%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP110^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n029.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 08:55:38 AM UTC 2026
% Result : Theorem 0.30s 1.68s
% Output : Refutation 0.30s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 5
% Syntax : Number of formulae : 32 ( 9 unt; 0 typ; 2 def)
% Number of atoms : 120 ( 32 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 125 ( 19 ~; 16 |; 0 &; 82 @)
% ( 2 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 3 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 5 ( 5 >; 0 *; 0 +; 0 <<)
% Number of symbols : 176 ( 173 usr; 20 con; 0-3 aty)
% Number of variables : 5 ( 0 ^; 5 !; 0 ?; 5 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_Extended_ereal_o: $tType ).
thf(type_def_6,type,
extended_ereal: $tType ).
thf(type_def_8,type,
set_a_o: $tType ).
thf(type_def_9,type,
a: $tType ).
thf(type_def_10,type,
set_Extended_ereal: $tType ).
thf(type_def_11,type,
set_real: $tType ).
thf(type_def_12,type,
real: $tType ).
thf(type_def_13,type,
set_se767749006_ereal: $tType ).
thf(type_def_14,type,
set_se1849684334_ereal: $tType ).
thf(type_def_15,type,
set_set_set_a: $tType ).
thf(type_def_16,type,
set_set_a: $tType ).
thf(type_def_17,type,
set_a: $tType ).
thf(type_def_18,type,
set_Pr1158285382l_real: $tType ).
thf(type_def_19,type,
set_Pr1928503567a_real: $tType ).
thf(type_def_20,type,
set_Pr597221087l_real: $tType ).
thf(type_def_21,type,
product_prod_a_real: $tType ).
thf(func_def_0,type,
set_Pr1158285382l_real: $tType ).
thf(func_def_1,type,
set_Pr597221087l_real: $tType ).
thf(func_def_2,type,
set_se1849684334_ereal: $tType ).
thf(func_def_3,type,
set_Pr1928503567a_real: $tType ).
thf(func_def_4,type,
set_se767749006_ereal: $tType ).
thf(func_def_5,type,
set_Extended_ereal_o: $tType ).
thf(func_def_6,type,
set_set_set_a: $tType ).
thf(func_def_7,type,
product_prod_a_real: $tType ).
thf(func_def_8,type,
set_Extended_ereal: $tType ).
thf(func_def_9,type,
set_set_a: $tType ).
thf(func_def_10,type,
set_a_o: $tType ).
thf(func_def_11,type,
set_real: $tType ).
thf(func_def_12,type,
extended_ereal: $tType ).
thf(func_def_13,type,
set_a: $tType ).
thf(func_def_14,type,
real: $tType ).
thf(func_def_15,type,
a: $tType ).
thf(func_def_16,type,
comple179807490real_o: set_Extended_ereal_o > extended_ereal > $o ).
thf(func_def_17,type,
complete_Sup_Sup_a_o: set_a_o > a > $o ).
thf(func_def_18,type,
comple1161760187_ereal: set_Extended_ereal > extended_ereal ).
thf(func_def_19,type,
comple2129349247p_real: set_real > real ).
thf(func_def_20,type,
comple767846299_ereal: set_se767749006_ereal > set_Extended_ereal ).
thf(func_def_21,type,
comple1517911419_ereal: set_se1849684334_ereal > set_se767749006_ereal ).
thf(func_def_22,type,
comple968993579_set_a: set_set_set_a > set_set_a ).
thf(func_def_23,type,
comple1766734283_set_a: set_set_a > set_a ).
thf(func_def_24,type,
convex291263275l_real: set_Pr1158285382l_real > $o ).
thf(func_def_25,type,
convex78185044a_real: set_Pr1928503567a_real > $o ).
thf(func_def_26,type,
convex_a: set_a > $o ).
thf(func_def_27,type,
extend1289208545_ereal: extended_ereal ).
thf(func_def_28,type,
extended_MInfty: extended_ereal ).
thf(func_def_29,type,
extended_PInfty: extended_ereal ).
thf(func_def_30,type,
extended_ereal2: real > extended_ereal ).
thf(func_def_31,type,
extend1716541707_ereal: extended_ereal > real ).
thf(func_def_32,type,
times_1966848393_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(func_def_33,type,
times_times_real: real > real > real ).
thf(func_def_34,type,
uminus1208298309_ereal: extended_ereal > extended_ereal ).
thf(func_def_35,type,
uminus_uminus_real: real > real ).
thf(func_def_36,type,
uminus360668453_ereal: set_Extended_ereal > set_Extended_ereal ).
thf(func_def_37,type,
uminus_uminus_set_a: set_a > set_a ).
thf(func_def_38,type,
zero_z163181189_ereal: extended_ereal ).
thf(func_def_39,type,
zero_zero_real: real ).
thf(func_def_40,type,
lower_331963542_ereal: set_Extended_ereal > ( extended_ereal > extended_ereal ) > set_Pr597221087l_real ).
thf(func_def_41,type,
lower_870348689a_real: set_Pr1928503567a_real > ( product_prod_a_real > extended_ereal ) > set_Pr1158285382l_real ).
thf(func_def_42,type,
lower_930854854raph_a: set_a > ( a > extended_ereal ) > set_Pr1928503567a_real ).
thf(func_def_43,type,
lower_128423320e_on_a: set_a > ( a > extended_ereal ) > $o ).
thf(func_def_44,type,
lower_1023700310e_on_a: set_a > ( a > extended_ereal ) > $o ).
thf(func_def_45,type,
lower_591730471a_real: set_Pr1928503567a_real > ( product_prod_a_real > extended_ereal ) > $o ).
thf(func_def_46,type,
lower_311861424x_on_a: set_a > ( a > extended_ereal ) > $o ).
thf(func_def_47,type,
lower_1359521925a_real: ( product_prod_a_real > extended_ereal ) > set_Pr1928503567a_real ).
thf(func_def_48,type,
lower_1391529426main_a: ( a > extended_ereal ) > set_a ).
thf(func_def_49,type,
lower_1486931688e_on_a: set_a > ( a > extended_ereal ) > $o ).
thf(func_def_50,type,
ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
thf(func_def_51,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_52,type,
ord_le88246606_ereal: set_Extended_ereal > set_Extended_ereal > $o ).
thf(func_def_53,type,
ord_le1586073967a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > $o ).
thf(func_def_54,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_55,type,
ord_le1153389358_ereal: set_se767749006_ereal > set_se767749006_ereal > $o ).
thf(func_def_56,type,
ord_le318720350_set_a: set_set_a > set_set_a > $o ).
thf(func_def_57,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_58,type,
order_1158471719_ereal: ( extended_ereal > $o ) > extended_ereal ).
thf(func_def_59,type,
order_Greatest_real: ( real > $o ) > real ).
thf(func_def_60,type,
top_to398855007real_o: extended_ereal > $o ).
thf(func_def_61,type,
top_top_a_o: a > $o ).
thf(func_def_63,type,
top_to802031902_ereal: extended_ereal ).
thf(func_def_64,type,
top_to772344469real_o: set_Extended_ereal_o ).
thf(func_def_65,type,
top_top_set_a_o: set_a_o ).
thf(func_def_66,type,
top_to1767659262_ereal: set_Extended_ereal ).
thf(func_def_67,type,
top_to2138011583a_real: set_Pr1928503567a_real ).
thf(func_def_68,type,
top_top_set_real: set_real ).
thf(func_def_69,type,
top_to692740318_ereal: set_se767749006_ereal ).
thf(func_def_70,type,
top_to1791633086_ereal: set_se1849684334_ereal ).
thf(func_def_71,type,
top_to1486684270_set_a: set_set_set_a ).
thf(func_def_72,type,
top_top_set_set_a: set_set_a ).
thf(func_def_73,type,
top_top_set_a: set_a ).
thf(func_def_74,type,
divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(func_def_75,type,
divide_divide_real: real > real > real ).
thf(func_def_76,type,
collec247695033_ereal: ( extended_ereal > $o ) > set_Extended_ereal ).
thf(func_def_77,type,
collec1714955950a_real: ( product_prod_a_real > $o ) > set_Pr1928503567a_real ).
thf(func_def_78,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_79,type,
member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(func_def_80,type,
member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).
thf(func_def_81,type,
member_real: real > set_real > $o ).
thf(func_def_82,type,
member450560855_ereal: set_Extended_ereal > set_se767749006_ereal > $o ).
thf(func_def_83,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_84,type,
member_a: a > set_a > $o ).
thf(func_def_85,type,
f: a > extended_ereal ).
thf(func_def_99,type,
sP0: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_100,type,
sP1: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_101,type,
sP2: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_102,type,
sP3: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_103,type,
sP4: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_104,type,
sP5: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_105,type,
sP6: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_106,type,
sP7: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_107,type,
sP8: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_108,type,
sP9: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_109,type,
sP10: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_110,type,
sP11: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_111,type,
sP12: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_112,type,
sP13: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_113,type,
sP14: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_114,type,
sP15: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_115,type,
sP16: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_116,type,
sP17: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_117,type,
sP18: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_118,type,
sP19: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_119,type,
sP20: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_120,type,
sP21: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_121,type,
sP22: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_122,type,
sP23: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_123,type,
sP24: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_124,type,
sP25: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_125,type,
sP26: extended_ereal > extended_ereal > $o ).
thf(func_def_126,type,
sP27: extended_ereal > extended_ereal > $o ).
thf(func_def_127,type,
sP28: extended_ereal > extended_ereal > $o ).
thf(func_def_128,type,
sP29: extended_ereal > extended_ereal > $o ).
thf(func_def_129,type,
sP30: extended_ereal > extended_ereal > $o ).
thf(func_def_130,type,
sP31: extended_ereal > extended_ereal > $o ).
thf(func_def_131,type,
sK32: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_132,type,
sK33: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_133,type,
sK34: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_134,type,
sK35: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_135,type,
sK36: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_136,type,
sK37: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_137,type,
sK38: ( set_a > real ) > set_a ).
thf(func_def_138,type,
sK39: ( set_a > real ) > set_a ).
thf(func_def_139,type,
sK40: ( extended_ereal > $o ) > real ).
thf(func_def_140,type,
sK41: extended_ereal > real ).
thf(func_def_141,type,
sK42: extended_ereal > real ).
thf(func_def_142,type,
sK43: extended_ereal > real ).
thf(func_def_143,type,
sK44: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_144,type,
sK45: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_145,type,
sK46: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_146,type,
sK47: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_147,type,
sK48: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_148,type,
sK49: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_149,type,
sK50: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_150,type,
sK51: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_151,type,
sK52: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_152,type,
sK53: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_153,type,
sK54: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_154,type,
sK55: ( extended_ereal > extended_ereal > $o ) > real ).
thf(func_def_155,type,
sK56: extended_ereal > real ).
thf(func_def_156,type,
sK57: extended_ereal > real ).
thf(func_def_157,type,
sK58: extended_ereal > real ).
thf(func_def_158,type,
sK59: extended_ereal > real ).
thf(func_def_159,type,
sK60: extended_ereal > real ).
thf(func_def_160,type,
sK61: extended_ereal > real ).
thf(func_def_161,type,
sK62: extended_ereal > real ).
thf(func_def_162,type,
sK63: extended_ereal > real ).
thf(func_def_163,type,
sK64: extended_ereal > real ).
thf(func_def_164,type,
sK65: extended_ereal > real ).
thf(func_def_165,type,
sK66: extended_ereal > real ).
thf(func_def_166,type,
sK67: extended_ereal > real ).
thf(func_def_167,type,
sK68: extended_ereal > real ).
thf(func_def_168,type,
sK69: extended_ereal > real ).
thf(func_def_169,type,
sK70: extended_ereal > real ).
thf(func_def_170,type,
sK71: extended_ereal > real ).
thf(func_def_171,type,
sK72: extended_ereal > real ).
thf(func_def_172,type,
sK73: extended_ereal > real ).
thf(func_def_173,type,
sK74: extended_ereal > real ).
thf(func_def_174,type,
sK75: extended_ereal > real ).
thf(func_def_175,type,
sK76: extended_ereal > real ).
thf(func_def_176,type,
sK77: extended_ereal > real ).
thf(func_def_177,type,
sK78: extended_ereal > real ).
thf(func_def_178,type,
sK79: extended_ereal > real ).
thf(func_def_179,type,
sK80: extended_ereal > real ).
thf(func_def_180,type,
sK81: extended_ereal > real ).
thf(func_def_181,type,
sK82: extended_ereal > real ).
thf(func_def_182,type,
sK83: extended_ereal > real ).
thf(func_def_183,type,
sK84: extended_ereal > real ).
thf(func_def_184,type,
sK85: extended_ereal > real ).
thf(func_def_185,type,
sK86: extended_ereal > real ).
thf(func_def_186,type,
sK87: extended_ereal > real ).
thf(func_def_187,type,
sK88: extended_ereal > real ).
thf(func_def_188,type,
sK89: extended_ereal > real ).
thf(func_def_189,type,
sK90: ( extended_ereal > set_a ) > extended_ereal ).
thf(func_def_190,type,
sK91: ( extended_ereal > set_a ) > extended_ereal ).
thf(func_def_191,type,
sK92: extended_ereal > real ).
thf(func_def_192,type,
sK93: ( a > extended_ereal ) > set_a ).
thf(func_def_193,type,
sK94: extended_ereal > real ).
thf(func_def_194,type,
sK95: ( real > set_a ) > real ).
thf(func_def_195,type,
sK96: ( real > set_a ) > real ).
thf(func_def_196,type,
sK97: ( set_a > extended_ereal ) > set_a ).
thf(func_def_197,type,
sK98: ( set_a > extended_ereal ) > set_a ).
thf(func_def_198,type,
sK99: ( a > extended_ereal ) > set_a ).
thf(func_def_199,type,
sK100: ( extended_ereal > $o ) > real ).
thf(func_def_201,type,
ph102:
!>[X0: $tType] : X0 ).
thf(f1388,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1351,f1377,f1387]) ).
thf(f1387,plain,
~ spl101_1,
inference(avatar_contradiction_clause,[status(thm)],[f1386]) ).
thf(f1386,plain,
( $false
| ~ spl101_1 ),
inference(trivial_inequality_removal,[status(thm)],[f1383]) ).
thf(f1383,plain,
( ( $false = $true )
| ~ spl101_1 ),
inference(superposition,[status(thm)],[f1346,f1353]) ).
thf(f1353,plain,
( ( convex_a @ ( lower_1391529426main_a @ f ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1352]) ).
thf(f1352,plain,
( ( $true != $true )
| ( ( convex_a @ ( lower_1391529426main_a @ f ) )
= $true ) ),
inference(superposition,[status(thm)],[f1319,f1342]) ).
thf(f1342,plain,
( ( lower_311861424x_on_a @ top_top_set_a @ f )
= $true ),
inference(subsumption_resolution,[status(thm)],[f1340,f1312]) ).
thf(f1312,plain,
( ( ( lower_311861424x_on_a @ top_top_set_a @ f )
= $true )
| ( ( convex_a @ ( lower_1391529426main_a @ f ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f916]) ).
thf(f916,plain,
( ( ( convex_a @ ( lower_1391529426main_a @ f ) )
!= $true )
| ( ( lower_311861424x_on_a @ top_top_set_a @ f )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f841]) ).
thf(f841,plain,
( ( ( convex_a @ ( lower_1391529426main_a @ f ) )
= $true )
=> ( ( lower_311861424x_on_a @ top_top_set_a @ f )
= $true ) ),
inference(fool_elimination,[status(thm)],[f840]) ).
thf(f840,plain,
( ( convex_a @ ( lower_1391529426main_a @ f ) )
=> ( lower_311861424x_on_a @ top_top_set_a @ f ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( convex_a @ ( lower_1391529426main_a @ f ) )
=> ( lower_311861424x_on_a @ top_top_set_a @ f ) ),
file('/export/starexec/sandbox/tmp/tmp.Lwp0UiQMvY/DTF2THF_9742.p',fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_AUNIV_Af_092_060close_062) ).
thf(f1340,plain,
( ( ( convex_a @ ( lower_1391529426main_a @ f ) )
= $true )
| ( ( lower_311861424x_on_a @ top_top_set_a @ f )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f1308]) ).
thf(f1308,plain,
( ( convex_a @ ( lower_1391529426main_a @ f ) )
!= ( lower_311861424x_on_a @ top_top_set_a @ f ) ),
inference(cnf_transformation,[status(thm)],[f888]) ).
thf(f888,plain,
( ( convex_a @ ( lower_1391529426main_a @ f ) )
!= ( lower_311861424x_on_a @ top_top_set_a @ f ) ),
inference(flattening,[status(thm)],[f356]) ).
thf(f356,negated_conjecture,
( ( convex_a @ ( lower_1391529426main_a @ f ) )
!= ( lower_311861424x_on_a @ top_top_set_a @ f ) ),
inference(negated_conjecture,[status(cth)],[f355]) ).
thf(f355,conjecture,
( ( convex_a @ ( lower_1391529426main_a @ f ) )
= ( lower_311861424x_on_a @ top_top_set_a @ f ) ),
file('/export/starexec/sandbox/tmp/tmp.Lwp0UiQMvY/DTF2THF_9742.p',conj_0) ).
thf(f1319,plain,
! [X0: a > extended_ereal] :
( ( ( lower_311861424x_on_a @ top_top_set_a @ X0 )
!= $true )
| ( ( convex_a @ ( lower_1391529426main_a @ X0 ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f912]) ).
thf(f912,plain,
! [X0: a > extended_ereal] :
( ( ( lower_311861424x_on_a @ top_top_set_a @ X0 )
!= $true )
| ( ( convex_a @ ( lower_1391529426main_a @ X0 ) )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f689]) ).
thf(f689,plain,
! [X0: a > extended_ereal] :
( ( ( lower_311861424x_on_a @ top_top_set_a @ X0 )
= $true )
=> ( ( convex_a @ ( lower_1391529426main_a @ X0 ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f688]) ).
thf(f688,plain,
! [X0: a > extended_ereal] :
( ( lower_311861424x_on_a @ top_top_set_a @ X0 )
=> ( convex_a @ ( lower_1391529426main_a @ X0 ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X0: a > extended_ereal] :
( ( lower_311861424x_on_a @ top_top_set_a @ X0 )
=> ( convex_a @ ( lower_1391529426main_a @ X0 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Lwp0UiQMvY/DTF2THF_9742.p',fact_2_convex__domain) ).
thf(f1346,plain,
( ( ( convex_a @ ( lower_1391529426main_a @ f ) )
= $false )
| ~ spl101_1 ),
inference(avatar_component_clause,[status(thm)],[f1344]) ).
thf(f1344,definition,
( spl101_1
<=> ( ( convex_a @ ( lower_1391529426main_a @ f ) )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl101_1])],[avatar_definition]) ).
thf(f1377,plain,
~ spl101_2,
inference(avatar_contradiction_clause,[status(thm)],[f1376]) ).
thf(f1376,plain,
( $false
| ~ spl101_2 ),
inference(subsumption_resolution,[status(thm)],[f1363,f1353]) ).
thf(f1363,plain,
( ( ( convex_a @ ( lower_1391529426main_a @ f ) )
!= $true )
| ~ spl101_2 ),
inference(trivial_inequality_removal,[status(thm)],[f1355]) ).
thf(f1355,plain,
( ( ( convex_a @ ( lower_1391529426main_a @ f ) )
!= $true )
| ( $false = $true )
| ~ spl101_2 ),
inference(superposition,[status(thm)],[f1350,f1312]) ).
thf(f1350,plain,
( ( ( lower_311861424x_on_a @ top_top_set_a @ f )
= $false )
| ~ spl101_2 ),
inference(avatar_component_clause,[status(thm)],[f1348]) ).
thf(f1348,definition,
( spl101_2
<=> ( ( lower_311861424x_on_a @ top_top_set_a @ f )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl101_2])],[avatar_definition]) ).
thf(f1351,plain,
( spl101_1
| spl101_2 ),
inference(avatar_split_clause,[status(thm)],[f1341,f1348,f1344]) ).
thf(f1341,plain,
( ( ( lower_311861424x_on_a @ top_top_set_a @ f )
= $false )
| ( ( convex_a @ ( lower_1391529426main_a @ f ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1308]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP110^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n029.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 08:26:14 EST 2026
% 0.14/0.33 % CPUTime :
% 0.14/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.18/0.45 thf(spec,logic,$$dhol).
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % File : ITP110^1 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.45 % Domain : Interactive Theorem Proving
% 0.18/0.45 % Problem : Sledgehammer Lower_Semicontinuous problem prob_1321__6263368_1
% 0.18/0.45 % Version : Especial.
% 0.18/0.45 % English :
% 0.18/0.45
% 0.18/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.45 % Source : [Des21]
% 0.18/0.45 % Names : Lower_Semicontinuous/prob_1321__6263368_1 [Des21]
% 0.18/0.45
% 0.18/0.45 % Status : Theorem
% 0.18/0.45 % Rating : 0.44 v9.1.0, 0.38 v9.0.0, 0.40 v8.2.0, 0.46 v8.1.0, 0.55 v7.5.0
% 0.18/0.45 % Syntax : Number of formulae : 441 ( 181 unt; 86 typ; 0 def)
% 0.18/0.45 % Number of atoms : 1033 ( 426 equ; 0 cnn)
% 0.18/0.45 % Maximal formula atoms : 81 ( 2 avg)
% 0.18/0.45 % Number of connectives : 2543 ( 109 ~; 19 |; 26 &;1862 @)
% 0.18/0.45 % ( 0 <=>; 527 =>; 0 <=; 0 <~>)
% 0.18/0.45 % Maximal formula depth : 34 ( 6 avg)
% 0.18/0.45 % Number of types : 17 ( 16 usr)
% 0.18/0.45 % Number of type conns : 177 ( 177 >; 0 *; 0 +; 0 <<)
% 0.18/0.45 % Number of symbols : 71 ( 70 usr; 17 con; 0-2 aty)
% 0.18/0.45 % Number of variables : 862 ( 72 ^; 756 !; 34 ?; 862 :)
% 0.18/0.45 % SPC : TH0_THM_EQU_NAR
% 0.18/0.45
% 0.18/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:44:43.435
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % Could-be-implicit typings (16)
% 0.18/0.45 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,
% 0.18/0.45 set_Pr1158285382l_real: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Real__Oreal_J_J,type,
% 0.18/0.45 set_Pr597221087l_real: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J_J,type,
% 0.18/0.45 set_se1849684334_ereal: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
% 0.18/0.45 set_Pr1928503567a_real: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
% 0.18/0.45 set_se767749006_ereal: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_I_062_It__Extended____Real__Oereal_M_Eo_J_J,type,
% 0.18/0.45 set_Extended_ereal_o: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.18/0.45 set_set_set_a: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.18/0.45 product_prod_a_real: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.18/0.45 set_Extended_ereal: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.18/0.45 set_set_a: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
% 0.18/0.45 set_a_o: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.18/0.45 set_real: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Extended____Real__Oereal,type,
% 0.18/0.45 extended_ereal: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 set_a: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Real__Oreal,type,
% 0.18/0.45 real: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_tf__a,type,
% 0.18/0.45 a: $tType ).
% 0.18/0.45
% 0.18/0.45 % Explicit typings (70)
% 0.18/0.45 thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
% 0.18/0.45 comple179807490real_o: set_Extended_ereal_o > extended_ereal > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
% 0.18/0.45 complete_Sup_Sup_a_o: set_a_o > a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Real__Oereal,type,
% 0.18/0.45 comple1161760187_ereal: set_Extended_ereal > extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
% 0.18/0.45 comple2129349247p_real: set_real > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.18/0.45 comple767846299_ereal: set_se767749006_ereal > set_Extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
% 0.18/0.45 comple1517911419_ereal: set_se1849684334_ereal > set_se767749006_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.18/0.45 comple968993579_set_a: set_set_set_a > set_set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 comple1766734283_set_a: set_set_a > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Convex_Oconvex_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
% 0.18/0.45 convex291263275l_real: set_Pr1158285382l_real > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Convex_Oconvex_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.18/0.45 convex78185044a_real: set_Pr1928503567a_real > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Convex_Oconvex_001tf__a,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
% 0.18/0.45 extend1289208545_ereal: extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Extended__Real_Oereal_OMInfty,type,
% 0.18/0.45 extended_MInfty: extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Extended__Real_Oereal_OPInfty,type,
% 0.18/0.45 extended_PInfty: extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 0.18/0.45 extended_ereal2: real > extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Extended__Real_Oreal__of__ereal,type,
% 0.18/0.45 extend1716541707_ereal: extended_ereal > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Real__Oereal,type,
% 0.18/0.45 times_1966848393_ereal: extended_ereal > extended_ereal > extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 0.18/0.45 times_times_real: real > real > real ).
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% 0.18/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
% 0.18/0.45 uminus1208298309_ereal: extended_ereal > extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.18/0.45 uminus_uminus_real: real > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.18/0.45 uminus360668453_ereal: set_Extended_ereal > set_Extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 uminus_uminus_set_a: set_a > set_a ).
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% 0.18/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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% 0.18/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.18/0.45 lower_870348689a_real: set_Pr1928503567a_real > ( product_prod_a_real > extended_ereal ) > set_Pr1158285382l_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001tf__a,type,
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% 0.18/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oaffine__on_001tf__a,type,
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% 0.18/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconcave__on_001tf__a,type,
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% 0.18/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_001tf__a,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Odomain_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Odomain_001tf__a,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ofinite__on_001tf__a,type,
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% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
% 0.18/0.45 ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
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% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.18/0.45 ord_le88246606_ereal: set_Extended_ereal > set_Extended_ereal > $o ).
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% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.18/0.45 ord_less_eq_set_real: set_real > set_real > $o ).
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% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
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% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.18/0.45 ord_le318720350_set_a: set_set_a > set_set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 ord_less_eq_set_a: set_a > set_a > $o ).
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% 0.18/0.45 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Real__Oereal,type,
% 0.18/0.45 order_1158471719_ereal: ( extended_ereal > $o ) > extended_ereal ).
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% 0.18/0.45 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
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% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
% 0.18/0.45 top_to398855007real_o: extended_ereal > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
% 0.18/0.45 top_top_a_o: a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
% 0.18/0.45 top_top_o: $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Real__Oereal,type,
% 0.18/0.45 top_to802031902_ereal: extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Extended____Real__Oereal_M_Eo_J_J,type,
% 0.18/0.45 top_to772344469real_o: set_Extended_ereal_o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
% 0.18/0.45 top_top_set_a_o: set_a_o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.18/0.45 top_to1767659262_ereal: set_Extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
% 0.18/0.45 top_to2138011583a_real: set_Pr1928503567a_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.18/0.45 top_top_set_real: set_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
% 0.18/0.45 top_to692740318_ereal: set_se767749006_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J_J,type,
% 0.18/0.45 top_to1791633086_ereal: set_se1849684334_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.18/0.45 top_to1486684270_set_a: set_set_set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.18/0.45 top_top_set_set_a: set_set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 top_top_set_a: set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Real__Oereal,type,
% 0.18/0.45 divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 0.18/0.45 divide_divide_real: real > real > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Extended____Real__Oereal,type,
% 0.18/0.45 collec247695033_ereal: ( extended_ereal > $o ) > set_Extended_ereal ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.18/0.45 collec1714955950a_real: ( product_prod_a_real > $o ) > set_Pr1928503567a_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.18/0.45 collect_a: ( a > $o ) > set_a ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Extended____Real__Oereal,type,
% 0.18/0.45 member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.18/0.45 member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Real__Oreal,type,
% 0.18/0.45 member_real: real > set_real > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.18/0.45 member450560855_ereal: set_Extended_ereal > set_se767749006_ereal > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.18/0.45 member_set_a: set_a > set_set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001tf__a,type,
% 0.18/0.45 member_a: a > set_a > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_f,type,
% 0.18/0.45 f: a > extended_ereal ).
% 0.18/0.45
% 0.18/0.45 % Relevant facts (354)
% 0.18/0.45 thf(fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_AUNIV_Af_092_060close_062,axiom,
% 0.18/0.45 ( ( convex_a @ ( lower_1391529426main_a @ f ) )
% 0.18/0.45 => ( lower_311861424x_on_a @ top_top_set_a @ f ) ) ).
% 0.18/0.45
% 0.18/0.45 % \<open>convex (domain f) \<Longrightarrow> Lower_Semicontinuous_Mirabelle_mxyexokbxt.convex_on UNIV f\<close>
% 0.18/0.45 thf(fact_1_convex__domain,axiom,
% 0.18/0.45 ! [F: product_prod_a_real > extended_ereal] :
% 0.18/0.45 ( ( lower_591730471a_real @ top_to2138011583a_real @ F )
% 0.18/0.45 => ( convex78185044a_real @ ( lower_1359521925a_real @ F ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_domain
% 0.18/0.45 thf(fact_2_convex__domain,axiom,
% 0.18/0.45 ! [F: a > extended_ereal] :
% 0.18/0.45 ( ( lower_311861424x_on_a @ top_top_set_a @ F )
% 0.18/0.45 => ( convex_a @ ( lower_1391529426main_a @ F ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_domain
% 0.18/0.45 thf(fact_3_convex__on__domain,axiom,
% 0.18/0.45 ! [F: a > extended_ereal] :
% 0.18/0.45 ( ( lower_311861424x_on_a @ ( lower_1391529426main_a @ F ) @ F )
% 0.18/0.45 = ( lower_311861424x_on_a @ top_top_set_a @ F ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_on_domain
% 0.18/0.45 thf(fact_4_convex__on__domain2,axiom,
% 0.18/0.45 ! [F: a > extended_ereal] :
% 0.18/0.45 ( ( lower_311861424x_on_a @ ( lower_1391529426main_a @ F ) @ F )
% 0.18/0.45 = ( ! [S: set_a] : ( lower_311861424x_on_a @ S @ F ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_on_domain2
% 0.18/0.45 thf(fact_5_convex__on__ereal__univ,axiom,
% 0.18/0.45 ! [F: a > extended_ereal] :
% 0.18/0.45 ( ( lower_311861424x_on_a @ top_top_set_a @ F )
% 0.18/0.45 = ( ! [S: set_a] : ( lower_311861424x_on_a @ S @ F ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_on_ereal_univ
% 0.18/0.45 thf(fact_6_convex__UNIV,axiom,
% 0.18/0.45 convex78185044a_real @ top_to2138011583a_real ).
% 0.18/0.45
% 0.18/0.45 % convex_UNIV
% 0.18/0.45 thf(fact_7_convex__UNIV,axiom,
% 0.18/0.45 convex_a @ top_top_set_a ).
% 0.18/0.45
% 0.18/0.45 % convex_UNIV
% 0.18/0.45 thf(fact_8_UNIV__I,axiom,
% 0.18/0.45 ! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_I
% 0.18/0.45 thf(fact_9_UNIV__I,axiom,
% 0.18/0.45 ! [X: set_Extended_ereal] : ( member450560855_ereal @ X @ top_to692740318_ereal ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_I
% 0.18/0.45 thf(fact_10_UNIV__I,axiom,
% 0.18/0.45 ! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_I
% 0.18/0.45 thf(fact_11_UNIV__I,axiom,
% 0.18/0.45 ! [X: extended_ereal] : ( member1900190071_ereal @ X @ top_to1767659262_ereal ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_I
% 0.18/0.45 thf(fact_12_UNIV__I,axiom,
% 0.18/0.45 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_I
% 0.18/0.45 thf(fact_13_iso__tuple__UNIV__I,axiom,
% 0.18/0.45 ! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% 0.18/0.45
% 0.18/0.45 % iso_tuple_UNIV_I
% 0.18/0.45 thf(fact_14_iso__tuple__UNIV__I,axiom,
% 0.18/0.45 ! [X: set_Extended_ereal] : ( member450560855_ereal @ X @ top_to692740318_ereal ) ).
% 0.18/0.45
% 0.18/0.45 % iso_tuple_UNIV_I
% 0.18/0.45 thf(fact_15_iso__tuple__UNIV__I,axiom,
% 0.18/0.45 ! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % iso_tuple_UNIV_I
% 0.18/0.45 thf(fact_16_iso__tuple__UNIV__I,axiom,
% 0.18/0.45 ! [X: extended_ereal] : ( member1900190071_ereal @ X @ top_to1767659262_ereal ) ).
% 0.18/0.45
% 0.18/0.45 % iso_tuple_UNIV_I
% 0.18/0.45 thf(fact_17_iso__tuple__UNIV__I,axiom,
% 0.18/0.45 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % iso_tuple_UNIV_I
% 0.18/0.45 thf(fact_18_top__apply,axiom,
% 0.18/0.45 ( top_to398855007real_o
% 0.18/0.45 = ( ^ [X2: extended_ereal] : top_top_o ) ) ).
% 0.18/0.45
% 0.18/0.45 % top_apply
% 0.18/0.45 thf(fact_19_top__apply,axiom,
% 0.18/0.45 ( top_top_a_o
% 0.18/0.45 = ( ^ [X2: a] : top_top_o ) ) ).
