%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP112^1 : 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 : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:39 AM UTC 2026
% Result : Theorem 3.04s 1.71s
% Output : Refutation 3.04s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 4
% Syntax : Number of formulae : 26 ( 18 unt; 0 typ; 0 def)
% Number of atoms : 74 ( 24 equ; 0 cnn)
% Maximal formula atoms : 2 ( 2 avg)
% Number of connectives : 234 ( 12 ~; 5 |; 0 &; 214 @)
% ( 0 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 147 ( 144 usr; 8 con; 0-3 aty)
% Number of variables : 45 ( 38 ^; 7 !; 0 ?; 45 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
extended_ereal: $tType ).
thf(type_def_6,type,
real: $tType ).
thf(type_def_8,type,
filter_real: $tType ).
thf(type_def_9,type,
filter_nat: $tType ).
thf(type_def_10,type,
nat: $tType ).
thf(type_def_11,type,
filter2049122004_ereal: $tType ).
thf(type_def_12,type,
int: $tType ).
thf(type_def_13,type,
filter_int: $tType ).
thf(type_def_14,type,
a: $tType ).
thf(type_def_15,type,
filter_a: $tType ).
thf(type_def_16,type,
set_real: $tType ).
thf(type_def_17,type,
set_Extended_ereal: $tType ).
thf(func_def_0,type,
filter2049122004_ereal: $tType ).
thf(func_def_1,type,
set_Extended_ereal: $tType ).
thf(func_def_2,type,
filter_real: $tType ).
thf(func_def_3,type,
filter_nat: $tType ).
thf(func_def_4,type,
filter_int: $tType ).
thf(func_def_5,type,
set_real: $tType ).
thf(func_def_6,type,
filter_a: $tType ).
thf(func_def_7,type,
extended_ereal: $tType ).
thf(func_def_8,type,
real: $tType ).
thf(func_def_9,type,
nat: $tType ).
thf(func_def_10,type,
int: $tType ).
thf(func_def_11,type,
a: $tType ).
thf(func_def_12,type,
extend1289208545_ereal: extended_ereal ).
thf(func_def_13,type,
extended_ereal2: real > extended_ereal ).
thf(func_def_14,type,
at_bot_real: filter_real ).
thf(func_def_15,type,
at_top_nat: filter_nat ).
thf(func_def_16,type,
at_top_real: filter_real ).
thf(func_def_17,type,
filter1531173832_ereal: ( nat > extended_ereal ) > filter2049122004_ereal > filter_nat > $o ).
thf(func_def_18,type,
filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
thf(func_def_19,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(func_def_20,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(func_def_21,type,
filterlim_nat_a: ( nat > a ) > filter_a > filter_nat > $o ).
thf(func_def_22,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(func_def_23,type,
comp_E1308517939al_nat: ( extended_ereal > extended_ereal ) > ( nat > extended_ereal ) > nat > extended_ereal ).
thf(func_def_24,type,
comp_E489644891real_a: ( extended_ereal > extended_ereal ) > ( a > extended_ereal ) > a > extended_ereal ).
thf(func_def_25,type,
comp_E1436437929nt_nat: ( extended_ereal > int ) > ( nat > extended_ereal ) > nat > int ).
thf(func_def_26,type,
comp_E1523169101at_nat: ( extended_ereal > nat ) > ( nat > extended_ereal ) > nat > nat ).
thf(func_def_27,type,
comp_E1477338153al_nat: ( extended_ereal > real ) > ( nat > extended_ereal ) > nat > real ).
thf(func_def_28,type,
comp_n1096781355al_nat: ( nat > extended_ereal ) > ( nat > nat ) > nat > extended_ereal ).
thf(func_def_29,type,
comp_nat_a_nat: ( nat > a ) > ( nat > nat ) > nat > a ).
thf(func_def_30,type,
comp_r1410008527al_nat: ( real > extended_ereal ) > ( nat > real ) > nat > extended_ereal ).
thf(func_def_31,type,
comp_real_int_nat: ( real > int ) > ( nat > real ) > nat > int ).
thf(func_def_32,type,
comp_real_nat_nat: ( real > nat ) > ( nat > real ) > nat > nat ).
thf(func_def_33,type,
comp_a1112243075al_nat: ( a > extended_ereal ) > ( nat > a ) > nat > extended_ereal ).
thf(func_def_34,type,
comp_a780206603real_a: ( a > extended_ereal ) > ( a > a ) > a > extended_ereal ).
thf(func_def_35,type,
comp_a_int_nat: ( a > int ) > ( nat > a ) > nat > int ).
thf(func_def_36,type,
comp_a_nat_nat: ( a > nat ) > ( nat > a ) > nat > nat ).
thf(func_def_37,type,
comp_a_real_nat: ( a > real ) > ( nat > a ) > nat > real ).
thf(func_def_38,type,
comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).
thf(func_def_39,type,
plus_p2118002693_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(func_def_40,type,
plus_plus_int: int > int > int ).
thf(func_def_41,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_42,type,
plus_plus_real: real > real > real ).
thf(func_def_43,type,
uminus1208298309_ereal: extended_ereal > extended_ereal ).
thf(func_def_44,type,
uminus_uminus_int: int > int ).
thf(func_def_45,type,
uminus_uminus_real: real > real ).
thf(func_def_46,type,
uniq_Extended_ereal: ( extended_ereal > $o ) > $o ).
thf(func_def_47,type,
uniq_real: ( real > $o ) > $o ).
thf(func_def_48,type,
uniq_a: ( a > $o ) > $o ).
thf(func_def_49,type,
liminf1045857232_ereal: filter_nat > ( nat > extended_ereal ) > extended_ereal ).
thf(func_def_50,type,
lower_1087098792_ereal: extended_ereal > ( extended_ereal > extended_ereal ) > $o ).
thf(func_def_51,type,
lower_48196818al_int: extended_ereal > ( extended_ereal > int ) > $o ).
thf(func_def_52,type,
lower_1558406774al_nat: extended_ereal > ( extended_ereal > nat ) > $o ).
thf(func_def_53,type,
lower_1165973074l_real: extended_ereal > ( extended_ereal > real ) > $o ).
thf(func_def_54,type,
lower_551915512_ereal: real > ( real > extended_ereal ) > $o ).
thf(func_def_55,type,
lower_153911426al_int: real > ( real > int ) > $o ).
thf(func_def_56,type,
lower_1664121382al_nat: real > ( real > nat ) > $o ).
thf(func_def_57,type,
lower_191460856_ereal: a > ( a > extended_ereal ) > $o ).
thf(func_def_58,type,
lower_956963458_a_int: a > ( a > int ) > $o ).
thf(func_def_59,type,
lower_319689766_a_nat: a > ( a > nat ) > $o ).
thf(func_def_60,type,
lower_231615490a_real: a > ( a > real ) > $o ).
thf(func_def_61,type,
lower_1071158961_ereal: extended_ereal > ( extended_ereal > extended_ereal ) > $o ).
thf(func_def_62,type,
lower_637387785al_int: extended_ereal > ( extended_ereal > int ) > $o ).
thf(func_def_63,type,
lower_114093al_nat: extended_ereal > ( extended_ereal > nat ) > $o ).
thf(func_def_64,type,
lower_737640969l_real: extended_ereal > ( extended_ereal > real ) > $o ).
thf(func_def_65,type,
lower_1075504779al_int: real > ( real > int ) > $o ).
thf(func_def_66,type,
lower_438231087al_nat: real > ( real > nat ) > $o ).
thf(func_def_67,type,
lower_534855297_ereal: a > ( a > extended_ereal ) > $o ).
thf(func_def_68,type,
lower_1672990777_a_int: a > ( a > int ) > $o ).
thf(func_def_69,type,
lower_1035717085_a_nat: a > ( a > nat ) > $o ).
thf(func_def_70,type,
lower_755922489a_real: a > ( a > real ) > $o ).
thf(func_def_71,type,
semiri2019852685at_int: nat > int ).
thf(func_def_72,type,
semiri2110766477t_real: nat > real ).
thf(func_def_73,type,
ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
thf(func_def_74,type,
ord_le1745708096er_nat: filter_nat > filter_nat > $o ).
thf(func_def_75,type,
ord_le132810396r_real: filter_real > filter_real > $o ).
thf(func_def_76,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_77,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_78,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_79,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_80,type,
topolo1069469409_ereal: ( nat > extended_ereal ) > $o ).
thf(func_def_81,type,
topolo411883481eq_int: ( nat > int ) > $o ).
thf(func_def_82,type,
topolo1922093437eq_nat: ( nat > nat ) > $o ).
thf(func_def_83,type,
topolo144289241q_real: ( nat > real ) > $o ).
thf(func_def_84,type,
topolo2140997059_ereal: extended_ereal > filter2049122004_ereal ).
thf(func_def_85,type,
topolo54776183ds_int: int > filter_int ).
thf(func_def_86,type,
topolo1564986139ds_nat: nat > filter_nat ).
thf(func_def_87,type,
topolo1664202871s_real: real > filter_real ).
thf(func_def_88,type,
topolo705128563nhds_a: a > filter_a ).
thf(func_def_89,type,
member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(func_def_90,type,
member_real: real > set_real > $o ).
thf(func_def_91,type,
a2: extended_ereal ).
thf(func_def_92,type,
f: a > extended_ereal ).
thf(func_def_93,type,
x0: a ).
thf(func_def_94,type,
x: nat > a ).
thf(func_def_110,type,
sP0: nat > nat > nat > $o ).
thf(func_def_111,type,
sP1: nat > nat > nat > $o ).
thf(func_def_112,type,
sP2: nat > nat > nat > $o ).
thf(func_def_113,type,
sP3: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_114,type,
sP4: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_115,type,
sP5: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_116,type,
sK6: ( nat > extended_ereal ) > extended_ereal ).
thf(func_def_117,type,
sK7: ( nat > extended_ereal ) > nat ).
thf(func_def_118,type,
sK8: ( nat > extended_ereal ) > nat ).
thf(func_def_119,type,
sK9: ( nat > nat ) > nat ).
thf(func_def_120,type,
sK10: ( nat > nat ) > nat ).
thf(func_def_121,type,
sK11: nat > ( nat > extended_ereal ) > ( nat > extended_ereal ) > nat ).
thf(func_def_122,type,
sK12: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_123,type,
sK13: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_124,type,
sK14: ( nat > extended_ereal ) > nat ).
thf(func_def_125,type,
sK15: ( nat > extended_ereal ) > nat ).
thf(func_def_126,type,
sK16: ( nat > extended_ereal ) > extended_ereal ).
thf(func_def_127,type,
sK17: ( nat > extended_ereal ) > nat ).
thf(func_def_128,type,
sK18: ( nat > extended_ereal ) > nat ).
thf(func_def_129,type,
sK19: ( nat > extended_ereal ) > nat ).
thf(func_def_130,type,
sK20: ( nat > extended_ereal ) > nat ).
thf(func_def_131,type,
sK21: ( nat > nat ) > nat ).
thf(func_def_132,type,
sK22: ( nat > nat ) > nat ).
thf(func_def_133,type,
sK23: ( nat > extended_ereal ) > nat ).
thf(func_def_134,type,
sK24: ( nat > extended_ereal ) > nat ).
thf(func_def_135,type,
sK25: ( nat > extended_ereal ) > nat ).
thf(func_def_136,type,
sK26: ( nat > extended_ereal ) > nat ).
thf(func_def_137,type,
sK27: nat > extended_ereal > ( nat > extended_ereal ) > nat ).
thf(func_def_138,type,
sK28: ( nat > nat > $o ) > nat ).
thf(func_def_139,type,
sK29: ( nat > nat > $o ) > nat ).
thf(func_def_140,type,
sK30: ( nat > nat > $o ) > nat ).
thf(func_def_141,type,
sK31: ( nat > nat > $o ) > nat ).
thf(func_def_142,type,
sK32: ( nat > nat ) > nat ).
thf(func_def_143,type,
sK33: ( nat > nat ) > nat ).
thf(func_def_144,type,
sK34: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_145,type,
sK35: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_146,type,
sK36: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_147,type,
sK37: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_148,type,
sK38: ( extended_ereal > nat ) > extended_ereal ).
thf(func_def_149,type,
sK39: ( extended_ereal > nat ) > extended_ereal ).
thf(func_def_150,type,
sK40: extended_ereal > nat > ( nat > extended_ereal ) > nat ).
thf(func_def_151,type,
sK41: nat > ( nat > extended_ereal ) > ( nat > extended_ereal ) > nat ).
thf(func_def_152,type,
sK42: nat > extended_ereal > ( nat > extended_ereal ) > nat ).
thf(func_def_153,type,
sK43: ( extended_ereal > nat ) > extended_ereal ).
thf(func_def_154,type,
sK44: ( extended_ereal > nat ) > extended_ereal ).
thf(func_def_155,type,
sK45: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_156,type,
sK46: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_157,type,
sK47: ( nat > nat ) > nat ).
thf(func_def_158,type,
sK48: ( nat > nat ) > nat ).
thf(func_def_159,type,
sK49: nat > extended_ereal > ( nat > extended_ereal ) > nat ).
thf(func_def_160,type,
sK50: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_161,type,
sK51: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_162,type,
sK52: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_163,type,
sK53: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_164,type,
sK54: ( nat > $o ) > nat > nat ).
thf(func_def_165,type,
sK55: ( nat > $o ) > nat ).
thf(func_def_166,type,
sK56: ( extended_ereal > nat ) > extended_ereal ).
thf(func_def_167,type,
sK57: ( extended_ereal > nat ) > extended_ereal ).
thf(func_def_168,type,
sK58: ( extended_ereal > nat ) > extended_ereal ).
thf(func_def_169,type,
sK59: ( extended_ereal > nat ) > extended_ereal ).
thf(func_def_171,type,
ph61:
!>[X0: $tType] : X0 ).
thf(f1414,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1413,f1411]) ).
thf(f1411,plain,
( $true
= ( filter1531173832_ereal
@ ^ [Y0: nat] : ( f @ ( x @ Y0 ) )
@ ( topolo2140997059_ereal @ a2 )
@ at_top_nat ) ),
inference(beta_eta_normalization,[status(thm)],[f1361]) ).
thf(f1361,plain,
( $true
= ( filter1531173832_ereal
@ ( ^ [Y0: a > extended_ereal,Y1: nat > a,Y2: nat] : ( Y0 @ ( Y1 @ Y2 ) )
@ f
@ x )
@ ( topolo2140997059_ereal @ a2 )
@ at_top_nat ) ),
inference(definition_unfolding,[status(thm)],[f1279,f1213]) ).
thf(f1213,plain,
( comp_a1112243075al_nat
= ( ^ [Y0: a > extended_ereal,Y1: nat > a,Y2: nat] : ( Y0 @ ( Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f752]) ).
thf(f752,plain,
( comp_a1112243075al_nat
= ( ^ [Y0: a > extended_ereal,Y1: nat > a,Y2: nat] : ( Y0 @ ( Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f751]) ).
thf(f751,plain,
( ( ^ [X0: a > extended_ereal,X1: nat > a,X2: nat] : ( X0 @ ( X1 @ X2 ) ) )
= comp_a1112243075al_nat ),
inference(rectify,[status(thm)],[f37]) ).
thf(f37,axiom,
( ( ^ [X13: a > extended_ereal,X14: nat > a,X1: nat] : ( X13 @ ( X14 @ X1 ) ) )
= comp_a1112243075al_nat ),
file('/export/starexec/sandbox/tmp/tmp.z9pBe0D4OI/DTF2THF_9897.p',fact_36_comp__apply) ).
thf(f1279,plain,
( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ at_top_nat )
= $true ),
inference(cnf_transformation,[status(thm)],[f894]) ).
thf(f894,plain,
( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ at_top_nat )
= $true ),
inference(fool_elimination,[status(thm)],[f893]) ).
thf(f893,plain,
filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ at_top_nat,
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ at_top_nat,
file('/export/starexec/sandbox/tmp/tmp.z9pBe0D4OI/DTF2THF_9897.p',fact_1_x__def_I2_J) ).
thf(f1413,plain,
( $true
!= ( filter1531173832_ereal
@ ^ [Y0: nat] : ( f @ ( x @ Y0 ) )
@ ( topolo2140997059_ereal @ a2 )
@ at_top_nat ) ),
inference(trivial_inequality_removal,[status(thm)],[f1412]) ).
thf(f1412,plain,
( ( $true != $true )
| ( $true
!= ( filter1531173832_ereal
@ ^ [Y0: nat] : ( f @ ( x @ Y0 ) )
@ ( topolo2140997059_ereal @ a2 )
@ at_top_nat ) ) ),
inference(superposition,[status(thm)],[f1261,f1406]) ).
thf(f1406,plain,
! [X0: filter_nat] :
( ( $true
= ( filter1531173832_ereal
@ ^ [Y0: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ Y0 ) ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ X0 ) )
| ( ( filter1531173832_ereal
@ ^ [Y0: nat] : ( f @ ( x @ Y0 ) )
@ ( topolo2140997059_ereal @ a2 )
@ X0 )
!= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1362]) ).
thf(f1362,plain,
! [X0: filter_nat] :
( ( ( filter1531173832_ereal
@ ^ [Y0: nat] :
( uminus1208298309_ereal
@ ( ^ [Y1: a > extended_ereal,Y2: nat > a,Y3: nat] : ( Y1 @ ( Y2 @ Y3 ) )
@ f
@ x
@ Y0 ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ X0 )
= $true )
| ( ( filter1531173832_ereal
@ ( ^ [Y0: a > extended_ereal,Y1: nat > a,Y2: nat] : ( Y0 @ ( Y1 @ Y2 ) )
@ f
@ x )
@ ( topolo2140997059_ereal @ a2 )
@ X0 )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f1280,f1213,f1213]) ).
thf(f1280,plain,
! [X0: filter_nat] :
( ( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ X0 )
!= $true )
| ( ( filter1531173832_ereal
@ ^ [Y0: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ Y0 ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ X0 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1060]) ).
thf(f1060,plain,
! [X0: filter_nat] :
( ( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ X0 )
!= $true )
| ( ( filter1531173832_ereal
@ ^ [Y0: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ Y0 ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ X0 )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f554]) ).
thf(f554,plain,
! [X0: filter_nat] :
( ( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ X0 )
= $true )
=> ( ( filter1531173832_ereal
@ ^ [Y0: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ Y0 ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ X0 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f553]) ).
thf(f553,plain,
! [X0: filter_nat] :
( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ X0 )
=> ( filter1531173832_ereal
@ ^ [X1: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X1 ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ X0 ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X0: filter_nat] :
( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ X0 )
=> ( filter1531173832_ereal
@ ^ [X1: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X1 ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ X0 ) ),
file('/export/starexec/sandbox/tmp/tmp.z9pBe0D4OI/DTF2THF_9897.p',fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062) ).
thf(f1261,plain,
( $true
!= ( filter1531173832_ereal
@ ^ [Y0: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ Y0 ) ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ at_top_nat ) ),
inference(cnf_transformation,[status(thm)],[f991]) ).
thf(f991,plain,
( $true
!= ( filter1531173832_ereal
@ ^ [Y0: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ Y0 ) ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ at_top_nat ) ),
inference(flattening,[status(thm)],[f782]) ).
thf(f782,plain,
( $true
!= ( filter1531173832_ereal
@ ^ [Y0: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ Y0 ) ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ at_top_nat ) ),
inference(fool_elimination,[status(thm)],[f781]) ).
thf(f781,plain,
~ ( filter1531173832_ereal
@ ^ [X0: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ X0 ) ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ at_top_nat ),
inference(rectify,[status(thm)],[f357]) ).
thf(f357,negated_conjecture,
~ ( filter1531173832_ereal
@ ^ [X85: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ X85 ) ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ at_top_nat ),
inference(negated_conjecture,[status(cth)],[f356]) ).
thf(f356,conjecture,
( filter1531173832_ereal
@ ^ [X85: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ X85 ) ) )
@ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
@ at_top_nat ),
file('/export/starexec/sandbox/tmp/tmp.z9pBe0D4OI/DTF2THF_9897.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP112^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n005.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:26:29 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : ITP112^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.45 % Domain : Interactive Theorem Proving
% 0.19/0.45 % Problem : Sledgehammer Lower_Semicontinuous problem prob_385__6250846_1
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English :
% 0.19/0.45
% 0.19/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.45 % Source : [Des21]
% 0.19/0.45 % Names : Lower_Semicontinuous/prob_385__6250846_1 [Des21]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.56 v9.1.0, 0.38 v9.0.0, 0.70 v8.2.0, 0.69 v8.1.0, 0.73 v7.5.0
% 0.19/0.45 % Syntax : Number of formulae : 451 ( 130 unt; 95 typ; 0 def)
% 0.19/0.45 % Number of atoms : 1312 ( 382 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 81 ( 3 avg)
% 0.19/0.45 % Number of connectives : 4092 ( 116 ~; 34 |; 57 &;3135 @)
% 0.19/0.45 % ( 0 <=>; 750 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 34 ( 8 avg)
% 0.19/0.45 % Number of types : 13 ( 12 usr)
% 0.19/0.45 % Number of type conns : 500 ( 500 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 84 ( 83 usr; 6 con; 0-3 aty)
% 0.19/0.45 % Number of variables : 1297 ( 136 ^;1117 !; 44 ?;1297 :)
% 0.19/0.45 % SPC : TH0_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:41:16.220
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % Could-be-implicit typings (12)
% 0.19/0.45 thf(ty_n_t__Filter__Ofilter_It__Extended____Real__Oereal_J,type,
% 0.19/0.45 filter2049122004_ereal: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.19/0.45 set_Extended_ereal: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 0.19/0.45 filter_real: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 0.19/0.45 filter_nat: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
% 0.19/0.45 filter_int: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.19/0.45 set_real: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Filter__Ofilter_Itf__a_J,type,
% 0.19/0.45 filter_a: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Extended____Real__Oereal,type,
% 0.19/0.45 extended_ereal: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Real__Oreal,type,
% 0.19/0.45 real: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Nat__Onat,type,
% 0.19/0.45 nat: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_t__Int__Oint,type,
% 0.19/0.45 int: $tType ).
% 0.19/0.45
% 0.19/0.45 thf(ty_n_tf__a,type,
% 0.19/0.45 a: $tType ).
% 0.19/0.45
% 0.19/0.45 % Explicit typings (83)
% 0.19/0.45 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
% 0.19/0.45 extend1289208545_ereal: extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 0.19/0.45 extended_ereal2: real > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 0.19/0.45 at_bot_real: filter_real ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 0.19/0.45 at_top_nat: filter_nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 0.19/0.45 at_top_real: filter_real ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
% 0.19/0.45 filter1531173832_ereal: ( nat > extended_ereal ) > filter2049122004_ereal > filter_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
% 0.19/0.45 filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.19/0.45 filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.19/0.45 filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001tf__a,type,
% 0.19/0.45 filterlim_nat_a: ( nat > a ) > filter_a > filter_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.19/0.45 filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 0.19/0.45 comp_E1308517939al_nat: ( extended_ereal > extended_ereal ) > ( nat > extended_ereal ) > nat > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001tf__a,type,
% 0.19/0.45 comp_E489644891real_a: ( extended_ereal > extended_ereal ) > ( a > extended_ereal ) > a > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Int__Oint_001t__Nat__Onat,type,
% 0.19/0.45 comp_E1436437929nt_nat: ( extended_ereal > int ) > ( nat > extended_ereal ) > nat > int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.19/0.45 comp_E1523169101at_nat: ( extended_ereal > nat ) > ( nat > extended_ereal ) > nat > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.19/0.45 comp_E1477338153al_nat: ( extended_ereal > real ) > ( nat > extended_ereal ) > nat > real ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 0.19/0.45 comp_n1096781355al_nat: ( nat > extended_ereal ) > ( nat > nat ) > nat > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001t__Nat__Onat,type,
% 0.19/0.45 comp_nat_a_nat: ( nat > a ) > ( nat > nat ) > nat > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 0.19/0.45 comp_r1410008527al_nat: ( real > extended_ereal ) > ( nat > real ) > nat > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Int__Oint_001t__Nat__Onat,type,
% 0.19/0.45 comp_real_int_nat: ( real > int ) > ( nat > real ) > nat > int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.19/0.45 comp_real_nat_nat: ( real > nat ) > ( nat > real ) > nat > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 0.19/0.45 comp_a1112243075al_nat: ( a > extended_ereal ) > ( nat > a ) > nat > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Real__Oereal_001tf__a,type,
% 0.19/0.45 comp_a780206603real_a: ( a > extended_ereal ) > ( a > a ) > a > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001tf__a_001t__Int__Oint_001t__Nat__Onat,type,
% 0.19/0.45 comp_a_int_nat: ( a > int ) > ( nat > a ) > nat > int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.19/0.45 comp_a_nat_nat: ( a > nat ) > ( nat > a ) > nat > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.19/0.45 comp_a_real_nat: ( a > real ) > ( nat > a ) > nat > real ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Nat__Onat,type,
% 0.19/0.45 comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
% 0.19/0.45 plus_p2118002693_ereal: extended_ereal > extended_ereal > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 0.19/0.45 plus_plus_int: int > int > int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.19/0.45 plus_plus_nat: nat > nat > nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 0.19/0.45 plus_plus_real: real > real > real ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
% 0.19/0.45 uminus1208298309_ereal: extended_ereal > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 0.19/0.45 uminus_uminus_int: int > int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.19/0.45 uminus_uminus_real: real > real ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HOL_OUniq_001t__Extended____Real__Oereal,type,
% 0.19/0.45 uniq_Extended_ereal: ( extended_ereal > $o ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HOL_OUniq_001t__Real__Oreal,type,
% 0.19/0.45 uniq_real: ( real > $o ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_HOL_OUniq_001tf__a,type,
% 0.19/0.45 uniq_a: ( a > $o ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Liminf__Limsup_OLiminf_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
% 0.19/0.45 liminf1045857232_ereal: filter_nat > ( nat > extended_ereal ) > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
% 0.19/0.45 lower_1087098792_ereal: extended_ereal > ( extended_ereal > extended_ereal ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Int__Oint,type,
% 0.19/0.45 lower_48196818al_int: extended_ereal > ( extended_ereal > int ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 0.19/0.45 lower_1558406774al_nat: extended_ereal > ( extended_ereal > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
% 0.19/0.45 lower_1165973074l_real: extended_ereal > ( extended_ereal > real ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Real__Oreal_001t__Extended____Real__Oereal,type,
% 0.19/0.45 lower_551915512_ereal: real > ( real > extended_ereal ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Real__Oreal_001t__Int__Oint,type,
% 0.19/0.45 lower_153911426al_int: real > ( real > int ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.19/0.45 lower_1664121382al_nat: real > ( real > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Extended____Real__Oereal,type,
% 0.19/0.45 lower_191460856_ereal: a > ( a > extended_ereal ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Int__Oint,type,
% 0.19/0.45 lower_956963458_a_int: a > ( a > int ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Nat__Onat,type,
% 0.19/0.45 lower_319689766_a_nat: a > ( a > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Real__Oreal,type,
% 0.19/0.45 lower_231615490a_real: a > ( a > real ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
% 0.19/0.45 lower_1071158961_ereal: extended_ereal > ( extended_ereal > extended_ereal ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Int__Oint,type,
% 0.19/0.45 lower_637387785al_int: extended_ereal > ( extended_ereal > int ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 0.19/0.45 lower_114093al_nat: extended_ereal > ( extended_ereal > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
% 0.19/0.45 lower_737640969l_real: extended_ereal > ( extended_ereal > real ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Real__Oreal_001t__Int__Oint,type,
% 0.19/0.45 lower_1075504779al_int: real > ( real > int ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Real__Oreal_001t__Nat__Onat,type,
% 0.19/0.45 lower_438231087al_nat: real > ( real > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Extended____Real__Oereal,type,
% 0.19/0.45 lower_534855297_ereal: a > ( a > extended_ereal ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Int__Oint,type,
% 0.19/0.45 lower_1672990777_a_int: a > ( a > int ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Nat__Onat,type,
% 0.19/0.45 lower_1035717085_a_nat: a > ( a > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Real__Oreal,type,
% 0.19/0.45 lower_755922489a_real: a > ( a > real ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.19/0.45 semiri2019852685at_int: nat > int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.19/0.45 semiri2110766477t_real: nat > real ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
% 0.19/0.45 ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 0.19/0.45 ord_le1745708096er_nat: filter_nat > filter_nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
% 0.19/0.45 ord_le132810396r_real: filter_real > filter_real > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.19/0.45 ord_less_eq_int: int > int > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.19/0.45 ord_less_eq_nat: nat > nat > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.19/0.45 ord_less_eq_real: real > real > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.19/0.45 collect_real: ( real > $o ) > set_real ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Topological__Spaces_Omonoseq_001t__Extended____Real__Oereal,type,
% 0.19/0.45 topolo1069469409_ereal: ( nat > extended_ereal ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 0.19/0.45 topolo411883481eq_int: ( nat > int ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 0.19/0.45 topolo1922093437eq_nat: ( nat > nat ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 0.19/0.45 topolo144289241q_real: ( nat > real ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Extended____Real__Oereal,type,
% 0.19/0.45 topolo2140997059_ereal: extended_ereal > filter2049122004_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Int__Oint,type,
% 0.19/0.45 topolo54776183ds_int: int > filter_int ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Nat__Onat,type,
% 0.19/0.45 topolo1564986139ds_nat: nat > filter_nat ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 0.19/0.45 topolo1664202871s_real: real > filter_real ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001tf__a,type,
% 0.19/0.45 topolo705128563nhds_a: a > filter_a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_member_001t__Extended____Real__Oereal,type,
% 0.19/0.45 member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_c_member_001t__Real__Oreal,type,
% 0.19/0.45 member_real: real > set_real > $o ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_A____,type,
% 0.19/0.45 a2: extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_f,type,
% 0.19/0.45 f: a > extended_ereal ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_x0,type,
% 0.19/0.45 x0: a ).
% 0.19/0.45
% 0.19/0.45 thf(sy_v_x____,type,
% 0.19/0.45 x: nat > a ).
% 0.19/0.45
% 0.19/0.45 % Relevant facts (355)
% 0.19/0.45 thf(fact_0_x__def_I1_J,axiom,
% 0.19/0.45 filterlim_nat_a @ x @ ( topolo705128563nhds_a @ x0 ) @ at_top_nat ).
% 0.19/0.45
% 0.19/0.45 % x_def(1)
% 0.19/0.45 thf(fact_1_x__def_I2_J,axiom,
% 0.19/0.45 filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ at_top_nat ).
% 0.19/0.45
% 0.19/0.45 % x_def(2)
% 0.19/0.45 thf(fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062,axiom,
% 0.19/0.45 ! [F: filter_nat] :
% 0.19/0.45 ( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ F )
% 0.19/0.45 => ( filter1531173832_ereal
% 0.19/0.45 @ ^ [X: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X ) )
% 0.19/0.45 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
% 0.19/0.45 @ F ) ) ).
% 0.19/0.45
% 0.19/0.45 % \<open>\<And>F. ((f \<circ> x) \<longlongrightarrow> A) F \<Longrightarrow> ((\<lambda>xa. - (f \<circ> x) xa) \<longlongrightarrow> - A) F\<close>
% 0.19/0.45 thf(fact_3_lsc,axiom,
% 0.19/0.45 ( lower_191460856_ereal @ x0
% 0.19/0.45 @ ^ [X: a] : ( uminus1208298309_ereal @ ( f @ X ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lsc
% 0.19/0.45 thf(fact_4_tendsto__uminus__ereal,axiom,
% 0.19/0.45 ! [F2: nat > extended_ereal,X2: extended_ereal,F: filter_nat] :
% 0.19/0.45 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ X2 ) @ F )
% 0.19/0.45 => ( filter1531173832_ereal
% 0.19/0.45 @ ^ [X: nat] : ( uminus1208298309_ereal @ ( F2 @ X ) )
% 0.19/0.45 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ X2 ) )
% 0.19/0.45 @ F ) ) ).
% 0.19/0.45
% 0.19/0.45 % tendsto_uminus_ereal
% 0.19/0.45 thf(fact_5_tendsto__const,axiom,
% 0.19/0.45 ! [K: real,F: filter_real] :
% 0.19/0.45 ( filterlim_real_real
% 0.19/0.45 @ ^ [X: real] : K
% 0.19/0.45 @ ( topolo1664202871s_real @ K )
% 0.19/0.45 @ F ) ).
% 0.19/0.45
% 0.19/0.45 % tendsto_const
% 0.19/0.45 thf(fact_6_tendsto__const,axiom,
% 0.19/0.45 ! [K: real,F: filter_nat] :
% 0.19/0.45 ( filterlim_nat_real
% 0.19/0.45 @ ^ [X: nat] : K
% 0.19/0.45 @ ( topolo1664202871s_real @ K )
% 0.19/0.45 @ F ) ).
% 0.19/0.45
% 0.19/0.45 % tendsto_const
% 0.19/0.45 thf(fact_7_tendsto__const,axiom,
% 0.19/0.45 ! [K: extended_ereal,F: filter_nat] :
% 0.19/0.45 ( filter1531173832_ereal
% 0.19/0.45 @ ^ [X: nat] : K
% 0.19/0.45 @ ( topolo2140997059_ereal @ K )
% 0.19/0.45 @ F ) ).
% 0.19/0.45
% 0.19/0.45 % tendsto_const
% 0.19/0.45 thf(fact_8_tendsto__const,axiom,
% 0.19/0.45 ! [K: a,F: filter_nat] :
% 0.19/0.45 ( filterlim_nat_a
% 0.19/0.45 @ ^ [X: nat] : K
% 0.19/0.45 @ ( topolo705128563nhds_a @ K )
% 0.19/0.45 @ F ) ).
