oexhaust) ).
thf(f2009,plain,
( ! [X0: real,X1: a] :
( ( $true
!= ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) )
| ( $true
!= ( member1103263856a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) ) ) )
| ~ spl151_1 ),
inference(avatar_component_clause,[status(thm)],[f2008]) ).
thf(f2008,definition,
( spl151_1
<=> ! [X0: real,X1: a] :
( ( $true
!= ( member1103263856a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) ) )
| ( $true
!= ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) ) ) ),
introduced(definition,[new_symbols(naming,[spl151_1])],[avatar_definition]) ).
thf(f2015,plain,
~ spl151_2,
inference(avatar_split_clause,[status(thm)],[f1640,f2011]) ).
thf(f2011,definition,
( spl151_2
<=> ( thesis = $true ) ),
introduced(definition,[new_symbols(naming,[spl151_2])],[avatar_definition]) ).
thf(f1640,plain,
thesis != $true,
inference(cnf_transformation,[status(thm)],[f989]) ).
thf(f989,plain,
thesis != $true,
inference(flattening,[status(thm)],[f910]) ).
thf(f910,plain,
thesis != $true,
inference(fool_elimination,[status(thm)],[f909]) ).
thf(f909,plain,
~ thesis,
inference(rectify,[status(thm)],[f347]) ).
thf(f347,negated_conjecture,
~ thesis,
inference(negated_conjecture,[status(cth)],[f346]) ).
thf(f346,conjecture,
thesis,
file('/export/starexec/sandbox2/tmp/tmp.5gsfs41n1a/DTF2THF_2567.p',conj_1) ).
thf(f2014,plain,
( spl151_1
| spl151_2 ),
inference(avatar_split_clause,[status(thm)],[f1694,f2011,f2008]) ).
thf(f1694,plain,
! [X0: real,X1: a] :
( ( $true
!= ( member1103263856a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) ) )
| ( $true
!= ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) )
| ( thesis = $true ) ),
inference(cnf_transformation,[status(thm)],[f1221]) ).
thf(f1221,plain,
! [X0: real,X1: a] :
( ( $true
!= ( member1103263856a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) ) )
| ( $true
!= ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) )
| ( thesis = $true ) ),
inference(flattening,[status(thm)],[f1220]) ).
thf(f1220,plain,
! [X0: real,X1: a] :
( ( thesis = $true )
| ( $true
!= ( member1103263856a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) ) )
| ( $true
!= ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) ) ),
inference(ennf_transformation,[status(thm)],[f700]) ).
thf(f700,plain,
! [X0: real,X1: a] :
( ( ( $true
= ( member1103263856a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) ) )
& ( $true
= ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) ) )
=> ( thesis = $true ) ),
inference(fool_elimination,[status(thm)],[f699]) ).
thf(f699,plain,
! [X0: real,X1: a] :
( ( ( member1103263856a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
& ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ X1 @ X0 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) )
=> thesis ),
inference(rectify,[status(thm)],[f345]) ).
thf(f345,axiom,
! [X59: real,X58: a] :
( ( ( member1103263856a_real @ ( product_Pair_a_real @ X58 @ X59 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
& ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ X58 @ X59 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) )
=> thesis ),
file('/export/starexec/sandbox2/tmp/tmp.5gsfs41n1a/DTF2THF_2567.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP113^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n008.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:26:39 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : ITP113^1 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.45 % Domain : Interactive Theorem Proving
% 0.29/0.45 % Problem : Sledgehammer Lower_Semicontinuous problem prob_663__6254006_1
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Des21]
% 0.29/0.45 % Names : Lower_Semicontinuous/prob_663__6254006_1 [Des21]
% 0.29/0.45
% 0.29/0.45 % Status : Theorem
% 0.29/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.40 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% 0.29/0.45 % Syntax : Number of formulae : 444 ( 140 unt; 98 typ; 0 def)
% 0.29/0.45 % Number of atoms : 1001 ( 251 equ; 0 cnn)
% 0.29/0.45 % Maximal formula atoms : 12 ( 2 avg)
% 0.29/0.45 % Number of connectives : 2614 ( 129 ~; 25 |; 41 &;1931 @)
% 0.29/0.45 % ( 0 <=>; 488 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 14 ( 7 avg)
% 0.29/0.45 % Number of types : 26 ( 25 usr)
% 0.29/0.45 % Number of type conns : 180 ( 180 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 74 ( 73 usr; 20 con; 0-2 aty)
% 0.29/0.45 % Number of variables : 963 ( 65 ^; 874 !; 24 ?; 963 :)
% 0.29/0.45 % SPC : TH0_THM_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:42:18.365
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % Could-be-implicit typings (25)
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
% 0.29/0.45 produc2021989178_set_a: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
% 0.29/0.45 produc211508562a_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
% 0.29/0.45 produc1312517202t_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
% 0.29/0.45 produc304417648_set_a: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
% 0.29/0.45 produc509513372a_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_Mt__Set__Oset_Itf__a_J_J,type,
% 0.29/0.45 produc25772804_set_a: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J_J,type,
% 0.29/0.45 set_Pr1158285382l_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
% 0.29/0.45 produc792423270l_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
% 0.29/0.45 produc1931999653t_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
% 0.29/0.45 produc232195311t_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_Itf__a_J_J,type,
% 0.29/0.45 produc1350454001_set_a: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.29/0.45 set_Pr147102617l_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 0.29/0.45 produc1691597095_set_a: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
% 0.29/0.45 set_Pr1928503567a_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.29/0.45 produc957004601l_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 0.29/0.45 produc581916509al_nat: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
% 0.29/0.45 produc845587165t_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.45 product_prod_nat_nat: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.29/0.45 product_prod_a_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.29/0.45 set_real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Extended____Real__Oereal,type,
% 0.29/0.45 extended_ereal: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.29/0.45 set_a: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Real__Oreal,type,
% 0.29/0.45 real: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_t__Nat__Onat,type,
% 0.29/0.45 nat: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_n_tf__a,type,
% 0.29/0.45 a: $tType ).
% 0.29/0.45
% 0.29/0.45 % Explicit typings (73)
% 0.29/0.45 thf(sy_c_Elementary__Topology_Oclosure_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.29/0.45 elemen1695521870a_real: set_Pr1928503567a_real > set_Pr1928503567a_real ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Elementary__Topology_Oclosure_001t__Real__Oreal,type,
% 0.29/0.45 elemen1168706977e_real: set_real > set_real ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Elementary__Topology_Oclosure_001tf__a,type,
% 0.29/0.45 elementary_closure_a: set_a > set_a ).
% 0.29/0.45
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% 0.29/0.45 % prod.inject
% 0.29/0.45 thf(fact_10_prod_Oinject,axiom,
% 0.29/0.45 ! [X1: a,X22: real,Y1: a,Y2: real] :
% 0.29/0.45 ( ( ( product_Pair_a_real @ X1 @ X22 )
% 0.29/0.45 = ( product_Pair_a_real @ Y1 @ Y2 ) )
% 0.29/0.45 = ( ( X1 = Y1 )
% 0.29/0.45 & ( X22 = Y2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % prod.inject
% 0.29/0.45 thf(fact_11_old_Oprod_Oinject,axiom,
% 0.29/0.45 ! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
% 0.29/0.45 ( ( ( produc1928581911_set_a @ A @ B )
% 0.29/0.45 = ( produc1928581911_set_a @ A2 @ B2 ) )
% 0.29/0.45 = ( ( A = A2 )
% 0.29/0.45 & ( B = B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.prod.inject
% 0.29/0.45 thf(fact_12_old_Oprod_Oinject,axiom,
% 0.29/0.45 ! [A: a,B: real,A2: a,B2: real] :
% 0.29/0.45 ( ( ( product_Pair_a_real @ A @ B )
% 0.29/0.45 = ( product_Pair_a_real @ A2 @ B2 ) )
% 0.29/0.45 = ( ( A = A2 )
% 0.29/0.45 & ( B = B2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % old.prod.inject
% 0.29/0.45 thf(fact_13_dist__commute__lessI,axiom,
% 0.29/0.45 ! [Y: product_prod_a_real,X2: product_prod_a_real,E: real] :
% 0.29/0.45 ( ( ord_less_real @ ( real_V404783528a_real @ Y @ X2 ) @ E )
% 0.29/0.45 => ( ord_less_real @ ( real_V404783528a_real @ X2 @ Y ) @ E ) ) ).
% 0.29/0.45
% 0.29/0.45 % dist_commute_lessI
% 0.29/0.45 thf(fact_14_top__prod__def,axiom,
% 0.29/0.45 ( top_to1477940855_set_a
% 0.29/0.45 = ( produc1928581911_set_a @ top_top_set_a @ top_top_set_a ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_15_top__prod__def,axiom,
% 0.29/0.45 ( top_to35724191t_real
% 0.29/0.45 = ( produc1266579561t_real @ top_top_set_a @ top_top_set_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_16_top__prod__def,axiom,
% 0.29/0.45 ( top_to1153982881_set_a
% 0.29/0.45 = ( produc190993315_set_a @ top_top_set_real @ top_top_set_a ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_17_top__prod__def,axiom,
% 0.29/0.45 ( top_to95743733t_real
% 0.29/0.45 = ( produc119681757t_real @ top_top_set_real @ top_top_set_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_18_top__prod__def,axiom,
% 0.29/0.45 ( top_to829322572a_real
% 0.29/0.45 = ( produc383769494a_real @ top_top_set_a @ top_to2138011583a_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_19_top__prod__def,axiom,
% 0.29/0.45 ( top_to2037809826a_real
% 0.29/0.45 = ( produc150798858a_real @ top_top_set_real @ top_to2138011583a_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_20_top__prod__def,axiom,
% 0.29/0.45 ( top_to345582004_set_a
% 0.29/0.45 = ( produc1220567734_set_a @ top_to2138011583a_real @ top_top_set_a ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_21_top__prod__def,axiom,
% 0.29/0.45 ( top_to991334818t_real
% 0.29/0.45 = ( produc471223946t_real @ top_to2138011583a_real @ top_top_set_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_22_top__prod__def,axiom,
% 0.29/0.45 ( top_to724687040_set_a
% 0.29/0.45 = ( produc272677600_set_a @ top_top_set_a @ top_to1477940855_set_a ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_23_top__prod__def,axiom,
% 0.29/0.45 ( top_to522338794_set_a
% 0.29/0.45 = ( produc1162518380_set_a @ top_top_set_real @ top_to1477940855_set_a ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_prod_def
% 0.29/0.45 thf(fact_24_top_Oextremum__strict,axiom,
% 0.29/0.45 ! [A: set_a] :
% 0.29/0.45 ~ ( ord_less_set_a @ top_top_set_a @ A ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_strict
% 0.29/0.45 thf(fact_25_top_Oextremum__strict,axiom,
% 0.29/0.45 ! [A: set_real] :
% 0.29/0.45 ~ ( ord_less_set_real @ top_top_set_real @ A ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_strict
% 0.29/0.45 thf(fact_26_top_Oextremum__strict,axiom,
% 0.29/0.45 ! [A: set_Pr1928503567a_real] :
% 0.29/0.45 ~ ( ord_le1379034467a_real @ top_to2138011583a_real @ A ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_strict
% 0.29/0.45 thf(fact_27_top_Oextremum__strict,axiom,
% 0.29/0.45 ! [A: produc1691597095_set_a] :
% 0.29/0.45 ~ ( ord_le297355219_set_a @ top_to1477940855_set_a @ A ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_strict
% 0.29/0.45 thf(fact_28_top_Onot__eq__extremum,axiom,
% 0.29/0.45 ! [A: set_a] :
% 0.29/0.45 ( ( A != top_top_set_a )
% 0.29/0.45 = ( ord_less_set_a @ A @ top_top_set_a ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.not_eq_extremum
% 0.29/0.45 thf(fact_29_top_Onot__eq__extremum,axiom,
% 0.29/0.45 ! [A: set_real] :
% 0.29/0.45 ( ( A != top_top_set_real )
% 0.29/0.45 = ( ord_less_set_real @ A @ top_top_set_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.not_eq_extremum
% 0.29/0.45 thf(fact_30_top_Onot__eq__extremum,axiom,
% 0.29/0.45 ! [A: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( A != top_to2138011583a_real )
% 0.29/0.45 = ( ord_le1379034467a_real @ A @ top_to2138011583a_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.not_eq_extremum
% 0.29/0.45 thf(fact_31_top_Onot__eq__extremum,axiom,
% 0.29/0.45 ! [A: produc1691597095_set_a] :
% 0.29/0.45 ( ( A != top_to1477940855_set_a )
% 0.29/0.45 = ( ord_le297355219_set_a @ A @ top_to1477940855_set_a ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.not_eq_extremum
% 0.29/0.45 thf(fact_32_epigraph__mono,axiom,
% 0.29/0.45 ! [X2: real,Y: real,F: real > extended_ereal,Z: real] :
% 0.29/0.45 ( ( ( member1068169442l_real @ ( produc705216881l_real @ X2 @ Y ) @ ( lower_1607476196h_real @ top_top_set_real @ F ) )
% 0.29/0.45 & ( ord_less_eq_real @ Y @ Z ) )
% 0.29/0.45 => ( member1068169442l_real @ ( produc705216881l_real @ X2 @ Z ) @ ( lower_1607476196h_real @ top_top_set_real @ F ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % epigraph_mono
% 0.29/0.45 thf(fact_33_epigraph__mono,axiom,
% 0.29/0.45 ! [X2: product_prod_a_real,Y: real,F: product_prod_a_real > extended_ereal,Z: real] :
% 0.29/0.45 ( ( ( member565909519l_real @ ( produc1214388894l_real @ X2 @ Y ) @ ( lower_870348689a_real @ top_to2138011583a_real @ F ) )
% 0.29/0.45 & ( ord_less_eq_real @ Y @ Z ) )
% 0.29/0.45 => ( member565909519l_real @ ( produc1214388894l_real @ X2 @ Z ) @ ( lower_870348689a_real @ top_to2138011583a_real @ F ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % epigraph_mono
% 0.29/0.45 thf(fact_34_epigraph__mono,axiom,
% 0.29/0.45 ! [X2: a,Y: real,F: a > extended_ereal,Z: real] :
% 0.29/0.45 ( ( ( member1103263856a_real @ ( product_Pair_a_real @ X2 @ Y ) @ ( lower_930854854raph_a @ top_top_set_a @ F ) )
% 0.29/0.45 & ( ord_less_eq_real @ Y @ Z ) )
% 0.29/0.45 => ( member1103263856a_real @ ( product_Pair_a_real @ X2 @ Z ) @ ( lower_930854854raph_a @ top_top_set_a @ F ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % epigraph_mono
% 0.29/0.45 thf(fact_35_metric__eq__thm,axiom,
% 0.29/0.45 ! [X2: real,S: set_real,Y: real] :
% 0.29/0.45 ( ( member_real @ X2 @ S )
% 0.29/0.45 => ( ( member_real @ Y @ S )
% 0.29/0.45 => ( ( X2 = Y )
% 0.29/0.45 = ( ! [X3: real] :
% 0.29/0.45 ( ( member_real @ X3 @ S )
% 0.29/0.45 => ( ( real_V1934908667t_real @ X2 @ X3 )
% 0.29/0.45 = ( real_V1934908667t_real @ Y @ X3 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % metric_eq_thm
% 0.29/0.45 thf(fact_36_metric__eq__thm,axiom,
% 0.29/0.45 ! [X2: a,S: set_a,Y: a] :
% 0.29/0.45 ( ( member_a @ X2 @ S )
% 0.29/0.45 => ( ( member_a @ Y @ S )
% 0.29/0.45 => ( ( X2 = Y )
% 0.29/0.45 = ( ! [X3: a] :
% 0.29/0.45 ( ( member_a @ X3 @ S )
% 0.29/0.45 => ( ( real_V1514887919dist_a @ X2 @ X3 )
% 0.29/0.45 = ( real_V1514887919dist_a @ Y @ X3 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % metric_eq_thm
% 0.29/0.45 thf(fact_37_metric__eq__thm,axiom,
% 0.29/0.45 ! [X2: product_prod_a_real,S: set_Pr1928503567a_real,Y: product_prod_a_real] :
% 0.29/0.45 ( ( member1103263856a_real @ X2 @ S )
% 0.29/0.45 => ( ( member1103263856a_real @ Y @ S )
% 0.29/0.45 => ( ( X2 = Y )
% 0.29/0.45 = ( ! [X3: product_prod_a_real] :
% 0.29/0.45 ( ( member1103263856a_real @ X3 @ S )
% 0.29/0.45 => ( ( real_V404783528a_real @ X2 @ X3 )
% 0.29/0.45 = ( real_V404783528a_real @ Y @ X3 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % metric_eq_thm
% 0.29/0.45 thf(fact_38_order__refl,axiom,
% 0.29/0.45 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 0.29/0.45
% 0.29/0.45 % order_refl
% 0.29/0.45 thf(fact_39_order__refl,axiom,
% 0.29/0.45 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.29/0.45
% 0.29/0.45 % order_refl
% 0.29/0.45 thf(fact_40_Pair__le,axiom,
% 0.29/0.45 ! [A: set_a,B: set_a,C: set_a,D: set_a] :
% 0.29/0.45 ( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A @ B ) @ ( produc1928581911_set_a @ C @ D ) )
% 0.29/0.45 = ( ( ord_less_eq_set_a @ A @ C )
% 0.29/0.45 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_le
% 0.29/0.45 thf(fact_41_Pair__le,axiom,
% 0.29/0.45 ! [A: real,B: real,C: real,D: real] :
% 0.29/0.45 ( ( ord_le1342644953l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
% 0.29/0.45 = ( ( ord_less_eq_real @ A @ C )
% 0.29/0.45 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_le
% 0.29/0.45 thf(fact_42_Pair__le,axiom,
% 0.29/0.45 ! [A: real,B: nat,C: real,D: nat] :
% 0.29/0.45 ( ( ord_le1686998781al_nat @ ( produc1541812501al_nat @ A @ B ) @ ( produc1541812501al_nat @ C @ D ) )
% 0.29/0.45 = ( ( ord_less_eq_real @ A @ C )
% 0.29/0.45 & ( ord_less_eq_nat @ B @ D ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_le
% 0.29/0.45 thf(fact_43_Pair__le,axiom,
% 0.29/0.45 ! [A: nat,B: real,C: nat,D: real] :
% 0.29/0.45 ( ( ord_le1950669437t_real @ ( produc49576085t_real @ A @ B ) @ ( produc49576085t_real @ C @ D ) )
% 0.29/0.45 = ( ( ord_less_eq_nat @ A @ C )
% 0.29/0.45 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_le
% 0.29/0.45 thf(fact_44_Pair__le,axiom,
% 0.29/0.45 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.45 ( ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
% 0.29/0.45 = ( ( ord_less_eq_nat @ A @ C )
% 0.29/0.45 & ( ord_less_eq_nat @ B @ D ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_le
% 0.29/0.45 thf(fact_45_dist__eq__0__iff,axiom,
% 0.29/0.45 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 0.29/0.45 ( ( ( real_V404783528a_real @ X2 @ Y )
% 0.29/0.45 = zero_zero_real )
% 0.29/0.45 = ( X2 = Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % dist_eq_0_iff
% 0.29/0.45 thf(fact_46_dist__self,axiom,
% 0.29/0.45 ! [X2: product_prod_a_real] :
% 0.29/0.45 ( ( real_V404783528a_real @ X2 @ X2 )
% 0.29/0.45 = zero_zero_real ) ).