% 0.18/0.45
% 0.18/0.45 % top_apply
% 0.18/0.45 thf(fact_20_assms,axiom,
% 0.18/0.45 ! [X3: a] :
% 0.18/0.45 ( ( ( f @ X3 )
% 0.18/0.45 = extend1289208545_ereal )
% 0.18/0.45 | ( ( f @ X3 )
% 0.18/0.45 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % assms
% 0.18/0.45 thf(fact_21_UNIV__eq__I,axiom,
% 0.18/0.45 ! [A: set_real] :
% 0.18/0.45 ( ! [X4: real] : ( member_real @ X4 @ A )
% 0.18/0.45 => ( top_top_set_real = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_eq_I
% 0.18/0.45 thf(fact_22_UNIV__eq__I,axiom,
% 0.18/0.45 ! [A: set_se767749006_ereal] :
% 0.18/0.45 ( ! [X4: set_Extended_ereal] : ( member450560855_ereal @ X4 @ A )
% 0.18/0.45 => ( top_to692740318_ereal = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_eq_I
% 0.18/0.45 thf(fact_23_UNIV__eq__I,axiom,
% 0.18/0.45 ! [A: set_set_a] :
% 0.18/0.45 ( ! [X4: set_a] : ( member_set_a @ X4 @ A )
% 0.18/0.45 => ( top_top_set_set_a = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_eq_I
% 0.18/0.45 thf(fact_24_UNIV__eq__I,axiom,
% 0.18/0.45 ! [A: set_a] :
% 0.18/0.45 ( ! [X4: a] : ( member_a @ X4 @ A )
% 0.18/0.45 => ( top_top_set_a = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_eq_I
% 0.18/0.45 thf(fact_25_UNIV__eq__I,axiom,
% 0.18/0.45 ! [A: set_Extended_ereal] :
% 0.18/0.45 ( ! [X4: extended_ereal] : ( member1900190071_ereal @ X4 @ A )
% 0.18/0.45 => ( top_to1767659262_ereal = A ) ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_eq_I
% 0.18/0.45 thf(fact_26_UNIV__witness,axiom,
% 0.18/0.45 ? [X4: real] : ( member_real @ X4 @ top_top_set_real ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_witness
% 0.18/0.45 thf(fact_27_UNIV__witness,axiom,
% 0.18/0.45 ? [X4: set_Extended_ereal] : ( member450560855_ereal @ X4 @ top_to692740318_ereal ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_witness
% 0.18/0.45 thf(fact_28_UNIV__witness,axiom,
% 0.18/0.45 ? [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_witness
% 0.18/0.45 thf(fact_29_UNIV__witness,axiom,
% 0.18/0.45 ? [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_witness
% 0.18/0.45 thf(fact_30_UNIV__witness,axiom,
% 0.18/0.45 ? [X4: extended_ereal] : ( member1900190071_ereal @ X4 @ top_to1767659262_ereal ) ).
% 0.18/0.45
% 0.18/0.45 % UNIV_witness
% 0.18/0.45 thf(fact_31_convex__Epigraph,axiom,
% 0.18/0.45 ! [S2: set_Pr1928503567a_real,F: product_prod_a_real > extended_ereal] :
% 0.18/0.45 ( ( convex78185044a_real @ S2 )
% 0.18/0.45 => ( ( convex291263275l_real @ ( lower_870348689a_real @ S2 @ F ) )
% 0.18/0.45 = ( lower_591730471a_real @ S2 @ F ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_Epigraph
% 0.18/0.45 thf(fact_32_convex__Epigraph,axiom,
% 0.18/0.45 ! [S2: set_a,F: a > extended_ereal] :
% 0.18/0.45 ( ( convex_a @ S2 )
% 0.18/0.45 => ( ( convex78185044a_real @ ( lower_930854854raph_a @ S2 @ F ) )
% 0.18/0.45 = ( lower_311861424x_on_a @ S2 @ F ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_Epigraph
% 0.18/0.45 thf(fact_33_convex__EpigraphI,axiom,
% 0.18/0.45 ! [S3: set_Pr1928503567a_real,F: product_prod_a_real > extended_ereal] :
% 0.18/0.45 ( ( lower_591730471a_real @ S3 @ F )
% 0.18/0.45 => ( ( convex78185044a_real @ S3 )
% 0.18/0.45 => ( convex291263275l_real @ ( lower_870348689a_real @ S3 @ F ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_EpigraphI
% 0.18/0.45 thf(fact_34_convex__EpigraphI,axiom,
% 0.18/0.45 ! [S3: set_a,F: a > extended_ereal] :
% 0.18/0.45 ( ( lower_311861424x_on_a @ S3 @ F )
% 0.18/0.45 => ( ( convex_a @ S3 )
% 0.18/0.45 => ( convex78185044a_real @ ( lower_930854854raph_a @ S3 @ F ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_EpigraphI
% 0.18/0.45 thf(fact_35_affine__on__def,axiom,
% 0.18/0.45 ( lower_128423320e_on_a
% 0.18/0.45 = ( ^ [S: set_a,F2: a > extended_ereal] :
% 0.18/0.45 ( ( lower_311861424x_on_a @ S @ F2 )
% 0.18/0.45 & ( lower_1023700310e_on_a @ S @ F2 )
% 0.18/0.45 & ( lower_1486931688e_on_a @ S @ F2 ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % affine_on_def
% 0.18/0.45 thf(fact_36_Sup__UNIV,axiom,
% 0.18/0.45 ( ( comple1517911419_ereal @ top_to1791633086_ereal )
% 0.18/0.45 = top_to692740318_ereal ) ).
% 0.18/0.45
% 0.18/0.45 % Sup_UNIV
% 0.18/0.45 thf(fact_37_Sup__UNIV,axiom,
% 0.18/0.45 ( ( comple968993579_set_a @ top_to1486684270_set_a )
% 0.18/0.45 = top_top_set_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % Sup_UNIV
% 0.18/0.45 thf(fact_38_Sup__UNIV,axiom,
% 0.18/0.45 ( ( comple179807490real_o @ top_to772344469real_o )
% 0.18/0.45 = top_to398855007real_o ) ).
% 0.18/0.45
% 0.18/0.45 % Sup_UNIV
% 0.18/0.45 thf(fact_39_Sup__UNIV,axiom,
% 0.18/0.45 ( ( complete_Sup_Sup_a_o @ top_top_set_a_o )
% 0.18/0.45 = top_top_a_o ) ).
% 0.18/0.45
% 0.18/0.45 % Sup_UNIV
% 0.18/0.45 thf(fact_40_Sup__UNIV,axiom,
% 0.18/0.45 ( ( comple1766734283_set_a @ top_top_set_set_a )
% 0.18/0.45 = top_top_set_a ) ).
% 0.18/0.45
% 0.18/0.45 % Sup_UNIV
% 0.18/0.45 thf(fact_41_Sup__UNIV,axiom,
% 0.18/0.45 ( ( comple767846299_ereal @ top_to692740318_ereal )
% 0.18/0.45 = top_to1767659262_ereal ) ).
% 0.18/0.45
% 0.18/0.45 % Sup_UNIV
% 0.18/0.45 thf(fact_42_Sup__UNIV,axiom,
% 0.18/0.45 ( ( comple1161760187_ereal @ top_to1767659262_ereal )
% 0.18/0.45 = top_to802031902_ereal ) ).
% 0.18/0.45
% 0.18/0.45 % Sup_UNIV
% 0.18/0.45 thf(fact_43_convex__on__ereal__subset,axiom,
% 0.18/0.45 ! [T: set_Pr1928503567a_real,F: product_prod_a_real > extended_ereal,S3: set_Pr1928503567a_real] :
% 0.18/0.45 ( ( lower_591730471a_real @ T @ F )
% 0.18/0.45 => ( ( ord_le1586073967a_real @ S3 @ T )
% 0.18/0.45 => ( lower_591730471a_real @ S3 @ F ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_on_ereal_subset
% 0.18/0.45 thf(fact_44_convex__on__ereal__subset,axiom,
% 0.18/0.45 ! [T: set_a,F: a > extended_ereal,S3: set_a] :
% 0.18/0.45 ( ( lower_311861424x_on_a @ T @ F )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ S3 @ T )
% 0.18/0.45 => ( lower_311861424x_on_a @ S3 @ F ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % convex_on_ereal_subset
% 0.18/0.45 thf(fact_45_order__refl,axiom,
% 0.18/0.45 ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % order_refl
% 0.18/0.45 thf(fact_46_order__refl,axiom,
% 0.18/0.45 ! [X: set_Extended_ereal] : ( ord_le88246606_ereal @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % order_refl
% 0.18/0.45 thf(fact_47_order__refl,axiom,
% 0.18/0.45 ! [X: set_Pr1928503567a_real] : ( ord_le1586073967a_real @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % order_refl
% 0.18/0.45 thf(fact_48_order__refl,axiom,
% 0.18/0.45 ! [X: extended_ereal] : ( ord_le824540014_ereal @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % order_refl
% 0.18/0.45 thf(fact_49_order__refl,axiom,
% 0.18/0.45 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 0.18/0.45
% 0.18/0.45 % order_refl
% 0.18/0.45 thf(fact_50_subsetI,axiom,
% 0.18/0.45 ! [A: set_set_a,B: set_set_a] :
% 0.18/0.45 ( ! [X4: set_a] :
% 0.18/0.45 ( ( member_set_a @ X4 @ A )
% 0.18/0.45 => ( member_set_a @ X4 @ B ) )
% 0.18/0.45 => ( ord_le318720350_set_a @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetI
% 0.18/0.45 thf(fact_51_subsetI,axiom,
% 0.18/0.45 ! [A: set_se767749006_ereal,B: set_se767749006_ereal] :
% 0.18/0.45 ( ! [X4: set_Extended_ereal] :
% 0.18/0.45 ( ( member450560855_ereal @ X4 @ A )
% 0.18/0.45 => ( member450560855_ereal @ X4 @ B ) )
% 0.18/0.45 => ( ord_le1153389358_ereal @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetI
% 0.18/0.45 thf(fact_52_subsetI,axiom,
% 0.18/0.45 ! [A: set_real,B: set_real] :
% 0.18/0.45 ( ! [X4: real] :
% 0.18/0.45 ( ( member_real @ X4 @ A )
% 0.18/0.45 => ( member_real @ X4 @ B ) )
% 0.18/0.45 => ( ord_less_eq_set_real @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetI
% 0.18/0.45 thf(fact_53_subsetI,axiom,
% 0.18/0.45 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.18/0.45 ( ! [X4: product_prod_a_real] :
% 0.18/0.45 ( ( member1103263856a_real @ X4 @ A )
% 0.18/0.45 => ( member1103263856a_real @ X4 @ B ) )
% 0.18/0.45 => ( ord_le1586073967a_real @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetI
% 0.18/0.45 thf(fact_54_subsetI,axiom,
% 0.18/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.18/0.45 ( ! [X4: extended_ereal] :
% 0.18/0.45 ( ( member1900190071_ereal @ X4 @ A )
% 0.18/0.45 => ( member1900190071_ereal @ X4 @ B ) )
% 0.18/0.45 => ( ord_le88246606_ereal @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetI
% 0.18/0.45 thf(fact_55_subsetI,axiom,
% 0.18/0.45 ! [A: set_a,B: set_a] :
% 0.18/0.45 ( ! [X4: a] :
% 0.18/0.45 ( ( member_a @ X4 @ A )
% 0.18/0.45 => ( member_a @ X4 @ B ) )
% 0.18/0.45 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.18/0.45
% 0.18/0.45 % subsetI
% 0.18/0.45 thf(fact_56_subset__antisym,axiom,
% 0.18/0.45 ! [A: set_a,B: set_a] :
% 0.18/0.45 ( ( ord_less_eq_set_a @ A @ B )
% 0.18/0.45 => ( ( ord_less_eq_set_a @ B @ A )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_antisym
% 0.18/0.45 thf(fact_57_subset__antisym,axiom,
% 0.18/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.18/0.45 ( ( ord_le88246606_ereal @ A @ B )
% 0.18/0.45 => ( ( ord_le88246606_ereal @ B @ A )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_antisym
% 0.18/0.45 thf(fact_58_subset__antisym,axiom,
% 0.18/0.45 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.18/0.45 ( ( ord_le1586073967a_real @ A @ B )
% 0.18/0.45 => ( ( ord_le1586073967a_real @ B @ A )
% 0.18/0.45 => ( A = B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % subset_antisym
% 0.18/0.45 thf(fact_59_in__mono,axiom,
% 0.18/0.45 ! [A: set_set_a,B: set_set_a,X: set_a] :
% 0.18/0.45 ( ( ord_le318720350_set_a @ A @ B )
% 0.18/0.45 => ( ( member_set_a @ X @ A )
% 0.18/0.45 => ( member_set_a @ X @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_mono
% 0.18/0.45 thf(fact_60_in__mono,axiom,
% 0.18/0.45 ! [A: set_se767749006_ereal,B: set_se767749006_ereal,X: set_Extended_ereal] :
% 0.18/0.45 ( ( ord_le1153389358_ereal @ A @ B )
% 0.18/0.45 => ( ( member450560855_ereal @ X @ A )
% 0.18/0.45 => ( member450560855_ereal @ X @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_mono
% 0.18/0.45 thf(fact_61_in__mono,axiom,
% 0.18/0.45 ! [A: set_real,B: set_real,X: real] :
% 0.18/0.45 ( ( ord_less_eq_set_real @ A @ B )
% 0.18/0.45 => ( ( member_real @ X @ A )
% 0.18/0.45 => ( member_real @ X @ B ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % in_mono
% 0.18/0.45 thf(fact_62_in__mono,axiom,
% 0.18/0.45 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real,X: product_prod_a_real] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ A @ B )
% 0.29/0.45 => ( ( member1103263856a_real @ X @ A )
% 0.29/0.45 => ( member1103263856a_real @ X @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % in_mono
% 0.29/0.45 thf(fact_63_in__mono,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal,X: extended_ereal] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ A @ B )
% 0.29/0.45 => ( ( member1900190071_ereal @ X @ A )
% 0.29/0.45 => ( member1900190071_ereal @ X @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % in_mono
% 0.29/0.45 thf(fact_64_in__mono,axiom,
% 0.29/0.45 ! [A: set_a,B: set_a,X: a] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.45 => ( ( member_a @ X @ A )
% 0.29/0.45 => ( member_a @ X @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % in_mono
% 0.29/0.45 thf(fact_65_subsetD,axiom,
% 0.29/0.45 ! [A: set_set_a,B: set_set_a,C: set_a] :
% 0.29/0.45 ( ( ord_le318720350_set_a @ A @ B )
% 0.29/0.45 => ( ( member_set_a @ C @ A )
% 0.29/0.45 => ( member_set_a @ C @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetD
% 0.29/0.45 thf(fact_66_subsetD,axiom,
% 0.29/0.45 ! [A: set_se767749006_ereal,B: set_se767749006_ereal,C: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le1153389358_ereal @ A @ B )
% 0.29/0.45 => ( ( member450560855_ereal @ C @ A )
% 0.29/0.45 => ( member450560855_ereal @ C @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetD
% 0.29/0.45 thf(fact_67_subsetD,axiom,
% 0.29/0.45 ! [A: set_real,B: set_real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_set_real @ A @ B )
% 0.29/0.45 => ( ( member_real @ C @ A )
% 0.29/0.45 => ( member_real @ C @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetD
% 0.29/0.45 thf(fact_68_subsetD,axiom,
% 0.29/0.45 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real,C: product_prod_a_real] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ A @ B )
% 0.29/0.45 => ( ( member1103263856a_real @ C @ A )
% 0.29/0.45 => ( member1103263856a_real @ C @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetD
% 0.29/0.45 thf(fact_69_subsetD,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ A @ B )
% 0.29/0.45 => ( ( member1900190071_ereal @ C @ A )
% 0.29/0.45 => ( member1900190071_ereal @ C @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetD
% 0.29/0.45 thf(fact_70_subsetD,axiom,
% 0.29/0.45 ! [A: set_a,B: set_a,C: a] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.45 => ( ( member_a @ C @ A )
% 0.29/0.45 => ( member_a @ C @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subsetD
% 0.29/0.45 thf(fact_71_equalityE,axiom,
% 0.29/0.45 ! [A: set_a,B: set_a] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ~ ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.45 => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityE
% 0.29/0.45 thf(fact_72_equalityE,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ~ ( ( ord_le88246606_ereal @ A @ B )
% 0.29/0.45 => ~ ( ord_le88246606_ereal @ B @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityE
% 0.29/0.45 thf(fact_73_equalityE,axiom,
% 0.29/0.45 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ~ ( ( ord_le1586073967a_real @ A @ B )
% 0.29/0.45 => ~ ( ord_le1586073967a_real @ B @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityE
% 0.29/0.45 thf(fact_74_subset__eq,axiom,
% 0.29/0.45 ( ord_le318720350_set_a
% 0.29/0.45 = ( ^ [A2: set_set_a,B2: set_set_a] :
% 0.29/0.45 ! [X2: set_a] :
% 0.29/0.45 ( ( member_set_a @ X2 @ A2 )
% 0.29/0.45 => ( member_set_a @ X2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_eq
% 0.29/0.45 thf(fact_75_subset__eq,axiom,
% 0.29/0.45 ( ord_le1153389358_ereal
% 0.29/0.45 = ( ^ [A2: set_se767749006_ereal,B2: set_se767749006_ereal] :
% 0.29/0.45 ! [X2: set_Extended_ereal] :
% 0.29/0.45 ( ( member450560855_ereal @ X2 @ A2 )
% 0.29/0.45 => ( member450560855_ereal @ X2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_eq
% 0.29/0.45 thf(fact_76_subset__eq,axiom,
% 0.29/0.45 ( ord_less_eq_set_real
% 0.29/0.45 = ( ^ [A2: set_real,B2: set_real] :
% 0.29/0.45 ! [X2: real] :
% 0.29/0.45 ( ( member_real @ X2 @ A2 )
% 0.29/0.45 => ( member_real @ X2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_eq
% 0.29/0.45 thf(fact_77_subset__eq,axiom,
% 0.29/0.45 ( ord_le1586073967a_real
% 0.29/0.45 = ( ^ [A2: set_Pr1928503567a_real,B2: set_Pr1928503567a_real] :
% 0.29/0.45 ! [X2: product_prod_a_real] :
% 0.29/0.45 ( ( member1103263856a_real @ X2 @ A2 )
% 0.29/0.45 => ( member1103263856a_real @ X2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_eq
% 0.29/0.45 thf(fact_78_subset__eq,axiom,
% 0.29/0.45 ( ord_le88246606_ereal
% 0.29/0.45 = ( ^ [A2: set_Extended_ereal,B2: set_Extended_ereal] :
% 0.29/0.45 ! [X2: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ X2 @ A2 )
% 0.29/0.45 => ( member1900190071_ereal @ X2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_eq
% 0.29/0.45 thf(fact_79_subset__eq,axiom,
% 0.29/0.45 ( ord_less_eq_set_a
% 0.29/0.45 = ( ^ [A2: set_a,B2: set_a] :
% 0.29/0.45 ! [X2: a] :
% 0.29/0.45 ( ( member_a @ X2 @ A2 )
% 0.29/0.45 => ( member_a @ X2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_eq
% 0.29/0.45 thf(fact_80_equalityD1,axiom,
% 0.29/0.45 ! [A: set_a,B: set_a] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD1
% 0.29/0.45 thf(fact_81_equalityD1,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ord_le88246606_ereal @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD1
% 0.29/0.45 thf(fact_82_equalityD1,axiom,
% 0.29/0.45 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ord_le1586073967a_real @ A @ B ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD1
% 0.29/0.45 thf(fact_83_equalityD2,axiom,
% 0.29/0.45 ! [A: set_a,B: set_a] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ord_less_eq_set_a @ B @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD2
% 0.29/0.45 thf(fact_84_equalityD2,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ord_le88246606_ereal @ B @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD2
% 0.29/0.45 thf(fact_85_equalityD2,axiom,
% 0.29/0.45 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ord_le1586073967a_real @ B @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % equalityD2
% 0.29/0.45 thf(fact_86_subset__iff,axiom,
% 0.29/0.45 ( ord_le318720350_set_a
% 0.29/0.45 = ( ^ [A2: set_set_a,B2: set_set_a] :
% 0.29/0.45 ! [T2: set_a] :
% 0.29/0.45 ( ( member_set_a @ T2 @ A2 )
% 0.29/0.45 => ( member_set_a @ T2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff
% 0.29/0.45 thf(fact_87_subset__iff,axiom,
% 0.29/0.45 ( ord_le1153389358_ereal
% 0.29/0.45 = ( ^ [A2: set_se767749006_ereal,B2: set_se767749006_ereal] :
% 0.29/0.45 ! [T2: set_Extended_ereal] :
% 0.29/0.45 ( ( member450560855_ereal @ T2 @ A2 )
% 0.29/0.45 => ( member450560855_ereal @ T2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff
% 0.29/0.45 thf(fact_88_subset__iff,axiom,
% 0.29/0.45 ( ord_less_eq_set_real
% 0.29/0.45 = ( ^ [A2: set_real,B2: set_real] :
% 0.29/0.45 ! [T2: real] :
% 0.29/0.45 ( ( member_real @ T2 @ A2 )
% 0.29/0.45 => ( member_real @ T2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff
% 0.29/0.45 thf(fact_89_subset__iff,axiom,
% 0.29/0.45 ( ord_le1586073967a_real
% 0.29/0.45 = ( ^ [A2: set_Pr1928503567a_real,B2: set_Pr1928503567a_real] :
% 0.29/0.45 ! [T2: product_prod_a_real] :
% 0.29/0.45 ( ( member1103263856a_real @ T2 @ A2 )
% 0.29/0.45 => ( member1103263856a_real @ T2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff
% 0.29/0.45 thf(fact_90_subset__iff,axiom,
% 0.29/0.45 ( ord_le88246606_ereal
% 0.29/0.45 = ( ^ [A2: set_Extended_ereal,B2: set_Extended_ereal] :
% 0.29/0.45 ! [T2: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ T2 @ A2 )
% 0.29/0.45 => ( member1900190071_ereal @ T2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff
% 0.29/0.45 thf(fact_91_subset__iff,axiom,
% 0.29/0.45 ( ord_less_eq_set_a
% 0.29/0.45 = ( ^ [A2: set_a,B2: set_a] :
% 0.29/0.45 ! [T2: a] :
% 0.29/0.45 ( ( member_a @ T2 @ A2 )
% 0.29/0.45 => ( member_a @ T2 @ B2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_iff
% 0.29/0.45 thf(fact_92_subset__refl,axiom,
% 0.29/0.45 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 0.29/0.45
% 0.29/0.45 % subset_refl
% 0.29/0.45 thf(fact_93_subset__refl,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal] : ( ord_le88246606_ereal @ A @ A ) ).
% 0.29/0.45
% 0.29/0.45 % subset_refl
% 0.29/0.45 thf(fact_94_subset__refl,axiom,
% 0.29/0.45 ! [A: set_Pr1928503567a_real] : ( ord_le1586073967a_real @ A @ A ) ).
% 0.29/0.45
% 0.29/0.45 % subset_refl
% 0.29/0.45 thf(fact_95_Collect__mono,axiom,
% 0.29/0.45 ! [P: a > $o,Q: a > $o] :
% 0.29/0.45 ( ! [X4: a] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.45 => ( Q @ X4 ) )
% 0.29/0.45 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono
% 0.29/0.45 thf(fact_96_Collect__mono,axiom,
% 0.29/0.45 ! [P: extended_ereal > $o,Q: extended_ereal > $o] :
% 0.29/0.45 ( ! [X4: extended_ereal] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.45 => ( Q @ X4 ) )
% 0.29/0.45 => ( ord_le88246606_ereal @ ( collec247695033_ereal @ P ) @ ( collec247695033_ereal @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono
% 0.29/0.45 thf(fact_97_Collect__mono,axiom,
% 0.29/0.45 ! [P: product_prod_a_real > $o,Q: product_prod_a_real > $o] :
% 0.29/0.45 ( ! [X4: product_prod_a_real] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.45 => ( Q @ X4 ) )
% 0.29/0.45 => ( ord_le1586073967a_real @ ( collec1714955950a_real @ P ) @ ( collec1714955950a_real @ Q ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono
% 0.29/0.45 thf(fact_98_subset__trans,axiom,
% 0.29/0.45 ! [A: set_a,B: set_a,C2: set_a] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ A @ B )
% 0.29/0.45 => ( ( ord_less_eq_set_a @ B @ C2 )
% 0.29/0.45 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_trans
% 0.29/0.45 thf(fact_99_subset__trans,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal,C2: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ A @ B )
% 0.29/0.45 => ( ( ord_le88246606_ereal @ B @ C2 )
% 0.29/0.45 => ( ord_le88246606_ereal @ A @ C2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_trans
% 0.29/0.45 thf(fact_100_subset__trans,axiom,
% 0.29/0.45 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real,C2: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ A @ B )
% 0.29/0.45 => ( ( ord_le1586073967a_real @ B @ C2 )
% 0.29/0.45 => ( ord_le1586073967a_real @ A @ C2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % subset_trans
% 0.29/0.45 thf(fact_101_set__eq__subset,axiom,
% 0.29/0.45 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 0.29/0.45 = ( ^ [A2: set_a,B2: set_a] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.29/0.45 & ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_eq_subset
% 0.29/0.45 thf(fact_102_set__eq__subset,axiom,
% 0.29/0.45 ( ( ^ [Y: set_Extended_ereal,Z: set_Extended_ereal] : ( Y = Z ) )
% 0.29/0.45 = ( ^ [A2: set_Extended_ereal,B2: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ A2 @ B2 )
% 0.29/0.45 & ( ord_le88246606_ereal @ B2 @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_eq_subset
% 0.29/0.45 thf(fact_103_set__eq__subset,axiom,
% 0.29/0.45 ( ( ^ [Y: set_Pr1928503567a_real,Z: set_Pr1928503567a_real] : ( Y = Z ) )
% 0.29/0.45 = ( ^ [A2: set_Pr1928503567a_real,B2: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ A2 @ B2 )
% 0.29/0.45 & ( ord_le1586073967a_real @ B2 @ A2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % set_eq_subset
% 0.29/0.45 thf(fact_104_Collect__mono__iff,axiom,
% 0.29/0.45 ! [P: a > $o,Q: a > $o] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.29/0.45 = ( ! [X2: a] :
% 0.29/0.45 ( ( P @ X2 )
% 0.29/0.45 => ( Q @ X2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono_iff
% 0.29/0.45 thf(fact_105_Collect__mono__iff,axiom,
% 0.29/0.45 ! [P: extended_ereal > $o,Q: extended_ereal > $o] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ ( collec247695033_ereal @ P ) @ ( collec247695033_ereal @ Q ) )
% 0.29/0.45 = ( ! [X2: extended_ereal] :
% 0.29/0.45 ( ( P @ X2 )
% 0.29/0.45 => ( Q @ X2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono_iff
% 0.29/0.45 thf(fact_106_Collect__mono__iff,axiom,
% 0.29/0.45 ! [P: product_prod_a_real > $o,Q: product_prod_a_real > $o] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ ( collec1714955950a_real @ P ) @ ( collec1714955950a_real @ Q ) )
% 0.29/0.45 = ( ! [X2: product_prod_a_real] :
% 0.29/0.45 ( ( P @ X2 )
% 0.29/0.45 => ( Q @ X2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mono_iff
% 0.29/0.45 thf(fact_107_order__subst1,axiom,
% 0.29/0.45 ! [A3: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.29/0.45 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_108_order__subst1,axiom,
% 0.29/0.45 ! [A3: extended_ereal,F: real > extended_ereal,B3: real,C: real] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ B3 @ C )
% 0.29/0.45 => ( ! [X4: real,Y2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_109_order__subst1,axiom,
% 0.29/0.45 ! [A3: real,F: extended_ereal > real,B3: extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.29/0.45 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_110_order__subst1,axiom,
% 0.29/0.45 ! [A3: real,F: real > real,B3: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ B3 @ C )
% 0.29/0.45 => ( ! [X4: real,Y2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_111_order__subst1,axiom,
% 0.29/0.45 ! [A3: extended_ereal,F: set_a > extended_ereal,B3: set_a,C: set_a] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq_set_a @ B3 @ C )
% 0.29/0.45 => ( ! [X4: set_a,Y2: set_a] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_112_order__subst1,axiom,
% 0.29/0.45 ! [A3: extended_ereal,F: set_Extended_ereal > extended_ereal,B3: set_Extended_ereal,C: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_le88246606_ereal @ B3 @ C )
% 0.29/0.45 => ( ! [X4: set_Extended_ereal,Y2: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_113_order__subst1,axiom,
% 0.29/0.45 ! [A3: real,F: set_a > real,B3: set_a,C: set_a] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq_set_a @ B3 @ C )
% 0.29/0.45 => ( ! [X4: set_a,Y2: set_a] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_114_order__subst1,axiom,
% 0.29/0.45 ! [A3: real,F: set_Extended_ereal > real,B3: set_Extended_ereal,C: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_le88246606_ereal @ B3 @ C )
% 0.29/0.45 => ( ! [X4: set_Extended_ereal,Y2: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_115_order__subst1,axiom,
% 0.29/0.45 ! [A3: set_a,F: extended_ereal > set_a,B3: extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.29/0.45 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_116_order__subst1,axiom,
% 0.29/0.45 ! [A3: set_a,F: real > set_a,B3: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ B3 @ C )
% 0.29/0.45 => ( ! [X4: real,Y2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst1
% 0.29/0.45 thf(fact_117_order__subst2,axiom,
% 0.29/0.45 ! [A3: set_Pr1928503567a_real,B3: set_Pr1928503567a_real,F: set_Pr1928503567a_real > set_a,C: set_a] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
% 0.29/0.45 => ( ! [X4: set_Pr1928503567a_real,Y2: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_118_order__subst2,axiom,
% 0.29/0.45 ! [A3: set_Pr1928503567a_real,B3: set_Pr1928503567a_real,F: set_Pr1928503567a_real > set_Extended_ereal,C: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_le88246606_ereal @ ( F @ B3 ) @ C )
% 0.29/0.45 => ( ! [X4: set_Pr1928503567a_real,Y2: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le88246606_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le88246606_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_119_order__subst2,axiom,
% 0.29/0.45 ! [A3: set_Pr1928503567a_real,B3: set_Pr1928503567a_real,F: set_Pr1928503567a_real > set_Pr1928503567a_real,C: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_le1586073967a_real @ ( F @ B3 ) @ C )
% 0.29/0.45 => ( ! [X4: set_Pr1928503567a_real,Y2: set_Pr1928503567a_real] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le1586073967a_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le1586073967a_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_120_order__subst2,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ ( F @ B3 ) @ C )
% 0.29/0.45 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_121_order__subst2,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > real,C: real] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 0.29/0.45 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_122_order__subst2,axiom,
% 0.29/0.45 ! [A3: real,B3: real,F: real > extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ ( F @ B3 ) @ C )
% 0.29/0.45 => ( ! [X4: real,Y2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_123_order__subst2,axiom,
% 0.29/0.45 ! [A3: real,B3: real,F: real > real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 0.29/0.45 => ( ! [X4: real,Y2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_subst2
% 0.29/0.45 thf(fact_124_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A3: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( A3
% 0.29/0.45 = ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.29/0.45 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_125_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A3: real,F: extended_ereal > real,B3: extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( A3
% 0.29/0.45 = ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.29/0.45 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_126_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A3: extended_ereal,F: real > extended_ereal,B3: real,C: real] :
% 0.29/0.45 ( ( A3
% 0.29/0.45 = ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ B3 @ C )
% 0.29/0.45 => ( ! [X4: real,Y2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_127_ord__eq__le__subst,axiom,
% 0.29/0.45 ! [A3: real,F: real > real,B3: real,C: real] :
% 0.29/0.45 ( ( A3
% 0.29/0.45 = ( F @ B3 ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ B3 @ C )
% 0.29/0.45 => ( ! [X4: real,Y2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_subst
% 0.29/0.45 thf(fact_128_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( ( ( F @ B3 )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_129_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > real,C: real] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( ( ( F @ B3 )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_130_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A3: real,B3: real,F: real > extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.45 => ( ( ( F @ B3 )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X4: real,Y2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_131_ord__le__eq__subst,axiom,
% 0.29/0.45 ! [A3: real,B3: real,F: real > real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.45 => ( ( ( F @ B3 )
% 0.29/0.45 = C )
% 0.29/0.45 => ( ! [X4: real,Y2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.29/0.45 => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_subst
% 0.29/0.45 thf(fact_132_eq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
% 0.29/0.45 = ( ^ [X2: extended_ereal,Y3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.29/0.45 & ( ord_le824540014_ereal @ Y3 @ X2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_iff
% 0.29/0.45 thf(fact_133_eq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y: real,Z: real] : ( Y = Z ) )
% 0.29/0.45 = ( ^ [X2: real,Y3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.29/0.45 & ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_iff
% 0.29/0.45 thf(fact_134_antisym,axiom,
% 0.29/0.45 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ Y4 @ X )
% 0.29/0.45 => ( X = Y4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym
% 0.29/0.45 thf(fact_135_antisym,axiom,
% 0.29/0.45 ! [X: real,Y4: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y4 )
% 0.29/0.45 => ( ( ord_less_eq_real @ Y4 @ X )
% 0.29/0.45 => ( X = Y4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym
% 0.29/0.45 thf(fact_136_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A3: extended_ereal,P: extended_ereal > $o] :
% 0.29/0.45 ( ( member1900190071_ereal @ A3 @ ( collec247695033_ereal @ P ) )
% 0.29/0.45 = ( P @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_137_mem__Collect__eq,axiom,
% 0.29/0.45 ! [A3: a,P: a > $o] :
% 0.29/0.45 ( ( member_a @ A3 @ ( collect_a @ P ) )
% 0.29/0.45 = ( P @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % mem_Collect_eq
% 0.29/0.45 thf(fact_138_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal] :
% 0.29/0.45 ( ( collec247695033_ereal
% 0.29/0.45 @ ^ [X2: extended_ereal] : ( member1900190071_ereal @ X2 @ A ) )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_139_Collect__mem__eq,axiom,
% 0.29/0.45 ! [A: set_a] :
% 0.29/0.45 ( ( collect_a
% 0.29/0.45 @ ^ [X2: a] : ( member_a @ X2 @ A ) )
% 0.29/0.45 = A ) ).