% 0.19/0.45
% 0.19/0.45 % tendsto_const
% 0.19/0.45 thf(fact_9_ereal__Lim__uminus,axiom,
% 0.19/0.45 ! [F2: nat > extended_ereal,F0: extended_ereal,Net: filter_nat] :
% 0.19/0.45 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ F0 ) @ Net )
% 0.19/0.45 = ( filter1531173832_ereal
% 0.19/0.45 @ ^ [X: nat] : ( uminus1208298309_ereal @ ( F2 @ X ) )
% 0.19/0.45 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ F0 ) )
% 0.19/0.45 @ Net ) ) ).
% 0.19/0.45
% 0.19/0.45 % ereal_Lim_uminus
% 0.19/0.45 thf(fact_10_LIMSEQ__const__iff,axiom,
% 0.19/0.45 ! [K: real,L: real] :
% 0.19/0.45 ( ( filterlim_nat_real
% 0.19/0.45 @ ^ [N: nat] : K
% 0.19/0.45 @ ( topolo1664202871s_real @ L )
% 0.19/0.45 @ at_top_nat )
% 0.19/0.45 = ( K = L ) ) ).
% 0.19/0.45
% 0.19/0.45 % LIMSEQ_const_iff
% 0.19/0.45 thf(fact_11_LIMSEQ__const__iff,axiom,
% 0.19/0.45 ! [K: extended_ereal,L: extended_ereal] :
% 0.19/0.45 ( ( filter1531173832_ereal
% 0.19/0.45 @ ^ [N: nat] : K
% 0.19/0.45 @ ( topolo2140997059_ereal @ L )
% 0.19/0.45 @ at_top_nat )
% 0.19/0.45 = ( K = L ) ) ).
% 0.19/0.45
% 0.19/0.45 % LIMSEQ_const_iff
% 0.19/0.45 thf(fact_12_LIMSEQ__const__iff,axiom,
% 0.19/0.45 ! [K: a,L: a] :
% 0.19/0.45 ( ( filterlim_nat_a
% 0.19/0.45 @ ^ [N: nat] : K
% 0.19/0.45 @ ( topolo705128563nhds_a @ L )
% 0.19/0.45 @ at_top_nat )
% 0.19/0.45 = ( K = L ) ) ).
% 0.19/0.45
% 0.19/0.45 % LIMSEQ_const_iff
% 0.19/0.45 thf(fact_13_tendsto__minus__cancel,axiom,
% 0.19/0.45 ! [F2: real > real,A: real,F: filter_real] :
% 0.19/0.45 ( ( filterlim_real_real
% 0.19/0.45 @ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
% 0.19/0.46 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 @ F )
% 0.19/0.46 => ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ A ) @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_minus_cancel
% 0.19/0.46 thf(fact_14_tendsto__minus__cancel,axiom,
% 0.19/0.46 ! [F2: nat > real,A: real,F: filter_nat] :
% 0.19/0.46 ( ( filterlim_nat_real
% 0.19/0.46 @ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) )
% 0.19/0.46 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 @ F )
% 0.19/0.46 => ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ A ) @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_minus_cancel
% 0.19/0.46 thf(fact_15_tendsto__minus__cancel__left,axiom,
% 0.19/0.46 ! [F2: real > real,Y: real,F: filter_real] :
% 0.19/0.46 ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ Y ) ) @ F )
% 0.19/0.46 = ( filterlim_real_real
% 0.19/0.46 @ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
% 0.19/0.46 @ ( topolo1664202871s_real @ Y )
% 0.19/0.46 @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_minus_cancel_left
% 0.19/0.46 thf(fact_16_tendsto__minus__cancel__left,axiom,
% 0.19/0.46 ! [F2: nat > real,Y: real,F: filter_nat] :
% 0.19/0.46 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ Y ) ) @ F )
% 0.19/0.46 = ( filterlim_nat_real
% 0.19/0.46 @ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) )
% 0.19/0.46 @ ( topolo1664202871s_real @ Y )
% 0.19/0.46 @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_minus_cancel_left
% 0.19/0.46 thf(fact_17_tendsto__minus,axiom,
% 0.19/0.46 ! [F2: real > real,A: real,F: filter_real] :
% 0.19/0.46 ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ A ) @ F )
% 0.19/0.46 => ( filterlim_real_real
% 0.19/0.46 @ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
% 0.19/0.46 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_minus
% 0.19/0.46 thf(fact_18_tendsto__minus,axiom,
% 0.19/0.46 ! [F2: nat > real,A: real,F: filter_nat] :
% 0.19/0.46 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ A ) @ F )
% 0.19/0.46 => ( filterlim_nat_real
% 0.19/0.46 @ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) )
% 0.19/0.46 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 @ F ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_minus
% 0.19/0.46 thf(fact_19_LIMSEQ__unique,axiom,
% 0.19/0.46 ! [X3: nat > extended_ereal,A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ A ) @ at_top_nat )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ B ) @ at_top_nat )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_unique
% 0.19/0.46 thf(fact_20_LIMSEQ__unique,axiom,
% 0.19/0.46 ! [X3: nat > a,A: a,B: a] :
% 0.19/0.46 ( ( filterlim_nat_a @ X3 @ ( topolo705128563nhds_a @ A ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_a @ X3 @ ( topolo705128563nhds_a @ B ) @ at_top_nat )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_unique
% 0.19/0.46 thf(fact_21_LIMSEQ__unique,axiom,
% 0.19/0.46 ! [X3: nat > real,A: real,B: real] :
% 0.19/0.46 ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ A ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ B ) @ at_top_nat )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_unique
% 0.19/0.46 thf(fact_22_tendsto__uminus__nhds,axiom,
% 0.19/0.46 ! [A: real] : ( filterlim_real_real @ uminus_uminus_real @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) ) @ ( topolo1664202871s_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_uminus_nhds
% 0.19/0.46 thf(fact_23_ereal__uminus__eq__iff,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ( uminus1208298309_ereal @ A )
% 0.19/0.46 = ( uminus1208298309_ereal @ B ) )
% 0.19/0.46 = ( A = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_uminus_eq_iff
% 0.19/0.46 thf(fact_24_ereal__uminus__uminus,axiom,
% 0.19/0.46 ! [A: extended_ereal] :
% 0.19/0.46 ( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A ) )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_uminus_uminus
% 0.19/0.46 thf(fact_25_verit__minus__simplify_I4_J,axiom,
% 0.19/0.46 ! [B: real] :
% 0.19/0.46 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % verit_minus_simplify(4)
% 0.19/0.46 thf(fact_26_verit__negate__coefficient_I3_J,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( uminus_uminus_real @ A )
% 0.19/0.46 = ( uminus_uminus_real @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_negate_coefficient(3)
% 0.19/0.46 thf(fact_27_ereal__uminus__eq__reorder,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ( uminus1208298309_ereal @ A )
% 0.19/0.46 = B )
% 0.19/0.46 = ( A
% 0.19/0.46 = ( uminus1208298309_ereal @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_uminus_eq_reorder
% 0.19/0.46 thf(fact_28_tendsto__eq__rhs,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal,X2: extended_ereal,F: filter_nat,Y: extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ X2 ) @ F )
% 0.19/0.46 => ( ( X2 = Y )
% 0.19/0.46 => ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ Y ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_eq_rhs
% 0.19/0.46 thf(fact_29_tendsto__eq__rhs,axiom,
% 0.19/0.46 ! [F2: nat > a,X2: a,F: filter_nat,Y: a] :
% 0.19/0.46 ( ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ X2 ) @ F )
% 0.19/0.46 => ( ( X2 = Y )
% 0.19/0.46 => ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ Y ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_eq_rhs
% 0.19/0.46 thf(fact_30_tendsto__eq__rhs,axiom,
% 0.19/0.46 ! [F2: real > real,X2: real,F: filter_real,Y: real] :
% 0.19/0.46 ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ X2 ) @ F )
% 0.19/0.46 => ( ( X2 = Y )
% 0.19/0.46 => ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ Y ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_eq_rhs
% 0.19/0.46 thf(fact_31_tendsto__eq__rhs,axiom,
% 0.19/0.46 ! [F2: nat > real,X2: real,F: filter_nat,Y: real] :
% 0.19/0.46 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ X2 ) @ F )
% 0.19/0.46 => ( ( X2 = Y )
% 0.19/0.46 => ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ Y ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_eq_rhs
% 0.19/0.46 thf(fact_32_tendsto__cong__limit,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal,L: extended_ereal,F: filter_nat,K: extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ F )
% 0.19/0.46 => ( ( K = L )
% 0.19/0.46 => ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ K ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_cong_limit
% 0.19/0.46 thf(fact_33_tendsto__cong__limit,axiom,
% 0.19/0.46 ! [F2: nat > a,L: a,F: filter_nat,K: a] :
% 0.19/0.46 ( ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ L ) @ F )
% 0.19/0.46 => ( ( K = L )
% 0.19/0.46 => ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ K ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_cong_limit
% 0.19/0.46 thf(fact_34_tendsto__cong__limit,axiom,
% 0.19/0.46 ! [F2: real > real,L: real,F: filter_real,K: real] :
% 0.19/0.46 ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ L ) @ F )
% 0.19/0.46 => ( ( K = L )
% 0.19/0.46 => ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ K ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_cong_limit
% 0.19/0.46 thf(fact_35_tendsto__cong__limit,axiom,
% 0.19/0.46 ! [F2: nat > real,L: real,F: filter_nat,K: real] :
% 0.19/0.46 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ F )
% 0.19/0.46 => ( ( K = L )
% 0.19/0.46 => ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ K ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_cong_limit
% 0.19/0.46 thf(fact_36_comp__apply,axiom,
% 0.19/0.46 ( comp_a1112243075al_nat
% 0.19/0.46 = ( ^ [F3: a > extended_ereal,G: nat > a,X: nat] : ( F3 @ ( G @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % comp_apply
% 0.19/0.46 thf(fact_37_neg__equal__iff__equal,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = ( uminus_uminus_real @ B ) )
% 0.19/0.46 = ( A = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_iff_equal
% 0.19/0.46 thf(fact_38_add_Oinverse__inverse,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add.inverse_inverse
% 0.19/0.46 thf(fact_39_lsc__sequentially__mem,axiom,
% 0.19/0.46 ! [X0: real,F2: real > extended_ereal,X2: nat > real,C: nat > extended_ereal,C0: extended_ereal] :
% 0.19/0.46 ( ( lower_551915512_ereal @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filter1531173832_ereal @ C @ ( topolo2140997059_ereal @ C0 ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] : ( ord_le824540014_ereal @ ( F2 @ ( X2 @ N2 ) ) @ ( C @ N2 ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ C0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_sequentially_mem
% 0.19/0.46 thf(fact_40_lsc__sequentially__mem,axiom,
% 0.19/0.46 ! [X0: a,F2: a > extended_ereal,X2: nat > a,C: nat > extended_ereal,C0: extended_ereal] :
% 0.19/0.46 ( ( lower_191460856_ereal @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filter1531173832_ereal @ C @ ( topolo2140997059_ereal @ C0 ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] : ( ord_le824540014_ereal @ ( F2 @ ( X2 @ N2 ) ) @ ( C @ N2 ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ C0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_sequentially_mem
% 0.19/0.46 thf(fact_41_lsc__sequentially__gen,axiom,
% 0.19/0.46 ( lower_551915512_ereal
% 0.19/0.46 = ( ^ [X02: real,F3: real > extended_ereal] :
% 0.19/0.46 ! [X: nat > real,C2: nat > extended_ereal,C02: extended_ereal] :
% 0.19/0.46 ( ( ( filterlim_nat_real @ X @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filter1531173832_ereal @ C2 @ ( topolo2140997059_ereal @ C02 ) @ at_top_nat )
% 0.19/0.46 & ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ ( C2 @ N ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C02 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_sequentially_gen
% 0.19/0.46 thf(fact_42_lsc__sequentially__gen,axiom,
% 0.19/0.46 ( lower_191460856_ereal
% 0.19/0.46 = ( ^ [X02: a,F3: a > extended_ereal] :
% 0.19/0.46 ! [X: nat > a,C2: nat > extended_ereal,C02: extended_ereal] :
% 0.19/0.46 ( ( ( filterlim_nat_a @ X @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filter1531173832_ereal @ C2 @ ( topolo2140997059_ereal @ C02 ) @ at_top_nat )
% 0.19/0.46 & ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ ( C2 @ N ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C02 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_sequentially_gen
% 0.19/0.46 thf(fact_43_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: extended_ereal,F2: extended_ereal > nat,X2: nat > extended_ereal,A2: nat] :
% 0.19/0.46 ( ( lower_1558406774al_nat @ X0 @ F2 )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_44_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: extended_ereal,F2: extended_ereal > int,X2: nat > extended_ereal,A2: int] :
% 0.19/0.46 ( ( lower_48196818al_int @ X0 @ F2 )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_45_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: extended_ereal,F2: extended_ereal > extended_ereal,X2: nat > extended_ereal,A2: extended_ereal] :
% 0.19/0.46 ( ( lower_1087098792_ereal @ X0 @ F2 )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_46_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: extended_ereal,F2: extended_ereal > real,X2: nat > extended_ereal,A2: real] :
% 0.19/0.46 ( ( lower_1165973074l_real @ X0 @ F2 )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_47_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: a,F2: a > nat,X2: nat > a,A2: nat] :
% 0.19/0.46 ( ( lower_319689766_a_nat @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_nat @ ( comp_a_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_48_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: a,F2: a > int,X2: nat > a,A2: int] :
% 0.19/0.46 ( ( lower_956963458_a_int @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_int @ ( comp_a_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_49_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: a,F2: a > real,X2: nat > a,A2: real] :
% 0.19/0.46 ( ( lower_231615490a_real @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_real @ ( comp_a_real_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_50_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: real,F2: real > nat,X2: nat > real,A2: nat] :
% 0.19/0.46 ( ( lower_1664121382al_nat @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_51_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: real,F2: real > int,X2: nat > real,A2: int] :
% 0.19/0.46 ( ( lower_153911426al_int @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_int @ ( comp_real_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_52_lsc__at__mem,axiom,
% 0.19/0.46 ! [X0: real,F2: real > extended_ereal,X2: nat > real,A2: extended_ereal] :
% 0.19/0.46 ( ( lower_551915512_ereal @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filter1531173832_ereal @ ( comp_r1410008527al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_mem
% 0.19/0.46 thf(fact_53_lsc__at__def,axiom,
% 0.19/0.46 ( lower_1558406774al_nat
% 0.19/0.46 = ( ^ [X02: extended_ereal,F3: extended_ereal > nat] :
% 0.19/0.46 ! [X4: nat > extended_ereal,L2: nat] :
% 0.19/0.46 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_54_lsc__at__def,axiom,
% 0.19/0.46 ( lower_48196818al_int
% 0.19/0.46 = ( ^ [X02: extended_ereal,F3: extended_ereal > int] :
% 0.19/0.46 ! [X4: nat > extended_ereal,L2: int] :
% 0.19/0.46 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_55_lsc__at__def,axiom,
% 0.19/0.46 ( lower_1087098792_ereal
% 0.19/0.46 = ( ^ [X02: extended_ereal,F3: extended_ereal > extended_ereal] :
% 0.19/0.46 ! [X4: nat > extended_ereal,L2: extended_ereal] :
% 0.19/0.46 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_56_lsc__at__def,axiom,
% 0.19/0.46 ( lower_1165973074l_real
% 0.19/0.46 = ( ^ [X02: extended_ereal,F3: extended_ereal > real] :
% 0.19/0.46 ! [X4: nat > extended_ereal,L2: real] :
% 0.19/0.46 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_57_lsc__at__def,axiom,
% 0.19/0.46 ( lower_319689766_a_nat
% 0.19/0.46 = ( ^ [X02: a,F3: a > nat] :
% 0.19/0.46 ! [X4: nat > a,L2: nat] :
% 0.19/0.46 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_nat @ ( comp_a_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_58_lsc__at__def,axiom,
% 0.19/0.46 ( lower_956963458_a_int
% 0.19/0.46 = ( ^ [X02: a,F3: a > int] :
% 0.19/0.46 ! [X4: nat > a,L2: int] :
% 0.19/0.46 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_int @ ( comp_a_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_59_lsc__at__def,axiom,
% 0.19/0.46 ( lower_231615490a_real
% 0.19/0.46 = ( ^ [X02: a,F3: a > real] :
% 0.19/0.46 ! [X4: nat > a,L2: real] :
% 0.19/0.46 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_real @ ( comp_a_real_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_60_lsc__at__def,axiom,
% 0.19/0.46 ( lower_1664121382al_nat
% 0.19/0.46 = ( ^ [X02: real,F3: real > nat] :
% 0.19/0.46 ! [X4: nat > real,L2: nat] :
% 0.19/0.46 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_nat @ ( comp_real_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_61_lsc__at__def,axiom,
% 0.19/0.46 ( lower_153911426al_int
% 0.19/0.46 = ( ^ [X02: real,F3: real > int] :
% 0.19/0.46 ! [X4: nat > real,L2: int] :
% 0.19/0.46 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_int @ ( comp_real_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_62_lsc__at__def,axiom,
% 0.19/0.46 ( lower_551915512_ereal
% 0.19/0.46 = ( ^ [X02: real,F3: real > extended_ereal] :
% 0.19/0.46 ! [X4: nat > real,L2: extended_ereal] :
% 0.19/0.46 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filter1531173832_ereal @ ( comp_r1410008527al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_at_def
% 0.19/0.46 thf(fact_63_neg__le__iff__le,axiom,
% 0.19/0.46 ! [B: real,A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_le_iff_le
% 0.19/0.46 thf(fact_64_neg__le__iff__le,axiom,
% 0.19/0.46 ! [B: int,A: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.19/0.46 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_le_iff_le
% 0.19/0.46 thf(fact_65_ereal__minus__le__minus,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ ( uminus1208298309_ereal @ B ) )
% 0.19/0.46 = ( ord_le824540014_ereal @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_minus_le_minus
% 0.19/0.46 thf(fact_66_verit__la__disequality,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 | ~ ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 | ~ ( ord_le824540014_ereal @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_la_disequality
% 0.19/0.46 thf(fact_67_verit__la__disequality,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 | ~ ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_la_disequality
% 0.19/0.46 thf(fact_68_verit__la__disequality,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 | ~ ( ord_less_eq_real @ A @ B )
% 0.19/0.46 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_la_disequality
% 0.19/0.46 thf(fact_69_verit__la__disequality,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 | ~ ( ord_less_eq_int @ A @ B )
% 0.19/0.46 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_la_disequality
% 0.19/0.46 thf(fact_70_ereal__complete__Inf,axiom,
% 0.19/0.46 ! [S: set_Extended_ereal] :
% 0.19/0.46 ? [X5: extended_ereal] :
% 0.19/0.46 ( ! [Xa: extended_ereal] :
% 0.19/0.46 ( ( member1900190071_ereal @ Xa @ S )
% 0.19/0.46 => ( ord_le824540014_ereal @ X5 @ Xa ) )
% 0.19/0.46 & ! [Z: extended_ereal] :
% 0.19/0.46 ( ! [Xa2: extended_ereal] :
% 0.19/0.46 ( ( member1900190071_ereal @ Xa2 @ S )
% 0.19/0.46 => ( ord_le824540014_ereal @ Z @ Xa2 ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ Z @ X5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_complete_Inf
% 0.19/0.46 thf(fact_71_ereal__complete__Sup,axiom,
% 0.19/0.46 ! [S: set_Extended_ereal] :
% 0.19/0.46 ? [X5: extended_ereal] :
% 0.19/0.46 ( ! [Xa: extended_ereal] :
% 0.19/0.46 ( ( member1900190071_ereal @ Xa @ S )
% 0.19/0.46 => ( ord_le824540014_ereal @ Xa @ X5 ) )
% 0.19/0.46 & ! [Z: extended_ereal] :
% 0.19/0.46 ( ! [Xa2: extended_ereal] :
% 0.19/0.46 ( ( member1900190071_ereal @ Xa2 @ S )
% 0.19/0.46 => ( ord_le824540014_ereal @ Xa2 @ Z ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ X5 @ Z ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_complete_Sup
% 0.19/0.46 thf(fact_72_le__minus__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.19/0.46 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_minus_iff
% 0.19/0.46 thf(fact_73_le__minus__iff,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.19/0.46 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_minus_iff
% 0.19/0.46 thf(fact_74_minus__le__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.19/0.46 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_le_iff
% 0.19/0.46 thf(fact_75_minus__le__iff,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.19/0.46 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_le_iff
% 0.19/0.46 thf(fact_76_le__imp__neg__le,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_imp_neg_le
% 0.19/0.46 thf(fact_77_le__imp__neg__le,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_imp_neg_le
% 0.19/0.46 thf(fact_78_ereal__uminus__le__reorder,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ B )
% 0.19/0.46 = ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ B ) @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_uminus_le_reorder
% 0.19/0.46 thf(fact_79_lim__mono,axiom,
% 0.19/0.46 ! [N3: nat,X3: nat > nat,Y2: nat > nat,X2: nat,Y: nat] :
% 0.19/0.46 ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N3 @ N2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.19/0.46 => ( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_nat @ Y2 @ ( topolo1564986139ds_nat @ Y ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lim_mono
% 0.19/0.46 thf(fact_80_lim__mono,axiom,
% 0.19/0.46 ! [N3: nat,X3: nat > int,Y2: nat > int,X2: int,Y: int] :
% 0.19/0.46 ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N3 @ N2 )
% 0.19/0.46 => ( ord_less_eq_int @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.19/0.46 => ( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_int @ Y2 @ ( topolo54776183ds_int @ Y ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_int @ X2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lim_mono
% 0.19/0.46 thf(fact_81_lim__mono,axiom,
% 0.19/0.46 ! [N3: nat,X3: nat > extended_ereal,Y2: nat > extended_ereal,X2: extended_ereal,Y: extended_ereal] :
% 0.19/0.46 ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N3 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( filter1531173832_ereal @ Y2 @ ( topolo2140997059_ereal @ Y ) @ at_top_nat )
% 0.19/0.46 => ( ord_le824540014_ereal @ X2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lim_mono
% 0.19/0.46 thf(fact_82_lim__mono,axiom,
% 0.19/0.46 ! [N3: nat,X3: nat > real,Y2: nat > real,X2: real,Y: real] :
% 0.19/0.46 ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N3 @ N2 )
% 0.19/0.46 => ( ord_less_eq_real @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.19/0.46 => ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_real @ Y2 @ ( topolo1664202871s_real @ Y ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_real @ X2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lim_mono
% 0.19/0.46 thf(fact_83_LIMSEQ__le,axiom,
% 0.19/0.46 ! [X3: nat > nat,X2: nat,Y2: nat > nat,Y: nat] :
% 0.19/0.46 ( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_nat @ Y2 @ ( topolo1564986139ds_nat @ Y ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le
% 0.19/0.46 thf(fact_84_LIMSEQ__le,axiom,
% 0.19/0.46 ! [X3: nat > int,X2: int,Y2: nat > int,Y: int] :
% 0.19/0.46 ( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_int @ Y2 @ ( topolo54776183ds_int @ Y ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_less_eq_int @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.19/0.46 => ( ord_less_eq_int @ X2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le
% 0.19/0.46 thf(fact_85_LIMSEQ__le,axiom,
% 0.19/0.46 ! [X3: nat > extended_ereal,X2: extended_ereal,Y2: nat > extended_ereal,Y: extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( filter1531173832_ereal @ Y2 @ ( topolo2140997059_ereal @ Y ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ X2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le
% 0.19/0.46 thf(fact_86_LIMSEQ__le,axiom,
% 0.19/0.46 ! [X3: nat > real,X2: real,Y2: nat > real,Y: real] :
% 0.19/0.46 ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_real @ Y2 @ ( topolo1664202871s_real @ Y ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_less_eq_real @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ X2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le
% 0.19/0.46 thf(fact_87_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: real,P: real > $o] :
% 0.19/0.46 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.19/0.46 = ( P @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_88_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A2: set_real] :
% 0.19/0.46 ( ( collect_real
% 0.19/0.46 @ ^ [X: real] : ( member_real @ X @ A2 ) )
% 0.19/0.46 = A2 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_89_Collect__cong,axiom,
% 0.19/0.46 ! [P: real > $o,Q: real > $o] :
% 0.19/0.46 ( ! [X5: real] :
% 0.19/0.46 ( ( P @ X5 )
% 0.19/0.46 = ( Q @ X5 ) )
% 0.19/0.46 => ( ( collect_real @ P )
% 0.19/0.46 = ( collect_real @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_cong
% 0.19/0.46 thf(fact_90_Lim__bounded,axiom,
% 0.19/0.46 ! [F2: nat > nat,L: nat,M: nat,C3: nat] :
% 0.19/0.46 ( ( filterlim_nat_nat @ F2 @ ( topolo1564986139ds_nat @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ N2 ) @ C3 ) )
% 0.19/0.46 => ( ord_less_eq_nat @ L @ C3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded
% 0.19/0.46 thf(fact_91_Lim__bounded,axiom,
% 0.19/0.46 ! [F2: nat > int,L: int,M: nat,C3: int] :
% 0.19/0.46 ( ( filterlim_nat_int @ F2 @ ( topolo54776183ds_int @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N2 )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ N2 ) @ C3 ) )
% 0.19/0.46 => ( ord_less_eq_int @ L @ C3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded
% 0.19/0.46 thf(fact_92_Lim__bounded,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal,L: extended_ereal,M: nat,C3: extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ N2 ) @ C3 ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ L @ C3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded
% 0.19/0.46 thf(fact_93_Lim__bounded,axiom,
% 0.19/0.46 ! [F2: nat > real,L: real,M: nat,C3: real] :
% 0.19/0.46 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N2 )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ N2 ) @ C3 ) )
% 0.19/0.46 => ( ord_less_eq_real @ L @ C3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded
% 0.19/0.46 thf(fact_94_Lim__bounded2,axiom,
% 0.19/0.46 ! [F2: nat > nat,L: nat,N3: nat,C3: nat] :
% 0.19/0.46 ( ( filterlim_nat_nat @ F2 @ ( topolo1564986139ds_nat @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N3 @ N2 )
% 0.19/0.46 => ( ord_less_eq_nat @ C3 @ ( F2 @ N2 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ C3 @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded2
% 0.19/0.46 thf(fact_95_Lim__bounded2,axiom,
% 0.19/0.46 ! [F2: nat > int,L: int,N3: nat,C3: int] :
% 0.19/0.46 ( ( filterlim_nat_int @ F2 @ ( topolo54776183ds_int @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N3 @ N2 )
% 0.19/0.46 => ( ord_less_eq_int @ C3 @ ( F2 @ N2 ) ) )
% 0.19/0.46 => ( ord_less_eq_int @ C3 @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded2
% 0.19/0.46 thf(fact_96_Lim__bounded2,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal,L: extended_ereal,N3: nat,C3: extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N3 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ C3 @ ( F2 @ N2 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ C3 @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded2
% 0.19/0.46 thf(fact_97_Lim__bounded2,axiom,
% 0.19/0.46 ! [F2: nat > real,L: real,N3: nat,C3: real] :
% 0.19/0.46 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N3 @ N2 )
% 0.19/0.46 => ( ord_less_eq_real @ C3 @ ( F2 @ N2 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ C3 @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded2
% 0.19/0.46 thf(fact_98_LIMSEQ__le__const,axiom,
% 0.19/0.46 ! [X3: nat > nat,X2: nat,A: nat] :
% 0.19/0.46 ( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( X3 @ N2 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le_const
% 0.19/0.46 thf(fact_99_LIMSEQ__le__const,axiom,
% 0.19/0.46 ! [X3: nat > int,X2: int,A: int] :
% 0.19/0.46 ( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_less_eq_int @ A @ ( X3 @ N2 ) ) )
% 0.19/0.46 => ( ord_less_eq_int @ A @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le_const
% 0.19/0.46 thf(fact_100_LIMSEQ__le__const,axiom,
% 0.19/0.46 ! [X3: nat > extended_ereal,X2: extended_ereal,A: extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ ( X3 @ N2 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le_const
% 0.19/0.46 thf(fact_101_LIMSEQ__le__const,axiom,
% 0.19/0.46 ! [X3: nat > real,X2: real,A: real] :
% 0.19/0.46 ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_less_eq_real @ A @ ( X3 @ N2 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ A @ X2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le_const
% 0.19/0.46 thf(fact_102_LIMSEQ__le__const2,axiom,
% 0.19/0.46 ! [X3: nat > nat,X2: nat,A: nat] :
% 0.19/0.46 ( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( X3 @ N2 ) @ A ) )
% 0.19/0.46 => ( ord_less_eq_nat @ X2 @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le_const2
% 0.19/0.46 thf(fact_103_LIMSEQ__le__const2,axiom,
% 0.19/0.46 ! [X3: nat > int,X2: int,A: int] :
% 0.19/0.46 ( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_less_eq_int @ ( X3 @ N2 ) @ A ) )
% 0.19/0.46 => ( ord_less_eq_int @ X2 @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le_const2
% 0.19/0.46 thf(fact_104_LIMSEQ__le__const2,axiom,
% 0.19/0.46 ! [X3: nat > extended_ereal,X2: extended_ereal,A: extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( X3 @ N2 ) @ A ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ X2 @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le_const2
% 0.19/0.46 thf(fact_105_LIMSEQ__le__const2,axiom,
% 0.19/0.46 ! [X3: nat > real,X2: real,A: real] :
% 0.19/0.46 ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ? [N4: nat] :
% 0.19/0.46 ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.19/0.46 => ( ord_less_eq_real @ ( X3 @ N2 ) @ A ) )
% 0.19/0.46 => ( ord_less_eq_real @ X2 @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_le_const2
% 0.19/0.46 thf(fact_106_tendsto__mono,axiom,
% 0.19/0.46 ! [F: filter_nat,F4: filter_nat,F2: nat > extended_ereal,L: extended_ereal] :
% 0.19/0.46 ( ( ord_le1745708096er_nat @ F @ F4 )
% 0.19/0.46 => ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ F4 )
% 0.19/0.46 => ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_mono
% 0.19/0.46 thf(fact_107_tendsto__mono,axiom,
% 0.19/0.46 ! [F: filter_nat,F4: filter_nat,F2: nat > a,L: a] :
% 0.19/0.46 ( ( ord_le1745708096er_nat @ F @ F4 )
% 0.19/0.46 => ( ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ L ) @ F4 )
% 0.19/0.46 => ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ L ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_mono
% 0.19/0.46 thf(fact_108_tendsto__mono,axiom,
% 0.19/0.46 ! [F: filter_real,F4: filter_real,F2: real > real,L: real] :
% 0.19/0.46 ( ( ord_le132810396r_real @ F @ F4 )
% 0.19/0.46 => ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ L ) @ F4 )
% 0.19/0.46 => ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ L ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_mono
% 0.19/0.46 thf(fact_109_tendsto__mono,axiom,
% 0.19/0.46 ! [F: filter_nat,F4: filter_nat,F2: nat > real,L: real] :
% 0.19/0.46 ( ( ord_le1745708096er_nat @ F @ F4 )
% 0.19/0.46 => ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ F4 )
% 0.19/0.46 => ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ F ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % tendsto_mono
% 0.19/0.46 thf(fact_110_equation__minus__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( uminus_uminus_real @ B ) )
% 0.19/0.46 = ( B
% 0.19/0.46 = ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % equation_minus_iff
% 0.19/0.46 thf(fact_111_minus__equation__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = B )
% 0.19/0.46 = ( ( uminus_uminus_real @ B )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_equation_iff
% 0.19/0.46 thf(fact_112_comp__eq__dest__lhs,axiom,
% 0.19/0.46 ! [A: a > extended_ereal,B: nat > a,C: nat > extended_ereal,V: nat] :
% 0.19/0.46 ( ( ( comp_a1112243075al_nat @ A @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ( A @ ( B @ V ) )
% 0.19/0.46 = ( C @ V ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % comp_eq_dest_lhs
% 0.19/0.46 thf(fact_113_comp__eq__elim,axiom,
% 0.19/0.46 ! [A: a > extended_ereal,B: nat > a,C: a > extended_ereal,D: nat > a] :
% 0.19/0.46 ( ( ( comp_a1112243075al_nat @ A @ B )
% 0.19/0.46 = ( comp_a1112243075al_nat @ C @ D ) )
% 0.19/0.46 => ! [V2: nat] :
% 0.19/0.46 ( ( A @ ( B @ V2 ) )
% 0.19/0.46 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % comp_eq_elim
% 0.19/0.46 thf(fact_114_comp__eq__dest,axiom,
% 0.19/0.46 ! [A: a > extended_ereal,B: nat > a,C: a > extended_ereal,D: nat > a,V: nat] :
% 0.19/0.46 ( ( ( comp_a1112243075al_nat @ A @ B )
% 0.19/0.46 = ( comp_a1112243075al_nat @ C @ D ) )
% 0.19/0.46 => ( ( A @ ( B @ V ) )
% 0.19/0.46 = ( C @ ( D @ V ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % comp_eq_dest
% 0.19/0.46 thf(fact_115_comp__assoc,axiom,
% 0.19/0.46 ! [F2: a > extended_ereal,G2: nat > a,H: nat > nat] :
% 0.19/0.46 ( ( comp_n1096781355al_nat @ ( comp_a1112243075al_nat @ F2 @ G2 ) @ H )
% 0.19/0.46 = ( comp_a1112243075al_nat @ F2 @ ( comp_nat_a_nat @ G2 @ H ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % comp_assoc
% 0.19/0.46 thf(fact_116_comp__assoc,axiom,
% 0.19/0.46 ! [F2: extended_ereal > extended_ereal,G2: a > extended_ereal,H: nat > a] :
% 0.19/0.46 ( ( comp_a1112243075al_nat @ ( comp_E489644891real_a @ F2 @ G2 ) @ H )
% 0.19/0.46 = ( comp_E1308517939al_nat @ F2 @ ( comp_a1112243075al_nat @ G2 @ H ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % comp_assoc
% 0.19/0.46 thf(fact_117_comp__assoc,axiom,
% 0.19/0.46 ! [F2: a > extended_ereal,G2: a > a,H: nat > a] :
% 0.19/0.46 ( ( comp_a1112243075al_nat @ ( comp_a780206603real_a @ F2 @ G2 ) @ H )
% 0.19/0.46 = ( comp_a1112243075al_nat @ F2 @ ( comp_a_a_nat @ G2 @ H ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % comp_assoc
% 0.19/0.46 thf(fact_118_comp__def,axiom,
% 0.19/0.46 ( comp_a1112243075al_nat
% 0.19/0.46 = ( ^ [F3: a > extended_ereal,G: nat > a,X: nat] : ( F3 @ ( G @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % comp_def
% 0.19/0.46 thf(fact_119_lsc__sequentially,axiom,
% 0.19/0.46 ( lower_551915512_ereal
% 0.19/0.46 = ( ^ [X02: real,F3: real > extended_ereal] :
% 0.19/0.46 ! [X: nat > real,C2: extended_ereal] :
% 0.19/0.46 ( ( ( filterlim_nat_real @ X @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 0.19/0.46 & ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ C2 ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_sequentially
% 0.19/0.46 thf(fact_120_lsc__sequentially,axiom,
% 0.19/0.46 ( lower_191460856_ereal
% 0.19/0.46 = ( ^ [X02: a,F3: a > extended_ereal] :
% 0.19/0.46 ! [X: nat > a,C2: extended_ereal] :
% 0.19/0.46 ( ( ( filterlim_nat_a @ X @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 & ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ C2 ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_sequentially
% 0.19/0.46 thf(fact_121_usc__at__def,axiom,
% 0.19/0.46 ( lower_114093al_nat
% 0.19/0.46 = ( ^ [X02: extended_ereal,F3: extended_ereal > nat] :
% 0.19/0.46 ! [X4: nat > extended_ereal,L2: nat] :
% 0.19/0.46 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_nat @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_122_usc__at__def,axiom,
% 0.19/0.46 ( lower_637387785al_int
% 0.19/0.46 = ( ^ [X02: extended_ereal,F3: extended_ereal > int] :
% 0.19/0.46 ! [X4: nat > extended_ereal,L2: int] :
% 0.19/0.46 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_int @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_123_usc__at__def,axiom,
% 0.19/0.46 ( lower_1071158961_ereal
% 0.19/0.46 = ( ^ [X02: extended_ereal,F3: extended_ereal > extended_ereal] :
% 0.19/0.46 ! [X4: nat > extended_ereal,L2: extended_ereal] :
% 0.19/0.46 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_124_usc__at__def,axiom,
% 0.19/0.46 ( lower_737640969l_real
% 0.19/0.46 = ( ^ [X02: extended_ereal,F3: extended_ereal > real] :
% 0.19/0.46 ! [X4: nat > extended_ereal,L2: real] :
% 0.19/0.46 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_real @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_125_usc__at__def,axiom,
% 0.19/0.46 ( lower_1035717085_a_nat
% 0.19/0.46 = ( ^ [X02: a,F3: a > nat] :
% 0.19/0.46 ! [X4: nat > a,L2: nat] :
% 0.19/0.46 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_nat @ ( comp_a_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_nat @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_126_usc__at__def,axiom,
% 0.19/0.46 ( lower_1672990777_a_int
% 0.19/0.46 = ( ^ [X02: a,F3: a > int] :
% 0.19/0.46 ! [X4: nat > a,L2: int] :
% 0.19/0.46 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_int @ ( comp_a_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_int @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_127_usc__at__def,axiom,
% 0.19/0.46 ( lower_534855297_ereal
% 0.19/0.46 = ( ^ [X02: a,F3: a > extended_ereal] :
% 0.19/0.46 ! [X4: nat > a,L2: extended_ereal] :
% 0.19/0.46 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_128_usc__at__def,axiom,
% 0.19/0.46 ( lower_755922489a_real
% 0.19/0.46 = ( ^ [X02: a,F3: a > real] :
% 0.19/0.46 ! [X4: nat > a,L2: real] :
% 0.19/0.46 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_real @ ( comp_a_real_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_real @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_129_usc__at__def,axiom,
% 0.19/0.46 ( lower_438231087al_nat
% 0.19/0.46 = ( ^ [X02: real,F3: real > nat] :
% 0.19/0.46 ! [X4: nat > real,L2: nat] :