% 0.29/0.45
% 0.29/0.45 % dist_self
% 0.29/0.45 thf(fact_47_dist__le__zero__iff,axiom,
% 0.29/0.45 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 0.29/0.45 ( ( ord_less_eq_real @ ( real_V404783528a_real @ X2 @ Y ) @ zero_zero_real )
% 0.29/0.45 = ( X2 = Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % dist_le_zero_iff
% 0.29/0.45 thf(fact_48_zero__less__dist__iff,axiom,
% 0.29/0.45 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 0.29/0.45 ( ( ord_less_real @ zero_zero_real @ ( real_V404783528a_real @ X2 @ Y ) )
% 0.29/0.45 = ( X2 != Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_less_dist_iff
% 0.29/0.45 thf(fact_49_order__subst1,axiom,
% 0.29/0.45 ! [A: real,F: real > real,B: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.45 => ( ! [X: real,Y3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_50_order__subst1,axiom,
% 0.29/0.45 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.29/0.45 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.45 => ( ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_51_order__subst1,axiom,
% 0.29/0.45 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.45 => ( ! [X: real,Y3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_52_order__subst1,axiom,
% 0.29/0.45 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.45 => ( ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_53_order__subst2,axiom,
% 0.29/0.45 ! [A: real,B: real,F: real > real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.45 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.29/0.45 => ( ! [X: real,Y3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_54_order__subst2,axiom,
% 0.29/0.45 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.29/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.29/0.45 => ( ! [X: real,Y3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_55_order__subst2,axiom,
% 0.29/0.45 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.45 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.29/0.45 => ( ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_56_order__subst2,axiom,
% 0.29/0.45 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.29/0.45 => ( ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_57_Pair__mono,axiom,
% 0.29/0.45 ! [X2: set_a,X4: set_a,Y: set_a,Y4: set_a] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ X2 @ X4 )
% 0.29/0.45 => ( ( ord_less_eq_set_a @ Y @ Y4 )
% 0.29/0.45 => ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ ( produc1928581911_set_a @ X4 @ Y4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_mono
% 0.29/0.45 thf(fact_58_Pair__mono,axiom,
% 0.29/0.45 ! [X2: real,X4: real,Y: real,Y4: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X2 @ X4 )
% 0.29/0.45 => ( ( ord_less_eq_real @ Y @ Y4 )
% 0.29/0.45 => ( ord_le1342644953l_real @ ( produc705216881l_real @ X2 @ Y ) @ ( produc705216881l_real @ X4 @ Y4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_mono
% 0.29/0.45 thf(fact_59_Pair__mono,axiom,
% 0.29/0.45 ! [X2: real,X4: real,Y: nat,Y4: nat] :
% 0.29/0.45 ( ( ord_less_eq_real @ X2 @ X4 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ Y @ Y4 )
% 0.29/0.45 => ( ord_le1686998781al_nat @ ( produc1541812501al_nat @ X2 @ Y ) @ ( produc1541812501al_nat @ X4 @ Y4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_mono
% 0.29/0.45 thf(fact_60_Pair__mono,axiom,
% 0.29/0.45 ! [X2: nat,X4: nat,Y: real,Y4: real] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X2 @ X4 )
% 0.29/0.45 => ( ( ord_less_eq_real @ Y @ Y4 )
% 0.29/0.45 => ( ord_le1950669437t_real @ ( produc49576085t_real @ X2 @ Y ) @ ( produc49576085t_real @ X4 @ Y4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_mono
% 0.29/0.45 thf(fact_61_Pair__mono,axiom,
% 0.29/0.45 ! [X2: nat,X4: nat,Y: nat,Y4: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X2 @ X4 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ Y @ Y4 )
% 0.29/0.45 => ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Pair_mono
% 0.29/0.45 thf(fact_62_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A: real,F: real > real,B: real,C: real] :
% 0.29/0.45 ( ( A
% 0.29/0.45 = ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.45 => ( ! [X: real,Y3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_63_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.29/0.45 ( ( A
% 0.29/0.45 = ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.45 => ( ! [X: real,Y3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_64_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.29/0.45 ( ( A
% 0.29/0.45 = ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.45 => ( ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_65_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.29/0.45 ( ( A
% 0.29/0.45 = ( F @ B ) )
% 0.29/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.45 => ( ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_66_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A: real,B: real,F: real > real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.45 => ( ( ( F @ B )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X: real,Y3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_67_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.29/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.45 => ( ( ( F @ B )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X: real,Y3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_68_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.45 => ( ( ( F @ B )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_69_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.45 => ( ( ( F @ B )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X: nat,Y3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_70_eq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
% 0.29/0.45 = ( ^ [X3: real,Y6: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X3 @ Y6 )
% 0.29/0.45 & ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_iff
% 0.29/0.45 thf(fact_71_eq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 0.29/0.45 = ( ^ [X3: nat,Y6: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 0.29/0.45 & ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_iff
% 0.29/0.45 thf(fact_72_antisym,axiom,
% 0.29/0.45 ! [X2: real,Y: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.45 => ( ( ord_less_eq_real @ Y @ X2 )
% 0.29/0.45 => ( X2 = Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym
% 0.29/0.45 thf(fact_73_antisym,axiom,
% 0.29/0.45 ! [X2: nat,Y: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.45 => ( ( ord_less_eq_nat @ Y @ X2 )
% 0.29/0.45 => ( X2 = Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym
% 0.29/0.45 thf(fact_74_linear,axiom,
% 0.29/0.45 ! [X2: real,Y: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.45 | ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % linear
% 0.29/0.45 thf(fact_75_linear,axiom,
% 0.29/0.45 ! [X2: nat,Y: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.45 | ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % linear
% 0.29/0.45 thf(fact_76_eq__refl,axiom,
% 0.29/0.45 ! [X2: real,Y: real] :
% 0.29/0.45 ( ( X2 = Y )
% 0.29/0.45 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_refl
% 0.29/0.45 thf(fact_77_eq__refl,axiom,
% 0.29/0.45 ! [X2: nat,Y: nat] :
% 0.29/0.45 ( ( X2 = Y )
% 0.29/0.45 => ( ord_less_eq_nat @ X2 @ Y ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_refl
% 0.29/0.45 thf(fact_78_le__cases,axiom,
% 0.29/0.45 ! [X2: real,Y: real] :
% 0.29/0.45 ( ~ ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.45 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases
% 0.29/0.45 thf(fact_79_le__cases,axiom,
% 0.29/0.45 ! [X2: nat,Y: nat] :
% 0.29/0.45 ( ~ ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.45 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases
% 0.29/0.45 thf(fact_80_order_Otrans,axiom,
% 0.29/0.45 ! [A: real,B: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.45 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.trans
% 0.29/0.45 thf(fact_81_order_Otrans,axiom,
% 0.29/0.45 ! [A: nat,B: nat,C: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.45 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.trans
% 0.29/0.45 thf(fact_82_le__cases3,axiom,
% 0.29/0.45 ! [X2: real,Y: real,Z: real] :
% 0.29/0.45 ( ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.45 => ~ ( ord_less_eq_real @ Y @ Z ) )
% 0.29/0.45 => ( ( ( ord_less_eq_real @ Y @ X2 )
% 0.29/0.45 => ~ ( ord_less_eq_real @ X2 @ Z ) )
% 0.29/0.45 => ( ( ( ord_less_eq_real @ X2 @ Z )
% 0.29/0.45 => ~ ( ord_less_eq_real @ Z @ Y ) )
% 0.29/0.45 => ( ( ( ord_less_eq_real @ Z @ Y )
% 0.29/0.45 => ~ ( ord_less_eq_real @ Y @ X2 ) )
% 0.29/0.45 => ( ( ( ord_less_eq_real @ Y @ Z )
% 0.29/0.45 => ~ ( ord_less_eq_real @ Z @ X2 ) )
% 0.29/0.45 => ~ ( ( ord_less_eq_real @ Z @ X2 )
% 0.29/0.45 => ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases3
% 0.29/0.45 thf(fact_83_le__cases3,axiom,
% 0.29/0.45 ! [X2: nat,Y: nat,Z: nat] :
% 0.29/0.45 ( ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.45 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 0.29/0.45 => ( ( ( ord_less_eq_nat @ Y @ X2 )
% 0.29/0.45 => ~ ( ord_less_eq_nat @ X2 @ Z ) )
% 0.29/0.45 => ( ( ( ord_less_eq_nat @ X2 @ Z )
% 0.29/0.45 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 0.29/0.45 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 0.29/0.45 => ~ ( ord_less_eq_nat @ Y @ X2 ) )
% 0.29/0.45 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 0.29/0.45 => ~ ( ord_less_eq_nat @ Z @ X2 ) )
% 0.29/0.45 => ~ ( ( ord_less_eq_nat @ Z @ X2 )
% 0.29/0.45 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases3
% 0.29/0.45 thf(fact_84_antisym__conv,axiom,
% 0.29/0.45 ! [Y: real,X2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ Y @ X2 )
% 0.29/0.45 => ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.45 = ( X2 = Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv
% 0.29/0.45 thf(fact_85_antisym__conv,axiom,
% 0.29/0.45 ! [Y: nat,X2: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ Y @ X2 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.45 = ( X2 = Y ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv
% 0.29/0.45 thf(fact_86_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
% 0.29/0.45 = ( ^ [A3: real,B3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.45 & ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.eq_iff
% 0.29/0.45 thf(fact_87_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 0.29/0.45 = ( ^ [A3: nat,B3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.29/0.45 & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.eq_iff
% 0.29/0.45 thf(fact_88_ord__eq__le__trans,axiom,
% 0.29/0.45 ! [A: real,B: real,C: real] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.45 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_trans
% 0.29/0.45 thf(fact_89_ord__eq__le__trans,axiom,
% 0.29/0.45 ! [A: nat,B: nat,C: nat] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.45 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_trans
% 0.29/0.45 thf(fact_90_ord__le__eq__trans,axiom,
% 0.29/0.45 ! [A: real,B: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.45 => ( ( B = C )
% 0.29/0.45 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_trans
% 0.29/0.45 thf(fact_91_ord__le__eq__trans,axiom,
% 0.29/0.45 ! [A: nat,B: nat,C: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.45 => ( ( B = C )
% 0.29/0.45 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_trans
% 0.29/0.45 thf(fact_92_order__class_Oorder_Oantisym,axiom,
% 0.29/0.45 ! [A: real,B: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.45 => ( ( ord_less_eq_real @ B @ A )
% 0.29/0.45 => ( A = B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.antisym
% 0.29/0.45 thf(fact_93_order__class_Oorder_Oantisym,axiom,
% 0.29/0.45 ! [A: nat,B: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.45 => ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.45 => ( A = B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.antisym
% 0.29/0.45 thf(fact_94_order__trans,axiom,
% 0.29/0.45 ! [X2: real,Y: real,Z: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.45 => ( ( ord_less_eq_real @ Y @ Z )
% 0.29/0.45 => ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_trans
% 0.29/0.45 thf(fact_95_order__trans,axiom,
% 0.29/0.45 ! [X2: nat,Y: nat,Z: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.45 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.29/0.45 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_trans
% 0.29/0.45 thf(fact_96_dual__order_Orefl,axiom,
% 0.29/0.45 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.refl
% 0.29/0.45 thf(fact_97_dual__order_Orefl,axiom,
% 0.29/0.45 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.refl
% 0.29/0.45 thf(fact_98_linorder__wlog,axiom,
% 0.29/0.45 ! [P: real > real > $o,A: real,B: real] :
% 0.29/0.45 ( ! [A4: real,B4: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A4 @ B4 )
% 0.29/0.45 => ( P @ A4 @ B4 ) )
% 0.29/0.45 => ( ! [A4: real,B4: real] :
% 0.29/0.45 ( ( P @ B4 @ A4 )
% 0.29/0.45 => ( P @ A4 @ B4 ) )
% 0.29/0.45 => ( P @ A @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_wlog
% 0.29/0.45 thf(fact_99_linorder__wlog,axiom,
% 0.29/0.45 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.29/0.45 ( ! [A4: nat,B4: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.29/0.45 => ( P @ A4 @ B4 ) )
% 0.29/0.45 => ( ! [A4: nat,B4: nat] :
% 0.29/0.45 ( ( P @ B4 @ A4 )
% 0.29/0.45 => ( P @ A4 @ B4 ) )
% 0.29/0.45 => ( P @ A @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_wlog
% 0.29/0.45 thf(fact_100_dual__order_Otrans,axiom,
% 0.29/0.45 ! [B: real,A: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ B @ A )
% 0.29/0.45 => ( ( ord_less_eq_real @ C @ B )
% 0.29/0.45 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.trans
% 0.29/0.45 thf(fact_101_dual__order_Otrans,axiom,
% 0.29/0.45 ! [B: nat,A: nat,C: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.45 => ( ( ord_less_eq_nat @ C @ B )
% 0.29/0.45 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.trans
% 0.29/0.45 thf(fact_102_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A: product_prod_a_real,P: product_prod_a_real > $o] :
% 0.29/0.45 ( ( member1103263856a_real @ A @ ( collec1714955950a_real @ P ) )
% 0.29/0.45 = ( P @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_103_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A: real,P: real > $o] :
% 0.29/0.45 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.29/0.45 = ( P @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_104_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A: a,P: a > $o] :
% 0.29/0.45 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.29/0.45 = ( P @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_105_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A5: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( collec1714955950a_real
% 0.29/0.45 @ ^ [X3: product_prod_a_real] : ( member1103263856a_real @ X3 @ A5 ) )
% 0.29/0.45 = A5 ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_106_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A5: set_real] :
% 0.29/0.45 ( ( collect_real
% 0.29/0.45 @ ^ [X3: real] : ( member_real @ X3 @ A5 ) )
% 0.29/0.45 = A5 ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_107_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A5: set_a] :
% 0.29/0.45 ( ( collect_a
% 0.29/0.45 @ ^ [X3: a] : ( member_a @ X3 @ A5 ) )
% 0.29/0.45 = A5 ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_108_dual__order_Oeq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
% 0.29/0.45 = ( ^ [A3: real,B3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ B3 @ A3 )
% 0.29/0.45 & ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.eq_iff
% 0.29/0.45 thf(fact_109_dual__order_Oeq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 0.29/0.45 = ( ^ [A3: nat,B3: nat] :
% 0.29/0.45 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.29/0.45 & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.eq_iff
% 0.29/0.46 thf(fact_110_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [B: real,A: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.29/0.46 => ( ( ord_less_eq_real @ A @ B )
% 0.29/0.46 => ( A = B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_111_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [B: nat,A: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( A = B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_112_zero__le__dist,axiom,
% 0.29/0.46 ! [X2: product_prod_a_real,Y: product_prod_a_real] : ( ord_less_eq_real @ zero_zero_real @ ( real_V404783528a_real @ X2 @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_dist
% 0.29/0.46 thf(fact_113_top__set__def,axiom,
% 0.29/0.46 ( top_top_set_a
% 0.29/0.46 = ( collect_a @ top_top_a_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_set_def
% 0.29/0.46 thf(fact_114_top__set__def,axiom,
% 0.29/0.46 ( top_top_set_real
% 0.29/0.46 = ( collect_real @ top_top_real_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_set_def
% 0.29/0.46 thf(fact_115_top__set__def,axiom,
% 0.29/0.46 ( top_to2138011583a_real
% 0.29/0.46 = ( collec1714955950a_real @ top_to1567080838real_o ) ) ).
% 0.29/0.46
% 0.29/0.46 % top_set_def
% 0.29/0.46 thf(fact_116_order_Onot__eq__order__implies__strict,axiom,
% 0.29/0.46 ! [A: real,B: real] :
% 0.29/0.46 ( ( A != B )
% 0.29/0.46 => ( ( ord_less_eq_real @ A @ B )
% 0.29/0.46 => ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.not_eq_order_implies_strict
% 0.29/0.46 thf(fact_117_order_Onot__eq__order__implies__strict,axiom,
% 0.29/0.46 ! [A: nat,B: nat] :
% 0.29/0.46 ( ( A != B )
% 0.29/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.not_eq_order_implies_strict
% 0.29/0.46 thf(fact_118_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.46 ! [B: real,A: real] :
% 0.29/0.46 ( ( ord_less_real @ B @ A )
% 0.29/0.46 => ( ord_less_eq_real @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_implies_order
% 0.29/0.46 thf(fact_119_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.46 ! [B: nat,A: nat] :
% 0.29/0.46 ( ( ord_less_nat @ B @ A )
% 0.29/0.46 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_implies_order
% 0.29/0.46 thf(fact_120_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.46 ( ord_less_real
% 0.29/0.46 = ( ^ [B3: real,A3: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ B3 @ A3 )
% 0.29/0.46 & ( A3 != B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_iff_order
% 0.29/0.46 thf(fact_121_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.46 ( ord_less_nat
% 0.29/0.46 = ( ^ [B3: nat,A3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.29/0.46 & ( A3 != B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_iff_order
% 0.29/0.46 thf(fact_122_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.46 ( ord_less_eq_real
% 0.29/0.46 = ( ^ [B3: real,A3: real] :
% 0.29/0.46 ( ( ord_less_real @ B3 @ A3 )
% 0.29/0.46 | ( A3 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.order_iff_strict
% 0.29/0.46 thf(fact_123_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.46 ( ord_less_eq_nat
% 0.29/0.46 = ( ^ [B3: nat,A3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ B3 @ A3 )
% 0.29/0.46 | ( A3 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.order_iff_strict
% 0.29/0.46 thf(fact_124_order_Ostrict__implies__order,axiom,
% 0.29/0.46 ! [A: real,B: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_implies_order
% 0.29/0.46 thf(fact_125_order_Ostrict__implies__order,axiom,
% 0.29/0.46 ! [A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_implies_order
% 0.29/0.46 thf(fact_126_dense__le__bounded,axiom,
% 0.29/0.46 ! [X2: real,Y: real,Z: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( ! [W: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ W )
% 0.29/0.46 => ( ( ord_less_real @ W @ Y )
% 0.29/0.46 => ( ord_less_eq_real @ W @ Z ) ) )
% 0.29/0.46 => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_le_bounded
% 0.29/0.46 thf(fact_127_dense__ge__bounded,axiom,
% 0.29/0.46 ! [Z: real,X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ Z @ X2 )
% 0.29/0.46 => ( ! [W: real] :
% 0.29/0.46 ( ( ord_less_real @ Z @ W )
% 0.29/0.46 => ( ( ord_less_real @ W @ X2 )
% 0.29/0.46 => ( ord_less_eq_real @ Y @ W ) ) )
% 0.29/0.46 => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_ge_bounded
% 0.29/0.46 thf(fact_128_dual__order_Ostrict__trans2,axiom,
% 0.29/0.46 ! [B: real,A: real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ B @ A )
% 0.29/0.46 => ( ( ord_less_eq_real @ C @ B )
% 0.29/0.46 => ( ord_less_real @ C @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans2
% 0.29/0.46 thf(fact_129_dual__order_Ostrict__trans2,axiom,
% 0.29/0.46 ! [B: nat,A: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ B @ A )
% 0.29/0.46 => ( ( ord_less_eq_nat @ C @ B )
% 0.29/0.46 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans2
% 0.29/0.46 thf(fact_130_dual__order_Ostrict__trans1,axiom,
% 0.29/0.46 ! [B: real,A: real,C: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.29/0.46 => ( ( ord_less_real @ C @ B )
% 0.29/0.46 => ( ord_less_real @ C @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans1
% 0.29/0.46 thf(fact_131_dual__order_Ostrict__trans1,axiom,
% 0.29/0.46 ! [B: nat,A: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.46 => ( ( ord_less_nat @ C @ B )
% 0.29/0.46 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans1
% 0.29/0.46 thf(fact_132_order_Ostrict__iff__order,axiom,
% 0.29/0.46 ( ord_less_real
% 0.29/0.46 = ( ^ [A3: real,B3: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.46 & ( A3 != B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_iff_order
% 0.29/0.46 thf(fact_133_order_Ostrict__iff__order,axiom,
% 0.29/0.46 ( ord_less_nat
% 0.29/0.46 = ( ^ [A3: nat,B3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.29/0.46 & ( A3 != B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_iff_order
% 0.29/0.46 thf(fact_134_order_Oorder__iff__strict,axiom,
% 0.29/0.46 ( ord_less_eq_real
% 0.29/0.46 = ( ^ [A3: real,B3: real] :
% 0.29/0.46 ( ( ord_less_real @ A3 @ B3 )
% 0.29/0.46 | ( A3 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.order_iff_strict
% 0.29/0.46 thf(fact_135_order_Oorder__iff__strict,axiom,
% 0.29/0.46 ( ord_less_eq_nat
% 0.29/0.46 = ( ^ [A3: nat,B3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A3 @ B3 )
% 0.29/0.46 | ( A3 = B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.order_iff_strict
% 0.29/0.46 thf(fact_136_order_Ostrict__trans2,axiom,
% 0.29/0.46 ! [A: real,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.46 => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans2
% 0.29/0.46 thf(fact_137_order_Ostrict__trans2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans2
% 0.29/0.46 thf(fact_138_order_Ostrict__trans1,axiom,
% 0.29/0.46 ! [A: real,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.46 => ( ( ord_less_real @ B @ C )
% 0.29/0.46 => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans1
% 0.29/0.46 thf(fact_139_order_Ostrict__trans1,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_nat @ B @ C )
% 0.29/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans1
% 0.29/0.46 thf(fact_140_not__le__imp__less,axiom,
% 0.29/0.46 ! [Y: real,X2: real] :
% 0.29/0.46 ( ~ ( ord_less_eq_real @ Y @ X2 )
% 0.29/0.46 => ( ord_less_real @ X2 @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_le_imp_less
% 0.29/0.46 thf(fact_141_not__le__imp__less,axiom,
% 0.29/0.46 ! [Y: nat,X2: nat] :
% 0.29/0.46 ( ~ ( ord_less_eq_nat @ Y @ X2 )
% 0.29/0.46 => ( ord_less_nat @ X2 @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_le_imp_less
% 0.29/0.46 thf(fact_142_less__le__not__le,axiom,
% 0.29/0.46 ( ord_less_real
% 0.29/0.46 = ( ^ [X3: real,Y6: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X3 @ Y6 )
% 0.29/0.46 & ~ ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le_not_le
% 0.29/0.46 thf(fact_143_less__le__not__le,axiom,
% 0.29/0.46 ( ord_less_nat
% 0.29/0.46 = ( ^ [X3: nat,Y6: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 0.29/0.46 & ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le_not_le
% 0.29/0.46 thf(fact_144_le__imp__less__or__eq,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 | ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_imp_less_or_eq
% 0.29/0.46 thf(fact_145_le__imp__less__or__eq,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 | ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_imp_less_or_eq
% 0.29/0.46 thf(fact_146_le__less__linear,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.46 | ( ord_less_real @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less_linear
% 0.29/0.46 thf(fact_147_le__less__linear,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.46 | ( ord_less_nat @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less_linear
% 0.29/0.46 thf(fact_148_dense__le,axiom,
% 0.29/0.46 ! [Y: real,Z: real] :
% 0.29/0.46 ( ! [X: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y )
% 0.29/0.46 => ( ord_less_eq_real @ X @ Z ) )
% 0.29/0.46 => ( ord_less_eq_real @ Y @ Z ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_le
% 0.29/0.46 thf(fact_149_dense__ge,axiom,
% 0.29/0.46 ! [Z: real,Y: real] :
% 0.29/0.46 ( ! [X: real] :
% 0.29/0.46 ( ( ord_less_real @ Z @ X )
% 0.29/0.46 => ( ord_less_eq_real @ Y @ X ) )
% 0.29/0.46 => ( ord_less_eq_real @ Y @ Z ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense_ge
% 0.29/0.46 thf(fact_150_less__le__trans,axiom,
% 0.29/0.46 ! [X2: real,Y: real,Z: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_eq_real @ Y @ Z )
% 0.29/0.46 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le_trans
% 0.29/0.46 thf(fact_151_less__le__trans,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat,Z: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.29/0.46 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le_trans
% 0.29/0.46 thf(fact_152_le__less__trans,axiom,
% 0.29/0.46 ! [X2: real,Y: real,Z: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_real @ Y @ Z )
% 0.29/0.46 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less_trans
% 0.29/0.46 thf(fact_153_le__less__trans,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat,Z: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_nat @ Y @ Z )
% 0.29/0.46 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less_trans
% 0.29/0.46 thf(fact_154_less__imp__le,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_le
% 0.29/0.46 thf(fact_155_less__imp__le,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( ord_less_eq_nat @ X2 @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_le
% 0.29/0.46 thf(fact_156_antisym__conv2,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.46 => ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 0.29/0.46 = ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv2
% 0.29/0.46 thf(fact_157_antisym__conv2,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.46 => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 0.29/0.46 = ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv2
% 0.29/0.46 thf(fact_158_antisym__conv1,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ~ ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_eq_real @ X2 @ Y )
% 0.29/0.46 = ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv1
% 0.29/0.46 thf(fact_159_antisym__conv1,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ~ ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_eq_nat @ X2 @ Y )
% 0.29/0.46 = ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv1
% 0.29/0.46 thf(fact_160_le__neq__trans,axiom,
% 0.29/0.46 ! [A: real,B: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.46 => ( ( A != B )
% 0.29/0.46 => ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_neq_trans
% 0.29/0.46 thf(fact_161_le__neq__trans,axiom,
% 0.29/0.46 ! [A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( A != B )
% 0.29/0.46 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_neq_trans
% 0.29/0.46 thf(fact_162_not__less,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 0.29/0.46 = ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less
% 0.29/0.46 thf(fact_163_not__less,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 0.29/0.46 = ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less
% 0.29/0.46 thf(fact_164_not__le,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ~ ( ord_less_eq_real @ X2 @ Y ) )
% 0.29/0.46 = ( ord_less_real @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_le
% 0.29/0.46 thf(fact_165_not__le,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
% 0.29/0.46 = ( ord_less_nat @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_le
% 0.29/0.46 thf(fact_166_order__less__le__subst2,axiom,
% 0.29/0.46 ! [A: real,B: real,F: real > real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst2
% 0.29/0.46 thf(fact_167_order__less__le__subst2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst2
% 0.29/0.46 thf(fact_168_order__less__le__subst2,axiom,
% 0.29/0.46 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst2
% 0.29/0.46 thf(fact_169_order__less__le__subst2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst2
% 0.29/0.46 thf(fact_170_order__less__le__subst1,axiom,
% 0.29/0.46 ! [A: real,F: real > real,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst1
% 0.29/0.46 thf(fact_171_order__less__le__subst1,axiom,
% 0.29/0.46 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst1
% 0.29/0.46 thf(fact_172_order__less__le__subst1,axiom,
% 0.29/0.46 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst1
% 0.29/0.46 thf(fact_173_order__less__le__subst1,axiom,
% 0.29/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_le_subst1
% 0.29/0.46 thf(fact_174_order__le__less__subst2,axiom,
% 0.29/0.46 ! [A: real,B: real,F: real > real,C: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.46 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst2
% 0.29/0.46 thf(fact_175_order__le__less__subst2,axiom,
% 0.29/0.46 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.29/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst2
% 0.29/0.46 thf(fact_176_order__le__less__subst2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst2
% 0.29/0.46 thf(fact_177_order__le__less__subst2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst2
% 0.29/0.46 thf(fact_178_order__le__less__subst1,axiom,
% 0.29/0.46 ! [A: real,F: real > real,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_real @ B @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst1
% 0.29/0.46 thf(fact_179_order__le__less__subst1,axiom,
% 0.29/0.46 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_nat @ B @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst1
% 0.29/0.46 thf(fact_180_order__le__less__subst1,axiom,
% 0.29/0.46 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_real @ B @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst1
% 0.29/0.46 thf(fact_181_order__le__less__subst1,axiom,
% 0.29/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_nat @ B @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_le_less_subst1
% 0.29/0.46 thf(fact_182_less__le,axiom,
% 0.29/0.46 ( ord_less_real
% 0.29/0.46 = ( ^ [X3: real,Y6: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X3 @ Y6 )
% 0.29/0.46 & ( X3 != Y6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le
% 0.29/0.46 thf(fact_183_less__le,axiom,
% 0.29/0.46 ( ord_less_nat
% 0.29/0.46 = ( ^ [X3: nat,Y6: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 0.29/0.46 & ( X3 != Y6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_le
% 0.29/0.46 thf(fact_184_le__less,axiom,
% 0.29/0.46 ( ord_less_eq_real
% 0.29/0.46 = ( ^ [X3: real,Y6: real] :
% 0.29/0.46 ( ( ord_less_real @ X3 @ Y6 )
% 0.29/0.46 | ( X3 = Y6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less
% 0.29/0.46 thf(fact_185_le__less,axiom,
% 0.29/0.46 ( ord_less_eq_nat
% 0.29/0.46 = ( ^ [X3: nat,Y6: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X3 @ Y6 )
% 0.29/0.46 | ( X3 = Y6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_less
% 0.29/0.46 thf(fact_186_leI,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ~ ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % leI
% 0.29/0.46 thf(fact_187_leI,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ~ ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % leI
% 0.29/0.46 thf(fact_188_leD,axiom,
% 0.29/0.46 ! [Y: real,X2: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ Y @ X2 )
% 0.29/0.46 => ~ ( ord_less_real @ X2 @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % leD
% 0.29/0.46 thf(fact_189_leD,axiom,
% 0.29/0.46 ! [Y: nat,X2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ Y @ X2 )
% 0.29/0.46 => ~ ( ord_less_nat @ X2 @ Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % leD
% 0.29/0.46 thf(fact_190_top_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ top_top_set_a @ A )
% 0.29/0.46 => ( A = top_top_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_uniqueI
% 0.29/0.46 thf(fact_191_top_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: set_real] :
% 0.29/0.46 ( ( ord_less_eq_set_real @ top_top_set_real @ A )
% 0.29/0.46 => ( A = top_top_set_real ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_uniqueI
% 0.29/0.46 thf(fact_192_top_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: set_Pr1928503567a_real] :
% 0.29/0.46 ( ( ord_le1586073967a_real @ top_to2138011583a_real @ A )
% 0.29/0.46 => ( A = top_to2138011583a_real ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_uniqueI
% 0.29/0.46 thf(fact_193_top_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: produc1691597095_set_a] :
% 0.29/0.46 ( ( ord_le486764743_set_a @ top_to1477940855_set_a @ A )
% 0.29/0.46 => ( A = top_to1477940855_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_uniqueI
% 0.29/0.46 thf(fact_194_top_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: set_a] :
% 0.29/0.46 ( ( ord_less_eq_set_a @ top_top_set_a @ A )
% 0.29/0.46 = ( A = top_top_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_unique
% 0.29/0.46 thf(fact_195_top_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: set_real] :
% 0.29/0.46 ( ( ord_less_eq_set_real @ top_top_set_real @ A )
% 0.29/0.46 = ( A = top_top_set_real ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_unique
% 0.29/0.46 thf(fact_196_top_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: set_Pr1928503567a_real] :
% 0.29/0.46 ( ( ord_le1586073967a_real @ top_to2138011583a_real @ A )
% 0.29/0.46 = ( A = top_to2138011583a_real ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_unique
% 0.29/0.46 thf(fact_197_top_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: produc1691597095_set_a] :
% 0.29/0.46 ( ( ord_le486764743_set_a @ top_to1477940855_set_a @ A )
% 0.29/0.46 = ( A = top_to1477940855_set_a ) ) ).
% 0.29/0.46
% 0.29/0.46 % top.extremum_unique
% 0.29/0.46 thf(fact_198_top__greatest,axiom,
% 0.29/0.46 ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % top_greatest
% 0.29/0.46 thf(fact_199_top__greatest,axiom,
% 0.29/0.46 ! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% 0.29/0.46
% 0.29/0.46 % top_greatest
% 0.29/0.46 thf(fact_200_top__greatest,axiom,
% 0.29/0.46 ! [A: set_Pr1928503567a_real] : ( ord_le1586073967a_real @ A @ top_to2138011583a_real ) ).
% 0.29/0.46
% 0.29/0.46 % top_greatest
% 0.29/0.46 thf(fact_201_top__greatest,axiom,
% 0.29/0.46 ! [A: produc1691597095_set_a] : ( ord_le486764743_set_a @ A @ top_to1477940855_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % top_greatest
% 0.29/0.46 thf(fact_202_dist__not__less__zero,axiom,
% 0.29/0.46 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 0.29/0.46 ~ ( ord_less_real @ ( real_V404783528a_real @ X2 @ Y ) @ zero_zero_real ) ).