% 0.29/0.45
% 0.29/0.45 % Collect_mem_eq
% 0.29/0.45 thf(fact_140_linear,axiom,
% 0.29/0.45 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.29/0.45 | ( ord_le824540014_ereal @ Y4 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % linear
% 0.29/0.45 thf(fact_141_linear,axiom,
% 0.29/0.45 ! [X: real,Y4: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y4 )
% 0.29/0.45 | ( ord_less_eq_real @ Y4 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % linear
% 0.29/0.45 thf(fact_142_eq__refl,axiom,
% 0.29/0.45 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.29/0.45 ( ( X = Y4 )
% 0.29/0.45 => ( ord_le824540014_ereal @ X @ Y4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_refl
% 0.29/0.45 thf(fact_143_eq__refl,axiom,
% 0.29/0.45 ! [X: real,Y4: real] :
% 0.29/0.45 ( ( X = Y4 )
% 0.29/0.45 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.29/0.45
% 0.29/0.45 % eq_refl
% 0.29/0.45 thf(fact_144_le__cases,axiom,
% 0.29/0.45 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.29/0.45 ( ~ ( ord_le824540014_ereal @ X @ Y4 )
% 0.29/0.45 => ( ord_le824540014_ereal @ Y4 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases
% 0.29/0.45 thf(fact_145_le__cases,axiom,
% 0.29/0.45 ! [X: real,Y4: real] :
% 0.29/0.45 ( ~ ( ord_less_eq_real @ X @ Y4 )
% 0.29/0.45 => ( ord_less_eq_real @ Y4 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases
% 0.29/0.45 thf(fact_146_order_Otrans,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.trans
% 0.29/0.45 thf(fact_147_order_Otrans,axiom,
% 0.29/0.45 ! [A3: real,B3: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_less_eq_real @ B3 @ C )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order.trans
% 0.29/0.45 thf(fact_148_le__cases3,axiom,
% 0.29/0.45 ! [X: extended_ereal,Y4: extended_ereal,Z2: extended_ereal] :
% 0.29/0.45 ( ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.29/0.45 => ~ ( ord_le824540014_ereal @ Y4 @ Z2 ) )
% 0.29/0.45 => ( ( ( ord_le824540014_ereal @ Y4 @ X )
% 0.29/0.45 => ~ ( ord_le824540014_ereal @ X @ Z2 ) )
% 0.29/0.45 => ( ( ( ord_le824540014_ereal @ X @ Z2 )
% 0.29/0.45 => ~ ( ord_le824540014_ereal @ Z2 @ Y4 ) )
% 0.29/0.45 => ( ( ( ord_le824540014_ereal @ Z2 @ Y4 )
% 0.29/0.45 => ~ ( ord_le824540014_ereal @ Y4 @ X ) )
% 0.29/0.45 => ( ( ( ord_le824540014_ereal @ Y4 @ Z2 )
% 0.29/0.45 => ~ ( ord_le824540014_ereal @ Z2 @ X ) )
% 0.29/0.45 => ~ ( ( ord_le824540014_ereal @ Z2 @ X )
% 0.29/0.45 => ~ ( ord_le824540014_ereal @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases3
% 0.29/0.45 thf(fact_149_le__cases3,axiom,
% 0.29/0.45 ! [X: real,Y4: real,Z2: real] :
% 0.29/0.45 ( ( ( ord_less_eq_real @ X @ Y4 )
% 0.29/0.45 => ~ ( ord_less_eq_real @ Y4 @ Z2 ) )
% 0.29/0.45 => ( ( ( ord_less_eq_real @ Y4 @ X )
% 0.29/0.45 => ~ ( ord_less_eq_real @ X @ Z2 ) )
% 0.29/0.45 => ( ( ( ord_less_eq_real @ X @ Z2 )
% 0.29/0.45 => ~ ( ord_less_eq_real @ Z2 @ Y4 ) )
% 0.29/0.45 => ( ( ( ord_less_eq_real @ Z2 @ Y4 )
% 0.29/0.45 => ~ ( ord_less_eq_real @ Y4 @ X ) )
% 0.29/0.45 => ( ( ( ord_less_eq_real @ Y4 @ Z2 )
% 0.29/0.45 => ~ ( ord_less_eq_real @ Z2 @ X ) )
% 0.29/0.45 => ~ ( ( ord_less_eq_real @ Z2 @ X )
% 0.29/0.45 => ~ ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_cases3
% 0.29/0.45 thf(fact_150_antisym__conv,axiom,
% 0.29/0.45 ! [Y4: extended_ereal,X: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ Y4 @ X )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.29/0.45 = ( X = Y4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv
% 0.29/0.45 thf(fact_151_antisym__conv,axiom,
% 0.29/0.45 ! [Y4: real,X: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ Y4 @ X )
% 0.29/0.45 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.29/0.45 = ( X = Y4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % antisym_conv
% 0.29/0.45 thf(fact_152_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
% 0.29/0.45 = ( ^ [A4: extended_ereal,B4: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A4 @ B4 )
% 0.29/0.45 & ( ord_le824540014_ereal @ B4 @ A4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.eq_iff
% 0.29/0.45 thf(fact_153_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y: real,Z: real] : ( Y = Z ) )
% 0.29/0.45 = ( ^ [A4: real,B4: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A4 @ B4 )
% 0.29/0.45 & ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.eq_iff
% 0.29/0.45 thf(fact_154_ord__eq__le__trans,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( A3 = B3 )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_trans
% 0.29/0.45 thf(fact_155_ord__eq__le__trans,axiom,
% 0.29/0.45 ! [A3: real,B3: real,C: real] :
% 0.29/0.45 ( ( A3 = B3 )
% 0.29/0.45 => ( ( ord_less_eq_real @ B3 @ C )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_eq_le_trans
% 0.29/0.45 thf(fact_156_ord__le__eq__trans,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( ( B3 = C )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_trans
% 0.29/0.45 thf(fact_157_ord__le__eq__trans,axiom,
% 0.29/0.45 ! [A3: real,B3: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.45 => ( ( B3 = C )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ C ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ord_le_eq_trans
% 0.29/0.45 thf(fact_158_order__class_Oorder_Oantisym,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ B3 @ A3 )
% 0.29/0.45 => ( A3 = B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.antisym
% 0.29/0.45 thf(fact_159_order__class_Oorder_Oantisym,axiom,
% 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 => ( A3 = B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_class.order.antisym
% 0.29/0.45 thf(fact_160_order__trans,axiom,
% 0.29/0.45 ! [X: extended_ereal,Y4: extended_ereal,Z2: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ Y4 @ Z2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ X @ Z2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_trans
% 0.29/0.45 thf(fact_161_order__trans,axiom,
% 0.29/0.45 ! [X: real,Y4: real,Z2: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ X @ Y4 )
% 0.29/0.45 => ( ( ord_less_eq_real @ Y4 @ Z2 )
% 0.29/0.45 => ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % order_trans
% 0.29/0.45 thf(fact_162_dual__order_Orefl,axiom,
% 0.29/0.45 ! [A3: extended_ereal] : ( ord_le824540014_ereal @ A3 @ A3 ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.refl
% 0.29/0.45 thf(fact_163_dual__order_Orefl,axiom,
% 0.29/0.45 ! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.refl
% 0.29/0.45 thf(fact_164_linorder__wlog,axiom,
% 0.29/0.45 ! [P: extended_ereal > extended_ereal > $o,A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ! [A5: extended_ereal,B5: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A5 @ B5 )
% 0.29/0.45 => ( P @ A5 @ B5 ) )
% 0.29/0.45 => ( ! [A5: extended_ereal,B5: extended_ereal] :
% 0.29/0.45 ( ( P @ B5 @ A5 )
% 0.29/0.45 => ( P @ A5 @ B5 ) )
% 0.29/0.45 => ( P @ A3 @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_wlog
% 0.29/0.45 thf(fact_165_linorder__wlog,axiom,
% 0.29/0.45 ! [P: real > real > $o,A3: real,B3: real] :
% 0.29/0.45 ( ! [A5: real,B5: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A5 @ B5 )
% 0.29/0.45 => ( P @ A5 @ B5 ) )
% 0.29/0.45 => ( ! [A5: real,B5: real] :
% 0.29/0.45 ( ( P @ B5 @ A5 )
% 0.29/0.45 => ( P @ A5 @ B5 ) )
% 0.29/0.45 => ( P @ A3 @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % linorder_wlog
% 0.29/0.45 thf(fact_166_dual__order_Otrans,axiom,
% 0.29/0.45 ! [B3: extended_ereal,A3: extended_ereal,C: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ B3 @ A3 )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ C @ B3 )
% 0.29/0.45 => ( ord_le824540014_ereal @ C @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.trans
% 0.29/0.45 thf(fact_167_dual__order_Otrans,axiom,
% 0.29/0.45 ! [B3: real,A3: real,C: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ B3 @ A3 )
% 0.29/0.45 => ( ( ord_less_eq_real @ C @ B3 )
% 0.29/0.45 => ( ord_less_eq_real @ C @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.trans
% 0.29/0.45 thf(fact_168_dual__order_Oeq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
% 0.29/0.45 = ( ^ [A4: extended_ereal,B4: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ B4 @ A4 )
% 0.29/0.45 & ( ord_le824540014_ereal @ A4 @ B4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.eq_iff
% 0.29/0.45 thf(fact_169_dual__order_Oeq__iff,axiom,
% 0.29/0.45 ( ( ^ [Y: real,Z: real] : ( Y = Z ) )
% 0.29/0.45 = ( ^ [A4: real,B4: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ B4 @ A4 )
% 0.29/0.45 & ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.eq_iff
% 0.29/0.45 thf(fact_170_dual__order_Oantisym,axiom,
% 0.29/0.45 ! [B3: extended_ereal,A3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ B3 @ A3 )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( A3 = B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.antisym
% 0.29/0.45 thf(fact_171_dual__order_Oantisym,axiom,
% 0.29/0.45 ! [B3: real,A3: 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 => ( A3 = B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % dual_order.antisym
% 0.29/0.45 thf(fact_172_Sup__eqI,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,X: extended_ereal] :
% 0.29/0.45 ( ! [Y2: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ Y2 @ A )
% 0.29/0.45 => ( ord_le824540014_ereal @ Y2 @ X ) )
% 0.29/0.45 => ( ! [Y2: extended_ereal] :
% 0.29/0.45 ( ! [Z3: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ Z3 @ A )
% 0.29/0.45 => ( ord_le824540014_ereal @ Z3 @ Y2 ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ X @ Y2 ) )
% 0.29/0.45 => ( ( comple1161760187_ereal @ A )
% 0.29/0.45 = X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup_eqI
% 0.29/0.45 thf(fact_173_Sup__mono,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.29/0.45 ( ! [A5: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ A5 @ A )
% 0.29/0.45 => ? [X3: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ X3 @ B )
% 0.29/0.45 & ( ord_le824540014_ereal @ A5 @ X3 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ ( comple1161760187_ereal @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup_mono
% 0.29/0.45 thf(fact_174_Sup__least,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,Z2: extended_ereal] :
% 0.29/0.45 ( ! [X4: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ X4 @ A )
% 0.29/0.45 => ( ord_le824540014_ereal @ X4 @ Z2 ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ Z2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup_least
% 0.29/0.45 thf(fact_175_Sup__upper,axiom,
% 0.29/0.45 ! [X: extended_ereal,A: set_Extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ X @ A )
% 0.29/0.45 => ( ord_le824540014_ereal @ X @ ( comple1161760187_ereal @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup_upper
% 0.29/0.45 thf(fact_176_Sup__le__iff,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ B3 )
% 0.29/0.45 = ( ! [X2: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ X2 @ A )
% 0.29/0.45 => ( ord_le824540014_ereal @ X2 @ B3 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup_le_iff
% 0.29/0.45 thf(fact_177_Sup__upper2,axiom,
% 0.29/0.45 ! [U: extended_ereal,A: set_Extended_ereal,V: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ U @ A )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ V @ U )
% 0.29/0.45 => ( ord_le824540014_ereal @ V @ ( comple1161760187_ereal @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup_upper2
% 0.29/0.45 thf(fact_178_Sup__subset__mono,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ A @ B )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ ( comple1161760187_ereal @ B ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup_subset_mono
% 0.29/0.45 thf(fact_179_top__set__def,axiom,
% 0.29/0.45 ( top_top_set_a
% 0.29/0.45 = ( collect_a @ top_top_a_o ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_set_def
% 0.29/0.45 thf(fact_180_top__set__def,axiom,
% 0.29/0.45 ( top_to1767659262_ereal
% 0.29/0.45 = ( collec247695033_ereal @ top_to398855007real_o ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_set_def
% 0.29/0.45 thf(fact_181_Union__UNIV,axiom,
% 0.29/0.45 ( ( comple1766734283_set_a @ top_top_set_set_a )
% 0.29/0.45 = top_top_set_a ) ).
% 0.29/0.45
% 0.29/0.45 % Union_UNIV
% 0.29/0.45 thf(fact_182_Union__UNIV,axiom,
% 0.29/0.45 ( ( comple767846299_ereal @ top_to692740318_ereal )
% 0.29/0.45 = top_to1767659262_ereal ) ).
% 0.29/0.45
% 0.29/0.45 % Union_UNIV
% 0.29/0.45 thf(fact_183_top_Oextremum__uniqueI,axiom,
% 0.29/0.45 ! [A3: set_a] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ top_top_set_a @ A3 )
% 0.29/0.45 => ( A3 = top_top_set_a ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_uniqueI
% 0.29/0.45 thf(fact_184_top_Oextremum__uniqueI,axiom,
% 0.29/0.45 ! [A3: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ top_to1767659262_ereal @ A3 )
% 0.29/0.45 => ( A3 = top_to1767659262_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_uniqueI
% 0.29/0.45 thf(fact_185_top_Oextremum__uniqueI,axiom,
% 0.29/0.45 ! [A3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ top_to802031902_ereal @ A3 )
% 0.29/0.45 => ( A3 = top_to802031902_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_uniqueI
% 0.29/0.45 thf(fact_186_top_Oextremum__unique,axiom,
% 0.29/0.45 ! [A3: set_a] :
% 0.29/0.45 ( ( ord_less_eq_set_a @ top_top_set_a @ A3 )
% 0.29/0.45 = ( A3 = top_top_set_a ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_unique
% 0.29/0.45 thf(fact_187_top_Oextremum__unique,axiom,
% 0.29/0.45 ! [A3: set_Extended_ereal] :
% 0.29/0.45 ( ( ord_le88246606_ereal @ top_to1767659262_ereal @ A3 )
% 0.29/0.45 = ( A3 = top_to1767659262_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_unique
% 0.29/0.45 thf(fact_188_top_Oextremum__unique,axiom,
% 0.29/0.45 ! [A3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ top_to802031902_ereal @ A3 )
% 0.29/0.45 = ( A3 = top_to802031902_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % top.extremum_unique
% 0.29/0.45 thf(fact_189_top__greatest,axiom,
% 0.29/0.45 ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).
% 0.29/0.45
% 0.29/0.45 % top_greatest
% 0.29/0.45 thf(fact_190_top__greatest,axiom,
% 0.29/0.45 ! [A3: set_Extended_ereal] : ( ord_le88246606_ereal @ A3 @ top_to1767659262_ereal ) ).
% 0.29/0.45
% 0.29/0.45 % top_greatest
% 0.29/0.45 thf(fact_191_top__greatest,axiom,
% 0.29/0.45 ! [A3: extended_ereal] : ( ord_le824540014_ereal @ A3 @ top_to802031902_ereal ) ).
% 0.29/0.45
% 0.29/0.45 % top_greatest
% 0.29/0.45 thf(fact_192_subset__UNIV,axiom,
% 0.29/0.45 ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% 0.29/0.45
% 0.29/0.45 % subset_UNIV
% 0.29/0.45 thf(fact_193_subset__UNIV,axiom,
% 0.29/0.45 ! [A: set_Extended_ereal] : ( ord_le88246606_ereal @ A @ top_to1767659262_ereal ) ).
% 0.29/0.45
% 0.29/0.45 % subset_UNIV
% 0.29/0.45 thf(fact_194_epigraph__invertible,axiom,
% 0.29/0.45 ! [F: a > extended_ereal,G: a > extended_ereal] :
% 0.29/0.45 ( ( ( lower_930854854raph_a @ top_top_set_a @ F )
% 0.29/0.45 = ( lower_930854854raph_a @ top_top_set_a @ G ) )
% 0.29/0.45 => ( F = G ) ) ).
% 0.29/0.45
% 0.29/0.45 % epigraph_invertible
% 0.29/0.45 thf(fact_195_epigraph__invertible,axiom,
% 0.29/0.45 ! [F: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
% 0.29/0.45 ( ( ( lower_331963542_ereal @ top_to1767659262_ereal @ F )
% 0.29/0.45 = ( lower_331963542_ereal @ top_to1767659262_ereal @ G ) )
% 0.29/0.45 => ( F = G ) ) ).
% 0.29/0.45
% 0.29/0.45 % epigraph_invertible
% 0.29/0.45 thf(fact_196_neg__le__iff__le,axiom,
% 0.29/0.45 ! [B3: real,A3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
% 0.29/0.45 = ( ord_less_eq_real @ A3 @ B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % neg_le_iff_le
% 0.29/0.45 thf(fact_197_ereal__uminus__eq__iff,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ( ( uminus1208298309_ereal @ A3 )
% 0.29/0.45 = ( uminus1208298309_ereal @ B3 ) )
% 0.29/0.45 = ( A3 = B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_uminus_eq_iff
% 0.29/0.45 thf(fact_198_ereal__uminus__uminus,axiom,
% 0.29/0.45 ! [A3: extended_ereal] :
% 0.29/0.45 ( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A3 ) )
% 0.29/0.45 = A3 ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_uminus_uminus
% 0.29/0.45 thf(fact_199_verit__minus__simplify_I4_J,axiom,
% 0.29/0.45 ! [B3: real] :
% 0.29/0.45 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B3 ) )
% 0.29/0.45 = B3 ) ).
% 0.29/0.45
% 0.29/0.45 % verit_minus_simplify(4)
% 0.29/0.45 thf(fact_200_add_Oinverse__inverse,axiom,
% 0.29/0.45 ! [A3: real] :
% 0.29/0.45 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A3 ) )
% 0.29/0.45 = A3 ) ).
% 0.29/0.45
% 0.29/0.45 % add.inverse_inverse
% 0.29/0.45 thf(fact_201_neg__equal__iff__equal,axiom,
% 0.29/0.45 ! [A3: real,B3: real] :
% 0.29/0.45 ( ( ( uminus_uminus_real @ A3 )
% 0.29/0.45 = ( uminus_uminus_real @ B3 ) )
% 0.29/0.45 = ( A3 = B3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % neg_equal_iff_equal
% 0.29/0.45 thf(fact_202_ereal__infty__less__eq_I1_J,axiom,
% 0.29/0.45 ! [X: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ extend1289208545_ereal @ X )
% 0.29/0.45 = ( X = extend1289208545_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_infty_less_eq(1)
% 0.29/0.45 thf(fact_203_UnionI,axiom,
% 0.29/0.45 ! [X5: set_Extended_ereal,C2: set_se767749006_ereal,A: extended_ereal] :
% 0.29/0.45 ( ( member450560855_ereal @ X5 @ C2 )
% 0.29/0.45 => ( ( member1900190071_ereal @ A @ X5 )
% 0.29/0.45 => ( member1900190071_ereal @ A @ ( comple767846299_ereal @ C2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnionI
% 0.29/0.45 thf(fact_204_UnionI,axiom,
% 0.29/0.45 ! [X5: set_a,C2: set_set_a,A: a] :
% 0.29/0.45 ( ( member_set_a @ X5 @ C2 )
% 0.29/0.45 => ( ( member_a @ A @ X5 )
% 0.29/0.45 => ( member_a @ A @ ( comple1766734283_set_a @ C2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnionI
% 0.29/0.45 thf(fact_205_Union__iff,axiom,
% 0.29/0.45 ! [A: extended_ereal,C2: set_se767749006_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ A @ ( comple767846299_ereal @ C2 ) )
% 0.29/0.45 = ( ? [X2: set_Extended_ereal] :
% 0.29/0.45 ( ( member450560855_ereal @ X2 @ C2 )
% 0.29/0.45 & ( member1900190071_ereal @ A @ X2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Union_iff
% 0.29/0.45 thf(fact_206_Union__iff,axiom,
% 0.29/0.45 ! [A: a,C2: set_set_a] :
% 0.29/0.45 ( ( member_a @ A @ ( comple1766734283_set_a @ C2 ) )
% 0.29/0.45 = ( ? [X2: set_a] :
% 0.29/0.45 ( ( member_set_a @ X2 @ C2 )
% 0.29/0.45 & ( member_a @ A @ X2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Union_iff
% 0.29/0.45 thf(fact_207_ereal__minus__le__minus,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A3 ) @ ( uminus1208298309_ereal @ B3 ) )
% 0.29/0.45 = ( ord_le824540014_ereal @ B3 @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_minus_le_minus
% 0.29/0.45 thf(fact_208_ereal__infty__less__eq_I2_J,axiom,
% 0.29/0.45 ! [X: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.45 = ( X
% 0.29/0.45 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_infty_less_eq(2)
% 0.29/0.45 thf(fact_209_UnionE,axiom,
% 0.29/0.45 ! [A: extended_ereal,C2: set_se767749006_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ A @ ( comple767846299_ereal @ C2 ) )
% 0.29/0.45 => ~ ! [X6: set_Extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ A @ X6 )
% 0.29/0.45 => ~ ( member450560855_ereal @ X6 @ C2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnionE
% 0.29/0.45 thf(fact_210_UnionE,axiom,
% 0.29/0.45 ! [A: a,C2: set_set_a] :
% 0.29/0.45 ( ( member_a @ A @ ( comple1766734283_set_a @ C2 ) )
% 0.29/0.45 => ~ ! [X6: set_a] :
% 0.29/0.45 ( ( member_a @ A @ X6 )
% 0.29/0.45 => ~ ( member_set_a @ X6 @ C2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % UnionE
% 0.29/0.45 thf(fact_211_neq__PInf__trans,axiom,
% 0.29/0.45 ! [Y4: extended_ereal,X: extended_ereal] :
% 0.29/0.45 ( ( Y4 != extend1289208545_ereal )
% 0.29/0.45 => ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.29/0.45 => ( X != extend1289208545_ereal ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % neq_PInf_trans
% 0.29/0.45 thf(fact_212_ereal__infty__less__eq2_I1_J,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( ( A3 = extend1289208545_ereal )
% 0.29/0.45 => ( B3 = extend1289208545_ereal ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_infty_less_eq2(1)
% 0.29/0.45 thf(fact_213_ereal__less__eq_I1_J,axiom,
% 0.29/0.45 ! [X: extended_ereal] : ( ord_le824540014_ereal @ X @ extend1289208545_ereal ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_less_eq(1)
% 0.29/0.45 thf(fact_214_ereal__uminus__le__reorder,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A3 ) @ B3 )
% 0.29/0.45 = ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ B3 ) @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_uminus_le_reorder
% 0.29/0.45 thf(fact_215_ereal__infty__less__eq2_I2_J,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 => ( ( B3
% 0.29/0.45 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.45 => ( A3
% 0.29/0.45 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_infty_less_eq2(2)
% 0.29/0.45 thf(fact_216_ereal__less__eq_I2_J,axiom,
% 0.29/0.45 ! [X: extended_ereal] : ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_less_eq(2)
% 0.29/0.45 thf(fact_217_epigraph__subset__iff,axiom,
% 0.29/0.45 ! [F: a > extended_ereal,G: a > extended_ereal] :
% 0.29/0.45 ( ( ord_le1586073967a_real @ ( lower_930854854raph_a @ top_top_set_a @ F ) @ ( lower_930854854raph_a @ top_top_set_a @ G ) )
% 0.29/0.45 = ( ! [X2: a] : ( ord_le824540014_ereal @ ( G @ X2 ) @ ( F @ X2 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % epigraph_subset_iff
% 0.29/0.45 thf(fact_218_verit__la__disequality,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ( A3 = B3 )
% 0.29/0.45 | ~ ( ord_le824540014_ereal @ A3 @ B3 )
% 0.29/0.45 | ~ ( ord_le824540014_ereal @ B3 @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % verit_la_disequality
% 0.29/0.45 thf(fact_219_verit__la__disequality,axiom,
% 0.29/0.45 ! [A3: real,B3: real] :
% 0.29/0.45 ( ( A3 = B3 )
% 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 % verit_la_disequality
% 0.29/0.45 thf(fact_220_minus__equation__iff,axiom,
% 0.29/0.45 ! [A3: real,B3: real] :
% 0.29/0.45 ( ( ( uminus_uminus_real @ A3 )
% 0.29/0.45 = B3 )
% 0.29/0.45 = ( ( uminus_uminus_real @ B3 )
% 0.29/0.45 = A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % minus_equation_iff
% 0.29/0.45 thf(fact_221_equation__minus__iff,axiom,
% 0.29/0.45 ! [A3: real,B3: real] :
% 0.29/0.45 ( ( A3
% 0.29/0.45 = ( uminus_uminus_real @ B3 ) )
% 0.29/0.45 = ( B3
% 0.29/0.45 = ( uminus_uminus_real @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % equation_minus_iff
% 0.29/0.45 thf(fact_222_verit__negate__coefficient_I3_J,axiom,
% 0.29/0.45 ! [A3: real,B3: real] :
% 0.29/0.45 ( ( A3 = B3 )
% 0.29/0.45 => ( ( uminus_uminus_real @ A3 )
% 0.29/0.45 = ( uminus_uminus_real @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % verit_negate_coefficient(3)
% 0.29/0.45 thf(fact_223_top__ereal__def,axiom,
% 0.29/0.45 top_to802031902_ereal = extend1289208545_ereal ).
% 0.29/0.45
% 0.29/0.45 % top_ereal_def
% 0.29/0.45 thf(fact_224_Sup__eq__PInfty,axiom,
% 0.29/0.45 ! [S2: set_Extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ extend1289208545_ereal @ S2 )
% 0.29/0.45 => ( ( comple1161760187_ereal @ S2 )
% 0.29/0.45 = extend1289208545_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % Sup_eq_PInfty
% 0.29/0.45 thf(fact_225_ereal__uminus__eq__reorder,axiom,
% 0.29/0.45 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.45 ( ( ( uminus1208298309_ereal @ A3 )
% 0.29/0.45 = B3 )
% 0.29/0.45 = ( A3
% 0.29/0.45 = ( uminus1208298309_ereal @ B3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_uminus_eq_reorder
% 0.29/0.45 thf(fact_226_le__imp__neg__le,axiom,
% 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 @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_imp_neg_le
% 0.29/0.45 thf(fact_227_minus__le__iff,axiom,
% 0.29/0.45 ! [A3: real,B3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 )
% 0.29/0.45 = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % minus_le_iff
% 0.29/0.45 thf(fact_228_le__minus__iff,axiom,
% 0.29/0.45 ! [A3: real,B3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ B3 ) )
% 0.29/0.45 = ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_minus_iff
% 0.29/0.45 thf(fact_229_MInfty__neq__PInfty_I1_J,axiom,
% 0.29/0.45 ( extend1289208545_ereal
% 0.29/0.45 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % MInfty_neq_PInfty(1)
% 0.29/0.45 thf(fact_230_top__empty__eq,axiom,
% 0.29/0.45 ( top_top_a_o
% 0.29/0.45 = ( ^ [X2: a] : ( member_a @ X2 @ top_top_set_a ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_empty_eq
% 0.29/0.45 thf(fact_231_top__empty__eq,axiom,
% 0.29/0.45 ( top_to398855007real_o
% 0.29/0.45 = ( ^ [X2: extended_ereal] : ( member1900190071_ereal @ X2 @ top_to1767659262_ereal ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % top_empty_eq
% 0.29/0.45 thf(fact_232_cSup__eq__maximum,axiom,
% 0.29/0.45 ! [Z2: real,X5: set_real] :
% 0.29/0.45 ( ( member_real @ Z2 @ X5 )
% 0.29/0.45 => ( ! [X4: real] :
% 0.29/0.45 ( ( member_real @ X4 @ X5 )
% 0.29/0.45 => ( ord_less_eq_real @ X4 @ Z2 ) )
% 0.29/0.45 => ( ( comple2129349247p_real @ X5 )
% 0.29/0.45 = Z2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % cSup_eq_maximum
% 0.29/0.45 thf(fact_233_cSup__eq__maximum,axiom,
% 0.29/0.45 ! [Z2: extended_ereal,X5: set_Extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ Z2 @ X5 )
% 0.29/0.45 => ( ! [X4: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ X4 @ X5 )
% 0.29/0.45 => ( ord_le824540014_ereal @ X4 @ Z2 ) )
% 0.29/0.45 => ( ( comple1161760187_ereal @ X5 )
% 0.29/0.45 = Z2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % cSup_eq_maximum
% 0.29/0.45 thf(fact_234_cSup__eq,axiom,
% 0.29/0.45 ! [X5: set_real,A3: real] :
% 0.29/0.45 ( ! [X4: real] :
% 0.29/0.45 ( ( member_real @ X4 @ X5 )
% 0.29/0.45 => ( ord_less_eq_real @ X4 @ A3 ) )
% 0.29/0.45 => ( ! [Y2: real] :
% 0.29/0.45 ( ! [X3: real] :
% 0.29/0.45 ( ( member_real @ X3 @ X5 )
% 0.29/0.45 => ( ord_less_eq_real @ X3 @ Y2 ) )
% 0.29/0.45 => ( ord_less_eq_real @ A3 @ Y2 ) )
% 0.29/0.45 => ( ( comple2129349247p_real @ X5 )
% 0.29/0.45 = A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % cSup_eq
% 0.29/0.45 thf(fact_235_MInfty__eq__minfinity,axiom,
% 0.29/0.45 ( extended_MInfty
% 0.29/0.45 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % MInfty_eq_minfinity
% 0.29/0.45 thf(fact_236_ComplI,axiom,
% 0.29/0.45 ! [C: extended_ereal,A: set_Extended_ereal] :
% 0.29/0.45 ( ~ ( member1900190071_ereal @ C @ A )
% 0.29/0.45 => ( member1900190071_ereal @ C @ ( uminus360668453_ereal @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ComplI
% 0.29/0.45 thf(fact_237_ComplI,axiom,
% 0.29/0.45 ! [C: a,A: set_a] :
% 0.29/0.45 ( ~ ( member_a @ C @ A )
% 0.29/0.45 => ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ComplI
% 0.29/0.45 thf(fact_238_Compl__iff,axiom,
% 0.29/0.45 ! [C: extended_ereal,A: set_Extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ C @ ( uminus360668453_ereal @ A ) )
% 0.29/0.45 = ( ~ ( member1900190071_ereal @ C @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Compl_iff
% 0.29/0.45 thf(fact_239_Compl__iff,axiom,
% 0.29/0.45 ! [C: a,A: set_a] :
% 0.29/0.45 ( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
% 0.29/0.45 = ( ~ ( member_a @ C @ A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Compl_iff
% 0.29/0.45 thf(fact_240_ComplD,axiom,
% 0.29/0.45 ! [C: extended_ereal,A: set_Extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ C @ ( uminus360668453_ereal @ A ) )
% 0.29/0.45 => ~ ( member1900190071_ereal @ C @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % ComplD
% 0.29/0.45 thf(fact_241_ComplD,axiom,
% 0.29/0.45 ! [C: a,A: set_a] :
% 0.29/0.45 ( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
% 0.29/0.45 => ~ ( member_a @ C @ A ) ) ).
% 0.29/0.45
% 0.29/0.45 % ComplD
% 0.29/0.45 thf(fact_242_ereal__complete__Inf,axiom,
% 0.29/0.45 ! [S2: set_Extended_ereal] :
% 0.29/0.45 ? [X4: extended_ereal] :
% 0.29/0.45 ( ! [Xa: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ Xa @ S2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ X4 @ Xa ) )
% 0.29/0.45 & ! [Z3: extended_ereal] :
% 0.29/0.45 ( ! [Xa2: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ Xa2 @ S2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ Z3 @ Xa2 ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ Z3 @ X4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_complete_Inf
% 0.29/0.45 thf(fact_243_ereal__complete__Sup,axiom,
% 0.29/0.45 ! [S2: set_Extended_ereal] :
% 0.29/0.45 ? [X4: extended_ereal] :
% 0.29/0.45 ( ! [Xa: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ Xa @ S2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ Xa @ X4 ) )
% 0.29/0.45 & ! [Z3: extended_ereal] :
% 0.29/0.45 ( ! [Xa2: extended_ereal] :
% 0.29/0.45 ( ( member1900190071_ereal @ Xa2 @ S2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ Xa2 @ Z3 ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ X4 @ Z3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_complete_Sup
% 0.29/0.45 thf(fact_244_uminus__ereal_Osimps_I3_J,axiom,
% 0.29/0.45 ( ( uminus1208298309_ereal @ extended_MInfty )
% 0.29/0.45 = extended_PInfty ) ).
% 0.29/0.45
% 0.29/0.45 % uminus_ereal.simps(3)
% 0.29/0.45 thf(fact_245_uminus__ereal_Osimps_I2_J,axiom,
% 0.29/0.45 ( ( uminus1208298309_ereal @ extended_PInfty )
% 0.29/0.45 = extended_MInfty ) ).
% 0.29/0.45
% 0.29/0.45 % uminus_ereal.simps(2)
% 0.29/0.45 thf(fact_246_GreatestI2__order,axiom,
% 0.29/0.45 ! [P: extended_ereal > $o,X: extended_ereal,Q: extended_ereal > $o] :
% 0.29/0.45 ( ( P @ X )
% 0.29/0.45 => ( ! [Y2: extended_ereal] :
% 0.29/0.45 ( ( P @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ Y2 @ X ) )
% 0.29/0.45 => ( ! [X4: extended_ereal] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.45 => ( ! [Y5: extended_ereal] :
% 0.29/0.45 ( ( P @ Y5 )
% 0.29/0.45 => ( ord_le824540014_ereal @ Y5 @ X4 ) )
% 0.29/0.45 => ( Q @ X4 ) ) )
% 0.29/0.45 => ( Q @ ( order_1158471719_ereal @ P ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % GreatestI2_order
% 0.29/0.45 thf(fact_247_GreatestI2__order,axiom,
% 0.29/0.45 ! [P: real > $o,X: real,Q: real > $o] :
% 0.29/0.45 ( ( P @ X )
% 0.29/0.45 => ( ! [Y2: real] :
% 0.29/0.45 ( ( P @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ Y2 @ X ) )
% 0.29/0.45 => ( ! [X4: real] :
% 0.29/0.45 ( ( P @ X4 )
% 0.29/0.45 => ( ! [Y5: real] :
% 0.29/0.45 ( ( P @ Y5 )
% 0.29/0.45 => ( ord_less_eq_real @ Y5 @ X4 ) )
% 0.29/0.45 => ( Q @ X4 ) ) )
% 0.29/0.45 => ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % GreatestI2_order
% 0.29/0.45 thf(fact_248_infinity__ereal__def,axiom,
% 0.29/0.45 extend1289208545_ereal = extended_PInfty ).