% 0.19/0.46 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_nat @ ( comp_real_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_nat @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_130_usc__at__def,axiom,
% 0.19/0.46 ( lower_1075504779al_int
% 0.19/0.46 = ( ^ [X02: real,F3: real > int] :
% 0.19/0.46 ! [X4: nat > real,L2: int] :
% 0.19/0.46 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 0.19/0.46 & ( filterlim_nat_int @ ( comp_real_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 0.19/0.46 => ( ord_less_eq_int @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_def
% 0.19/0.46 thf(fact_131_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: extended_ereal,F2: extended_ereal > nat,X2: nat > extended_ereal,A2: nat] :
% 0.19/0.46 ( ( lower_114093al_nat @ X0 @ F2 )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_nat @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_132_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: extended_ereal,F2: extended_ereal > int,X2: nat > extended_ereal,A2: int] :
% 0.19/0.46 ( ( lower_637387785al_int @ X0 @ F2 )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_int @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_133_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: extended_ereal,F2: extended_ereal > extended_ereal,X2: nat > extended_ereal,A2: extended_ereal] :
% 0.19/0.46 ( ( lower_1071158961_ereal @ X0 @ F2 )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_le824540014_ereal @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_134_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: extended_ereal,F2: extended_ereal > real,X2: nat > extended_ereal,A2: real] :
% 0.19/0.46 ( ( lower_737640969l_real @ X0 @ F2 )
% 0.19/0.46 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_real @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_135_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: a,F2: a > nat,X2: nat > a,A2: nat] :
% 0.19/0.46 ( ( lower_1035717085_a_nat @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_nat @ ( comp_a_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_nat @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_136_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: a,F2: a > int,X2: nat > a,A2: int] :
% 0.19/0.46 ( ( lower_1672990777_a_int @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_int @ ( comp_a_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_int @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_137_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: a,F2: a > extended_ereal,X2: nat > a,A2: extended_ereal] :
% 0.19/0.46 ( ( lower_534855297_ereal @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_le824540014_ereal @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_138_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: a,F2: a > real,X2: nat > a,A2: real] :
% 0.19/0.46 ( ( lower_755922489a_real @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_real @ ( comp_a_real_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_real @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_139_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: real,F2: real > nat,X2: nat > real,A2: nat] :
% 0.19/0.46 ( ( lower_438231087al_nat @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_nat @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_140_usc__at__mem,axiom,
% 0.19/0.46 ! [X0: real,F2: real > int,X2: nat > real,A2: int] :
% 0.19/0.46 ( ( lower_1075504779al_int @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ( filterlim_nat_int @ ( comp_real_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 0.19/0.46 => ( ord_less_eq_int @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % usc_at_mem
% 0.19/0.46 thf(fact_141_lim__decreasing__cl,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal] :
% 0.19/0.46 ( ! [N2: nat,M2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ N2 ) @ ( F2 @ M2 ) ) )
% 0.19/0.46 => ~ ! [L3: extended_ereal] :
% 0.19/0.46 ~ ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L3 ) @ at_top_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % lim_decreasing_cl
% 0.19/0.46 thf(fact_142_lim__increasing__cl,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal] :
% 0.19/0.46 ( ! [N2: nat,M2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ M2 ) @ ( F2 @ N2 ) ) )
% 0.19/0.46 => ~ ! [L3: extended_ereal] :
% 0.19/0.46 ~ ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L3 ) @ at_top_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % lim_increasing_cl
% 0.19/0.46 thf(fact_143_order__refl,axiom,
% 0.19/0.46 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ X2 ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_144_order__refl,axiom,
% 0.19/0.46 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_145_order__refl,axiom,
% 0.19/0.46 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_146_order__refl,axiom,
% 0.19/0.46 ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_147_monoseq__le,axiom,
% 0.19/0.46 ! [A: nat > nat,X2: nat] :
% 0.19/0.46 ( ( topolo1922093437eq_nat @ A )
% 0.19/0.46 => ( ( filterlim_nat_nat @ A @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( ! [N5: nat] : ( ord_less_eq_nat @ ( A @ N5 ) @ X2 )
% 0.19/0.46 & ! [M3: nat,N5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M3 @ N5 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
% 0.19/0.46 | ( ! [N5: nat] : ( ord_less_eq_nat @ X2 @ ( A @ N5 ) )
% 0.19/0.46 & ! [M3: nat,N5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M3 @ N5 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoseq_le
% 0.19/0.46 thf(fact_148_monoseq__le,axiom,
% 0.19/0.46 ! [A: nat > int,X2: int] :
% 0.19/0.46 ( ( topolo411883481eq_int @ A )
% 0.19/0.46 => ( ( filterlim_nat_int @ A @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( ! [N5: nat] : ( ord_less_eq_int @ ( A @ N5 ) @ X2 )
% 0.19/0.46 & ! [M3: nat,N5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M3 @ N5 )
% 0.19/0.46 => ( ord_less_eq_int @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
% 0.19/0.46 | ( ! [N5: nat] : ( ord_less_eq_int @ X2 @ ( A @ N5 ) )
% 0.19/0.46 & ! [M3: nat,N5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M3 @ N5 )
% 0.19/0.46 => ( ord_less_eq_int @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoseq_le
% 0.19/0.46 thf(fact_149_monoseq__le,axiom,
% 0.19/0.46 ! [A: nat > extended_ereal,X2: extended_ereal] :
% 0.19/0.46 ( ( topolo1069469409_ereal @ A )
% 0.19/0.46 => ( ( filter1531173832_ereal @ A @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( ! [N5: nat] : ( ord_le824540014_ereal @ ( A @ N5 ) @ X2 )
% 0.19/0.46 & ! [M3: nat,N5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M3 @ N5 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
% 0.19/0.46 | ( ! [N5: nat] : ( ord_le824540014_ereal @ X2 @ ( A @ N5 ) )
% 0.19/0.46 & ! [M3: nat,N5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M3 @ N5 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoseq_le
% 0.19/0.46 thf(fact_150_monoseq__le,axiom,
% 0.19/0.46 ! [A: nat > real,X2: real] :
% 0.19/0.46 ( ( topolo144289241q_real @ A )
% 0.19/0.46 => ( ( filterlim_nat_real @ A @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 0.19/0.46 => ( ( ! [N5: nat] : ( ord_less_eq_real @ ( A @ N5 ) @ X2 )
% 0.19/0.46 & ! [M3: nat,N5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M3 @ N5 )
% 0.19/0.46 => ( ord_less_eq_real @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
% 0.19/0.46 | ( ! [N5: nat] : ( ord_less_eq_real @ X2 @ ( A @ N5 ) )
% 0.19/0.46 & ! [M3: nat,N5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M3 @ N5 )
% 0.19/0.46 => ( ord_less_eq_real @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoseq_le
% 0.19/0.46 thf(fact_151_lsc__liminf,axiom,
% 0.19/0.46 ( lower_551915512_ereal
% 0.19/0.46 = ( ^ [X02: real,F3: real > extended_ereal] :
% 0.19/0.46 ! [X: nat > real] :
% 0.19/0.46 ( ( filterlim_nat_real @ X @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_r1410008527al_nat @ F3 @ X ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_liminf
% 0.19/0.46 thf(fact_152_lsc__liminf,axiom,
% 0.19/0.46 ( lower_191460856_ereal
% 0.19/0.46 = ( ^ [X02: a,F3: a > extended_ereal] :
% 0.19/0.46 ! [X: nat > a] :
% 0.19/0.46 ( ( filterlim_nat_a @ X @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_a1112243075al_nat @ F3 @ X ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_liminf
% 0.19/0.46 thf(fact_153_lsc__imp__liminf,axiom,
% 0.19/0.46 ! [X0: real,F2: real > extended_ereal,X2: nat > real] :
% 0.19/0.46 ( ( lower_551915512_ereal @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_r1410008527al_nat @ F2 @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_imp_liminf
% 0.19/0.46 thf(fact_154_lsc__imp__liminf,axiom,
% 0.19/0.46 ! [X0: a,F2: a > extended_ereal,X2: nat > a] :
% 0.19/0.46 ( ( lower_191460856_ereal @ X0 @ F2 )
% 0.19/0.46 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_a1112243075al_nat @ F2 @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % lsc_imp_liminf
% 0.19/0.46 thf(fact_155_LIMSEQ__Uniq,axiom,
% 0.19/0.46 ! [X3: nat > extended_ereal] :
% 0.19/0.46 ( uniq_Extended_ereal
% 0.19/0.46 @ ^ [L2: extended_ereal] : ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_Uniq
% 0.19/0.46 thf(fact_156_LIMSEQ__Uniq,axiom,
% 0.19/0.46 ! [X3: nat > a] :
% 0.19/0.46 ( uniq_a
% 0.19/0.46 @ ^ [L2: a] : ( filterlim_nat_a @ X3 @ ( topolo705128563nhds_a @ L2 ) @ at_top_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_Uniq
% 0.19/0.46 thf(fact_157_LIMSEQ__Uniq,axiom,
% 0.19/0.46 ! [X3: nat > real] :
% 0.19/0.46 ( uniq_real
% 0.19/0.46 @ ^ [L2: real] : ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % LIMSEQ_Uniq
% 0.19/0.46 thf(fact_158_monoseq__def,axiom,
% 0.19/0.46 ( topolo1069469409_ereal
% 0.19/0.46 = ( ^ [X4: nat > extended_ereal] :
% 0.19/0.46 ( ! [M4: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M4 @ N )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 0.19/0.46 | ! [M4: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M4 @ N )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoseq_def
% 0.19/0.46 thf(fact_159_monoseq__def,axiom,
% 0.19/0.46 ( topolo1922093437eq_nat
% 0.19/0.46 = ( ^ [X4: nat > nat] :
% 0.19/0.46 ( ! [M4: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M4 @ N )
% 0.19/0.46 => ( ord_less_eq_nat @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 0.19/0.46 | ! [M4: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M4 @ N )
% 0.19/0.46 => ( ord_less_eq_nat @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoseq_def
% 0.19/0.46 thf(fact_160_monoseq__def,axiom,
% 0.19/0.46 ( topolo144289241q_real
% 0.19/0.46 = ( ^ [X4: nat > real] :
% 0.19/0.46 ( ! [M4: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M4 @ N )
% 0.19/0.46 => ( ord_less_eq_real @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 0.19/0.46 | ! [M4: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M4 @ N )
% 0.19/0.46 => ( ord_less_eq_real @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoseq_def
% 0.19/0.46 thf(fact_161_monoseq__def,axiom,
% 0.19/0.46 ( topolo411883481eq_int
% 0.19/0.46 = ( ^ [X4: nat > int] :
% 0.19/0.46 ( ! [M4: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M4 @ N )
% 0.19/0.46 => ( ord_less_eq_int @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 0.19/0.46 | ! [M4: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M4 @ N )
% 0.19/0.46 => ( ord_less_eq_int @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoseq_def
% 0.19/0.46 thf(fact_162_monoI2,axiom,
% 0.19/0.46 ! [X3: nat > extended_ereal] :
% 0.19/0.46 ( ! [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 0.19/0.46 => ( topolo1069469409_ereal @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoI2
% 0.19/0.46 thf(fact_163_monoI2,axiom,
% 0.19/0.46 ! [X3: nat > nat] :
% 0.19/0.46 ( ! [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 0.19/0.46 => ( topolo1922093437eq_nat @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoI2
% 0.19/0.46 thf(fact_164_monoI2,axiom,
% 0.19/0.46 ! [X3: nat > real] :
% 0.19/0.46 ( ! [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_less_eq_real @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 0.19/0.46 => ( topolo144289241q_real @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoI2
% 0.19/0.46 thf(fact_165_monoI2,axiom,
% 0.19/0.46 ! [X3: nat > int] :
% 0.19/0.46 ( ! [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_less_eq_int @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 0.19/0.46 => ( topolo411883481eq_int @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoI2
% 0.19/0.46 thf(fact_166_monoI1,axiom,
% 0.19/0.46 ! [X3: nat > extended_ereal] :
% 0.19/0.46 ( ! [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 0.19/0.46 => ( topolo1069469409_ereal @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoI1
% 0.19/0.46 thf(fact_167_monoI1,axiom,
% 0.19/0.46 ! [X3: nat > nat] :
% 0.19/0.46 ( ! [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 0.19/0.46 => ( topolo1922093437eq_nat @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoI1
% 0.19/0.46 thf(fact_168_monoI1,axiom,
% 0.19/0.46 ! [X3: nat > real] :
% 0.19/0.46 ( ! [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_less_eq_real @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 0.19/0.46 => ( topolo144289241q_real @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoI1
% 0.19/0.46 thf(fact_169_monoI1,axiom,
% 0.19/0.46 ! [X3: nat > int] :
% 0.19/0.46 ( ! [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 => ( ord_less_eq_int @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 0.19/0.46 => ( topolo411883481eq_int @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoI1
% 0.19/0.46 thf(fact_170_monoseq__minus,axiom,
% 0.19/0.46 ! [A: nat > real] :
% 0.19/0.46 ( ( topolo144289241q_real @ A )
% 0.19/0.46 => ( topolo144289241q_real
% 0.19/0.46 @ ^ [N: nat] : ( uminus_uminus_real @ ( A @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % monoseq_minus
% 0.19/0.46 thf(fact_171_order__subst1,axiom,
% 0.19/0.46 ! [A: extended_ereal,F2: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_172_order__subst1,axiom,
% 0.19/0.46 ! [A: extended_ereal,F2: nat > extended_ereal,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_173_order__subst1,axiom,
% 0.19/0.46 ! [A: extended_ereal,F2: real > extended_ereal,B: real,C: real] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ! [X5: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_174_order__subst1,axiom,
% 0.19/0.46 ! [A: extended_ereal,F2: int > extended_ereal,B: int,C: int] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.19/0.46 => ( ! [X5: int,Y3: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_175_order__subst1,axiom,
% 0.19/0.46 ! [A: nat,F2: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_176_order__subst1,axiom,
% 0.19/0.46 ! [A: nat,F2: nat > nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_177_order__subst1,axiom,
% 0.19/0.46 ! [A: nat,F2: real > nat,B: real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ! [X5: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_178_order__subst1,axiom,
% 0.19/0.46 ! [A: nat,F2: int > nat,B: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.19/0.46 => ( ! [X5: int,Y3: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_179_order__subst1,axiom,
% 0.19/0.46 ! [A: real,F2: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_180_order__subst1,axiom,
% 0.19/0.46 ! [A: real,F2: nat > real,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_181_order__subst2,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_182_order__subst2,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > nat,C: nat] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_183_order__subst2,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > real,C: real] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_184_order__subst2,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > int,C: int] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_185_order__subst2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F2: nat > extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_186_order__subst2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F2: nat > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_187_order__subst2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F2: nat > real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_188_order__subst2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F2: nat > int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_189_order__subst2,axiom,
% 0.19/0.46 ! [A: real,B: real,F2: real > extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_190_order__subst2,axiom,
% 0.19/0.46 ! [A: real,B: real,F2: real > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 0.19/0.46 => ( ! [X5: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_191_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: extended_ereal,F2: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_192_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: nat,F2: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_193_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: real,F2: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_194_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: int,F2: extended_ereal > int,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_195_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: extended_ereal,F2: nat > extended_ereal,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_196_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: nat,F2: nat > nat,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_197_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: real,F2: nat > real,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_198_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: int,F2: nat > int,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_199_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: extended_ereal,F2: real > extended_ereal,B: real,C: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ! [X5: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_200_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: nat,F2: real > nat,B: real,C: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F2 @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ! [X5: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_201_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_202_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > nat,C: nat] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_203_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > real,C: real] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_204_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > int,C: int] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_205_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F2: nat > extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_206_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F2: nat > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_207_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F2: nat > real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_208_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F2: nat > int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: nat,Y3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_209_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: real,B: real,F2: real > extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X5 @ Y3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_210_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: real,B: real,F2: real > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ( F2 @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X5: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X5 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_211_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: extended_ereal,Z2: extended_ereal] : ( Y4 = Z2 ) )
% 0.19/0.46 = ( ^ [X: extended_ereal,Y5: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X @ Y5 )
% 0.19/0.46 & ( ord_le824540014_ereal @ Y5 @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_212_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 0.19/0.46 = ( ^ [X: nat,Y5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y5 )
% 0.19/0.46 & ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_213_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
% 0.19/0.46 = ( ^ [X: real,Y5: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X @ Y5 )
% 0.19/0.46 & ( ord_less_eq_real @ Y5 @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_214_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
% 0.19/0.46 = ( ^ [X: int,Y5: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ X @ Y5 )
% 0.19/0.46 & ( ord_less_eq_int @ Y5 @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_215_antisym,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Y: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X2 @ Y )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ Y @ X2 )
% 0.19/0.46 => ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_216_antisym,axiom,
% 0.19/0.46 ! [X2: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq_nat @ Y @ X2 )
% 0.19/0.46 => ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_217_antisym,axiom,
% 0.19/0.46 ! [X2: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq_real @ Y @ X2 )
% 0.19/0.46 => ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_218_antisym,axiom,
% 0.19/0.46 ! [X2: int,Y: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq_int @ Y @ X2 )
% 0.19/0.46 => ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_219_linear,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Y: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X2 @ Y )
% 0.19/0.46 | ( ord_le824540014_ereal @ Y @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_220_linear,axiom,
% 0.19/0.46 ! [X2: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.19/0.46 | ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_221_linear,axiom,
% 0.19/0.46 ! [X2: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X2 @ Y )
% 0.19/0.46 | ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_222_linear,axiom,
% 0.19/0.46 ! [X2: int,Y: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ X2 @ Y )
% 0.19/0.46 | ( ord_less_eq_int @ Y @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_223_eq__refl,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Y: extended_ereal] :
% 0.19/0.46 ( ( X2 = Y )
% 0.19/0.46 => ( ord_le824540014_ereal @ X2 @ Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_224_eq__refl,axiom,
% 0.19/0.46 ! [X2: nat,Y: nat] :
% 0.19/0.46 ( ( X2 = Y )
% 0.19/0.46 => ( ord_less_eq_nat @ X2 @ Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_225_eq__refl,axiom,
% 0.19/0.46 ! [X2: real,Y: real] :
% 0.19/0.46 ( ( X2 = Y )
% 0.19/0.46 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_226_eq__refl,axiom,
% 0.19/0.46 ! [X2: int,Y: int] :
% 0.19/0.46 ( ( X2 = Y )
% 0.19/0.46 => ( ord_less_eq_int @ X2 @ Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_227_le__cases,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Y: extended_ereal] :
% 0.19/0.46 ( ~ ( ord_le824540014_ereal @ X2 @ Y )
% 0.19/0.46 => ( ord_le824540014_ereal @ Y @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_228_le__cases,axiom,
% 0.19/0.46 ! [X2: nat,Y: nat] :
% 0.19/0.46 ( ~ ( ord_less_eq_nat @ X2 @ Y )
% 0.19/0.46 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_229_le__cases,axiom,
% 0.19/0.46 ! [X2: real,Y: real] :
% 0.19/0.46 ( ~ ( ord_less_eq_real @ X2 @ Y )
% 0.19/0.46 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_230_le__cases,axiom,
% 0.19/0.46 ! [X2: int,Y: int] :
% 0.19/0.46 ( ~ ( ord_less_eq_int @ X2 @ Y )
% 0.19/0.46 => ( ord_less_eq_int @ Y @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_231_order_Otrans,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_232_order_Otrans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_233_order_Otrans,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_234_order_Otrans,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.19/0.46 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_235_le__cases3,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
% 0.19/0.46 ( ( ( ord_le824540014_ereal @ X2 @ Y )
% 0.19/0.46 => ~ ( ord_le824540014_ereal @ Y @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_le824540014_ereal @ Y @ X2 )
% 0.19/0.46 => ~ ( ord_le824540014_ereal @ X2 @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_le824540014_ereal @ X2 @ Z3 )
% 0.19/0.46 => ~ ( ord_le824540014_ereal @ Z3 @ Y ) )
% 0.19/0.46 => ( ( ( ord_le824540014_ereal @ Z3 @ Y )
% 0.19/0.46 => ~ ( ord_le824540014_ereal @ Y @ X2 ) )
% 0.19/0.46 => ( ( ( ord_le824540014_ereal @ Y @ Z3 )
% 0.19/0.46 => ~ ( ord_le824540014_ereal @ Z3 @ X2 ) )
% 0.19/0.46 => ~ ( ( ord_le824540014_ereal @ Z3 @ X2 )
% 0.19/0.46 => ~ ( ord_le824540014_ereal @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_236_le__cases3,axiom,
% 0.19/0.46 ! [X2: nat,Y: nat,Z3: nat] :
% 0.19/0.46 ( ( ( ord_less_eq_nat @ X2 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ Y @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ X2 @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ Z3 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Y @ X2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ Y @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
% 0.19/0.46 => ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_237_le__cases3,axiom,
% 0.19/0.46 ! [X2: real,Y: real,Z3: real] :
% 0.19/0.46 ( ( ( ord_less_eq_real @ X2 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_real @ Y @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_real @ Y @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ X2 @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_real @ X2 @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ Z3 @ Y ) )
% 0.19/0.46 => ( ( ( ord_less_eq_real @ Z3 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_real @ Y @ X2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_real @ Y @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ Z3 @ X2 ) )
% 0.19/0.46 => ~ ( ( ord_less_eq_real @ Z3 @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_238_le__cases3,axiom,
% 0.19/0.46 ! [X2: int,Y: int,Z3: int] :
% 0.19/0.46 ( ( ( ord_less_eq_int @ X2 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_int @ Y @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_int @ Y @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq_int @ X2 @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_int @ X2 @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_int @ Z3 @ Y ) )
% 0.19/0.46 => ( ( ( ord_less_eq_int @ Z3 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_int @ Y @ X2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_int @ Y @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_int @ Z3 @ X2 ) )
% 0.19/0.46 => ~ ( ( ord_less_eq_int @ Z3 @ X2 )
% 0.19/0.46 => ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_239_antisym__conv,axiom,
% 0.19/0.46 ! [Y: extended_ereal,X2: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ Y @ X2 )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ X2 @ Y )
% 0.19/0.46 = ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_240_antisym__conv,axiom,
% 0.19/0.46 ! [Y: nat,X2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ Y @ X2 )
% 0.19/0.46 => ( ( ord_less_eq_nat @ X2 @ Y )
% 0.19/0.46 = ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_241_antisym__conv,axiom,
% 0.19/0.46 ! [Y: real,X2: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ Y @ X2 )
% 0.19/0.46 => ( ( ord_less_eq_real @ X2 @ Y )
% 0.19/0.46 = ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_242_antisym__conv,axiom,
% 0.19/0.46 ! [Y: int,X2: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ Y @ X2 )
% 0.19/0.46 => ( ( ord_less_eq_int @ X2 @ Y )
% 0.19/0.46 = ( X2 = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_243_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: extended_ereal,Z2: extended_ereal] : ( Y4 = Z2 ) )
% 0.19/0.46 = ( ^ [A3: extended_ereal,B2: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A3 @ B2 )
% 0.19/0.46 & ( ord_le824540014_ereal @ B2 @ A3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_244_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 0.19/0.46 = ( ^ [A3: nat,B2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.19/0.46 & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_245_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
% 0.19/0.46 = ( ^ [A3: real,B2: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A3 @ B2 )
% 0.19/0.46 & ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_246_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
% 0.19/0.46 = ( ^ [A3: int,B2: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.19/0.46 & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_247_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_248_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_249_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_250_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.19/0.46 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_251_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_252_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_253_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_254_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: int,B: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_255_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_256_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_257_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_258_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: int,B: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_int @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_259_order__trans,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X2 @ Y )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ Y @ Z3 )
% 0.19/0.46 => ( ord_le824540014_ereal @ X2 @ Z3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_260_order__trans,axiom,
% 0.19/0.46 ! [X2: nat,Y: nat,Z3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq_nat @ Y @ Z3 )
% 0.19/0.46 => ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_261_order__trans,axiom,
% 0.19/0.46 ! [X2: real,Y: real,Z3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq_real @ Y @ Z3 )
% 0.19/0.46 => ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_262_order__trans,axiom,
% 0.19/0.46 ! [X2: int,Y: int,Z3: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ X2 @ Y )
% 0.19/0.46 => ( ( ord_less_eq_int @ Y @ Z3 )
% 0.19/0.46 => ( ord_less_eq_int @ X2 @ Z3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_263_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: extended_ereal] : ( ord_le824540014_ereal @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_264_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_265_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_266_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_267_linorder__wlog,axiom,
% 0.19/0.46 ! [P: extended_ereal > extended_ereal > $o,A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ! [A4: extended_ereal,B3: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A4 @ B3 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( ! [A4: extended_ereal,B3: extended_ereal] :
% 0.19/0.46 ( ( P @ B3 @ A4 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( P @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_268_linorder__wlog,axiom,
% 0.19/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.19/0.46 ( ! [A4: nat,B3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A4 @ B3 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( ! [A4: nat,B3: nat] :
% 0.19/0.46 ( ( P @ B3 @ A4 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( P @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_269_linorder__wlog,axiom,
% 0.19/0.46 ! [P: real > real > $o,A: real,B: real] :
% 0.19/0.46 ( ! [A4: real,B3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A4 @ B3 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( ! [A4: real,B3: real] :
% 0.19/0.46 ( ( P @ B3 @ A4 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( P @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_270_linorder__wlog,axiom,
% 0.19/0.46 ! [P: int > int > $o,A: int,B: int] :
% 0.19/0.46 ( ! [A4: int,B3: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ A4 @ B3 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( ! [A4: int,B3: int] :
% 0.19/0.46 ( ( P @ B3 @ A4 )
% 0.19/0.46 => ( P @ A4 @ B3 ) )
% 0.19/0.46 => ( P @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_271_dual__order_Otrans,axiom,
% 0.19/0.46 ! [B: nat,A: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.46 => ( ( ord_less_eq_nat @ C @ B )
% 0.19/0.46 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.trans
% 0.19/0.46 thf(fact_272_dual__order_Otrans,axiom,
% 0.19/0.46 ! [B: real,A: real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.19/0.46 => ( ( ord_less_eq_real @ C @ B )
% 0.19/0.46 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.trans
% 0.19/0.46 thf(fact_273_dual__order_Otrans,axiom,
% 0.19/0.46 ! [B: int,A: int,C: int] :
% 0.19/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.19/0.46 => ( ( ord_less_eq_int @ C @ B )
% 0.19/0.46 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.trans
% 0.19/0.46 thf(fact_274_liminf__PInfty,axiom,
% 0.19/0.46 ! [X3: nat > extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ extend1289208545_ereal ) @ at_top_nat )
% 0.19/0.46 = ( ( liminf1045857232_ereal @ at_top_nat @ X3 )
% 0.19/0.46 = extend1289208545_ereal ) ) ).
% 0.19/0.46
% 0.19/0.46 % liminf_PInfty
% 0.19/0.46 thf(fact_275_ereal__infty__less__eq_I1_J,axiom,
% 0.19/0.46 ! [X2: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ extend1289208545_ereal @ X2 )
% 0.19/0.46 = ( X2 = extend1289208545_ereal ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_infty_less_eq(1)
% 0.19/0.46 thf(fact_276_ereal__infty__less__eq_I2_J,axiom,
% 0.19/0.46 ! [X2: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 = ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_infty_less_eq(2)
% 0.19/0.46 thf(fact_277_MInfty__neq__PInfty_I1_J,axiom,
% 0.19/0.46 ( extend1289208545_ereal
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.19/0.46
% 0.19/0.46 % MInfty_neq_PInfty(1)
% 0.19/0.46 thf(fact_278_ereal__less__eq_I1_J,axiom,
% 0.19/0.46 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ extend1289208545_ereal ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_less_eq(1)
% 0.19/0.46 thf(fact_279_ereal__infty__less__eq2_I1_J,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( A = extend1289208545_ereal )
% 0.19/0.46 => ( B = extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_infty_less_eq2(1)
% 0.19/0.46 thf(fact_280_neq__PInf__trans,axiom,
% 0.19/0.46 ! [Y: extended_ereal,X2: extended_ereal] :
% 0.19/0.46 ( ( Y != extend1289208545_ereal )
% 0.19/0.46 => ( ( ord_le824540014_ereal @ X2 @ Y )
% 0.19/0.46 => ( X2 != extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % neq_PInf_trans
% 0.19/0.46 thf(fact_281_ereal__infty__less__eq2_I2_J,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 => ( ( B
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( A
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_infty_less_eq2(2)
% 0.19/0.46 thf(fact_282_ereal__less__eq_I2_J,axiom,
% 0.19/0.46 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X2 ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_less_eq(2)
% 0.19/0.46 thf(fact_283_Lim__MInfty,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) @ at_top_nat )
% 0.19/0.46 = ( ! [B4: real] :
% 0.19/0.46 ? [N6: nat] :
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N6 @ N )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ N ) @ ( extended_ereal2 @ B4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_MInfty
% 0.19/0.46 thf(fact_284_Lim__bounded__MInfty,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal,L: extended_ereal,B5: real] :
% 0.19/0.46 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B5 ) @ ( F2 @ N2 ) )
% 0.19/0.46 => ( L
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded_MInfty
% 0.19/0.46 thf(fact_285_Lim__PInfty,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal] :
% 0.19/0.46 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ extend1289208545_ereal ) @ at_top_nat )
% 0.19/0.46 = ( ! [B4: real] :
% 0.19/0.46 ? [N6: nat] :
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N6 @ N )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( extended_ereal2 @ B4 ) @ ( F2 @ N ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_PInfty
% 0.19/0.46 thf(fact_286_Lim__bounded__PInfty2,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal,L: extended_ereal,N3: nat,B5: real] :
% 0.19/0.46 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N3 @ N2 )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B5 ) ) )
% 0.19/0.46 => ( L != extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded_PInfty2
% 0.19/0.46 thf(fact_287_ereal_Oinject,axiom,
% 0.19/0.46 ! [X1: real,Y1: real] :
% 0.19/0.46 ( ( ( extended_ereal2 @ X1 )
% 0.19/0.46 = ( extended_ereal2 @ Y1 ) )
% 0.19/0.46 = ( X1 = Y1 ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal.inject
% 0.19/0.46 thf(fact_288_ereal__cong,axiom,
% 0.19/0.46 ! [X2: real,Y: real] :
% 0.19/0.46 ( ( X2 = Y )
% 0.19/0.46 => ( ( extended_ereal2 @ X2 )
% 0.19/0.46 = ( extended_ereal2 @ Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_cong
% 0.19/0.46 thf(fact_289_ereal__less__eq_I3_J,axiom,
% 0.19/0.46 ! [R: real,P2: real] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
% 0.19/0.46 = ( ord_less_eq_real @ R @ P2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_less_eq(3)
% 0.19/0.46 thf(fact_290_PInfty__neq__ereal_I1_J,axiom,
% 0.19/0.46 ! [R: real] :
% 0.19/0.46 ( ( extended_ereal2 @ R )
% 0.19/0.46 != extend1289208545_ereal ) ).