% 0.29/0.46
% 0.29/0.46 % dist_not_less_zero
% 0.29/0.46 thf(fact_203_dist__pos__lt,axiom,
% 0.29/0.46 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 0.29/0.46 ( ( X2 != Y )
% 0.29/0.46 => ( ord_less_real @ zero_zero_real @ ( real_V404783528a_real @ X2 @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dist_pos_lt
% 0.29/0.46 thf(fact_204_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.46 ! [B: real,A: real] :
% 0.29/0.46 ( ( ord_less_real @ B @ A )
% 0.29/0.46 => ( A != B ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_implies_not_eq
% 0.29/0.46 thf(fact_205_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.46 ! [B: nat,A: nat] :
% 0.29/0.46 ( ( ord_less_nat @ B @ A )
% 0.29/0.46 => ( A != B ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_implies_not_eq
% 0.29/0.46 thf(fact_206_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.46 ! [A: real,B: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( A != B ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_implies_not_eq
% 0.29/0.46 thf(fact_207_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.46 ! [A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( A != B ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_implies_not_eq
% 0.29/0.46 thf(fact_208_not__less__iff__gr__or__eq,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 0.29/0.46 = ( ( ord_less_real @ Y @ X2 )
% 0.29/0.46 | ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less_iff_gr_or_eq
% 0.29/0.46 thf(fact_209_not__less__iff__gr__or__eq,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 0.29/0.46 = ( ( ord_less_nat @ Y @ X2 )
% 0.29/0.46 | ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_less_iff_gr_or_eq
% 0.29/0.46 thf(fact_210_dual__order_Ostrict__trans,axiom,
% 0.29/0.46 ! [B: real,A: real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ B @ A )
% 0.29/0.46 => ( ( ord_less_real @ C @ B )
% 0.29/0.46 => ( ord_less_real @ C @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans
% 0.29/0.46 thf(fact_211_dual__order_Ostrict__trans,axiom,
% 0.29/0.46 ! [B: nat,A: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ B @ A )
% 0.29/0.46 => ( ( ord_less_nat @ C @ B )
% 0.29/0.46 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.strict_trans
% 0.29/0.46 thf(fact_212_linorder__less__wlog,axiom,
% 0.29/0.46 ! [P: real > real > $o,A: real,B: real] :
% 0.29/0.46 ( ! [A4: real,B4: real] :
% 0.29/0.46 ( ( ord_less_real @ A4 @ B4 )
% 0.29/0.46 => ( P @ A4 @ B4 ) )
% 0.29/0.46 => ( ! [A4: real] : ( P @ A4 @ A4 )
% 0.29/0.46 => ( ! [A4: real,B4: real] :
% 0.29/0.46 ( ( P @ B4 @ A4 )
% 0.29/0.46 => ( P @ A4 @ B4 ) )
% 0.29/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_less_wlog
% 0.29/0.46 thf(fact_213_linorder__less__wlog,axiom,
% 0.29/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.29/0.46 ( ! [A4: nat,B4: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A4 @ B4 )
% 0.29/0.46 => ( P @ A4 @ B4 ) )
% 0.29/0.46 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 0.29/0.46 => ( ! [A4: nat,B4: nat] :
% 0.29/0.46 ( ( P @ B4 @ A4 )
% 0.29/0.46 => ( P @ A4 @ B4 ) )
% 0.29/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_less_wlog
% 0.29/0.46 thf(fact_214_exists__least__iff,axiom,
% 0.29/0.46 ( ( ^ [P2: nat > $o] :
% 0.29/0.46 ? [X5: nat] : ( P2 @ X5 ) )
% 0.29/0.46 = ( ^ [P3: nat > $o] :
% 0.29/0.46 ? [N: nat] :
% 0.29/0.46 ( ( P3 @ N )
% 0.29/0.46 & ! [M: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M @ N )
% 0.29/0.46 => ~ ( P3 @ M ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % exists_least_iff
% 0.29/0.46 thf(fact_215_less__imp__not__less,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_less
% 0.29/0.46 thf(fact_216_less__imp__not__less,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_less
% 0.29/0.46 thf(fact_217_order_Ostrict__trans,axiom,
% 0.29/0.46 ! [A: real,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ( ord_less_real @ B @ C )
% 0.29/0.46 => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans
% 0.29/0.46 thf(fact_218_order_Ostrict__trans,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_nat @ B @ C )
% 0.29/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.strict_trans
% 0.29/0.46 thf(fact_219_dual__order_Oirrefl,axiom,
% 0.29/0.46 ! [A: real] :
% 0.29/0.46 ~ ( ord_less_real @ A @ A ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.irrefl
% 0.29/0.46 thf(fact_220_dual__order_Oirrefl,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ A @ A ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.irrefl
% 0.29/0.46 thf(fact_221_linorder__cases,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ~ ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( ( X2 != Y )
% 0.29/0.46 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_cases
% 0.29/0.46 thf(fact_222_linorder__cases,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ~ ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( ( X2 != Y )
% 0.29/0.46 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_cases
% 0.29/0.46 thf(fact_223_less__imp__triv,axiom,
% 0.29/0.46 ! [X2: real,Y: real,P: $o] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_real @ Y @ X2 )
% 0.29/0.46 => P ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_triv
% 0.29/0.46 thf(fact_224_less__imp__triv,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat,P: $o] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_nat @ Y @ X2 )
% 0.29/0.46 => P ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_triv
% 0.29/0.46 thf(fact_225_less__imp__not__eq2,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( Y != X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_eq2
% 0.29/0.46 thf(fact_226_less__imp__not__eq2,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( Y != X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_eq2
% 0.29/0.46 thf(fact_227_antisym__conv3,axiom,
% 0.29/0.46 ! [Y: real,X2: real] :
% 0.29/0.46 ( ~ ( ord_less_real @ Y @ X2 )
% 0.29/0.46 => ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 0.29/0.46 = ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv3
% 0.29/0.46 thf(fact_228_antisym__conv3,axiom,
% 0.29/0.46 ! [Y: nat,X2: nat] :
% 0.29/0.46 ( ~ ( ord_less_nat @ Y @ X2 )
% 0.29/0.46 => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 0.29/0.46 = ( X2 = Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv3
% 0.29/0.46 thf(fact_229_less__induct,axiom,
% 0.29/0.46 ! [P: nat > $o,A: nat] :
% 0.29/0.46 ( ! [X: nat] :
% 0.29/0.46 ( ! [Y7: nat] :
% 0.29/0.46 ( ( ord_less_nat @ Y7 @ X )
% 0.29/0.46 => ( P @ Y7 ) )
% 0.29/0.46 => ( P @ X ) )
% 0.29/0.46 => ( P @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_induct
% 0.29/0.46 thf(fact_230_less__not__sym,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_sym
% 0.29/0.46 thf(fact_231_less__not__sym,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_sym
% 0.29/0.46 thf(fact_232_less__imp__not__eq,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( X2 != Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_eq
% 0.29/0.46 thf(fact_233_less__imp__not__eq,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( X2 != Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_not_eq
% 0.29/0.46 thf(fact_234_dual__order_Oasym,axiom,
% 0.29/0.46 ! [B: real,A: real] :
% 0.29/0.46 ( ( ord_less_real @ B @ A )
% 0.29/0.46 => ~ ( ord_less_real @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.asym
% 0.29/0.46 thf(fact_235_dual__order_Oasym,axiom,
% 0.29/0.46 ! [B: nat,A: nat] :
% 0.29/0.46 ( ( ord_less_nat @ B @ A )
% 0.29/0.46 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.asym
% 0.29/0.46 thf(fact_236_ord__less__eq__trans,axiom,
% 0.29/0.46 ! [A: real,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ( B = C )
% 0.29/0.46 => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_less_eq_trans
% 0.29/0.46 thf(fact_237_ord__less__eq__trans,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( B = C )
% 0.29/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_less_eq_trans
% 0.29/0.46 thf(fact_238_ord__eq__less__trans,axiom,
% 0.29/0.46 ! [A: real,B: real,C: real] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ( ord_less_real @ B @ C )
% 0.29/0.46 => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_less_trans
% 0.29/0.46 thf(fact_239_ord__eq__less__trans,axiom,
% 0.29/0.46 ! [A: nat,B: nat,C: nat] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ( ord_less_nat @ B @ C )
% 0.29/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_less_trans
% 0.29/0.46 thf(fact_240_less__irrefl,axiom,
% 0.29/0.46 ! [X2: real] :
% 0.29/0.46 ~ ( ord_less_real @ X2 @ X2 ) ).
% 0.29/0.46
% 0.29/0.46 % less_irrefl
% 0.29/0.46 thf(fact_241_less__irrefl,axiom,
% 0.29/0.46 ! [X2: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 0.29/0.46
% 0.29/0.46 % less_irrefl
% 0.29/0.46 thf(fact_242_less__linear,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 | ( X2 = Y )
% 0.29/0.46 | ( ord_less_real @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_linear
% 0.29/0.46 thf(fact_243_less__linear,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 | ( X2 = Y )
% 0.29/0.46 | ( ord_less_nat @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_linear
% 0.29/0.46 thf(fact_244_less__trans,axiom,
% 0.29/0.46 ! [X2: real,Y: real,Z: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_real @ Y @ Z )
% 0.29/0.46 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_trans
% 0.29/0.46 thf(fact_245_less__trans,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat,Z: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( ( ord_less_nat @ Y @ Z )
% 0.29/0.46 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_trans
% 0.29/0.46 thf(fact_246_less__asym_H,axiom,
% 0.29/0.46 ! [A: real,B: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_asym'
% 0.29/0.46 thf(fact_247_less__asym_H,axiom,
% 0.29/0.46 ! [A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_asym'
% 0.29/0.46 thf(fact_248_less__asym,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_asym
% 0.29/0.46 thf(fact_249_less__asym,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_asym
% 0.29/0.46 thf(fact_250_less__imp__neq,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( X2 != Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_neq
% 0.29/0.46 thf(fact_251_less__imp__neq,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( X2 != Y ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_neq
% 0.29/0.46 thf(fact_252_dense,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ? [Z3: real] :
% 0.29/0.46 ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.46 & ( ord_less_real @ Z3 @ Y ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dense
% 0.29/0.46 thf(fact_253_order_Oasym,axiom,
% 0.29/0.46 ! [A: real,B: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.asym
% 0.29/0.46 thf(fact_254_order_Oasym,axiom,
% 0.29/0.46 ! [A: nat,B: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.asym
% 0.29/0.46 thf(fact_255_neq__iff,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( X2 != Y )
% 0.29/0.46 = ( ( ord_less_real @ X2 @ Y )
% 0.29/0.46 | ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neq_iff
% 0.29/0.46 thf(fact_256_neq__iff,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( X2 != Y )
% 0.29/0.46 = ( ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 | ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neq_iff
% 0.29/0.46 thf(fact_257_neqE,axiom,
% 0.29/0.46 ! [X2: real,Y: real] :
% 0.29/0.46 ( ( X2 != Y )
% 0.29/0.46 => ( ~ ( ord_less_real @ X2 @ Y )
% 0.29/0.46 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neqE
% 0.29/0.46 thf(fact_258_neqE,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( X2 != Y )
% 0.29/0.46 => ( ~ ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % neqE
% 0.29/0.46 thf(fact_259_gt__ex,axiom,
% 0.29/0.46 ! [X2: real] :
% 0.29/0.46 ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 0.29/0.46
% 0.29/0.46 % gt_ex
% 0.29/0.46 thf(fact_260_gt__ex,axiom,
% 0.29/0.46 ! [X2: nat] :
% 0.29/0.46 ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% 0.29/0.46
% 0.29/0.46 % gt_ex
% 0.29/0.46 thf(fact_261_lt__ex,axiom,
% 0.29/0.46 ! [X2: real] :
% 0.29/0.46 ? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).
% 0.29/0.46
% 0.29/0.46 % lt_ex
% 0.29/0.46 thf(fact_262_order__less__subst2,axiom,
% 0.29/0.46 ! [A: real,B: real,F: real > real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst2
% 0.29/0.46 thf(fact_263_order__less__subst2,axiom,
% 0.29/0.46 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst2
% 0.29/0.46 thf(fact_264_order__less__subst2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst2
% 0.29/0.46 thf(fact_265_order__less__subst2,axiom,
% 0.29/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst2
% 0.29/0.46 thf(fact_266_order__less__subst1,axiom,
% 0.29/0.46 ! [A: real,F: real > real,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_real @ B @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst1
% 0.29/0.46 thf(fact_267_order__less__subst1,axiom,
% 0.29/0.46 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_nat @ B @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst1
% 0.29/0.46 thf(fact_268_order__less__subst1,axiom,
% 0.29/0.46 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.29/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_real @ B @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst1
% 0.29/0.46 thf(fact_269_order__less__subst1,axiom,
% 0.29/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_nat @ B @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_less_subst1
% 0.29/0.46 thf(fact_270_ord__less__eq__subst,axiom,
% 0.29/0.46 ! [A: real,B: real,F: real > real,C: real] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ( ( F @ B )
% 0.29/0.46 = C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_less_eq_subst
% 0.29/0.46 thf(fact_271_ord__less__eq__subst,axiom,
% 0.29/0.46 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.29/0.46 ( ( ord_less_real @ A @ B )
% 0.29/0.46 => ( ( ( F @ B )
% 0.29/0.46 = C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_less_eq_subst
% 0.29/0.46 thf(fact_272_ord__less__eq__subst,axiom,
% 0.29/0.46 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ( F @ B )
% 0.29/0.46 = C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_less_eq_subst
% 0.29/0.46 thf(fact_273_ord__less__eq__subst,axiom,
% 0.29/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.29/0.46 ( ( ord_less_nat @ A @ B )
% 0.29/0.46 => ( ( ( F @ B )
% 0.29/0.46 = C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_less_eq_subst
% 0.29/0.46 thf(fact_274_ord__eq__less__subst,axiom,
% 0.29/0.46 ! [A: real,F: real > real,B: real,C: real] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_real @ B @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_less_subst
% 0.29/0.46 thf(fact_275_ord__eq__less__subst,axiom,
% 0.29/0.46 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_real @ B @ C )
% 0.29/0.46 => ( ! [X: real,Y3: real] :
% 0.29/0.46 ( ( ord_less_real @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_less_subst
% 0.29/0.46 thf(fact_276_ord__eq__less__subst,axiom,
% 0.29/0.46 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_nat @ B @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_less_subst
% 0.29/0.46 thf(fact_277_ord__eq__less__subst,axiom,
% 0.29/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.29/0.46 ( ( A
% 0.29/0.46 = ( F @ B ) )
% 0.29/0.46 => ( ( ord_less_nat @ B @ C )
% 0.29/0.46 => ( ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.29/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_less_subst
% 0.29/0.46 thf(fact_278_old_Oprod_Oinducts,axiom,
% 0.29/0.46 ! [P: product_prod_a_real > $o,Prod: product_prod_a_real] :
% 0.29/0.46 ( ! [A4: a,B4: real] : ( P @ ( product_Pair_a_real @ A4 @ B4 ) )
% 0.29/0.46 => ( P @ Prod ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.inducts
% 0.29/0.46 thf(fact_279_old_Oprod_Oinducts,axiom,
% 0.29/0.46 ! [P: produc1691597095_set_a > $o,Prod: produc1691597095_set_a] :
% 0.29/0.46 ( ! [A4: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A4 @ B4 ) )
% 0.29/0.46 => ( P @ Prod ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.inducts
% 0.29/0.46 thf(fact_280_old_Oprod_Oexhaust,axiom,
% 0.29/0.46 ! [Y: product_prod_a_real] :
% 0.29/0.46 ~ ! [A4: a,B4: real] :
% 0.29/0.46 ( Y
% 0.29/0.46 != ( product_Pair_a_real @ A4 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.exhaust
% 0.29/0.46 thf(fact_281_old_Oprod_Oexhaust,axiom,
% 0.29/0.46 ! [Y: produc1691597095_set_a] :
% 0.29/0.46 ~ ! [A4: set_a,B4: set_a] :
% 0.29/0.46 ( Y
% 0.29/0.46 != ( produc1928581911_set_a @ A4 @ B4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.exhaust
% 0.29/0.46 thf(fact_282_Pair__inject,axiom,
% 0.29/0.46 ! [A: a,B: real,A2: a,B2: real] :
% 0.29/0.46 ( ( ( product_Pair_a_real @ A @ B )
% 0.29/0.46 = ( product_Pair_a_real @ A2 @ B2 ) )
% 0.29/0.46 => ~ ( ( A = A2 )
% 0.29/0.46 => ( B != B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Pair_inject
% 0.29/0.46 thf(fact_283_Pair__inject,axiom,
% 0.29/0.46 ! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
% 0.29/0.46 ( ( ( produc1928581911_set_a @ A @ B )
% 0.29/0.46 = ( produc1928581911_set_a @ A2 @ B2 ) )
% 0.29/0.46 => ~ ( ( A = A2 )
% 0.29/0.46 => ( B != B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Pair_inject
% 0.29/0.46 thf(fact_284_prod__cases,axiom,
% 0.29/0.46 ! [P: product_prod_a_real > $o,P4: product_prod_a_real] :
% 0.29/0.46 ( ! [A4: a,B4: real] : ( P @ ( product_Pair_a_real @ A4 @ B4 ) )
% 0.29/0.46 => ( P @ P4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases
% 0.29/0.46 thf(fact_285_prod__cases,axiom,
% 0.29/0.46 ! [P: produc1691597095_set_a > $o,P4: produc1691597095_set_a] :
% 0.29/0.46 ( ! [A4: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A4 @ B4 ) )
% 0.29/0.46 => ( P @ P4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases
% 0.29/0.46 thf(fact_286_surj__pair,axiom,
% 0.29/0.46 ! [P4: product_prod_a_real] :
% 0.29/0.46 ? [X: a,Y3: real] :
% 0.29/0.46 ( P4
% 0.29/0.46 = ( product_Pair_a_real @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % surj_pair
% 0.29/0.46 thf(fact_287_surj__pair,axiom,
% 0.29/0.46 ! [P4: produc1691597095_set_a] :
% 0.29/0.46 ? [X: set_a,Y3: set_a] :
% 0.29/0.46 ( P4
% 0.29/0.46 = ( produc1928581911_set_a @ X @ Y3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % surj_pair
% 0.29/0.46 thf(fact_288_UNIV__witness,axiom,
% 0.29/0.46 ? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_witness
% 0.29/0.46 thf(fact_289_UNIV__witness,axiom,
% 0.29/0.46 ? [X: real] : ( member_real @ X @ top_top_set_real ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_witness
% 0.29/0.46 thf(fact_290_UNIV__witness,axiom,
% 0.29/0.46 ? [X: product_prod_a_real] : ( member1103263856a_real @ X @ top_to2138011583a_real ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_witness
% 0.29/0.46 thf(fact_291_UNIV__eq__I,axiom,
% 0.29/0.46 ! [A5: set_a] :
% 0.29/0.46 ( ! [X: a] : ( member_a @ X @ A5 )
% 0.29/0.46 => ( top_top_set_a = A5 ) ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_eq_I
% 0.29/0.46 thf(fact_292_UNIV__eq__I,axiom,
% 0.29/0.46 ! [A5: set_real] :
% 0.29/0.46 ( ! [X: real] : ( member_real @ X @ A5 )
% 0.29/0.46 => ( top_top_set_real = A5 ) ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_eq_I
% 0.29/0.46 thf(fact_293_UNIV__eq__I,axiom,
% 0.29/0.46 ! [A5: set_Pr1928503567a_real] :
% 0.29/0.46 ( ! [X: product_prod_a_real] : ( member1103263856a_real @ X @ A5 )
% 0.29/0.46 => ( top_to2138011583a_real = A5 ) ) ).
% 0.29/0.46
% 0.29/0.46 % UNIV_eq_I
% 0.29/0.46 thf(fact_294_dist__commute,axiom,
% 0.29/0.46 ( real_V404783528a_real
% 0.29/0.46 = ( ^ [X3: product_prod_a_real,Y6: product_prod_a_real] : ( real_V404783528a_real @ Y6 @ X3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dist_commute
% 0.29/0.46 thf(fact_295_closure__UNIV,axiom,
% 0.29/0.46 ( ( elementary_closure_a @ top_top_set_a )
% 0.29/0.46 = top_top_set_a ) ).
% 0.29/0.46
% 0.29/0.46 % closure_UNIV
% 0.29/0.46 thf(fact_296_closure__UNIV,axiom,
% 0.29/0.46 ( ( elemen1168706977e_real @ top_top_set_real )
% 0.29/0.46 = top_top_set_real ) ).
% 0.29/0.46
% 0.29/0.46 % closure_UNIV
% 0.29/0.46 thf(fact_297_closure__UNIV,axiom,
% 0.29/0.46 ( ( elemen1695521870a_real @ top_to2138011583a_real )
% 0.29/0.46 = top_to2138011583a_real ) ).
% 0.29/0.46
% 0.29/0.46 % closure_UNIV
% 0.29/0.46 thf(fact_298_closure__approachable__le,axiom,
% 0.29/0.46 ! [X2: real,S2: set_real] :
% 0.29/0.46 ( ( member_real @ X2 @ ( elemen1168706977e_real @ S2 ) )
% 0.29/0.46 = ( ! [E2: real] :
% 0.29/0.46 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.29/0.46 => ? [X3: real] :
% 0.29/0.46 ( ( member_real @ X3 @ S2 )
% 0.29/0.46 & ( ord_less_eq_real @ ( real_V1934908667t_real @ X3 @ X2 ) @ E2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closure_approachable_le
% 0.29/0.46 thf(fact_299_closure__approachable__le,axiom,
% 0.29/0.46 ! [X2: a,S2: set_a] :
% 0.29/0.46 ( ( member_a @ X2 @ ( elementary_closure_a @ S2 ) )
% 0.29/0.46 = ( ! [E2: real] :
% 0.29/0.46 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.29/0.46 => ? [X3: a] :
% 0.29/0.46 ( ( member_a @ X3 @ S2 )
% 0.29/0.46 & ( ord_less_eq_real @ ( real_V1514887919dist_a @ X3 @ X2 ) @ E2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closure_approachable_le
% 0.29/0.46 thf(fact_300_closure__approachable__le,axiom,
% 0.29/0.46 ! [X2: product_prod_a_real,S2: set_Pr1928503567a_real] :
% 0.29/0.46 ( ( member1103263856a_real @ X2 @ ( elemen1695521870a_real @ S2 ) )
% 0.29/0.46 = ( ! [E2: real] :
% 0.29/0.46 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.29/0.46 => ? [X3: product_prod_a_real] :
% 0.29/0.46 ( ( member1103263856a_real @ X3 @ S2 )
% 0.29/0.46 & ( ord_less_eq_real @ ( real_V404783528a_real @ X3 @ X2 ) @ E2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % closure_approachable_le
% 0.29/0.46 thf(fact_301_not__gr__zero,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.29/0.46 = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_gr_zero
% 0.29/0.46 thf(fact_302_le__zero__eq,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.29/0.46 = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_zero_eq
% 0.29/0.46 thf(fact_303_seq__mono__lemma,axiom,
% 0.29/0.46 ! [M2: nat,D: nat > real,E: nat > real] :
% 0.29/0.46 ( ! [N3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M2 @ N3 )
% 0.29/0.46 => ( ord_less_real @ ( D @ N3 ) @ ( E @ N3 ) ) )
% 0.29/0.46 => ( ! [N3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M2 @ N3 )
% 0.29/0.46 => ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M2 ) ) )
% 0.29/0.46 => ! [N4: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.29/0.46 => ( ord_less_real @ ( D @ N4 ) @ ( E @ M2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % seq_mono_lemma
% 0.29/0.46 thf(fact_304_le0,axiom,
% 0.29/0.46 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.29/0.46
% 0.29/0.46 % le0
% 0.29/0.46 thf(fact_305_bot__nat__0_Oextremum,axiom,
% 0.29/0.46 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum
% 0.29/0.46 thf(fact_306_neq0__conv,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ( ( N2 != zero_zero_nat )
% 0.29/0.46 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % neq0_conv
% 0.29/0.46 thf(fact_307_less__nat__zero__code,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % less_nat_zero_code
% 0.29/0.46 thf(fact_308_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ( ( A != zero_zero_nat )
% 0.29/0.46 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.not_eq_extremum
% 0.29/0.46 thf(fact_309_gr0I,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ( ( N2 != zero_zero_nat )
% 0.29/0.46 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr0I
% 0.29/0.46 thf(fact_310_not__gr0,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.29/0.46 = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % not_gr0
% 0.29/0.46 thf(fact_311_not__less0,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % not_less0
% 0.29/0.46 thf(fact_312_less__zeroE,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % less_zeroE
% 0.29/0.46 thf(fact_313_gr__implies__not0,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.46 => ( N2 != zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % gr_implies_not0
% 0.29/0.46 thf(fact_314_infinite__descent0,axiom,
% 0.29/0.46 ! [P: nat > $o,N2: nat] :
% 0.29/0.46 ( ( P @ zero_zero_nat )
% 0.29/0.46 => ( ! [N3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.29/0.46 => ( ~ ( P @ N3 )
% 0.29/0.46 => ? [M3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.29/0.46 & ~ ( P @ M3 ) ) ) )
% 0.29/0.46 => ( P @ N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent0
% 0.29/0.46 thf(fact_315_bot__nat__0_Oextremum__strict,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_strict
% 0.29/0.46 thf(fact_316_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.46 => ( A = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_uniqueI
% 0.29/0.46 thf(fact_317_bot__nat__0_Oextremum__unique,axiom,
% 0.29/0.46 ! [A: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.46 = ( A = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % bot_nat_0.extremum_unique
% 0.29/0.46 thf(fact_318_less__mono__imp__le__mono,axiom,
% 0.29/0.46 ! [F: nat > nat,I: nat,J: nat] :
% 0.29/0.46 ( ! [I2: nat,J2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ I2 @ J2 )
% 0.29/0.46 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_mono_imp_le_mono
% 0.29/0.46 thf(fact_319_le__neq__implies__less,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.29/0.46 => ( ( M2 != N2 )
% 0.29/0.46 => ( ord_less_nat @ M2 @ N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_neq_implies_less
% 0.29/0.46 thf(fact_320_Nat_Oex__has__greatest__nat,axiom,
% 0.29/0.46 ! [P: nat > $o,K: nat,B: nat] :
% 0.29/0.46 ( ( P @ K )
% 0.29/0.46 => ( ! [Y3: nat] :
% 0.29/0.46 ( ( P @ Y3 )
% 0.29/0.46 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.29/0.46 => ? [X: nat] :
% 0.29/0.46 ( ( P @ X )
% 0.29/0.46 & ! [Y7: nat] :
% 0.29/0.46 ( ( P @ Y7 )
% 0.29/0.46 => ( ord_less_eq_nat @ Y7 @ X ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.ex_has_greatest_nat
% 0.29/0.46 thf(fact_321_less__or__eq__imp__le,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.46 | ( M2 = N2 ) )
% 0.29/0.46 => ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_or_eq_imp_le
% 0.29/0.46 thf(fact_322_le__eq__less__or__eq,axiom,
% 0.29/0.46 ( ord_less_eq_nat
% 0.29/0.46 = ( ^ [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M @ N )
% 0.29/0.46 | ( M = N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_eq_less_or_eq
% 0.29/0.46 thf(fact_323_less__imp__le__nat,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.46 => ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_le_nat
% 0.29/0.46 thf(fact_324_ex__least__nat__le,axiom,
% 0.29/0.46 ! [P: nat > $o,N2: nat] :
% 0.29/0.46 ( ( P @ N2 )
% 0.29/0.46 => ( ~ ( P @ zero_zero_nat )
% 0.29/0.46 => ? [K2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.29/0.46 & ! [I3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ I3 @ K2 )
% 0.29/0.46 => ~ ( P @ I3 ) )
% 0.29/0.46 & ( P @ K2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_least_nat_le
% 0.29/0.46 thf(fact_325_nat__le__linear,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.29/0.46 | ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_linear
% 0.29/0.46 thf(fact_326_nat__less__le,axiom,
% 0.29/0.46 ( ord_less_nat
% 0.29/0.46 = ( ^ [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 & ( M != N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_le
% 0.29/0.46 thf(fact_327_le__antisym,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N2 @ M2 )
% 0.29/0.46 => ( M2 = N2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_antisym
% 0.29/0.46 thf(fact_328_eq__imp__le,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( M2 = N2 )
% 0.29/0.46 => ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_imp_le
% 0.29/0.46 thf(fact_329_le__trans,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.46 => ( ( ord_less_eq_nat @ J @ K )
% 0.29/0.46 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_trans
% 0.29/0.46 thf(fact_330_le__refl,axiom,
% 0.29/0.46 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 0.29/0.46
% 0.29/0.46 % le_refl
% 0.29/0.46 thf(fact_331_le__0__eq,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.29/0.46 = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_0_eq
% 0.29/0.46 thf(fact_332_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.46 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_nat.simps(1)
% 0.29/0.46 thf(fact_333_linorder__neqE__nat,axiom,
% 0.29/0.46 ! [X2: nat,Y: nat] :
% 0.29/0.46 ( ( X2 != Y )
% 0.29/0.46 => ( ~ ( ord_less_nat @ X2 @ Y )
% 0.29/0.46 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_neqE_nat
% 0.29/0.46 thf(fact_334_infinite__descent,axiom,
% 0.29/0.46 ! [P: nat > $o,N2: nat] :
% 0.29/0.46 ( ! [N3: nat] :
% 0.29/0.46 ( ~ ( P @ N3 )
% 0.29/0.46 => ? [M3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.29/0.46 & ~ ( P @ M3 ) ) )
% 0.29/0.46 => ( P @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent
% 0.29/0.46 thf(fact_335_nat__less__induct,axiom,
% 0.29/0.46 ! [P: nat > $o,N2: nat] :
% 0.29/0.46 ( ! [N3: nat] :
% 0.29/0.46 ( ! [M3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.29/0.46 => ( P @ M3 ) )
% 0.29/0.46 => ( P @ N3 ) )
% 0.29/0.46 => ( P @ N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_induct
% 0.29/0.46 thf(fact_336_less__irrefl__nat,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N2 @ N2 ) ).