% 0.29/0.45
% 0.29/0.45 % infinity_ereal_def
% 0.29/0.45 thf(fact_249_ereal_Odistinct_I5_J,axiom,
% 0.29/0.45 extended_PInfty != extended_MInfty ).
% 0.29/0.45
% 0.29/0.45 % ereal.distinct(5)
% 0.29/0.45 thf(fact_250_Greatest__equality,axiom,
% 0.29/0.45 ! [P: extended_ereal > $o,X: extended_ereal] :
% 0.29/0.45 ( ( P @ X )
% 0.29/0.45 => ( ! [Y2: extended_ereal] :
% 0.29/0.45 ( ( P @ Y2 )
% 0.29/0.45 => ( ord_le824540014_ereal @ Y2 @ X ) )
% 0.29/0.45 => ( ( order_1158471719_ereal @ P )
% 0.29/0.45 = X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Greatest_equality
% 0.29/0.45 thf(fact_251_Greatest__equality,axiom,
% 0.29/0.45 ! [P: real > $o,X: real] :
% 0.29/0.45 ( ( P @ X )
% 0.29/0.45 => ( ! [Y2: real] :
% 0.29/0.45 ( ( P @ Y2 )
% 0.29/0.45 => ( ord_less_eq_real @ Y2 @ X ) )
% 0.29/0.45 => ( ( order_Greatest_real @ P )
% 0.29/0.45 = X ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % Greatest_equality
% 0.29/0.45 thf(fact_252_uminus__ereal_Oelims,axiom,
% 0.29/0.45 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.29/0.45 ( ( ( uminus1208298309_ereal @ X )
% 0.29/0.45 = Y4 )
% 0.29/0.45 => ( ! [R: real] :
% 0.29/0.45 ( ( X
% 0.29/0.45 = ( extended_ereal2 @ R ) )
% 0.29/0.45 => ( Y4
% 0.29/0.45 != ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) )
% 0.29/0.45 => ( ( ( X = extended_PInfty )
% 0.29/0.45 => ( Y4 != extended_MInfty ) )
% 0.29/0.45 => ~ ( ( X = extended_MInfty )
% 0.29/0.45 => ( Y4 != extended_PInfty ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % uminus_ereal.elims
% 0.29/0.45 thf(fact_253_not__MInfty__nonneg,axiom,
% 0.29/0.45 ! [X: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X )
% 0.29/0.45 => ( X
% 0.29/0.45 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % not_MInfty_nonneg
% 0.29/0.45 thf(fact_254_ereal_Oinject,axiom,
% 0.29/0.45 ! [X1: real,Y1: real] :
% 0.29/0.45 ( ( ( extended_ereal2 @ X1 )
% 0.29/0.45 = ( extended_ereal2 @ Y1 ) )
% 0.29/0.45 = ( X1 = Y1 ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal.inject
% 0.29/0.45 thf(fact_255_ereal__cong,axiom,
% 0.29/0.45 ! [X: real,Y4: real] :
% 0.29/0.45 ( ( X = Y4 )
% 0.29/0.45 => ( ( extended_ereal2 @ X )
% 0.29/0.45 = ( extended_ereal2 @ Y4 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_cong
% 0.29/0.45 thf(fact_256_neg__equal__zero,axiom,
% 0.29/0.45 ! [A3: real] :
% 0.29/0.45 ( ( ( uminus_uminus_real @ A3 )
% 0.29/0.45 = A3 )
% 0.29/0.45 = ( A3 = zero_zero_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % neg_equal_zero
% 0.29/0.45 thf(fact_257_equal__neg__zero,axiom,
% 0.29/0.45 ! [A3: real] :
% 0.29/0.45 ( ( A3
% 0.29/0.45 = ( uminus_uminus_real @ A3 ) )
% 0.29/0.45 = ( A3 = zero_zero_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % equal_neg_zero
% 0.29/0.45 thf(fact_258_neg__equal__0__iff__equal,axiom,
% 0.29/0.45 ! [A3: real] :
% 0.29/0.45 ( ( ( uminus_uminus_real @ A3 )
% 0.29/0.45 = zero_zero_real )
% 0.29/0.45 = ( A3 = zero_zero_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % neg_equal_0_iff_equal
% 0.29/0.45 thf(fact_259_neg__0__equal__iff__equal,axiom,
% 0.29/0.45 ! [A3: real] :
% 0.29/0.45 ( ( zero_zero_real
% 0.29/0.45 = ( uminus_uminus_real @ A3 ) )
% 0.29/0.45 = ( zero_zero_real = A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % neg_0_equal_iff_equal
% 0.29/0.45 thf(fact_260_add_Oinverse__neutral,axiom,
% 0.29/0.45 ( ( uminus_uminus_real @ zero_zero_real )
% 0.29/0.45 = zero_zero_real ) ).
% 0.29/0.45
% 0.29/0.45 % add.inverse_neutral
% 0.29/0.45 thf(fact_261_ereal__less__eq_I3_J,axiom,
% 0.29/0.45 ! [R2: real,P2: real] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 0.29/0.45 = ( ord_less_eq_real @ R2 @ P2 ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_less_eq(3)
% 0.29/0.45 thf(fact_262_ereal__uminus__zero,axiom,
% 0.29/0.45 ( ( uminus1208298309_ereal @ zero_z163181189_ereal )
% 0.29/0.45 = zero_z163181189_ereal ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_uminus_zero
% 0.29/0.45 thf(fact_263_ereal__uminus__zero__iff,axiom,
% 0.29/0.45 ! [A3: extended_ereal] :
% 0.29/0.45 ( ( ( uminus1208298309_ereal @ A3 )
% 0.29/0.45 = zero_z163181189_ereal )
% 0.29/0.45 = ( A3 = zero_z163181189_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_uminus_zero_iff
% 0.29/0.45 thf(fact_264_neg__0__le__iff__le,axiom,
% 0.29/0.45 ! [A3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
% 0.29/0.45 = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % neg_0_le_iff_le
% 0.29/0.45 thf(fact_265_neg__le__0__iff__le,axiom,
% 0.29/0.45 ! [A3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
% 0.29/0.45 = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % neg_le_0_iff_le
% 0.29/0.45 thf(fact_266_less__eq__neg__nonpos,axiom,
% 0.29/0.45 ! [A3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ A3 ) )
% 0.29/0.45 = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % less_eq_neg_nonpos
% 0.29/0.45 thf(fact_267_neg__less__eq__nonneg,axiom,
% 0.29/0.45 ! [A3: real] :
% 0.29/0.45 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ A3 )
% 0.29/0.45 = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % neg_less_eq_nonneg
% 0.29/0.45 thf(fact_268_ereal__uminus__le__0__iff,axiom,
% 0.29/0.45 ! [A3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A3 ) @ zero_z163181189_ereal )
% 0.29/0.45 = ( ord_le824540014_ereal @ zero_z163181189_ereal @ A3 ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_uminus_le_0_iff
% 0.29/0.45 thf(fact_269_ereal__0__le__uminus__iff,axiom,
% 0.29/0.45 ! [A3: extended_ereal] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( uminus1208298309_ereal @ A3 ) )
% 0.29/0.45 = ( ord_le824540014_ereal @ A3 @ zero_z163181189_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_0_le_uminus_iff
% 0.29/0.45 thf(fact_270_ereal_Odistinct_I1_J,axiom,
% 0.29/0.45 ! [X1: real] :
% 0.29/0.45 ( ( extended_ereal2 @ X1 )
% 0.29/0.45 != extended_PInfty ) ).
% 0.29/0.45
% 0.29/0.45 % ereal.distinct(1)
% 0.29/0.45 thf(fact_271_zero__reorient,axiom,
% 0.29/0.45 ! [X: extended_ereal] :
% 0.29/0.45 ( ( zero_z163181189_ereal = X )
% 0.29/0.45 = ( X = zero_z163181189_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_reorient
% 0.29/0.45 thf(fact_272_zero__reorient,axiom,
% 0.29/0.45 ! [X: real] :
% 0.29/0.45 ( ( zero_zero_real = X )
% 0.29/0.45 = ( X = zero_zero_real ) ) ).
% 0.29/0.45
% 0.29/0.45 % zero_reorient
% 0.29/0.45 thf(fact_273_Infty__neq__0_I1_J,axiom,
% 0.29/0.45 extend1289208545_ereal != zero_z163181189_ereal ).
% 0.29/0.45
% 0.29/0.45 % Infty_neq_0(1)
% 0.29/0.45 thf(fact_274_uminus__ereal_Osimps_I1_J,axiom,
% 0.29/0.45 ! [R2: real] :
% 0.29/0.45 ( ( uminus1208298309_ereal @ ( extended_ereal2 @ R2 ) )
% 0.29/0.45 = ( extended_ereal2 @ ( uminus_uminus_real @ R2 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % uminus_ereal.simps(1)
% 0.29/0.45 thf(fact_275_PInfty__neq__ereal_I1_J,axiom,
% 0.29/0.45 ! [R2: real] :
% 0.29/0.45 ( ( extended_ereal2 @ R2 )
% 0.29/0.45 != extend1289208545_ereal ) ).
% 0.29/0.45
% 0.29/0.45 % PInfty_neq_ereal(1)
% 0.29/0.45 thf(fact_276_ereal_Odistinct_I3_J,axiom,
% 0.29/0.45 ! [X1: real] :
% 0.29/0.45 ( ( extended_ereal2 @ X1 )
% 0.29/0.45 != extended_MInfty ) ).
% 0.29/0.45
% 0.29/0.45 % ereal.distinct(3)
% 0.29/0.45 thf(fact_277_ereal__le__real,axiom,
% 0.29/0.45 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.29/0.45 ( ! [Z4: real] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ X @ ( extended_ereal2 @ Z4 ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ Y4 @ ( extended_ereal2 @ Z4 ) ) )
% 0.29/0.45 => ( ord_le824540014_ereal @ Y4 @ X ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_le_real
% 0.29/0.45 thf(fact_278_le__ereal__le,axiom,
% 0.29/0.45 ! [A3: extended_ereal,X: real,Y4: real] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ A3 @ ( extended_ereal2 @ X ) )
% 0.29/0.45 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.29/0.45 => ( ord_le824540014_ereal @ A3 @ ( extended_ereal2 @ Y4 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % le_ereal_le
% 0.29/0.45 thf(fact_279_ereal__le__le,axiom,
% 0.29/0.45 ! [Y4: real,A3: extended_ereal,X: real] :
% 0.29/0.45 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ Y4 ) @ A3 )
% 0.29/0.45 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.29/0.45 => ( ord_le824540014_ereal @ ( extended_ereal2 @ X ) @ A3 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_le_le
% 0.29/0.45 thf(fact_280_MInfty__neq__ereal_I1_J,axiom,
% 0.29/0.45 ! [R2: real] :
% 0.29/0.45 ( ( extended_ereal2 @ R2 )
% 0.29/0.45 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.29/0.45
% 0.29/0.45 % MInfty_neq_ereal(1)
% 0.29/0.45 thf(fact_281_ereal__cases,axiom,
% 0.29/0.45 ! [X: extended_ereal] :
% 0.29/0.45 ( ! [R: real] :
% 0.29/0.45 ( X
% 0.29/0.45 != ( extended_ereal2 @ R ) )
% 0.29/0.45 => ( ( X != extend1289208545_ereal )
% 0.29/0.45 => ( X
% 0.29/0.45 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 % ereal_cases
% 0.29/0.45 thf(fact_282_ereal2__cases,axiom,
% 0.29/0.45 ! [X: extended_ereal,Xa3: extended_ereal] :
% 0.29/0.45 ( ( ? [R: real] :
% 0.29/0.45 ( X
% 0.29/0.45 = ( extended_ereal2 @ R ) )
% 0.29/0.45 => ! [Ra: real] :
% 0.29/0.45 ( Xa3
% 0.29/0.45 != ( extended_ereal2 @ Ra ) ) )
% 0.29/0.45 => ( ( ? [R: real] :
% 0.29/0.45 ( X
% 0.29/0.45 = ( extended_ereal2 @ R ) )
% 0.29/0.45 => ( Xa3 != extend1289208545_ereal ) )
% 0.29/0.45 => ( ( ? [R: real] :
% 0.29/0.45 ( X
% 0.29/0.45 = ( extended_ereal2 @ R ) )
% 0.29/0.45 => ( Xa3
% 0.29/0.45 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ! [R: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 != ( extended_ereal2 @ R ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( Xa3 != extend1289208545_ereal ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( Xa3
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ! [R: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 != ( extended_ereal2 @ R ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( Xa3 != extend1289208545_ereal ) )
% 0.29/0.46 => ~ ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( Xa3
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal2_cases
% 0.29/0.46 thf(fact_283_ereal3__cases,axiom,
% 0.29/0.46 ! [X: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 0.29/0.46 ( ( ? [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ? [Ra: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 = ( extended_ereal2 @ Ra ) )
% 0.29/0.46 => ! [Rb: real] :
% 0.29/0.46 ( Xb
% 0.29/0.46 != ( extended_ereal2 @ Rb ) ) ) )
% 0.29/0.46 => ( ( ? [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ? [Ra: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 = ( extended_ereal2 @ Ra ) )
% 0.29/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ? [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ? [Ra: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 = ( extended_ereal2 @ Ra ) )
% 0.29/0.46 => ( Xb
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.29/0.46 => ( ( ? [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.29/0.46 => ! [Ra: real] :
% 0.29/0.46 ( Xb
% 0.29/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.29/0.46 => ( ( ? [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.29/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ? [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.29/0.46 => ( Xb
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.29/0.46 => ( ( ? [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( Xa3
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ! [Ra: real] :
% 0.29/0.46 ( Xb
% 0.29/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.29/0.46 => ( ( ? [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( Xa3
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ? [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( Xa3
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( Xb
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( ? [R: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ! [Ra: real] :
% 0.29/0.46 ( Xb
% 0.29/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( ? [R: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( ? [R: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( Xb
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.29/0.46 => ! [R: real] :
% 0.29/0.46 ( Xb
% 0.29/0.46 != ( extended_ereal2 @ R ) ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.29/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.29/0.46 => ( Xb
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( ( Xa3
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ! [R: real] :
% 0.29/0.46 ( Xb
% 0.29/0.46 != ( extended_ereal2 @ R ) ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( ( Xa3
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ( X = extend1289208545_ereal )
% 0.29/0.46 => ( ( Xa3
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( Xb
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ? [R: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ! [Ra: real] :
% 0.29/0.46 ( Xb
% 0.29/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ? [R: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ? [R: real] :
% 0.29/0.46 ( Xa3
% 0.29/0.46 = ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( Xb
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.29/0.46 => ! [R: real] :
% 0.29/0.46 ( Xb
% 0.29/0.46 != ( extended_ereal2 @ R ) ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.29/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.29/0.46 => ( Xb
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ( Xa3
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ! [R: real] :
% 0.29/0.46 ( Xb
% 0.29/0.46 != ( extended_ereal2 @ R ) ) ) )
% 0.29/0.46 => ( ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ( Xa3
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.29/0.46 => ~ ( ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ( Xa3
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( Xb
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal3_cases
% 0.29/0.46 thf(fact_284_ereal__ex__split,axiom,
% 0.29/0.46 ( ( ^ [P3: extended_ereal > $o] :
% 0.29/0.46 ? [X7: extended_ereal] : ( P3 @ X7 ) )
% 0.29/0.46 = ( ^ [P4: extended_ereal > $o] :
% 0.29/0.46 ( ( P4 @ extend1289208545_ereal )
% 0.29/0.46 | ? [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
% 0.29/0.46 | ( P4 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_ex_split
% 0.29/0.46 thf(fact_285_abs__ereal_Ocases,axiom,
% 0.29/0.46 ! [X: extended_ereal] :
% 0.29/0.46 ( ! [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 != ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( X
% 0.29/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( X = extend1289208545_ereal ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_ereal.cases
% 0.29/0.46 thf(fact_286_ereal__all__split,axiom,
% 0.29/0.46 ( ( ^ [P3: extended_ereal > $o] :
% 0.29/0.46 ! [X7: extended_ereal] : ( P3 @ X7 ) )
% 0.29/0.46 = ( ^ [P4: extended_ereal > $o] :
% 0.29/0.46 ( ( P4 @ extend1289208545_ereal )
% 0.29/0.46 & ! [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
% 0.29/0.46 & ( P4 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_all_split
% 0.29/0.46 thf(fact_287_abs__ereal_Oinduct,axiom,
% 0.29/0.46 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.29/0.46 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ( P @ extend1289208545_ereal )
% 0.29/0.46 => ( P @ A0 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % abs_ereal.induct
% 0.29/0.46 thf(fact_288_less__ereal_Oinduct,axiom,
% 0.29/0.46 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.29/0.46 ( ! [X4: real,Y2: real] : ( P @ ( extended_ereal2 @ X4 ) @ ( extended_ereal2 @ Y2 ) )
% 0.29/0.46 => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
% 0.29/0.46 => ( ! [A5: extended_ereal] : ( P @ A5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) @ extend1289208545_ereal )
% 0.29/0.46 => ( ! [R: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 0.29/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_ereal.induct
% 0.29/0.46 thf(fact_289_plus__ereal_Oinduct,axiom,
% 0.29/0.46 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.29/0.46 ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
% 0.29/0.46 => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
% 0.29/0.46 => ( ! [A5: extended_ereal] : ( P @ A5 @ extend1289208545_ereal )
% 0.29/0.46 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ! [P5: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P5 ) )
% 0.29/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % plus_ereal.induct
% 0.29/0.46 thf(fact_290_times__ereal_Oinduct,axiom,
% 0.29/0.46 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.29/0.46 ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
% 0.29/0.46 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ extend1289208545_ereal )
% 0.29/0.46 => ( ! [R: real] : ( P @ extend1289208545_ereal @ ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ! [R: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( P @ extend1289208545_ereal @ extend1289208545_ereal )
% 0.29/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 0.29/0.46 => ( ( P @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % times_ereal.induct
% 0.29/0.46 thf(fact_291_real__of__ereal_Ocases,axiom,
% 0.29/0.46 ! [X: extended_ereal] :
% 0.29/0.46 ( ! [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 != ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( X != extend1289208545_ereal )
% 0.29/0.46 => ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % real_of_ereal.cases
% 0.29/0.46 thf(fact_292_real__of__ereal_Oinduct,axiom,
% 0.29/0.46 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.29/0.46 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( P @ extend1289208545_ereal )
% 0.29/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 => ( P @ A0 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % real_of_ereal.induct
% 0.29/0.46 thf(fact_293_domain__Epigraph__aux,axiom,
% 0.29/0.46 ! [X: extended_ereal] :
% 0.29/0.46 ( ( X != extend1289208545_ereal )
% 0.29/0.46 => ? [R: real] : ( ord_le824540014_ereal @ X @ ( extended_ereal2 @ R ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % domain_Epigraph_aux
% 0.29/0.46 thf(fact_294_ereal__top,axiom,
% 0.29/0.46 ! [X: extended_ereal] :
% 0.29/0.46 ( ! [B6: real] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B6 ) @ X )
% 0.29/0.46 => ( X = extend1289208545_ereal ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_top
% 0.29/0.46 thf(fact_295_Infty__neq__0_I3_J,axiom,
% 0.29/0.46 ( ( uminus1208298309_ereal @ extend1289208545_ereal )
% 0.29/0.46 != zero_z163181189_ereal ) ).
% 0.29/0.46
% 0.29/0.46 % Infty_neq_0(3)
% 0.29/0.46 thf(fact_296_ereal_Oinduct,axiom,
% 0.29/0.46 ! [P: extended_ereal > $o,Ereal: extended_ereal] :
% 0.29/0.46 ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) )
% 0.29/0.46 => ( ( P @ extended_PInfty )
% 0.29/0.46 => ( ( P @ extended_MInfty )
% 0.29/0.46 => ( P @ Ereal ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal.induct
% 0.29/0.46 thf(fact_297_ereal_Oexhaust,axiom,
% 0.29/0.46 ! [Y4: extended_ereal] :
% 0.29/0.46 ( ! [X12: real] :
% 0.29/0.46 ( Y4
% 0.29/0.46 != ( extended_ereal2 @ X12 ) )
% 0.29/0.46 => ( ( Y4 != extended_PInfty )
% 0.29/0.46 => ( Y4 = extended_MInfty ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal.exhaust
% 0.29/0.46 thf(fact_298_uminus__ereal_Ocases,axiom,
% 0.29/0.46 ! [X: extended_ereal] :
% 0.29/0.46 ( ! [R: real] :
% 0.29/0.46 ( X
% 0.29/0.46 != ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( X != extended_PInfty )
% 0.29/0.46 => ( X = extended_MInfty ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % uminus_ereal.cases
% 0.29/0.46 thf(fact_299_uminus__ereal_Oinduct,axiom,
% 0.29/0.46 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.29/0.46 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 0.29/0.46 => ( ( P @ extended_PInfty )
% 0.29/0.46 => ( ( P @ extended_MInfty )
% 0.29/0.46 => ( P @ A0 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % uminus_ereal.induct
% 0.29/0.46 thf(fact_300_ereal__bot,axiom,
% 0.29/0.46 ! [X: extended_ereal] :
% 0.29/0.46 ( ! [B6: real] : ( ord_le824540014_ereal @ X @ ( extended_ereal2 @ B6 ) )
% 0.29/0.46 => ( X
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_bot
% 0.29/0.46 thf(fact_301_le__numeral__extra_I3_J,axiom,
% 0.29/0.46 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.29/0.46
% 0.29/0.46 % le_numeral_extra(3)
% 0.29/0.46 thf(fact_302_ereal__less__eq_I4_J,axiom,
% 0.29/0.46 ! [R2: real] :
% 0.29/0.46 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R2 ) @ zero_z163181189_ereal )
% 0.29/0.46 = ( ord_less_eq_real @ R2 @ zero_zero_real ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_less_eq(4)
% 0.29/0.46 thf(fact_303_ereal__less__eq_I5_J,axiom,
% 0.29/0.46 ! [R2: real] :
% 0.29/0.46 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extended_ereal2 @ R2 ) )
% 0.29/0.46 = ( ord_less_eq_real @ zero_zero_real @ R2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_less_eq(5)
% 0.29/0.46 thf(fact_304_ereal__eq__0_I2_J,axiom,
% 0.29/0.46 ! [R2: real] :
% 0.29/0.46 ( ( zero_z163181189_ereal
% 0.29/0.46 = ( extended_ereal2 @ R2 ) )
% 0.29/0.46 = ( R2 = zero_zero_real ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_eq_0(2)
% 0.29/0.46 thf(fact_305_ereal__eq__0_I1_J,axiom,
% 0.29/0.46 ! [R2: real] :
% 0.29/0.46 ( ( ( extended_ereal2 @ R2 )
% 0.29/0.46 = zero_z163181189_ereal )
% 0.29/0.46 = ( R2 = zero_zero_real ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_eq_0(1)
% 0.29/0.46 thf(fact_306_zero__ereal__def,axiom,
% 0.29/0.46 ( zero_z163181189_ereal
% 0.29/0.46 = ( extended_ereal2 @ zero_zero_real ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_ereal_def
% 0.29/0.46 thf(fact_307_real__eq__0__iff__le__ge__0,axiom,
% 0.29/0.46 ! [X: real] :
% 0.29/0.46 ( ( X = zero_zero_real )
% 0.29/0.46 = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.46 & ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % real_eq_0_iff_le_ge_0
% 0.29/0.46 thf(fact_308_ereal__divide__ereal,axiom,
% 0.29/0.46 ! [R2: real] :
% 0.29/0.46 ( ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.29/0.46 => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
% 0.29/0.46 = extend1289208545_ereal ) )
% 0.29/0.46 & ( ~ ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.29/0.46 => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
% 0.29/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_divide_ereal
% 0.29/0.46 thf(fact_309_ereal__uminus__divide,axiom,
% 0.29/0.46 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.29/0.46 ( ( divide595620860_ereal @ ( uminus1208298309_ereal @ X ) @ Y4 )
% 0.29/0.46 = ( uminus1208298309_ereal @ ( divide595620860_ereal @ X @ Y4 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_uminus_divide
% 0.29/0.46 thf(fact_310_ereal__divide__zero__left,axiom,
% 0.29/0.46 ! [A3: extended_ereal] :
% 0.29/0.46 ( ( divide595620860_ereal @ zero_z163181189_ereal @ A3 )
% 0.29/0.46 = zero_z163181189_ereal ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_divide_zero_left
% 0.29/0.46 thf(fact_311_ereal__divide__Infty_I1_J,axiom,
% 0.29/0.46 ! [X: extended_ereal] :
% 0.29/0.46 ( ( divide595620860_ereal @ X @ extend1289208545_ereal )
% 0.29/0.46 = zero_z163181189_ereal ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_divide_Infty(1)
% 0.29/0.46 thf(fact_312_ereal__divide__Infty_I2_J,axiom,
% 0.29/0.46 ! [X: extended_ereal] :
% 0.29/0.46 ( ( divide595620860_ereal @ X @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.29/0.46 = zero_z163181189_ereal ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_divide_Infty(2)
% 0.29/0.46 thf(fact_313_zero__le__divide__ereal,axiom,
% 0.29/0.46 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.46 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ A3 )
% 0.29/0.46 => ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ B3 )
% 0.29/0.46 => ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( divide595620860_ereal @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_divide_ereal
% 0.29/0.46 thf(fact_314_div__0,axiom,
% 0.29/0.46 ! [A3: real] :
% 0.29/0.46 ( ( divide_divide_real @ zero_zero_real @ A3 )
% 0.29/0.46 = zero_zero_real ) ).
% 0.29/0.46
% 0.29/0.46 % div_0
% 0.29/0.46 thf(fact_315_div__by__0,axiom,
% 0.29/0.46 ! [A3: real] :
% 0.29/0.46 ( ( divide_divide_real @ A3 @ zero_zero_real )
% 0.29/0.46 = zero_zero_real ) ).
% 0.29/0.46
% 0.29/0.46 % div_by_0
% 0.29/0.46 thf(fact_316_division__ring__divide__zero,axiom,
% 0.29/0.46 ! [A3: real] :
% 0.29/0.46 ( ( divide_divide_real @ A3 @ zero_zero_real )
% 0.29/0.46 = zero_zero_real ) ).
% 0.29/0.46
% 0.29/0.46 % division_ring_divide_zero
% 0.29/0.46 thf(fact_317_divide__eq__0__iff,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( ( divide_divide_real @ A3 @ B3 )
% 0.29/0.46 = zero_zero_real )
% 0.29/0.46 = ( ( A3 = zero_zero_real )
% 0.29/0.46 | ( B3 = zero_zero_real ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_eq_0_iff
% 0.29/0.46 thf(fact_318_divide__cancel__left,axiom,
% 0.29/0.46 ! [C: real,A3: real,B3: real] :
% 0.29/0.46 ( ( ( divide_divide_real @ C @ A3 )
% 0.29/0.46 = ( divide_divide_real @ C @ B3 ) )
% 0.29/0.46 = ( ( C = zero_zero_real )
% 0.29/0.46 | ( A3 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_cancel_left
% 0.29/0.46 thf(fact_319_divide__cancel__right,axiom,
% 0.29/0.46 ! [A3: real,C: real,B3: real] :
% 0.29/0.46 ( ( ( divide_divide_real @ A3 @ C )
% 0.29/0.46 = ( divide_divide_real @ B3 @ C ) )
% 0.29/0.46 = ( ( C = zero_zero_real )
% 0.29/0.46 | ( A3 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_cancel_right
% 0.29/0.46 thf(fact_320_minus__divide__left,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
% 0.29/0.46 = ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_divide_left
% 0.29/0.46 thf(fact_321_minus__divide__divide,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
% 0.29/0.46 = ( divide_divide_real @ A3 @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_divide_divide
% 0.29/0.46 thf(fact_322_minus__divide__right,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
% 0.29/0.46 = ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_divide_right
% 0.29/0.46 thf(fact_323_divide__right__mono__neg,axiom,
% 0.29/0.46 ! [A3: real,B3: real,C: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.46 => ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( divide_divide_real @ A3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_right_mono_neg
% 0.29/0.46 thf(fact_324_divide__nonpos__nonpos,axiom,
% 0.29/0.46 ! [X: real,Y4: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.46 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 0.29/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_nonpos
% 0.29/0.46 thf(fact_325_divide__nonpos__nonneg,axiom,
% 0.29/0.46 ! [X: real,Y4: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.29/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.29/0.46 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonpos_nonneg
% 0.29/0.46 thf(fact_326_divide__nonneg__nonpos,axiom,
% 0.29/0.46 ! [X: real,Y4: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.46 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 0.29/0.46 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_nonpos
% 0.29/0.46 thf(fact_327_divide__nonneg__nonneg,axiom,
% 0.29/0.46 ! [X: real,Y4: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.29/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.29/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_nonneg_nonneg
% 0.29/0.46 thf(fact_328_zero__le__divide__iff,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A3 @ B3 ) )
% 0.29/0.46 = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
% 0.29/0.46 & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
% 0.29/0.46 | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
% 0.29/0.46 & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % zero_le_divide_iff
% 0.29/0.46 thf(fact_329_divide__right__mono,axiom,
% 0.29/0.46 ! [A3: real,B3: real,C: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.46 => ( ord_less_eq_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_right_mono
% 0.29/0.46 thf(fact_330_divide__le__0__iff,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ B3 ) @ zero_zero_real )
% 0.29/0.46 = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
% 0.29/0.46 & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
% 0.29/0.46 | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
% 0.29/0.46 & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % divide_le_0_iff
% 0.29/0.46 thf(fact_331_nonzero__minus__divide__divide,axiom,
% 0.29/0.46 ! [B3: real,A3: real] :
% 0.29/0.46 ( ( B3 != zero_zero_real )
% 0.29/0.46 => ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
% 0.29/0.46 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nonzero_minus_divide_divide
% 0.29/0.46 thf(fact_332_nonzero__minus__divide__right,axiom,
% 0.29/0.46 ! [B3: real,A3: real] :
% 0.29/0.46 ( ( B3 != zero_zero_real )
% 0.29/0.46 => ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
% 0.29/0.46 = ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nonzero_minus_divide_right
% 0.29/0.46 thf(fact_333_real__of__ereal__le__0,axiom,
% 0.29/0.46 ! [X: extended_ereal] :
% 0.29/0.46 ( ( ord_less_eq_real @ ( extend1716541707_ereal @ X ) @ zero_zero_real )
% 0.29/0.46 = ( ( ord_le824540014_ereal @ X @ zero_z163181189_ereal )
% 0.29/0.46 | ( X = extend1289208545_ereal ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % real_of_ereal_le_0
% 0.29/0.46 thf(fact_334_ereal__divide,axiom,
% 0.29/0.46 ! [P2: real,R2: real] :
% 0.29/0.46 ( ( ( P2 = zero_zero_real )
% 0.29/0.46 => ( ( divide595620860_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 0.29/0.46 = ( times_1966848393_ereal @ ( extended_ereal2 @ R2 ) @ extend1289208545_ereal ) ) )
% 0.29/0.46 & ( ( P2 != zero_zero_real )
% 0.29/0.46 => ( ( divide595620860_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 0.29/0.46 = ( extended_ereal2 @ ( divide_divide_real @ R2 @ P2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_divide
% 0.29/0.46 thf(fact_335_mult__zero__left,axiom,
% 0.29/0.46 ! [A3: real] :
% 0.29/0.46 ( ( times_times_real @ zero_zero_real @ A3 )
% 0.29/0.46 = zero_zero_real ) ).
% 0.29/0.46
% 0.29/0.46 % mult_zero_left
% 0.29/0.46 thf(fact_336_mult__zero__right,axiom,
% 0.29/0.46 ! [A3: real] :
% 0.29/0.46 ( ( times_times_real @ A3 @ zero_zero_real )
% 0.29/0.46 = zero_zero_real ) ).
% 0.29/0.46
% 0.29/0.46 % mult_zero_right
% 0.29/0.46 thf(fact_337_mult__eq__0__iff,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( ( times_times_real @ A3 @ B3 )
% 0.29/0.46 = zero_zero_real )
% 0.29/0.46 = ( ( A3 = zero_zero_real )
% 0.29/0.46 | ( B3 = zero_zero_real ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult_eq_0_iff
% 0.29/0.46 thf(fact_338_mult__cancel__left,axiom,
% 0.29/0.46 ! [C: real,A3: real,B3: real] :
% 0.29/0.46 ( ( ( times_times_real @ C @ A3 )
% 0.29/0.46 = ( times_times_real @ C @ B3 ) )
% 0.29/0.46 = ( ( C = zero_zero_real )
% 0.29/0.46 | ( A3 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult_cancel_left
% 0.29/0.46 thf(fact_339_mult__cancel__right,axiom,
% 0.29/0.46 ! [A3: real,C: real,B3: real] :
% 0.29/0.46 ( ( ( times_times_real @ A3 @ C )
% 0.29/0.46 = ( times_times_real @ B3 @ C ) )
% 0.29/0.46 = ( ( C = zero_zero_real )
% 0.29/0.46 | ( A3 = B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult_cancel_right
% 0.29/0.46 thf(fact_340_mult__minus__right,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) )
% 0.29/0.46 = ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult_minus_right
% 0.29/0.46 thf(fact_341_minus__mult__minus,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
% 0.29/0.46 = ( times_times_real @ A3 @ B3 ) ) ).