% 0.19/0.46
% 0.19/0.46 % PInfty_neq_ereal(1)
% 0.19/0.46 thf(fact_291_uminus__ereal_Osimps_I1_J,axiom,
% 0.19/0.46 ! [R: real] :
% 0.19/0.46 ( ( uminus1208298309_ereal @ ( extended_ereal2 @ R ) )
% 0.19/0.46 = ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_ereal.simps(1)
% 0.19/0.46 thf(fact_292_ereal__le__le,axiom,
% 0.19/0.46 ! [Y: real,A: extended_ereal,X2: real] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ Y ) @ A )
% 0.19/0.46 => ( ( ord_less_eq_real @ X2 @ Y )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_le_le
% 0.19/0.46 thf(fact_293_le__ereal__le,axiom,
% 0.19/0.46 ! [A: extended_ereal,X2: real,Y: real] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ X2 ) )
% 0.19/0.46 => ( ( ord_less_eq_real @ X2 @ Y )
% 0.19/0.46 => ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ Y ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_ereal_le
% 0.19/0.46 thf(fact_294_ereal__le__real,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Y: extended_ereal] :
% 0.19/0.46 ( ! [Z4: real] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ Y @ ( extended_ereal2 @ Z4 ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ Y @ X2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_le_real
% 0.19/0.46 thf(fact_295_ereal__semiline__unique,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ( collect_real
% 0.19/0.46 @ ^ [Y5: real] : ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ Y5 ) ) )
% 0.19/0.46 = ( collect_real
% 0.19/0.46 @ ^ [Y5: real] : ( ord_le824540014_ereal @ B @ ( extended_ereal2 @ Y5 ) ) ) )
% 0.19/0.46 = ( A = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_semiline_unique
% 0.19/0.46 thf(fact_296_real__of__ereal_Oinduct,axiom,
% 0.19/0.46 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.19/0.46 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( P @ extend1289208545_ereal )
% 0.19/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( P @ A0 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % real_of_ereal.induct
% 0.19/0.46 thf(fact_297_real__of__ereal_Ocases,axiom,
% 0.19/0.46 ! [X2: extended_ereal] :
% 0.19/0.46 ( ! [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 != ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( X2 != extend1289208545_ereal )
% 0.19/0.46 => ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % real_of_ereal.cases
% 0.19/0.46 thf(fact_298_times__ereal_Oinduct,axiom,
% 0.19/0.46 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.19/0.46 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 0.19/0.46 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ extend1289208545_ereal )
% 0.19/0.46 => ( ! [R2: real] : ( P @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( P @ extend1289208545_ereal @ extend1289208545_ereal )
% 0.19/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 0.19/0.46 => ( ( P @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % times_ereal.induct
% 0.19/0.46 thf(fact_299_plus__ereal_Oinduct,axiom,
% 0.19/0.46 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.19/0.46 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 0.19/0.46 => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
% 0.19/0.46 => ( ! [A4: extended_ereal] : ( P @ A4 @ extend1289208545_ereal )
% 0.19/0.46 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ! [P3: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P3 ) )
% 0.19/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % plus_ereal.induct
% 0.19/0.46 thf(fact_300_less__ereal_Oinduct,axiom,
% 0.19/0.46 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.19/0.46 ( ! [X5: real,Y3: real] : ( P @ ( extended_ereal2 @ X5 ) @ ( extended_ereal2 @ Y3 ) )
% 0.19/0.46 => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
% 0.19/0.46 => ( ! [A4: extended_ereal] : ( P @ A4 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ! [X5: real] : ( P @ ( extended_ereal2 @ X5 ) @ extend1289208545_ereal )
% 0.19/0.46 => ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 0.19/0.46 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_ereal.induct
% 0.19/0.46 thf(fact_301_abs__ereal_Oinduct,axiom,
% 0.19/0.46 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.19/0.46 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( P @ extend1289208545_ereal )
% 0.19/0.46 => ( P @ A0 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_ereal.induct
% 0.19/0.46 thf(fact_302_ereal__all__split,axiom,
% 0.19/0.46 ( ( ^ [P4: extended_ereal > $o] :
% 0.19/0.46 ! [X6: extended_ereal] : ( P4 @ X6 ) )
% 0.19/0.46 = ( ^ [P5: extended_ereal > $o] :
% 0.19/0.46 ( ( P5 @ extend1289208545_ereal )
% 0.19/0.46 & ! [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
% 0.19/0.46 & ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_all_split
% 0.19/0.46 thf(fact_303_abs__ereal_Ocases,axiom,
% 0.19/0.46 ! [X2: extended_ereal] :
% 0.19/0.46 ( ! [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 != ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( X2
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( X2 = extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % abs_ereal.cases
% 0.19/0.46 thf(fact_304_ereal__ex__split,axiom,
% 0.19/0.46 ( ( ^ [P4: extended_ereal > $o] :
% 0.19/0.46 ? [X6: extended_ereal] : ( P4 @ X6 ) )
% 0.19/0.46 = ( ^ [P5: extended_ereal > $o] :
% 0.19/0.46 ( ( P5 @ extend1289208545_ereal )
% 0.19/0.46 | ? [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
% 0.19/0.46 | ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_ex_split
% 0.19/0.46 thf(fact_305_ereal3__cases,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 0.19/0.46 ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ? [Ra: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ Ra ) )
% 0.19/0.46 => ! [Rb: real] :
% 0.19/0.46 ( Xb
% 0.19/0.46 != ( extended_ereal2 @ Rb ) ) ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ? [Ra: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ Ra ) )
% 0.19/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ? [Ra: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ Ra ) )
% 0.19/0.46 => ( Xb
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ! [Ra: real] :
% 0.19/0.46 ( Xb
% 0.19/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ( Xb
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ! [Ra: real] :
% 0.19/0.46 ( Xb
% 0.19/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Xb
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( ? [R2: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ! [Ra: real] :
% 0.19/0.46 ( Xb
% 0.19/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( ? [R2: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( ? [R2: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( Xb
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ! [R2: real] :
% 0.19/0.46 ( Xb
% 0.19/0.46 != ( extended_ereal2 @ R2 ) ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ( Xb
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ! [R2: real] :
% 0.19/0.46 ( Xb
% 0.19/0.46 != ( extended_ereal2 @ R2 ) ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Xb
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ? [R2: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ! [Ra: real] :
% 0.19/0.46 ( Xb
% 0.19/0.46 != ( extended_ereal2 @ Ra ) ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ? [R2: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ? [R2: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( Xb
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ! [R2: real] :
% 0.19/0.46 ( Xb
% 0.19/0.46 != ( extended_ereal2 @ R2 ) ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ( Xb
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ! [R2: real] :
% 0.19/0.46 ( Xb
% 0.19/0.46 != ( extended_ereal2 @ R2 ) ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Xb != extend1289208545_ereal ) ) )
% 0.19/0.46 => ~ ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Xb
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal3_cases
% 0.19/0.46 thf(fact_306_ereal2__cases,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Xa3: extended_ereal] :
% 0.19/0.46 ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ! [Ra: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 != ( extended_ereal2 @ Ra ) ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( Xa3 != extend1289208545_ereal ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( Xa3
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ! [R2: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 != ( extended_ereal2 @ R2 ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( Xa3 != extend1289208545_ereal ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( Xa3
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ! [R2: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 != ( extended_ereal2 @ R2 ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Xa3 != extend1289208545_ereal ) )
% 0.19/0.46 => ~ ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Xa3
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal2_cases
% 0.19/0.46 thf(fact_307_ereal__cases,axiom,
% 0.19/0.46 ! [X2: extended_ereal] :
% 0.19/0.46 ( ! [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 != ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( X2 != extend1289208545_ereal )
% 0.19/0.46 => ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_cases
% 0.19/0.46 thf(fact_308_MInfty__neq__ereal_I1_J,axiom,
% 0.19/0.46 ! [R: real] :
% 0.19/0.46 ( ( extended_ereal2 @ R )
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.19/0.46
% 0.19/0.46 % MInfty_neq_ereal(1)
% 0.19/0.46 thf(fact_309_ereal__top,axiom,
% 0.19/0.46 ! [X2: extended_ereal] :
% 0.19/0.46 ( ! [B6: real] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B6 ) @ X2 )
% 0.19/0.46 => ( X2 = extend1289208545_ereal ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_top
% 0.19/0.46 thf(fact_310_ereal__bot,axiom,
% 0.19/0.46 ! [X2: extended_ereal] :
% 0.19/0.46 ( ! [B6: real] : ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ B6 ) )
% 0.19/0.46 => ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_bot
% 0.19/0.46 thf(fact_311_Lim__bounded__PInfty,axiom,
% 0.19/0.46 ! [F2: nat > extended_ereal,L: extended_ereal,B5: real] :
% 0.19/0.46 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 0.19/0.46 => ( ! [N2: nat] : ( ord_le824540014_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B5 ) )
% 0.19/0.46 => ( L != extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Lim_bounded_PInfty
% 0.19/0.46 thf(fact_312_ereal__minus__real__tendsto__MInf,axiom,
% 0.19/0.46 ( filter1531173832_ereal
% 0.19/0.46 @ ^ [X: nat] : ( extended_ereal2 @ ( uminus_uminus_real @ ( semiri2110766477t_real @ X ) ) )
% 0.19/0.46 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 @ at_top_nat ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_minus_real_tendsto_MInf
% 0.19/0.46 thf(fact_313_ereal__PInfty__eq__plus,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( extend1289208545_ereal
% 0.19/0.46 = ( plus_p2118002693_ereal @ A @ B ) )
% 0.19/0.46 = ( ( A = extend1289208545_ereal )
% 0.19/0.46 | ( B = extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_PInfty_eq_plus
% 0.19/0.46 thf(fact_314_ereal__plus__eq__PInfty,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ( plus_p2118002693_ereal @ A @ B )
% 0.19/0.46 = extend1289208545_ereal )
% 0.19/0.46 = ( ( A = extend1289208545_ereal )
% 0.19/0.46 | ( B = extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_plus_eq_PInfty
% 0.19/0.46 thf(fact_315_ereal__MInfty__eq__plus,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ( uminus1208298309_ereal @ extend1289208545_ereal )
% 0.19/0.46 = ( plus_p2118002693_ereal @ A @ B ) )
% 0.19/0.46 = ( ( ( A
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 & ( B != extend1289208545_ereal ) )
% 0.19/0.46 | ( ( B
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 & ( A != extend1289208545_ereal ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_MInfty_eq_plus
% 0.19/0.46 thf(fact_316_ereal__plus__eq__MInfty,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ( plus_p2118002693_ereal @ A @ B )
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 = ( ( ( A
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 | ( B
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.19/0.46 & ( A != extend1289208545_ereal )
% 0.19/0.46 & ( B != extend1289208545_ereal ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_plus_eq_MInfty
% 0.19/0.46 thf(fact_317_plus__ereal_Osimps_I2_J,axiom,
% 0.19/0.46 ! [A: extended_ereal] :
% 0.19/0.46 ( ( plus_p2118002693_ereal @ extend1289208545_ereal @ A )
% 0.19/0.46 = extend1289208545_ereal ) ).
% 0.19/0.46
% 0.19/0.46 % plus_ereal.simps(2)
% 0.19/0.46 thf(fact_318_plus__ereal_Osimps_I3_J,axiom,
% 0.19/0.46 ! [A: extended_ereal] :
% 0.19/0.46 ( ( plus_p2118002693_ereal @ A @ extend1289208545_ereal )
% 0.19/0.46 = extend1289208545_ereal ) ).
% 0.19/0.46
% 0.19/0.46 % plus_ereal.simps(3)
% 0.19/0.46 thf(fact_319_plus__ereal_Osimps_I6_J,axiom,
% 0.19/0.46 ( ( plus_p2118002693_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.19/0.46
% 0.19/0.46 % plus_ereal.simps(6)
% 0.19/0.46 thf(fact_320_ereal__add__cancel__left,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( A
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( ( plus_p2118002693_ereal @ A @ B )
% 0.19/0.46 = ( plus_p2118002693_ereal @ A @ C ) )
% 0.19/0.46 = ( ( A = extend1289208545_ereal )
% 0.19/0.46 | ( B = C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_add_cancel_left
% 0.19/0.46 thf(fact_321_ereal__add__cancel__right,axiom,
% 0.19/0.46 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.19/0.46 ( ( A
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( ( plus_p2118002693_ereal @ B @ A )
% 0.19/0.46 = ( plus_p2118002693_ereal @ C @ A ) )
% 0.19/0.46 = ( ( A = extend1289208545_ereal )
% 0.19/0.46 | ( B = C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_add_cancel_right
% 0.19/0.46 thf(fact_322_plus__ereal_Osimps_I5_J,axiom,
% 0.19/0.46 ! [P2: real] :
% 0.19/0.46 ( ( plus_p2118002693_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P2 ) )
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.19/0.46
% 0.19/0.46 % plus_ereal.simps(5)
% 0.19/0.46 thf(fact_323_plus__ereal_Osimps_I4_J,axiom,
% 0.19/0.46 ! [R: real] :
% 0.19/0.46 ( ( plus_p2118002693_ereal @ ( extended_ereal2 @ R ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.19/0.46
% 0.19/0.46 % plus_ereal.simps(4)
% 0.19/0.46 thf(fact_324_ereal__add__le__add__iff,axiom,
% 0.19/0.46 ! [C: extended_ereal,A: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ ( plus_p2118002693_ereal @ C @ A ) @ ( plus_p2118002693_ereal @ C @ B ) )
% 0.19/0.46 = ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 | ( C = extend1289208545_ereal )
% 0.19/0.46 | ( ( C
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 & ( A != extend1289208545_ereal )
% 0.19/0.46 & ( B != extend1289208545_ereal ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_add_le_add_iff
% 0.19/0.46 thf(fact_325_ereal__add__le__add__iff2,axiom,
% 0.19/0.46 ! [A: extended_ereal,C: extended_ereal,B: extended_ereal] :
% 0.19/0.46 ( ( ord_le824540014_ereal @ ( plus_p2118002693_ereal @ A @ C ) @ ( plus_p2118002693_ereal @ B @ C ) )
% 0.19/0.46 = ( ( ord_le824540014_ereal @ A @ B )
% 0.19/0.46 | ( C = extend1289208545_ereal )
% 0.19/0.46 | ( ( C
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 & ( A != extend1289208545_ereal )
% 0.19/0.46 & ( B != extend1289208545_ereal ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_add_le_add_iff2
% 0.19/0.46 thf(fact_326_id__nat__ereal__tendsto__PInf,axiom,
% 0.19/0.46 ( filter1531173832_ereal
% 0.19/0.46 @ ^ [X: nat] : ( extended_ereal2 @ ( semiri2110766477t_real @ X ) )
% 0.19/0.46 @ ( topolo2140997059_ereal @ extend1289208545_ereal )
% 0.19/0.46 @ at_top_nat ) ).
% 0.19/0.46
% 0.19/0.46 % id_nat_ereal_tendsto_PInf
% 0.19/0.46 thf(fact_327_ereal__liminf__add__mono,axiom,
% 0.19/0.46 ! [U: nat > extended_ereal,V: nat > extended_ereal] :
% 0.19/0.46 ( ~ ( ( ( ( liminf1045857232_ereal @ at_top_nat @ U )
% 0.19/0.46 = extend1289208545_ereal )
% 0.19/0.46 & ( ( liminf1045857232_ereal @ at_top_nat @ V )
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.19/0.46 | ( ( ( liminf1045857232_ereal @ at_top_nat @ U )
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 & ( ( liminf1045857232_ereal @ at_top_nat @ V )
% 0.19/0.46 = extend1289208545_ereal ) ) )
% 0.19/0.46 => ( ord_le824540014_ereal @ ( plus_p2118002693_ereal @ ( liminf1045857232_ereal @ at_top_nat @ U ) @ ( liminf1045857232_ereal @ at_top_nat @ V ) )
% 0.19/0.46 @ ( liminf1045857232_ereal @ at_top_nat
% 0.19/0.46 @ ^ [N: nat] : ( plus_p2118002693_ereal @ ( U @ N ) @ ( V @ N ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ereal_liminf_add_mono
% 0.19/0.46 thf(fact_328_nat__add__left__cancel__le,axiom,
% 0.19/0.46 ! [K: nat,M5: nat,N7: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M5 ) @ ( plus_plus_nat @ K @ N7 ) )
% 0.19/0.46 = ( ord_less_eq_nat @ M5 @ N7 ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_add_left_cancel_le
% 0.19/0.46 thf(fact_329_nat__int__comparison_I3_J,axiom,
% 0.19/0.46 ( ord_less_eq_nat
% 0.19/0.46 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_int_comparison(3)
% 0.19/0.46 thf(fact_330_add__leE,axiom,
% 0.19/0.46 ! [M5: nat,K: nat,N7: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N7 )
% 0.19/0.46 => ~ ( ( ord_less_eq_nat @ M5 @ N7 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ K @ N7 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_leE
% 0.19/0.46 thf(fact_331_le__add1,axiom,
% 0.19/0.46 ! [N7: nat,M5: nat] : ( ord_less_eq_nat @ N7 @ ( plus_plus_nat @ N7 @ M5 ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add1
% 0.19/0.46 thf(fact_332_le__add2,axiom,
% 0.19/0.46 ! [N7: nat,M5: nat] : ( ord_less_eq_nat @ N7 @ ( plus_plus_nat @ M5 @ N7 ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add2
% 0.19/0.46 thf(fact_333_add__leD1,axiom,
% 0.19/0.46 ! [M5: nat,K: nat,N7: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N7 )
% 0.19/0.46 => ( ord_less_eq_nat @ M5 @ N7 ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_leD1
% 0.19/0.46 thf(fact_334_add__leD2,axiom,
% 0.19/0.46 ! [M5: nat,K: nat,N7: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N7 )
% 0.19/0.46 => ( ord_less_eq_nat @ K @ N7 ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_leD2
% 0.19/0.46 thf(fact_335_le__Suc__ex,axiom,
% 0.19/0.46 ! [K: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ L )
% 0.19/0.46 => ? [N2: nat] :
% 0.19/0.46 ( L
% 0.19/0.46 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_Suc_ex
% 0.19/0.46 thf(fact_336_add__le__mono,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ( ord_less_eq_nat @ K @ L )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_mono
% 0.19/0.46 thf(fact_337_add__le__mono1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_mono1
% 0.19/0.46 thf(fact_338_trans__le__add1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M5 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_le_add1
% 0.19/0.46 thf(fact_339_trans__le__add2,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M5 @ J ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_le_add2
% 0.19/0.46 thf(fact_340_nat__le__iff__add,axiom,
% 0.19/0.46 ( ord_less_eq_nat
% 0.19/0.46 = ( ^ [M4: nat,N: nat] :
% 0.19/0.46 ? [K2: nat] :
% 0.19/0.46 ( N
% 0.19/0.46 = ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_le_iff_add
% 0.19/0.46 thf(fact_341_nat__leq__as__int,axiom,
% 0.19/0.46 ( ord_less_eq_nat
% 0.19/0.46 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_leq_as_int
% 0.19/0.46 thf(fact_342_plus__ereal_Osimps_I1_J,axiom,
% 0.19/0.46 ! [R: real,P2: real] :
% 0.19/0.46 ( ( plus_p2118002693_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
% 0.19/0.46 = ( extended_ereal2 @ ( plus_plus_real @ R @ P2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % plus_ereal.simps(1)
% 0.19/0.46 thf(fact_343_filterlim__uminus__at__bot__at__top,axiom,
% 0.19/0.46 filterlim_real_real @ uminus_uminus_real @ at_bot_real @ at_top_real ).
% 0.19/0.46
% 0.19/0.46 % filterlim_uminus_at_bot_at_top
% 0.19/0.46 thf(fact_344_filterlim__uminus__at__top__at__bot,axiom,
% 0.19/0.46 filterlim_real_real @ uminus_uminus_real @ at_top_real @ at_bot_real ).
% 0.19/0.46
% 0.19/0.46 % filterlim_uminus_at_top_at_bot
% 0.19/0.46 thf(fact_345_Nat_Oex__has__greatest__nat,axiom,
% 0.19/0.46 ! [P: nat > $o,K: nat,B: nat] :
% 0.19/0.46 ( ( P @ K )
% 0.19/0.46 => ( ! [Y3: nat] :
% 0.19/0.46 ( ( P @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.19/0.46 => ? [X5: nat] :
% 0.19/0.46 ( ( P @ X5 )
% 0.19/0.46 & ! [Y6: nat] :
% 0.19/0.46 ( ( P @ Y6 )
% 0.19/0.46 => ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.ex_has_greatest_nat
% 0.19/0.46 thf(fact_346_nat__le__linear,axiom,
% 0.19/0.46 ! [M5: nat,N7: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M5 @ N7 )
% 0.19/0.46 | ( ord_less_eq_nat @ N7 @ M5 ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_le_linear
% 0.19/0.46 thf(fact_347_le__antisym,axiom,
% 0.19/0.46 ! [M5: nat,N7: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M5 @ N7 )
% 0.19/0.46 => ( ( ord_less_eq_nat @ N7 @ M5 )
% 0.19/0.46 => ( M5 = N7 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_antisym
% 0.19/0.46 thf(fact_348_eq__imp__le,axiom,
% 0.19/0.46 ! [M5: nat,N7: nat] :
% 0.19/0.46 ( ( M5 = N7 )
% 0.19/0.46 => ( ord_less_eq_nat @ M5 @ N7 ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_imp_le
% 0.19/0.46 thf(fact_349_le__trans,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ( ord_less_eq_nat @ J @ K )
% 0.19/0.46 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_trans
% 0.19/0.46 thf(fact_350_le__refl,axiom,
% 0.19/0.46 ! [N7: nat] : ( ord_less_eq_nat @ N7 @ N7 ) ).
% 0.19/0.46
% 0.19/0.46 % le_refl
% 0.19/0.46 thf(fact_351_plus__ereal_Oelims,axiom,
% 0.19/0.46 ! [X2: extended_ereal,Xa3: extended_ereal,Y: extended_ereal] :
% 0.19/0.46 ( ( ( plus_p2118002693_ereal @ X2 @ Xa3 )
% 0.19/0.46 = Y )
% 0.19/0.46 => ( ! [R2: real] :
% 0.19/0.46 ( ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ! [P3: real] :
% 0.19/0.46 ( ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ P3 ) )
% 0.19/0.46 => ( Y
% 0.19/0.46 != ( extended_ereal2 @ ( plus_plus_real @ R2 @ P3 ) ) ) ) )
% 0.19/0.46 => ( ( ( X2 = extend1289208545_ereal )
% 0.19/0.46 => ( Y != extend1289208545_ereal ) )
% 0.19/0.46 => ( ( ( Xa3 = extend1289208545_ereal )
% 0.19/0.46 => ( Y != extend1289208545_ereal ) )
% 0.19/0.46 => ( ( ? [R2: real] :
% 0.19/0.46 ( X2
% 0.19/0.46 = ( extended_ereal2 @ R2 ) )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Y
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ( ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ? [P3: real] :
% 0.19/0.46 ( Xa3
% 0.19/0.46 = ( extended_ereal2 @ P3 ) )
% 0.19/0.46 => ( Y
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.19/0.46 => ~ ( ( X2
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( ( Xa3
% 0.19/0.46 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.19/0.46 => ( Y
% 0.19/0.46 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % plus_ereal.elims
% 0.19/0.46 thf(fact_352_filterlim__real__sequentially,axiom,
% 0.19/0.46 filterlim_nat_real @ semiri2110766477t_real @ at_top_real @ at_top_nat ).
% 0.19/0.46
% 0.19/0.46 % filterlim_real_sequentially
% 0.19/0.46 thf(fact_353_int__ops_I5_J,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( semiri2019852685at_int @ ( plus_plus_nat @ A @ B ) )
% 0.19/0.46 = ( plus_plus_int @ ( semiri2019852685at_int @ A ) @ ( semiri2019852685at_int @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % int_ops(5)
% 0.19/0.47 thf(fact_354_nat__int__comparison_I1_J,axiom,
% 0.19/0.47 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 0.19/0.47 = ( ^ [A3: nat,B2: nat] :
% 0.19/0.47 ( ( semiri2019852685at_int @ A3 )
% 0.19/0.47 = ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % nat_int_comparison(1)
% 0.19/0.47
% 0.19/0.47 % Conjectures (1)
% 0.19/0.47 thf(conj_0,conjecture,
% 0.19/0.47 ( filter1531173832_ereal
% 0.19/0.47 @ ^ [I2: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ I2 ) ) )
% 0.19/0.47 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
% 0.19/0.47 @ at_top_nat ) ).
% 0.19/0.47
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 ------------------------
% 2.72/1.59 ---- Embedded in THF ---
% 2.72/1.60 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.72/1.60 %%% Generated on Tue Feb 24 08:26:30 EST 2026
% 2.72/1.60 %%% using '$$dhol' embedding, version 1.3.0.
% 2.72/1.60 %%% Logic specification used:
% 2.72/1.60 %%% thf(spec, logic, $$dhol).
% 2.72/1.60
% 2.72/1.60 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.z9pBe0D4OI/DTF2DTF9897.p
% See solution above
% 3.00/1.60 ------------------------
% 3.00/1.63 ---- Cleaned THF ---
% 3.00/1.63 thf(ty_n_t__Filter__Ofilter_It__Extended____Real__Oereal_J,type,
% 3.00/1.63 filter2049122004_ereal: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 3.00/1.63 set_Extended_ereal: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 3.00/1.63 filter_real: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 3.00/1.63 filter_nat: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
% 3.00/1.63 filter_int: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 3.00/1.63 set_real: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Filter__Ofilter_Itf__a_J,type,
% 3.00/1.63 filter_a: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Extended____Real__Oereal,type,
% 3.00/1.63 extended_ereal: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Real__Oreal,type,
% 3.00/1.63 real: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Nat__Onat,type,
% 3.00/1.63 nat: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_t__Int__Oint,type,
% 3.00/1.63 int: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(ty_n_tf__a,type,
% 3.00/1.63 a: $tType ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
% 3.00/1.63 extend1289208545_ereal: extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 3.00/1.63 extended_ereal2: real > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 3.00/1.63 at_bot_real: filter_real ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 3.00/1.63 at_top_nat: filter_nat ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 3.00/1.63 at_top_real: filter_real ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
% 3.00/1.63 filter1531173832_ereal: ( nat > extended_ereal ) > filter2049122004_ereal > filter_nat > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
% 3.00/1.63 filterlim_nat_int: ( nat > int ) > filter_int > filter_nat > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 3.00/1.63 filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 3.00/1.63 filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001tf__a,type,
% 3.00/1.63 filterlim_nat_a: ( nat > a ) > filter_a > filter_nat > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 3.00/1.63 filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 3.00/1.63 comp_E1308517939al_nat: ( extended_ereal > extended_ereal ) > ( nat > extended_ereal ) > nat > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal_001tf__a,type,
% 3.00/1.63 comp_E489644891real_a: ( extended_ereal > extended_ereal ) > ( a > extended_ereal ) > a > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Int__Oint_001t__Nat__Onat,type,
% 3.00/1.63 comp_E1436437929nt_nat: ( extended_ereal > int ) > ( nat > extended_ereal ) > nat > int ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Nat__Onat_001t__Nat__Onat,type,
% 3.00/1.63 comp_E1523169101at_nat: ( extended_ereal > nat ) > ( nat > extended_ereal ) > nat > nat ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Extended____Real__Oereal_001t__Real__Oreal_001t__Nat__Onat,type,
% 3.00/1.63 comp_E1477338153al_nat: ( extended_ereal > real ) > ( nat > extended_ereal ) > nat > real ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 3.00/1.63 comp_n1096781355al_nat: ( nat > extended_ereal ) > ( nat > nat ) > nat > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001tf__a_001t__Nat__Onat,type,
% 3.00/1.63 comp_nat_a_nat: ( nat > a ) > ( nat > nat ) > nat > a ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 3.00/1.63 comp_r1410008527al_nat: ( real > extended_ereal ) > ( nat > real ) > nat > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Int__Oint_001t__Nat__Onat,type,
% 3.00/1.63 comp_real_int_nat: ( real > int ) > ( nat > real ) > nat > int ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
% 3.00/1.63 comp_real_nat_nat: ( real > nat ) > ( nat > real ) > nat > nat ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 3.00/1.63 comp_a1112243075al_nat: ( a > extended_ereal ) > ( nat > a ) > nat > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001tf__a_001t__Extended____Real__Oereal_001tf__a,type,
% 3.00/1.63 comp_a780206603real_a: ( a > extended_ereal ) > ( a > a ) > a > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001tf__a_001t__Int__Oint_001t__Nat__Onat,type,
% 3.00/1.63 comp_a_int_nat: ( a > int ) > ( nat > a ) > nat > int ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
% 3.00/1.63 comp_a_nat_nat: ( a > nat ) > ( nat > a ) > nat > nat ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001t__Nat__Onat,type,
% 3.00/1.63 comp_a_real_nat: ( a > real ) > ( nat > a ) > nat > real ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Nat__Onat,type,
% 3.00/1.63 comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
% 3.00/1.63 plus_p2118002693_ereal: extended_ereal > extended_ereal > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 3.00/1.63 plus_plus_int: int > int > int ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 3.00/1.63 plus_plus_nat: nat > nat > nat ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 3.00/1.63 plus_plus_real: real > real > real ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
% 3.00/1.63 uminus1208298309_ereal: extended_ereal > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 3.00/1.63 uminus_uminus_int: int > int ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 3.00/1.63 uminus_uminus_real: real > real ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_HOL_OUniq_001t__Extended____Real__Oereal,type,
% 3.00/1.63 uniq_Extended_ereal: ( extended_ereal > $o ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_HOL_OUniq_001t__Real__Oreal,type,
% 3.00/1.63 uniq_real: ( real > $o ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_HOL_OUniq_001tf__a,type,
% 3.00/1.63 uniq_a: ( a > $o ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Liminf__Limsup_OLiminf_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
% 3.00/1.63 liminf1045857232_ereal: filter_nat > ( nat > extended_ereal ) > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
% 3.00/1.63 lower_1087098792_ereal: extended_ereal > ( extended_ereal > extended_ereal ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Int__Oint,type,
% 3.00/1.63 lower_48196818al_int: extended_ereal > ( extended_ereal > int ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 3.00/1.63 lower_1558406774al_nat: extended_ereal > ( extended_ereal > nat ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
% 3.00/1.63 lower_1165973074l_real: extended_ereal > ( extended_ereal > real ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Real__Oreal_001t__Extended____Real__Oereal,type,
% 3.00/1.63 lower_551915512_ereal: real > ( real > extended_ereal ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Real__Oreal_001t__Int__Oint,type,
% 3.00/1.63 lower_153911426al_int: real > ( real > int ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001t__Real__Oreal_001t__Nat__Onat,type,
% 3.00/1.63 lower_1664121382al_nat: real > ( real > nat ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Extended____Real__Oereal,type,
% 3.00/1.63 lower_191460856_ereal: a > ( a > extended_ereal ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Int__Oint,type,
% 3.00/1.63 lower_956963458_a_int: a > ( a > int ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Nat__Onat,type,
% 3.00/1.63 lower_319689766_a_nat: a > ( a > nat ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__at_001tf__a_001t__Real__Oreal,type,
% 3.00/1.63 lower_231615490a_real: a > ( a > real ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
% 3.00/1.63 lower_1071158961_ereal: extended_ereal > ( extended_ereal > extended_ereal ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Int__Oint,type,
% 3.00/1.63 lower_637387785al_int: extended_ereal > ( extended_ereal > int ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 3.00/1.63 lower_114093al_nat: extended_ereal > ( extended_ereal > nat ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
% 3.00/1.63 lower_737640969l_real: extended_ereal > ( extended_ereal > real ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Real__Oreal_001t__Int__Oint,type,
% 3.00/1.63 lower_1075504779al_int: real > ( real > int ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001t__Real__Oreal_001t__Nat__Onat,type,
% 3.00/1.63 lower_438231087al_nat: real > ( real > nat ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Extended____Real__Oereal,type,
% 3.00/1.63 lower_534855297_ereal: a > ( a > extended_ereal ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Int__Oint,type,
% 3.00/1.63 lower_1672990777_a_int: a > ( a > int ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Nat__Onat,type,
% 3.00/1.63 lower_1035717085_a_nat: a > ( a > nat ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Ousc__at_001tf__a_001t__Real__Oreal,type,
% 3.00/1.63 lower_755922489a_real: a > ( a > real ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 3.00/1.63 semiri2019852685at_int: nat > int ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 3.00/1.63 semiri2110766477t_real: nat > real ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
% 3.00/1.63 ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 3.00/1.63 ord_le1745708096er_nat: filter_nat > filter_nat > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
% 3.00/1.63 ord_le132810396r_real: filter_real > filter_real > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 3.00/1.63 ord_less_eq_int: int > int > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 3.00/1.63 ord_less_eq_nat: nat > nat > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 3.00/1.63 ord_less_eq_real: real > real > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 3.00/1.63 collect_real: ( real > $o ) > set_real ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Topological__Spaces_Omonoseq_001t__Extended____Real__Oereal,type,
% 3.00/1.63 topolo1069469409_ereal: ( nat > extended_ereal ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 3.00/1.63 topolo411883481eq_int: ( nat > int ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 3.00/1.63 topolo1922093437eq_nat: ( nat > nat ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 3.00/1.63 topolo144289241q_real: ( nat > real ) > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Extended____Real__Oereal,type,
% 3.00/1.63 topolo2140997059_ereal: extended_ereal > filter2049122004_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Int__Oint,type,
% 3.00/1.63 topolo54776183ds_int: int > filter_int ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Nat__Onat,type,
% 3.00/1.63 topolo1564986139ds_nat: nat > filter_nat ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 3.00/1.63 topolo1664202871s_real: real > filter_real ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001tf__a,type,
% 3.00/1.63 topolo705128563nhds_a: a > filter_a ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_member_001t__Extended____Real__Oereal,type,
% 3.00/1.63 member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_c_member_001t__Real__Oreal,type,
% 3.00/1.63 member_real: real > set_real > $o ).