% 0.29/0.46
% 0.29/0.46 % less_irrefl_nat
% 0.29/0.46 thf(fact_337_less__not__refl3,axiom,
% 0.29/0.46 ! [S: nat,T: nat] :
% 0.29/0.46 ( ( ord_less_nat @ S @ T )
% 0.29/0.46 => ( S != T ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl3
% 0.29/0.46 thf(fact_338_less__not__refl2,axiom,
% 0.29/0.46 ! [N2: nat,M2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ N2 @ M2 )
% 0.29/0.46 => ( M2 != N2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl2
% 0.29/0.46 thf(fact_339_less__not__refl,axiom,
% 0.29/0.46 ! [N2: nat] :
% 0.29/0.46 ~ ( ord_less_nat @ N2 @ N2 ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl
% 0.29/0.46 thf(fact_340_nat__neq__iff,axiom,
% 0.29/0.46 ! [M2: nat,N2: nat] :
% 0.29/0.46 ( ( M2 != N2 )
% 0.29/0.46 = ( ( ord_less_nat @ M2 @ N2 )
% 0.29/0.46 | ( ord_less_nat @ N2 @ M2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_neq_iff
% 0.29/0.46 thf(fact_341_nat__descend__induct,axiom,
% 0.29/0.46 ! [N2: nat,P: nat > $o,M2: nat] :
% 0.29/0.46 ( ! [K2: nat] :
% 0.29/0.46 ( ( ord_less_nat @ N2 @ K2 )
% 0.29/0.46 => ( P @ K2 ) )
% 0.29/0.46 => ( ! [K2: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.29/0.46 => ( ! [I3: nat] :
% 0.29/0.46 ( ( ord_less_nat @ K2 @ I3 )
% 0.29/0.46 => ( P @ I3 ) )
% 0.29/0.46 => ( P @ K2 ) ) )
% 0.29/0.46 => ( P @ M2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_descend_induct
% 0.29/0.46 thf(fact_342_less__eq__real__def,axiom,
% 0.29/0.46 ( ord_less_eq_real
% 0.29/0.46 = ( ^ [X3: real,Y6: real] :
% 0.29/0.46 ( ( ord_less_real @ X3 @ Y6 )
% 0.29/0.46 | ( X3 = Y6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eq_real_def
% 0.29/0.46 thf(fact_343_complete__real,axiom,
% 0.29/0.46 ! [S2: set_real] :
% 0.29/0.46 ( ? [X6: real] : ( member_real @ X6 @ S2 )
% 0.29/0.46 => ( ? [Z4: real] :
% 0.29/0.46 ! [X: real] :
% 0.29/0.46 ( ( member_real @ X @ S2 )
% 0.29/0.46 => ( ord_less_eq_real @ X @ Z4 ) )
% 0.29/0.46 => ? [Y3: real] :
% 0.29/0.46 ( ! [X6: real] :
% 0.29/0.46 ( ( member_real @ X6 @ S2 )
% 0.29/0.46 => ( ord_less_eq_real @ X6 @ Y3 ) )
% 0.29/0.46 & ! [Z4: real] :
% 0.29/0.46 ( ! [X: real] :
% 0.29/0.46 ( ( member_real @ X @ S2 )
% 0.29/0.46 => ( ord_less_eq_real @ X @ Z4 ) )
% 0.29/0.46 => ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % complete_real
% 0.29/0.46
% 0.29/0.46 % Conjectures (2)
% 0.29/0.46 thf(conj_0,hypothesis,
% 0.29/0.46 ! [A6: a,B5: real] :
% 0.29/0.46 ( ( ( member1103263856a_real @ ( product_Pair_a_real @ A6 @ B5 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
% 0.29/0.46 & ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ A6 @ B5 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) )
% 0.29/0.46 => thesis ) ).
% 0.29/0.46
% 0.29/0.46 thf(conj_1,conjecture,
% 0.29/0.46 thesis ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.58/1.44 ---- Embedded in THF ---
% 2.58/1.44 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.58/1.44 %%% Generated on Tue Feb 24 08:26:40 EST 2026
% 2.58/1.44 %%% using '$$dhol' embedding, version 1.3.0.
% 2.58/1.44 %%% Logic specification used:
% 2.58/1.44 %%% thf(spec, logic, $$dhol).
% 2.58/1.44
% 2.58/1.44 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.5gsfs41n1a/DTF2DTF2567.p
% See solution above
% 2.58/1.44 ------------------------
% 2.58/1.49 ---- Cleaned THF ---
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
% 2.58/1.49 produc2021989178_set_a: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
% 2.58/1.49 produc211508562a_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
% 2.58/1.49 produc1312517202t_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J_J,type,
% 2.58/1.49 produc304417648_set_a: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
% 2.58/1.49 produc509513372a_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_Mt__Set__Oset_Itf__a_J_J,type,
% 2.58/1.49 produc25772804_set_a: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J_J,type,
% 2.58/1.49 set_Pr1158285382l_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
% 2.58/1.49 produc792423270l_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
% 2.58/1.49 produc1931999653t_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
% 2.58/1.49 produc232195311t_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_Itf__a_J_J,type,
% 2.58/1.49 produc1350454001_set_a: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 2.58/1.49 set_Pr147102617l_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
% 2.58/1.49 produc1691597095_set_a: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
% 2.58/1.49 set_Pr1928503567a_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 2.58/1.49 produc957004601l_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 2.58/1.49 produc581916509al_nat: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
% 2.58/1.49 produc845587165t_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.58/1.49 product_prod_nat_nat: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 2.58/1.49 product_prod_a_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 2.58/1.49 set_real: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Extended____Real__Oereal,type,
% 2.58/1.49 extended_ereal: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.58/1.49 set_a: $tType ).
% 2.58/1.49
% 2.58/1.49 thf(ty_n_t__Real__Oreal,type,
% 2.58/1.49 real: $tType ).
% 2.58/1.49
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% 2.58/1.49 ! [X1: set_a,X22: set_a,Y1: set_a,Y2: set_a] :
% 2.58/1.49 ( ( ( produc1928581911_set_a @ X1 @ X22 )
% 2.58/1.49 = ( produc1928581911_set_a @ Y1 @ Y2 ) )
% 2.58/1.49 = ( ( X1 = Y1 )
% 2.58/1.49 & ( X22 = Y2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_10_prod_Oinject,axiom,
% 2.58/1.49 ! [X1: a,X22: real,Y1: a,Y2: real] :
% 2.58/1.49 ( ( ( product_Pair_a_real @ X1 @ X22 )
% 2.58/1.49 = ( product_Pair_a_real @ Y1 @ Y2 ) )
% 2.58/1.49 = ( ( X1 = Y1 )
% 2.58/1.49 & ( X22 = Y2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_11_old_Oprod_Oinject,axiom,
% 2.58/1.49 ! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
% 2.58/1.49 ( ( ( produc1928581911_set_a @ A @ B )
% 2.58/1.49 = ( produc1928581911_set_a @ A2 @ B2 ) )
% 2.58/1.49 = ( ( A = A2 )
% 2.58/1.49 & ( B = B2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_12_old_Oprod_Oinject,axiom,
% 2.58/1.49 ! [A: a,B: real,A2: a,B2: real] :
% 2.58/1.49 ( ( ( product_Pair_a_real @ A @ B )
% 2.58/1.49 = ( product_Pair_a_real @ A2 @ B2 ) )
% 2.58/1.49 = ( ( A = A2 )
% 2.58/1.49 & ( B = B2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_13_dist__commute__lessI,axiom,
% 2.58/1.49 ! [Y: product_prod_a_real,X2: product_prod_a_real,E: real] :
% 2.58/1.49 ( ( ord_less_real @ ( real_V404783528a_real @ Y @ X2 ) @ E )
% 2.58/1.49 => ( ord_less_real @ ( real_V404783528a_real @ X2 @ Y ) @ E ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_14_top__prod__def,axiom,
% 2.58/1.49 ( top_to1477940855_set_a
% 2.58/1.49 = ( produc1928581911_set_a @ top_top_set_a @ top_top_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_15_top__prod__def,axiom,
% 2.58/1.49 ( top_to35724191t_real
% 2.58/1.49 = ( produc1266579561t_real @ top_top_set_a @ top_top_set_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_16_top__prod__def,axiom,
% 2.58/1.49 ( top_to1153982881_set_a
% 2.58/1.49 = ( produc190993315_set_a @ top_top_set_real @ top_top_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_17_top__prod__def,axiom,
% 2.58/1.49 ( top_to95743733t_real
% 2.58/1.49 = ( produc119681757t_real @ top_top_set_real @ top_top_set_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_18_top__prod__def,axiom,
% 2.58/1.49 ( top_to829322572a_real
% 2.58/1.49 = ( produc383769494a_real @ top_top_set_a @ top_to2138011583a_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_19_top__prod__def,axiom,
% 2.58/1.49 ( top_to2037809826a_real
% 2.58/1.49 = ( produc150798858a_real @ top_top_set_real @ top_to2138011583a_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_20_top__prod__def,axiom,
% 2.58/1.49 ( top_to345582004_set_a
% 2.58/1.49 = ( produc1220567734_set_a @ top_to2138011583a_real @ top_top_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_21_top__prod__def,axiom,
% 2.58/1.49 ( top_to991334818t_real
% 2.58/1.49 = ( produc471223946t_real @ top_to2138011583a_real @ top_top_set_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_22_top__prod__def,axiom,
% 2.58/1.49 ( top_to724687040_set_a
% 2.58/1.49 = ( produc272677600_set_a @ top_top_set_a @ top_to1477940855_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_23_top__prod__def,axiom,
% 2.58/1.49 ( top_to522338794_set_a
% 2.58/1.49 = ( produc1162518380_set_a @ top_top_set_real @ top_to1477940855_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_24_top_Oextremum__strict,axiom,
% 2.58/1.49 ! [A: set_a] :
% 2.58/1.49 ~ ( ord_less_set_a @ top_top_set_a @ A ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_25_top_Oextremum__strict,axiom,
% 2.58/1.49 ! [A: set_real] :
% 2.58/1.49 ~ ( ord_less_set_real @ top_top_set_real @ A ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_26_top_Oextremum__strict,axiom,
% 2.58/1.49 ! [A: set_Pr1928503567a_real] :
% 2.58/1.49 ~ ( ord_le1379034467a_real @ top_to2138011583a_real @ A ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_27_top_Oextremum__strict,axiom,
% 2.58/1.49 ! [A: produc1691597095_set_a] :
% 2.58/1.49 ~ ( ord_le297355219_set_a @ top_to1477940855_set_a @ A ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_28_top_Onot__eq__extremum,axiom,
% 2.58/1.49 ! [A: set_a] :
% 2.58/1.49 ( ( A != top_top_set_a )
% 2.58/1.49 = ( ord_less_set_a @ A @ top_top_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_29_top_Onot__eq__extremum,axiom,
% 2.58/1.49 ! [A: set_real] :
% 2.58/1.49 ( ( A != top_top_set_real )
% 2.58/1.49 = ( ord_less_set_real @ A @ top_top_set_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_30_top_Onot__eq__extremum,axiom,
% 2.58/1.49 ! [A: set_Pr1928503567a_real] :
% 2.58/1.49 ( ( A != top_to2138011583a_real )
% 2.58/1.49 = ( ord_le1379034467a_real @ A @ top_to2138011583a_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_31_top_Onot__eq__extremum,axiom,
% 2.58/1.49 ! [A: produc1691597095_set_a] :
% 2.58/1.49 ( ( A != top_to1477940855_set_a )
% 2.58/1.49 = ( ord_le297355219_set_a @ A @ top_to1477940855_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_32_epigraph__mono,axiom,
% 2.58/1.49 ! [X2: real,Y: real,F: real > extended_ereal,Z: real] :
% 2.58/1.49 ( ( ( member1068169442l_real @ ( produc705216881l_real @ X2 @ Y ) @ ( lower_1607476196h_real @ top_top_set_real @ F ) )
% 2.58/1.49 & ( ord_less_eq_real @ Y @ Z ) )
% 2.58/1.49 => ( member1068169442l_real @ ( produc705216881l_real @ X2 @ Z ) @ ( lower_1607476196h_real @ top_top_set_real @ F ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_33_epigraph__mono,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real,Y: real,F: product_prod_a_real > extended_ereal,Z: real] :
% 2.58/1.49 ( ( ( member565909519l_real @ ( produc1214388894l_real @ X2 @ Y ) @ ( lower_870348689a_real @ top_to2138011583a_real @ F ) )
% 2.58/1.49 & ( ord_less_eq_real @ Y @ Z ) )
% 2.58/1.49 => ( member565909519l_real @ ( produc1214388894l_real @ X2 @ Z ) @ ( lower_870348689a_real @ top_to2138011583a_real @ F ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_34_epigraph__mono,axiom,
% 2.58/1.49 ! [X2: a,Y: real,F: a > extended_ereal,Z: real] :
% 2.58/1.49 ( ( ( member1103263856a_real @ ( product_Pair_a_real @ X2 @ Y ) @ ( lower_930854854raph_a @ top_top_set_a @ F ) )
% 2.58/1.49 & ( ord_less_eq_real @ Y @ Z ) )
% 2.58/1.49 => ( member1103263856a_real @ ( product_Pair_a_real @ X2 @ Z ) @ ( lower_930854854raph_a @ top_top_set_a @ F ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_35_metric__eq__thm,axiom,
% 2.58/1.49 ! [X2: real,S: set_real,Y: real] :
% 2.58/1.49 ( ( member_real @ X2 @ S )
% 2.58/1.49 => ( ( member_real @ Y @ S )
% 2.58/1.49 => ( ( X2 = Y )
% 2.58/1.49 = ( ! [X3: real] :
% 2.58/1.49 ( ( member_real @ X3 @ S )
% 2.58/1.49 => ( ( real_V1934908667t_real @ X2 @ X3 )
% 2.58/1.49 = ( real_V1934908667t_real @ Y @ X3 ) ) ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_36_metric__eq__thm,axiom,
% 2.58/1.49 ! [X2: a,S: set_a,Y: a] :
% 2.58/1.49 ( ( member_a @ X2 @ S )
% 2.58/1.49 => ( ( member_a @ Y @ S )
% 2.58/1.49 => ( ( X2 = Y )
% 2.58/1.49 = ( ! [X3: a] :
% 2.58/1.49 ( ( member_a @ X3 @ S )
% 2.58/1.49 => ( ( real_V1514887919dist_a @ X2 @ X3 )
% 2.58/1.49 = ( real_V1514887919dist_a @ Y @ X3 ) ) ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_37_metric__eq__thm,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real,S: set_Pr1928503567a_real,Y: product_prod_a_real] :
% 2.58/1.49 ( ( member1103263856a_real @ X2 @ S )
% 2.58/1.49 => ( ( member1103263856a_real @ Y @ S )
% 2.58/1.49 => ( ( X2 = Y )
% 2.58/1.49 = ( ! [X3: product_prod_a_real] :
% 2.58/1.49 ( ( member1103263856a_real @ X3 @ S )
% 2.58/1.49 => ( ( real_V404783528a_real @ X2 @ X3 )
% 2.58/1.49 = ( real_V404783528a_real @ Y @ X3 ) ) ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_38_order__refl,axiom,
% 2.58/1.49 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_39_order__refl,axiom,
% 2.58/1.49 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_40_Pair__le,axiom,
% 2.58/1.49 ! [A: set_a,B: set_a,C: set_a,D: set_a] :
% 2.58/1.49 ( ( ord_le486764743_set_a @ ( produc1928581911_set_a @ A @ B ) @ ( produc1928581911_set_a @ C @ D ) )
% 2.58/1.49 = ( ( ord_less_eq_set_a @ A @ C )
% 2.58/1.49 & ( ord_less_eq_set_a @ B @ D ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_41_Pair__le,axiom,
% 2.58/1.49 ! [A: real,B: real,C: real,D: real] :
% 2.58/1.49 ( ( ord_le1342644953l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
% 2.58/1.49 = ( ( ord_less_eq_real @ A @ C )
% 2.58/1.49 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_42_Pair__le,axiom,
% 2.58/1.49 ! [A: real,B: nat,C: real,D: nat] :
% 2.58/1.49 ( ( ord_le1686998781al_nat @ ( produc1541812501al_nat @ A @ B ) @ ( produc1541812501al_nat @ C @ D ) )
% 2.58/1.49 = ( ( ord_less_eq_real @ A @ C )
% 2.58/1.49 & ( ord_less_eq_nat @ B @ D ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_43_Pair__le,axiom,
% 2.58/1.49 ! [A: nat,B: real,C: nat,D: real] :
% 2.58/1.49 ( ( ord_le1950669437t_real @ ( produc49576085t_real @ A @ B ) @ ( produc49576085t_real @ C @ D ) )
% 2.58/1.49 = ( ( ord_less_eq_nat @ A @ C )
% 2.58/1.49 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_44_Pair__le,axiom,
% 2.58/1.49 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.58/1.49 ( ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
% 2.58/1.49 = ( ( ord_less_eq_nat @ A @ C )
% 2.58/1.49 & ( ord_less_eq_nat @ B @ D ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_45_dist__eq__0__iff,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 2.58/1.49 ( ( ( real_V404783528a_real @ X2 @ Y )
% 2.58/1.49 = zero_zero_real )
% 2.58/1.49 = ( X2 = Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_46_dist__self,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real] :
% 2.58/1.49 ( ( real_V404783528a_real @ X2 @ X2 )
% 2.58/1.49 = zero_zero_real ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_47_dist__le__zero__iff,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 2.58/1.49 ( ( ord_less_eq_real @ ( real_V404783528a_real @ X2 @ Y ) @ zero_zero_real )
% 2.58/1.49 = ( X2 = Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_48_zero__less__dist__iff,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 2.58/1.49 ( ( ord_less_real @ zero_zero_real @ ( real_V404783528a_real @ X2 @ Y ) )
% 2.58/1.49 = ( X2 != Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_49_order__subst1,axiom,
% 2.58/1.49 ! [A: real,F: real > real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_50_order__subst1,axiom,
% 2.58/1.49 ! [A: real,F: nat > real,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_51_order__subst1,axiom,
% 2.58/1.49 ! [A: nat,F: real > nat,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_52_order__subst1,axiom,
% 2.58/1.49 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_53_order__subst2,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_54_order__subst2,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_55_order__subst2,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_56_order__subst2,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_57_Pair__mono,axiom,
% 2.58/1.49 ! [X2: set_a,X4: set_a,Y: set_a,Y4: set_a] :
% 2.58/1.49 ( ( ord_less_eq_set_a @ X2 @ X4 )
% 2.58/1.49 => ( ( ord_less_eq_set_a @ Y @ Y4 )
% 2.58/1.49 => ( ord_le486764743_set_a @ ( produc1928581911_set_a @ X2 @ Y ) @ ( produc1928581911_set_a @ X4 @ Y4 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_58_Pair__mono,axiom,
% 2.58/1.49 ! [X2: real,X4: real,Y: real,Y4: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X2 @ X4 )
% 2.58/1.49 => ( ( ord_less_eq_real @ Y @ Y4 )
% 2.58/1.49 => ( ord_le1342644953l_real @ ( produc705216881l_real @ X2 @ Y ) @ ( produc705216881l_real @ X4 @ Y4 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_59_Pair__mono,axiom,
% 2.58/1.49 ! [X2: real,X4: real,Y: nat,Y4: nat] :
% 2.58/1.49 ( ( ord_less_eq_real @ X2 @ X4 )
% 2.58/1.49 => ( ( ord_less_eq_nat @ Y @ Y4 )
% 2.58/1.49 => ( ord_le1686998781al_nat @ ( produc1541812501al_nat @ X2 @ Y ) @ ( produc1541812501al_nat @ X4 @ Y4 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_60_Pair__mono,axiom,
% 2.58/1.49 ! [X2: nat,X4: nat,Y: real,Y4: real] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X2 @ X4 )
% 2.58/1.49 => ( ( ord_less_eq_real @ Y @ Y4 )
% 2.58/1.49 => ( ord_le1950669437t_real @ ( produc49576085t_real @ X2 @ Y ) @ ( produc49576085t_real @ X4 @ Y4 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_61_Pair__mono,axiom,
% 2.58/1.49 ! [X2: nat,X4: nat,Y: nat,Y4: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X2 @ X4 )
% 2.58/1.49 => ( ( ord_less_eq_nat @ Y @ Y4 )
% 2.58/1.49 => ( ord_le1302190241at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_62_ord__eq__le__subst,axiom,
% 2.58/1.49 ! [A: real,F: real > real,B: real,C: real] :
% 2.58/1.49 ( ( A
% 2.58/1.49 = ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_63_ord__eq__le__subst,axiom,
% 2.58/1.49 ! [A: nat,F: real > nat,B: real,C: real] :
% 2.58/1.49 ( ( A
% 2.58/1.49 = ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_64_ord__eq__le__subst,axiom,
% 2.58/1.49 ! [A: real,F: nat > real,B: nat,C: nat] :
% 2.58/1.49 ( ( A
% 2.58/1.49 = ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_65_ord__eq__le__subst,axiom,
% 2.58/1.49 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.58/1.49 ( ( A
% 2.58/1.49 = ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_66_ord__le__eq__subst,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( ( F @ B )
% 2.58/1.49 = C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_67_ord__le__eq__subst,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( ( F @ B )
% 2.58/1.49 = C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_68_ord__le__eq__subst,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( ( F @ B )
% 2.58/1.49 = C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_69_ord__le__eq__subst,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( ( F @ B )
% 2.58/1.49 = C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_70_eq__iff,axiom,
% 2.58/1.49 ( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
% 2.58/1.49 = ( ^ [X3: real,Y6: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X3 @ Y6 )
% 2.58/1.49 & ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_71_eq__iff,axiom,
% 2.58/1.49 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 2.58/1.49 = ( ^ [X3: nat,Y6: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 2.58/1.49 & ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_72_antisym,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_eq_real @ Y @ X2 )
% 2.58/1.49 => ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_73_antisym,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_eq_nat @ Y @ X2 )
% 2.58/1.49 => ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_74_linear,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 | ( ord_less_eq_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_75_linear,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 | ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_76_eq__refl,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( X2 = Y )
% 2.58/1.49 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_77_eq__refl,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( X2 = Y )
% 2.58/1.49 => ( ord_less_eq_nat @ X2 @ Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_78_le__cases,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ~ ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_79_le__cases,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ~ ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_80_order_Otrans,axiom,
% 2.58/1.49 ! [A: real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ C )
% 2.58/1.49 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_81_order_Otrans,axiom,
% 2.58/1.49 ! [A: nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ C )
% 2.58/1.49 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_82_le__cases3,axiom,
% 2.58/1.49 ! [X2: real,Y: real,Z: real] :
% 2.58/1.49 ( ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 => ~ ( ord_less_eq_real @ Y @ Z ) )
% 2.58/1.49 => ( ( ( ord_less_eq_real @ Y @ X2 )
% 2.58/1.49 => ~ ( ord_less_eq_real @ X2 @ Z ) )
% 2.58/1.49 => ( ( ( ord_less_eq_real @ X2 @ Z )
% 2.58/1.49 => ~ ( ord_less_eq_real @ Z @ Y ) )
% 2.58/1.49 => ( ( ( ord_less_eq_real @ Z @ Y )
% 2.58/1.49 => ~ ( ord_less_eq_real @ Y @ X2 ) )
% 2.58/1.49 => ( ( ( ord_less_eq_real @ Y @ Z )
% 2.58/1.49 => ~ ( ord_less_eq_real @ Z @ X2 ) )
% 2.58/1.49 => ~ ( ( ord_less_eq_real @ Z @ X2 )
% 2.58/1.49 => ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_83_le__cases3,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat,Z: nat] :
% 2.58/1.49 ( ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 2.58/1.49 => ( ( ( ord_less_eq_nat @ Y @ X2 )
% 2.58/1.49 => ~ ( ord_less_eq_nat @ X2 @ Z ) )
% 2.58/1.49 => ( ( ( ord_less_eq_nat @ X2 @ Z )
% 2.58/1.49 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 2.58/1.49 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 2.58/1.49 => ~ ( ord_less_eq_nat @ Y @ X2 ) )
% 2.58/1.49 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 2.58/1.49 => ~ ( ord_less_eq_nat @ Z @ X2 ) )
% 2.58/1.49 => ~ ( ( ord_less_eq_nat @ Z @ X2 )
% 2.58/1.49 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_84_antisym__conv,axiom,
% 2.58/1.49 ! [Y: real,X2: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ Y @ X2 )
% 2.58/1.49 => ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 = ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_85_antisym__conv,axiom,
% 2.58/1.49 ! [Y: nat,X2: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ Y @ X2 )
% 2.58/1.49 => ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 = ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_86_order__class_Oorder_Oeq__iff,axiom,
% 2.58/1.49 ( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
% 2.58/1.49 = ( ^ [A3: real,B3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A3 @ B3 )
% 2.58/1.49 & ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_87_order__class_Oorder_Oeq__iff,axiom,
% 2.58/1.49 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 2.58/1.49 = ( ^ [A3: nat,B3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A3 @ B3 )
% 2.58/1.49 & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_88_ord__eq__le__trans,axiom,
% 2.58/1.49 ! [A: real,B: real,C: real] :
% 2.58/1.49 ( ( A = B )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ C )
% 2.58/1.49 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_89_ord__eq__le__trans,axiom,
% 2.58/1.49 ! [A: nat,B: nat,C: nat] :
% 2.58/1.49 ( ( A = B )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ C )
% 2.58/1.49 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_90_ord__le__eq__trans,axiom,
% 2.58/1.49 ! [A: real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( B = C )
% 2.58/1.49 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_91_ord__le__eq__trans,axiom,
% 2.58/1.49 ! [A: nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( B = C )
% 2.58/1.49 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_92_order__class_Oorder_Oantisym,axiom,
% 2.58/1.49 ! [A: real,B: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ A )
% 2.58/1.49 => ( A = B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_93_order__class_Oorder_Oantisym,axiom,
% 2.58/1.49 ! [A: nat,B: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ A )
% 2.58/1.49 => ( A = B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_94_order__trans,axiom,
% 2.58/1.49 ! [X2: real,Y: real,Z: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_eq_real @ Y @ Z )
% 2.58/1.49 => ( ord_less_eq_real @ X2 @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_95_order__trans,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat,Z: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_eq_nat @ Y @ Z )