% 0.29/0.46
% 0.29/0.46 % minus_mult_minus
% 0.29/0.46 thf(fact_342_mult__minus__left,axiom,
% 0.29/0.46 ! [A3: real,B3: real] :
% 0.29/0.46 ( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
% 0.29/0.46 = ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult_minus_left
% 0.29/0.46 thf(fact_343_ereal__mult__minus__right,axiom,
% 0.29/0.46 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.46 ( ( times_1966848393_ereal @ A3 @ ( uminus1208298309_ereal @ B3 ) )
% 0.29/0.46 = ( uminus1208298309_ereal @ ( times_1966848393_ereal @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_mult_minus_right
% 0.29/0.46 thf(fact_344_ereal__mult__minus__left,axiom,
% 0.29/0.46 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.46 ( ( times_1966848393_ereal @ ( uminus1208298309_ereal @ A3 ) @ B3 )
% 0.29/0.46 = ( uminus1208298309_ereal @ ( times_1966848393_ereal @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_mult_minus_left
% 0.29/0.46 thf(fact_345_ereal__mult__zero,axiom,
% 0.29/0.46 ! [A3: extended_ereal] :
% 0.29/0.46 ( ( times_1966848393_ereal @ A3 @ zero_z163181189_ereal )
% 0.29/0.46 = zero_z163181189_ereal ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_mult_zero
% 0.29/0.46 thf(fact_346_ereal__zero__mult,axiom,
% 0.29/0.46 ! [A3: extended_ereal] :
% 0.29/0.46 ( ( times_1966848393_ereal @ zero_z163181189_ereal @ A3 )
% 0.29/0.46 = zero_z163181189_ereal ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_zero_mult
% 0.29/0.46 thf(fact_347_ereal__zero__times,axiom,
% 0.29/0.46 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.29/0.46 ( ( ( times_1966848393_ereal @ A3 @ B3 )
% 0.29/0.46 = zero_z163181189_ereal )
% 0.29/0.46 = ( ( A3 = zero_z163181189_ereal )
% 0.29/0.46 | ( B3 = zero_z163181189_ereal ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_zero_times
% 0.29/0.46 thf(fact_348_ereal__times__divide__eq__left,axiom,
% 0.29/0.46 ! [B3: extended_ereal,C: extended_ereal,A3: extended_ereal] :
% 0.29/0.46 ( ( times_1966848393_ereal @ ( divide595620860_ereal @ B3 @ C ) @ A3 )
% 0.29/0.46 = ( divide595620860_ereal @ ( times_1966848393_ereal @ B3 @ A3 ) @ C ) ) ).
% 0.29/0.46
% 0.29/0.46 % ereal_times_divide_eq_left
% 0.29/0.46 thf(fact_349_mult__divide__mult__cancel__left__if,axiom,
% 0.29/0.46 ! [C: real,A3: real,B3: real] :
% 0.29/0.46 ( ( ( C = zero_zero_real )
% 0.29/0.46 => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
% 0.29/0.46 = zero_zero_real ) )
% 0.29/0.46 & ( ( C != zero_zero_real )
% 0.29/0.46 => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
% 0.29/0.46 = ( divide_divide_real @ A3 @ B3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mult_divide_mult_cancel_left_if
% 0.29/0.46 thf(fact_350_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.29/0.46 ! [C: real,A3: real,B3: real] :
% 0.29/0.46 ( ( C != zero_zero_real )
% 0.29/0.46 => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
% 0.29/0.46 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nonzero_mult_divide_mult_cancel_left
% 0.29/0.46 thf(fact_351_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.29/0.46 ! [C: real,A3: real,B3: real] :
% 0.29/0.46 ( ( C != zero_zero_real )
% 0.29/0.46 => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ B3 @ C ) )
% 0.29/0.46 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nonzero_mult_divide_mult_cancel_left2
% 0.29/0.46 thf(fact_352_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.29/0.46 ! [C: real,A3: real,B3: real] :
% 0.29/0.46 ( ( C != zero_zero_real )
% 0.29/0.46 => ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
% 0.29/0.46 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nonzero_mult_divide_mult_cancel_right
% 0.29/0.46 thf(fact_353_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.29/0.46 ! [C: real,A3: real,B3: real] :
% 0.29/0.46 ( ( C != zero_zero_real )
% 0.29/0.46 => ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ C @ B3 ) )
% 0.29/0.46 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nonzero_mult_divide_mult_cancel_right2
% 0.29/0.46
% 0.29/0.46 % Conjectures (1)
% 0.29/0.46 thf(conj_0,conjecture,
% 0.29/0.46 ( ( lower_311861424x_on_a @ top_top_set_a @ f )
% 0.29/0.46 = ( convex_a @ ( lower_1391529426main_a @ f ) ) ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 0.30/1.60 ---- Embedded in THF ---
% 0.30/1.60 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.60 %%% Generated on Tue Feb 24 08:26:15 EST 2026
% 0.30/1.60 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.60 %%% Logic specification used:
% 0.30/1.60 %%% thf(spec, logic, $$dhol).
% 0.30/1.60
% 0.30/1.60 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Lwp0UiQMvY/DTF2DTF9742.p
% See solution above
% 0.30/1.60 ------------------------
% 0.30/1.63 ---- Cleaned THF ---
% 0.30/1.63 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,
% 0.30/1.63 set_Pr1158285382l_real: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Extended____Real__Oereal_Mt__Real__Oreal_J_J,type,
% 0.30/1.63 set_Pr597221087l_real: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J_J,type,
% 0.30/1.63 set_se1849684334_ereal: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
% 0.30/1.63 set_Pr1928503567a_real: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
% 0.30/1.63 set_se767749006_ereal: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_I_062_It__Extended____Real__Oereal_M_Eo_J_J,type,
% 0.30/1.63 set_Extended_ereal_o: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
% 0.30/1.63 set_set_set_a: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.30/1.63 product_prod_a_real: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.30/1.63 set_Extended_ereal: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.30/1.63 set_set_a: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
% 0.30/1.63 set_a_o: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.30/1.63 set_real: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Extended____Real__Oereal,type,
% 0.30/1.63 extended_ereal: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.30/1.63 set_a: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_t__Real__Oreal,type,
% 0.30/1.63 real: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(ty_n_tf__a,type,
% 0.30/1.63 a: $tType ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
% 0.30/1.63 comple179807490real_o: set_Extended_ereal_o > extended_ereal > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
% 0.30/1.63 complete_Sup_Sup_a_o: set_a_o > a > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Real__Oereal,type,
% 0.30/1.63 comple1161760187_ereal: set_Extended_ereal > extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
% 0.30/1.63 comple2129349247p_real: set_real > real ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.30/1.63 comple767846299_ereal: set_se767749006_ereal > set_Extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Extended____Real__Oereal_J_J,type,
% 0.30/1.63 comple1517911419_ereal: set_se1849684334_ereal > set_se767749006_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
% 0.30/1.63 comple968993579_set_a: set_set_set_a > set_set_a ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
% 0.30/1.63 comple1766734283_set_a: set_set_a > set_a ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Convex_Oconvex_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
% 0.30/1.63 convex291263275l_real: set_Pr1158285382l_real > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Convex_Oconvex_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.30/1.63 convex78185044a_real: set_Pr1928503567a_real > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Convex_Oconvex_001tf__a,type,
% 0.30/1.63 convex_a: set_a > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
% 0.30/1.63 extend1289208545_ereal: extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Extended__Real_Oereal_OMInfty,type,
% 0.30/1.63 extended_MInfty: extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Extended__Real_Oereal_OPInfty,type,
% 0.30/1.63 extended_PInfty: extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 0.30/1.63 extended_ereal2: real > extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Extended__Real_Oreal__of__ereal,type,
% 0.30/1.63 extend1716541707_ereal: extended_ereal > real ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Real__Oereal,type,
% 0.30/1.63 times_1966848393_ereal: extended_ereal > extended_ereal > extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 0.30/1.63 times_times_real: real > real > real ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
% 0.30/1.63 uminus1208298309_ereal: extended_ereal > extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.30/1.63 uminus_uminus_real: real > real ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.30/1.63 uminus360668453_ereal: set_Extended_ereal > set_Extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
% 0.30/1.63 uminus_uminus_set_a: set_a > set_a ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
% 0.30/1.63 zero_z163181189_ereal: extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.30/1.63 zero_zero_real: real ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001t__Extended____Real__Oereal,type,
% 0.30/1.63 lower_331963542_ereal: set_Extended_ereal > ( extended_ereal > extended_ereal ) > set_Pr597221087l_real ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.30/1.63 lower_870348689a_real: set_Pr1928503567a_real > ( product_prod_a_real > extended_ereal ) > set_Pr1158285382l_real ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_OEpigraph_001tf__a,type,
% 0.30/1.63 lower_930854854raph_a: set_a > ( a > extended_ereal ) > set_Pr1928503567a_real ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oaffine__on_001tf__a,type,
% 0.30/1.63 lower_128423320e_on_a: set_a > ( a > extended_ereal ) > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconcave__on_001tf__a,type,
% 0.30/1.63 lower_1023700310e_on_a: set_a > ( a > extended_ereal ) > $o ).
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% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
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% 0.30/1.63
% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_001tf__a,type,
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% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Odomain_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
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% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Odomain_001tf__a,type,
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% 0.30/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ofinite__on_001tf__a,type,
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% 0.30/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
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% 0.30/1.63
% 0.30/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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% 0.30/1.63
% 0.30/1.63 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.30/1.63 collec1714955950a_real: ( product_prod_a_real > $o ) > set_Pr1928503567a_real ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.30/1.63 collect_a: ( a > $o ) > set_a ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_member_001t__Extended____Real__Oereal,type,
% 0.30/1.63 member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.30/1.63 member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_member_001t__Real__Oreal,type,
% 0.30/1.63 member_real: real > set_real > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_member_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.30/1.63 member450560855_ereal: set_Extended_ereal > set_se767749006_ereal > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.30/1.63 member_set_a: set_a > set_set_a > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_c_member_001tf__a,type,
% 0.30/1.63 member_a: a > set_a > $o ).
% 0.30/1.63
% 0.30/1.63 thf(sy_v_f,type,
% 0.30/1.63 f: a > extended_ereal ).
% 0.30/1.63
% 0.30/1.63 thf(fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_AUNIV_Af_092_060close_062,axiom,
% 0.30/1.63 ( ( convex_a @ ( lower_1391529426main_a @ f ) )
% 0.30/1.63 => ( lower_311861424x_on_a @ top_top_set_a @ f ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_1_convex__domain,axiom,
% 0.30/1.63 ! [F: product_prod_a_real > extended_ereal] :
% 0.30/1.63 ( ( lower_591730471a_real @ top_to2138011583a_real @ F )
% 0.30/1.63 => ( convex78185044a_real @ ( lower_1359521925a_real @ F ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_2_convex__domain,axiom,
% 0.30/1.63 ! [F: a > extended_ereal] :
% 0.30/1.63 ( ( lower_311861424x_on_a @ top_top_set_a @ F )
% 0.30/1.63 => ( convex_a @ ( lower_1391529426main_a @ F ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_3_convex__on__domain,axiom,
% 0.30/1.63 ! [F: a > extended_ereal] :
% 0.30/1.63 ( ( lower_311861424x_on_a @ ( lower_1391529426main_a @ F ) @ F )
% 0.30/1.63 = ( lower_311861424x_on_a @ top_top_set_a @ F ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_4_convex__on__domain2,axiom,
% 0.30/1.63 ! [F: a > extended_ereal] :
% 0.30/1.63 ( ( lower_311861424x_on_a @ ( lower_1391529426main_a @ F ) @ F )
% 0.30/1.63 = ( ! [S: set_a] : ( lower_311861424x_on_a @ S @ F ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_5_convex__on__ereal__univ,axiom,
% 0.30/1.63 ! [F: a > extended_ereal] :
% 0.30/1.63 ( ( lower_311861424x_on_a @ top_top_set_a @ F )
% 0.30/1.63 = ( ! [S: set_a] : ( lower_311861424x_on_a @ S @ F ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_6_convex__UNIV,axiom,
% 0.30/1.63 convex78185044a_real @ top_to2138011583a_real ).
% 0.30/1.63
% 0.30/1.63 thf(fact_7_convex__UNIV,axiom,
% 0.30/1.63 convex_a @ top_top_set_a ).
% 0.30/1.63
% 0.30/1.63 thf(fact_8_UNIV__I,axiom,
% 0.30/1.63 ! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_9_UNIV__I,axiom,
% 0.30/1.63 ! [X: set_Extended_ereal] : ( member450560855_ereal @ X @ top_to692740318_ereal ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_10_UNIV__I,axiom,
% 0.30/1.63 ! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_11_UNIV__I,axiom,
% 0.30/1.63 ! [X: extended_ereal] : ( member1900190071_ereal @ X @ top_to1767659262_ereal ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_12_UNIV__I,axiom,
% 0.30/1.63 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_13_iso__tuple__UNIV__I,axiom,
% 0.30/1.63 ! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_14_iso__tuple__UNIV__I,axiom,
% 0.30/1.63 ! [X: set_Extended_ereal] : ( member450560855_ereal @ X @ top_to692740318_ereal ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_15_iso__tuple__UNIV__I,axiom,
% 0.30/1.63 ! [X: set_a] : ( member_set_a @ X @ top_top_set_set_a ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_16_iso__tuple__UNIV__I,axiom,
% 0.30/1.63 ! [X: extended_ereal] : ( member1900190071_ereal @ X @ top_to1767659262_ereal ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_17_iso__tuple__UNIV__I,axiom,
% 0.30/1.63 ! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_18_top__apply,axiom,
% 0.30/1.63 ( top_to398855007real_o
% 0.30/1.63 = ( ^ [X2: extended_ereal] : top_top_o ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_19_top__apply,axiom,
% 0.30/1.63 ( top_top_a_o
% 0.30/1.63 = ( ^ [X2: a] : top_top_o ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_20_assms,axiom,
% 0.30/1.63 ! [X3: a] :
% 0.30/1.63 ( ( ( f @ X3 )
% 0.30/1.63 = extend1289208545_ereal )
% 0.30/1.63 | ( ( f @ X3 )
% 0.30/1.63 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_21_UNIV__eq__I,axiom,
% 0.30/1.63 ! [A: set_real] :
% 0.30/1.63 ( ! [X4: real] : ( member_real @ X4 @ A )
% 0.30/1.63 => ( top_top_set_real = A ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_22_UNIV__eq__I,axiom,
% 0.30/1.63 ! [A: set_se767749006_ereal] :
% 0.30/1.63 ( ! [X4: set_Extended_ereal] : ( member450560855_ereal @ X4 @ A )
% 0.30/1.63 => ( top_to692740318_ereal = A ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_23_UNIV__eq__I,axiom,
% 0.30/1.63 ! [A: set_set_a] :
% 0.30/1.63 ( ! [X4: set_a] : ( member_set_a @ X4 @ A )
% 0.30/1.63 => ( top_top_set_set_a = A ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_24_UNIV__eq__I,axiom,
% 0.30/1.63 ! [A: set_a] :
% 0.30/1.63 ( ! [X4: a] : ( member_a @ X4 @ A )
% 0.30/1.63 => ( top_top_set_a = A ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_25_UNIV__eq__I,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal] :
% 0.30/1.63 ( ! [X4: extended_ereal] : ( member1900190071_ereal @ X4 @ A )
% 0.30/1.63 => ( top_to1767659262_ereal = A ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_26_UNIV__witness,axiom,
% 0.30/1.63 ? [X4: real] : ( member_real @ X4 @ top_top_set_real ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_27_UNIV__witness,axiom,
% 0.30/1.63 ? [X4: set_Extended_ereal] : ( member450560855_ereal @ X4 @ top_to692740318_ereal ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_28_UNIV__witness,axiom,
% 0.30/1.63 ? [X4: set_a] : ( member_set_a @ X4 @ top_top_set_set_a ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_29_UNIV__witness,axiom,
% 0.30/1.63 ? [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_30_UNIV__witness,axiom,
% 0.30/1.63 ? [X4: extended_ereal] : ( member1900190071_ereal @ X4 @ top_to1767659262_ereal ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_31_convex__Epigraph,axiom,
% 0.30/1.63 ! [S2: set_Pr1928503567a_real,F: product_prod_a_real > extended_ereal] :
% 0.30/1.63 ( ( convex78185044a_real @ S2 )
% 0.30/1.63 => ( ( convex291263275l_real @ ( lower_870348689a_real @ S2 @ F ) )
% 0.30/1.63 = ( lower_591730471a_real @ S2 @ F ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_32_convex__Epigraph,axiom,
% 0.30/1.63 ! [S2: set_a,F: a > extended_ereal] :
% 0.30/1.63 ( ( convex_a @ S2 )
% 0.30/1.63 => ( ( convex78185044a_real @ ( lower_930854854raph_a @ S2 @ F ) )
% 0.30/1.63 = ( lower_311861424x_on_a @ S2 @ F ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_33_convex__EpigraphI,axiom,
% 0.30/1.63 ! [S3: set_Pr1928503567a_real,F: product_prod_a_real > extended_ereal] :
% 0.30/1.63 ( ( lower_591730471a_real @ S3 @ F )
% 0.30/1.63 => ( ( convex78185044a_real @ S3 )
% 0.30/1.63 => ( convex291263275l_real @ ( lower_870348689a_real @ S3 @ F ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_34_convex__EpigraphI,axiom,
% 0.30/1.63 ! [S3: set_a,F: a > extended_ereal] :
% 0.30/1.63 ( ( lower_311861424x_on_a @ S3 @ F )
% 0.30/1.63 => ( ( convex_a @ S3 )
% 0.30/1.63 => ( convex78185044a_real @ ( lower_930854854raph_a @ S3 @ F ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_35_affine__on__def,axiom,
% 0.30/1.63 ( lower_128423320e_on_a
% 0.30/1.63 = ( ^ [S: set_a,F2: a > extended_ereal] :
% 0.30/1.63 ( ( lower_311861424x_on_a @ S @ F2 )
% 0.30/1.63 & ( lower_1023700310e_on_a @ S @ F2 )
% 0.30/1.63 & ( lower_1486931688e_on_a @ S @ F2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_36_Sup__UNIV,axiom,
% 0.30/1.63 ( ( comple1517911419_ereal @ top_to1791633086_ereal )
% 0.30/1.63 = top_to692740318_ereal ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_37_Sup__UNIV,axiom,
% 0.30/1.63 ( ( comple968993579_set_a @ top_to1486684270_set_a )
% 0.30/1.63 = top_top_set_set_a ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_38_Sup__UNIV,axiom,
% 0.30/1.63 ( ( comple179807490real_o @ top_to772344469real_o )
% 0.30/1.63 = top_to398855007real_o ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_39_Sup__UNIV,axiom,
% 0.30/1.63 ( ( complete_Sup_Sup_a_o @ top_top_set_a_o )
% 0.30/1.63 = top_top_a_o ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_40_Sup__UNIV,axiom,
% 0.30/1.63 ( ( comple1766734283_set_a @ top_top_set_set_a )
% 0.30/1.63 = top_top_set_a ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_41_Sup__UNIV,axiom,
% 0.30/1.63 ( ( comple767846299_ereal @ top_to692740318_ereal )
% 0.30/1.63 = top_to1767659262_ereal ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_42_Sup__UNIV,axiom,
% 0.30/1.63 ( ( comple1161760187_ereal @ top_to1767659262_ereal )
% 0.30/1.63 = top_to802031902_ereal ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_43_convex__on__ereal__subset,axiom,
% 0.30/1.63 ! [T: set_Pr1928503567a_real,F: product_prod_a_real > extended_ereal,S3: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( lower_591730471a_real @ T @ F )
% 0.30/1.63 => ( ( ord_le1586073967a_real @ S3 @ T )
% 0.30/1.63 => ( lower_591730471a_real @ S3 @ F ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_44_convex__on__ereal__subset,axiom,
% 0.30/1.63 ! [T: set_a,F: a > extended_ereal,S3: set_a] :
% 0.30/1.63 ( ( lower_311861424x_on_a @ T @ F )
% 0.30/1.63 => ( ( ord_less_eq_set_a @ S3 @ T )
% 0.30/1.63 => ( lower_311861424x_on_a @ S3 @ F ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_45_order__refl,axiom,
% 0.30/1.63 ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_46_order__refl,axiom,
% 0.30/1.63 ! [X: set_Extended_ereal] : ( ord_le88246606_ereal @ X @ X ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_47_order__refl,axiom,
% 0.30/1.63 ! [X: set_Pr1928503567a_real] : ( ord_le1586073967a_real @ X @ X ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_48_order__refl,axiom,
% 0.30/1.63 ! [X: extended_ereal] : ( ord_le824540014_ereal @ X @ X ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_49_order__refl,axiom,
% 0.30/1.63 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_50_subsetI,axiom,
% 0.30/1.63 ! [A: set_set_a,B: set_set_a] :
% 0.30/1.63 ( ! [X4: set_a] :
% 0.30/1.63 ( ( member_set_a @ X4 @ A )
% 0.30/1.63 => ( member_set_a @ X4 @ B ) )
% 0.30/1.63 => ( ord_le318720350_set_a @ A @ B ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_51_subsetI,axiom,
% 0.30/1.63 ! [A: set_se767749006_ereal,B: set_se767749006_ereal] :
% 0.30/1.63 ( ! [X4: set_Extended_ereal] :
% 0.30/1.63 ( ( member450560855_ereal @ X4 @ A )
% 0.30/1.63 => ( member450560855_ereal @ X4 @ B ) )
% 0.30/1.63 => ( ord_le1153389358_ereal @ A @ B ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_52_subsetI,axiom,
% 0.30/1.63 ! [A: set_real,B: set_real] :
% 0.30/1.63 ( ! [X4: real] :
% 0.30/1.63 ( ( member_real @ X4 @ A )
% 0.30/1.63 => ( member_real @ X4 @ B ) )
% 0.30/1.63 => ( ord_less_eq_set_real @ A @ B ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_53_subsetI,axiom,
% 0.30/1.63 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.30/1.63 ( ! [X4: product_prod_a_real] :
% 0.30/1.63 ( ( member1103263856a_real @ X4 @ A )
% 0.30/1.63 => ( member1103263856a_real @ X4 @ B ) )
% 0.30/1.63 => ( ord_le1586073967a_real @ A @ B ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_54_subsetI,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.30/1.63 ( ! [X4: extended_ereal] :
% 0.30/1.63 ( ( member1900190071_ereal @ X4 @ A )
% 0.30/1.63 => ( member1900190071_ereal @ X4 @ B ) )
% 0.30/1.63 => ( ord_le88246606_ereal @ A @ B ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_55_subsetI,axiom,
% 0.30/1.63 ! [A: set_a,B: set_a] :
% 0.30/1.63 ( ! [X4: a] :
% 0.30/1.63 ( ( member_a @ X4 @ A )
% 0.30/1.63 => ( member_a @ X4 @ B ) )
% 0.30/1.63 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_56_subset__antisym,axiom,
% 0.30/1.63 ! [A: set_a,B: set_a] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ A @ B )
% 0.30/1.63 => ( ( ord_less_eq_set_a @ B @ A )
% 0.30/1.63 => ( A = B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_57_subset__antisym,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_le88246606_ereal @ A @ B )
% 0.30/1.63 => ( ( ord_le88246606_ereal @ B @ A )
% 0.30/1.63 => ( A = B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_58_subset__antisym,axiom,
% 0.30/1.63 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ A @ B )
% 0.30/1.63 => ( ( ord_le1586073967a_real @ B @ A )
% 0.30/1.63 => ( A = B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_59_in__mono,axiom,
% 0.30/1.63 ! [A: set_set_a,B: set_set_a,X: set_a] :
% 0.30/1.63 ( ( ord_le318720350_set_a @ A @ B )
% 0.30/1.63 => ( ( member_set_a @ X @ A )
% 0.30/1.63 => ( member_set_a @ X @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_60_in__mono,axiom,
% 0.30/1.63 ! [A: set_se767749006_ereal,B: set_se767749006_ereal,X: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_le1153389358_ereal @ A @ B )
% 0.30/1.63 => ( ( member450560855_ereal @ X @ A )
% 0.30/1.63 => ( member450560855_ereal @ X @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_61_in__mono,axiom,
% 0.30/1.63 ! [A: set_real,B: set_real,X: real] :
% 0.30/1.63 ( ( ord_less_eq_set_real @ A @ B )
% 0.30/1.63 => ( ( member_real @ X @ A )
% 0.30/1.63 => ( member_real @ X @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_62_in__mono,axiom,
% 0.30/1.63 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real,X: product_prod_a_real] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ A @ B )
% 0.30/1.63 => ( ( member1103263856a_real @ X @ A )
% 0.30/1.63 => ( member1103263856a_real @ X @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_63_in__mono,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal,B: set_Extended_ereal,X: extended_ereal] :
% 0.30/1.63 ( ( ord_le88246606_ereal @ A @ B )
% 0.30/1.63 => ( ( member1900190071_ereal @ X @ A )
% 0.30/1.63 => ( member1900190071_ereal @ X @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_64_in__mono,axiom,
% 0.30/1.63 ! [A: set_a,B: set_a,X: a] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ A @ B )
% 0.30/1.63 => ( ( member_a @ X @ A )
% 0.30/1.63 => ( member_a @ X @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_65_subsetD,axiom,
% 0.30/1.63 ! [A: set_set_a,B: set_set_a,C: set_a] :
% 0.30/1.63 ( ( ord_le318720350_set_a @ A @ B )
% 0.30/1.63 => ( ( member_set_a @ C @ A )
% 0.30/1.63 => ( member_set_a @ C @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_66_subsetD,axiom,
% 0.30/1.63 ! [A: set_se767749006_ereal,B: set_se767749006_ereal,C: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_le1153389358_ereal @ A @ B )
% 0.30/1.63 => ( ( member450560855_ereal @ C @ A )
% 0.30/1.63 => ( member450560855_ereal @ C @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_67_subsetD,axiom,
% 0.30/1.63 ! [A: set_real,B: set_real,C: real] :
% 0.30/1.63 ( ( ord_less_eq_set_real @ A @ B )
% 0.30/1.63 => ( ( member_real @ C @ A )
% 0.30/1.63 => ( member_real @ C @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_68_subsetD,axiom,
% 0.30/1.63 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real,C: product_prod_a_real] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ A @ B )
% 0.30/1.63 => ( ( member1103263856a_real @ C @ A )
% 0.30/1.63 => ( member1103263856a_real @ C @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_69_subsetD,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal,B: set_Extended_ereal,C: extended_ereal] :
% 0.30/1.63 ( ( ord_le88246606_ereal @ A @ B )
% 0.30/1.63 => ( ( member1900190071_ereal @ C @ A )
% 0.30/1.63 => ( member1900190071_ereal @ C @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_70_subsetD,axiom,
% 0.30/1.63 ! [A: set_a,B: set_a,C: a] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ A @ B )
% 0.30/1.63 => ( ( member_a @ C @ A )
% 0.30/1.63 => ( member_a @ C @ B ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_71_equalityE,axiom,
% 0.30/1.63 ! [A: set_a,B: set_a] :
% 0.30/1.63 ( ( A = B )
% 0.30/1.63 => ~ ( ( ord_less_eq_set_a @ A @ B )
% 0.30/1.63 => ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_72_equalityE,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.30/1.63 ( ( A = B )
% 0.30/1.63 => ~ ( ( ord_le88246606_ereal @ A @ B )
% 0.30/1.63 => ~ ( ord_le88246606_ereal @ B @ A ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_73_equalityE,axiom,
% 0.30/1.63 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( A = B )
% 0.30/1.63 => ~ ( ( ord_le1586073967a_real @ A @ B )
% 0.30/1.63 => ~ ( ord_le1586073967a_real @ B @ A ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_74_subset__eq,axiom,
% 0.30/1.63 ( ord_le318720350_set_a
% 0.30/1.63 = ( ^ [A2: set_set_a,B2: set_set_a] :
% 0.30/1.63 ! [X2: set_a] :
% 0.30/1.63 ( ( member_set_a @ X2 @ A2 )
% 0.30/1.63 => ( member_set_a @ X2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_75_subset__eq,axiom,
% 0.30/1.63 ( ord_le1153389358_ereal
% 0.30/1.63 = ( ^ [A2: set_se767749006_ereal,B2: set_se767749006_ereal] :
% 0.30/1.63 ! [X2: set_Extended_ereal] :
% 0.30/1.63 ( ( member450560855_ereal @ X2 @ A2 )
% 0.30/1.63 => ( member450560855_ereal @ X2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_76_subset__eq,axiom,
% 0.30/1.63 ( ord_less_eq_set_real
% 0.30/1.63 = ( ^ [A2: set_real,B2: set_real] :
% 0.30/1.63 ! [X2: real] :
% 0.30/1.63 ( ( member_real @ X2 @ A2 )
% 0.30/1.63 => ( member_real @ X2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_77_subset__eq,axiom,
% 0.30/1.63 ( ord_le1586073967a_real
% 0.30/1.63 = ( ^ [A2: set_Pr1928503567a_real,B2: set_Pr1928503567a_real] :
% 0.30/1.63 ! [X2: product_prod_a_real] :
% 0.30/1.63 ( ( member1103263856a_real @ X2 @ A2 )
% 0.30/1.63 => ( member1103263856a_real @ X2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_78_subset__eq,axiom,
% 0.30/1.63 ( ord_le88246606_ereal
% 0.30/1.63 = ( ^ [A2: set_Extended_ereal,B2: set_Extended_ereal] :
% 0.30/1.63 ! [X2: extended_ereal] :
% 0.30/1.63 ( ( member1900190071_ereal @ X2 @ A2 )
% 0.30/1.63 => ( member1900190071_ereal @ X2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_79_subset__eq,axiom,
% 0.30/1.63 ( ord_less_eq_set_a
% 0.30/1.63 = ( ^ [A2: set_a,B2: set_a] :
% 0.30/1.63 ! [X2: a] :
% 0.30/1.63 ( ( member_a @ X2 @ A2 )
% 0.30/1.63 => ( member_a @ X2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_80_equalityD1,axiom,
% 0.30/1.63 ! [A: set_a,B: set_a] :
% 0.30/1.63 ( ( A = B )
% 0.30/1.63 => ( ord_less_eq_set_a @ A @ B ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_81_equalityD1,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.30/1.63 ( ( A = B )
% 0.30/1.63 => ( ord_le88246606_ereal @ A @ B ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_82_equalityD1,axiom,
% 0.30/1.63 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( A = B )
% 0.30/1.63 => ( ord_le1586073967a_real @ A @ B ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_83_equalityD2,axiom,
% 0.30/1.63 ! [A: set_a,B: set_a] :
% 0.30/1.63 ( ( A = B )
% 0.30/1.63 => ( ord_less_eq_set_a @ B @ A ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_84_equalityD2,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.30/1.63 ( ( A = B )
% 0.30/1.63 => ( ord_le88246606_ereal @ B @ A ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_85_equalityD2,axiom,
% 0.30/1.63 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( A = B )
% 0.30/1.63 => ( ord_le1586073967a_real @ B @ A ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_86_subset__iff,axiom,
% 0.30/1.63 ( ord_le318720350_set_a
% 0.30/1.63 = ( ^ [A2: set_set_a,B2: set_set_a] :
% 0.30/1.63 ! [T2: set_a] :
% 0.30/1.63 ( ( member_set_a @ T2 @ A2 )
% 0.30/1.63 => ( member_set_a @ T2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_87_subset__iff,axiom,
% 0.30/1.63 ( ord_le1153389358_ereal
% 0.30/1.63 = ( ^ [A2: set_se767749006_ereal,B2: set_se767749006_ereal] :
% 0.30/1.63 ! [T2: set_Extended_ereal] :
% 0.30/1.63 ( ( member450560855_ereal @ T2 @ A2 )
% 0.30/1.63 => ( member450560855_ereal @ T2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_88_subset__iff,axiom,
% 0.30/1.63 ( ord_less_eq_set_real
% 0.30/1.63 = ( ^ [A2: set_real,B2: set_real] :
% 0.30/1.63 ! [T2: real] :
% 0.30/1.63 ( ( member_real @ T2 @ A2 )
% 0.30/1.63 => ( member_real @ T2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_89_subset__iff,axiom,
% 0.30/1.63 ( ord_le1586073967a_real
% 0.30/1.63 = ( ^ [A2: set_Pr1928503567a_real,B2: set_Pr1928503567a_real] :
% 0.30/1.63 ! [T2: product_prod_a_real] :
% 0.30/1.63 ( ( member1103263856a_real @ T2 @ A2 )
% 0.30/1.63 => ( member1103263856a_real @ T2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_90_subset__iff,axiom,
% 0.30/1.63 ( ord_le88246606_ereal
% 0.30/1.63 = ( ^ [A2: set_Extended_ereal,B2: set_Extended_ereal] :
% 0.30/1.63 ! [T2: extended_ereal] :
% 0.30/1.63 ( ( member1900190071_ereal @ T2 @ A2 )
% 0.30/1.63 => ( member1900190071_ereal @ T2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_91_subset__iff,axiom,
% 0.30/1.63 ( ord_less_eq_set_a
% 0.30/1.63 = ( ^ [A2: set_a,B2: set_a] :
% 0.30/1.63 ! [T2: a] :
% 0.30/1.63 ( ( member_a @ T2 @ A2 )
% 0.30/1.63 => ( member_a @ T2 @ B2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_92_subset__refl,axiom,
% 0.30/1.63 ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_93_subset__refl,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal] : ( ord_le88246606_ereal @ A @ A ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_94_subset__refl,axiom,
% 0.30/1.63 ! [A: set_Pr1928503567a_real] : ( ord_le1586073967a_real @ A @ A ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_95_Collect__mono,axiom,
% 0.30/1.63 ! [P: a > $o,Q: a > $o] :
% 0.30/1.63 ( ! [X4: a] :
% 0.30/1.63 ( ( P @ X4 )
% 0.30/1.63 => ( Q @ X4 ) )
% 0.30/1.63 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_96_Collect__mono,axiom,
% 0.30/1.63 ! [P: extended_ereal > $o,Q: extended_ereal > $o] :
% 0.30/1.63 ( ! [X4: extended_ereal] :
% 0.30/1.63 ( ( P @ X4 )
% 0.30/1.63 => ( Q @ X4 ) )
% 0.30/1.63 => ( ord_le88246606_ereal @ ( collec247695033_ereal @ P ) @ ( collec247695033_ereal @ Q ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_97_Collect__mono,axiom,
% 0.30/1.63 ! [P: product_prod_a_real > $o,Q: product_prod_a_real > $o] :
% 0.30/1.63 ( ! [X4: product_prod_a_real] :
% 0.30/1.63 ( ( P @ X4 )
% 0.30/1.63 => ( Q @ X4 ) )
% 0.30/1.63 => ( ord_le1586073967a_real @ ( collec1714955950a_real @ P ) @ ( collec1714955950a_real @ Q ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_98_subset__trans,axiom,
% 0.30/1.63 ! [A: set_a,B: set_a,C2: set_a] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ A @ B )
% 0.30/1.63 => ( ( ord_less_eq_set_a @ B @ C2 )
% 0.30/1.63 => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_99_subset__trans,axiom,
% 0.30/1.63 ! [A: set_Extended_ereal,B: set_Extended_ereal,C2: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_le88246606_ereal @ A @ B )