% 3.00/1.63
% 3.00/1.63 thf(sy_v_A____,type,
% 3.00/1.63 a2: extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_v_f,type,
% 3.00/1.63 f: a > extended_ereal ).
% 3.00/1.63
% 3.00/1.63 thf(sy_v_x0,type,
% 3.00/1.63 x0: a ).
% 3.00/1.63
% 3.00/1.63 thf(sy_v_x____,type,
% 3.00/1.63 x: nat > a ).
% 3.00/1.63
% 3.00/1.63 thf(fact_0_x__def_I1_J,axiom,
% 3.00/1.63 filterlim_nat_a @ x @ ( topolo705128563nhds_a @ x0 ) @ at_top_nat ).
% 3.00/1.63
% 3.00/1.63 thf(fact_1_x__def_I2_J,axiom,
% 3.00/1.63 filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ at_top_nat ).
% 3.00/1.63
% 3.00/1.63 thf(fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062,axiom,
% 3.00/1.63 ! [F: filter_nat] :
% 3.00/1.63 ( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ f @ x ) @ ( topolo2140997059_ereal @ a2 ) @ F )
% 3.00/1.63 => ( filter1531173832_ereal
% 3.00/1.63 @ ^ [X: nat] : ( uminus1208298309_ereal @ ( comp_a1112243075al_nat @ f @ x @ X ) )
% 3.00/1.63 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
% 3.00/1.63 @ F ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_3_lsc,axiom,
% 3.00/1.63 ( lower_191460856_ereal @ x0
% 3.00/1.63 @ ^ [X: a] : ( uminus1208298309_ereal @ ( f @ X ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_4_tendsto__uminus__ereal,axiom,
% 3.00/1.63 ! [F2: nat > extended_ereal,X2: extended_ereal,F: filter_nat] :
% 3.00/1.63 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ X2 ) @ F )
% 3.00/1.63 => ( filter1531173832_ereal
% 3.00/1.63 @ ^ [X: nat] : ( uminus1208298309_ereal @ ( F2 @ X ) )
% 3.00/1.63 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ X2 ) )
% 3.00/1.63 @ F ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_5_tendsto__const,axiom,
% 3.00/1.63 ! [K: real,F: filter_real] :
% 3.00/1.63 ( filterlim_real_real
% 3.00/1.63 @ ^ [X: real] : K
% 3.00/1.63 @ ( topolo1664202871s_real @ K )
% 3.00/1.63 @ F ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_6_tendsto__const,axiom,
% 3.00/1.63 ! [K: real,F: filter_nat] :
% 3.00/1.63 ( filterlim_nat_real
% 3.00/1.63 @ ^ [X: nat] : K
% 3.00/1.63 @ ( topolo1664202871s_real @ K )
% 3.00/1.63 @ F ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_7_tendsto__const,axiom,
% 3.00/1.63 ! [K: extended_ereal,F: filter_nat] :
% 3.00/1.63 ( filter1531173832_ereal
% 3.00/1.63 @ ^ [X: nat] : K
% 3.00/1.63 @ ( topolo2140997059_ereal @ K )
% 3.00/1.63 @ F ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_8_tendsto__const,axiom,
% 3.00/1.63 ! [K: a,F: filter_nat] :
% 3.00/1.63 ( filterlim_nat_a
% 3.00/1.63 @ ^ [X: nat] : K
% 3.00/1.63 @ ( topolo705128563nhds_a @ K )
% 3.00/1.63 @ F ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_9_ereal__Lim__uminus,axiom,
% 3.00/1.63 ! [F2: nat > extended_ereal,F0: extended_ereal,Net: filter_nat] :
% 3.00/1.63 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ F0 ) @ Net )
% 3.00/1.63 = ( filter1531173832_ereal
% 3.00/1.63 @ ^ [X: nat] : ( uminus1208298309_ereal @ ( F2 @ X ) )
% 3.00/1.63 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ F0 ) )
% 3.00/1.63 @ Net ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_10_LIMSEQ__const__iff,axiom,
% 3.00/1.63 ! [K: real,L: real] :
% 3.00/1.63 ( ( filterlim_nat_real
% 3.00/1.63 @ ^ [N: nat] : K
% 3.00/1.63 @ ( topolo1664202871s_real @ L )
% 3.00/1.63 @ at_top_nat )
% 3.00/1.63 = ( K = L ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_11_LIMSEQ__const__iff,axiom,
% 3.00/1.63 ! [K: extended_ereal,L: extended_ereal] :
% 3.00/1.63 ( ( filter1531173832_ereal
% 3.00/1.63 @ ^ [N: nat] : K
% 3.00/1.63 @ ( topolo2140997059_ereal @ L )
% 3.00/1.63 @ at_top_nat )
% 3.00/1.63 = ( K = L ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_12_LIMSEQ__const__iff,axiom,
% 3.00/1.63 ! [K: a,L: a] :
% 3.00/1.63 ( ( filterlim_nat_a
% 3.00/1.63 @ ^ [N: nat] : K
% 3.00/1.63 @ ( topolo705128563nhds_a @ L )
% 3.00/1.63 @ at_top_nat )
% 3.00/1.63 = ( K = L ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_13_tendsto__minus__cancel,axiom,
% 3.00/1.63 ! [F2: real > real,A: real,F: filter_real] :
% 3.00/1.63 ( ( filterlim_real_real
% 3.00/1.63 @ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
% 3.00/1.63 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
% 3.00/1.63 @ F )
% 3.00/1.63 => ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ A ) @ F ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_14_tendsto__minus__cancel,axiom,
% 3.00/1.63 ! [F2: nat > real,A: real,F: filter_nat] :
% 3.00/1.63 ( ( filterlim_nat_real
% 3.00/1.63 @ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) )
% 3.00/1.63 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
% 3.00/1.63 @ F )
% 3.00/1.63 => ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ A ) @ F ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_15_tendsto__minus__cancel__left,axiom,
% 3.00/1.63 ! [F2: real > real,Y: real,F: filter_real] :
% 3.00/1.63 ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ Y ) ) @ F )
% 3.00/1.63 = ( filterlim_real_real
% 3.00/1.63 @ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
% 3.00/1.63 @ ( topolo1664202871s_real @ Y )
% 3.00/1.63 @ F ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_16_tendsto__minus__cancel__left,axiom,
% 3.00/1.63 ! [F2: nat > real,Y: real,F: filter_nat] :
% 3.00/1.63 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ Y ) ) @ F )
% 3.00/1.63 = ( filterlim_nat_real
% 3.00/1.63 @ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) )
% 3.00/1.63 @ ( topolo1664202871s_real @ Y )
% 3.00/1.63 @ F ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_17_tendsto__minus,axiom,
% 3.00/1.63 ! [F2: real > real,A: real,F: filter_real] :
% 3.00/1.63 ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ A ) @ F )
% 3.00/1.63 => ( filterlim_real_real
% 3.00/1.63 @ ^ [X: real] : ( uminus_uminus_real @ ( F2 @ X ) )
% 3.00/1.63 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
% 3.00/1.63 @ F ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_18_tendsto__minus,axiom,
% 3.00/1.63 ! [F2: nat > real,A: real,F: filter_nat] :
% 3.00/1.63 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ A ) @ F )
% 3.00/1.63 => ( filterlim_nat_real
% 3.00/1.63 @ ^ [X: nat] : ( uminus_uminus_real @ ( F2 @ X ) )
% 3.00/1.63 @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) )
% 3.00/1.63 @ F ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_19_LIMSEQ__unique,axiom,
% 3.00/1.63 ! [X3: nat > extended_ereal,A: extended_ereal,B: extended_ereal] :
% 3.00/1.63 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ A ) @ at_top_nat )
% 3.00/1.63 => ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ B ) @ at_top_nat )
% 3.00/1.63 => ( A = B ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_20_LIMSEQ__unique,axiom,
% 3.00/1.63 ! [X3: nat > a,A: a,B: a] :
% 3.00/1.63 ( ( filterlim_nat_a @ X3 @ ( topolo705128563nhds_a @ A ) @ at_top_nat )
% 3.00/1.63 => ( ( filterlim_nat_a @ X3 @ ( topolo705128563nhds_a @ B ) @ at_top_nat )
% 3.00/1.63 => ( A = B ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_21_LIMSEQ__unique,axiom,
% 3.00/1.63 ! [X3: nat > real,A: real,B: real] :
% 3.00/1.63 ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ A ) @ at_top_nat )
% 3.00/1.63 => ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ B ) @ at_top_nat )
% 3.00/1.63 => ( A = B ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_22_tendsto__uminus__nhds,axiom,
% 3.00/1.63 ! [A: real] : ( filterlim_real_real @ uminus_uminus_real @ ( topolo1664202871s_real @ ( uminus_uminus_real @ A ) ) @ ( topolo1664202871s_real @ A ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_23_ereal__uminus__eq__iff,axiom,
% 3.00/1.63 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.63 ( ( ( uminus1208298309_ereal @ A )
% 3.00/1.63 = ( uminus1208298309_ereal @ B ) )
% 3.00/1.63 = ( A = B ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_24_ereal__uminus__uminus,axiom,
% 3.00/1.63 ! [A: extended_ereal] :
% 3.00/1.63 ( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A ) )
% 3.00/1.63 = A ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_25_verit__minus__simplify_I4_J,axiom,
% 3.00/1.63 ! [B: real] :
% 3.00/1.63 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 3.00/1.63 = B ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_26_verit__negate__coefficient_I3_J,axiom,
% 3.00/1.63 ! [A: real,B: real] :
% 3.00/1.63 ( ( A = B )
% 3.00/1.63 => ( ( uminus_uminus_real @ A )
% 3.00/1.63 = ( uminus_uminus_real @ B ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_27_ereal__uminus__eq__reorder,axiom,
% 3.00/1.63 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.63 ( ( ( uminus1208298309_ereal @ A )
% 3.00/1.63 = B )
% 3.00/1.63 = ( A
% 3.00/1.63 = ( uminus1208298309_ereal @ B ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_28_tendsto__eq__rhs,axiom,
% 3.00/1.63 ! [F2: nat > extended_ereal,X2: extended_ereal,F: filter_nat,Y: extended_ereal] :
% 3.00/1.63 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ X2 ) @ F )
% 3.00/1.63 => ( ( X2 = Y )
% 3.00/1.63 => ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ Y ) @ F ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_29_tendsto__eq__rhs,axiom,
% 3.00/1.63 ! [F2: nat > a,X2: a,F: filter_nat,Y: a] :
% 3.00/1.63 ( ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ X2 ) @ F )
% 3.00/1.63 => ( ( X2 = Y )
% 3.00/1.63 => ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ Y ) @ F ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_30_tendsto__eq__rhs,axiom,
% 3.00/1.63 ! [F2: real > real,X2: real,F: filter_real,Y: real] :
% 3.00/1.63 ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ X2 ) @ F )
% 3.00/1.63 => ( ( X2 = Y )
% 3.00/1.63 => ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ Y ) @ F ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_31_tendsto__eq__rhs,axiom,
% 3.00/1.63 ! [F2: nat > real,X2: real,F: filter_nat,Y: real] :
% 3.00/1.63 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ X2 ) @ F )
% 3.00/1.63 => ( ( X2 = Y )
% 3.00/1.63 => ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ Y ) @ F ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_32_tendsto__cong__limit,axiom,
% 3.00/1.63 ! [F2: nat > extended_ereal,L: extended_ereal,F: filter_nat,K: extended_ereal] :
% 3.00/1.63 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ F )
% 3.00/1.63 => ( ( K = L )
% 3.00/1.63 => ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ K ) @ F ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_33_tendsto__cong__limit,axiom,
% 3.00/1.63 ! [F2: nat > a,L: a,F: filter_nat,K: a] :
% 3.00/1.63 ( ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ L ) @ F )
% 3.00/1.63 => ( ( K = L )
% 3.00/1.63 => ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ K ) @ F ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_34_tendsto__cong__limit,axiom,
% 3.00/1.63 ! [F2: real > real,L: real,F: filter_real,K: real] :
% 3.00/1.63 ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ L ) @ F )
% 3.00/1.63 => ( ( K = L )
% 3.00/1.63 => ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ K ) @ F ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_35_tendsto__cong__limit,axiom,
% 3.00/1.63 ! [F2: nat > real,L: real,F: filter_nat,K: real] :
% 3.00/1.63 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ F )
% 3.00/1.63 => ( ( K = L )
% 3.00/1.63 => ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ K ) @ F ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_36_comp__apply,axiom,
% 3.00/1.63 ( comp_a1112243075al_nat
% 3.00/1.63 = ( ^ [F3: a > extended_ereal,G: nat > a,X: nat] : ( F3 @ ( G @ X ) ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_37_neg__equal__iff__equal,axiom,
% 3.00/1.63 ! [A: real,B: real] :
% 3.00/1.63 ( ( ( uminus_uminus_real @ A )
% 3.00/1.63 = ( uminus_uminus_real @ B ) )
% 3.00/1.63 = ( A = B ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_38_add_Oinverse__inverse,axiom,
% 3.00/1.63 ! [A: real] :
% 3.00/1.63 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 3.00/1.63 = A ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_39_lsc__sequentially__mem,axiom,
% 3.00/1.63 ! [X0: real,F2: real > extended_ereal,X2: nat > real,C: nat > extended_ereal,C0: extended_ereal] :
% 3.00/1.63 ( ( lower_551915512_ereal @ X0 @ F2 )
% 3.00/1.63 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 3.00/1.63 => ( ( filter1531173832_ereal @ C @ ( topolo2140997059_ereal @ C0 ) @ at_top_nat )
% 3.00/1.63 => ( ! [N2: nat] : ( ord_le824540014_ereal @ ( F2 @ ( X2 @ N2 ) ) @ ( C @ N2 ) )
% 3.00/1.63 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ C0 ) ) ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_40_lsc__sequentially__mem,axiom,
% 3.00/1.63 ! [X0: a,F2: a > extended_ereal,X2: nat > a,C: nat > extended_ereal,C0: extended_ereal] :
% 3.00/1.63 ( ( lower_191460856_ereal @ X0 @ F2 )
% 3.00/1.63 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 3.00/1.63 => ( ( filter1531173832_ereal @ C @ ( topolo2140997059_ereal @ C0 ) @ at_top_nat )
% 3.00/1.63 => ( ! [N2: nat] : ( ord_le824540014_ereal @ ( F2 @ ( X2 @ N2 ) ) @ ( C @ N2 ) )
% 3.00/1.63 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ C0 ) ) ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_41_lsc__sequentially__gen,axiom,
% 3.00/1.63 ( lower_551915512_ereal
% 3.00/1.63 = ( ^ [X02: real,F3: real > extended_ereal] :
% 3.00/1.63 ! [X: nat > real,C2: nat > extended_ereal,C02: extended_ereal] :
% 3.00/1.63 ( ( ( filterlim_nat_real @ X @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 3.00/1.63 & ( filter1531173832_ereal @ C2 @ ( topolo2140997059_ereal @ C02 ) @ at_top_nat )
% 3.00/1.63 & ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ ( C2 @ N ) ) )
% 3.00/1.63 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C02 ) ) ) ) ).
% 3.00/1.63
% 3.00/1.63 thf(fact_42_lsc__sequentially__gen,axiom,
% 3.00/1.63 ( lower_191460856_ereal
% 3.00/1.64 = ( ^ [X02: a,F3: a > extended_ereal] :
% 3.00/1.64 ! [X: nat > a,C2: nat > extended_ereal,C02: extended_ereal] :
% 3.00/1.64 ( ( ( filterlim_nat_a @ X @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filter1531173832_ereal @ C2 @ ( topolo2140997059_ereal @ C02 ) @ at_top_nat )
% 3.00/1.64 & ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ ( C2 @ N ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C02 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_43_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: extended_ereal,F2: extended_ereal > nat,X2: nat > extended_ereal,A2: nat] :
% 3.00/1.64 ( ( lower_1558406774al_nat @ X0 @ F2 )
% 3.00/1.64 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_44_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: extended_ereal,F2: extended_ereal > int,X2: nat > extended_ereal,A2: int] :
% 3.00/1.64 ( ( lower_48196818al_int @ X0 @ F2 )
% 3.00/1.64 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_45_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: extended_ereal,F2: extended_ereal > extended_ereal,X2: nat > extended_ereal,A2: extended_ereal] :
% 3.00/1.64 ( ( lower_1087098792_ereal @ X0 @ F2 )
% 3.00/1.64 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_46_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: extended_ereal,F2: extended_ereal > real,X2: nat > extended_ereal,A2: real] :
% 3.00/1.64 ( ( lower_1165973074l_real @ X0 @ F2 )
% 3.00/1.64 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_47_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: a,F2: a > nat,X2: nat > a,A2: nat] :
% 3.00/1.64 ( ( lower_319689766_a_nat @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_nat @ ( comp_a_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_48_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: a,F2: a > int,X2: nat > a,A2: int] :
% 3.00/1.64 ( ( lower_956963458_a_int @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_int @ ( comp_a_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_49_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: a,F2: a > real,X2: nat > a,A2: real] :
% 3.00/1.64 ( ( lower_231615490a_real @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_real @ ( comp_a_real_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_50_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: real,F2: real > nat,X2: nat > real,A2: nat] :
% 3.00/1.64 ( ( lower_1664121382al_nat @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_51_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: real,F2: real > int,X2: nat > real,A2: int] :
% 3.00/1.64 ( ( lower_153911426al_int @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_int @ ( comp_real_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_52_lsc__at__mem,axiom,
% 3.00/1.64 ! [X0: real,F2: real > extended_ereal,X2: nat > real,A2: extended_ereal] :
% 3.00/1.64 ( ( lower_551915512_ereal @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filter1531173832_ereal @ ( comp_r1410008527al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ A2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_53_lsc__at__def,axiom,
% 3.00/1.64 ( lower_1558406774al_nat
% 3.00/1.64 = ( ^ [X02: extended_ereal,F3: extended_ereal > nat] :
% 3.00/1.64 ! [X4: nat > extended_ereal,L2: nat] :
% 3.00/1.64 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_54_lsc__at__def,axiom,
% 3.00/1.64 ( lower_48196818al_int
% 3.00/1.64 = ( ^ [X02: extended_ereal,F3: extended_ereal > int] :
% 3.00/1.64 ! [X4: nat > extended_ereal,L2: int] :
% 3.00/1.64 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_int @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_55_lsc__at__def,axiom,
% 3.00/1.64 ( lower_1087098792_ereal
% 3.00/1.64 = ( ^ [X02: extended_ereal,F3: extended_ereal > extended_ereal] :
% 3.00/1.64 ! [X4: nat > extended_ereal,L2: extended_ereal] :
% 3.00/1.64 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_56_lsc__at__def,axiom,
% 3.00/1.64 ( lower_1165973074l_real
% 3.00/1.64 = ( ^ [X02: extended_ereal,F3: extended_ereal > real] :
% 3.00/1.64 ! [X4: nat > extended_ereal,L2: real] :
% 3.00/1.64 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_real @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_57_lsc__at__def,axiom,
% 3.00/1.64 ( lower_319689766_a_nat
% 3.00/1.64 = ( ^ [X02: a,F3: a > nat] :
% 3.00/1.64 ! [X4: nat > a,L2: nat] :
% 3.00/1.64 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_nat @ ( comp_a_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_58_lsc__at__def,axiom,
% 3.00/1.64 ( lower_956963458_a_int
% 3.00/1.64 = ( ^ [X02: a,F3: a > int] :
% 3.00/1.64 ! [X4: nat > a,L2: int] :
% 3.00/1.64 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_int @ ( comp_a_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_int @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_59_lsc__at__def,axiom,
% 3.00/1.64 ( lower_231615490a_real
% 3.00/1.64 = ( ^ [X02: a,F3: a > real] :
% 3.00/1.64 ! [X4: nat > a,L2: real] :
% 3.00/1.64 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_real @ ( comp_a_real_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_real @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_60_lsc__at__def,axiom,
% 3.00/1.64 ( lower_1664121382al_nat
% 3.00/1.64 = ( ^ [X02: real,F3: real > nat] :
% 3.00/1.64 ! [X4: nat > real,L2: nat] :
% 3.00/1.64 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_nat @ ( comp_real_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_61_lsc__at__def,axiom,
% 3.00/1.64 ( lower_153911426al_int
% 3.00/1.64 = ( ^ [X02: real,F3: real > int] :
% 3.00/1.64 ! [X4: nat > real,L2: int] :
% 3.00/1.64 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_int @ ( comp_real_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_int @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_62_lsc__at__def,axiom,
% 3.00/1.64 ( lower_551915512_ereal
% 3.00/1.64 = ( ^ [X02: real,F3: real > extended_ereal] :
% 3.00/1.64 ! [X4: nat > real,L2: extended_ereal] :
% 3.00/1.64 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filter1531173832_ereal @ ( comp_r1410008527al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ L2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_63_neg__le__iff__le,axiom,
% 3.00/1.64 ! [B: real,A: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 3.00/1.64 = ( ord_less_eq_real @ A @ B ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_64_neg__le__iff__le,axiom,
% 3.00/1.64 ! [B: int,A: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 3.00/1.64 = ( ord_less_eq_int @ A @ B ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_65_ereal__minus__le__minus,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ ( uminus1208298309_ereal @ B ) )
% 3.00/1.64 = ( ord_le824540014_ereal @ B @ A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_66_verit__la__disequality,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( A = B )
% 3.00/1.64 | ~ ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 | ~ ( ord_le824540014_ereal @ B @ A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_67_verit__la__disequality,axiom,
% 3.00/1.64 ! [A: nat,B: nat] :
% 3.00/1.64 ( ( A = B )
% 3.00/1.64 | ~ ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_68_verit__la__disequality,axiom,
% 3.00/1.64 ! [A: real,B: real] :
% 3.00/1.64 ( ( A = B )
% 3.00/1.64 | ~ ( ord_less_eq_real @ A @ B )
% 3.00/1.64 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_69_verit__la__disequality,axiom,
% 3.00/1.64 ! [A: int,B: int] :
% 3.00/1.64 ( ( A = B )
% 3.00/1.64 | ~ ( ord_less_eq_int @ A @ B )
% 3.00/1.64 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_70_ereal__complete__Inf,axiom,
% 3.00/1.64 ! [S: set_Extended_ereal] :
% 3.00/1.64 ? [X5: extended_ereal] :
% 3.00/1.64 ( ! [Xa: extended_ereal] :
% 3.00/1.64 ( ( member1900190071_ereal @ Xa @ S )
% 3.00/1.64 => ( ord_le824540014_ereal @ X5 @ Xa ) )
% 3.00/1.64 & ! [Z: extended_ereal] :
% 3.00/1.64 ( ! [Xa2: extended_ereal] :
% 3.00/1.64 ( ( member1900190071_ereal @ Xa2 @ S )
% 3.00/1.64 => ( ord_le824540014_ereal @ Z @ Xa2 ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ Z @ X5 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_71_ereal__complete__Sup,axiom,
% 3.00/1.64 ! [S: set_Extended_ereal] :
% 3.00/1.64 ? [X5: extended_ereal] :
% 3.00/1.64 ( ! [Xa: extended_ereal] :
% 3.00/1.64 ( ( member1900190071_ereal @ Xa @ S )
% 3.00/1.64 => ( ord_le824540014_ereal @ Xa @ X5 ) )
% 3.00/1.64 & ! [Z: extended_ereal] :
% 3.00/1.64 ( ! [Xa2: extended_ereal] :
% 3.00/1.64 ( ( member1900190071_ereal @ Xa2 @ S )
% 3.00/1.64 => ( ord_le824540014_ereal @ Xa2 @ Z ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ X5 @ Z ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_72_le__minus__iff,axiom,
% 3.00/1.64 ! [A: real,B: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 3.00/1.64 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_73_le__minus__iff,axiom,
% 3.00/1.64 ! [A: int,B: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 3.00/1.64 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_74_minus__le__iff,axiom,
% 3.00/1.64 ! [A: real,B: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 3.00/1.64 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_75_minus__le__iff,axiom,
% 3.00/1.64 ! [A: int,B: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 3.00/1.64 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_76_le__imp__neg__le,axiom,
% 3.00/1.64 ! [A: real,B: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ B )
% 3.00/1.64 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_77_le__imp__neg__le,axiom,
% 3.00/1.64 ! [A: int,B: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ A @ B )
% 3.00/1.64 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_78_ereal__uminus__le__reorder,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ B )
% 3.00/1.64 = ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ B ) @ A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_79_lim__mono,axiom,
% 3.00/1.64 ! [N3: nat,X3: nat > nat,Y2: nat > nat,X2: nat,Y: nat] :
% 3.00/1.64 ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N3 @ N2 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 3.00/1.64 => ( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_nat @ Y2 @ ( topolo1564986139ds_nat @ Y ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_80_lim__mono,axiom,
% 3.00/1.64 ! [N3: nat,X3: nat > int,Y2: nat > int,X2: int,Y: int] :
% 3.00/1.64 ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N3 @ N2 )
% 3.00/1.64 => ( ord_less_eq_int @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 3.00/1.64 => ( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_int @ Y2 @ ( topolo54776183ds_int @ Y ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_int @ X2 @ Y ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_81_lim__mono,axiom,
% 3.00/1.64 ! [N3: nat,X3: nat > extended_ereal,Y2: nat > extended_ereal,X2: extended_ereal,Y: extended_ereal] :
% 3.00/1.64 ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N3 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 3.00/1.64 => ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( filter1531173832_ereal @ Y2 @ ( topolo2140997059_ereal @ Y ) @ at_top_nat )
% 3.00/1.64 => ( ord_le824540014_ereal @ X2 @ Y ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_82_lim__mono,axiom,
% 3.00/1.64 ! [N3: nat,X3: nat > real,Y2: nat > real,X2: real,Y: real] :
% 3.00/1.64 ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N3 @ N2 )
% 3.00/1.64 => ( ord_less_eq_real @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 3.00/1.64 => ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_real @ Y2 @ ( topolo1664202871s_real @ Y ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_real @ X2 @ Y ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_83_LIMSEQ__le,axiom,
% 3.00/1.64 ! [X3: nat > nat,X2: nat,Y2: nat > nat,Y: nat] :
% 3.00/1.64 ( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_nat @ Y2 @ ( topolo1564986139ds_nat @ Y ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_84_LIMSEQ__le,axiom,
% 3.00/1.64 ! [X3: nat > int,X2: int,Y2: nat > int,Y: int] :
% 3.00/1.64 ( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_int @ Y2 @ ( topolo54776183ds_int @ Y ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_less_eq_int @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 3.00/1.64 => ( ord_less_eq_int @ X2 @ Y ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_85_LIMSEQ__le,axiom,
% 3.00/1.64 ! [X3: nat > extended_ereal,X2: extended_ereal,Y2: nat > extended_ereal,Y: extended_ereal] :
% 3.00/1.64 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( filter1531173832_ereal @ Y2 @ ( topolo2140997059_ereal @ Y ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ X2 @ Y ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_86_LIMSEQ__le,axiom,
% 3.00/1.64 ! [X3: nat > real,X2: real,Y2: nat > real,Y: real] :
% 3.00/1.64 ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_real @ Y2 @ ( topolo1664202871s_real @ Y ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_less_eq_real @ ( X3 @ N2 ) @ ( Y2 @ N2 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ X2 @ Y ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_87_mem__Collect__eq,axiom,
% 3.00/1.64 ! [A: real,P: real > $o] :
% 3.00/1.64 ( ( member_real @ A @ ( collect_real @ P ) )
% 3.00/1.64 = ( P @ A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_88_Collect__mem__eq,axiom,
% 3.00/1.64 ! [A2: set_real] :
% 3.00/1.64 ( ( collect_real
% 3.00/1.64 @ ^ [X: real] : ( member_real @ X @ A2 ) )
% 3.00/1.64 = A2 ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_89_Collect__cong,axiom,
% 3.00/1.64 ! [P: real > $o,Q: real > $o] :
% 3.00/1.64 ( ! [X5: real] :
% 3.00/1.64 ( ( P @ X5 )
% 3.00/1.64 = ( Q @ X5 ) )
% 3.00/1.64 => ( ( collect_real @ P )
% 3.00/1.64 = ( collect_real @ Q ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_90_Lim__bounded,axiom,
% 3.00/1.64 ! [F2: nat > nat,L: nat,M: nat,C3: nat] :
% 3.00/1.64 ( ( filterlim_nat_nat @ F2 @ ( topolo1564986139ds_nat @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M @ N2 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ N2 ) @ C3 ) )
% 3.00/1.64 => ( ord_less_eq_nat @ L @ C3 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_91_Lim__bounded,axiom,
% 3.00/1.64 ! [F2: nat > int,L: int,M: nat,C3: int] :
% 3.00/1.64 ( ( filterlim_nat_int @ F2 @ ( topolo54776183ds_int @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M @ N2 )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ N2 ) @ C3 ) )
% 3.00/1.64 => ( ord_less_eq_int @ L @ C3 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_92_Lim__bounded,axiom,
% 3.00/1.64 ! [F2: nat > extended_ereal,L: extended_ereal,M: nat,C3: extended_ereal] :
% 3.00/1.64 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ N2 ) @ C3 ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ L @ C3 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_93_Lim__bounded,axiom,
% 3.00/1.64 ! [F2: nat > real,L: real,M: nat,C3: real] :
% 3.00/1.64 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M @ N2 )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ N2 ) @ C3 ) )
% 3.00/1.64 => ( ord_less_eq_real @ L @ C3 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_94_Lim__bounded2,axiom,
% 3.00/1.64 ! [F2: nat > nat,L: nat,N3: nat,C3: nat] :
% 3.00/1.64 ( ( filterlim_nat_nat @ F2 @ ( topolo1564986139ds_nat @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N3 @ N2 )
% 3.00/1.64 => ( ord_less_eq_nat @ C3 @ ( F2 @ N2 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ C3 @ L ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_95_Lim__bounded2,axiom,
% 3.00/1.64 ! [F2: nat > int,L: int,N3: nat,C3: int] :
% 3.00/1.64 ( ( filterlim_nat_int @ F2 @ ( topolo54776183ds_int @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N3 @ N2 )
% 3.00/1.64 => ( ord_less_eq_int @ C3 @ ( F2 @ N2 ) ) )
% 3.00/1.64 => ( ord_less_eq_int @ C3 @ L ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_96_Lim__bounded2,axiom,
% 3.00/1.64 ! [F2: nat > extended_ereal,L: extended_ereal,N3: nat,C3: extended_ereal] :
% 3.00/1.64 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N3 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ C3 @ ( F2 @ N2 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ C3 @ L ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_97_Lim__bounded2,axiom,
% 3.00/1.64 ! [F2: nat > real,L: real,N3: nat,C3: real] :
% 3.00/1.64 ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N3 @ N2 )
% 3.00/1.64 => ( ord_less_eq_real @ C3 @ ( F2 @ N2 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ C3 @ L ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_98_LIMSEQ__le__const,axiom,
% 3.00/1.64 ! [X3: nat > nat,X2: nat,A: nat] :
% 3.00/1.64 ( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ ( X3 @ N2 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ X2 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_99_LIMSEQ__le__const,axiom,
% 3.00/1.64 ! [X3: nat > int,X2: int,A: int] :
% 3.00/1.64 ( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_less_eq_int @ A @ ( X3 @ N2 ) ) )
% 3.00/1.64 => ( ord_less_eq_int @ A @ X2 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_100_LIMSEQ__le__const,axiom,
% 3.00/1.64 ! [X3: nat > extended_ereal,X2: extended_ereal,A: extended_ereal] :
% 3.00/1.64 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ ( X3 @ N2 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ X2 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_101_LIMSEQ__le__const,axiom,
% 3.00/1.64 ! [X3: nat > real,X2: real,A: real] :
% 3.00/1.64 ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_less_eq_real @ A @ ( X3 @ N2 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ A @ X2 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_102_LIMSEQ__le__const2,axiom,
% 3.00/1.64 ! [X3: nat > nat,X2: nat,A: nat] :
% 3.00/1.64 ( ( filterlim_nat_nat @ X3 @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( X3 @ N2 ) @ A ) )
% 3.00/1.64 => ( ord_less_eq_nat @ X2 @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_103_LIMSEQ__le__const2,axiom,
% 3.00/1.64 ! [X3: nat > int,X2: int,A: int] :
% 3.00/1.64 ( ( filterlim_nat_int @ X3 @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_less_eq_int @ ( X3 @ N2 ) @ A ) )
% 3.00/1.64 => ( ord_less_eq_int @ X2 @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_104_LIMSEQ__le__const2,axiom,
% 3.00/1.64 ! [X3: nat > extended_ereal,X2: extended_ereal,A: extended_ereal] :
% 3.00/1.64 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( X3 @ N2 ) @ A ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ X2 @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_105_LIMSEQ__le__const2,axiom,
% 3.00/1.64 ! [X3: nat > real,X2: real,A: real] :
% 3.00/1.64 ( ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ? [N4: nat] :
% 3.00/1.64 ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N4 @ N2 )
% 3.00/1.64 => ( ord_less_eq_real @ ( X3 @ N2 ) @ A ) )
% 3.00/1.64 => ( ord_less_eq_real @ X2 @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_106_tendsto__mono,axiom,
% 3.00/1.64 ! [F: filter_nat,F4: filter_nat,F2: nat > extended_ereal,L: extended_ereal] :
% 3.00/1.64 ( ( ord_le1745708096er_nat @ F @ F4 )
% 3.00/1.64 => ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ F4 )
% 3.00/1.64 => ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ F ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_107_tendsto__mono,axiom,
% 3.00/1.64 ! [F: filter_nat,F4: filter_nat,F2: nat > a,L: a] :
% 3.00/1.64 ( ( ord_le1745708096er_nat @ F @ F4 )
% 3.00/1.64 => ( ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ L ) @ F4 )
% 3.00/1.64 => ( filterlim_nat_a @ F2 @ ( topolo705128563nhds_a @ L ) @ F ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_108_tendsto__mono,axiom,
% 3.00/1.64 ! [F: filter_real,F4: filter_real,F2: real > real,L: real] :