% 2.58/1.49 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_96_dual__order_Orefl,axiom,
% 2.58/1.49 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_97_dual__order_Orefl,axiom,
% 2.58/1.49 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_98_linorder__wlog,axiom,
% 2.58/1.49 ! [P: real > real > $o,A: real,B: real] :
% 2.58/1.49 ( ! [A4: real,B4: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A4 @ B4 )
% 2.58/1.49 => ( P @ A4 @ B4 ) )
% 2.58/1.49 => ( ! [A4: real,B4: real] :
% 2.58/1.49 ( ( P @ B4 @ A4 )
% 2.58/1.49 => ( P @ A4 @ B4 ) )
% 2.58/1.49 => ( P @ A @ B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_99_linorder__wlog,axiom,
% 2.58/1.49 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.58/1.49 ( ! [A4: nat,B4: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A4 @ B4 )
% 2.58/1.49 => ( P @ A4 @ B4 ) )
% 2.58/1.49 => ( ! [A4: nat,B4: nat] :
% 2.58/1.49 ( ( P @ B4 @ A4 )
% 2.58/1.49 => ( P @ A4 @ B4 ) )
% 2.58/1.49 => ( P @ A @ B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_100_dual__order_Otrans,axiom,
% 2.58/1.49 ! [B: real,A: real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ B @ A )
% 2.58/1.49 => ( ( ord_less_eq_real @ C @ B )
% 2.58/1.49 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_101_dual__order_Otrans,axiom,
% 2.58/1.49 ! [B: nat,A: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ B @ A )
% 2.58/1.49 => ( ( ord_less_eq_nat @ C @ B )
% 2.58/1.49 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_102_mem__Collect__eq,axiom,
% 2.58/1.49 ! [A: product_prod_a_real,P: product_prod_a_real > $o] :
% 2.58/1.49 ( ( member1103263856a_real @ A @ ( collec1714955950a_real @ P ) )
% 2.58/1.49 = ( P @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_103_mem__Collect__eq,axiom,
% 2.58/1.49 ! [A: real,P: real > $o] :
% 2.58/1.49 ( ( member_real @ A @ ( collect_real @ P ) )
% 2.58/1.49 = ( P @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_104_mem__Collect__eq,axiom,
% 2.58/1.49 ! [A: a,P: a > $o] :
% 2.58/1.49 ( ( member_a @ A @ ( collect_a @ P ) )
% 2.58/1.49 = ( P @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_105_Collect__mem__eq,axiom,
% 2.58/1.49 ! [A5: set_Pr1928503567a_real] :
% 2.58/1.49 ( ( collec1714955950a_real
% 2.58/1.49 @ ^ [X3: product_prod_a_real] : ( member1103263856a_real @ X3 @ A5 ) )
% 2.58/1.49 = A5 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_106_Collect__mem__eq,axiom,
% 2.58/1.49 ! [A5: set_real] :
% 2.58/1.49 ( ( collect_real
% 2.58/1.49 @ ^ [X3: real] : ( member_real @ X3 @ A5 ) )
% 2.58/1.49 = A5 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_107_Collect__mem__eq,axiom,
% 2.58/1.49 ! [A5: set_a] :
% 2.58/1.49 ( ( collect_a
% 2.58/1.49 @ ^ [X3: a] : ( member_a @ X3 @ A5 ) )
% 2.58/1.49 = A5 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_108_dual__order_Oeq__iff,axiom,
% 2.58/1.49 ( ( ^ [Y5: real,Z2: real] : ( Y5 = Z2 ) )
% 2.58/1.49 = ( ^ [A3: real,B3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ B3 @ A3 )
% 2.58/1.49 & ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_109_dual__order_Oeq__iff,axiom,
% 2.58/1.49 ( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
% 2.58/1.49 = ( ^ [A3: nat,B3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ B3 @ A3 )
% 2.58/1.49 & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_110_dual__order_Oantisym,axiom,
% 2.58/1.49 ! [B: real,A: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ B @ A )
% 2.58/1.49 => ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( A = B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_111_dual__order_Oantisym,axiom,
% 2.58/1.49 ! [B: nat,A: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ B @ A )
% 2.58/1.49 => ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( A = B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_112_zero__le__dist,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real,Y: product_prod_a_real] : ( ord_less_eq_real @ zero_zero_real @ ( real_V404783528a_real @ X2 @ Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_113_top__set__def,axiom,
% 2.58/1.49 ( top_top_set_a
% 2.58/1.49 = ( collect_a @ top_top_a_o ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_114_top__set__def,axiom,
% 2.58/1.49 ( top_top_set_real
% 2.58/1.49 = ( collect_real @ top_top_real_o ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_115_top__set__def,axiom,
% 2.58/1.49 ( top_to2138011583a_real
% 2.58/1.49 = ( collec1714955950a_real @ top_to1567080838real_o ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_116_order_Onot__eq__order__implies__strict,axiom,
% 2.58/1.49 ! [A: real,B: real] :
% 2.58/1.49 ( ( A != B )
% 2.58/1.49 => ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ord_less_real @ A @ B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_117_order_Onot__eq__order__implies__strict,axiom,
% 2.58/1.49 ! [A: nat,B: nat] :
% 2.58/1.49 ( ( A != B )
% 2.58/1.49 => ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_118_dual__order_Ostrict__implies__order,axiom,
% 2.58/1.49 ! [B: real,A: real] :
% 2.58/1.49 ( ( ord_less_real @ B @ A )
% 2.58/1.49 => ( ord_less_eq_real @ B @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_119_dual__order_Ostrict__implies__order,axiom,
% 2.58/1.49 ! [B: nat,A: nat] :
% 2.58/1.49 ( ( ord_less_nat @ B @ A )
% 2.58/1.49 => ( ord_less_eq_nat @ B @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_120_dual__order_Ostrict__iff__order,axiom,
% 2.58/1.49 ( ord_less_real
% 2.58/1.49 = ( ^ [B3: real,A3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ B3 @ A3 )
% 2.58/1.49 & ( A3 != B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_121_dual__order_Ostrict__iff__order,axiom,
% 2.58/1.49 ( ord_less_nat
% 2.58/1.49 = ( ^ [B3: nat,A3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ B3 @ A3 )
% 2.58/1.49 & ( A3 != B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_122_dual__order_Oorder__iff__strict,axiom,
% 2.58/1.49 ( ord_less_eq_real
% 2.58/1.49 = ( ^ [B3: real,A3: real] :
% 2.58/1.49 ( ( ord_less_real @ B3 @ A3 )
% 2.58/1.49 | ( A3 = B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_123_dual__order_Oorder__iff__strict,axiom,
% 2.58/1.49 ( ord_less_eq_nat
% 2.58/1.49 = ( ^ [B3: nat,A3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ B3 @ A3 )
% 2.58/1.49 | ( A3 = B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_124_order_Ostrict__implies__order,axiom,
% 2.58/1.49 ! [A: real,B: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ord_less_eq_real @ A @ B ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_125_order_Ostrict__implies__order,axiom,
% 2.58/1.49 ! [A: nat,B: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_126_dense__le__bounded,axiom,
% 2.58/1.49 ! [X2: real,Y: real,Z: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( ! [W: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ W )
% 2.58/1.49 => ( ( ord_less_real @ W @ Y )
% 2.58/1.49 => ( ord_less_eq_real @ W @ Z ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_127_dense__ge__bounded,axiom,
% 2.58/1.49 ! [Z: real,X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ Z @ X2 )
% 2.58/1.49 => ( ! [W: real] :
% 2.58/1.49 ( ( ord_less_real @ Z @ W )
% 2.58/1.49 => ( ( ord_less_real @ W @ X2 )
% 2.58/1.49 => ( ord_less_eq_real @ Y @ W ) ) )
% 2.58/1.49 => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_128_dual__order_Ostrict__trans2,axiom,
% 2.58/1.49 ! [B: real,A: real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ B @ A )
% 2.58/1.49 => ( ( ord_less_eq_real @ C @ B )
% 2.58/1.49 => ( ord_less_real @ C @ A ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_129_dual__order_Ostrict__trans2,axiom,
% 2.58/1.49 ! [B: nat,A: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ B @ A )
% 2.58/1.49 => ( ( ord_less_eq_nat @ C @ B )
% 2.58/1.49 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_130_dual__order_Ostrict__trans1,axiom,
% 2.58/1.49 ! [B: real,A: real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ B @ A )
% 2.58/1.49 => ( ( ord_less_real @ C @ B )
% 2.58/1.49 => ( ord_less_real @ C @ A ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_131_dual__order_Ostrict__trans1,axiom,
% 2.58/1.49 ! [B: nat,A: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ B @ A )
% 2.58/1.49 => ( ( ord_less_nat @ C @ B )
% 2.58/1.49 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_132_order_Ostrict__iff__order,axiom,
% 2.58/1.49 ( ord_less_real
% 2.58/1.49 = ( ^ [A3: real,B3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A3 @ B3 )
% 2.58/1.49 & ( A3 != B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_133_order_Ostrict__iff__order,axiom,
% 2.58/1.49 ( ord_less_nat
% 2.58/1.49 = ( ^ [A3: nat,B3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A3 @ B3 )
% 2.58/1.49 & ( A3 != B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_134_order_Oorder__iff__strict,axiom,
% 2.58/1.49 ( ord_less_eq_real
% 2.58/1.49 = ( ^ [A3: real,B3: real] :
% 2.58/1.49 ( ( ord_less_real @ A3 @ B3 )
% 2.58/1.49 | ( A3 = B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_135_order_Oorder__iff__strict,axiom,
% 2.58/1.49 ( ord_less_eq_nat
% 2.58/1.49 = ( ^ [A3: nat,B3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A3 @ B3 )
% 2.58/1.49 | ( A3 = B3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_136_order_Ostrict__trans2,axiom,
% 2.58/1.49 ! [A: real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ C )
% 2.58/1.49 => ( ord_less_real @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_137_order_Ostrict__trans2,axiom,
% 2.58/1.49 ! [A: nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ C )
% 2.58/1.49 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_138_order_Ostrict__trans1,axiom,
% 2.58/1.49 ! [A: real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_real @ B @ C )
% 2.58/1.49 => ( ord_less_real @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_139_order_Ostrict__trans1,axiom,
% 2.58/1.49 ! [A: nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_nat @ B @ C )
% 2.58/1.49 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_140_not__le__imp__less,axiom,
% 2.58/1.49 ! [Y: real,X2: real] :
% 2.58/1.49 ( ~ ( ord_less_eq_real @ Y @ X2 )
% 2.58/1.49 => ( ord_less_real @ X2 @ Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_141_not__le__imp__less,axiom,
% 2.58/1.49 ! [Y: nat,X2: nat] :
% 2.58/1.49 ( ~ ( ord_less_eq_nat @ Y @ X2 )
% 2.58/1.49 => ( ord_less_nat @ X2 @ Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_142_less__le__not__le,axiom,
% 2.58/1.49 ( ord_less_real
% 2.58/1.49 = ( ^ [X3: real,Y6: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X3 @ Y6 )
% 2.58/1.49 & ~ ( ord_less_eq_real @ Y6 @ X3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_143_less__le__not__le,axiom,
% 2.58/1.49 ( ord_less_nat
% 2.58/1.49 = ( ^ [X3: nat,Y6: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 2.58/1.49 & ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_144_le__imp__less__or__eq,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 | ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_145_le__imp__less__or__eq,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 | ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_146_le__less__linear,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 | ( ord_less_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_147_le__less__linear,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 | ( ord_less_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_148_dense__le,axiom,
% 2.58/1.49 ! [Y: real,Z: real] :
% 2.58/1.49 ( ! [X: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y )
% 2.58/1.49 => ( ord_less_eq_real @ X @ Z ) )
% 2.58/1.49 => ( ord_less_eq_real @ Y @ Z ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_149_dense__ge,axiom,
% 2.58/1.49 ! [Z: real,Y: real] :
% 2.58/1.49 ( ! [X: real] :
% 2.58/1.49 ( ( ord_less_real @ Z @ X )
% 2.58/1.49 => ( ord_less_eq_real @ Y @ X ) )
% 2.58/1.49 => ( ord_less_eq_real @ Y @ Z ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_150_less__le__trans,axiom,
% 2.58/1.49 ! [X2: real,Y: real,Z: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_eq_real @ Y @ Z )
% 2.58/1.49 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_151_less__le__trans,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat,Z: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_eq_nat @ Y @ Z )
% 2.58/1.49 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_152_le__less__trans,axiom,
% 2.58/1.49 ! [X2: real,Y: real,Z: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_real @ Y @ Z )
% 2.58/1.49 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_153_le__less__trans,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat,Z: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_nat @ Y @ Z )
% 2.58/1.49 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_154_less__imp__le,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_155_less__imp__le,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( ord_less_eq_nat @ X2 @ Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_156_antisym__conv2,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 => ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 2.58/1.49 = ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_157_antisym__conv2,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 2.58/1.49 = ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_158_antisym__conv1,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ~ ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_eq_real @ X2 @ Y )
% 2.58/1.49 = ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_159_antisym__conv1,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ~ ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_eq_nat @ X2 @ Y )
% 2.58/1.49 = ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_160_le__neq__trans,axiom,
% 2.58/1.49 ! [A: real,B: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( A != B )
% 2.58/1.49 => ( ord_less_real @ A @ B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_161_le__neq__trans,axiom,
% 2.58/1.49 ! [A: nat,B: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( A != B )
% 2.58/1.49 => ( ord_less_nat @ A @ B ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_162_not__less,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 2.58/1.49 = ( ord_less_eq_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_163_not__less,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 2.58/1.49 = ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_164_not__le,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ~ ( ord_less_eq_real @ X2 @ Y ) )
% 2.58/1.49 = ( ord_less_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_165_not__le,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
% 2.58/1.49 = ( ord_less_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_166_order__less__le__subst2,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_167_order__less__le__subst2,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > real,C: real] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_168_order__less__le__subst2,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_169_order__less__le__subst2,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_170_order__less__le__subst1,axiom,
% 2.58/1.49 ! [A: real,F: real > real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_171_order__less__le__subst1,axiom,
% 2.58/1.49 ! [A: nat,F: real > nat,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_172_order__less__le__subst1,axiom,
% 2.58/1.49 ! [A: real,F: nat > real,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_real @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_173_order__less__le__subst1,axiom,
% 2.58/1.49 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_eq_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_174_order__le__less__subst2,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_real @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_175_order__le__less__subst2,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_176_order__le__less__subst2,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_real @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_177_order__le__less__subst2,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_178_order__le__less__subst1,axiom,
% 2.58/1.49 ! [A: real,F: real > real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_179_order__le__less__subst1,axiom,
% 2.58/1.49 ! [A: real,F: nat > real,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_180_order__le__less__subst1,axiom,
% 2.58/1.49 ! [A: nat,F: real > nat,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_181_order__le__less__subst1,axiom,
% 2.58/1.49 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_182_less__le,axiom,
% 2.58/1.49 ( ord_less_real
% 2.58/1.49 = ( ^ [X3: real,Y6: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ X3 @ Y6 )
% 2.58/1.49 & ( X3 != Y6 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_183_less__le,axiom,
% 2.58/1.49 ( ord_less_nat
% 2.58/1.49 = ( ^ [X3: nat,Y6: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ X3 @ Y6 )
% 2.58/1.49 & ( X3 != Y6 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_184_le__less,axiom,
% 2.58/1.49 ( ord_less_eq_real
% 2.58/1.49 = ( ^ [X3: real,Y6: real] :
% 2.58/1.49 ( ( ord_less_real @ X3 @ Y6 )
% 2.58/1.49 | ( X3 = Y6 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_185_le__less,axiom,
% 2.58/1.49 ( ord_less_eq_nat
% 2.58/1.49 = ( ^ [X3: nat,Y6: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X3 @ Y6 )
% 2.58/1.49 | ( X3 = Y6 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_186_leI,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ~ ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_187_leI,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ~ ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_188_leD,axiom,
% 2.58/1.49 ! [Y: real,X2: real] :
% 2.58/1.49 ( ( ord_less_eq_real @ Y @ X2 )
% 2.58/1.49 => ~ ( ord_less_real @ X2 @ Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_189_leD,axiom,
% 2.58/1.49 ! [Y: nat,X2: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ Y @ X2 )
% 2.58/1.49 => ~ ( ord_less_nat @ X2 @ Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_190_top_Oextremum__uniqueI,axiom,
% 2.58/1.49 ! [A: set_a] :
% 2.58/1.49 ( ( ord_less_eq_set_a @ top_top_set_a @ A )
% 2.58/1.49 => ( A = top_top_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_191_top_Oextremum__uniqueI,axiom,
% 2.58/1.49 ! [A: set_real] :
% 2.58/1.49 ( ( ord_less_eq_set_real @ top_top_set_real @ A )
% 2.58/1.49 => ( A = top_top_set_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_192_top_Oextremum__uniqueI,axiom,
% 2.58/1.49 ! [A: set_Pr1928503567a_real] :
% 2.58/1.49 ( ( ord_le1586073967a_real @ top_to2138011583a_real @ A )
% 2.58/1.49 => ( A = top_to2138011583a_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_193_top_Oextremum__uniqueI,axiom,
% 2.58/1.49 ! [A: produc1691597095_set_a] :
% 2.58/1.49 ( ( ord_le486764743_set_a @ top_to1477940855_set_a @ A )
% 2.58/1.49 => ( A = top_to1477940855_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_194_top_Oextremum__unique,axiom,
% 2.58/1.49 ! [A: set_a] :
% 2.58/1.49 ( ( ord_less_eq_set_a @ top_top_set_a @ A )
% 2.58/1.49 = ( A = top_top_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_195_top_Oextremum__unique,axiom,
% 2.58/1.49 ! [A: set_real] :
% 2.58/1.49 ( ( ord_less_eq_set_real @ top_top_set_real @ A )
% 2.58/1.49 = ( A = top_top_set_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_196_top_Oextremum__unique,axiom,
% 2.58/1.49 ! [A: set_Pr1928503567a_real] :
% 2.58/1.49 ( ( ord_le1586073967a_real @ top_to2138011583a_real @ A )
% 2.58/1.49 = ( A = top_to2138011583a_real ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_197_top_Oextremum__unique,axiom,
% 2.58/1.49 ! [A: produc1691597095_set_a] :
% 2.58/1.49 ( ( ord_le486764743_set_a @ top_to1477940855_set_a @ A )
% 2.58/1.49 = ( A = top_to1477940855_set_a ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_198_top__greatest,axiom,
% 2.58/1.49 ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_199_top__greatest,axiom,
% 2.58/1.49 ! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_200_top__greatest,axiom,
% 2.58/1.49 ! [A: set_Pr1928503567a_real] : ( ord_le1586073967a_real @ A @ top_to2138011583a_real ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_201_top__greatest,axiom,
% 2.58/1.49 ! [A: produc1691597095_set_a] : ( ord_le486764743_set_a @ A @ top_to1477940855_set_a ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_202_dist__not__less__zero,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 2.58/1.49 ~ ( ord_less_real @ ( real_V404783528a_real @ X2 @ Y ) @ zero_zero_real ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_203_dist__pos__lt,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real,Y: product_prod_a_real] :
% 2.58/1.49 ( ( X2 != Y )
% 2.58/1.49 => ( ord_less_real @ zero_zero_real @ ( real_V404783528a_real @ X2 @ Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_204_dual__order_Ostrict__implies__not__eq,axiom,
% 2.58/1.49 ! [B: real,A: real] :
% 2.58/1.49 ( ( ord_less_real @ B @ A )
% 2.58/1.49 => ( A != B ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_205_dual__order_Ostrict__implies__not__eq,axiom,
% 2.58/1.49 ! [B: nat,A: nat] :
% 2.58/1.49 ( ( ord_less_nat @ B @ A )
% 2.58/1.49 => ( A != B ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_206_order_Ostrict__implies__not__eq,axiom,
% 2.58/1.49 ! [A: real,B: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( A != B ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_207_order_Ostrict__implies__not__eq,axiom,
% 2.58/1.49 ! [A: nat,B: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( A != B ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_208_not__less__iff__gr__or__eq,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 2.58/1.49 = ( ( ord_less_real @ Y @ X2 )
% 2.58/1.49 | ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_209_not__less__iff__gr__or__eq,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 2.58/1.49 = ( ( ord_less_nat @ Y @ X2 )
% 2.58/1.49 | ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_210_dual__order_Ostrict__trans,axiom,
% 2.58/1.49 ! [B: real,A: real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ B @ A )
% 2.58/1.49 => ( ( ord_less_real @ C @ B )
% 2.58/1.49 => ( ord_less_real @ C @ A ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_211_dual__order_Ostrict__trans,axiom,
% 2.58/1.49 ! [B: nat,A: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ B @ A )
% 2.58/1.49 => ( ( ord_less_nat @ C @ B )
% 2.58/1.49 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_212_linorder__less__wlog,axiom,
% 2.58/1.49 ! [P: real > real > $o,A: real,B: real] :
% 2.58/1.49 ( ! [A4: real,B4: real] :
% 2.58/1.49 ( ( ord_less_real @ A4 @ B4 )
% 2.58/1.49 => ( P @ A4 @ B4 ) )
% 2.58/1.49 => ( ! [A4: real] : ( P @ A4 @ A4 )
% 2.58/1.49 => ( ! [A4: real,B4: real] :
% 2.58/1.49 ( ( P @ B4 @ A4 )
% 2.58/1.49 => ( P @ A4 @ B4 ) )
% 2.58/1.49 => ( P @ A @ B ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_213_linorder__less__wlog,axiom,
% 2.58/1.49 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.58/1.49 ( ! [A4: nat,B4: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A4 @ B4 )
% 2.58/1.49 => ( P @ A4 @ B4 ) )
% 2.58/1.49 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 2.58/1.49 => ( ! [A4: nat,B4: nat] :
% 2.58/1.49 ( ( P @ B4 @ A4 )
% 2.58/1.49 => ( P @ A4 @ B4 ) )
% 2.58/1.49 => ( P @ A @ B ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_214_exists__least__iff,axiom,
% 2.58/1.49 ( ( ^ [P2: nat > $o] :
% 2.58/1.49 ? [X5: nat] : ( P2 @ X5 ) )
% 2.58/1.49 = ( ^ [P3: nat > $o] :
% 2.58/1.49 ? [N: nat] :
% 2.58/1.49 ( ( P3 @ N )
% 2.58/1.49 & ! [M: nat] :
% 2.58/1.49 ( ( ord_less_nat @ M @ N )
% 2.58/1.49 => ~ ( P3 @ M ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_215_less__imp__not__less,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_216_less__imp__not__less,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_217_order_Ostrict__trans,axiom,
% 2.58/1.49 ! [A: real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_real @ B @ C )
% 2.58/1.49 => ( ord_less_real @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_218_order_Ostrict__trans,axiom,
% 2.58/1.49 ! [A: nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_nat @ B @ C )
% 2.58/1.49 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_219_dual__order_Oirrefl,axiom,
% 2.58/1.49 ! [A: real] :
% 2.58/1.49 ~ ( ord_less_real @ A @ A ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_220_dual__order_Oirrefl,axiom,
% 2.58/1.49 ! [A: nat] :
% 2.58/1.49 ~ ( ord_less_nat @ A @ A ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_221_linorder__cases,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ~ ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( ( X2 != Y )