% 0.30/1.63 => ( ( ord_le88246606_ereal @ B @ C2 )
% 0.30/1.63 => ( ord_le88246606_ereal @ A @ C2 ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_100_subset__trans,axiom,
% 0.30/1.63 ! [A: set_Pr1928503567a_real,B: set_Pr1928503567a_real,C2: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ A @ B )
% 0.30/1.63 => ( ( ord_le1586073967a_real @ B @ C2 )
% 0.30/1.63 => ( ord_le1586073967a_real @ A @ C2 ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_101_set__eq__subset,axiom,
% 0.30/1.63 ( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
% 0.30/1.63 = ( ^ [A2: set_a,B2: set_a] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ A2 @ B2 )
% 0.30/1.63 & ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_102_set__eq__subset,axiom,
% 0.30/1.63 ( ( ^ [Y: set_Extended_ereal,Z: set_Extended_ereal] : ( Y = Z ) )
% 0.30/1.63 = ( ^ [A2: set_Extended_ereal,B2: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_le88246606_ereal @ A2 @ B2 )
% 0.30/1.63 & ( ord_le88246606_ereal @ B2 @ A2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_103_set__eq__subset,axiom,
% 0.30/1.63 ( ( ^ [Y: set_Pr1928503567a_real,Z: set_Pr1928503567a_real] : ( Y = Z ) )
% 0.30/1.63 = ( ^ [A2: set_Pr1928503567a_real,B2: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ A2 @ B2 )
% 0.30/1.63 & ( ord_le1586073967a_real @ B2 @ A2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_104_Collect__mono__iff,axiom,
% 0.30/1.63 ! [P: a > $o,Q: a > $o] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.30/1.63 = ( ! [X2: a] :
% 0.30/1.63 ( ( P @ X2 )
% 0.30/1.63 => ( Q @ X2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_105_Collect__mono__iff,axiom,
% 0.30/1.63 ! [P: extended_ereal > $o,Q: extended_ereal > $o] :
% 0.30/1.63 ( ( ord_le88246606_ereal @ ( collec247695033_ereal @ P ) @ ( collec247695033_ereal @ Q ) )
% 0.30/1.63 = ( ! [X2: extended_ereal] :
% 0.30/1.63 ( ( P @ X2 )
% 0.30/1.63 => ( Q @ X2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_106_Collect__mono__iff,axiom,
% 0.30/1.63 ! [P: product_prod_a_real > $o,Q: product_prod_a_real > $o] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ ( collec1714955950a_real @ P ) @ ( collec1714955950a_real @ Q ) )
% 0.30/1.63 = ( ! [X2: product_prod_a_real] :
% 0.30/1.63 ( ( P @ X2 )
% 0.30/1.63 => ( Q @ X2 ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_107_order__subst1,axiom,
% 0.30/1.63 ! [A3: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.30/1.63 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_108_order__subst1,axiom,
% 0.30/1.63 ! [A3: extended_ereal,F: real > extended_ereal,B3: real,C: real] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_less_eq_real @ B3 @ C )
% 0.30/1.63 => ( ! [X4: real,Y2: real] :
% 0.30/1.63 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_109_order__subst1,axiom,
% 0.30/1.63 ! [A3: real,F: extended_ereal > real,B3: extended_ereal,C: extended_ereal] :
% 0.30/1.63 ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.30/1.63 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_110_order__subst1,axiom,
% 0.30/1.63 ! [A3: real,F: real > real,B3: real,C: real] :
% 0.30/1.63 ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_less_eq_real @ B3 @ C )
% 0.30/1.63 => ( ! [X4: real,Y2: real] :
% 0.30/1.63 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_111_order__subst1,axiom,
% 0.30/1.63 ! [A3: extended_ereal,F: set_a > extended_ereal,B3: set_a,C: set_a] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_less_eq_set_a @ B3 @ C )
% 0.30/1.63 => ( ! [X4: set_a,Y2: set_a] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_112_order__subst1,axiom,
% 0.30/1.63 ! [A3: extended_ereal,F: set_Extended_ereal > extended_ereal,B3: set_Extended_ereal,C: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_le88246606_ereal @ B3 @ C )
% 0.30/1.63 => ( ! [X4: set_Extended_ereal,Y2: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_le88246606_ereal @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_113_order__subst1,axiom,
% 0.30/1.63 ! [A3: real,F: set_a > real,B3: set_a,C: set_a] :
% 0.30/1.63 ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_less_eq_set_a @ B3 @ C )
% 0.30/1.63 => ( ! [X4: set_a,Y2: set_a] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_114_order__subst1,axiom,
% 0.30/1.63 ! [A3: real,F: set_Extended_ereal > real,B3: set_Extended_ereal,C: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_le88246606_ereal @ B3 @ C )
% 0.30/1.63 => ( ! [X4: set_Extended_ereal,Y2: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_le88246606_ereal @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_115_order__subst1,axiom,
% 0.30/1.63 ! [A3: set_a,F: extended_ereal > set_a,B3: extended_ereal,C: extended_ereal] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.30/1.63 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_116_order__subst1,axiom,
% 0.30/1.63 ! [A3: set_a,F: real > set_a,B3: real,C: real] :
% 0.30/1.63 ( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_less_eq_real @ B3 @ C )
% 0.30/1.63 => ( ! [X4: real,Y2: real] :
% 0.30/1.63 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_set_a @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_117_order__subst2,axiom,
% 0.30/1.63 ! [A3: set_Pr1928503567a_real,B3: set_Pr1928503567a_real,F: set_Pr1928503567a_real > set_a,C: set_a] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ A3 @ B3 )
% 0.30/1.63 => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
% 0.30/1.63 => ( ! [X4: set_Pr1928503567a_real,Y2: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_set_a @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_118_order__subst2,axiom,
% 0.30/1.63 ! [A3: set_Pr1928503567a_real,B3: set_Pr1928503567a_real,F: set_Pr1928503567a_real > set_Extended_ereal,C: set_Extended_ereal] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ A3 @ B3 )
% 0.30/1.63 => ( ( ord_le88246606_ereal @ ( F @ B3 ) @ C )
% 0.30/1.63 => ( ! [X4: set_Pr1928503567a_real,Y2: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le88246606_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le88246606_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_119_order__subst2,axiom,
% 0.30/1.63 ! [A3: set_Pr1928503567a_real,B3: set_Pr1928503567a_real,F: set_Pr1928503567a_real > set_Pr1928503567a_real,C: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ A3 @ B3 )
% 0.30/1.63 => ( ( ord_le1586073967a_real @ ( F @ B3 ) @ C )
% 0.30/1.63 => ( ! [X4: set_Pr1928503567a_real,Y2: set_Pr1928503567a_real] :
% 0.30/1.63 ( ( ord_le1586073967a_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le1586073967a_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le1586073967a_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_120_order__subst2,axiom,
% 0.30/1.63 ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.63 => ( ( ord_le824540014_ereal @ ( F @ B3 ) @ C )
% 0.30/1.63 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_121_order__subst2,axiom,
% 0.30/1.63 ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > real,C: real] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.63 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 0.30/1.63 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_122_order__subst2,axiom,
% 0.30/1.63 ! [A3: real,B3: real,F: real > extended_ereal,C: extended_ereal] :
% 0.30/1.63 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.63 => ( ( ord_le824540014_ereal @ ( F @ B3 ) @ C )
% 0.30/1.63 => ( ! [X4: real,Y2: real] :
% 0.30/1.63 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_123_order__subst2,axiom,
% 0.30/1.63 ! [A3: real,B3: real,F: real > real,C: real] :
% 0.30/1.63 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.63 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 0.30/1.63 => ( ! [X4: real,Y2: real] :
% 0.30/1.63 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_124_ord__eq__le__subst,axiom,
% 0.30/1.63 ! [A3: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.30/1.63 ( ( A3
% 0.30/1.63 = ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.30/1.63 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_125_ord__eq__le__subst,axiom,
% 0.30/1.63 ! [A3: real,F: extended_ereal > real,B3: extended_ereal,C: extended_ereal] :
% 0.30/1.63 ( ( A3
% 0.30/1.63 = ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.30/1.63 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_126_ord__eq__le__subst,axiom,
% 0.30/1.63 ! [A3: extended_ereal,F: real > extended_ereal,B3: real,C: real] :
% 0.30/1.63 ( ( A3
% 0.30/1.63 = ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_less_eq_real @ B3 @ C )
% 0.30/1.63 => ( ! [X4: real,Y2: real] :
% 0.30/1.63 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_le824540014_ereal @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_127_ord__eq__le__subst,axiom,
% 0.30/1.63 ! [A3: real,F: real > real,B3: real,C: real] :
% 0.30/1.63 ( ( A3
% 0.30/1.63 = ( F @ B3 ) )
% 0.30/1.63 => ( ( ord_less_eq_real @ B3 @ C )
% 0.30/1.63 => ( ! [X4: real,Y2: real] :
% 0.30/1.63 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.30/1.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.63 => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% 0.30/1.63
% 0.30/1.63 thf(fact_128_ord__le__eq__subst,axiom,
% 0.30/1.63 ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
% 0.30/1.63 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.63 => ( ( ( F @ B3 )
% 0.30/1.63 = C )
% 0.30/1.63 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.30/1.64 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.64 => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_129_ord__le__eq__subst,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal,F: extended_ereal > real,C: real] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.64 => ( ( ( F @ B3 )
% 0.30/1.64 = C )
% 0.30/1.64 => ( ! [X4: extended_ereal,Y2: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ X4 @ Y2 )
% 0.30/1.64 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.64 => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_130_ord__le__eq__subst,axiom,
% 0.30/1.64 ! [A3: real,B3: real,F: real > extended_ereal,C: extended_ereal] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 => ( ( ( F @ B3 )
% 0.30/1.64 = C )
% 0.30/1.64 => ( ! [X4: real,Y2: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.30/1.64 => ( ord_le824540014_ereal @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.64 => ( ord_le824540014_ereal @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_131_ord__le__eq__subst,axiom,
% 0.30/1.64 ! [A3: real,B3: real,F: real > real,C: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 => ( ( ( F @ B3 )
% 0.30/1.64 = C )
% 0.30/1.64 => ( ! [X4: real,Y2: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ X4 @ Y2 )
% 0.30/1.64 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y2 ) ) )
% 0.30/1.64 => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_132_eq__iff,axiom,
% 0.30/1.64 ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
% 0.30/1.64 = ( ^ [X2: extended_ereal,Y3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.30/1.64 & ( ord_le824540014_ereal @ Y3 @ X2 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_133_eq__iff,axiom,
% 0.30/1.64 ( ( ^ [Y: real,Z: real] : ( Y = Z ) )
% 0.30/1.64 = ( ^ [X2: real,Y3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.30/1.64 & ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_134_antisym,axiom,
% 0.30/1.64 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ Y4 @ X )
% 0.30/1.64 => ( X = Y4 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_135_antisym,axiom,
% 0.30/1.64 ! [X: real,Y4: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ X @ Y4 )
% 0.30/1.64 => ( ( ord_less_eq_real @ Y4 @ X )
% 0.30/1.64 => ( X = Y4 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_136_mem__Collect__eq,axiom,
% 0.30/1.64 ! [A3: extended_ereal,P: extended_ereal > $o] :
% 0.30/1.64 ( ( member1900190071_ereal @ A3 @ ( collec247695033_ereal @ P ) )
% 0.30/1.64 = ( P @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_137_mem__Collect__eq,axiom,
% 0.30/1.64 ! [A3: a,P: a > $o] :
% 0.30/1.64 ( ( member_a @ A3 @ ( collect_a @ P ) )
% 0.30/1.64 = ( P @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_138_Collect__mem__eq,axiom,
% 0.30/1.64 ! [A: set_Extended_ereal] :
% 0.30/1.64 ( ( collec247695033_ereal
% 0.30/1.64 @ ^ [X2: extended_ereal] : ( member1900190071_ereal @ X2 @ A ) )
% 0.30/1.64 = A ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_139_Collect__mem__eq,axiom,
% 0.30/1.64 ! [A: set_a] :
% 0.30/1.64 ( ( collect_a
% 0.30/1.64 @ ^ [X2: a] : ( member_a @ X2 @ A ) )
% 0.30/1.64 = A ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_140_linear,axiom,
% 0.30/1.64 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.30/1.64 | ( ord_le824540014_ereal @ Y4 @ X ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_141_linear,axiom,
% 0.30/1.64 ! [X: real,Y4: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ X @ Y4 )
% 0.30/1.64 | ( ord_less_eq_real @ Y4 @ X ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_142_eq__refl,axiom,
% 0.30/1.64 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.30/1.64 ( ( X = Y4 )
% 0.30/1.64 => ( ord_le824540014_ereal @ X @ Y4 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_143_eq__refl,axiom,
% 0.30/1.64 ! [X: real,Y4: real] :
% 0.30/1.64 ( ( X = Y4 )
% 0.30/1.64 => ( ord_less_eq_real @ X @ Y4 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_144_le__cases,axiom,
% 0.30/1.64 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.30/1.64 ( ~ ( ord_le824540014_ereal @ X @ Y4 )
% 0.30/1.64 => ( ord_le824540014_ereal @ Y4 @ X ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_145_le__cases,axiom,
% 0.30/1.64 ! [X: real,Y4: real] :
% 0.30/1.64 ( ~ ( ord_less_eq_real @ X @ Y4 )
% 0.30/1.64 => ( ord_less_eq_real @ Y4 @ X ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_146_order_Otrans,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.30/1.64 => ( ord_le824540014_ereal @ A3 @ C ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_147_order_Otrans,axiom,
% 0.30/1.64 ! [A3: real,B3: real,C: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 => ( ( ord_less_eq_real @ B3 @ C )
% 0.30/1.64 => ( ord_less_eq_real @ A3 @ C ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_148_le__cases3,axiom,
% 0.30/1.64 ! [X: extended_ereal,Y4: extended_ereal,Z2: extended_ereal] :
% 0.30/1.64 ( ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.30/1.64 => ~ ( ord_le824540014_ereal @ Y4 @ Z2 ) )
% 0.30/1.64 => ( ( ( ord_le824540014_ereal @ Y4 @ X )
% 0.30/1.64 => ~ ( ord_le824540014_ereal @ X @ Z2 ) )
% 0.30/1.64 => ( ( ( ord_le824540014_ereal @ X @ Z2 )
% 0.30/1.64 => ~ ( ord_le824540014_ereal @ Z2 @ Y4 ) )
% 0.30/1.64 => ( ( ( ord_le824540014_ereal @ Z2 @ Y4 )
% 0.30/1.64 => ~ ( ord_le824540014_ereal @ Y4 @ X ) )
% 0.30/1.64 => ( ( ( ord_le824540014_ereal @ Y4 @ Z2 )
% 0.30/1.64 => ~ ( ord_le824540014_ereal @ Z2 @ X ) )
% 0.30/1.64 => ~ ( ( ord_le824540014_ereal @ Z2 @ X )
% 0.30/1.64 => ~ ( ord_le824540014_ereal @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_149_le__cases3,axiom,
% 0.30/1.64 ! [X: real,Y4: real,Z2: real] :
% 0.30/1.64 ( ( ( ord_less_eq_real @ X @ Y4 )
% 0.30/1.64 => ~ ( ord_less_eq_real @ Y4 @ Z2 ) )
% 0.30/1.64 => ( ( ( ord_less_eq_real @ Y4 @ X )
% 0.30/1.64 => ~ ( ord_less_eq_real @ X @ Z2 ) )
% 0.30/1.64 => ( ( ( ord_less_eq_real @ X @ Z2 )
% 0.30/1.64 => ~ ( ord_less_eq_real @ Z2 @ Y4 ) )
% 0.30/1.64 => ( ( ( ord_less_eq_real @ Z2 @ Y4 )
% 0.30/1.64 => ~ ( ord_less_eq_real @ Y4 @ X ) )
% 0.30/1.64 => ( ( ( ord_less_eq_real @ Y4 @ Z2 )
% 0.30/1.64 => ~ ( ord_less_eq_real @ Z2 @ X ) )
% 0.30/1.64 => ~ ( ( ord_less_eq_real @ Z2 @ X )
% 0.30/1.64 => ~ ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_150_antisym__conv,axiom,
% 0.30/1.64 ! [Y4: extended_ereal,X: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ Y4 @ X )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.30/1.64 = ( X = Y4 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_151_antisym__conv,axiom,
% 0.30/1.64 ! [Y4: real,X: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ Y4 @ X )
% 0.30/1.64 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.30/1.64 = ( X = Y4 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_152_order__class_Oorder_Oeq__iff,axiom,
% 0.30/1.64 ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
% 0.30/1.64 = ( ^ [A4: extended_ereal,B4: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ A4 @ B4 )
% 0.30/1.64 & ( ord_le824540014_ereal @ B4 @ A4 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_153_order__class_Oorder_Oeq__iff,axiom,
% 0.30/1.64 ( ( ^ [Y: real,Z: real] : ( Y = Z ) )
% 0.30/1.64 = ( ^ [A4: real,B4: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A4 @ B4 )
% 0.30/1.64 & ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_154_ord__eq__le__trans,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.30/1.64 ( ( A3 = B3 )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ B3 @ C )
% 0.30/1.64 => ( ord_le824540014_ereal @ A3 @ C ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_155_ord__eq__le__trans,axiom,
% 0.30/1.64 ! [A3: real,B3: real,C: real] :
% 0.30/1.64 ( ( A3 = B3 )
% 0.30/1.64 => ( ( ord_less_eq_real @ B3 @ C )
% 0.30/1.64 => ( ord_less_eq_real @ A3 @ C ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_156_ord__le__eq__trans,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal,C: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.64 => ( ( B3 = C )
% 0.30/1.64 => ( ord_le824540014_ereal @ A3 @ C ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_157_ord__le__eq__trans,axiom,
% 0.30/1.64 ! [A3: real,B3: real,C: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 => ( ( B3 = C )
% 0.30/1.64 => ( ord_less_eq_real @ A3 @ C ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_158_order__class_Oorder_Oantisym,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ B3 @ A3 )
% 0.30/1.64 => ( A3 = B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_159_order__class_Oorder_Oantisym,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 => ( ( ord_less_eq_real @ B3 @ A3 )
% 0.30/1.64 => ( A3 = B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_160_order__trans,axiom,
% 0.30/1.64 ! [X: extended_ereal,Y4: extended_ereal,Z2: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ Y4 @ Z2 )
% 0.30/1.64 => ( ord_le824540014_ereal @ X @ Z2 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_161_order__trans,axiom,
% 0.30/1.64 ! [X: real,Y4: real,Z2: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ X @ Y4 )
% 0.30/1.64 => ( ( ord_less_eq_real @ Y4 @ Z2 )
% 0.30/1.64 => ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_162_dual__order_Orefl,axiom,
% 0.30/1.64 ! [A3: extended_ereal] : ( ord_le824540014_ereal @ A3 @ A3 ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_163_dual__order_Orefl,axiom,
% 0.30/1.64 ! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_164_linorder__wlog,axiom,
% 0.30/1.64 ! [P: extended_ereal > extended_ereal > $o,A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ! [A5: extended_ereal,B5: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ A5 @ B5 )
% 0.30/1.64 => ( P @ A5 @ B5 ) )
% 0.30/1.64 => ( ! [A5: extended_ereal,B5: extended_ereal] :
% 0.30/1.64 ( ( P @ B5 @ A5 )
% 0.30/1.64 => ( P @ A5 @ B5 ) )
% 0.30/1.64 => ( P @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_165_linorder__wlog,axiom,
% 0.30/1.64 ! [P: real > real > $o,A3: real,B3: real] :
% 0.30/1.64 ( ! [A5: real,B5: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A5 @ B5 )
% 0.30/1.64 => ( P @ A5 @ B5 ) )
% 0.30/1.64 => ( ! [A5: real,B5: real] :
% 0.30/1.64 ( ( P @ B5 @ A5 )
% 0.30/1.64 => ( P @ A5 @ B5 ) )
% 0.30/1.64 => ( P @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_166_dual__order_Otrans,axiom,
% 0.30/1.64 ! [B3: extended_ereal,A3: extended_ereal,C: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ B3 @ A3 )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ C @ B3 )
% 0.30/1.64 => ( ord_le824540014_ereal @ C @ A3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_167_dual__order_Otrans,axiom,
% 0.30/1.64 ! [B3: real,A3: real,C: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ B3 @ A3 )
% 0.30/1.64 => ( ( ord_less_eq_real @ C @ B3 )
% 0.30/1.64 => ( ord_less_eq_real @ C @ A3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_168_dual__order_Oeq__iff,axiom,
% 0.30/1.64 ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
% 0.30/1.64 = ( ^ [A4: extended_ereal,B4: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ B4 @ A4 )
% 0.30/1.64 & ( ord_le824540014_ereal @ A4 @ B4 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_169_dual__order_Oeq__iff,axiom,
% 0.30/1.64 ( ( ^ [Y: real,Z: real] : ( Y = Z ) )
% 0.30/1.64 = ( ^ [A4: real,B4: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ B4 @ A4 )
% 0.30/1.64 & ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_170_dual__order_Oantisym,axiom,
% 0.30/1.64 ! [B3: extended_ereal,A3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ B3 @ A3 )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.64 => ( A3 = B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_171_dual__order_Oantisym,axiom,
% 0.30/1.64 ! [B3: real,A3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ B3 @ A3 )
% 0.30/1.64 => ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 => ( A3 = B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_172_Sup__eqI,axiom,
% 0.30/1.64 ! [A: set_Extended_ereal,X: extended_ereal] :
% 0.30/1.64 ( ! [Y2: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ Y2 @ A )
% 0.30/1.64 => ( ord_le824540014_ereal @ Y2 @ X ) )
% 0.30/1.64 => ( ! [Y2: extended_ereal] :
% 0.30/1.64 ( ! [Z3: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ Z3 @ A )
% 0.30/1.64 => ( ord_le824540014_ereal @ Z3 @ Y2 ) )
% 0.30/1.64 => ( ord_le824540014_ereal @ X @ Y2 ) )
% 0.30/1.64 => ( ( comple1161760187_ereal @ A )
% 0.30/1.64 = X ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_173_Sup__mono,axiom,
% 0.30/1.64 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.30/1.64 ( ! [A5: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ A5 @ A )
% 0.30/1.64 => ? [X3: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ X3 @ B )
% 0.30/1.64 & ( ord_le824540014_ereal @ A5 @ X3 ) ) )
% 0.30/1.64 => ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ ( comple1161760187_ereal @ B ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_174_Sup__least,axiom,
% 0.30/1.64 ! [A: set_Extended_ereal,Z2: extended_ereal] :
% 0.30/1.64 ( ! [X4: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ X4 @ A )
% 0.30/1.64 => ( ord_le824540014_ereal @ X4 @ Z2 ) )
% 0.30/1.64 => ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ Z2 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_175_Sup__upper,axiom,
% 0.30/1.64 ! [X: extended_ereal,A: set_Extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ X @ A )
% 0.30/1.64 => ( ord_le824540014_ereal @ X @ ( comple1161760187_ereal @ A ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_176_Sup__le__iff,axiom,
% 0.30/1.64 ! [A: set_Extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ B3 )
% 0.30/1.64 = ( ! [X2: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ X2 @ A )
% 0.30/1.64 => ( ord_le824540014_ereal @ X2 @ B3 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_177_Sup__upper2,axiom,
% 0.30/1.64 ! [U: extended_ereal,A: set_Extended_ereal,V: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ U @ A )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ V @ U )
% 0.30/1.64 => ( ord_le824540014_ereal @ V @ ( comple1161760187_ereal @ A ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_178_Sup__subset__mono,axiom,
% 0.30/1.64 ! [A: set_Extended_ereal,B: set_Extended_ereal] :
% 0.30/1.64 ( ( ord_le88246606_ereal @ A @ B )
% 0.30/1.64 => ( ord_le824540014_ereal @ ( comple1161760187_ereal @ A ) @ ( comple1161760187_ereal @ B ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_179_top__set__def,axiom,
% 0.30/1.64 ( top_top_set_a
% 0.30/1.64 = ( collect_a @ top_top_a_o ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_180_top__set__def,axiom,
% 0.30/1.64 ( top_to1767659262_ereal
% 0.30/1.64 = ( collec247695033_ereal @ top_to398855007real_o ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_181_Union__UNIV,axiom,
% 0.30/1.64 ( ( comple1766734283_set_a @ top_top_set_set_a )
% 0.30/1.64 = top_top_set_a ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_182_Union__UNIV,axiom,
% 0.30/1.64 ( ( comple767846299_ereal @ top_to692740318_ereal )
% 0.30/1.64 = top_to1767659262_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_183_top_Oextremum__uniqueI,axiom,
% 0.30/1.64 ! [A3: set_a] :
% 0.30/1.64 ( ( ord_less_eq_set_a @ top_top_set_a @ A3 )
% 0.30/1.64 => ( A3 = top_top_set_a ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_184_top_Oextremum__uniqueI,axiom,
% 0.30/1.64 ! [A3: set_Extended_ereal] :
% 0.30/1.64 ( ( ord_le88246606_ereal @ top_to1767659262_ereal @ A3 )
% 0.30/1.64 => ( A3 = top_to1767659262_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_185_top_Oextremum__uniqueI,axiom,
% 0.30/1.64 ! [A3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ top_to802031902_ereal @ A3 )
% 0.30/1.64 => ( A3 = top_to802031902_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_186_top_Oextremum__unique,axiom,
% 0.30/1.64 ! [A3: set_a] :
% 0.30/1.64 ( ( ord_less_eq_set_a @ top_top_set_a @ A3 )
% 0.30/1.64 = ( A3 = top_top_set_a ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_187_top_Oextremum__unique,axiom,
% 0.30/1.64 ! [A3: set_Extended_ereal] :
% 0.30/1.64 ( ( ord_le88246606_ereal @ top_to1767659262_ereal @ A3 )
% 0.30/1.64 = ( A3 = top_to1767659262_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_188_top_Oextremum__unique,axiom,
% 0.30/1.64 ! [A3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ top_to802031902_ereal @ A3 )
% 0.30/1.64 = ( A3 = top_to802031902_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_189_top__greatest,axiom,
% 0.30/1.64 ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ top_top_set_a ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_190_top__greatest,axiom,
% 0.30/1.64 ! [A3: set_Extended_ereal] : ( ord_le88246606_ereal @ A3 @ top_to1767659262_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_191_top__greatest,axiom,
% 0.30/1.64 ! [A3: extended_ereal] : ( ord_le824540014_ereal @ A3 @ top_to802031902_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_192_subset__UNIV,axiom,
% 0.30/1.64 ! [A: set_a] : ( ord_less_eq_set_a @ A @ top_top_set_a ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_193_subset__UNIV,axiom,
% 0.30/1.64 ! [A: set_Extended_ereal] : ( ord_le88246606_ereal @ A @ top_to1767659262_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_194_epigraph__invertible,axiom,
% 0.30/1.64 ! [F: a > extended_ereal,G: a > extended_ereal] :
% 0.30/1.64 ( ( ( lower_930854854raph_a @ top_top_set_a @ F )
% 0.30/1.64 = ( lower_930854854raph_a @ top_top_set_a @ G ) )
% 0.30/1.64 => ( F = G ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_195_epigraph__invertible,axiom,
% 0.30/1.64 ! [F: extended_ereal > extended_ereal,G: extended_ereal > extended_ereal] :
% 0.30/1.64 ( ( ( lower_331963542_ereal @ top_to1767659262_ereal @ F )
% 0.30/1.64 = ( lower_331963542_ereal @ top_to1767659262_ereal @ G ) )
% 0.30/1.64 => ( F = G ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_196_neg__le__iff__le,axiom,
% 0.30/1.64 ! [B3: real,A3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
% 0.30/1.64 = ( ord_less_eq_real @ A3 @ B3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_197_ereal__uminus__eq__iff,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ( uminus1208298309_ereal @ A3 )
% 0.30/1.64 = ( uminus1208298309_ereal @ B3 ) )
% 0.30/1.64 = ( A3 = B3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_198_ereal__uminus__uminus,axiom,
% 0.30/1.64 ! [A3: extended_ereal] :
% 0.30/1.64 ( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A3 ) )
% 0.30/1.64 = A3 ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_199_verit__minus__simplify_I4_J,axiom,
% 0.30/1.64 ! [B3: real] :
% 0.30/1.64 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B3 ) )
% 0.30/1.64 = B3 ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_200_add_Oinverse__inverse,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A3 ) )
% 0.30/1.64 = A3 ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_201_neg__equal__iff__equal,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ( uminus_uminus_real @ A3 )
% 0.30/1.64 = ( uminus_uminus_real @ B3 ) )
% 0.30/1.64 = ( A3 = B3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_202_ereal__infty__less__eq_I1_J,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ extend1289208545_ereal @ X )
% 0.30/1.64 = ( X = extend1289208545_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_203_UnionI,axiom,
% 0.30/1.64 ! [X5: set_Extended_ereal,C2: set_se767749006_ereal,A: extended_ereal] :
% 0.30/1.64 ( ( member450560855_ereal @ X5 @ C2 )
% 0.30/1.64 => ( ( member1900190071_ereal @ A @ X5 )
% 0.30/1.64 => ( member1900190071_ereal @ A @ ( comple767846299_ereal @ C2 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_204_UnionI,axiom,
% 0.30/1.64 ! [X5: set_a,C2: set_set_a,A: a] :
% 0.30/1.64 ( ( member_set_a @ X5 @ C2 )
% 0.30/1.64 => ( ( member_a @ A @ X5 )
% 0.30/1.64 => ( member_a @ A @ ( comple1766734283_set_a @ C2 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_205_Union__iff,axiom,
% 0.30/1.64 ! [A: extended_ereal,C2: set_se767749006_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ A @ ( comple767846299_ereal @ C2 ) )
% 0.30/1.64 = ( ? [X2: set_Extended_ereal] :
% 0.30/1.64 ( ( member450560855_ereal @ X2 @ C2 )
% 0.30/1.64 & ( member1900190071_ereal @ A @ X2 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_206_Union__iff,axiom,
% 0.30/1.64 ! [A: a,C2: set_set_a] :
% 0.30/1.64 ( ( member_a @ A @ ( comple1766734283_set_a @ C2 ) )
% 0.30/1.64 = ( ? [X2: set_a] :
% 0.30/1.64 ( ( member_set_a @ X2 @ C2 )
% 0.30/1.64 & ( member_a @ A @ X2 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_207_ereal__minus__le__minus,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A3 ) @ ( uminus1208298309_ereal @ B3 ) )
% 0.30/1.64 = ( ord_le824540014_ereal @ B3 @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_208_ereal__infty__less__eq_I2_J,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ X @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 = ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_209_UnionE,axiom,
% 0.30/1.64 ! [A: extended_ereal,C2: set_se767749006_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ A @ ( comple767846299_ereal @ C2 ) )
% 0.30/1.64 => ~ ! [X6: set_Extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ A @ X6 )
% 0.30/1.64 => ~ ( member450560855_ereal @ X6 @ C2 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_210_UnionE,axiom,
% 0.30/1.64 ! [A: a,C2: set_set_a] :
% 0.30/1.64 ( ( member_a @ A @ ( comple1766734283_set_a @ C2 ) )
% 0.30/1.64 => ~ ! [X6: set_a] :
% 0.30/1.64 ( ( member_a @ A @ X6 )
% 0.30/1.64 => ~ ( member_set_a @ X6 @ C2 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_211_neq__PInf__trans,axiom,
% 0.30/1.64 ! [Y4: extended_ereal,X: extended_ereal] :
% 0.30/1.64 ( ( Y4 != extend1289208545_ereal )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ X @ Y4 )
% 0.30/1.64 => ( X != extend1289208545_ereal ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_212_ereal__infty__less__eq2_I1_J,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.64 => ( ( A3 = extend1289208545_ereal )
% 0.30/1.64 => ( B3 = extend1289208545_ereal ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_213_ereal__less__eq_I1_J,axiom,
% 0.30/1.64 ! [X: extended_ereal] : ( ord_le824540014_ereal @ X @ extend1289208545_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_214_ereal__uminus__le__reorder,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A3 ) @ B3 )
% 0.30/1.64 = ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ B3 ) @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_215_ereal__infty__less__eq2_I2_J,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.64 => ( ( B3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( A3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_216_ereal__less__eq_I2_J,axiom,
% 0.30/1.64 ! [X: extended_ereal] : ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_217_epigraph__subset__iff,axiom,
% 0.30/1.64 ! [F: a > extended_ereal,G: a > extended_ereal] :
% 0.30/1.64 ( ( ord_le1586073967a_real @ ( lower_930854854raph_a @ top_top_set_a @ F ) @ ( lower_930854854raph_a @ top_top_set_a @ G ) )
% 0.30/1.64 = ( ! [X2: a] : ( ord_le824540014_ereal @ ( G @ X2 ) @ ( F @ X2 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_218_verit__la__disequality,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( A3 = B3 )
% 0.30/1.64 | ~ ( ord_le824540014_ereal @ A3 @ B3 )
% 0.30/1.64 | ~ ( ord_le824540014_ereal @ B3 @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_219_verit__la__disequality,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( A3 = B3 )
% 0.30/1.64 | ~ ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 | ~ ( ord_less_eq_real @ B3 @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_220_minus__equation__iff,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ( uminus_uminus_real @ A3 )
% 0.30/1.64 = B3 )
% 0.30/1.64 = ( ( uminus_uminus_real @ B3 )
% 0.30/1.64 = A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_221_equation__minus__iff,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( A3
% 0.30/1.64 = ( uminus_uminus_real @ B3 ) )
% 0.30/1.64 = ( B3
% 0.30/1.64 = ( uminus_uminus_real @ A3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_222_verit__negate__coefficient_I3_J,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( A3 = B3 )
% 0.30/1.64 => ( ( uminus_uminus_real @ A3 )
% 0.30/1.64 = ( uminus_uminus_real @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_223_top__ereal__def,axiom,
% 0.30/1.64 top_to802031902_ereal = extend1289208545_ereal ).