% 3.00/1.64 ( ( ord_le132810396r_real @ F @ F4 )
% 3.00/1.64 => ( ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ L ) @ F4 )
% 3.00/1.64 => ( filterlim_real_real @ F2 @ ( topolo1664202871s_real @ L ) @ F ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_109_tendsto__mono,axiom,
% 3.00/1.64 ! [F: filter_nat,F4: filter_nat,F2: nat > real,L: real] :
% 3.00/1.64 ( ( ord_le1745708096er_nat @ F @ F4 )
% 3.00/1.64 => ( ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ F4 )
% 3.00/1.64 => ( filterlim_nat_real @ F2 @ ( topolo1664202871s_real @ L ) @ F ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_110_equation__minus__iff,axiom,
% 3.00/1.64 ! [A: real,B: real] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( uminus_uminus_real @ B ) )
% 3.00/1.64 = ( B
% 3.00/1.64 = ( uminus_uminus_real @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_111_minus__equation__iff,axiom,
% 3.00/1.64 ! [A: real,B: real] :
% 3.00/1.64 ( ( ( uminus_uminus_real @ A )
% 3.00/1.64 = B )
% 3.00/1.64 = ( ( uminus_uminus_real @ B )
% 3.00/1.64 = A ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_112_comp__eq__dest__lhs,axiom,
% 3.00/1.64 ! [A: a > extended_ereal,B: nat > a,C: nat > extended_ereal,V: nat] :
% 3.00/1.64 ( ( ( comp_a1112243075al_nat @ A @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ( A @ ( B @ V ) )
% 3.00/1.64 = ( C @ V ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_113_comp__eq__elim,axiom,
% 3.00/1.64 ! [A: a > extended_ereal,B: nat > a,C: a > extended_ereal,D: nat > a] :
% 3.00/1.64 ( ( ( comp_a1112243075al_nat @ A @ B )
% 3.00/1.64 = ( comp_a1112243075al_nat @ C @ D ) )
% 3.00/1.64 => ! [V2: nat] :
% 3.00/1.64 ( ( A @ ( B @ V2 ) )
% 3.00/1.64 = ( C @ ( D @ V2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_114_comp__eq__dest,axiom,
% 3.00/1.64 ! [A: a > extended_ereal,B: nat > a,C: a > extended_ereal,D: nat > a,V: nat] :
% 3.00/1.64 ( ( ( comp_a1112243075al_nat @ A @ B )
% 3.00/1.64 = ( comp_a1112243075al_nat @ C @ D ) )
% 3.00/1.64 => ( ( A @ ( B @ V ) )
% 3.00/1.64 = ( C @ ( D @ V ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_115_comp__assoc,axiom,
% 3.00/1.64 ! [F2: a > extended_ereal,G2: nat > a,H: nat > nat] :
% 3.00/1.64 ( ( comp_n1096781355al_nat @ ( comp_a1112243075al_nat @ F2 @ G2 ) @ H )
% 3.00/1.64 = ( comp_a1112243075al_nat @ F2 @ ( comp_nat_a_nat @ G2 @ H ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_116_comp__assoc,axiom,
% 3.00/1.64 ! [F2: extended_ereal > extended_ereal,G2: a > extended_ereal,H: nat > a] :
% 3.00/1.64 ( ( comp_a1112243075al_nat @ ( comp_E489644891real_a @ F2 @ G2 ) @ H )
% 3.00/1.64 = ( comp_E1308517939al_nat @ F2 @ ( comp_a1112243075al_nat @ G2 @ H ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_117_comp__assoc,axiom,
% 3.00/1.64 ! [F2: a > extended_ereal,G2: a > a,H: nat > a] :
% 3.00/1.64 ( ( comp_a1112243075al_nat @ ( comp_a780206603real_a @ F2 @ G2 ) @ H )
% 3.00/1.64 = ( comp_a1112243075al_nat @ F2 @ ( comp_a_a_nat @ G2 @ H ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_118_comp__def,axiom,
% 3.00/1.64 ( comp_a1112243075al_nat
% 3.00/1.64 = ( ^ [F3: a > extended_ereal,G: nat > a,X: nat] : ( F3 @ ( G @ X ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_119_lsc__sequentially,axiom,
% 3.00/1.64 ( lower_551915512_ereal
% 3.00/1.64 = ( ^ [X02: real,F3: real > extended_ereal] :
% 3.00/1.64 ! [X: nat > real,C2: extended_ereal] :
% 3.00/1.64 ( ( ( filterlim_nat_real @ X @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 3.00/1.64 & ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ C2 ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_120_lsc__sequentially,axiom,
% 3.00/1.64 ( lower_191460856_ereal
% 3.00/1.64 = ( ^ [X02: a,F3: a > extended_ereal] :
% 3.00/1.64 ! [X: nat > a,C2: extended_ereal] :
% 3.00/1.64 ( ( ( filterlim_nat_a @ X @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 & ! [N: nat] : ( ord_le824540014_ereal @ ( F3 @ ( X @ N ) ) @ C2 ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ C2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_121_usc__at__def,axiom,
% 3.00/1.64 ( lower_114093al_nat
% 3.00/1.64 = ( ^ [X02: extended_ereal,F3: extended_ereal > nat] :
% 3.00/1.64 ! [X4: nat > extended_ereal,L2: nat] :
% 3.00/1.64 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_nat @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_122_usc__at__def,axiom,
% 3.00/1.64 ( lower_637387785al_int
% 3.00/1.64 = ( ^ [X02: extended_ereal,F3: extended_ereal > int] :
% 3.00/1.64 ! [X4: nat > extended_ereal,L2: int] :
% 3.00/1.64 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_int @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_123_usc__at__def,axiom,
% 3.00/1.64 ( lower_1071158961_ereal
% 3.00/1.64 = ( ^ [X02: extended_ereal,F3: extended_ereal > extended_ereal] :
% 3.00/1.64 ! [X4: nat > extended_ereal,L2: extended_ereal] :
% 3.00/1.64 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_124_usc__at__def,axiom,
% 3.00/1.64 ( lower_737640969l_real
% 3.00/1.64 = ( ^ [X02: extended_ereal,F3: extended_ereal > real] :
% 3.00/1.64 ! [X4: nat > extended_ereal,L2: real] :
% 3.00/1.64 ( ( ( filter1531173832_ereal @ X4 @ ( topolo2140997059_ereal @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_real @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_125_usc__at__def,axiom,
% 3.00/1.64 ( lower_1035717085_a_nat
% 3.00/1.64 = ( ^ [X02: a,F3: a > nat] :
% 3.00/1.64 ! [X4: nat > a,L2: nat] :
% 3.00/1.64 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_nat @ ( comp_a_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_nat @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_126_usc__at__def,axiom,
% 3.00/1.64 ( lower_1672990777_a_int
% 3.00/1.64 = ( ^ [X02: a,F3: a > int] :
% 3.00/1.64 ! [X4: nat > a,L2: int] :
% 3.00/1.64 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_int @ ( comp_a_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_int @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_127_usc__at__def,axiom,
% 3.00/1.64 ( lower_534855297_ereal
% 3.00/1.64 = ( ^ [X02: a,F3: a > extended_ereal] :
% 3.00/1.64 ! [X4: nat > a,L2: extended_ereal] :
% 3.00/1.64 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ F3 @ X4 ) @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_128_usc__at__def,axiom,
% 3.00/1.64 ( lower_755922489a_real
% 3.00/1.64 = ( ^ [X02: a,F3: a > real] :
% 3.00/1.64 ! [X4: nat > a,L2: real] :
% 3.00/1.64 ( ( ( filterlim_nat_a @ X4 @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_real @ ( comp_a_real_nat @ F3 @ X4 ) @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_real @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_129_usc__at__def,axiom,
% 3.00/1.64 ( lower_438231087al_nat
% 3.00/1.64 = ( ^ [X02: real,F3: real > nat] :
% 3.00/1.64 ! [X4: nat > real,L2: nat] :
% 3.00/1.64 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_nat @ ( comp_real_nat_nat @ F3 @ X4 ) @ ( topolo1564986139ds_nat @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_nat @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_130_usc__at__def,axiom,
% 3.00/1.64 ( lower_1075504779al_int
% 3.00/1.64 = ( ^ [X02: real,F3: real > int] :
% 3.00/1.64 ! [X4: nat > real,L2: int] :
% 3.00/1.64 ( ( ( filterlim_nat_real @ X4 @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 3.00/1.64 & ( filterlim_nat_int @ ( comp_real_int_nat @ F3 @ X4 ) @ ( topolo54776183ds_int @ L2 ) @ at_top_nat ) )
% 3.00/1.64 => ( ord_less_eq_int @ L2 @ ( F3 @ X02 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_131_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: extended_ereal,F2: extended_ereal > nat,X2: nat > extended_ereal,A2: nat] :
% 3.00/1.64 ( ( lower_114093al_nat @ X0 @ F2 )
% 3.00/1.64 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_nat @ ( comp_E1523169101at_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_nat @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_132_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: extended_ereal,F2: extended_ereal > int,X2: nat > extended_ereal,A2: int] :
% 3.00/1.64 ( ( lower_637387785al_int @ X0 @ F2 )
% 3.00/1.64 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_int @ ( comp_E1436437929nt_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_int @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_133_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: extended_ereal,F2: extended_ereal > extended_ereal,X2: nat > extended_ereal,A2: extended_ereal] :
% 3.00/1.64 ( ( lower_1071158961_ereal @ X0 @ F2 )
% 3.00/1.64 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filter1531173832_ereal @ ( comp_E1308517939al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_le824540014_ereal @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_134_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: extended_ereal,F2: extended_ereal > real,X2: nat > extended_ereal,A2: real] :
% 3.00/1.64 ( ( lower_737640969l_real @ X0 @ F2 )
% 3.00/1.64 => ( ( filter1531173832_ereal @ X2 @ ( topolo2140997059_ereal @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_real @ ( comp_E1477338153al_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_real @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_135_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: a,F2: a > nat,X2: nat > a,A2: nat] :
% 3.00/1.64 ( ( lower_1035717085_a_nat @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_nat @ ( comp_a_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_nat @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_136_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: a,F2: a > int,X2: nat > a,A2: int] :
% 3.00/1.64 ( ( lower_1672990777_a_int @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_int @ ( comp_a_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_int @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_137_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: a,F2: a > extended_ereal,X2: nat > a,A2: extended_ereal] :
% 3.00/1.64 ( ( lower_534855297_ereal @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filter1531173832_ereal @ ( comp_a1112243075al_nat @ F2 @ X2 ) @ ( topolo2140997059_ereal @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_le824540014_ereal @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_138_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: a,F2: a > real,X2: nat > a,A2: real] :
% 3.00/1.64 ( ( lower_755922489a_real @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_real @ ( comp_a_real_nat @ F2 @ X2 ) @ ( topolo1664202871s_real @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_real @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_139_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: real,F2: real > nat,X2: nat > real,A2: nat] :
% 3.00/1.64 ( ( lower_438231087al_nat @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_nat @ ( comp_real_nat_nat @ F2 @ X2 ) @ ( topolo1564986139ds_nat @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_nat @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_140_usc__at__mem,axiom,
% 3.00/1.64 ! [X0: real,F2: real > int,X2: nat > real,A2: int] :
% 3.00/1.64 ( ( lower_1075504779al_int @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ( filterlim_nat_int @ ( comp_real_int_nat @ F2 @ X2 ) @ ( topolo54776183ds_int @ A2 ) @ at_top_nat )
% 3.00/1.64 => ( ord_less_eq_int @ A2 @ ( F2 @ X0 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_141_lim__decreasing__cl,axiom,
% 3.00/1.64 ! [F2: nat > extended_ereal] :
% 3.00/1.64 ( ! [N2: nat,M2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ N2 ) @ ( F2 @ M2 ) ) )
% 3.00/1.64 => ~ ! [L3: extended_ereal] :
% 3.00/1.64 ~ ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L3 ) @ at_top_nat ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_142_lim__increasing__cl,axiom,
% 3.00/1.64 ! [F2: nat > extended_ereal] :
% 3.00/1.64 ( ! [N2: nat,M2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ M2 ) @ ( F2 @ N2 ) ) )
% 3.00/1.64 => ~ ! [L3: extended_ereal] :
% 3.00/1.64 ~ ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L3 ) @ at_top_nat ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_143_order__refl,axiom,
% 3.00/1.64 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ X2 ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_144_order__refl,axiom,
% 3.00/1.64 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_145_order__refl,axiom,
% 3.00/1.64 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_146_order__refl,axiom,
% 3.00/1.64 ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_147_monoseq__le,axiom,
% 3.00/1.64 ! [A: nat > nat,X2: nat] :
% 3.00/1.64 ( ( topolo1922093437eq_nat @ A )
% 3.00/1.64 => ( ( filterlim_nat_nat @ A @ ( topolo1564986139ds_nat @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( ! [N5: nat] : ( ord_less_eq_nat @ ( A @ N5 ) @ X2 )
% 3.00/1.64 & ! [M3: nat,N5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M3 @ N5 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
% 3.00/1.64 | ( ! [N5: nat] : ( ord_less_eq_nat @ X2 @ ( A @ N5 ) )
% 3.00/1.64 & ! [M3: nat,N5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M3 @ N5 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_148_monoseq__le,axiom,
% 3.00/1.64 ! [A: nat > int,X2: int] :
% 3.00/1.64 ( ( topolo411883481eq_int @ A )
% 3.00/1.64 => ( ( filterlim_nat_int @ A @ ( topolo54776183ds_int @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( ! [N5: nat] : ( ord_less_eq_int @ ( A @ N5 ) @ X2 )
% 3.00/1.64 & ! [M3: nat,N5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M3 @ N5 )
% 3.00/1.64 => ( ord_less_eq_int @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
% 3.00/1.64 | ( ! [N5: nat] : ( ord_less_eq_int @ X2 @ ( A @ N5 ) )
% 3.00/1.64 & ! [M3: nat,N5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M3 @ N5 )
% 3.00/1.64 => ( ord_less_eq_int @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_149_monoseq__le,axiom,
% 3.00/1.64 ! [A: nat > extended_ereal,X2: extended_ereal] :
% 3.00/1.64 ( ( topolo1069469409_ereal @ A )
% 3.00/1.64 => ( ( filter1531173832_ereal @ A @ ( topolo2140997059_ereal @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( ! [N5: nat] : ( ord_le824540014_ereal @ ( A @ N5 ) @ X2 )
% 3.00/1.64 & ! [M3: nat,N5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M3 @ N5 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
% 3.00/1.64 | ( ! [N5: nat] : ( ord_le824540014_ereal @ X2 @ ( A @ N5 ) )
% 3.00/1.64 & ! [M3: nat,N5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M3 @ N5 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_150_monoseq__le,axiom,
% 3.00/1.64 ! [A: nat > real,X2: real] :
% 3.00/1.64 ( ( topolo144289241q_real @ A )
% 3.00/1.64 => ( ( filterlim_nat_real @ A @ ( topolo1664202871s_real @ X2 ) @ at_top_nat )
% 3.00/1.64 => ( ( ! [N5: nat] : ( ord_less_eq_real @ ( A @ N5 ) @ X2 )
% 3.00/1.64 & ! [M3: nat,N5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M3 @ N5 )
% 3.00/1.64 => ( ord_less_eq_real @ ( A @ M3 ) @ ( A @ N5 ) ) ) )
% 3.00/1.64 | ( ! [N5: nat] : ( ord_less_eq_real @ X2 @ ( A @ N5 ) )
% 3.00/1.64 & ! [M3: nat,N5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M3 @ N5 )
% 3.00/1.64 => ( ord_less_eq_real @ ( A @ N5 ) @ ( A @ M3 ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_151_lsc__liminf,axiom,
% 3.00/1.64 ( lower_551915512_ereal
% 3.00/1.64 = ( ^ [X02: real,F3: real > extended_ereal] :
% 3.00/1.64 ! [X: nat > real] :
% 3.00/1.64 ( ( filterlim_nat_real @ X @ ( topolo1664202871s_real @ X02 ) @ at_top_nat )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_r1410008527al_nat @ F3 @ X ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_152_lsc__liminf,axiom,
% 3.00/1.64 ( lower_191460856_ereal
% 3.00/1.64 = ( ^ [X02: a,F3: a > extended_ereal] :
% 3.00/1.64 ! [X: nat > a] :
% 3.00/1.64 ( ( filterlim_nat_a @ X @ ( topolo705128563nhds_a @ X02 ) @ at_top_nat )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F3 @ X02 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_a1112243075al_nat @ F3 @ X ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_153_lsc__imp__liminf,axiom,
% 3.00/1.64 ! [X0: real,F2: real > extended_ereal,X2: nat > real] :
% 3.00/1.64 ( ( lower_551915512_ereal @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_real @ X2 @ ( topolo1664202871s_real @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_r1410008527al_nat @ F2 @ X2 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_154_lsc__imp__liminf,axiom,
% 3.00/1.64 ! [X0: a,F2: a > extended_ereal,X2: nat > a] :
% 3.00/1.64 ( ( lower_191460856_ereal @ X0 @ F2 )
% 3.00/1.64 => ( ( filterlim_nat_a @ X2 @ ( topolo705128563nhds_a @ X0 ) @ at_top_nat )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X0 ) @ ( liminf1045857232_ereal @ at_top_nat @ ( comp_a1112243075al_nat @ F2 @ X2 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_155_LIMSEQ__Uniq,axiom,
% 3.00/1.64 ! [X3: nat > extended_ereal] :
% 3.00/1.64 ( uniq_Extended_ereal
% 3.00/1.64 @ ^ [L2: extended_ereal] : ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ L2 ) @ at_top_nat ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_156_LIMSEQ__Uniq,axiom,
% 3.00/1.64 ! [X3: nat > a] :
% 3.00/1.64 ( uniq_a
% 3.00/1.64 @ ^ [L2: a] : ( filterlim_nat_a @ X3 @ ( topolo705128563nhds_a @ L2 ) @ at_top_nat ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_157_LIMSEQ__Uniq,axiom,
% 3.00/1.64 ! [X3: nat > real] :
% 3.00/1.64 ( uniq_real
% 3.00/1.64 @ ^ [L2: real] : ( filterlim_nat_real @ X3 @ ( topolo1664202871s_real @ L2 ) @ at_top_nat ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_158_monoseq__def,axiom,
% 3.00/1.64 ( topolo1069469409_ereal
% 3.00/1.64 = ( ^ [X4: nat > extended_ereal] :
% 3.00/1.64 ( ! [M4: nat,N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M4 @ N )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 3.00/1.64 | ! [M4: nat,N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M4 @ N )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_159_monoseq__def,axiom,
% 3.00/1.64 ( topolo1922093437eq_nat
% 3.00/1.64 = ( ^ [X4: nat > nat] :
% 3.00/1.64 ( ! [M4: nat,N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M4 @ N )
% 3.00/1.64 => ( ord_less_eq_nat @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 3.00/1.64 | ! [M4: nat,N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M4 @ N )
% 3.00/1.64 => ( ord_less_eq_nat @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_160_monoseq__def,axiom,
% 3.00/1.64 ( topolo144289241q_real
% 3.00/1.64 = ( ^ [X4: nat > real] :
% 3.00/1.64 ( ! [M4: nat,N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M4 @ N )
% 3.00/1.64 => ( ord_less_eq_real @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 3.00/1.64 | ! [M4: nat,N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M4 @ N )
% 3.00/1.64 => ( ord_less_eq_real @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_161_monoseq__def,axiom,
% 3.00/1.64 ( topolo411883481eq_int
% 3.00/1.64 = ( ^ [X4: nat > int] :
% 3.00/1.64 ( ! [M4: nat,N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M4 @ N )
% 3.00/1.64 => ( ord_less_eq_int @ ( X4 @ M4 ) @ ( X4 @ N ) ) )
% 3.00/1.64 | ! [M4: nat,N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M4 @ N )
% 3.00/1.64 => ( ord_less_eq_int @ ( X4 @ N ) @ ( X4 @ M4 ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_162_monoI2,axiom,
% 3.00/1.64 ! [X3: nat > extended_ereal] :
% 3.00/1.64 ( ! [M2: nat,N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 3.00/1.64 => ( topolo1069469409_ereal @ X3 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_163_monoI2,axiom,
% 3.00/1.64 ! [X3: nat > nat] :
% 3.00/1.64 ( ! [M2: nat,N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 3.00/1.64 => ( topolo1922093437eq_nat @ X3 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_164_monoI2,axiom,
% 3.00/1.64 ! [X3: nat > real] :
% 3.00/1.64 ( ! [M2: nat,N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_less_eq_real @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 3.00/1.64 => ( topolo144289241q_real @ X3 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_165_monoI2,axiom,
% 3.00/1.64 ! [X3: nat > int] :
% 3.00/1.64 ( ! [M2: nat,N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_less_eq_int @ ( X3 @ N2 ) @ ( X3 @ M2 ) ) )
% 3.00/1.64 => ( topolo411883481eq_int @ X3 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_166_monoI1,axiom,
% 3.00/1.64 ! [X3: nat > extended_ereal] :
% 3.00/1.64 ( ! [M2: nat,N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 3.00/1.64 => ( topolo1069469409_ereal @ X3 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_167_monoI1,axiom,
% 3.00/1.64 ! [X3: nat > nat] :
% 3.00/1.64 ( ! [M2: nat,N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 3.00/1.64 => ( topolo1922093437eq_nat @ X3 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_168_monoI1,axiom,
% 3.00/1.64 ! [X3: nat > real] :
% 3.00/1.64 ( ! [M2: nat,N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_less_eq_real @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 3.00/1.64 => ( topolo144289241q_real @ X3 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_169_monoI1,axiom,
% 3.00/1.64 ! [X3: nat > int] :
% 3.00/1.64 ( ! [M2: nat,N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M2 @ N2 )
% 3.00/1.64 => ( ord_less_eq_int @ ( X3 @ M2 ) @ ( X3 @ N2 ) ) )
% 3.00/1.64 => ( topolo411883481eq_int @ X3 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_170_monoseq__minus,axiom,
% 3.00/1.64 ! [A: nat > real] :
% 3.00/1.64 ( ( topolo144289241q_real @ A )
% 3.00/1.64 => ( topolo144289241q_real
% 3.00/1.64 @ ^ [N: nat] : ( uminus_uminus_real @ ( A @ N ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_171_order__subst1,axiom,
% 3.00/1.64 ! [A: extended_ereal,F2: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_172_order__subst1,axiom,
% 3.00/1.64 ! [A: extended_ereal,F2: nat > extended_ereal,B: nat,C: nat] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_173_order__subst1,axiom,
% 3.00/1.64 ! [A: extended_ereal,F2: real > extended_ereal,B: real,C: real] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_real @ B @ C )
% 3.00/1.64 => ( ! [X5: real,Y3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_174_order__subst1,axiom,
% 3.00/1.64 ! [A: extended_ereal,F2: int > extended_ereal,B: int,C: int] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_int @ B @ C )
% 3.00/1.64 => ( ! [X5: int,Y3: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_175_order__subst1,axiom,
% 3.00/1.64 ! [A: nat,F2: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_176_order__subst1,axiom,
% 3.00/1.64 ! [A: nat,F2: nat > nat,B: nat,C: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_177_order__subst1,axiom,
% 3.00/1.64 ! [A: nat,F2: real > nat,B: real,C: real] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_real @ B @ C )
% 3.00/1.64 => ( ! [X5: real,Y3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_178_order__subst1,axiom,
% 3.00/1.64 ! [A: nat,F2: int > nat,B: int,C: int] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_int @ B @ C )
% 3.00/1.64 => ( ! [X5: int,Y3: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_179_order__subst1,axiom,
% 3.00/1.64 ! [A: real,F2: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_180_order__subst1,axiom,
% 3.00/1.64 ! [A: real,F2: nat > real,B: nat,C: nat] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_181_order__subst2,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_182_order__subst2,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > nat,C: nat] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_183_order__subst2,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > real,C: real] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_184_order__subst2,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > int,C: int] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_185_order__subst2,axiom,
% 3.00/1.64 ! [A: nat,B: nat,F2: nat > extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_186_order__subst2,axiom,
% 3.00/1.64 ! [A: nat,B: nat,F2: nat > nat,C: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_187_order__subst2,axiom,
% 3.00/1.64 ! [A: nat,B: nat,F2: nat > real,C: real] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_188_order__subst2,axiom,
% 3.00/1.64 ! [A: nat,B: nat,F2: nat > int,C: int] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_189_order__subst2,axiom,
% 3.00/1.64 ! [A: real,B: real,F2: real > extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ B )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: real,Y3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_190_order__subst2,axiom,
% 3.00/1.64 ! [A: real,B: real,F2: real > nat,C: nat] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
% 3.00/1.64 => ( ! [X5: real,Y3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_191_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: extended_ereal,F2: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_192_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: nat,F2: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_193_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: real,F2: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_194_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: int,F2: extended_ereal > int,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_195_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: extended_ereal,F2: nat > extended_ereal,B: nat,C: nat] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_196_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: nat,F2: nat > nat,B: nat,C: nat] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_197_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: real,F2: nat > real,B: nat,C: nat] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_198_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: int,F2: nat > int,B: nat,C: nat] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_199_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: extended_ereal,F2: real > extended_ereal,B: real,C: real] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_real @ B @ C )
% 3.00/1.64 => ( ! [X5: real,Y3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_200_ord__eq__le__subst,axiom,
% 3.00/1.64 ! [A: nat,F2: real > nat,B: real,C: real] :
% 3.00/1.64 ( ( A
% 3.00/1.64 = ( F2 @ B ) )
% 3.00/1.64 => ( ( ord_less_eq_real @ B @ C )
% 3.00/1.64 => ( ! [X5: real,Y3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_201_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_202_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > nat,C: nat] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_203_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > real,C: real] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_204_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,F2: extended_ereal > int,C: int] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: extended_ereal,Y3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_205_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: nat,B: nat,F2: nat > extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_206_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: nat,B: nat,F2: nat > nat,C: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_207_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: nat,B: nat,F2: nat > real,C: real] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_208_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: nat,B: nat,F2: nat > int,C: int] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: nat,Y3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_209_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: real,B: real,F2: real > extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: real,Y3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X5 @ Y3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_210_ord__le__eq__subst,axiom,
% 3.00/1.64 ! [A: real,B: real,F2: real > nat,C: nat] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ B )
% 3.00/1.64 => ( ( ( F2 @ B )
% 3.00/1.64 = C )
% 3.00/1.64 => ( ! [X5: real,Y3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X5 @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) )
% 3.00/1.64 => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_211_eq__iff,axiom,
% 3.00/1.64 ( ( ^ [Y4: extended_ereal,Z2: extended_ereal] : ( Y4 = Z2 ) )
% 3.00/1.64 = ( ^ [X: extended_ereal,Y5: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X @ Y5 )
% 3.00/1.64 & ( ord_le824540014_ereal @ Y5 @ X ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_212_eq__iff,axiom,
% 3.00/1.64 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 3.00/1.64 = ( ^ [X: nat,Y5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X @ Y5 )
% 3.00/1.64 & ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_213_eq__iff,axiom,
% 3.00/1.64 ( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
% 3.00/1.64 = ( ^ [X: real,Y5: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X @ Y5 )
% 3.00/1.64 & ( ord_less_eq_real @ Y5 @ X ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_214_eq__iff,axiom,
% 3.00/1.64 ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
% 3.00/1.64 = ( ^ [X: int,Y5: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ X @ Y5 )
% 3.00/1.64 & ( ord_less_eq_int @ Y5 @ X ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_215_antisym,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Y: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X2 @ Y )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ Y @ X2 )
% 3.00/1.64 => ( X2 = Y ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_216_antisym,axiom,
% 3.00/1.64 ! [X2: nat,Y: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.00/1.64 => ( ( ord_less_eq_nat @ Y @ X2 )
% 3.00/1.64 => ( X2 = Y ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_217_antisym,axiom,
% 3.00/1.64 ! [X2: real,Y: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X2 @ Y )
% 3.00/1.64 => ( ( ord_less_eq_real @ Y @ X2 )
% 3.00/1.64 => ( X2 = Y ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_218_antisym,axiom,
% 3.00/1.64 ! [X2: int,Y: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ X2 @ Y )
% 3.00/1.64 => ( ( ord_less_eq_int @ Y @ X2 )
% 3.00/1.64 => ( X2 = Y ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_219_linear,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Y: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X2 @ Y )
% 3.00/1.64 | ( ord_le824540014_ereal @ Y @ X2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_220_linear,axiom,
% 3.00/1.64 ! [X2: nat,Y: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.00/1.64 | ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_221_linear,axiom,
% 3.00/1.64 ! [X2: real,Y: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X2 @ Y )
% 3.00/1.64 | ( ord_less_eq_real @ Y @ X2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_222_linear,axiom,
% 3.00/1.64 ! [X2: int,Y: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ X2 @ Y )
% 3.00/1.64 | ( ord_less_eq_int @ Y @ X2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_223_eq__refl,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Y: extended_ereal] :
% 3.00/1.64 ( ( X2 = Y )
% 3.00/1.64 => ( ord_le824540014_ereal @ X2 @ Y ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_224_eq__refl,axiom,
% 3.00/1.64 ! [X2: nat,Y: nat] :
% 3.00/1.64 ( ( X2 = Y )
% 3.00/1.64 => ( ord_less_eq_nat @ X2 @ Y ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_225_eq__refl,axiom,
% 3.00/1.64 ! [X2: real,Y: real] :
% 3.00/1.64 ( ( X2 = Y )
% 3.00/1.64 => ( ord_less_eq_real @ X2 @ Y ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_226_eq__refl,axiom,
% 3.00/1.64 ! [X2: int,Y: int] :
% 3.00/1.64 ( ( X2 = Y )
% 3.00/1.64 => ( ord_less_eq_int @ X2 @ Y ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_227_le__cases,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Y: extended_ereal] :
% 3.00/1.64 ( ~ ( ord_le824540014_ereal @ X2 @ Y )
% 3.00/1.64 => ( ord_le824540014_ereal @ Y @ X2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_228_le__cases,axiom,
% 3.00/1.64 ! [X2: nat,Y: nat] :
% 3.00/1.64 ( ~ ( ord_less_eq_nat @ X2 @ Y )
% 3.00/1.64 => ( ord_less_eq_nat @ Y @ X2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_229_le__cases,axiom,
% 3.00/1.64 ! [X2: real,Y: real] :
% 3.00/1.64 ( ~ ( ord_less_eq_real @ X2 @ Y )
% 3.00/1.64 => ( ord_less_eq_real @ Y @ X2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_230_le__cases,axiom,
% 3.00/1.64 ! [X2: int,Y: int] :
% 3.00/1.64 ( ~ ( ord_less_eq_int @ X2 @ Y )
% 3.00/1.64 => ( ord_less_eq_int @ Y @ X2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_231_order_Otrans,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ C )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_232_order_Otrans,axiom,
% 3.00/1.64 ! [A: nat,B: nat,C: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_233_order_Otrans,axiom,
% 3.00/1.64 ! [A: real,B: real,C: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_real @ B @ C )
% 3.00/1.64 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_234_order_Otrans,axiom,
% 3.00/1.64 ! [A: int,B: int,C: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_int @ B @ C )
% 3.00/1.64 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_235_le__cases3,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
% 3.00/1.64 ( ( ( ord_le824540014_ereal @ X2 @ Y )
% 3.00/1.64 => ~ ( ord_le824540014_ereal @ Y @ Z3 ) )
% 3.00/1.64 => ( ( ( ord_le824540014_ereal @ Y @ X2 )
% 3.00/1.64 => ~ ( ord_le824540014_ereal @ X2 @ Z3 ) )
% 3.00/1.64 => ( ( ( ord_le824540014_ereal @ X2 @ Z3 )
% 3.00/1.64 => ~ ( ord_le824540014_ereal @ Z3 @ Y ) )
% 3.00/1.64 => ( ( ( ord_le824540014_ereal @ Z3 @ Y )