% 2.58/1.49 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_222_linorder__cases,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ~ ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( ( X2 != Y )
% 2.58/1.49 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_223_less__imp__triv,axiom,
% 2.58/1.49 ! [X2: real,Y: real,P: $o] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_real @ Y @ X2 )
% 2.58/1.49 => P ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_224_less__imp__triv,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat,P: $o] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_nat @ Y @ X2 )
% 2.58/1.49 => P ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_225_less__imp__not__eq2,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( Y != X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_226_less__imp__not__eq2,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( Y != X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_227_antisym__conv3,axiom,
% 2.58/1.49 ! [Y: real,X2: real] :
% 2.58/1.49 ( ~ ( ord_less_real @ Y @ X2 )
% 2.58/1.49 => ( ( ~ ( ord_less_real @ X2 @ Y ) )
% 2.58/1.49 = ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_228_antisym__conv3,axiom,
% 2.58/1.49 ! [Y: nat,X2: nat] :
% 2.58/1.49 ( ~ ( ord_less_nat @ Y @ X2 )
% 2.58/1.49 => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
% 2.58/1.49 = ( X2 = Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_229_less__induct,axiom,
% 2.58/1.49 ! [P: nat > $o,A: nat] :
% 2.58/1.49 ( ! [X: nat] :
% 2.58/1.49 ( ! [Y7: nat] :
% 2.58/1.49 ( ( ord_less_nat @ Y7 @ X )
% 2.58/1.49 => ( P @ Y7 ) )
% 2.58/1.49 => ( P @ X ) )
% 2.58/1.49 => ( P @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_230_less__not__sym,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_231_less__not__sym,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_232_less__imp__not__eq,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( X2 != Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_233_less__imp__not__eq,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( X2 != Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_234_dual__order_Oasym,axiom,
% 2.58/1.49 ! [B: real,A: real] :
% 2.58/1.49 ( ( ord_less_real @ B @ A )
% 2.58/1.49 => ~ ( ord_less_real @ A @ B ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_235_dual__order_Oasym,axiom,
% 2.58/1.49 ! [B: nat,A: nat] :
% 2.58/1.49 ( ( ord_less_nat @ B @ A )
% 2.58/1.49 => ~ ( ord_less_nat @ A @ B ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_236_ord__less__eq__trans,axiom,
% 2.58/1.49 ! [A: real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ( B = C )
% 2.58/1.49 => ( ord_less_real @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_237_ord__less__eq__trans,axiom,
% 2.58/1.49 ! [A: nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ( B = C )
% 2.58/1.49 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_238_ord__eq__less__trans,axiom,
% 2.58/1.49 ! [A: real,B: real,C: real] :
% 2.58/1.49 ( ( A = B )
% 2.58/1.49 => ( ( ord_less_real @ B @ C )
% 2.58/1.49 => ( ord_less_real @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_239_ord__eq__less__trans,axiom,
% 2.58/1.49 ! [A: nat,B: nat,C: nat] :
% 2.58/1.49 ( ( A = B )
% 2.58/1.49 => ( ( ord_less_nat @ B @ C )
% 2.58/1.49 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_240_less__irrefl,axiom,
% 2.58/1.49 ! [X2: real] :
% 2.58/1.49 ~ ( ord_less_real @ X2 @ X2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_241_less__irrefl,axiom,
% 2.58/1.49 ! [X2: nat] :
% 2.58/1.49 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_242_less__linear,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 | ( X2 = Y )
% 2.58/1.49 | ( ord_less_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_243_less__linear,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 | ( X2 = Y )
% 2.58/1.49 | ( ord_less_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_244_less__trans,axiom,
% 2.58/1.49 ! [X2: real,Y: real,Z: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_real @ Y @ Z )
% 2.58/1.49 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_245_less__trans,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat,Z: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( ( ord_less_nat @ Y @ Z )
% 2.58/1.49 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_246_less__asym_H,axiom,
% 2.58/1.49 ! [A: real,B: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ~ ( ord_less_real @ B @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_247_less__asym_H,axiom,
% 2.58/1.49 ! [A: nat,B: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_248_less__asym,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ~ ( ord_less_real @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_249_less__asym,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ~ ( ord_less_nat @ Y @ X2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_250_less__imp__neq,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( X2 != Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_251_less__imp__neq,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( X2 != Y ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_252_dense,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ? [Z3: real] :
% 2.58/1.49 ( ( ord_less_real @ X2 @ Z3 )
% 2.58/1.49 & ( ord_less_real @ Z3 @ Y ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_253_order_Oasym,axiom,
% 2.58/1.49 ! [A: real,B: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ~ ( ord_less_real @ B @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_254_order_Oasym,axiom,
% 2.58/1.49 ! [A: nat,B: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ~ ( ord_less_nat @ B @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_255_neq__iff,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( X2 != Y )
% 2.58/1.49 = ( ( ord_less_real @ X2 @ Y )
% 2.58/1.49 | ( ord_less_real @ Y @ X2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_256_neq__iff,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( X2 != Y )
% 2.58/1.49 = ( ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 | ( ord_less_nat @ Y @ X2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_257_neqE,axiom,
% 2.58/1.49 ! [X2: real,Y: real] :
% 2.58/1.49 ( ( X2 != Y )
% 2.58/1.49 => ( ~ ( ord_less_real @ X2 @ Y )
% 2.58/1.49 => ( ord_less_real @ Y @ X2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_258_neqE,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( X2 != Y )
% 2.58/1.49 => ( ~ ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_259_gt__ex,axiom,
% 2.58/1.49 ! [X2: real] :
% 2.58/1.49 ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_260_gt__ex,axiom,
% 2.58/1.49 ! [X2: nat] :
% 2.58/1.49 ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_261_lt__ex,axiom,
% 2.58/1.49 ! [X2: real] :
% 2.58/1.49 ? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_262_order__less__subst2,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_real @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_263_order__less__subst2,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_264_order__less__subst2,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > real,C: real] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_real @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_265_order__less__subst2,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_266_order__less__subst1,axiom,
% 2.58/1.49 ! [A: real,F: real > real,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_267_order__less__subst1,axiom,
% 2.58/1.49 ! [A: real,F: nat > real,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_real @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_268_order__less__subst1,axiom,
% 2.58/1.49 ! [A: nat,F: real > nat,B: real,C: real] :
% 2.58/1.49 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_269_order__less__subst1,axiom,
% 2.58/1.49 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_270_ord__less__eq__subst,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > real,C: real] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ( ( F @ B )
% 2.58/1.49 = C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_271_ord__less__eq__subst,axiom,
% 2.58/1.49 ! [A: real,B: real,F: real > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_real @ A @ B )
% 2.58/1.49 => ( ( ( F @ B )
% 2.58/1.49 = C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_272_ord__less__eq__subst,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > real,C: real] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ( ( F @ B )
% 2.58/1.49 = C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_273_ord__less__eq__subst,axiom,
% 2.58/1.49 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.58/1.49 ( ( ord_less_nat @ A @ B )
% 2.58/1.49 => ( ( ( F @ B )
% 2.58/1.49 = C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_274_ord__eq__less__subst,axiom,
% 2.58/1.49 ! [A: real,F: real > real,B: real,C: real] :
% 2.58/1.49 ( ( A
% 2.58/1.49 = ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_275_ord__eq__less__subst,axiom,
% 2.58/1.49 ! [A: nat,F: real > nat,B: real,C: real] :
% 2.58/1.49 ( ( A
% 2.58/1.49 = ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_real @ B @ C )
% 2.58/1.49 => ( ! [X: real,Y3: real] :
% 2.58/1.49 ( ( ord_less_real @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_276_ord__eq__less__subst,axiom,
% 2.58/1.49 ! [A: real,F: nat > real,B: nat,C: nat] :
% 2.58/1.49 ( ( A
% 2.58/1.49 = ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_real @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_277_ord__eq__less__subst,axiom,
% 2.58/1.49 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.58/1.49 ( ( A
% 2.58/1.49 = ( F @ B ) )
% 2.58/1.49 => ( ( ord_less_nat @ B @ C )
% 2.58/1.49 => ( ! [X: nat,Y3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ X @ Y3 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.58/1.49 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_278_old_Oprod_Oinducts,axiom,
% 2.58/1.49 ! [P: product_prod_a_real > $o,Prod: product_prod_a_real] :
% 2.58/1.49 ( ! [A4: a,B4: real] : ( P @ ( product_Pair_a_real @ A4 @ B4 ) )
% 2.58/1.49 => ( P @ Prod ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_279_old_Oprod_Oinducts,axiom,
% 2.58/1.49 ! [P: produc1691597095_set_a > $o,Prod: produc1691597095_set_a] :
% 2.58/1.49 ( ! [A4: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A4 @ B4 ) )
% 2.58/1.49 => ( P @ Prod ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_280_old_Oprod_Oexhaust,axiom,
% 2.58/1.49 ! [Y: product_prod_a_real] :
% 2.58/1.49 ~ ! [A4: a,B4: real] :
% 2.58/1.49 ( Y
% 2.58/1.49 != ( product_Pair_a_real @ A4 @ B4 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_281_old_Oprod_Oexhaust,axiom,
% 2.58/1.49 ! [Y: produc1691597095_set_a] :
% 2.58/1.49 ~ ! [A4: set_a,B4: set_a] :
% 2.58/1.49 ( Y
% 2.58/1.49 != ( produc1928581911_set_a @ A4 @ B4 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_282_Pair__inject,axiom,
% 2.58/1.49 ! [A: a,B: real,A2: a,B2: real] :
% 2.58/1.49 ( ( ( product_Pair_a_real @ A @ B )
% 2.58/1.49 = ( product_Pair_a_real @ A2 @ B2 ) )
% 2.58/1.49 => ~ ( ( A = A2 )
% 2.58/1.49 => ( B != B2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_283_Pair__inject,axiom,
% 2.58/1.49 ! [A: set_a,B: set_a,A2: set_a,B2: set_a] :
% 2.58/1.49 ( ( ( produc1928581911_set_a @ A @ B )
% 2.58/1.49 = ( produc1928581911_set_a @ A2 @ B2 ) )
% 2.58/1.49 => ~ ( ( A = A2 )
% 2.58/1.49 => ( B != B2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_284_prod__cases,axiom,
% 2.58/1.49 ! [P: product_prod_a_real > $o,P4: product_prod_a_real] :
% 2.58/1.49 ( ! [A4: a,B4: real] : ( P @ ( product_Pair_a_real @ A4 @ B4 ) )
% 2.58/1.49 => ( P @ P4 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_285_prod__cases,axiom,
% 2.58/1.49 ! [P: produc1691597095_set_a > $o,P4: produc1691597095_set_a] :
% 2.58/1.49 ( ! [A4: set_a,B4: set_a] : ( P @ ( produc1928581911_set_a @ A4 @ B4 ) )
% 2.58/1.49 => ( P @ P4 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_286_surj__pair,axiom,
% 2.58/1.49 ! [P4: product_prod_a_real] :
% 2.58/1.49 ? [X: a,Y3: real] :
% 2.58/1.49 ( P4
% 2.58/1.49 = ( product_Pair_a_real @ X @ Y3 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_287_surj__pair,axiom,
% 2.58/1.49 ! [P4: produc1691597095_set_a] :
% 2.58/1.49 ? [X: set_a,Y3: set_a] :
% 2.58/1.49 ( P4
% 2.58/1.49 = ( produc1928581911_set_a @ X @ Y3 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_288_UNIV__witness,axiom,
% 2.58/1.49 ? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_289_UNIV__witness,axiom,
% 2.58/1.49 ? [X: real] : ( member_real @ X @ top_top_set_real ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_290_UNIV__witness,axiom,
% 2.58/1.49 ? [X: product_prod_a_real] : ( member1103263856a_real @ X @ top_to2138011583a_real ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_291_UNIV__eq__I,axiom,
% 2.58/1.49 ! [A5: set_a] :
% 2.58/1.49 ( ! [X: a] : ( member_a @ X @ A5 )
% 2.58/1.49 => ( top_top_set_a = A5 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_292_UNIV__eq__I,axiom,
% 2.58/1.49 ! [A5: set_real] :
% 2.58/1.49 ( ! [X: real] : ( member_real @ X @ A5 )
% 2.58/1.49 => ( top_top_set_real = A5 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_293_UNIV__eq__I,axiom,
% 2.58/1.49 ! [A5: set_Pr1928503567a_real] :
% 2.58/1.49 ( ! [X: product_prod_a_real] : ( member1103263856a_real @ X @ A5 )
% 2.58/1.49 => ( top_to2138011583a_real = A5 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_294_dist__commute,axiom,
% 2.58/1.49 ( real_V404783528a_real
% 2.58/1.49 = ( ^ [X3: product_prod_a_real,Y6: product_prod_a_real] : ( real_V404783528a_real @ Y6 @ X3 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_295_closure__UNIV,axiom,
% 2.58/1.49 ( ( elementary_closure_a @ top_top_set_a )
% 2.58/1.49 = top_top_set_a ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_296_closure__UNIV,axiom,
% 2.58/1.49 ( ( elemen1168706977e_real @ top_top_set_real )
% 2.58/1.49 = top_top_set_real ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_297_closure__UNIV,axiom,
% 2.58/1.49 ( ( elemen1695521870a_real @ top_to2138011583a_real )
% 2.58/1.49 = top_to2138011583a_real ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_298_closure__approachable__le,axiom,
% 2.58/1.49 ! [X2: real,S2: set_real] :
% 2.58/1.49 ( ( member_real @ X2 @ ( elemen1168706977e_real @ S2 ) )
% 2.58/1.49 = ( ! [E2: real] :
% 2.58/1.49 ( ( ord_less_real @ zero_zero_real @ E2 )
% 2.58/1.49 => ? [X3: real] :
% 2.58/1.49 ( ( member_real @ X3 @ S2 )
% 2.58/1.49 & ( ord_less_eq_real @ ( real_V1934908667t_real @ X3 @ X2 ) @ E2 ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_299_closure__approachable__le,axiom,
% 2.58/1.49 ! [X2: a,S2: set_a] :
% 2.58/1.49 ( ( member_a @ X2 @ ( elementary_closure_a @ S2 ) )
% 2.58/1.49 = ( ! [E2: real] :
% 2.58/1.49 ( ( ord_less_real @ zero_zero_real @ E2 )
% 2.58/1.49 => ? [X3: a] :
% 2.58/1.49 ( ( member_a @ X3 @ S2 )
% 2.58/1.49 & ( ord_less_eq_real @ ( real_V1514887919dist_a @ X3 @ X2 ) @ E2 ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_300_closure__approachable__le,axiom,
% 2.58/1.49 ! [X2: product_prod_a_real,S2: set_Pr1928503567a_real] :
% 2.58/1.49 ( ( member1103263856a_real @ X2 @ ( elemen1695521870a_real @ S2 ) )
% 2.58/1.49 = ( ! [E2: real] :
% 2.58/1.49 ( ( ord_less_real @ zero_zero_real @ E2 )
% 2.58/1.49 => ? [X3: product_prod_a_real] :
% 2.58/1.49 ( ( member1103263856a_real @ X3 @ S2 )
% 2.58/1.49 & ( ord_less_eq_real @ ( real_V404783528a_real @ X3 @ X2 ) @ E2 ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_301_not__gr__zero,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 2.58/1.49 = ( N2 = zero_zero_nat ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_302_le__zero__eq,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 2.58/1.49 = ( N2 = zero_zero_nat ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_303_seq__mono__lemma,axiom,
% 2.58/1.49 ! [M2: nat,D: nat > real,E: nat > real] :
% 2.58/1.49 ( ! [N3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ M2 @ N3 )
% 2.58/1.49 => ( ord_less_real @ ( D @ N3 ) @ ( E @ N3 ) ) )
% 2.58/1.49 => ( ! [N3: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ M2 @ N3 )
% 2.58/1.49 => ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M2 ) ) )
% 2.58/1.49 => ! [N4: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ M2 @ N4 )
% 2.58/1.49 => ( ord_less_real @ ( D @ N4 ) @ ( E @ M2 ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_304_le0,axiom,
% 2.58/1.49 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_305_bot__nat__0_Oextremum,axiom,
% 2.58/1.49 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_306_neq0__conv,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ( ( N2 != zero_zero_nat )
% 2.58/1.49 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_307_less__nat__zero__code,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_308_bot__nat__0_Onot__eq__extremum,axiom,
% 2.58/1.49 ! [A: nat] :
% 2.58/1.49 ( ( A != zero_zero_nat )
% 2.58/1.49 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_309_gr0I,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ( ( N2 != zero_zero_nat )
% 2.58/1.49 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_310_not__gr0,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 2.58/1.49 = ( N2 = zero_zero_nat ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_311_not__less0,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_312_less__zeroE,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_313_gr__implies__not0,axiom,
% 2.58/1.49 ! [M2: nat,N2: nat] :
% 2.58/1.49 ( ( ord_less_nat @ M2 @ N2 )
% 2.58/1.49 => ( N2 != zero_zero_nat ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_314_infinite__descent0,axiom,
% 2.58/1.49 ! [P: nat > $o,N2: nat] :
% 2.58/1.49 ( ( P @ zero_zero_nat )
% 2.58/1.49 => ( ! [N3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 2.58/1.49 => ( ~ ( P @ N3 )
% 2.58/1.49 => ? [M3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ M3 @ N3 )
% 2.58/1.49 & ~ ( P @ M3 ) ) ) )
% 2.58/1.49 => ( P @ N2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_315_bot__nat__0_Oextremum__strict,axiom,
% 2.58/1.49 ! [A: nat] :
% 2.58/1.49 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_316_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.58/1.49 ! [A: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.58/1.49 => ( A = zero_zero_nat ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_317_bot__nat__0_Oextremum__unique,axiom,
% 2.58/1.49 ! [A: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.58/1.49 = ( A = zero_zero_nat ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_318_less__mono__imp__le__mono,axiom,
% 2.58/1.49 ! [F: nat > nat,I: nat,J: nat] :
% 2.58/1.49 ( ! [I2: nat,J2: nat] :
% 2.58/1.49 ( ( ord_less_nat @ I2 @ J2 )
% 2.58/1.49 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.58/1.49 => ( ( ord_less_eq_nat @ I @ J )
% 2.58/1.49 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_319_le__neq__implies__less,axiom,
% 2.58/1.49 ! [M2: nat,N2: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.58/1.49 => ( ( M2 != N2 )
% 2.58/1.49 => ( ord_less_nat @ M2 @ N2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_320_Nat_Oex__has__greatest__nat,axiom,
% 2.58/1.49 ! [P: nat > $o,K: nat,B: nat] :
% 2.58/1.49 ( ( P @ K )
% 2.58/1.49 => ( ! [Y3: nat] :
% 2.58/1.49 ( ( P @ Y3 )
% 2.58/1.49 => ( ord_less_eq_nat @ Y3 @ B ) )
% 2.58/1.49 => ? [X: nat] :
% 2.58/1.49 ( ( P @ X )
% 2.58/1.49 & ! [Y7: nat] :
% 2.58/1.49 ( ( P @ Y7 )
% 2.58/1.49 => ( ord_less_eq_nat @ Y7 @ X ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_321_less__or__eq__imp__le,axiom,
% 2.58/1.49 ! [M2: nat,N2: nat] :
% 2.58/1.49 ( ( ( ord_less_nat @ M2 @ N2 )
% 2.58/1.49 | ( M2 = N2 ) )
% 2.58/1.49 => ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_322_le__eq__less__or__eq,axiom,
% 2.58/1.49 ( ord_less_eq_nat
% 2.58/1.49 = ( ^ [M: nat,N: nat] :
% 2.58/1.49 ( ( ord_less_nat @ M @ N )
% 2.58/1.49 | ( M = N ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_323_less__imp__le__nat,axiom,
% 2.58/1.49 ! [M2: nat,N2: nat] :
% 2.58/1.49 ( ( ord_less_nat @ M2 @ N2 )
% 2.58/1.49 => ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_324_ex__least__nat__le,axiom,
% 2.58/1.49 ! [P: nat > $o,N2: nat] :
% 2.58/1.49 ( ( P @ N2 )
% 2.58/1.49 => ( ~ ( P @ zero_zero_nat )
% 2.58/1.49 => ? [K2: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ K2 @ N2 )
% 2.58/1.49 & ! [I3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ I3 @ K2 )
% 2.58/1.49 => ~ ( P @ I3 ) )
% 2.58/1.49 & ( P @ K2 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_325_nat__le__linear,axiom,
% 2.58/1.49 ! [M2: nat,N2: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.58/1.49 | ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_326_nat__less__le,axiom,
% 2.58/1.49 ( ord_less_nat
% 2.58/1.49 = ( ^ [M: nat,N: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ M @ N )
% 2.58/1.49 & ( M != N ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_327_le__antisym,axiom,
% 2.58/1.49 ! [M2: nat,N2: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.58/1.49 => ( ( ord_less_eq_nat @ N2 @ M2 )
% 2.58/1.49 => ( M2 = N2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_328_eq__imp__le,axiom,
% 2.58/1.49 ! [M2: nat,N2: nat] :
% 2.58/1.49 ( ( M2 = N2 )
% 2.58/1.49 => ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_329_le__trans,axiom,
% 2.58/1.49 ! [I: nat,J: nat,K: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ I @ J )
% 2.58/1.49 => ( ( ord_less_eq_nat @ J @ K )
% 2.58/1.49 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_330_le__refl,axiom,
% 2.58/1.49 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_331_le__0__eq,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 2.58/1.49 = ( N2 = zero_zero_nat ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_332_less__eq__nat_Osimps_I1_J,axiom,
% 2.58/1.49 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_333_linorder__neqE__nat,axiom,
% 2.58/1.49 ! [X2: nat,Y: nat] :
% 2.58/1.49 ( ( X2 != Y )
% 2.58/1.49 => ( ~ ( ord_less_nat @ X2 @ Y )
% 2.58/1.49 => ( ord_less_nat @ Y @ X2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_334_infinite__descent,axiom,
% 2.58/1.49 ! [P: nat > $o,N2: nat] :
% 2.58/1.49 ( ! [N3: nat] :
% 2.58/1.49 ( ~ ( P @ N3 )
% 2.58/1.49 => ? [M3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ M3 @ N3 )
% 2.58/1.49 & ~ ( P @ M3 ) ) )
% 2.58/1.49 => ( P @ N2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_335_nat__less__induct,axiom,
% 2.58/1.49 ! [P: nat > $o,N2: nat] :
% 2.58/1.49 ( ! [N3: nat] :
% 2.58/1.49 ( ! [M3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ M3 @ N3 )
% 2.58/1.49 => ( P @ M3 ) )
% 2.58/1.49 => ( P @ N3 ) )
% 2.58/1.49 => ( P @ N2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_336_less__irrefl__nat,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ~ ( ord_less_nat @ N2 @ N2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_337_less__not__refl3,axiom,
% 2.58/1.49 ! [S: nat,T: nat] :
% 2.58/1.49 ( ( ord_less_nat @ S @ T )
% 2.58/1.49 => ( S != T ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_338_less__not__refl2,axiom,
% 2.58/1.49 ! [N2: nat,M2: nat] :
% 2.58/1.49 ( ( ord_less_nat @ N2 @ M2 )
% 2.58/1.49 => ( M2 != N2 ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_339_less__not__refl,axiom,
% 2.58/1.49 ! [N2: nat] :
% 2.58/1.49 ~ ( ord_less_nat @ N2 @ N2 ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_340_nat__neq__iff,axiom,
% 2.58/1.49 ! [M2: nat,N2: nat] :
% 2.58/1.49 ( ( M2 != N2 )
% 2.58/1.49 = ( ( ord_less_nat @ M2 @ N2 )
% 2.58/1.49 | ( ord_less_nat @ N2 @ M2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_341_nat__descend__induct,axiom,
% 2.58/1.49 ! [N2: nat,P: nat > $o,M2: nat] :
% 2.58/1.49 ( ! [K2: nat] :
% 2.58/1.49 ( ( ord_less_nat @ N2 @ K2 )
% 2.58/1.49 => ( P @ K2 ) )
% 2.58/1.49 => ( ! [K2: nat] :
% 2.58/1.49 ( ( ord_less_eq_nat @ K2 @ N2 )
% 2.58/1.49 => ( ! [I3: nat] :
% 2.58/1.49 ( ( ord_less_nat @ K2 @ I3 )
% 2.58/1.49 => ( P @ I3 ) )
% 2.58/1.49 => ( P @ K2 ) ) )
% 2.58/1.49 => ( P @ M2 ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_342_less__eq__real__def,axiom,
% 2.58/1.49 ( ord_less_eq_real
% 2.58/1.49 = ( ^ [X3: real,Y6: real] :
% 2.58/1.49 ( ( ord_less_real @ X3 @ Y6 )
% 2.58/1.49 | ( X3 = Y6 ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(fact_343_complete__real,axiom,
% 2.58/1.49 ! [S2: set_real] :
% 2.58/1.49 ( ? [X6: real] : ( member_real @ X6 @ S2 )
% 2.58/1.49 => ( ? [Z4: real] :
% 2.58/1.49 ! [X: real] :
% 2.58/1.49 ( ( member_real @ X @ S2 )
% 2.58/1.49 => ( ord_less_eq_real @ X @ Z4 ) )
% 2.58/1.49 => ? [Y3: real] :
% 2.58/1.49 ( ! [X6: real] :
% 2.58/1.49 ( ( member_real @ X6 @ S2 )
% 2.58/1.49 => ( ord_less_eq_real @ X6 @ Y3 ) )
% 2.58/1.49 & ! [Z4: real] :
% 2.58/1.49 ( ! [X: real] :
% 2.58/1.49 ( ( member_real @ X @ S2 )
% 2.58/1.49 => ( ord_less_eq_real @ X @ Z4 ) )
% 2.58/1.49 => ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).