% 0.30/1.64
% 0.30/1.64 thf(fact_224_Sup__eq__PInfty,axiom,
% 0.30/1.64 ! [S2: set_Extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ extend1289208545_ereal @ S2 )
% 0.30/1.64 => ( ( comple1161760187_ereal @ S2 )
% 0.30/1.64 = extend1289208545_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_225_ereal__uminus__eq__reorder,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ( uminus1208298309_ereal @ A3 )
% 0.30/1.64 = B3 )
% 0.30/1.64 = ( A3
% 0.30/1.64 = ( uminus1208298309_ereal @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_226_le__imp__neg__le,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_227_minus__le__iff,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 )
% 0.30/1.64 = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_228_le__minus__iff,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ B3 ) )
% 0.30/1.64 = ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_229_MInfty__neq__PInfty_I1_J,axiom,
% 0.30/1.64 ( extend1289208545_ereal
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_230_top__empty__eq,axiom,
% 0.30/1.64 ( top_top_a_o
% 0.30/1.64 = ( ^ [X2: a] : ( member_a @ X2 @ top_top_set_a ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_231_top__empty__eq,axiom,
% 0.30/1.64 ( top_to398855007real_o
% 0.30/1.64 = ( ^ [X2: extended_ereal] : ( member1900190071_ereal @ X2 @ top_to1767659262_ereal ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_232_cSup__eq__maximum,axiom,
% 0.30/1.64 ! [Z2: real,X5: set_real] :
% 0.30/1.64 ( ( member_real @ Z2 @ X5 )
% 0.30/1.64 => ( ! [X4: real] :
% 0.30/1.64 ( ( member_real @ X4 @ X5 )
% 0.30/1.64 => ( ord_less_eq_real @ X4 @ Z2 ) )
% 0.30/1.64 => ( ( comple2129349247p_real @ X5 )
% 0.30/1.64 = Z2 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_233_cSup__eq__maximum,axiom,
% 0.30/1.64 ! [Z2: extended_ereal,X5: set_Extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ Z2 @ X5 )
% 0.30/1.64 => ( ! [X4: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ X4 @ X5 )
% 0.30/1.64 => ( ord_le824540014_ereal @ X4 @ Z2 ) )
% 0.30/1.64 => ( ( comple1161760187_ereal @ X5 )
% 0.30/1.64 = Z2 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_234_cSup__eq,axiom,
% 0.30/1.64 ! [X5: set_real,A3: real] :
% 0.30/1.64 ( ! [X4: real] :
% 0.30/1.64 ( ( member_real @ X4 @ X5 )
% 0.30/1.64 => ( ord_less_eq_real @ X4 @ A3 ) )
% 0.30/1.64 => ( ! [Y2: real] :
% 0.30/1.64 ( ! [X3: real] :
% 0.30/1.64 ( ( member_real @ X3 @ X5 )
% 0.30/1.64 => ( ord_less_eq_real @ X3 @ Y2 ) )
% 0.30/1.64 => ( ord_less_eq_real @ A3 @ Y2 ) )
% 0.30/1.64 => ( ( comple2129349247p_real @ X5 )
% 0.30/1.64 = A3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_235_MInfty__eq__minfinity,axiom,
% 0.30/1.64 ( extended_MInfty
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_236_ComplI,axiom,
% 0.30/1.64 ! [C: extended_ereal,A: set_Extended_ereal] :
% 0.30/1.64 ( ~ ( member1900190071_ereal @ C @ A )
% 0.30/1.64 => ( member1900190071_ereal @ C @ ( uminus360668453_ereal @ A ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_237_ComplI,axiom,
% 0.30/1.64 ! [C: a,A: set_a] :
% 0.30/1.64 ( ~ ( member_a @ C @ A )
% 0.30/1.64 => ( member_a @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_238_Compl__iff,axiom,
% 0.30/1.64 ! [C: extended_ereal,A: set_Extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ C @ ( uminus360668453_ereal @ A ) )
% 0.30/1.64 = ( ~ ( member1900190071_ereal @ C @ A ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_239_Compl__iff,axiom,
% 0.30/1.64 ! [C: a,A: set_a] :
% 0.30/1.64 ( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
% 0.30/1.64 = ( ~ ( member_a @ C @ A ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_240_ComplD,axiom,
% 0.30/1.64 ! [C: extended_ereal,A: set_Extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ C @ ( uminus360668453_ereal @ A ) )
% 0.30/1.64 => ~ ( member1900190071_ereal @ C @ A ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_241_ComplD,axiom,
% 0.30/1.64 ! [C: a,A: set_a] :
% 0.30/1.64 ( ( member_a @ C @ ( uminus_uminus_set_a @ A ) )
% 0.30/1.64 => ~ ( member_a @ C @ A ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_242_ereal__complete__Inf,axiom,
% 0.30/1.64 ! [S2: set_Extended_ereal] :
% 0.30/1.64 ? [X4: extended_ereal] :
% 0.30/1.64 ( ! [Xa: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ Xa @ S2 )
% 0.30/1.64 => ( ord_le824540014_ereal @ X4 @ Xa ) )
% 0.30/1.64 & ! [Z3: extended_ereal] :
% 0.30/1.64 ( ! [Xa2: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ Xa2 @ S2 )
% 0.30/1.64 => ( ord_le824540014_ereal @ Z3 @ Xa2 ) )
% 0.30/1.64 => ( ord_le824540014_ereal @ Z3 @ X4 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_243_ereal__complete__Sup,axiom,
% 0.30/1.64 ! [S2: set_Extended_ereal] :
% 0.30/1.64 ? [X4: extended_ereal] :
% 0.30/1.64 ( ! [Xa: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ Xa @ S2 )
% 0.30/1.64 => ( ord_le824540014_ereal @ Xa @ X4 ) )
% 0.30/1.64 & ! [Z3: extended_ereal] :
% 0.30/1.64 ( ! [Xa2: extended_ereal] :
% 0.30/1.64 ( ( member1900190071_ereal @ Xa2 @ S2 )
% 0.30/1.64 => ( ord_le824540014_ereal @ Xa2 @ Z3 ) )
% 0.30/1.64 => ( ord_le824540014_ereal @ X4 @ Z3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_244_uminus__ereal_Osimps_I3_J,axiom,
% 0.30/1.64 ( ( uminus1208298309_ereal @ extended_MInfty )
% 0.30/1.64 = extended_PInfty ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_245_uminus__ereal_Osimps_I2_J,axiom,
% 0.30/1.64 ( ( uminus1208298309_ereal @ extended_PInfty )
% 0.30/1.64 = extended_MInfty ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_246_GreatestI2__order,axiom,
% 0.30/1.64 ! [P: extended_ereal > $o,X: extended_ereal,Q: extended_ereal > $o] :
% 0.30/1.64 ( ( P @ X )
% 0.30/1.64 => ( ! [Y2: extended_ereal] :
% 0.30/1.64 ( ( P @ Y2 )
% 0.30/1.64 => ( ord_le824540014_ereal @ Y2 @ X ) )
% 0.30/1.64 => ( ! [X4: extended_ereal] :
% 0.30/1.64 ( ( P @ X4 )
% 0.30/1.64 => ( ! [Y5: extended_ereal] :
% 0.30/1.64 ( ( P @ Y5 )
% 0.30/1.64 => ( ord_le824540014_ereal @ Y5 @ X4 ) )
% 0.30/1.64 => ( Q @ X4 ) ) )
% 0.30/1.64 => ( Q @ ( order_1158471719_ereal @ P ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_247_GreatestI2__order,axiom,
% 0.30/1.64 ! [P: real > $o,X: real,Q: real > $o] :
% 0.30/1.64 ( ( P @ X )
% 0.30/1.64 => ( ! [Y2: real] :
% 0.30/1.64 ( ( P @ Y2 )
% 0.30/1.64 => ( ord_less_eq_real @ Y2 @ X ) )
% 0.30/1.64 => ( ! [X4: real] :
% 0.30/1.64 ( ( P @ X4 )
% 0.30/1.64 => ( ! [Y5: real] :
% 0.30/1.64 ( ( P @ Y5 )
% 0.30/1.64 => ( ord_less_eq_real @ Y5 @ X4 ) )
% 0.30/1.64 => ( Q @ X4 ) ) )
% 0.30/1.64 => ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_248_infinity__ereal__def,axiom,
% 0.30/1.64 extend1289208545_ereal = extended_PInfty ).
% 0.30/1.64
% 0.30/1.64 thf(fact_249_ereal_Odistinct_I5_J,axiom,
% 0.30/1.64 extended_PInfty != extended_MInfty ).
% 0.30/1.64
% 0.30/1.64 thf(fact_250_Greatest__equality,axiom,
% 0.30/1.64 ! [P: extended_ereal > $o,X: extended_ereal] :
% 0.30/1.64 ( ( P @ X )
% 0.30/1.64 => ( ! [Y2: extended_ereal] :
% 0.30/1.64 ( ( P @ Y2 )
% 0.30/1.64 => ( ord_le824540014_ereal @ Y2 @ X ) )
% 0.30/1.64 => ( ( order_1158471719_ereal @ P )
% 0.30/1.64 = X ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_251_Greatest__equality,axiom,
% 0.30/1.64 ! [P: real > $o,X: real] :
% 0.30/1.64 ( ( P @ X )
% 0.30/1.64 => ( ! [Y2: real] :
% 0.30/1.64 ( ( P @ Y2 )
% 0.30/1.64 => ( ord_less_eq_real @ Y2 @ X ) )
% 0.30/1.64 => ( ( order_Greatest_real @ P )
% 0.30/1.64 = X ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_252_uminus__ereal_Oelims,axiom,
% 0.30/1.64 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.30/1.64 ( ( ( uminus1208298309_ereal @ X )
% 0.30/1.64 = Y4 )
% 0.30/1.64 => ( ! [R: real] :
% 0.30/1.64 ( ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( Y4
% 0.30/1.64 != ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) )
% 0.30/1.64 => ( ( ( X = extended_PInfty )
% 0.30/1.64 => ( Y4 != extended_MInfty ) )
% 0.30/1.64 => ~ ( ( X = extended_MInfty )
% 0.30/1.64 => ( Y4 != extended_PInfty ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_253_not__MInfty__nonneg,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X )
% 0.30/1.64 => ( X
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_254_ereal_Oinject,axiom,
% 0.30/1.64 ! [X1: real,Y1: real] :
% 0.30/1.64 ( ( ( extended_ereal2 @ X1 )
% 0.30/1.64 = ( extended_ereal2 @ Y1 ) )
% 0.30/1.64 = ( X1 = Y1 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_255_ereal__cong,axiom,
% 0.30/1.64 ! [X: real,Y4: real] :
% 0.30/1.64 ( ( X = Y4 )
% 0.30/1.64 => ( ( extended_ereal2 @ X )
% 0.30/1.64 = ( extended_ereal2 @ Y4 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_256_neg__equal__zero,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( ( uminus_uminus_real @ A3 )
% 0.30/1.64 = A3 )
% 0.30/1.64 = ( A3 = zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_257_equal__neg__zero,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( A3
% 0.30/1.64 = ( uminus_uminus_real @ A3 ) )
% 0.30/1.64 = ( A3 = zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_258_neg__equal__0__iff__equal,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( ( uminus_uminus_real @ A3 )
% 0.30/1.64 = zero_zero_real )
% 0.30/1.64 = ( A3 = zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_259_neg__0__equal__iff__equal,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( zero_zero_real
% 0.30/1.64 = ( uminus_uminus_real @ A3 ) )
% 0.30/1.64 = ( zero_zero_real = A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_260_add_Oinverse__neutral,axiom,
% 0.30/1.64 ( ( uminus_uminus_real @ zero_zero_real )
% 0.30/1.64 = zero_zero_real ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_261_ereal__less__eq_I3_J,axiom,
% 0.30/1.64 ! [R2: real,P2: real] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 0.30/1.64 = ( ord_less_eq_real @ R2 @ P2 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_262_ereal__uminus__zero,axiom,
% 0.30/1.64 ( ( uminus1208298309_ereal @ zero_z163181189_ereal )
% 0.30/1.64 = zero_z163181189_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_263_ereal__uminus__zero__iff,axiom,
% 0.30/1.64 ! [A3: extended_ereal] :
% 0.30/1.64 ( ( ( uminus1208298309_ereal @ A3 )
% 0.30/1.64 = zero_z163181189_ereal )
% 0.30/1.64 = ( A3 = zero_z163181189_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_264_neg__0__le__iff__le,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
% 0.30/1.64 = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_265_neg__le__0__iff__le,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
% 0.30/1.64 = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_266_less__eq__neg__nonpos,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ A3 ) )
% 0.30/1.64 = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_267_neg__less__eq__nonneg,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ A3 )
% 0.30/1.64 = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_268_ereal__uminus__le__0__iff,axiom,
% 0.30/1.64 ! [A3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A3 ) @ zero_z163181189_ereal )
% 0.30/1.64 = ( ord_le824540014_ereal @ zero_z163181189_ereal @ A3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_269_ereal__0__le__uminus__iff,axiom,
% 0.30/1.64 ! [A3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( uminus1208298309_ereal @ A3 ) )
% 0.30/1.64 = ( ord_le824540014_ereal @ A3 @ zero_z163181189_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_270_ereal_Odistinct_I1_J,axiom,
% 0.30/1.64 ! [X1: real] :
% 0.30/1.64 ( ( extended_ereal2 @ X1 )
% 0.30/1.64 != extended_PInfty ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_271_zero__reorient,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ( zero_z163181189_ereal = X )
% 0.30/1.64 = ( X = zero_z163181189_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_272_zero__reorient,axiom,
% 0.30/1.64 ! [X: real] :
% 0.30/1.64 ( ( zero_zero_real = X )
% 0.30/1.64 = ( X = zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_273_Infty__neq__0_I1_J,axiom,
% 0.30/1.64 extend1289208545_ereal != zero_z163181189_ereal ).
% 0.30/1.64
% 0.30/1.64 thf(fact_274_uminus__ereal_Osimps_I1_J,axiom,
% 0.30/1.64 ! [R2: real] :
% 0.30/1.64 ( ( uminus1208298309_ereal @ ( extended_ereal2 @ R2 ) )
% 0.30/1.64 = ( extended_ereal2 @ ( uminus_uminus_real @ R2 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_275_PInfty__neq__ereal_I1_J,axiom,
% 0.30/1.64 ! [R2: real] :
% 0.30/1.64 ( ( extended_ereal2 @ R2 )
% 0.30/1.64 != extend1289208545_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_276_ereal_Odistinct_I3_J,axiom,
% 0.30/1.64 ! [X1: real] :
% 0.30/1.64 ( ( extended_ereal2 @ X1 )
% 0.30/1.64 != extended_MInfty ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_277_ereal__le__real,axiom,
% 0.30/1.64 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.30/1.64 ( ! [Z4: real] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ X @ ( extended_ereal2 @ Z4 ) )
% 0.30/1.64 => ( ord_le824540014_ereal @ Y4 @ ( extended_ereal2 @ Z4 ) ) )
% 0.30/1.64 => ( ord_le824540014_ereal @ Y4 @ X ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_278_le__ereal__le,axiom,
% 0.30/1.64 ! [A3: extended_ereal,X: real,Y4: real] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ A3 @ ( extended_ereal2 @ X ) )
% 0.30/1.64 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.30/1.64 => ( ord_le824540014_ereal @ A3 @ ( extended_ereal2 @ Y4 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_279_ereal__le__le,axiom,
% 0.30/1.64 ! [Y4: real,A3: extended_ereal,X: real] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ Y4 ) @ A3 )
% 0.30/1.64 => ( ( ord_less_eq_real @ X @ Y4 )
% 0.30/1.64 => ( ord_le824540014_ereal @ ( extended_ereal2 @ X ) @ A3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_280_MInfty__neq__ereal_I1_J,axiom,
% 0.30/1.64 ! [R2: real] :
% 0.30/1.64 ( ( extended_ereal2 @ R2 )
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_281_ereal__cases,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ! [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 != ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( X != extend1289208545_ereal )
% 0.30/1.64 => ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_282_ereal2__cases,axiom,
% 0.30/1.64 ! [X: extended_ereal,Xa3: extended_ereal] :
% 0.30/1.64 ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ! [Ra: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 != ( extended_ereal2 @ Ra ) ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( Xa3 != extend1289208545_ereal ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( Xa3
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ! [R: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 != ( extended_ereal2 @ R ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( Xa3 != extend1289208545_ereal ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( Xa3
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ! [R: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 != ( extended_ereal2 @ R ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( Xa3 != extend1289208545_ereal ) )
% 0.30/1.64 => ~ ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( Xa3
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_283_ereal3__cases,axiom,
% 0.30/1.64 ! [X: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 0.30/1.64 ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ? [Ra: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 = ( extended_ereal2 @ Ra ) )
% 0.30/1.64 => ! [Rb: real] :
% 0.30/1.64 ( Xb
% 0.30/1.64 != ( extended_ereal2 @ Rb ) ) ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ? [Ra: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 = ( extended_ereal2 @ Ra ) )
% 0.30/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ? [Ra: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 = ( extended_ereal2 @ Ra ) )
% 0.30/1.64 => ( Xb
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 0.30/1.64 => ! [Ra: real] :
% 0.30/1.64 ( Xb
% 0.30/1.64 != ( extended_ereal2 @ Ra ) ) ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 0.30/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 0.30/1.64 => ( Xb
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( Xa3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ! [Ra: real] :
% 0.30/1.64 ( Xb
% 0.30/1.64 != ( extended_ereal2 @ Ra ) ) ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( Xa3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ? [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( Xa3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( Xb
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( ? [R: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ! [Ra: real] :
% 0.30/1.64 ( Xb
% 0.30/1.64 != ( extended_ereal2 @ Ra ) ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( ? [R: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( ? [R: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( Xb
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 0.30/1.64 => ! [R: real] :
% 0.30/1.64 ( Xb
% 0.30/1.64 != ( extended_ereal2 @ R ) ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 0.30/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 0.30/1.64 => ( Xb
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( ( Xa3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ! [R: real] :
% 0.30/1.64 ( Xb
% 0.30/1.64 != ( extended_ereal2 @ R ) ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( ( Xa3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ( X = extend1289208545_ereal )
% 0.30/1.64 => ( ( Xa3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( Xb
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ? [R: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ! [Ra: real] :
% 0.30/1.64 ( Xb
% 0.30/1.64 != ( extended_ereal2 @ Ra ) ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ? [R: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ? [R: real] :
% 0.30/1.64 ( Xa3
% 0.30/1.64 = ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( Xb
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 0.30/1.64 => ! [R: real] :
% 0.30/1.64 ( Xb
% 0.30/1.64 != ( extended_ereal2 @ R ) ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 0.30/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 0.30/1.64 => ( Xb
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ( Xa3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ! [R: real] :
% 0.30/1.64 ( Xb
% 0.30/1.64 != ( extended_ereal2 @ R ) ) ) )
% 0.30/1.64 => ( ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ( Xa3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 0.30/1.64 => ~ ( ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ( Xa3
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( Xb
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_284_ereal__ex__split,axiom,
% 0.30/1.64 ( ( ^ [P3: extended_ereal > $o] :
% 0.30/1.64 ? [X7: extended_ereal] : ( P3 @ X7 ) )
% 0.30/1.64 = ( ^ [P4: extended_ereal > $o] :
% 0.30/1.64 ( ( P4 @ extend1289208545_ereal )
% 0.30/1.64 | ? [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
% 0.30/1.64 | ( P4 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_285_abs__ereal_Ocases,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ! [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 != ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( X
% 0.30/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( X = extend1289208545_ereal ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_286_ereal__all__split,axiom,
% 0.30/1.64 ( ( ^ [P3: extended_ereal > $o] :
% 0.30/1.64 ! [X7: extended_ereal] : ( P3 @ X7 ) )
% 0.30/1.64 = ( ^ [P4: extended_ereal > $o] :
% 0.30/1.64 ( ( P4 @ extend1289208545_ereal )
% 0.30/1.64 & ! [X2: real] : ( P4 @ ( extended_ereal2 @ X2 ) )
% 0.30/1.64 & ( P4 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_287_abs__ereal_Oinduct,axiom,
% 0.30/1.64 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.30/1.64 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ( P @ extend1289208545_ereal )
% 0.30/1.64 => ( P @ A0 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_288_less__ereal_Oinduct,axiom,
% 0.30/1.64 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.30/1.64 ( ! [X4: real,Y2: real] : ( P @ ( extended_ereal2 @ X4 ) @ ( extended_ereal2 @ Y2 ) )
% 0.30/1.64 => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
% 0.30/1.64 => ( ! [A5: extended_ereal] : ( P @ A5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) @ extend1289208545_ereal )
% 0.30/1.64 => ( ! [R: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 0.30/1.64 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_289_plus__ereal_Oinduct,axiom,
% 0.30/1.64 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.30/1.64 ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
% 0.30/1.64 => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
% 0.30/1.64 => ( ! [A5: extended_ereal] : ( P @ A5 @ extend1289208545_ereal )
% 0.30/1.64 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ! [P5: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P5 ) )
% 0.30/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_290_times__ereal_Oinduct,axiom,
% 0.30/1.64 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.30/1.64 ( ! [R: real,P5: real] : ( P @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P5 ) )
% 0.30/1.64 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ extend1289208545_ereal )
% 0.30/1.64 => ( ! [R: real] : ( P @ extend1289208545_ereal @ ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ! [R: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( P @ extend1289208545_ereal @ extend1289208545_ereal )
% 0.30/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 0.30/1.64 => ( ( P @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_291_real__of__ereal_Ocases,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ! [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 != ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( X != extend1289208545_ereal )
% 0.30/1.64 => ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_292_real__of__ereal_Oinduct,axiom,
% 0.30/1.64 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.30/1.64 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( P @ extend1289208545_ereal )
% 0.30/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 => ( P @ A0 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_293_domain__Epigraph__aux,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ( X != extend1289208545_ereal )
% 0.30/1.64 => ? [R: real] : ( ord_le824540014_ereal @ X @ ( extended_ereal2 @ R ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_294_ereal__top,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ! [B6: real] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B6 ) @ X )
% 0.30/1.64 => ( X = extend1289208545_ereal ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_295_Infty__neq__0_I3_J,axiom,
% 0.30/1.64 ( ( uminus1208298309_ereal @ extend1289208545_ereal )
% 0.30/1.64 != zero_z163181189_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_296_ereal_Oinduct,axiom,
% 0.30/1.64 ! [P: extended_ereal > $o,Ereal: extended_ereal] :
% 0.30/1.64 ( ! [X4: real] : ( P @ ( extended_ereal2 @ X4 ) )
% 0.30/1.64 => ( ( P @ extended_PInfty )
% 0.30/1.64 => ( ( P @ extended_MInfty )
% 0.30/1.64 => ( P @ Ereal ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_297_ereal_Oexhaust,axiom,
% 0.30/1.64 ! [Y4: extended_ereal] :
% 0.30/1.64 ( ! [X12: real] :
% 0.30/1.64 ( Y4
% 0.30/1.64 != ( extended_ereal2 @ X12 ) )
% 0.30/1.64 => ( ( Y4 != extended_PInfty )
% 0.30/1.64 => ( Y4 = extended_MInfty ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_298_uminus__ereal_Ocases,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ! [R: real] :
% 0.30/1.64 ( X
% 0.30/1.64 != ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( X != extended_PInfty )
% 0.30/1.64 => ( X = extended_MInfty ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_299_uminus__ereal_Oinduct,axiom,
% 0.30/1.64 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.30/1.64 ( ! [R: real] : ( P @ ( extended_ereal2 @ R ) )
% 0.30/1.64 => ( ( P @ extended_PInfty )
% 0.30/1.64 => ( ( P @ extended_MInfty )
% 0.30/1.64 => ( P @ A0 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_300_ereal__bot,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ! [B6: real] : ( ord_le824540014_ereal @ X @ ( extended_ereal2 @ B6 ) )
% 0.30/1.64 => ( X
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_301_le__numeral__extra_I3_J,axiom,
% 0.30/1.64 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.30/1.64
% 0.30/1.64 thf(fact_302_ereal__less__eq_I4_J,axiom,
% 0.30/1.64 ! [R2: real] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R2 ) @ zero_z163181189_ereal )
% 0.30/1.64 = ( ord_less_eq_real @ R2 @ zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_303_ereal__less__eq_I5_J,axiom,
% 0.30/1.64 ! [R2: real] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extended_ereal2 @ R2 ) )
% 0.30/1.64 = ( ord_less_eq_real @ zero_zero_real @ R2 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_304_ereal__eq__0_I2_J,axiom,
% 0.30/1.64 ! [R2: real] :
% 0.30/1.64 ( ( zero_z163181189_ereal
% 0.30/1.64 = ( extended_ereal2 @ R2 ) )
% 0.30/1.64 = ( R2 = zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_305_ereal__eq__0_I1_J,axiom,
% 0.30/1.64 ! [R2: real] :
% 0.30/1.64 ( ( ( extended_ereal2 @ R2 )
% 0.30/1.64 = zero_z163181189_ereal )
% 0.30/1.64 = ( R2 = zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_306_zero__ereal__def,axiom,
% 0.30/1.64 ( zero_z163181189_ereal
% 0.30/1.64 = ( extended_ereal2 @ zero_zero_real ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_307_real__eq__0__iff__le__ge__0,axiom,
% 0.30/1.64 ! [X: real] :
% 0.30/1.64 ( ( X = zero_zero_real )
% 0.30/1.64 = ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/1.64 & ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_308_ereal__divide__ereal,axiom,
% 0.30/1.64 ! [R2: real] :
% 0.30/1.64 ( ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.30/1.64 => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
% 0.30/1.64 = extend1289208545_ereal ) )
% 0.30/1.64 & ( ~ ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.30/1.64 => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
% 0.30/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_309_ereal__uminus__divide,axiom,
% 0.30/1.64 ! [X: extended_ereal,Y4: extended_ereal] :
% 0.30/1.64 ( ( divide595620860_ereal @ ( uminus1208298309_ereal @ X ) @ Y4 )
% 0.30/1.64 = ( uminus1208298309_ereal @ ( divide595620860_ereal @ X @ Y4 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_310_ereal__divide__zero__left,axiom,
% 0.30/1.64 ! [A3: extended_ereal] :
% 0.30/1.64 ( ( divide595620860_ereal @ zero_z163181189_ereal @ A3 )
% 0.30/1.64 = zero_z163181189_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_311_ereal__divide__Infty_I1_J,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ( divide595620860_ereal @ X @ extend1289208545_ereal )
% 0.30/1.64 = zero_z163181189_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_312_ereal__divide__Infty_I2_J,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ( divide595620860_ereal @ X @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.30/1.64 = zero_z163181189_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_313_zero__le__divide__ereal,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ A3 )
% 0.30/1.64 => ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ B3 )
% 0.30/1.64 => ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( divide595620860_ereal @ A3 @ B3 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_314_div__0,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( divide_divide_real @ zero_zero_real @ A3 )
% 0.30/1.64 = zero_zero_real ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_315_div__by__0,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( divide_divide_real @ A3 @ zero_zero_real )
% 0.30/1.64 = zero_zero_real ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_316_division__ring__divide__zero,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( divide_divide_real @ A3 @ zero_zero_real )
% 0.30/1.64 = zero_zero_real ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_317_divide__eq__0__iff,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ( divide_divide_real @ A3 @ B3 )
% 0.30/1.64 = zero_zero_real )
% 0.30/1.64 = ( ( A3 = zero_zero_real )
% 0.30/1.64 | ( B3 = zero_zero_real ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_318_divide__cancel__left,axiom,
% 0.30/1.64 ! [C: real,A3: real,B3: real] :
% 0.30/1.64 ( ( ( divide_divide_real @ C @ A3 )
% 0.30/1.64 = ( divide_divide_real @ C @ B3 ) )
% 0.30/1.64 = ( ( C = zero_zero_real )
% 0.30/1.64 | ( A3 = B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_319_divide__cancel__right,axiom,
% 0.30/1.64 ! [A3: real,C: real,B3: real] :
% 0.30/1.64 ( ( ( divide_divide_real @ A3 @ C )
% 0.30/1.64 = ( divide_divide_real @ B3 @ C ) )
% 0.30/1.64 = ( ( C = zero_zero_real )
% 0.30/1.64 | ( A3 = B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_320_minus__divide__left,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
% 0.30/1.64 = ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_321_minus__divide__divide,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
% 0.30/1.64 = ( divide_divide_real @ A3 @ B3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_322_minus__divide__right,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
% 0.30/1.64 = ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_323_divide__right__mono__neg,axiom,
% 0.30/1.64 ! [A3: real,B3: real,C: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/1.64 => ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( divide_divide_real @ A3 @ C ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_324_divide__nonpos__nonpos,axiom,
% 0.30/1.64 ! [X: real,Y4: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/1.64 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 0.30/1.64 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_325_divide__nonpos__nonneg,axiom,
% 0.30/1.64 ! [X: real,Y4: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/1.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.30/1.64 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_326_divide__nonneg__nonpos,axiom,
% 0.30/1.64 ! [X: real,Y4: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/1.64 => ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
% 0.30/1.64 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_327_divide__nonneg__nonneg,axiom,
% 0.30/1.64 ! [X: real,Y4: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/1.64 => ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
% 0.30/1.64 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_328_zero__le__divide__iff,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A3 @ B3 ) )
% 0.30/1.64 = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
% 0.30/1.64 & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
% 0.30/1.64 | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
% 0.30/1.64 & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_329_divide__right__mono,axiom,
% 0.30/1.64 ! [A3: real,B3: real,C: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/1.64 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/1.64 => ( ord_less_eq_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_330_divide__le__0__iff,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ B3 ) @ zero_zero_real )
% 0.30/1.64 = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
% 0.30/1.64 & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
% 0.30/1.64 | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
% 0.30/1.64 & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_331_nonzero__minus__divide__divide,axiom,
% 0.30/1.64 ! [B3: real,A3: real] :
% 0.30/1.64 ( ( B3 != zero_zero_real )
% 0.30/1.64 => ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
% 0.30/1.64 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_332_nonzero__minus__divide__right,axiom,
% 0.30/1.64 ! [B3: real,A3: real] :
% 0.30/1.64 ( ( B3 != zero_zero_real )
% 0.30/1.64 => ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
% 0.30/1.64 = ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_333_real__of__ereal__le__0,axiom,
% 0.30/1.64 ! [X: extended_ereal] :
% 0.30/1.64 ( ( ord_less_eq_real @ ( extend1716541707_ereal @ X ) @ zero_zero_real )
% 0.30/1.64 = ( ( ord_le824540014_ereal @ X @ zero_z163181189_ereal )
% 0.30/1.64 | ( X = extend1289208545_ereal ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_334_ereal__divide,axiom,
% 0.30/1.64 ! [P2: real,R2: real] :
% 0.30/1.64 ( ( ( P2 = zero_zero_real )
% 0.30/1.64 => ( ( divide595620860_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 0.30/1.64 = ( times_1966848393_ereal @ ( extended_ereal2 @ R2 ) @ extend1289208545_ereal ) ) )
% 0.30/1.64 & ( ( P2 != zero_zero_real )
% 0.30/1.64 => ( ( divide595620860_ereal @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P2 ) )
% 0.30/1.64 = ( extended_ereal2 @ ( divide_divide_real @ R2 @ P2 ) ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_335_mult__zero__left,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( times_times_real @ zero_zero_real @ A3 )
% 0.30/1.64 = zero_zero_real ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_336_mult__zero__right,axiom,
% 0.30/1.64 ! [A3: real] :
% 0.30/1.64 ( ( times_times_real @ A3 @ zero_zero_real )
% 0.30/1.64 = zero_zero_real ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_337_mult__eq__0__iff,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( ( times_times_real @ A3 @ B3 )
% 0.30/1.64 = zero_zero_real )
% 0.30/1.64 = ( ( A3 = zero_zero_real )
% 0.30/1.64 | ( B3 = zero_zero_real ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_338_mult__cancel__left,axiom,
% 0.30/1.64 ! [C: real,A3: real,B3: real] :
% 0.30/1.64 ( ( ( times_times_real @ C @ A3 )
% 0.30/1.64 = ( times_times_real @ C @ B3 ) )
% 0.30/1.64 = ( ( C = zero_zero_real )
% 0.30/1.64 | ( A3 = B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_339_mult__cancel__right,axiom,
% 0.30/1.64 ! [A3: real,C: real,B3: real] :
% 0.30/1.64 ( ( ( times_times_real @ A3 @ C )
% 0.30/1.64 = ( times_times_real @ B3 @ C ) )
% 0.30/1.64 = ( ( C = zero_zero_real )
% 0.30/1.64 | ( A3 = B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_340_mult__minus__right,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) )
% 0.30/1.64 = ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_341_minus__mult__minus,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
% 0.30/1.64 = ( times_times_real @ A3 @ B3 ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_342_mult__minus__left,axiom,
% 0.30/1.64 ! [A3: real,B3: real] :
% 0.30/1.64 ( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
% 0.30/1.64 = ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_343_ereal__mult__minus__right,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( times_1966848393_ereal @ A3 @ ( uminus1208298309_ereal @ B3 ) )
% 0.30/1.64 = ( uminus1208298309_ereal @ ( times_1966848393_ereal @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_344_ereal__mult__minus__left,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( times_1966848393_ereal @ ( uminus1208298309_ereal @ A3 ) @ B3 )
% 0.30/1.64 = ( uminus1208298309_ereal @ ( times_1966848393_ereal @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_345_ereal__mult__zero,axiom,
% 0.30/1.64 ! [A3: extended_ereal] :
% 0.30/1.64 ( ( times_1966848393_ereal @ A3 @ zero_z163181189_ereal )
% 0.30/1.64 = zero_z163181189_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_346_ereal__zero__mult,axiom,
% 0.30/1.64 ! [A3: extended_ereal] :
% 0.30/1.64 ( ( times_1966848393_ereal @ zero_z163181189_ereal @ A3 )
% 0.30/1.64 = zero_z163181189_ereal ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_347_ereal__zero__times,axiom,
% 0.30/1.64 ! [A3: extended_ereal,B3: extended_ereal] :
% 0.30/1.64 ( ( ( times_1966848393_ereal @ A3 @ B3 )
% 0.30/1.64 = zero_z163181189_ereal )
% 0.30/1.64 = ( ( A3 = zero_z163181189_ereal )
% 0.30/1.64 | ( B3 = zero_z163181189_ereal ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_348_ereal__times__divide__eq__left,axiom,
% 0.30/1.64 ! [B3: extended_ereal,C: extended_ereal,A3: extended_ereal] :
% 0.30/1.64 ( ( times_1966848393_ereal @ ( divide595620860_ereal @ B3 @ C ) @ A3 )
% 0.30/1.64 = ( divide595620860_ereal @ ( times_1966848393_ereal @ B3 @ A3 ) @ C ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_349_mult__divide__mult__cancel__left__if,axiom,
% 0.30/1.64 ! [C: real,A3: real,B3: real] :
% 0.30/1.64 ( ( ( C = zero_zero_real )
% 0.30/1.64 => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
% 0.30/1.64 = zero_zero_real ) )
% 0.30/1.64 & ( ( C != zero_zero_real )
% 0.30/1.64 => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
% 0.30/1.64 = ( divide_divide_real @ A3 @ B3 ) ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_350_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/1.64 ! [C: real,A3: real,B3: real] :
% 0.30/1.64 ( ( C != zero_zero_real )
% 0.30/1.64 => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
% 0.30/1.64 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_351_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/1.64 ! [C: real,A3: real,B3: real] :
% 0.30/1.64 ( ( C != zero_zero_real )
% 0.30/1.64 => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ B3 @ C ) )
% 0.30/1.64 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_352_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/1.64 ! [C: real,A3: real,B3: real] :
% 0.30/1.64 ( ( C != zero_zero_real )
% 0.30/1.64 => ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
% 0.30/1.64 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(fact_353_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/1.64 ! [C: real,A3: real,B3: real] :
% 0.30/1.64 ( ( C != zero_zero_real )
% 0.30/1.64 => ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ C @ B3 ) )
% 0.30/1.64 = ( divide_divide_real @ A3 @ B3 ) ) ) ).