% 3.00/1.64 => ~ ( ord_le824540014_ereal @ Y @ X2 ) )
% 3.00/1.64 => ( ( ( ord_le824540014_ereal @ Y @ Z3 )
% 3.00/1.64 => ~ ( ord_le824540014_ereal @ Z3 @ X2 ) )
% 3.00/1.64 => ~ ( ( ord_le824540014_ereal @ Z3 @ X2 )
% 3.00/1.64 => ~ ( ord_le824540014_ereal @ X2 @ Y ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_236_le__cases3,axiom,
% 3.00/1.64 ! [X2: nat,Y: nat,Z3: nat] :
% 3.00/1.64 ( ( ( ord_less_eq_nat @ X2 @ Y )
% 3.00/1.64 => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
% 3.00/1.64 => ( ( ( ord_less_eq_nat @ Y @ X2 )
% 3.00/1.64 => ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
% 3.00/1.64 => ( ( ( ord_less_eq_nat @ X2 @ Z3 )
% 3.00/1.64 => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
% 3.00/1.64 => ( ( ( ord_less_eq_nat @ Z3 @ Y )
% 3.00/1.64 => ~ ( ord_less_eq_nat @ Y @ X2 ) )
% 3.00/1.64 => ( ( ( ord_less_eq_nat @ Y @ Z3 )
% 3.00/1.64 => ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
% 3.00/1.64 => ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
% 3.00/1.64 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_237_le__cases3,axiom,
% 3.00/1.64 ! [X2: real,Y: real,Z3: real] :
% 3.00/1.64 ( ( ( ord_less_eq_real @ X2 @ Y )
% 3.00/1.64 => ~ ( ord_less_eq_real @ Y @ Z3 ) )
% 3.00/1.64 => ( ( ( ord_less_eq_real @ Y @ X2 )
% 3.00/1.64 => ~ ( ord_less_eq_real @ X2 @ Z3 ) )
% 3.00/1.64 => ( ( ( ord_less_eq_real @ X2 @ Z3 )
% 3.00/1.64 => ~ ( ord_less_eq_real @ Z3 @ Y ) )
% 3.00/1.64 => ( ( ( ord_less_eq_real @ Z3 @ Y )
% 3.00/1.64 => ~ ( ord_less_eq_real @ Y @ X2 ) )
% 3.00/1.64 => ( ( ( ord_less_eq_real @ Y @ Z3 )
% 3.00/1.64 => ~ ( ord_less_eq_real @ Z3 @ X2 ) )
% 3.00/1.64 => ~ ( ( ord_less_eq_real @ Z3 @ X2 )
% 3.00/1.64 => ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_238_le__cases3,axiom,
% 3.00/1.64 ! [X2: int,Y: int,Z3: int] :
% 3.00/1.64 ( ( ( ord_less_eq_int @ X2 @ Y )
% 3.00/1.64 => ~ ( ord_less_eq_int @ Y @ Z3 ) )
% 3.00/1.64 => ( ( ( ord_less_eq_int @ Y @ X2 )
% 3.00/1.64 => ~ ( ord_less_eq_int @ X2 @ Z3 ) )
% 3.00/1.64 => ( ( ( ord_less_eq_int @ X2 @ Z3 )
% 3.00/1.64 => ~ ( ord_less_eq_int @ Z3 @ Y ) )
% 3.00/1.64 => ( ( ( ord_less_eq_int @ Z3 @ Y )
% 3.00/1.64 => ~ ( ord_less_eq_int @ Y @ X2 ) )
% 3.00/1.64 => ( ( ( ord_less_eq_int @ Y @ Z3 )
% 3.00/1.64 => ~ ( ord_less_eq_int @ Z3 @ X2 ) )
% 3.00/1.64 => ~ ( ( ord_less_eq_int @ Z3 @ X2 )
% 3.00/1.64 => ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_239_antisym__conv,axiom,
% 3.00/1.64 ! [Y: extended_ereal,X2: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ Y @ X2 )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ X2 @ Y )
% 3.00/1.64 = ( X2 = Y ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_240_antisym__conv,axiom,
% 3.00/1.64 ! [Y: nat,X2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ Y @ X2 )
% 3.00/1.64 => ( ( ord_less_eq_nat @ X2 @ Y )
% 3.00/1.64 = ( X2 = Y ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_241_antisym__conv,axiom,
% 3.00/1.64 ! [Y: real,X2: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ Y @ X2 )
% 3.00/1.64 => ( ( ord_less_eq_real @ X2 @ Y )
% 3.00/1.64 = ( X2 = Y ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_242_antisym__conv,axiom,
% 3.00/1.64 ! [Y: int,X2: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ Y @ X2 )
% 3.00/1.64 => ( ( ord_less_eq_int @ X2 @ Y )
% 3.00/1.64 = ( X2 = Y ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_243_order__class_Oorder_Oeq__iff,axiom,
% 3.00/1.64 ( ( ^ [Y4: extended_ereal,Z2: extended_ereal] : ( Y4 = Z2 ) )
% 3.00/1.64 = ( ^ [A3: extended_ereal,B2: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A3 @ B2 )
% 3.00/1.64 & ( ord_le824540014_ereal @ B2 @ A3 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_244_order__class_Oorder_Oeq__iff,axiom,
% 3.00/1.64 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 3.00/1.64 = ( ^ [A3: nat,B2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A3 @ B2 )
% 3.00/1.64 & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_245_order__class_Oorder_Oeq__iff,axiom,
% 3.00/1.64 ( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
% 3.00/1.64 = ( ^ [A3: real,B2: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ A3 @ B2 )
% 3.00/1.64 & ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_246_order__class_Oorder_Oeq__iff,axiom,
% 3.00/1.64 ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
% 3.00/1.64 = ( ^ [A3: int,B2: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ A3 @ B2 )
% 3.00/1.64 & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_247_ord__eq__le__trans,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( A = B )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ C )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_248_ord__eq__le__trans,axiom,
% 3.00/1.64 ! [A: nat,B: nat,C: nat] :
% 3.00/1.64 ( ( A = B )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ C )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_249_ord__eq__le__trans,axiom,
% 3.00/1.64 ! [A: real,B: real,C: real] :
% 3.00/1.64 ( ( A = B )
% 3.00/1.64 => ( ( ord_less_eq_real @ B @ C )
% 3.00/1.64 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_250_ord__eq__le__trans,axiom,
% 3.00/1.64 ! [A: int,B: int,C: int] :
% 3.00/1.64 ( ( A = B )
% 3.00/1.64 => ( ( ord_less_eq_int @ B @ C )
% 3.00/1.64 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_251_ord__le__eq__trans,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( B = C )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_252_ord__le__eq__trans,axiom,
% 3.00/1.64 ! [A: nat,B: nat,C: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( B = C )
% 3.00/1.64 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_253_ord__le__eq__trans,axiom,
% 3.00/1.64 ! [A: real,B: real,C: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ B )
% 3.00/1.64 => ( ( B = C )
% 3.00/1.64 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_254_ord__le__eq__trans,axiom,
% 3.00/1.64 ! [A: int,B: int,C: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ A @ B )
% 3.00/1.64 => ( ( B = C )
% 3.00/1.64 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_255_order__class_Oorder_Oantisym,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ B @ A )
% 3.00/1.64 => ( A = B ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_256_order__class_Oorder_Oantisym,axiom,
% 3.00/1.64 ! [A: nat,B: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_nat @ B @ A )
% 3.00/1.64 => ( A = B ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_257_order__class_Oorder_Oantisym,axiom,
% 3.00/1.64 ! [A: real,B: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_real @ B @ A )
% 3.00/1.64 => ( A = B ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_258_order__class_Oorder_Oantisym,axiom,
% 3.00/1.64 ! [A: int,B: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ A @ B )
% 3.00/1.64 => ( ( ord_less_eq_int @ B @ A )
% 3.00/1.64 => ( A = B ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_259_order__trans,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Y: extended_ereal,Z3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X2 @ Y )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ Y @ Z3 )
% 3.00/1.64 => ( ord_le824540014_ereal @ X2 @ Z3 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_260_order__trans,axiom,
% 3.00/1.64 ! [X2: nat,Y: nat,Z3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ X2 @ Y )
% 3.00/1.64 => ( ( ord_less_eq_nat @ Y @ Z3 )
% 3.00/1.64 => ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_261_order__trans,axiom,
% 3.00/1.64 ! [X2: real,Y: real,Z3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ X2 @ Y )
% 3.00/1.64 => ( ( ord_less_eq_real @ Y @ Z3 )
% 3.00/1.64 => ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_262_order__trans,axiom,
% 3.00/1.64 ! [X2: int,Y: int,Z3: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ X2 @ Y )
% 3.00/1.64 => ( ( ord_less_eq_int @ Y @ Z3 )
% 3.00/1.64 => ( ord_less_eq_int @ X2 @ Z3 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_263_dual__order_Orefl,axiom,
% 3.00/1.64 ! [A: extended_ereal] : ( ord_le824540014_ereal @ A @ A ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_264_dual__order_Orefl,axiom,
% 3.00/1.64 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_265_dual__order_Orefl,axiom,
% 3.00/1.64 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_266_dual__order_Orefl,axiom,
% 3.00/1.64 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_267_linorder__wlog,axiom,
% 3.00/1.64 ! [P: extended_ereal > extended_ereal > $o,A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ! [A4: extended_ereal,B3: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A4 @ B3 )
% 3.00/1.64 => ( P @ A4 @ B3 ) )
% 3.00/1.64 => ( ! [A4: extended_ereal,B3: extended_ereal] :
% 3.00/1.64 ( ( P @ B3 @ A4 )
% 3.00/1.64 => ( P @ A4 @ B3 ) )
% 3.00/1.64 => ( P @ A @ B ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_268_linorder__wlog,axiom,
% 3.00/1.64 ! [P: nat > nat > $o,A: nat,B: nat] :
% 3.00/1.64 ( ! [A4: nat,B3: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ A4 @ B3 )
% 3.00/1.64 => ( P @ A4 @ B3 ) )
% 3.00/1.64 => ( ! [A4: nat,B3: nat] :
% 3.00/1.64 ( ( P @ B3 @ A4 )
% 3.00/1.64 => ( P @ A4 @ B3 ) )
% 3.00/1.64 => ( P @ A @ B ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_269_linorder__wlog,axiom,
% 3.00/1.64 ! [P: real > real > $o,A: real,B: real] :
% 3.00/1.64 ( ! [A4: real,B3: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ A4 @ B3 )
% 3.00/1.64 => ( P @ A4 @ B3 ) )
% 3.00/1.64 => ( ! [A4: real,B3: real] :
% 3.00/1.64 ( ( P @ B3 @ A4 )
% 3.00/1.64 => ( P @ A4 @ B3 ) )
% 3.00/1.64 => ( P @ A @ B ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_270_linorder__wlog,axiom,
% 3.00/1.64 ! [P: int > int > $o,A: int,B: int] :
% 3.00/1.64 ( ! [A4: int,B3: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ A4 @ B3 )
% 3.00/1.64 => ( P @ A4 @ B3 ) )
% 3.00/1.64 => ( ! [A4: int,B3: int] :
% 3.00/1.64 ( ( P @ B3 @ A4 )
% 3.00/1.64 => ( P @ A4 @ B3 ) )
% 3.00/1.64 => ( P @ A @ B ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_271_dual__order_Otrans,axiom,
% 3.00/1.64 ! [B: nat,A: nat,C: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ B @ A )
% 3.00/1.64 => ( ( ord_less_eq_nat @ C @ B )
% 3.00/1.64 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_272_dual__order_Otrans,axiom,
% 3.00/1.64 ! [B: real,A: real,C: real] :
% 3.00/1.64 ( ( ord_less_eq_real @ B @ A )
% 3.00/1.64 => ( ( ord_less_eq_real @ C @ B )
% 3.00/1.64 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_273_dual__order_Otrans,axiom,
% 3.00/1.64 ! [B: int,A: int,C: int] :
% 3.00/1.64 ( ( ord_less_eq_int @ B @ A )
% 3.00/1.64 => ( ( ord_less_eq_int @ C @ B )
% 3.00/1.64 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_274_liminf__PInfty,axiom,
% 3.00/1.64 ! [X3: nat > extended_ereal] :
% 3.00/1.64 ( ( filter1531173832_ereal @ X3 @ ( topolo2140997059_ereal @ extend1289208545_ereal ) @ at_top_nat )
% 3.00/1.64 = ( ( liminf1045857232_ereal @ at_top_nat @ X3 )
% 3.00/1.64 = extend1289208545_ereal ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_275_ereal__infty__less__eq_I1_J,axiom,
% 3.00/1.64 ! [X2: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ extend1289208545_ereal @ X2 )
% 3.00/1.64 = ( X2 = extend1289208545_ereal ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_276_ereal__infty__less__eq_I2_J,axiom,
% 3.00/1.64 ! [X2: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 = ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_277_MInfty__neq__PInfty_I1_J,axiom,
% 3.00/1.64 ( extend1289208545_ereal
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_278_ereal__less__eq_I1_J,axiom,
% 3.00/1.64 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ extend1289208545_ereal ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_279_ereal__infty__less__eq2_I1_J,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( A = extend1289208545_ereal )
% 3.00/1.64 => ( B = extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_280_neq__PInf__trans,axiom,
% 3.00/1.64 ! [Y: extended_ereal,X2: extended_ereal] :
% 3.00/1.64 ( ( Y != extend1289208545_ereal )
% 3.00/1.64 => ( ( ord_le824540014_ereal @ X2 @ Y )
% 3.00/1.64 => ( X2 != extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_281_ereal__infty__less__eq2_I2_J,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 => ( ( B
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( A
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_282_ereal__less__eq_I2_J,axiom,
% 3.00/1.64 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X2 ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_283_Lim__MInfty,axiom,
% 3.00/1.64 ! [F2: nat > extended_ereal] :
% 3.00/1.64 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) @ at_top_nat )
% 3.00/1.64 = ( ! [B4: real] :
% 3.00/1.64 ? [N6: nat] :
% 3.00/1.64 ! [N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N6 @ N )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ N ) @ ( extended_ereal2 @ B4 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_284_Lim__bounded__MInfty,axiom,
% 3.00/1.64 ! [F2: nat > extended_ereal,L: extended_ereal,B5: real] :
% 3.00/1.64 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B5 ) @ ( F2 @ N2 ) )
% 3.00/1.64 => ( L
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_285_Lim__PInfty,axiom,
% 3.00/1.64 ! [F2: nat > extended_ereal] :
% 3.00/1.64 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ extend1289208545_ereal ) @ at_top_nat )
% 3.00/1.64 = ( ! [B4: real] :
% 3.00/1.64 ? [N6: nat] :
% 3.00/1.64 ! [N: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N6 @ N )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( extended_ereal2 @ B4 ) @ ( F2 @ N ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_286_Lim__bounded__PInfty2,axiom,
% 3.00/1.64 ! [F2: nat > extended_ereal,L: extended_ereal,N3: nat,B5: real] :
% 3.00/1.64 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ N3 @ N2 )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B5 ) ) )
% 3.00/1.64 => ( L != extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_287_ereal_Oinject,axiom,
% 3.00/1.64 ! [X1: real,Y1: real] :
% 3.00/1.64 ( ( ( extended_ereal2 @ X1 )
% 3.00/1.64 = ( extended_ereal2 @ Y1 ) )
% 3.00/1.64 = ( X1 = Y1 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_288_ereal__cong,axiom,
% 3.00/1.64 ! [X2: real,Y: real] :
% 3.00/1.64 ( ( X2 = Y )
% 3.00/1.64 => ( ( extended_ereal2 @ X2 )
% 3.00/1.64 = ( extended_ereal2 @ Y ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_289_ereal__less__eq_I3_J,axiom,
% 3.00/1.64 ! [R: real,P2: real] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
% 3.00/1.64 = ( ord_less_eq_real @ R @ P2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_290_PInfty__neq__ereal_I1_J,axiom,
% 3.00/1.64 ! [R: real] :
% 3.00/1.64 ( ( extended_ereal2 @ R )
% 3.00/1.64 != extend1289208545_ereal ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_291_uminus__ereal_Osimps_I1_J,axiom,
% 3.00/1.64 ! [R: real] :
% 3.00/1.64 ( ( uminus1208298309_ereal @ ( extended_ereal2 @ R ) )
% 3.00/1.64 = ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_292_ereal__le__le,axiom,
% 3.00/1.64 ! [Y: real,A: extended_ereal,X2: real] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ Y ) @ A )
% 3.00/1.64 => ( ( ord_less_eq_real @ X2 @ Y )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_293_le__ereal__le,axiom,
% 3.00/1.64 ! [A: extended_ereal,X2: real,Y: real] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ X2 ) )
% 3.00/1.64 => ( ( ord_less_eq_real @ X2 @ Y )
% 3.00/1.64 => ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ Y ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_294_ereal__le__real,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Y: extended_ereal] :
% 3.00/1.64 ( ! [Z4: real] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ Y @ ( extended_ereal2 @ Z4 ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ Y @ X2 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_295_ereal__semiline__unique,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ( collect_real
% 3.00/1.64 @ ^ [Y5: real] : ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ Y5 ) ) )
% 3.00/1.64 = ( collect_real
% 3.00/1.64 @ ^ [Y5: real] : ( ord_le824540014_ereal @ B @ ( extended_ereal2 @ Y5 ) ) ) )
% 3.00/1.64 = ( A = B ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_296_real__of__ereal_Oinduct,axiom,
% 3.00/1.64 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 3.00/1.64 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( P @ extend1289208545_ereal )
% 3.00/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( P @ A0 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_297_real__of__ereal_Ocases,axiom,
% 3.00/1.64 ! [X2: extended_ereal] :
% 3.00/1.64 ( ! [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 != ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( X2 != extend1289208545_ereal )
% 3.00/1.64 => ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_298_times__ereal_Oinduct,axiom,
% 3.00/1.64 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 3.00/1.64 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 3.00/1.64 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ extend1289208545_ereal )
% 3.00/1.64 => ( ! [R2: real] : ( P @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( P @ extend1289208545_ereal @ extend1289208545_ereal )
% 3.00/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 3.00/1.64 => ( ( P @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_299_plus__ereal_Oinduct,axiom,
% 3.00/1.64 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 3.00/1.64 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 3.00/1.64 => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
% 3.00/1.64 => ( ! [A4: extended_ereal] : ( P @ A4 @ extend1289208545_ereal )
% 3.00/1.64 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ! [P3: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P3 ) )
% 3.00/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_300_less__ereal_Oinduct,axiom,
% 3.00/1.64 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 3.00/1.64 ( ! [X5: real,Y3: real] : ( P @ ( extended_ereal2 @ X5 ) @ ( extended_ereal2 @ Y3 ) )
% 3.00/1.64 => ( ! [X_1: extended_ereal] : ( P @ extend1289208545_ereal @ X_1 )
% 3.00/1.64 => ( ! [A4: extended_ereal] : ( P @ A4 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ! [X5: real] : ( P @ ( extended_ereal2 @ X5 ) @ extend1289208545_ereal )
% 3.00/1.64 => ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 3.00/1.64 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_301_abs__ereal_Oinduct,axiom,
% 3.00/1.64 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 3.00/1.64 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( P @ extend1289208545_ereal )
% 3.00/1.64 => ( P @ A0 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_302_ereal__all__split,axiom,
% 3.00/1.64 ( ( ^ [P4: extended_ereal > $o] :
% 3.00/1.64 ! [X6: extended_ereal] : ( P4 @ X6 ) )
% 3.00/1.64 = ( ^ [P5: extended_ereal > $o] :
% 3.00/1.64 ( ( P5 @ extend1289208545_ereal )
% 3.00/1.64 & ! [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
% 3.00/1.64 & ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_303_abs__ereal_Ocases,axiom,
% 3.00/1.64 ! [X2: extended_ereal] :
% 3.00/1.64 ( ! [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 != ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( X2
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( X2 = extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_304_ereal__ex__split,axiom,
% 3.00/1.64 ( ( ^ [P4: extended_ereal > $o] :
% 3.00/1.64 ? [X6: extended_ereal] : ( P4 @ X6 ) )
% 3.00/1.64 = ( ^ [P5: extended_ereal > $o] :
% 3.00/1.64 ( ( P5 @ extend1289208545_ereal )
% 3.00/1.64 | ? [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
% 3.00/1.64 | ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_305_ereal3__cases,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 3.00/1.64 ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ? [Ra: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ Ra ) )
% 3.00/1.64 => ! [Rb: real] :
% 3.00/1.64 ( Xb
% 3.00/1.64 != ( extended_ereal2 @ Rb ) ) ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ? [Ra: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ Ra ) )
% 3.00/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ? [Ra: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ Ra ) )
% 3.00/1.64 => ( Xb
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ! [Ra: real] :
% 3.00/1.64 ( Xb
% 3.00/1.64 != ( extended_ereal2 @ Ra ) ) ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ( Xb
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ! [Ra: real] :
% 3.00/1.64 ( Xb
% 3.00/1.64 != ( extended_ereal2 @ Ra ) ) ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Xb
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( ? [R2: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ! [Ra: real] :
% 3.00/1.64 ( Xb
% 3.00/1.64 != ( extended_ereal2 @ Ra ) ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( ? [R2: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( ? [R2: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( Xb
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ! [R2: real] :
% 3.00/1.64 ( Xb
% 3.00/1.64 != ( extended_ereal2 @ R2 ) ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ( Xb
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ! [R2: real] :
% 3.00/1.64 ( Xb
% 3.00/1.64 != ( extended_ereal2 @ R2 ) ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Xb
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ? [R2: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ! [Ra: real] :
% 3.00/1.64 ( Xb
% 3.00/1.64 != ( extended_ereal2 @ Ra ) ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ? [R2: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ? [R2: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( Xb
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ! [R2: real] :
% 3.00/1.64 ( Xb
% 3.00/1.64 != ( extended_ereal2 @ R2 ) ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ( Xb
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ! [R2: real] :
% 3.00/1.64 ( Xb
% 3.00/1.64 != ( extended_ereal2 @ R2 ) ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Xb != extend1289208545_ereal ) ) )
% 3.00/1.64 => ~ ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Xb
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_306_ereal2__cases,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Xa3: extended_ereal] :
% 3.00/1.64 ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ! [Ra: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 != ( extended_ereal2 @ Ra ) ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( Xa3 != extend1289208545_ereal ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( Xa3
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ! [R2: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 != ( extended_ereal2 @ R2 ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( Xa3 != extend1289208545_ereal ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( Xa3
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ! [R2: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 != ( extended_ereal2 @ R2 ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Xa3 != extend1289208545_ereal ) )
% 3.00/1.64 => ~ ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Xa3
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_307_ereal__cases,axiom,
% 3.00/1.64 ! [X2: extended_ereal] :
% 3.00/1.64 ( ! [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 != ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( X2 != extend1289208545_ereal )
% 3.00/1.64 => ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_308_MInfty__neq__ereal_I1_J,axiom,
% 3.00/1.64 ! [R: real] :
% 3.00/1.64 ( ( extended_ereal2 @ R )
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_309_ereal__top,axiom,
% 3.00/1.64 ! [X2: extended_ereal] :
% 3.00/1.64 ( ! [B6: real] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B6 ) @ X2 )
% 3.00/1.64 => ( X2 = extend1289208545_ereal ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_310_ereal__bot,axiom,
% 3.00/1.64 ! [X2: extended_ereal] :
% 3.00/1.64 ( ! [B6: real] : ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ B6 ) )
% 3.00/1.64 => ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_311_Lim__bounded__PInfty,axiom,
% 3.00/1.64 ! [F2: nat > extended_ereal,L: extended_ereal,B5: real] :
% 3.00/1.64 ( ( filter1531173832_ereal @ F2 @ ( topolo2140997059_ereal @ L ) @ at_top_nat )
% 3.00/1.64 => ( ! [N2: nat] : ( ord_le824540014_ereal @ ( F2 @ N2 ) @ ( extended_ereal2 @ B5 ) )
% 3.00/1.64 => ( L != extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_312_ereal__minus__real__tendsto__MInf,axiom,
% 3.00/1.64 ( filter1531173832_ereal
% 3.00/1.64 @ ^ [X: nat] : ( extended_ereal2 @ ( uminus_uminus_real @ ( semiri2110766477t_real @ X ) ) )
% 3.00/1.64 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 @ at_top_nat ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_313_ereal__PInfty__eq__plus,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( extend1289208545_ereal
% 3.00/1.64 = ( plus_p2118002693_ereal @ A @ B ) )
% 3.00/1.64 = ( ( A = extend1289208545_ereal )
% 3.00/1.64 | ( B = extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_314_ereal__plus__eq__PInfty,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ( plus_p2118002693_ereal @ A @ B )
% 3.00/1.64 = extend1289208545_ereal )
% 3.00/1.64 = ( ( A = extend1289208545_ereal )
% 3.00/1.64 | ( B = extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_315_ereal__MInfty__eq__plus,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ( uminus1208298309_ereal @ extend1289208545_ereal )
% 3.00/1.64 = ( plus_p2118002693_ereal @ A @ B ) )
% 3.00/1.64 = ( ( ( A
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 & ( B != extend1289208545_ereal ) )
% 3.00/1.64 | ( ( B
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 & ( A != extend1289208545_ereal ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_316_ereal__plus__eq__MInfty,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ( plus_p2118002693_ereal @ A @ B )
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 = ( ( ( A
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 | ( B
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 3.00/1.64 & ( A != extend1289208545_ereal )
% 3.00/1.64 & ( B != extend1289208545_ereal ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_317_plus__ereal_Osimps_I2_J,axiom,
% 3.00/1.64 ! [A: extended_ereal] :
% 3.00/1.64 ( ( plus_p2118002693_ereal @ extend1289208545_ereal @ A )
% 3.00/1.64 = extend1289208545_ereal ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_318_plus__ereal_Osimps_I3_J,axiom,
% 3.00/1.64 ! [A: extended_ereal] :
% 3.00/1.64 ( ( plus_p2118002693_ereal @ A @ extend1289208545_ereal )
% 3.00/1.64 = extend1289208545_ereal ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_319_plus__ereal_Osimps_I6_J,axiom,
% 3.00/1.64 ( ( plus_p2118002693_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_320_ereal__add__cancel__left,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( A
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( ( plus_p2118002693_ereal @ A @ B )
% 3.00/1.64 = ( plus_p2118002693_ereal @ A @ C ) )
% 3.00/1.64 = ( ( A = extend1289208545_ereal )
% 3.00/1.64 | ( B = C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_321_ereal__add__cancel__right,axiom,
% 3.00/1.64 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 3.00/1.64 ( ( A
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( ( plus_p2118002693_ereal @ B @ A )
% 3.00/1.64 = ( plus_p2118002693_ereal @ C @ A ) )
% 3.00/1.64 = ( ( A = extend1289208545_ereal )
% 3.00/1.64 | ( B = C ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_322_plus__ereal_Osimps_I5_J,axiom,
% 3.00/1.64 ! [P2: real] :
% 3.00/1.64 ( ( plus_p2118002693_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P2 ) )
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_323_plus__ereal_Osimps_I4_J,axiom,
% 3.00/1.64 ! [R: real] :
% 3.00/1.64 ( ( plus_p2118002693_ereal @ ( extended_ereal2 @ R ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_324_ereal__add__le__add__iff,axiom,
% 3.00/1.64 ! [C: extended_ereal,A: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ ( plus_p2118002693_ereal @ C @ A ) @ ( plus_p2118002693_ereal @ C @ B ) )
% 3.00/1.64 = ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 | ( C = extend1289208545_ereal )
% 3.00/1.64 | ( ( C
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 & ( A != extend1289208545_ereal )
% 3.00/1.64 & ( B != extend1289208545_ereal ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_325_ereal__add__le__add__iff2,axiom,
% 3.00/1.64 ! [A: extended_ereal,C: extended_ereal,B: extended_ereal] :
% 3.00/1.64 ( ( ord_le824540014_ereal @ ( plus_p2118002693_ereal @ A @ C ) @ ( plus_p2118002693_ereal @ B @ C ) )
% 3.00/1.64 = ( ( ord_le824540014_ereal @ A @ B )
% 3.00/1.64 | ( C = extend1289208545_ereal )
% 3.00/1.64 | ( ( C
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 & ( A != extend1289208545_ereal )
% 3.00/1.64 & ( B != extend1289208545_ereal ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_326_id__nat__ereal__tendsto__PInf,axiom,
% 3.00/1.64 ( filter1531173832_ereal
% 3.00/1.64 @ ^ [X: nat] : ( extended_ereal2 @ ( semiri2110766477t_real @ X ) )
% 3.00/1.64 @ ( topolo2140997059_ereal @ extend1289208545_ereal )
% 3.00/1.64 @ at_top_nat ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_327_ereal__liminf__add__mono,axiom,
% 3.00/1.64 ! [U: nat > extended_ereal,V: nat > extended_ereal] :
% 3.00/1.64 ( ~ ( ( ( ( liminf1045857232_ereal @ at_top_nat @ U )
% 3.00/1.64 = extend1289208545_ereal )
% 3.00/1.64 & ( ( liminf1045857232_ereal @ at_top_nat @ V )
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 3.00/1.64 | ( ( ( liminf1045857232_ereal @ at_top_nat @ U )
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 & ( ( liminf1045857232_ereal @ at_top_nat @ V )
% 3.00/1.64 = extend1289208545_ereal ) ) )
% 3.00/1.64 => ( ord_le824540014_ereal @ ( plus_p2118002693_ereal @ ( liminf1045857232_ereal @ at_top_nat @ U ) @ ( liminf1045857232_ereal @ at_top_nat @ V ) )
% 3.00/1.64 @ ( liminf1045857232_ereal @ at_top_nat
% 3.00/1.64 @ ^ [N: nat] : ( plus_p2118002693_ereal @ ( U @ N ) @ ( V @ N ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_328_nat__add__left__cancel__le,axiom,
% 3.00/1.64 ! [K: nat,M5: nat,N7: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M5 ) @ ( plus_plus_nat @ K @ N7 ) )
% 3.00/1.64 = ( ord_less_eq_nat @ M5 @ N7 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_329_nat__int__comparison_I3_J,axiom,
% 3.00/1.64 ( ord_less_eq_nat
% 3.00/1.64 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_330_add__leE,axiom,
% 3.00/1.64 ! [M5: nat,K: nat,N7: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N7 )
% 3.00/1.64 => ~ ( ( ord_less_eq_nat @ M5 @ N7 )
% 3.00/1.64 => ~ ( ord_less_eq_nat @ K @ N7 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_331_le__add1,axiom,
% 3.00/1.64 ! [N7: nat,M5: nat] : ( ord_less_eq_nat @ N7 @ ( plus_plus_nat @ N7 @ M5 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_332_le__add2,axiom,
% 3.00/1.64 ! [N7: nat,M5: nat] : ( ord_less_eq_nat @ N7 @ ( plus_plus_nat @ M5 @ N7 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_333_add__leD1,axiom,
% 3.00/1.64 ! [M5: nat,K: nat,N7: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N7 )
% 3.00/1.64 => ( ord_less_eq_nat @ M5 @ N7 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_334_add__leD2,axiom,
% 3.00/1.64 ! [M5: nat,K: nat,N7: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N7 )
% 3.00/1.64 => ( ord_less_eq_nat @ K @ N7 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_335_le__Suc__ex,axiom,
% 3.00/1.64 ! [K: nat,L: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ K @ L )
% 3.00/1.64 => ? [N2: nat] :
% 3.00/1.64 ( L
% 3.00/1.64 = ( plus_plus_nat @ K @ N2 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_336_add__le__mono,axiom,
% 3.00/1.64 ! [I: nat,J: nat,K: nat,L: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ I @ J )
% 3.00/1.64 => ( ( ord_less_eq_nat @ K @ L )
% 3.00/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_337_add__le__mono1,axiom,
% 3.00/1.64 ! [I: nat,J: nat,K: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ I @ J )
% 3.00/1.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_338_trans__le__add1,axiom,
% 3.00/1.64 ! [I: nat,J: nat,M5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ I @ J )
% 3.00/1.64 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M5 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_339_trans__le__add2,axiom,
% 3.00/1.64 ! [I: nat,J: nat,M5: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ I @ J )
% 3.00/1.64 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M5 @ J ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_340_nat__le__iff__add,axiom,
% 3.00/1.64 ( ord_less_eq_nat
% 3.00/1.64 = ( ^ [M4: nat,N: nat] :
% 3.00/1.64 ? [K2: nat] :
% 3.00/1.64 ( N
% 3.00/1.64 = ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_341_nat__leq__as__int,axiom,
% 3.00/1.64 ( ord_less_eq_nat
% 3.00/1.64 = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_342_plus__ereal_Osimps_I1_J,axiom,
% 3.00/1.64 ! [R: real,P2: real] :
% 3.00/1.64 ( ( plus_p2118002693_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
% 3.00/1.64 = ( extended_ereal2 @ ( plus_plus_real @ R @ P2 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_343_filterlim__uminus__at__bot__at__top,axiom,
% 3.00/1.64 filterlim_real_real @ uminus_uminus_real @ at_bot_real @ at_top_real ).