% 2.58/1.49
% 2.58/1.49 thf(conj_0,hypothesis,
% 2.58/1.49 ! [A6: a,B5: real] :
% 2.58/1.49 ( ( ( member1103263856a_real @ ( product_Pair_a_real @ A6 @ B5 ) @ ( lower_930854854raph_a @ top_top_set_a @ f ) )
% 2.58/1.49 & ( ord_less_real @ ( real_V404783528a_real @ ( product_Pair_a_real @ A6 @ B5 ) @ ( product_Pair_a_real @ x @ y ) ) @ e ) )
% 2.58/1.49 => thesis ) ).
% 2.58/1.49
% 2.58/1.49 thf(conj_1,conjecture,
% 2.58/1.49 thesis ).
% 2.58/1.49 ------------------------
% 2.70/1.52 % (2668)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_2567 for (2999ds/4Mi)
% 2.70/1.52 % (2669)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_2567 for (2999ds/27Mi)
% 2.70/1.52 % (2667)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/183Mi)
% 2.70/1.52 % (2668)Instruction limit reached!
% 2.70/1.52 % (2668)------------------------------
% 2.70/1.52 % (2668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.52 % (2668)Termination reason: Unknown
% 2.70/1.52 % (2668)Termination phase: shuffling
% 2.70/1.52
% 2.70/1.52 % (2668)Memory used [KB]: 1407
% 2.70/1.52 % (2668)Time elapsed: 0.003 s
% 2.70/1.52 % (2668)Instructions burned: 5 (million)
% 2.70/1.52 % (2668)------------------------------
% 2.70/1.52 % (2668)------------------------------
% 2.70/1.53 % (2670)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/2Mi)
% 2.70/1.53 % (2671)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_2567 for (2999ds/2Mi)
% 2.70/1.53 % (2673)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/18Mi)
% 2.70/1.53 % (2672)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_2567 for (2999ds/275Mi)
% 2.70/1.53 % (2669)Instruction limit reached!
% 2.70/1.53 % (2669)------------------------------
% 2.70/1.53 % (2669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.53 % (2669)Termination reason: Unknown
% 2.70/1.53 % (2669)Termination phase: Property scanning
% 2.70/1.53
% 2.70/1.53 % (2669)Memory used [KB]: 1791
% 2.70/1.53 % (2669)Time elapsed: 0.010 s
% 2.70/1.53 % (2669)Instructions burned: 27 (million)
% 2.70/1.53 % (2669)------------------------------
% 2.70/1.53 % (2669)------------------------------
% 2.70/1.53 % (2670)Instruction limit reached!
% 2.70/1.53 % (2670)------------------------------
% 2.70/1.53 % (2670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.53 % (2670)Termination reason: Unknown
% 2.70/1.53 % (2670)Termination phase: shuffling
% 2.70/1.53
% 2.70/1.53 % (2670)Memory used [KB]: 1407
% 2.70/1.53 % (2671)Instruction limit reached!
% 2.70/1.53 % (2671)------------------------------
% 2.70/1.53 % (2671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.53 % (2671)Termination reason: Unknown
% 2.70/1.53 % (2671)Termination phase: shuffling
% 2.70/1.53
% 2.70/1.53 % (2671)Memory used [KB]: 1407
% 2.70/1.53 % (2671)Time elapsed: 0.004 s
% 2.70/1.53 % (2671)Instructions burned: 3 (million)
% 2.70/1.53 % (2671)------------------------------
% 2.70/1.53 % (2671)------------------------------
% 2.70/1.53 % (2670)Time elapsed: 0.004 s
% 2.70/1.53 % (2670)Instructions burned: 3 (million)
% 2.70/1.53 % (2670)------------------------------
% 2.70/1.53 % (2670)------------------------------
% 2.70/1.53 % (2674)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/3Mi)
% 2.70/1.53 % (2674)Instruction limit reached!
% 2.70/1.53 % (2674)------------------------------
% 2.70/1.53 % (2674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.53 % (2674)Termination reason: Unknown
% 2.70/1.53 % (2674)Termination phase: shuffling
% 2.70/1.53
% 2.70/1.53 % (2674)Memory used [KB]: 1407
% 2.70/1.53 % (2674)Time elapsed: 0.002 s
% 2.70/1.53 % (2674)Instructions burned: 3 (million)
% 2.70/1.53 % (2674)------------------------------
% 2.70/1.53 % (2674)------------------------------
% 2.70/1.54 % (2673)Instruction limit reached!
% 2.70/1.54 % (2673)------------------------------
% 2.70/1.54 % (2673)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.54 % (2673)Termination reason: Unknown
% 2.70/1.54 % (2673)Termination phase: shuffling
% 2.70/1.54
% 2.70/1.54 % (2673)Memory used [KB]: 1663
% 2.70/1.54 % (2673)Time elapsed: 0.013 s
% 2.70/1.54 % (2673)Instructions burned: 18 (million)
% 2.70/1.54 % (2673)------------------------------
% 2.70/1.54 % (2673)------------------------------
% 2.70/1.54 % (2675)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_2567 for (2999ds/37Mi)
% 2.70/1.54 % (2678)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_2567 for (2999ds/1041Mi)
% 2.70/1.55 % (2677)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_2567 for (2999ds/3Mi)
% 2.70/1.55 % (2676)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_2567 for (2999ds/15Mi)
% 2.70/1.55 % (2677)Instruction limit reached!
% 2.70/1.55 % (2677)------------------------------
% 2.70/1.55 % (2677)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.55 % (2677)Termination reason: Unknown
% 2.70/1.55 % (2677)Termination phase: shuffling
% 2.70/1.55
% 2.70/1.55 % (2677)Memory used [KB]: 1407
% 2.70/1.55 % (2677)Time elapsed: 0.004 s
% 2.70/1.55 % (2677)Instructions burned: 3 (million)
% 2.70/1.55 % (2677)------------------------------
% 2.70/1.55 % (2677)------------------------------
% 2.70/1.55 % (2675)Instruction limit reached!
% 2.70/1.55 % (2675)------------------------------
% 2.70/1.55 % (2675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.55 % (2675)Termination reason: Unknown
% 2.70/1.55 % (2675)Termination phase: shuffling
% 2.70/1.55
% 2.70/1.55 % (2675)Memory used [KB]: 2174
% 2.70/1.55 % (2675)Time elapsed: 0.013 s
% 2.70/1.55 % (2675)Instructions burned: 38 (million)
% 2.70/1.55 % (2675)------------------------------
% 2.70/1.55 % (2675)------------------------------
% 2.70/1.56 % (2679)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/7Mi)
% 2.70/1.56 % (2676)Instruction limit reached!
% 2.70/1.56 % (2676)------------------------------
% 2.70/1.56 % (2676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.56 % (2676)Termination reason: Unknown
% 2.70/1.56 % (2676)Termination phase: shuffling
% 2.70/1.56
% 2.70/1.56 % (2676)Memory used [KB]: 1663
% 2.70/1.56 % (2676)Time elapsed: 0.011 s
% 2.70/1.56 % (2676)Instructions burned: 15 (million)
% 2.70/1.56 % (2676)------------------------------
% 2.70/1.56 % (2676)------------------------------
% 2.70/1.56 % (2679)Instruction limit reached!
% 2.70/1.56 % (2679)------------------------------
% 2.70/1.56 % (2679)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.56 % (2679)Termination reason: Unknown
% 2.70/1.56 % (2679)Termination phase: shuffling
% 2.70/1.56
% 2.70/1.56 % (2679)Memory used [KB]: 1535
% 2.70/1.56 % (2679)Time elapsed: 0.006 s
% 2.70/1.56 % (2679)Instructions burned: 7 (million)
% 2.70/1.56 % (2679)------------------------------
% 2.70/1.56 % (2679)------------------------------
% 2.70/1.56 % (2681)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_2567 for (2999ds/3Mi)
% 2.70/1.56 % (2681)Instruction limit reached!
% 2.70/1.56 % (2681)------------------------------
% 2.70/1.56 % (2681)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.56 % (2681)Termination reason: Unknown
% 2.70/1.56 % (2681)Termination phase: shuffling
% 2.70/1.56
% 2.70/1.56 % (2681)Memory used [KB]: 1407
% 2.70/1.56 % (2681)Time elapsed: 0.003 s
% 2.70/1.56 % (2681)Instructions burned: 3 (million)
% 2.70/1.56 % (2681)------------------------------
% 2.70/1.56 % (2681)------------------------------
% 2.70/1.57 % (2680)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/16Mi)
% 2.70/1.57 % (2684)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/3Mi)
% 2.70/1.57 % (2684)Instruction limit reached!
% 2.70/1.57 % (2684)------------------------------
% 2.70/1.57 % (2684)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.57 % (2684)Termination reason: Unknown
% 2.70/1.57 % (2684)Termination phase: shuffling
% 2.70/1.57
% 2.70/1.57 % (2684)Memory used [KB]: 1407
% 2.70/1.57 % (2684)Time elapsed: 0.003 s
% 2.70/1.57 % (2684)Instructions burned: 3 (million)
% 2.70/1.57 % (2684)------------------------------
% 2.70/1.57 % (2684)------------------------------
% 2.70/1.57 % (2683)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/7Mi)
% 2.70/1.58 % (2667)Instruction limit reached!
% 2.70/1.58 % (2667)------------------------------
% 2.70/1.58 % (2667)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.58 % (2667)Termination reason: Unknown
% 2.70/1.58 % (2667)Termination phase: Saturation
% 2.70/1.58
% 2.70/1.58 % (2667)Memory used [KB]: 7419
% 2.70/1.58 % (2667)Time elapsed: 0.056 s
% 2.70/1.58 % (2667)Instructions burned: 184 (million)
% 2.70/1.58 % (2667)------------------------------
% 2.70/1.58 % (2667)------------------------------
% 2.70/1.58 % (2682)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_2567 for (2999ds/3Mi)
% 2.70/1.58 % (2680)Instruction limit reached!
% 2.70/1.58 % (2680)------------------------------
% 2.70/1.58 % (2680)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.58 % (2680)Termination reason: Unknown
% 2.70/1.58 % (2680)Termination phase: shuffling
% 2.70/1.58
% 2.70/1.58 % (2680)Memory used [KB]: 1663
% 2.70/1.58 % (2680)Time elapsed: 0.011 s
% 2.70/1.58 % (2680)Instructions burned: 16 (million)
% 2.70/1.58 % (2680)------------------------------
% 2.70/1.58 % (2680)------------------------------
% 2.70/1.58 % (2683)Instruction limit reached!
% 2.70/1.58 % (2683)------------------------------
% 2.70/1.58 % (2683)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.58 % (2683)Termination reason: Unknown
% 2.70/1.58 % (2683)Termination phase: shuffling
% 2.70/1.58
% 2.70/1.58 % (2683)Memory used [KB]: 1535
% 2.70/1.58 % (2683)Time elapsed: 0.005 s
% 2.70/1.58 % (2683)Instructions burned: 9 (million)
% 2.70/1.58 % (2683)------------------------------
% 2.70/1.58 % (2683)------------------------------
% 2.70/1.58 % (2682)Instruction limit reached!
% 2.70/1.58 % (2682)------------------------------
% 2.70/1.58 % (2682)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.58 % (2682)Termination reason: Unknown
% 2.70/1.58 % (2682)Termination phase: shuffling
% 2.70/1.58
% 2.70/1.58 % (2682)Memory used [KB]: 1407
% 2.70/1.58 % (2682)Time elapsed: 0.004 s
% 2.70/1.58 % (2682)Instructions burned: 3 (million)
% 2.70/1.58 % (2682)------------------------------
% 2.70/1.58 % (2682)------------------------------
% 2.70/1.58 % (2685)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/4Mi)
% 2.70/1.59 % (2685)Instruction limit reached!
% 2.70/1.59 % (2685)------------------------------
% 2.70/1.59 % (2685)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (2685)Termination reason: Unknown
% 2.70/1.59 % (2685)Termination phase: shuffling
% 2.70/1.59
% 2.70/1.59 % (2685)Memory used [KB]: 1407
% 2.70/1.59 % (2685)Time elapsed: 0.003 s
% 2.70/1.59 % (2685)Instructions burned: 4 (million)
% 2.70/1.59 % (2685)------------------------------
% 2.70/1.59 % (2685)------------------------------
% 2.70/1.59 % (2686)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/18Mi)
% 2.70/1.59 % (2688)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/6Mi)
% 2.70/1.59 % (2687)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_2567 for (2999ds/710Mi)
% 2.70/1.59 % (2672)First to succeed.
% 2.70/1.59 % (2688)Instruction limit reached!
% 2.70/1.59 % (2688)------------------------------
% 2.70/1.59 % (2688)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (2688)Termination reason: Unknown
% 2.70/1.59 % (2688)Termination phase: shuffling
% 2.70/1.59
% 2.70/1.59 % (2688)Memory used [KB]: 1535
% 2.70/1.59 % (2688)Time elapsed: 0.004 s
% 2.70/1.59 % (2688)Instructions burned: 7 (million)
% 2.70/1.59 % (2688)------------------------------
% 2.70/1.59 % (2688)------------------------------
% 2.70/1.59 % (2672)Refutation found. Thanks to Tanya!
% 2.70/1.59 % SZS status Theorem for DTF2THF_2567
% 2.70/1.59 % SZS output start Proof for DTF2THF_2567
% 2.70/1.59 thf(type_def_5, type, set_Pr1928503567a_real: $tType).
% 2.70/1.59 thf(type_def_7, type, set_real: $tType).
% 2.70/1.59 thf(type_def_8, type, set_a: $tType).
% 2.70/1.59 thf(type_def_9, type, nat: $tType).
% 2.70/1.59 thf(type_def_10, type, real: $tType).
% 2.70/1.59 thf(type_def_11, type, product_prod_a_real: $tType).
% 2.70/1.59 thf(type_def_12, type, extended_ereal: $tType).
% 2.70/1.59 thf(type_def_13, type, set_Pr1158285382l_real: $tType).
% 2.70/1.59 thf(type_def_14, type, set_Pr147102617l_real: $tType).
% 2.70/1.59 thf(type_def_15, type, a: $tType).
% 2.70/1.59 thf(type_def_16, type, produc1691597095_set_a: $tType).
% 2.70/1.59 thf(type_def_17, type, product_prod_nat_nat: $tType).
% 2.70/1.59 thf(type_def_18, type, produc845587165t_real: $tType).
% 2.70/1.59 thf(type_def_19, type, produc581916509al_nat: $tType).
% 2.70/1.59 thf(type_def_20, type, produc957004601l_real: $tType).
% 2.70/1.59 thf(type_def_21, type, produc1312517202t_real: $tType).
% 2.70/1.59 thf(type_def_22, type, produc25772804_set_a: $tType).
% 2.70/1.59 thf(type_def_23, type, produc2021989178_set_a: $tType).
% 2.70/1.59 thf(type_def_24, type, produc211508562a_real: $tType).
% 2.70/1.59 thf(type_def_25, type, produc1931999653t_real: $tType).
% 2.70/1.59 thf(type_def_26, type, produc1350454001_set_a: $tType).
% 2.70/1.59 thf(type_def_27, type, produc304417648_set_a: $tType).
% 2.70/1.59 thf(type_def_28, type, produc509513372a_real: $tType).
% 2.70/1.59 thf(type_def_29, type, produc232195311t_real: $tType).
% 2.70/1.59 thf(type_def_30, type, produc792423270l_real: $tType).
% 2.70/1.59 thf(func_def_0, type, produc2021989178_set_a: $tType).
% 2.70/1.59 thf(func_def_1, type, produc211508562a_real: $tType).
% 2.70/1.59 thf(func_def_2, type, produc1312517202t_real: $tType).
% 2.70/1.59 thf(func_def_3, type, produc304417648_set_a: $tType).
% 2.70/1.59 thf(func_def_4, type, produc509513372a_real: $tType).
% 2.70/1.59 thf(func_def_5, type, produc25772804_set_a: $tType).
% 2.70/1.59 thf(func_def_6, type, set_Pr1158285382l_real: $tType).
% 2.70/1.59 thf(func_def_7, type, produc792423270l_real: $tType).
% 2.70/1.59 thf(func_def_8, type, produc1931999653t_real: $tType).
% 2.70/1.59 thf(func_def_9, type, produc232195311t_real: $tType).
% 2.70/1.59 thf(func_def_10, type, produc1350454001_set_a: $tType).
% 2.70/1.59 thf(func_def_11, type, set_Pr147102617l_real: $tType).
% 2.70/1.59 thf(func_def_12, type, produc1691597095_set_a: $tType).
% 2.70/1.59 thf(func_def_13, type, set_Pr1928503567a_real: $tType).
% 2.70/1.59 thf(func_def_14, type, produc957004601l_real: $tType).
% 2.70/1.59 thf(func_def_15, type, produc581916509al_nat: $tType).
% 2.70/1.59 thf(func_def_16, type, produc845587165t_real: $tType).
% 2.70/1.59 thf(func_def_17, type, product_prod_nat_nat: $tType).
% 2.70/1.59 thf(func_def_18, type, product_prod_a_real: $tType).
% 2.70/1.59 thf(func_def_19, type, set_real: $tType).
% 2.70/1.59 thf(func_def_20, type, extended_ereal: $tType).
% 2.70/1.59 thf(func_def_21, type, set_a: $tType).
% 2.70/1.59 thf(func_def_22, type, real: $tType).
% 2.70/1.59 thf(func_def_23, type, nat: $tType).
% 2.70/1.59 thf(func_def_24, type, a: $tType).
% 2.70/1.59 thf(func_def_25, type, elemen1695521870a_real: set_Pr1928503567a_real > set_Pr1928503567a_real).
% 2.70/1.59 thf(func_def_26, type, elemen1168706977e_real: set_real > set_real).
% 2.70/1.59 thf(func_def_27, type, elementary_closure_a: set_a > set_a).
% 2.70/1.59 thf(func_def_28, type, zero_zero_nat: nat).
% 2.70/1.59 thf(func_def_29, type, zero_zero_real: real).
% 2.70/1.59 thf(func_def_30, type, lower_870348689a_real: set_Pr1928503567a_real > (product_prod_a_real > extended_ereal) > set_Pr1158285382l_real).
% 2.70/1.59 thf(func_def_31, type, lower_1607476196h_real: set_real > (real > extended_ereal) > set_Pr147102617l_real).
% 2.70/1.59 thf(func_def_32, type, lower_930854854raph_a: set_a > (a > extended_ereal) > set_Pr1928503567a_real).
% 2.70/1.59 thf(func_def_33, type, ord_less_nat: nat > nat > $o).
% 2.70/1.59 thf(func_def_34, type, ord_le297355219_set_a: produc1691597095_set_a > produc1691597095_set_a > $o).
% 2.70/1.59 thf(func_def_35, type, ord_less_real: real > real > $o).
% 2.70/1.59 thf(func_def_36, type, ord_le1379034467a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > $o).
% 2.70/1.59 thf(func_def_37, type, ord_less_set_real: set_real > set_real > $o).
% 2.70/1.59 thf(func_def_38, type, ord_less_set_a: set_a > set_a > $o).
% 2.70/1.59 thf(func_def_39, type, ord_less_eq_nat: nat > nat > $o).
% 2.70/1.59 thf(func_def_40, type, ord_le1302190241at_nat: product_prod_nat_nat > product_prod_nat_nat > $o).
% 2.70/1.59 thf(func_def_41, type, ord_le1950669437t_real: produc845587165t_real > produc845587165t_real > $o).
% 2.70/1.59 thf(func_def_42, type, ord_le1686998781al_nat: produc581916509al_nat > produc581916509al_nat > $o).
% 2.70/1.59 thf(func_def_43, type, ord_le1342644953l_real: produc957004601l_real > produc957004601l_real > $o).
% 2.70/1.59 thf(func_def_44, type, ord_le486764743_set_a: produc1691597095_set_a > produc1691597095_set_a > $o).
% 2.70/1.59 thf(func_def_45, type, ord_less_eq_real: real > real > $o).
% 2.70/1.59 thf(func_def_46, type, ord_le1586073967a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > $o).
% 2.70/1.59 thf(func_def_47, type, ord_less_eq_set_real: set_real > set_real > $o).
% 2.70/1.59 thf(func_def_48, type, ord_less_eq_set_a: set_a > set_a > $o).
% 2.70/1.59 thf(func_def_49, type, top_to1567080838real_o: product_prod_a_real > $o).
% 2.70/1.59 thf(func_def_50, type, top_top_real_o: real > $o).
% 2.70/1.59 thf(func_def_51, type, top_top_a_o: a > $o).
% 2.70/1.59 thf(func_def_52, type, top_to991334818t_real: produc1312517202t_real).
% 2.70/1.59 thf(func_def_53, type, top_to345582004_set_a: produc25772804_set_a).
% 2.70/1.59 thf(func_def_54, type, top_to522338794_set_a: produc2021989178_set_a).
% 2.70/1.59 thf(func_def_55, type, top_to2037809826a_real: produc211508562a_real).
% 2.70/1.59 thf(func_def_56, type, top_to95743733t_real: produc1931999653t_real).
% 2.70/1.59 thf(func_def_57, type, top_to1153982881_set_a: produc1350454001_set_a).
% 2.70/1.59 thf(func_def_58, type, top_to724687040_set_a: produc304417648_set_a).
% 2.70/1.59 thf(func_def_59, type, top_to829322572a_real: produc509513372a_real).
% 2.70/1.59 thf(func_def_60, type, top_to35724191t_real: produc232195311t_real).
% 2.70/1.59 thf(func_def_61, type, top_to1477940855_set_a: produc1691597095_set_a).
% 2.70/1.59 thf(func_def_62, type, top_to2138011583a_real: set_Pr1928503567a_real).
% 2.70/1.59 thf(func_def_63, type, top_top_set_real: set_real).
% 2.70/1.59 thf(func_def_64, type, top_top_set_a: set_a).
% 2.70/1.59 thf(func_def_65, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 2.70/1.59 thf(func_def_66, type, produc49576085t_real: nat > real > produc845587165t_real).
% 2.70/1.59 thf(func_def_67, type, produc1214388894l_real: product_prod_a_real > real > produc792423270l_real).
% 2.70/1.59 thf(func_def_68, type, produc1541812501al_nat: real > nat > produc581916509al_nat).
% 2.70/1.59 thf(func_def_69, type, produc705216881l_real: real > real > produc957004601l_real).
% 2.70/1.59 thf(func_def_70, type, produc471223946t_real: set_Pr1928503567a_real > set_real > produc1312517202t_real).
% 2.70/1.59 thf(func_def_71, type, produc1220567734_set_a: set_Pr1928503567a_real > set_a > produc25772804_set_a).
% 2.70/1.59 thf(func_def_72, type, produc1162518380_set_a: set_real > produc1691597095_set_a > produc2021989178_set_a).
% 2.70/1.59 thf(func_def_73, type, produc150798858a_real: set_real > set_Pr1928503567a_real > produc211508562a_real).
% 2.70/1.59 thf(func_def_74, type, produc119681757t_real: set_real > set_real > produc1931999653t_real).
% 2.70/1.59 thf(func_def_75, type, produc190993315_set_a: set_real > set_a > produc1350454001_set_a).
% 2.70/1.59 thf(func_def_76, type, produc272677600_set_a: set_a > produc1691597095_set_a > produc304417648_set_a).
% 2.70/1.59 thf(func_def_77, type, produc383769494a_real: set_a > set_Pr1928503567a_real > produc509513372a_real).
% 2.70/1.59 thf(func_def_78, type, produc1266579561t_real: set_a > set_real > produc232195311t_real).
% 2.70/1.59 thf(func_def_79, type, produc1928581911_set_a: set_a > set_a > produc1691597095_set_a).
% 2.70/1.59 thf(func_def_80, type, product_Pair_a_real: a > real > product_prod_a_real).
% 2.70/1.59 thf(func_def_81, type, real_V404783528a_real: product_prod_a_real > product_prod_a_real > real).
% 2.70/1.59 thf(func_def_82, type, real_V1934908667t_real: real > real > real).
% 2.70/1.59 thf(func_def_83, type, real_V1514887919dist_a: a > a > real).
% 2.70/1.59 thf(func_def_84, type, collec1714955950a_real: (product_prod_a_real > $o) > set_Pr1928503567a_real).
% 2.70/1.59 thf(func_def_85, type, collect_real: (real > $o) > set_real).
% 2.70/1.59 thf(func_def_86, type, collect_a: (a > $o) > set_a).
% 2.70/1.59 thf(func_def_87, type, member565909519l_real: produc792423270l_real > set_Pr1158285382l_real > $o).
% 2.70/1.59 thf(func_def_88, type, member1068169442l_real: produc957004601l_real > set_Pr147102617l_real > $o).
% 2.70/1.59 thf(func_def_89, type, member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o).