% 0.30/1.64
% 0.30/1.64 thf(conj_0,conjecture,
% 0.30/1.64 ( ( lower_311861424x_on_a @ top_top_set_a @ f )
% 0.30/1.64 = ( convex_a @ ( lower_1391529426main_a @ f ) ) ) ).
% 0.30/1.64 ------------------------
% 0.30/1.66 % (9940)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9742 for (2999ds/183Mi)
% 0.30/1.66 % (9944)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_9742 for (2999ds/2Mi)
% 0.30/1.66 % (9945)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_9742 for (2999ds/275Mi)
% 0.30/1.66 % (9942)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_9742 for (2999ds/27Mi)
% 0.30/1.66 % (9941)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_9742 for (2999ds/4Mi)
% 0.30/1.66 % (9944)Instruction limit reached!
% 0.30/1.66 % (9944)------------------------------
% 0.30/1.66 % (9944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.66 % (9944)Termination reason: Unknown
% 0.30/1.66 % (9944)Termination phase: shuffling
% 0.30/1.66
% 0.30/1.66 % (9944)Memory used [KB]: 1407
% 0.30/1.66 % (9944)Time elapsed: 0.002 s
% 0.30/1.66 % (9944)Instructions burned: 3 (million)
% 0.30/1.66 % (9944)------------------------------
% 0.30/1.66 % (9944)------------------------------
% 0.30/1.66 % (9941)Instruction limit reached!
% 0.30/1.66 % (9941)------------------------------
% 0.30/1.66 % (9941)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.66 % (9941)Termination reason: Unknown
% 0.30/1.66 % (9941)Termination phase: shuffling
% 0.30/1.66
% 0.30/1.66 % (9941)Memory used [KB]: 1535
% 0.30/1.66 % (9941)Time elapsed: 0.003 s
% 0.30/1.66 % (9941)Instructions burned: 6 (million)
% 0.30/1.66 % (9941)------------------------------
% 0.30/1.66 % (9941)------------------------------
% 0.30/1.66 % (9946)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9742 for (2999ds/18Mi)
% 0.30/1.66 % (9943)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9742 for (2999ds/2Mi)
% 0.30/1.66 % (9943)Instruction limit reached!
% 0.30/1.66 % (9943)------------------------------
% 0.30/1.66 % (9943)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.66 % (9943)Termination reason: Unknown
% 0.30/1.66 % (9943)Termination phase: shuffling
% 0.30/1.66
% 0.30/1.66 % (9943)Memory used [KB]: 1407
% 0.30/1.66 % (9943)Time elapsed: 0.004 s
% 0.30/1.66 % (9943)Instructions burned: 3 (million)
% 0.30/1.66 % (9943)------------------------------
% 0.30/1.66 % (9943)------------------------------
% 0.30/1.67 % (9942)Instruction limit reached!
% 0.30/1.67 % (9942)------------------------------
% 0.30/1.67 % (9942)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.67 % (9942)Termination reason: Unknown
% 0.30/1.67 % (9942)Termination phase: Property scanning
% 0.30/1.67
% 0.30/1.67 % (9942)Memory used [KB]: 1791
% 0.30/1.67 % (9942)Time elapsed: 0.010 s
% 0.30/1.67 % (9942)Instructions burned: 29 (million)
% 0.30/1.67 % (9942)------------------------------
% 0.30/1.67 % (9942)------------------------------
% 0.30/1.67 % (9946)Instruction limit reached!
% 0.30/1.67 % (9946)------------------------------
% 0.30/1.67 % (9946)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.67 % (9946)Termination reason: Unknown
% 0.30/1.67 % (9946)Termination phase: shuffling
% 0.30/1.67
% 0.30/1.67 % (9946)Memory used [KB]: 1791
% 0.30/1.67 % (9946)Time elapsed: 0.007 s
% 0.30/1.67 % (9946)Instructions burned: 19 (million)
% 0.30/1.67 % (9946)------------------------------
% 0.30/1.67 % (9946)------------------------------
% 0.30/1.67 % (9947)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9742 for (2999ds/3Mi)
% 0.30/1.67 % (9947)Instruction limit reached!
% 0.30/1.67 % (9947)------------------------------
% 0.30/1.67 % (9947)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.67 % (9947)Termination reason: Unknown
% 0.30/1.67 % (9947)Termination phase: shuffling
% 0.30/1.67
% 0.30/1.67 % (9947)Memory used [KB]: 1407
% 0.30/1.67 % (9947)Time elapsed: 0.002 s
% 0.30/1.67 % (9947)Instructions burned: 3 (million)
% 0.30/1.67 % (9947)------------------------------
% 0.30/1.67 % (9947)------------------------------
% 0.30/1.67 % (9948)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_9742 for (2999ds/37Mi)
% 0.30/1.68 % (9951)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_9742 for (2999ds/1041Mi)
% 0.30/1.68 % (9945)First to succeed.
% 0.30/1.68 % (9952)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9742 for (2999ds/7Mi)
% 0.30/1.68 % (9945)Refutation found. Thanks to Tanya!
% 0.30/1.68 % SZS status Theorem for DTF2THF_9742
% 0.30/1.68 % SZS output start Proof for DTF2THF_9742
% 0.30/1.68 thf(type_def_5, type, set_Extended_ereal_o: $tType).
% 0.30/1.68 thf(type_def_6, type, extended_ereal: $tType).
% 0.30/1.68 thf(type_def_8, type, set_a_o: $tType).
% 0.30/1.68 thf(type_def_9, type, a: $tType).
% 0.30/1.68 thf(type_def_10, type, set_Extended_ereal: $tType).
% 0.30/1.68 thf(type_def_11, type, set_real: $tType).
% 0.30/1.68 thf(type_def_12, type, real: $tType).
% 0.30/1.68 thf(type_def_13, type, set_se767749006_ereal: $tType).
% 0.30/1.68 thf(type_def_14, type, set_se1849684334_ereal: $tType).
% 0.30/1.68 thf(type_def_15, type, set_set_set_a: $tType).
% 0.30/1.68 thf(type_def_16, type, set_set_a: $tType).
% 0.30/1.68 thf(type_def_17, type, set_a: $tType).
% 0.30/1.68 thf(type_def_18, type, set_Pr1158285382l_real: $tType).
% 0.30/1.68 thf(type_def_19, type, set_Pr1928503567a_real: $tType).
% 0.30/1.68 thf(type_def_20, type, set_Pr597221087l_real: $tType).
% 0.30/1.68 thf(type_def_21, type, product_prod_a_real: $tType).
% 0.30/1.68 thf(func_def_0, type, set_Pr1158285382l_real: $tType).
% 0.30/1.68 thf(func_def_1, type, set_Pr597221087l_real: $tType).
% 0.30/1.68 thf(func_def_2, type, set_se1849684334_ereal: $tType).
% 0.30/1.68 thf(func_def_3, type, set_Pr1928503567a_real: $tType).
% 0.30/1.68 thf(func_def_4, type, set_se767749006_ereal: $tType).
% 0.30/1.68 thf(func_def_5, type, set_Extended_ereal_o: $tType).
% 0.30/1.68 thf(func_def_6, type, set_set_set_a: $tType).
% 0.30/1.68 thf(func_def_7, type, product_prod_a_real: $tType).
% 0.30/1.68 thf(func_def_8, type, set_Extended_ereal: $tType).
% 0.30/1.68 thf(func_def_9, type, set_set_a: $tType).
% 0.30/1.68 thf(func_def_10, type, set_a_o: $tType).
% 0.30/1.68 thf(func_def_11, type, set_real: $tType).
% 0.30/1.68 thf(func_def_12, type, extended_ereal: $tType).
% 0.30/1.68 thf(func_def_13, type, set_a: $tType).
% 0.30/1.68 thf(func_def_14, type, real: $tType).
% 0.30/1.68 thf(func_def_15, type, a: $tType).
% 0.30/1.68 thf(func_def_16, type, comple179807490real_o: set_Extended_ereal_o > extended_ereal > $o).
% 0.30/1.68 thf(func_def_17, type, complete_Sup_Sup_a_o: set_a_o > a > $o).
% 0.30/1.68 thf(func_def_18, type, comple1161760187_ereal: set_Extended_ereal > extended_ereal).
% 0.30/1.68 thf(func_def_19, type, comple2129349247p_real: set_real > real).
% 0.30/1.68 thf(func_def_20, type, comple767846299_ereal: set_se767749006_ereal > set_Extended_ereal).
% 0.30/1.68 thf(func_def_21, type, comple1517911419_ereal: set_se1849684334_ereal > set_se767749006_ereal).
% 0.30/1.68 thf(func_def_22, type, comple968993579_set_a: set_set_set_a > set_set_a).
% 0.30/1.68 thf(func_def_23, type, comple1766734283_set_a: set_set_a > set_a).
% 0.30/1.68 thf(func_def_24, type, convex291263275l_real: set_Pr1158285382l_real > $o).
% 0.30/1.68 thf(func_def_25, type, convex78185044a_real: set_Pr1928503567a_real > $o).
% 0.30/1.68 thf(func_def_26, type, convex_a: set_a > $o).
% 0.30/1.68 thf(func_def_27, type, extend1289208545_ereal: extended_ereal).
% 0.30/1.68 thf(func_def_28, type, extended_MInfty: extended_ereal).
% 0.30/1.68 thf(func_def_29, type, extended_PInfty: extended_ereal).
% 0.30/1.68 thf(func_def_30, type, extended_ereal2: real > extended_ereal).
% 0.30/1.68 thf(func_def_31, type, extend1716541707_ereal: extended_ereal > real).
% 0.30/1.68 thf(func_def_32, type, times_1966848393_ereal: extended_ereal > extended_ereal > extended_ereal).
% 0.30/1.68 thf(func_def_33, type, times_times_real: real > real > real).
% 0.30/1.68 thf(func_def_34, type, uminus1208298309_ereal: extended_ereal > extended_ereal).
% 0.30/1.68 thf(func_def_35, type, uminus_uminus_real: real > real).
% 0.30/1.68 thf(func_def_36, type, uminus360668453_ereal: set_Extended_ereal > set_Extended_ereal).
% 0.30/1.68 thf(func_def_37, type, uminus_uminus_set_a: set_a > set_a).
% 0.30/1.68 thf(func_def_38, type, zero_z163181189_ereal: extended_ereal).
% 0.30/1.68 thf(func_def_39, type, zero_zero_real: real).
% 0.30/1.68 thf(func_def_40, type, lower_331963542_ereal: set_Extended_ereal > (extended_ereal > extended_ereal) > set_Pr597221087l_real).
% 0.30/1.68 thf(func_def_41, type, lower_870348689a_real: set_Pr1928503567a_real > (product_prod_a_real > extended_ereal) > set_Pr1158285382l_real).
% 0.30/1.68 thf(func_def_42, type, lower_930854854raph_a: set_a > (a > extended_ereal) > set_Pr1928503567a_real).
% 0.30/1.68 thf(func_def_43, type, lower_128423320e_on_a: set_a > (a > extended_ereal) > $o).
% 0.30/1.68 thf(func_def_44, type, lower_1023700310e_on_a: set_a > (a > extended_ereal) > $o).
% 0.30/1.68 thf(func_def_45, type, lower_591730471a_real: set_Pr1928503567a_real > (product_prod_a_real > extended_ereal) > $o).
% 0.30/1.68 thf(func_def_46, type, lower_311861424x_on_a: set_a > (a > extended_ereal) > $o).
% 0.30/1.68 thf(func_def_47, type, lower_1359521925a_real: (product_prod_a_real > extended_ereal) > set_Pr1928503567a_real).
% 0.30/1.68 thf(func_def_48, type, lower_1391529426main_a: (a > extended_ereal) > set_a).
% 0.30/1.68 thf(func_def_49, type, lower_1486931688e_on_a: set_a > (a > extended_ereal) > $o).
% 0.30/1.68 thf(func_def_50, type, ord_le824540014_ereal: extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_51, type, ord_less_eq_real: real > real > $o).
% 0.30/1.68 thf(func_def_52, type, ord_le88246606_ereal: set_Extended_ereal > set_Extended_ereal > $o).
% 0.30/1.68 thf(func_def_53, type, ord_le1586073967a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > $o).
% 0.30/1.68 thf(func_def_54, type, ord_less_eq_set_real: set_real > set_real > $o).
% 0.30/1.68 thf(func_def_55, type, ord_le1153389358_ereal: set_se767749006_ereal > set_se767749006_ereal > $o).
% 0.30/1.68 thf(func_def_56, type, ord_le318720350_set_a: set_set_a > set_set_a > $o).
% 0.30/1.68 thf(func_def_57, type, ord_less_eq_set_a: set_a > set_a > $o).
% 0.30/1.68 thf(func_def_58, type, order_1158471719_ereal: (extended_ereal > $o) > extended_ereal).
% 0.30/1.68 thf(func_def_59, type, order_Greatest_real: (real > $o) > real).
% 0.30/1.68 thf(func_def_60, type, top_to398855007real_o: extended_ereal > $o).
% 0.30/1.68 thf(func_def_61, type, top_top_a_o: a > $o).
% 0.30/1.68 thf(func_def_63, type, top_to802031902_ereal: extended_ereal).
% 0.30/1.68 thf(func_def_64, type, top_to772344469real_o: set_Extended_ereal_o).
% 0.30/1.68 thf(func_def_65, type, top_top_set_a_o: set_a_o).
% 0.30/1.68 thf(func_def_66, type, top_to1767659262_ereal: set_Extended_ereal).
% 0.30/1.68 thf(func_def_67, type, top_to2138011583a_real: set_Pr1928503567a_real).
% 0.30/1.68 thf(func_def_68, type, top_top_set_real: set_real).
% 0.30/1.68 thf(func_def_69, type, top_to692740318_ereal: set_se767749006_ereal).
% 0.30/1.68 thf(func_def_70, type, top_to1791633086_ereal: set_se1849684334_ereal).
% 0.30/1.68 thf(func_def_71, type, top_to1486684270_set_a: set_set_set_a).
% 0.30/1.68 thf(func_def_72, type, top_top_set_set_a: set_set_a).
% 0.30/1.68 thf(func_def_73, type, top_top_set_a: set_a).
% 0.30/1.68 thf(func_def_74, type, divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal).
% 0.30/1.68 thf(func_def_75, type, divide_divide_real: real > real > real).
% 0.30/1.68 thf(func_def_76, type, collec247695033_ereal: (extended_ereal > $o) > set_Extended_ereal).
% 0.30/1.68 thf(func_def_77, type, collec1714955950a_real: (product_prod_a_real > $o) > set_Pr1928503567a_real).
% 0.30/1.68 thf(func_def_78, type, collect_a: (a > $o) > set_a).
% 0.30/1.68 thf(func_def_79, type, member1900190071_ereal: extended_ereal > set_Extended_ereal > $o).
% 0.30/1.68 thf(func_def_80, type, member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o).
% 0.30/1.68 thf(func_def_81, type, member_real: real > set_real > $o).
% 0.30/1.68 thf(func_def_82, type, member450560855_ereal: set_Extended_ereal > set_se767749006_ereal > $o).
% 0.30/1.68 thf(func_def_83, type, member_set_a: set_a > set_set_a > $o).
% 0.30/1.68 thf(func_def_84, type, member_a: a > set_a > $o).
% 0.30/1.68 thf(func_def_85, type, f: a > extended_ereal).
% 0.30/1.68 thf(func_def_99, type, sP0: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_100, type, sP1: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_101, type, sP2: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_102, type, sP3: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_103, type, sP4: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_104, type, sP5: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_105, type, sP6: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_106, type, sP7: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_107, type, sP8: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_108, type, sP9: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_109, type, sP10: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_110, type, sP11: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_111, type, sP12: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_112, type, sP13: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_113, type, sP14: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_114, type, sP15: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_115, type, sP16: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_116, type, sP17: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_117, type, sP18: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_118, type, sP19: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_119, type, sP20: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_120, type, sP21: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_121, type, sP22: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_122, type, sP23: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_123, type, sP24: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_124, type, sP25: extended_ereal > extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_125, type, sP26: extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_126, type, sP27: extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_127, type, sP28: extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_128, type, sP29: extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_129, type, sP30: extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_130, type, sP31: extended_ereal > extended_ereal > $o).
% 0.30/1.68 thf(func_def_131, type, sK32: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 0.30/1.68 thf(func_def_132, type, sK33: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_133, type, sK34: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_134, type, sK35: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 0.30/1.68 thf(func_def_135, type, sK36: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_136, type, sK37: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_137, type, sK38: (set_a > real) > set_a).
% 0.30/1.68 thf(func_def_138, type, sK39: (set_a > real) > set_a).
% 0.30/1.68 thf(func_def_139, type, sK40: (extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_140, type, sK41: extended_ereal > real).
% 0.30/1.68 thf(func_def_141, type, sK42: extended_ereal > real).
% 0.30/1.68 thf(func_def_142, type, sK43: extended_ereal > real).
% 0.30/1.68 thf(func_def_143, type, sK44: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 0.30/1.68 thf(func_def_144, type, sK45: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_145, type, sK46: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_146, type, sK47: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_147, type, sK48: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 0.30/1.68 thf(func_def_148, type, sK49: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_149, type, sK50: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_150, type, sK51: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_151, type, sK52: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_152, type, sK53: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_153, type, sK54: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_154, type, sK55: (extended_ereal > extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_155, type, sK56: extended_ereal > real).
% 0.30/1.68 thf(func_def_156, type, sK57: extended_ereal > real).
% 0.30/1.68 thf(func_def_157, type, sK58: extended_ereal > real).
% 0.30/1.68 thf(func_def_158, type, sK59: extended_ereal > real).
% 0.30/1.68 thf(func_def_159, type, sK60: extended_ereal > real).
% 0.30/1.68 thf(func_def_160, type, sK61: extended_ereal > real).
% 0.30/1.68 thf(func_def_161, type, sK62: extended_ereal > real).
% 0.30/1.68 thf(func_def_162, type, sK63: extended_ereal > real).
% 0.30/1.68 thf(func_def_163, type, sK64: extended_ereal > real).
% 0.30/1.68 thf(func_def_164, type, sK65: extended_ereal > real).
% 0.30/1.68 thf(func_def_165, type, sK66: extended_ereal > real).
% 0.30/1.68 thf(func_def_166, type, sK67: extended_ereal > real).
% 0.30/1.68 thf(func_def_167, type, sK68: extended_ereal > real).
% 0.30/1.68 thf(func_def_168, type, sK69: extended_ereal > real).
% 0.30/1.68 thf(func_def_169, type, sK70: extended_ereal > real).
% 0.30/1.68 thf(func_def_170, type, sK71: extended_ereal > real).
% 0.30/1.68 thf(func_def_171, type, sK72: extended_ereal > real).
% 0.30/1.68 thf(func_def_172, type, sK73: extended_ereal > real).
% 0.30/1.68 thf(func_def_173, type, sK74: extended_ereal > real).
% 0.30/1.68 thf(func_def_174, type, sK75: extended_ereal > real).
% 0.30/1.68 thf(func_def_175, type, sK76: extended_ereal > real).
% 0.30/1.68 thf(func_def_176, type, sK77: extended_ereal > real).
% 0.30/1.68 thf(func_def_177, type, sK78: extended_ereal > real).
% 0.30/1.68 thf(func_def_178, type, sK79: extended_ereal > real).
% 0.30/1.68 thf(func_def_179, type, sK80: extended_ereal > real).
% 0.30/1.68 thf(func_def_180, type, sK81: extended_ereal > real).
% 0.30/1.68 thf(func_def_181, type, sK82: extended_ereal > real).
% 0.30/1.68 thf(func_def_182, type, sK83: extended_ereal > real).
% 0.30/1.68 thf(func_def_183, type, sK84: extended_ereal > real).
% 0.30/1.68 thf(func_def_184, type, sK85: extended_ereal > real).
% 0.30/1.68 thf(func_def_185, type, sK86: extended_ereal > real).
% 0.30/1.68 thf(func_def_186, type, sK87: extended_ereal > real).
% 0.30/1.68 thf(func_def_187, type, sK88: extended_ereal > real).
% 0.30/1.68 thf(func_def_188, type, sK89: extended_ereal > real).
% 0.30/1.68 thf(func_def_189, type, sK90: (extended_ereal > set_a) > extended_ereal).
% 0.30/1.68 thf(func_def_190, type, sK91: (extended_ereal > set_a) > extended_ereal).
% 0.30/1.68 thf(func_def_191, type, sK92: extended_ereal > real).
% 0.30/1.68 thf(func_def_192, type, sK93: (a > extended_ereal) > set_a).
% 0.30/1.68 thf(func_def_193, type, sK94: extended_ereal > real).
% 0.30/1.68 thf(func_def_194, type, sK95: (real > set_a) > real).
% 0.30/1.68 thf(func_def_195, type, sK96: (real > set_a) > real).
% 0.30/1.68 thf(func_def_196, type, sK97: (set_a > extended_ereal) > set_a).
% 0.30/1.68 thf(func_def_197, type, sK98: (set_a > extended_ereal) > set_a).
% 0.30/1.68 thf(func_def_198, type, sK99: (a > extended_ereal) > set_a).
% 0.30/1.68 thf(func_def_199, type, sK100: (extended_ereal > $o) > real).
% 0.30/1.68 thf(func_def_201, type, ph102: !>[X0: $tType]:(X0)).
% 0.30/1.68 thf(f1388,plain,(
% 0.30/1.68 $false),
% 0.30/1.68 inference(avatar_sat_refutation,[status(thm)],[f1351,f1377,f1387])).
% 0.30/1.68 thf(f1387,plain,(
% 0.30/1.68 ~spl101_1),
% 0.30/1.68 inference(avatar_contradiction_clause,[status(thm)],[f1386])).
% 0.30/1.68 thf(f1386,plain,(
% 0.30/1.68 $false | ~spl101_1),
% 0.30/1.68 inference(trivial_inequality_removal,[status(thm)],[f1383])).
% 0.30/1.68 thf(f1383,plain,(
% 0.30/1.68 ($false = $true) | ~spl101_1),
% 0.30/1.68 inference(superposition,[status(thm)],[f1346,f1353])).
% 0.30/1.68 thf(f1353,plain,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) = $true)),
% 0.30/1.68 inference(trivial_inequality_removal,[status(thm)],[f1352])).
% 0.30/1.68 thf(f1352,plain,(
% 0.30/1.68 ($true != $true) | ((convex_a @ (lower_1391529426main_a @ f)) = $true)),
% 0.30/1.68 inference(superposition,[status(thm)],[f1319,f1342])).
% 0.30/1.68 thf(f1342,plain,(
% 0.30/1.68 ((lower_311861424x_on_a @ top_top_set_a @ f) = $true)),
% 0.30/1.68 inference(subsumption_resolution,[status(thm)],[f1340,f1312])).
% 0.30/1.68 thf(f1312,plain,(
% 0.30/1.68 ((lower_311861424x_on_a @ top_top_set_a @ f) = $true) | ((convex_a @ (lower_1391529426main_a @ f)) != $true)),
% 0.30/1.68 inference(cnf_transformation,[status(thm)],[f916])).
% 0.30/1.68 thf(f916,plain,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) != $true) | ((lower_311861424x_on_a @ top_top_set_a @ f) = $true)),
% 0.30/1.68 inference(ennf_transformation,[status(thm)],[f841])).
% 0.30/1.68 thf(f841,plain,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) = $true) => ((lower_311861424x_on_a @ top_top_set_a @ f) = $true)),
% 0.30/1.68 inference(fool_elimination,[status(thm)],[f840])).
% 0.30/1.68 thf(f840,plain,(
% 0.30/1.68 (convex_a @ (lower_1391529426main_a @ f)) => (lower_311861424x_on_a @ top_top_set_a @ f)),
% 0.30/1.68 inference(rectify,[status(thm)],[f1])).
% 0.30/1.68 thf(f1,axiom,(
% 0.30/1.68 (convex_a @ (lower_1391529426main_a @ f)) => (lower_311861424x_on_a @ top_top_set_a @ f)),
% 0.30/1.68 file('/export/starexec/sandbox/tmp/tmp.Lwp0UiQMvY/DTF2THF_9742.p',fact_0__092_060open_062convex_A_Idomain_Af_J_A_092_060Longrightarrow_062_ALower__Semicontinuous__Mirabelle__mxyexokbxt_Oconvex__on_AUNIV_Af_092_060close_062)).
% 0.30/1.68 thf(f1340,plain,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) = $true) | ((lower_311861424x_on_a @ top_top_set_a @ f) = $true)),
% 0.30/1.68 inference(binary_proxy_clausification,[status(thm)],[f1308])).
% 0.30/1.68 thf(f1308,plain,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) != (lower_311861424x_on_a @ top_top_set_a @ f))),
% 0.30/1.68 inference(cnf_transformation,[status(thm)],[f888])).
% 0.30/1.68 thf(f888,plain,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) != (lower_311861424x_on_a @ top_top_set_a @ f))),
% 0.30/1.68 inference(flattening,[status(thm)],[f356])).
% 0.30/1.68 thf(f356,negated_conjecture,(
% 0.30/1.68 ~((convex_a @ (lower_1391529426main_a @ f)) = (lower_311861424x_on_a @ top_top_set_a @ f))),
% 0.30/1.68 inference(negated_conjecture,[status(cth)],[f355])).
% 0.30/1.68 thf(f355,conjecture,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) = (lower_311861424x_on_a @ top_top_set_a @ f))),
% 0.30/1.68 file('/export/starexec/sandbox/tmp/tmp.Lwp0UiQMvY/DTF2THF_9742.p',conj_0)).
% 0.30/1.68 thf(f1319,plain,(
% 0.30/1.68 ( ! [X0 : a > extended_ereal] : (((lower_311861424x_on_a @ top_top_set_a @ X0) != $true) | ((convex_a @ (lower_1391529426main_a @ X0)) = $true)) )),
% 0.30/1.68 inference(cnf_transformation,[status(thm)],[f912])).
% 0.30/1.68 thf(f912,plain,(
% 0.30/1.68 ! [X0 : a > extended_ereal] : (((lower_311861424x_on_a @ top_top_set_a @ X0) != $true) | ((convex_a @ (lower_1391529426main_a @ X0)) = $true))),
% 0.30/1.68 inference(ennf_transformation,[status(thm)],[f689])).
% 0.30/1.68 thf(f689,plain,(
% 0.30/1.68 ! [X0 : a > extended_ereal] : (((lower_311861424x_on_a @ top_top_set_a @ X0) = $true) => ((convex_a @ (lower_1391529426main_a @ X0)) = $true))),
% 0.30/1.68 inference(fool_elimination,[status(thm)],[f688])).
% 0.30/1.68 thf(f688,plain,(
% 0.30/1.68 ! [X0 : a > extended_ereal] : ((lower_311861424x_on_a @ top_top_set_a @ X0) => (convex_a @ (lower_1391529426main_a @ X0)))),
% 0.30/1.68 inference(rectify,[status(thm)],[f3])).
% 0.30/1.68 thf(f3,axiom,(
% 0.30/1.68 ! [X0 : a > extended_ereal] : ((lower_311861424x_on_a @ top_top_set_a @ X0) => (convex_a @ (lower_1391529426main_a @ X0)))),
% 0.30/1.68 file('/export/starexec/sandbox/tmp/tmp.Lwp0UiQMvY/DTF2THF_9742.p',fact_2_convex__domain)).
% 0.30/1.68 thf(f1346,plain,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) = $false) | ~spl101_1),
% 0.30/1.68 inference(avatar_component_clause,[status(thm)],[f1344])).
% 0.30/1.68 thf(f1344,definition,(
% 0.30/1.68 spl101_1 <=> ((convex_a @ (lower_1391529426main_a @ f)) = $false)),
% 0.30/1.68 introduced(definition,[new_symbols(naming,[spl101_1])],[avatar_definition])).
% 0.30/1.68 thf(f1377,plain,(
% 0.30/1.68 ~spl101_2),
% 0.30/1.68 inference(avatar_contradiction_clause,[status(thm)],[f1376])).
% 0.30/1.68 thf(f1376,plain,(
% 0.30/1.68 $false | ~spl101_2),
% 0.30/1.68 inference(subsumption_resolution,[status(thm)],[f1363,f1353])).
% 0.30/1.68 thf(f1363,plain,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) != $true) | ~spl101_2),
% 0.30/1.68 inference(trivial_inequality_removal,[status(thm)],[f1355])).
% 0.30/1.68 thf(f1355,plain,(
% 0.30/1.68 ((convex_a @ (lower_1391529426main_a @ f)) != $true) | ($false = $true) | ~spl101_2),
% 0.30/1.68 inference(superposition,[status(thm)],[f1350,f1312])).
% 0.30/1.68 thf(f1350,plain,(
% 0.30/1.68 ((lower_311861424x_on_a @ top_top_set_a @ f) = $false) | ~spl101_2),
% 0.30/1.68 inference(avatar_component_clause,[status(thm)],[f1348])).
% 0.30/1.68 thf(f1348,definition,(
% 0.30/1.68 spl101_2 <=> ((lower_311861424x_on_a @ top_top_set_a @ f) = $false)),
% 0.30/1.68 introduced(definition,[new_symbols(naming,[spl101_2])],[avatar_definition])).
% 0.30/1.68 thf(f1351,plain,(
% 0.30/1.68 spl101_1 | spl101_2),
% 0.30/1.68 inference(avatar_split_clause,[status(thm)],[f1341,f1348,f1344])).
% 0.30/1.68 thf(f1341,plain,(
% 0.30/1.68 ((lower_311861424x_on_a @ top_top_set_a @ f) = $false) | ((convex_a @ (lower_1391529426main_a @ f)) = $false)),
% 0.30/1.68 inference(binary_proxy_clausification,[status(thm)],[f1308])).
% 0.30/1.68 % SZS output end Proof for DTF2THF_9742
% 0.30/1.68 % (9945)------------------------------
% 0.30/1.68 % (9945)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.68 % (9945)Termination reason: Refutation
% 0.30/1.68
% 0.30/1.68 % (9945)Memory used [KB]: 6780
% 0.30/1.68 % (9945)Time elapsed: 0.021 s
% 0.30/1.68 % (9945)Instructions burned: 67 (million)
% 0.30/1.68 % (9945)------------------------------
% 0.30/1.68 % (9945)------------------------------
% 0.30/1.68 % (9939)Success in time 0.032 s
%------------------------------------------------------------------------------