% 2.70/1.59 thf(func_def_90, type, member_real: real > set_real > $o).
% 2.70/1.59 thf(func_def_91, type, member_a: a > set_a > $o).
% 2.70/1.59 thf(func_def_92, type, e: real).
% 2.70/1.59 thf(func_def_93, type, f: a > extended_ereal).
% 2.70/1.59 thf(func_def_95, type, x: a).
% 2.70/1.59 thf(func_def_96, type, y: real).
% 2.70/1.59 thf(func_def_97, type, z: real).
% 2.70/1.59 thf(func_def_112, type, sP0: nat > nat > nat > $o).
% 2.70/1.59 thf(func_def_113, type, sP1: nat > nat > nat > $o).
% 2.70/1.59 thf(func_def_114, type, sP2: nat > nat > nat > $o).
% 2.70/1.59 thf(func_def_115, type, sP3: real > real > real > $o).
% 2.70/1.59 thf(func_def_116, type, sP4: real > real > real > $o).
% 2.70/1.59 thf(func_def_117, type, sP5: real > real > real > $o).
% 2.70/1.59 thf(func_def_118, type, sK6: (real > real) > real).
% 2.70/1.59 thf(func_def_119, type, sK7: (real > real) > real).
% 2.70/1.59 thf(func_def_120, type, sK8: (real > nat) > real).
% 2.70/1.59 thf(func_def_121, type, sK9: (real > nat) > real).
% 2.70/1.59 thf(func_def_122, type, sK10: (real > real) > real).
% 2.70/1.59 thf(func_def_123, type, sK11: (real > real) > real).
% 2.70/1.59 thf(func_def_124, type, sK12: product_prod_a_real > a).
% 2.70/1.59 thf(func_def_125, type, sK13: product_prod_a_real > real).
% 2.70/1.59 thf(func_def_126, type, sK14: (real > nat) > real).
% 2.70/1.59 thf(func_def_127, type, sK15: (real > nat) > real).
% 2.70/1.59 thf(func_def_128, type, sK16: product_prod_a_real).
% 2.70/1.59 thf(func_def_129, type, sK17: (nat > real) > nat).
% 2.70/1.59 thf(func_def_130, type, sK18: (nat > real) > nat).
% 2.70/1.59 thf(func_def_131, type, sK19: (real > real) > real).
% 2.70/1.59 thf(func_def_132, type, sK20: (real > real) > real).
% 2.70/1.59 thf(func_def_133, type, sK21: real > real > real).
% 2.70/1.59 thf(func_def_134, type, sK22: product_prod_a_real).
% 2.70/1.59 thf(func_def_135, type, sK23: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_136, type, sK24: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_137, type, sK25: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_138, type, sK26: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_139, type, sK27: real > real).
% 2.70/1.59 thf(func_def_140, type, sK28: (nat > $o) > nat).
% 2.70/1.59 thf(func_def_141, type, sK29: real > real > real).
% 2.70/1.59 thf(func_def_142, type, sK30: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_143, type, sK31: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_144, type, sK32: (real > nat) > real).
% 2.70/1.59 thf(func_def_145, type, sK33: (real > nat) > real).
% 2.70/1.59 thf(func_def_146, type, sK34: (real > nat) > real).
% 2.70/1.59 thf(func_def_147, type, sK35: (real > nat) > real).
% 2.70/1.59 thf(func_def_148, type, sK36: (nat > real) > nat).
% 2.70/1.59 thf(func_def_149, type, sK37: (nat > real) > nat).
% 2.70/1.59 thf(func_def_150, type, sK38: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_151, type, sK39: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_152, type, sK40: (real > nat) > real).
% 2.70/1.59 thf(func_def_153, type, sK41: (real > nat) > real).
% 2.70/1.59 thf(func_def_154, type, sK42: (nat > real) > nat).
% 2.70/1.59 thf(func_def_155, type, sK43: (nat > real) > nat).
% 2.70/1.59 thf(func_def_156, type, sK44: real > real > real).
% 2.70/1.59 thf(func_def_157, type, sK45: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_158, type, sK46: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_159, type, sK47: (real > nat) > real).
% 2.70/1.59 thf(func_def_160, type, sK48: (real > nat) > real).
% 2.70/1.59 thf(func_def_161, type, sK49: (nat > nat > $o) > nat).
% 2.70/1.59 thf(func_def_162, type, sK50: (nat > nat > $o) > nat).
% 2.70/1.59 thf(func_def_163, type, sK51: (nat > nat > $o) > nat).
% 2.70/1.59 thf(func_def_164, type, sK52: (nat > nat > $o) > nat).
% 2.70/1.59 thf(func_def_165, type, sK53: (nat > nat > $o) > nat).
% 2.70/1.59 thf(func_def_166, type, sK54: real > real).
% 2.70/1.59 thf(func_def_167, type, sK55: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_168, type, sK56: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_169, type, sK57: (product_prod_a_real > $o) > real).
% 2.70/1.59 thf(func_def_170, type, sK58: (product_prod_a_real > $o) > a).
% 2.70/1.59 thf(func_def_171, type, sK59: (real > real) > real).
% 2.70/1.59 thf(func_def_172, type, sK60: (real > real) > real).
% 2.70/1.59 thf(func_def_173, type, sK61: (real > nat) > real).
% 2.70/1.59 thf(func_def_174, type, sK62: (real > nat) > real).
% 2.70/1.59 thf(func_def_175, type, sK63: (nat > nat > $o) > nat).
% 2.70/1.59 thf(func_def_176, type, sK64: (nat > nat > $o) > nat).
% 2.70/1.59 thf(func_def_177, type, sK65: (nat > nat > $o) > nat).
% 2.70/1.59 thf(func_def_178, type, sK66: (nat > nat > $o) > nat).
% 2.70/1.59 thf(func_def_179, type, sK67: (nat > real) > nat).
% 2.70/1.59 thf(func_def_180, type, sK68: (nat > real) > nat).
% 2.70/1.59 thf(func_def_181, type, sK69: (nat > $o) > nat > nat).
% 2.70/1.59 thf(func_def_182, type, sK70: (nat > $o) > nat).
% 2.70/1.59 thf(func_def_183, type, sK71: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_184, type, sK72: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_185, type, sK73: set_Pr1928503567a_real > product_prod_a_real).
% 2.70/1.59 thf(func_def_186, type, sK74: (nat > real) > nat).
% 2.70/1.59 thf(func_def_187, type, sK75: (nat > real) > nat).
% 2.70/1.59 thf(func_def_188, type, sK76: real > real > real > real).
% 2.70/1.59 thf(func_def_189, type, sK77: nat > (nat > real) > nat).
% 2.70/1.59 thf(func_def_190, type, sK78: (nat > real) > nat > (nat > real) > nat).
% 2.70/1.59 thf(func_def_191, type, sK79: (real > nat) > real).
% 2.70/1.59 thf(func_def_192, type, sK80: (real > nat) > real).
% 2.70/1.59 thf(func_def_193, type, sK81: (nat > real) > nat).
% 2.70/1.59 thf(func_def_194, type, sK82: (nat > real) > nat).
% 2.70/1.59 thf(func_def_195, type, sK83: (nat > real) > nat).
% 2.70/1.59 thf(func_def_196, type, sK84: (nat > real) > nat).
% 2.70/1.59 thf(func_def_197, type, sK85: (nat > $o) > nat).
% 2.70/1.59 thf(func_def_198, type, sK86: (real > real) > real).
% 2.70/1.59 thf(func_def_199, type, sK87: (real > real) > real).
% 2.70/1.59 thf(func_def_200, type, sK88: (nat > real) > nat).
% 2.70/1.59 thf(func_def_201, type, sK89: (nat > real) > nat).
% 2.70/1.59 thf(func_def_202, type, sK90: (product_prod_a_real > $o) > real).
% 2.70/1.59 thf(func_def_203, type, sK91: (product_prod_a_real > $o) > a).
% 2.70/1.59 thf(func_def_204, type, sK92: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_205, type, sK93: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_206, type, sK94: (real > real > $o) > real).
% 2.70/1.59 thf(func_def_207, type, sK95: (real > real > $o) > real).
% 2.70/1.59 thf(func_def_208, type, sK96: (real > real > $o) > real).
% 2.70/1.59 thf(func_def_209, type, sK97: (real > real > $o) > real).
% 2.70/1.59 thf(func_def_210, type, sK98: (real > real > $o) > real).
% 2.70/1.59 thf(func_def_211, type, sK99: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_212, type, sK100: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_213, type, sK101: (real > real > $o) > real).
% 2.70/1.59 thf(func_def_214, type, sK102: (real > real > $o) > real).
% 2.70/1.59 thf(func_def_215, type, sK103: (real > real > $o) > real).
% 2.70/1.59 thf(func_def_216, type, sK104: (real > real > $o) > real).
% 2.70/1.59 thf(func_def_217, type, sK105: (real > real) > real).
% 2.70/1.59 thf(func_def_218, type, sK106: (real > real) > real).
% 2.70/1.59 thf(func_def_219, type, sK107: (nat > real) > nat).
% 2.70/1.59 thf(func_def_220, type, sK108: (nat > real) > nat).
% 2.70/1.59 thf(func_def_221, type, sK109: (real > real) > real).
% 2.70/1.59 thf(func_def_222, type, sK110: (real > real) > real).
% 2.70/1.59 thf(func_def_223, type, sK111: product_prod_a_real > product_prod_a_real > set_Pr1928503567a_real > product_prod_a_real).
% 2.70/1.59 thf(func_def_224, type, sK112: (real > real) > real).
% 2.70/1.59 thf(func_def_225, type, sK113: (real > real) > real).
% 2.70/1.59 thf(func_def_226, type, sK114: (real > nat) > real).
% 2.70/1.59 thf(func_def_227, type, sK115: (real > nat) > real).
% 2.70/1.59 thf(func_def_228, type, sK116: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_229, type, sK117: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_230, type, sK118: nat > nat).
% 2.70/1.59 thf(func_def_231, type, sK119: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_232, type, sK120: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_233, type, sK121: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_234, type, sK122: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_235, type, sK123: (nat > real) > nat).
% 2.70/1.59 thf(func_def_236, type, sK124: (nat > real) > nat).
% 2.70/1.59 thf(func_def_237, type, sK125: (nat > real) > nat).
% 2.70/1.59 thf(func_def_238, type, sK126: (nat > real) > nat).
% 2.70/1.59 thf(func_def_239, type, sK127: (nat > real) > nat).
% 2.70/1.59 thf(func_def_240, type, sK128: (nat > real) > nat).
% 2.70/1.59 thf(func_def_241, type, sK129: real > real > real > real).
% 2.70/1.59 thf(func_def_242, type, sK130: (real > real) > real).
% 2.70/1.59 thf(func_def_243, type, sK131: (real > real) > real).
% 2.70/1.59 thf(func_def_244, type, sK132: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_245, type, sK133: (nat > nat) > nat).
% 2.70/1.59 thf(func_def_246, type, sK134: (real > nat) > real).
% 2.70/1.59 thf(func_def_247, type, sK135: (real > nat) > real).
% 2.70/1.59 thf(func_def_248, type, sK136: (nat > $o) > nat > nat).
% 2.70/1.59 thf(func_def_249, type, sK137: (nat > $o) > nat > nat).
% 2.70/1.59 thf(func_def_250, type, sK138: (real > nat) > real).
% 2.70/1.59 thf(func_def_251, type, sK139: (real > nat) > real).
% 2.70/1.59 thf(func_def_252, type, sK140: (real > real) > real).
% 2.70/1.59 thf(func_def_253, type, sK141: (real > real) > real).
% 2.70/1.59 thf(func_def_254, type, sK142: (nat > $o) > nat).
% 2.70/1.59 thf(func_def_255, type, sK143: (real > real) > real).
% 2.70/1.59 thf(func_def_256, type, sK144: (real > real) > real).
% 2.70/1.59 thf(func_def_257, type, sK145: (real > nat) > real).
% 2.70/1.59 thf(func_def_258, type, sK146: (real > nat) > real).
% 2.70/1.59 thf(func_def_259, type, sK147: product_prod_a_real > a).
% 2.70/1.59 thf(func_def_260, type, sK148: product_prod_a_real > real).
% 2.70/1.59 thf(func_def_261, type, sK149: (real > real) > real).
% 2.70/1.59 thf(func_def_262, type, sK150: (real > real) > real).
% 2.70/1.59 thf(func_def_264, type, ph152: !>[X0: $tType]:(X0)).
% 2.70/1.59 thf(f2113,plain,(
% 2.70/1.59 $false),
% 2.70/1.59 inference(avatar_sat_refutation,[status(thm)],[f2014,f2015,f2112])).
% 2.70/1.59 thf(f2112,plain,(
% 2.70/1.59 ~spl151_1),
% 2.70/1.59 inference(avatar_contradiction_clause,[status(thm)],[f2111])).
% 2.70/1.59 thf(f2111,plain,(
% 2.70/1.59 $false | ~spl151_1),
% 2.70/1.59 inference(subsumption_resolution,[status(thm)],[f2097,f1598])).
% 2.70/1.59 thf(f1598,plain,(
% 2.70/1.59 ((member1103263856a_real @ sK16 @ (lower_930854854raph_a @ top_top_set_a @ f)) = $true)),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f1367])).
% 2.70/1.59 thf(f1367,plain,(
% 2.70/1.59 ((ord_less_real @ (real_V404783528a_real @ sK16 @ (product_Pair_a_real @ x @ y)) @ e) = $true) & ((member1103263856a_real @ sK16 @ (lower_930854854raph_a @ top_top_set_a @ f)) = $true)),
% 2.70/1.59 inference(skolemisation,[status(esa),new_symbols(skolem,[sK16])],[f534,f1366])).
% 2.70/1.59 thf(f1366,plain,(
% 2.70/1.59 ? [X0 : product_prod_a_real] : (((ord_less_real @ (real_V404783528a_real @ X0 @ (product_Pair_a_real @ x @ y)) @ e) = $true) & ((member1103263856a_real @ X0 @ (lower_930854854raph_a @ top_top_set_a @ f)) = $true)) => (((ord_less_real @ (real_V404783528a_real @ sK16 @ (product_Pair_a_real @ x @ y)) @ e) = $true) & ((member1103263856a_real @ sK16 @ (lower_930854854raph_a @ top_top_set_a @ f)) = $true))),
% 2.70/1.59 introduced(definition,[],[choice_axiom])).
% 2.70/1.59 thf(f534,plain,(
% 2.70/1.59 ? [X0 : product_prod_a_real] : (((ord_less_real @ (real_V404783528a_real @ X0 @ (product_Pair_a_real @ x @ y)) @ e) = $true) & ((member1103263856a_real @ X0 @ (lower_930854854raph_a @ top_top_set_a @ f)) = $true))),
% 2.70/1.59 inference(fool_elimination,[status(thm)],[f533])).
% 2.70/1.59 thf(f533,plain,(
% 2.70/1.59 ? [X0 : product_prod_a_real] : ((member1103263856a_real @ X0 @ (lower_930854854raph_a @ top_top_set_a @ f)) & (ord_less_real @ (real_V404783528a_real @ X0 @ (product_Pair_a_real @ x @ y)) @ e))),
% 2.70/1.59 inference(rectify,[status(thm)],[f1])).
% 2.70/1.59 thf(f1,axiom,(
% 2.70/1.59 ? [X0 : product_prod_a_real] : ((member1103263856a_real @ X0 @ (lower_930854854raph_a @ top_top_set_a @ f)) & (ord_less_real @ (real_V404783528a_real @ X0 @ (product_Pair_a_real @ x @ y)) @ e))),
% 2.70/1.59 file('/export/starexec/sandbox2/tmp/tmp.5gsfs41n1a/DTF2THF_2567.p',fact_0__092_060open_062_092_060exists_062ya_092_060in_062Epigraph_AUNIV_Af_O_Adist_Aya_A_Ix_M_Ay_J_A_060_Ae_092_060close_062)).
% 2.70/1.59 thf(f2097,plain,(
% 2.70/1.59 ((member1103263856a_real @ sK16 @ (lower_930854854raph_a @ top_top_set_a @ f)) != $true) | ~spl151_1),
% 2.70/1.59 inference(trivial_inequality_removal,[status(thm)],[f2067])).
% 2.70/1.59 thf(f2067,plain,(
% 2.70/1.59 ((member1103263856a_real @ sK16 @ (lower_930854854raph_a @ top_top_set_a @ f)) != $true) | ($true != $true) | ~spl151_1),
% 2.70/1.59 inference(superposition,[status(thm)],[f2017,f1599])).
% 2.70/1.59 thf(f1599,plain,(
% 2.70/1.59 ((ord_less_real @ (real_V404783528a_real @ sK16 @ (product_Pair_a_real @ x @ y)) @ e) = $true)),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f1367])).
% 2.70/1.59 thf(f2017,plain,(
% 2.70/1.59 ( ! [X0 : product_prod_a_real] : (((ord_less_real @ (real_V404783528a_real @ X0 @ (product_Pair_a_real @ x @ y)) @ e) != $true) | ((member1103263856a_real @ X0 @ (lower_930854854raph_a @ top_top_set_a @ f)) != $true)) ) | ~spl151_1),
% 2.70/1.59 inference(superposition,[status(thm)],[f2009,f1588])).
% 2.70/1.59 thf(f1588,plain,(
% 2.70/1.59 ( ! [X0 : product_prod_a_real] : (((product_Pair_a_real @ (sK12 @ X0) @ (sK13 @ X0)) = X0)) )),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f1362])).
% 2.70/1.59 thf(f1362,plain,(
% 2.70/1.59 ! [X0 : product_prod_a_real] : ((product_Pair_a_real @ (sK12 @ X0) @ (sK13 @ X0)) = X0)),
% 2.70/1.59 inference(skolemisation,[status(esa),new_symbols(skolem,[sK12,sK13])],[f1283,f1361])).
% 2.70/1.59 thf(f1361,plain,(
% 2.70/1.59 ! [X0 : product_prod_a_real] : (? [X1 : a,X2 : real] : ((product_Pair_a_real @ X1 @ X2) = X0) => ((product_Pair_a_real @ (sK12 @ X0) @ (sK13 @ X0)) = X0))),
% 2.70/1.59 introduced(definition,[],[choice_axiom])).
% 2.70/1.59 thf(f1283,plain,(
% 2.70/1.59 ! [X0 : product_prod_a_real] : ? [X1 : a,X2 : real] : ((product_Pair_a_real @ X1 @ X2) = X0)),
% 2.70/1.59 inference(ennf_transformation,[status(thm)],[f1004])).
% 2.70/1.59 thf(f1004,plain,(
% 2.70/1.59 ! [X0 : product_prod_a_real] : ~! [X1 : a,X2 : real] : ((product_Pair_a_real @ X1 @ X2) != X0)),
% 2.70/1.59 inference(rectify,[status(thm)],[f281])).
% 2.70/1.59 thf(f281,axiom,(
% 2.70/1.59 ! [X10 : product_prod_a_real] : ~! [X27 : a,X28 : real] : ((product_Pair_a_real @ X27 @ X28) != X10)),
% 2.70/1.59 file('/export/starexec/sandbox2/tmp/tmp.5gsfs41n1a/DTF2THF_2567.p',fact_280_old_Oprod_Oexhaust)).
% 2.70/1.59 thf(f2009,plain,(
% 2.70/1.59 ( ! [X0 : real,X1 : a] : (($true != (ord_less_real @ (real_V404783528a_real @ (product_Pair_a_real @ X1 @ X0) @ (product_Pair_a_real @ x @ y)) @ e)) | ($true != (member1103263856a_real @ (product_Pair_a_real @ X1 @ X0) @ (lower_930854854raph_a @ top_top_set_a @ f)))) ) | ~spl151_1),
% 2.70/1.59 inference(avatar_component_clause,[status(thm)],[f2008])).
% 2.70/1.59 thf(f2008,definition,(
% 2.70/1.59 spl151_1 <=> ! [X0 : real,X1 : a] : (($true != (member1103263856a_real @ (product_Pair_a_real @ X1 @ X0) @ (lower_930854854raph_a @ top_top_set_a @ f))) | ($true != (ord_less_real @ (real_V404783528a_real @ (product_Pair_a_real @ X1 @ X0) @ (product_Pair_a_real @ x @ y)) @ e)))),
% 2.70/1.59 introduced(definition,[new_symbols(naming,[spl151_1])],[avatar_definition])).
% 2.70/1.59 thf(f2015,plain,(
% 2.70/1.59 ~spl151_2),
% 2.70/1.59 inference(avatar_split_clause,[status(thm)],[f1640,f2011])).
% 2.70/1.59 thf(f2011,definition,(
% 2.70/1.59 spl151_2 <=> (thesis = $true)),
% 2.70/1.59 introduced(definition,[new_symbols(naming,[spl151_2])],[avatar_definition])).
% 2.70/1.59 thf(f1640,plain,(
% 2.70/1.59 (thesis != $true)),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f989])).
% 2.70/1.59 thf(f989,plain,(
% 2.70/1.59 (thesis != $true)),
% 2.70/1.59 inference(flattening,[status(thm)],[f910])).
% 2.70/1.59 thf(f910,plain,(
% 2.70/1.59 ~(thesis = $true)),
% 2.70/1.59 inference(fool_elimination,[status(thm)],[f909])).
% 2.70/1.59 thf(f909,plain,(
% 2.70/1.59 ~thesis),
% 2.70/1.59 inference(rectify,[status(thm)],[f347])).
% 2.70/1.59 thf(f347,negated_conjecture,(
% 2.70/1.59 ~thesis),
% 2.70/1.59 inference(negated_conjecture,[status(cth)],[f346])).
% 2.70/1.59 thf(f346,conjecture,(
% 2.70/1.59 thesis),
% 2.70/1.59 file('/export/starexec/sandbox2/tmp/tmp.5gsfs41n1a/DTF2THF_2567.p',conj_1)).
% 2.70/1.59 thf(f2014,plain,(
% 2.70/1.59 spl151_1 | spl151_2),
% 2.70/1.59 inference(avatar_split_clause,[status(thm)],[f1694,f2011,f2008])).
% 2.70/1.59 thf(f1694,plain,(
% 2.70/1.59 ( ! [X0 : real,X1 : a] : (($true != (member1103263856a_real @ (product_Pair_a_real @ X1 @ X0) @ (lower_930854854raph_a @ top_top_set_a @ f))) | ($true != (ord_less_real @ (real_V404783528a_real @ (product_Pair_a_real @ X1 @ X0) @ (product_Pair_a_real @ x @ y)) @ e)) | (thesis = $true)) )),
% 2.70/1.59 inference(cnf_transformation,[status(thm)],[f1221])).
% 2.70/1.59 thf(f1221,plain,(
% 2.70/1.59 ! [X0 : real,X1 : a] : (($true != (member1103263856a_real @ (product_Pair_a_real @ X1 @ X0) @ (lower_930854854raph_a @ top_top_set_a @ f))) | ($true != (ord_less_real @ (real_V404783528a_real @ (product_Pair_a_real @ X1 @ X0) @ (product_Pair_a_real @ x @ y)) @ e)) | (thesis = $true))),
% 2.70/1.59 inference(flattening,[status(thm)],[f1220])).
% 2.70/1.59 thf(f1220,plain,(
% 2.70/1.59 ! [X0 : real,X1 : a] : ((thesis = $true) | (($true != (member1103263856a_real @ (product_Pair_a_real @ X1 @ X0) @ (lower_930854854raph_a @ top_top_set_a @ f))) | ($true != (ord_less_real @ (real_V404783528a_real @ (product_Pair_a_real @ X1 @ X0) @ (product_Pair_a_real @ x @ y)) @ e))))),
% 2.70/1.59 inference(ennf_transformation,[status(thm)],[f700])).
% 2.70/1.59 thf(f700,plain,(
% 2.70/1.59 ! [X0 : real,X1 : a] : ((($true = (member1103263856a_real @ (product_Pair_a_real @ X1 @ X0) @ (lower_930854854raph_a @ top_top_set_a @ f))) & ($true = (ord_less_real @ (real_V404783528a_real @ (product_Pair_a_real @ X1 @ X0) @ (product_Pair_a_real @ x @ y)) @ e))) => (thesis = $true))),
% 2.70/1.59 inference(fool_elimination,[status(thm)],[f699])).
% 2.70/1.59 thf(f699,plain,(
% 2.70/1.59 ! [X0 : real,X1 : a] : (((member1103263856a_real @ (product_Pair_a_real @ X1 @ X0) @ (lower_930854854raph_a @ top_top_set_a @ f)) & (ord_less_real @ (real_V404783528a_real @ (product_Pair_a_real @ X1 @ X0) @ (product_Pair_a_real @ x @ y)) @ e)) => thesis)),
% 2.70/1.59 inference(rectify,[status(thm)],[f345])).
% 2.70/1.59 thf(f345,axiom,(
% 2.70/1.59 ! [X59 : real,X58 : a] : (((member1103263856a_real @ (product_Pair_a_real @ X58 @ X59) @ (lower_930854854raph_a @ top_top_set_a @ f)) & (ord_less_real @ (real_V404783528a_real @ (product_Pair_a_real @ X58 @ X59) @ (product_Pair_a_real @ x @ y)) @ e)) => thesis)),
% 2.70/1.59 file('/export/starexec/sandbox2/tmp/tmp.5gsfs41n1a/DTF2THF_2567.p',conj_0)).
% 2.70/1.59 % SZS output end Proof for DTF2THF_2567
% 2.70/1.59 % (2672)------------------------------
% 2.70/1.59 % (2672)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.59 % (2672)Termination reason: Refutation
% 2.70/1.59
% 2.70/1.59 % (2672)Memory used [KB]: 7164
% 2.70/1.59 % (2672)Time elapsed: 0.064 s
% 2.70/1.59 % (2672)Instructions burned: 141 (million)
% 2.70/1.59 % (2672)------------------------------
% 2.70/1.59 % (2672)------------------------------
% 2.70/1.59 % (2666)Success in time 0.082 s
%------------------------------------------------------------------------------