% 3.00/1.64
% 3.00/1.64 thf(fact_344_filterlim__uminus__at__top__at__bot,axiom,
% 3.00/1.64 filterlim_real_real @ uminus_uminus_real @ at_top_real @ at_bot_real ).
% 3.00/1.64
% 3.00/1.64 thf(fact_345_Nat_Oex__has__greatest__nat,axiom,
% 3.00/1.64 ! [P: nat > $o,K: nat,B: nat] :
% 3.00/1.64 ( ( P @ K )
% 3.00/1.64 => ( ! [Y3: nat] :
% 3.00/1.64 ( ( P @ Y3 )
% 3.00/1.64 => ( ord_less_eq_nat @ Y3 @ B ) )
% 3.00/1.64 => ? [X5: nat] :
% 3.00/1.64 ( ( P @ X5 )
% 3.00/1.64 & ! [Y6: nat] :
% 3.00/1.64 ( ( P @ Y6 )
% 3.00/1.64 => ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_346_nat__le__linear,axiom,
% 3.00/1.64 ! [M5: nat,N7: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M5 @ N7 )
% 3.00/1.64 | ( ord_less_eq_nat @ N7 @ M5 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_347_le__antisym,axiom,
% 3.00/1.64 ! [M5: nat,N7: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ M5 @ N7 )
% 3.00/1.64 => ( ( ord_less_eq_nat @ N7 @ M5 )
% 3.00/1.64 => ( M5 = N7 ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_348_eq__imp__le,axiom,
% 3.00/1.64 ! [M5: nat,N7: nat] :
% 3.00/1.64 ( ( M5 = N7 )
% 3.00/1.64 => ( ord_less_eq_nat @ M5 @ N7 ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_349_le__trans,axiom,
% 3.00/1.64 ! [I: nat,J: nat,K: nat] :
% 3.00/1.64 ( ( ord_less_eq_nat @ I @ J )
% 3.00/1.64 => ( ( ord_less_eq_nat @ J @ K )
% 3.00/1.64 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_350_le__refl,axiom,
% 3.00/1.64 ! [N7: nat] : ( ord_less_eq_nat @ N7 @ N7 ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_351_plus__ereal_Oelims,axiom,
% 3.00/1.64 ! [X2: extended_ereal,Xa3: extended_ereal,Y: extended_ereal] :
% 3.00/1.64 ( ( ( plus_p2118002693_ereal @ X2 @ Xa3 )
% 3.00/1.64 = Y )
% 3.00/1.64 => ( ! [R2: real] :
% 3.00/1.64 ( ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ! [P3: real] :
% 3.00/1.64 ( ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ P3 ) )
% 3.00/1.64 => ( Y
% 3.00/1.64 != ( extended_ereal2 @ ( plus_plus_real @ R2 @ P3 ) ) ) ) )
% 3.00/1.64 => ( ( ( X2 = extend1289208545_ereal )
% 3.00/1.64 => ( Y != extend1289208545_ereal ) )
% 3.00/1.64 => ( ( ( Xa3 = extend1289208545_ereal )
% 3.00/1.64 => ( Y != extend1289208545_ereal ) )
% 3.00/1.64 => ( ( ? [R2: real] :
% 3.00/1.64 ( X2
% 3.00/1.64 = ( extended_ereal2 @ R2 ) )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Y
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ( ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ? [P3: real] :
% 3.00/1.64 ( Xa3
% 3.00/1.64 = ( extended_ereal2 @ P3 ) )
% 3.00/1.64 => ( Y
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 3.00/1.64 => ~ ( ( X2
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( ( Xa3
% 3.00/1.64 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 3.00/1.64 => ( Y
% 3.00/1.64 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_352_filterlim__real__sequentially,axiom,
% 3.00/1.64 filterlim_nat_real @ semiri2110766477t_real @ at_top_real @ at_top_nat ).
% 3.00/1.64
% 3.00/1.64 thf(fact_353_int__ops_I5_J,axiom,
% 3.00/1.64 ! [A: nat,B: nat] :
% 3.00/1.64 ( ( semiri2019852685at_int @ ( plus_plus_nat @ A @ B ) )
% 3.00/1.64 = ( plus_plus_int @ ( semiri2019852685at_int @ A ) @ ( semiri2019852685at_int @ B ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(fact_354_nat__int__comparison_I1_J,axiom,
% 3.00/1.64 ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
% 3.00/1.64 = ( ^ [A3: nat,B2: nat] :
% 3.00/1.64 ( ( semiri2019852685at_int @ A3 )
% 3.00/1.64 = ( semiri2019852685at_int @ B2 ) ) ) ) ).
% 3.00/1.64
% 3.00/1.64 thf(conj_0,conjecture,
% 3.00/1.64 ( filter1531173832_ereal
% 3.00/1.64 @ ^ [I2: nat] : ( uminus1208298309_ereal @ ( f @ ( x @ I2 ) ) )
% 3.00/1.64 @ ( topolo2140997059_ereal @ ( uminus1208298309_ereal @ a2 ) )
% 3.00/1.64 @ at_top_nat ) ).
% 3.00/1.64 ------------------------
% 3.00/1.68 % (10227)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_9897 for (2999ds/4Mi)
% 3.00/1.68 % (10231)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_9897 for (2999ds/275Mi)
% 3.00/1.68 % (10227)Instruction limit reached!
% 3.00/1.68 % (10227)------------------------------
% 3.00/1.68 % (10227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (10227)Termination reason: Unknown
% 3.00/1.68 % (10227)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (10227)Memory used [KB]: 1663
% 3.00/1.68 % (10227)Time elapsed: 0.003 s
% 3.00/1.68 % (10227)Instructions burned: 5 (million)
% 3.00/1.68 % (10227)------------------------------
% 3.00/1.68 % (10227)------------------------------
% 3.00/1.69 % (10226)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9897 for (2999ds/183Mi)
% 3.00/1.69 % (10228)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_9897 for (2999ds/27Mi)
% 3.00/1.69 % (10229)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9897 for (2999ds/2Mi)
% 3.00/1.69 % (10230)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_9897 for (2999ds/2Mi)
% 3.00/1.69 % (10232)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9897 for (2999ds/18Mi)
% 3.00/1.69 % (10229)Instruction limit reached!
% 3.00/1.69 % (10229)------------------------------
% 3.00/1.69 % (10229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.69 % (10229)Termination reason: Unknown
% 3.00/1.69 % (10229)Termination phase: shuffling
% 3.00/1.69
% 3.00/1.69 % (10229)Memory used [KB]: 1535
% 3.00/1.69 % (10229)Time elapsed: 0.004 s
% 3.00/1.69 % (10229)Instructions burned: 2 (million)
% 3.00/1.69 % (10229)------------------------------
% 3.00/1.69 % (10229)------------------------------
% 3.00/1.69 % (10230)Instruction limit reached!
% 3.00/1.69 % (10230)------------------------------
% 3.00/1.69 % (10230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.69 % (10230)Termination reason: Unknown
% 3.00/1.69 % (10230)Termination phase: shuffling
% 3.00/1.69
% 3.00/1.69 % (10230)Memory used [KB]: 1535
% 3.00/1.69 % (10230)Time elapsed: 0.004 s
% 3.00/1.69 % (10230)Instructions burned: 3 (million)
% 3.00/1.69 % (10230)------------------------------
% 3.00/1.69 % (10230)------------------------------
% 3.00/1.69 % (10233)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9897 for (2999ds/3Mi)
% 3.00/1.69 % (10233)Instruction limit reached!
% 3.00/1.69 % (10233)------------------------------
% 3.00/1.69 % (10233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.69 % (10233)Termination reason: Unknown
% 3.00/1.69 % (10233)Termination phase: shuffling
% 3.00/1.69
% 3.00/1.69 % (10233)Memory used [KB]: 1535
% 3.00/1.69 % (10233)Time elapsed: 0.002 s
% 3.00/1.69 % (10233)Instructions burned: 3 (million)
% 3.00/1.69 % (10233)------------------------------
% 3.00/1.69 % (10233)------------------------------
% 3.04/1.70 % (10232)Instruction limit reached!
% 3.04/1.70 % (10232)------------------------------
% 3.04/1.70 % (10232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.70 % (10232)Termination reason: Unknown
% 3.04/1.70 % (10232)Termination phase: shuffling
% 3.04/1.70
% 3.04/1.70 % (10232)Memory used [KB]: 1918
% 3.04/1.70 % (10232)Time elapsed: 0.014 s
% 3.04/1.70 % (10232)Instructions burned: 19 (million)
% 3.04/1.70 % (10232)------------------------------
% 3.04/1.70 % (10232)------------------------------
% 3.04/1.70 % (10236)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9897 for (2999ds/3Mi)
% 3.04/1.70 % (10236)Instruction limit reached!
% 3.04/1.70 % (10236)------------------------------
% 3.04/1.70 % (10236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.70 % (10236)Termination reason: Unknown
% 3.04/1.70 % (10236)Termination phase: shuffling
% 3.04/1.70
% 3.04/1.70 % (10236)Memory used [KB]: 1535
% 3.04/1.70 % (10236)Time elapsed: 0.002 s
% 3.04/1.70 % (10236)Instructions burned: 3 (million)
% 3.04/1.70 % (10236)------------------------------
% 3.04/1.70 % (10236)------------------------------
% 3.04/1.70 % (10228)Instruction limit reached!
% 3.04/1.70 % (10228)------------------------------
% 3.04/1.70 % (10228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.70 % (10228)Termination reason: Unknown
% 3.04/1.70 % (10228)Termination phase: shuffling
% 3.04/1.70
% 3.04/1.70 % (10228)Memory used [KB]: 2046
% 3.04/1.70 % (10228)Time elapsed: 0.019 s
% 3.04/1.70 % (10228)Instructions burned: 28 (million)
% 3.04/1.70 % (10228)------------------------------
% 3.04/1.70 % (10228)------------------------------
% 3.04/1.71 % (10234)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_9897 for (2999ds/37Mi)
% 3.04/1.71 % (10235)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_9897 for (2999ds/15Mi)
% 3.04/1.71 % (10231)First to succeed.
% 3.04/1.71 % (10231)Refutation found. Thanks to Tanya!
% 3.04/1.71 % SZS status Theorem for DTF2THF_9897
% 3.04/1.71 % SZS output start Proof for DTF2THF_9897
% 3.04/1.71 thf(type_def_5, type, extended_ereal: $tType).
% 3.04/1.71 thf(type_def_6, type, real: $tType).
% 3.04/1.71 thf(type_def_8, type, filter_real: $tType).
% 3.04/1.71 thf(type_def_9, type, filter_nat: $tType).
% 3.04/1.71 thf(type_def_10, type, nat: $tType).
% 3.04/1.71 thf(type_def_11, type, filter2049122004_ereal: $tType).
% 3.04/1.71 thf(type_def_12, type, int: $tType).
% 3.04/1.71 thf(type_def_13, type, filter_int: $tType).
% 3.04/1.71 thf(type_def_14, type, a: $tType).
% 3.04/1.71 thf(type_def_15, type, filter_a: $tType).
% 3.04/1.71 thf(type_def_16, type, set_real: $tType).
% 3.04/1.71 thf(type_def_17, type, set_Extended_ereal: $tType).
% 3.04/1.71 thf(func_def_0, type, filter2049122004_ereal: $tType).
% 3.04/1.71 thf(func_def_1, type, set_Extended_ereal: $tType).
% 3.04/1.71 thf(func_def_2, type, filter_real: $tType).
% 3.04/1.71 thf(func_def_3, type, filter_nat: $tType).
% 3.04/1.71 thf(func_def_4, type, filter_int: $tType).
% 3.04/1.71 thf(func_def_5, type, set_real: $tType).
% 3.04/1.71 thf(func_def_6, type, filter_a: $tType).
% 3.04/1.71 thf(func_def_7, type, extended_ereal: $tType).
% 3.04/1.71 thf(func_def_8, type, real: $tType).
% 3.04/1.71 thf(func_def_9, type, nat: $tType).
% 3.04/1.71 thf(func_def_10, type, int: $tType).
% 3.04/1.71 thf(func_def_11, type, a: $tType).
% 3.04/1.71 thf(func_def_12, type, extend1289208545_ereal: extended_ereal).
% 3.04/1.71 thf(func_def_13, type, extended_ereal2: real > extended_ereal).
% 3.04/1.71 thf(func_def_14, type, at_bot_real: filter_real).
% 3.04/1.71 thf(func_def_15, type, at_top_nat: filter_nat).
% 3.04/1.71 thf(func_def_16, type, at_top_real: filter_real).
% 3.04/1.71 thf(func_def_17, type, filter1531173832_ereal: (nat > extended_ereal) > filter2049122004_ereal > filter_nat > $o).
% 3.04/1.71 thf(func_def_18, type, filterlim_nat_int: (nat > int) > filter_int > filter_nat > $o).
% 3.04/1.71 thf(func_def_19, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 3.04/1.71 thf(func_def_20, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 3.04/1.71 thf(func_def_21, type, filterlim_nat_a: (nat > a) > filter_a > filter_nat > $o).
% 3.04/1.71 thf(func_def_22, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 3.04/1.71 thf(func_def_23, type, comp_E1308517939al_nat: (extended_ereal > extended_ereal) > (nat > extended_ereal) > nat > extended_ereal).
% 3.04/1.71 thf(func_def_24, type, comp_E489644891real_a: (extended_ereal > extended_ereal) > (a > extended_ereal) > a > extended_ereal).
% 3.04/1.71 thf(func_def_25, type, comp_E1436437929nt_nat: (extended_ereal > int) > (nat > extended_ereal) > nat > int).
% 3.04/1.71 thf(func_def_26, type, comp_E1523169101at_nat: (extended_ereal > nat) > (nat > extended_ereal) > nat > nat).
% 3.04/1.71 thf(func_def_27, type, comp_E1477338153al_nat: (extended_ereal > real) > (nat > extended_ereal) > nat > real).
% 3.04/1.71 thf(func_def_28, type, comp_n1096781355al_nat: (nat > extended_ereal) > (nat > nat) > nat > extended_ereal).
% 3.04/1.71 thf(func_def_29, type, comp_nat_a_nat: (nat > a) > (nat > nat) > nat > a).
% 3.04/1.71 thf(func_def_30, type, comp_r1410008527al_nat: (real > extended_ereal) > (nat > real) > nat > extended_ereal).
% 3.04/1.71 thf(func_def_31, type, comp_real_int_nat: (real > int) > (nat > real) > nat > int).
% 3.04/1.71 thf(func_def_32, type, comp_real_nat_nat: (real > nat) > (nat > real) > nat > nat).
% 3.04/1.71 thf(func_def_33, type, comp_a1112243075al_nat: (a > extended_ereal) > (nat > a) > nat > extended_ereal).
% 3.04/1.71 thf(func_def_34, type, comp_a780206603real_a: (a > extended_ereal) > (a > a) > a > extended_ereal).
% 3.04/1.71 thf(func_def_35, type, comp_a_int_nat: (a > int) > (nat > a) > nat > int).
% 3.04/1.71 thf(func_def_36, type, comp_a_nat_nat: (a > nat) > (nat > a) > nat > nat).
% 3.04/1.71 thf(func_def_37, type, comp_a_real_nat: (a > real) > (nat > a) > nat > real).
% 3.04/1.71 thf(func_def_38, type, comp_a_a_nat: (a > a) > (nat > a) > nat > a).
% 3.04/1.71 thf(func_def_39, type, plus_p2118002693_ereal: extended_ereal > extended_ereal > extended_ereal).
% 3.04/1.71 thf(func_def_40, type, plus_plus_int: int > int > int).
% 3.04/1.71 thf(func_def_41, type, plus_plus_nat: nat > nat > nat).
% 3.04/1.71 thf(func_def_42, type, plus_plus_real: real > real > real).
% 3.04/1.71 thf(func_def_43, type, uminus1208298309_ereal: extended_ereal > extended_ereal).
% 3.04/1.71 thf(func_def_44, type, uminus_uminus_int: int > int).
% 3.04/1.71 thf(func_def_45, type, uminus_uminus_real: real > real).
% 3.04/1.71 thf(func_def_46, type, uniq_Extended_ereal: (extended_ereal > $o) > $o).
% 3.04/1.71 thf(func_def_47, type, uniq_real: (real > $o) > $o).
% 3.04/1.71 thf(func_def_48, type, uniq_a: (a > $o) > $o).
% 3.04/1.71 thf(func_def_49, type, liminf1045857232_ereal: filter_nat > (nat > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_50, type, lower_1087098792_ereal: extended_ereal > (extended_ereal > extended_ereal) > $o).
% 3.04/1.71 thf(func_def_51, type, lower_48196818al_int: extended_ereal > (extended_ereal > int) > $o).
% 3.04/1.71 thf(func_def_52, type, lower_1558406774al_nat: extended_ereal > (extended_ereal > nat) > $o).
% 3.04/1.71 thf(func_def_53, type, lower_1165973074l_real: extended_ereal > (extended_ereal > real) > $o).
% 3.04/1.71 thf(func_def_54, type, lower_551915512_ereal: real > (real > extended_ereal) > $o).
% 3.04/1.71 thf(func_def_55, type, lower_153911426al_int: real > (real > int) > $o).
% 3.04/1.71 thf(func_def_56, type, lower_1664121382al_nat: real > (real > nat) > $o).
% 3.04/1.71 thf(func_def_57, type, lower_191460856_ereal: a > (a > extended_ereal) > $o).
% 3.04/1.71 thf(func_def_58, type, lower_956963458_a_int: a > (a > int) > $o).
% 3.04/1.71 thf(func_def_59, type, lower_319689766_a_nat: a > (a > nat) > $o).
% 3.04/1.71 thf(func_def_60, type, lower_231615490a_real: a > (a > real) > $o).
% 3.04/1.71 thf(func_def_61, type, lower_1071158961_ereal: extended_ereal > (extended_ereal > extended_ereal) > $o).
% 3.04/1.71 thf(func_def_62, type, lower_637387785al_int: extended_ereal > (extended_ereal > int) > $o).
% 3.04/1.71 thf(func_def_63, type, lower_114093al_nat: extended_ereal > (extended_ereal > nat) > $o).
% 3.04/1.71 thf(func_def_64, type, lower_737640969l_real: extended_ereal > (extended_ereal > real) > $o).
% 3.04/1.71 thf(func_def_65, type, lower_1075504779al_int: real > (real > int) > $o).
% 3.04/1.71 thf(func_def_66, type, lower_438231087al_nat: real > (real > nat) > $o).
% 3.04/1.71 thf(func_def_67, type, lower_534855297_ereal: a > (a > extended_ereal) > $o).
% 3.04/1.71 thf(func_def_68, type, lower_1672990777_a_int: a > (a > int) > $o).
% 3.04/1.71 thf(func_def_69, type, lower_1035717085_a_nat: a > (a > nat) > $o).
% 3.04/1.71 thf(func_def_70, type, lower_755922489a_real: a > (a > real) > $o).
% 3.04/1.71 thf(func_def_71, type, semiri2019852685at_int: nat > int).
% 3.04/1.71 thf(func_def_72, type, semiri2110766477t_real: nat > real).
% 3.04/1.71 thf(func_def_73, type, ord_le824540014_ereal: extended_ereal > extended_ereal > $o).
% 3.04/1.71 thf(func_def_74, type, ord_le1745708096er_nat: filter_nat > filter_nat > $o).
% 3.04/1.71 thf(func_def_75, type, ord_le132810396r_real: filter_real > filter_real > $o).
% 3.04/1.71 thf(func_def_76, type, ord_less_eq_int: int > int > $o).
% 3.04/1.71 thf(func_def_77, type, ord_less_eq_nat: nat > nat > $o).
% 3.04/1.71 thf(func_def_78, type, ord_less_eq_real: real > real > $o).
% 3.04/1.71 thf(func_def_79, type, collect_real: (real > $o) > set_real).
% 3.04/1.71 thf(func_def_80, type, topolo1069469409_ereal: (nat > extended_ereal) > $o).
% 3.04/1.71 thf(func_def_81, type, topolo411883481eq_int: (nat > int) > $o).
% 3.04/1.71 thf(func_def_82, type, topolo1922093437eq_nat: (nat > nat) > $o).
% 3.04/1.71 thf(func_def_83, type, topolo144289241q_real: (nat > real) > $o).
% 3.04/1.71 thf(func_def_84, type, topolo2140997059_ereal: extended_ereal > filter2049122004_ereal).
% 3.04/1.71 thf(func_def_85, type, topolo54776183ds_int: int > filter_int).
% 3.04/1.71 thf(func_def_86, type, topolo1564986139ds_nat: nat > filter_nat).
% 3.04/1.71 thf(func_def_87, type, topolo1664202871s_real: real > filter_real).
% 3.04/1.71 thf(func_def_88, type, topolo705128563nhds_a: a > filter_a).
% 3.04/1.71 thf(func_def_89, type, member1900190071_ereal: extended_ereal > set_Extended_ereal > $o).
% 3.04/1.71 thf(func_def_90, type, member_real: real > set_real > $o).
% 3.04/1.71 thf(func_def_91, type, a2: extended_ereal).
% 3.04/1.71 thf(func_def_92, type, f: a > extended_ereal).
% 3.04/1.71 thf(func_def_93, type, x0: a).
% 3.04/1.71 thf(func_def_94, type, x: nat > a).
% 3.04/1.71 thf(func_def_110, type, sP0: nat > nat > nat > $o).
% 3.04/1.71 thf(func_def_111, type, sP1: nat > nat > nat > $o).
% 3.04/1.71 thf(func_def_112, type, sP2: nat > nat > nat > $o).
% 3.04/1.71 thf(func_def_113, type, sP3: extended_ereal > extended_ereal > extended_ereal > $o).
% 3.04/1.71 thf(func_def_114, type, sP4: extended_ereal > extended_ereal > extended_ereal > $o).
% 3.04/1.71 thf(func_def_115, type, sP5: extended_ereal > extended_ereal > extended_ereal > $o).
% 3.04/1.71 thf(func_def_116, type, sK6: (nat > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_117, type, sK7: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_118, type, sK8: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_119, type, sK9: (nat > nat) > nat).
% 3.04/1.71 thf(func_def_120, type, sK10: (nat > nat) > nat).
% 3.04/1.71 thf(func_def_121, type, sK11: nat > (nat > extended_ereal) > (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_122, type, sK12: (extended_ereal > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_123, type, sK13: (extended_ereal > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_124, type, sK14: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_125, type, sK15: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_126, type, sK16: (nat > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_127, type, sK17: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_128, type, sK18: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_129, type, sK19: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_130, type, sK20: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_131, type, sK21: (nat > nat) > nat).
% 3.04/1.71 thf(func_def_132, type, sK22: (nat > nat) > nat).
% 3.04/1.71 thf(func_def_133, type, sK23: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_134, type, sK24: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_135, type, sK25: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_136, type, sK26: (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_137, type, sK27: nat > extended_ereal > (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_138, type, sK28: (nat > nat > $o) > nat).
% 3.04/1.71 thf(func_def_139, type, sK29: (nat > nat > $o) > nat).
% 3.04/1.71 thf(func_def_140, type, sK30: (nat > nat > $o) > nat).
% 3.04/1.71 thf(func_def_141, type, sK31: (nat > nat > $o) > nat).
% 3.04/1.71 thf(func_def_142, type, sK32: (nat > nat) > nat).
% 3.04/1.71 thf(func_def_143, type, sK33: (nat > nat) > nat).
% 3.04/1.71 thf(func_def_144, type, sK34: (extended_ereal > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_145, type, sK35: (extended_ereal > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_146, type, sK36: (extended_ereal > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_147, type, sK37: (extended_ereal > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_148, type, sK38: (extended_ereal > nat) > extended_ereal).
% 3.04/1.71 thf(func_def_149, type, sK39: (extended_ereal > nat) > extended_ereal).
% 3.04/1.71 thf(func_def_150, type, sK40: extended_ereal > nat > (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_151, type, sK41: nat > (nat > extended_ereal) > (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_152, type, sK42: nat > extended_ereal > (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_153, type, sK43: (extended_ereal > nat) > extended_ereal).
% 3.04/1.71 thf(func_def_154, type, sK44: (extended_ereal > nat) > extended_ereal).
% 3.04/1.71 thf(func_def_155, type, sK45: (extended_ereal > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_156, type, sK46: (extended_ereal > extended_ereal) > extended_ereal).
% 3.04/1.71 thf(func_def_157, type, sK47: (nat > nat) > nat).
% 3.04/1.71 thf(func_def_158, type, sK48: (nat > nat) > nat).
% 3.04/1.71 thf(func_def_159, type, sK49: nat > extended_ereal > (nat > extended_ereal) > nat).
% 3.04/1.71 thf(func_def_160, type, sK50: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 3.04/1.71 thf(func_def_161, type, sK51: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 3.04/1.71 thf(func_def_162, type, sK52: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 3.04/1.71 thf(func_def_163, type, sK53: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 3.04/1.71 thf(func_def_164, type, sK54: (nat > $o) > nat > nat).
% 3.04/1.71 thf(func_def_165, type, sK55: (nat > $o) > nat).
% 3.04/1.71 thf(func_def_166, type, sK56: (extended_ereal > nat) > extended_ereal).
% 3.04/1.71 thf(func_def_167, type, sK57: (extended_ereal > nat) > extended_ereal).
% 3.04/1.71 thf(func_def_168, type, sK58: (extended_ereal > nat) > extended_ereal).
% 3.04/1.71 thf(func_def_169, type, sK59: (extended_ereal > nat) > extended_ereal).
% 3.04/1.71 thf(func_def_171, type, ph61: !>[X0: $tType]:(X0)).
% 3.04/1.71 thf(f1414,plain,(
% 3.04/1.71 $false),
% 3.04/1.71 inference(subsumption_resolution,[status(thm)],[f1413,f1411])).
% 3.04/1.71 thf(f1411,plain,(
% 3.04/1.71 ($true = (filter1531173832_ereal @ (^[Y0 : nat]: (f @ (x @ Y0))) @ (topolo2140997059_ereal @ a2) @ at_top_nat))),
% 3.04/1.71 inference(beta_eta_normalization,[status(thm)],[f1361])).
% 3.04/1.71 thf(f1361,plain,(
% 3.04/1.71 ($true = (filter1531173832_ereal @ ((^[Y0 : a > extended_ereal]: ((^[Y1 : nat > a]: ((^[Y2 : nat]: (Y0 @ (Y1 @ Y2))))))) @ f @ x) @ (topolo2140997059_ereal @ a2) @ at_top_nat))),
% 3.04/1.71 inference(definition_unfolding,[status(thm)],[f1279,f1213])).
% 3.04/1.71 thf(f1213,plain,(
% 3.04/1.71 (comp_a1112243075al_nat = (^[Y0 : a > extended_ereal]: ((^[Y1 : nat > a]: ((^[Y2 : nat]: (Y0 @ (Y1 @ Y2))))))))),
% 3.04/1.71 inference(cnf_transformation,[status(thm)],[f752])).
% 3.04/1.71 thf(f752,plain,(
% 3.04/1.71 (comp_a1112243075al_nat = (^[Y0 : a > extended_ereal]: ((^[Y1 : nat > a]: ((^[Y2 : nat]: (Y0 @ (Y1 @ Y2))))))))),
% 3.04/1.71 inference(fool_elimination,[status(thm)],[f751])).
% 3.04/1.71 thf(f751,plain,(
% 3.04/1.71 ((^[X0 : a > extended_ereal, X1 : nat > a, X2 : nat] : (X0 @ (X1 @ X2))) = comp_a1112243075al_nat)),
% 3.04/1.71 inference(rectify,[status(thm)],[f37])).
% 3.04/1.71 thf(f37,axiom,(
% 3.04/1.71 ((^[X13 : a > extended_ereal, X14 : nat > a, X1 : nat] : (X13 @ (X14 @ X1))) = comp_a1112243075al_nat)),
% 3.04/1.71 file('/export/starexec/sandbox/tmp/tmp.z9pBe0D4OI/DTF2THF_9897.p',fact_36_comp__apply)).
% 3.04/1.71 thf(f1279,plain,(
% 3.04/1.71 ((filter1531173832_ereal @ (comp_a1112243075al_nat @ f @ x) @ (topolo2140997059_ereal @ a2) @ at_top_nat) = $true)),
% 3.04/1.71 inference(cnf_transformation,[status(thm)],[f894])).
% 3.04/1.71 thf(f894,plain,(
% 3.04/1.71 ((filter1531173832_ereal @ (comp_a1112243075al_nat @ f @ x) @ (topolo2140997059_ereal @ a2) @ at_top_nat) = $true)),
% 3.04/1.71 inference(fool_elimination,[status(thm)],[f893])).
% 3.04/1.71 thf(f893,plain,(
% 3.04/1.71 (filter1531173832_ereal @ (comp_a1112243075al_nat @ f @ x) @ (topolo2140997059_ereal @ a2) @ at_top_nat)),
% 3.04/1.71 inference(rectify,[status(thm)],[f2])).
% 3.04/1.71 thf(f2,axiom,(
% 3.04/1.71 (filter1531173832_ereal @ (comp_a1112243075al_nat @ f @ x) @ (topolo2140997059_ereal @ a2) @ at_top_nat)),
% 3.04/1.71 file('/export/starexec/sandbox/tmp/tmp.z9pBe0D4OI/DTF2THF_9897.p',fact_1_x__def_I2_J)).
% 3.04/1.71 thf(f1413,plain,(
% 3.04/1.71 ($true != (filter1531173832_ereal @ (^[Y0 : nat]: (f @ (x @ Y0))) @ (topolo2140997059_ereal @ a2) @ at_top_nat))),
% 3.04/1.71 inference(trivial_inequality_removal,[status(thm)],[f1412])).
% 3.04/1.71 thf(f1412,plain,(
% 3.04/1.71 ($true != $true) | ($true != (filter1531173832_ereal @ (^[Y0 : nat]: (f @ (x @ Y0))) @ (topolo2140997059_ereal @ a2) @ at_top_nat))),
% 3.04/1.71 inference(superposition,[status(thm)],[f1261,f1406])).
% 3.04/1.71 thf(f1406,plain,(
% 3.04/1.71 ( ! [X0 : filter_nat] : (($true = (filter1531173832_ereal @ (^[Y0 : nat]: (uminus1208298309_ereal @ (f @ (x @ Y0)))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ X0)) | ((filter1531173832_ereal @ (^[Y0 : nat]: (f @ (x @ Y0))) @ (topolo2140997059_ereal @ a2) @ X0) != $true)) )),
% 3.04/1.71 inference(beta_eta_normalization,[status(thm)],[f1362])).
% 3.04/1.71 thf(f1362,plain,(
% 3.04/1.71 ( ! [X0 : filter_nat] : (((filter1531173832_ereal @ (^[Y0 : nat]: (uminus1208298309_ereal @ ((^[Y1 : a > extended_ereal]: ((^[Y2 : nat > a]: ((^[Y3 : nat]: (Y1 @ (Y2 @ Y3))))))) @ f @ x @ Y0))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ X0) = $true) | ((filter1531173832_ereal @ ((^[Y0 : a > extended_ereal]: ((^[Y1 : nat > a]: ((^[Y2 : nat]: (Y0 @ (Y1 @ Y2))))))) @ f @ x) @ (topolo2140997059_ereal @ a2) @ X0) != $true)) )),
% 3.04/1.71 inference(definition_unfolding,[status(thm)],[f1280,f1213,f1213])).
% 3.04/1.71 thf(f1280,plain,(
% 3.04/1.71 ( ! [X0 : filter_nat] : (((filter1531173832_ereal @ (comp_a1112243075al_nat @ f @ x) @ (topolo2140997059_ereal @ a2) @ X0) != $true) | ((filter1531173832_ereal @ (^[Y0 : nat]: (uminus1208298309_ereal @ (comp_a1112243075al_nat @ f @ x @ Y0))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ X0) = $true)) )),
% 3.04/1.71 inference(cnf_transformation,[status(thm)],[f1060])).
% 3.04/1.71 thf(f1060,plain,(
% 3.04/1.71 ! [X0 : filter_nat] : (((filter1531173832_ereal @ (comp_a1112243075al_nat @ f @ x) @ (topolo2140997059_ereal @ a2) @ X0) != $true) | ((filter1531173832_ereal @ (^[Y0 : nat]: (uminus1208298309_ereal @ (comp_a1112243075al_nat @ f @ x @ Y0))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ X0) = $true))),
% 3.04/1.71 inference(ennf_transformation,[status(thm)],[f554])).
% 3.04/1.71 thf(f554,plain,(
% 3.04/1.71 ! [X0 : filter_nat] : (((filter1531173832_ereal @ (comp_a1112243075al_nat @ f @ x) @ (topolo2140997059_ereal @ a2) @ X0) = $true) => ((filter1531173832_ereal @ (^[Y0 : nat]: (uminus1208298309_ereal @ (comp_a1112243075al_nat @ f @ x @ Y0))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ X0) = $true))),
% 3.04/1.71 inference(fool_elimination,[status(thm)],[f553])).
% 3.04/1.71 thf(f553,plain,(
% 3.04/1.71 ! [X0 : filter_nat] : ((filter1531173832_ereal @ (comp_a1112243075al_nat @ f @ x) @ (topolo2140997059_ereal @ a2) @ X0) => (filter1531173832_ereal @ (^[X1 : nat] : (uminus1208298309_ereal @ (comp_a1112243075al_nat @ f @ x @ X1))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ X0))),
% 3.04/1.71 inference(rectify,[status(thm)],[f3])).
% 3.04/1.71 thf(f3,axiom,(
% 3.04/1.71 ! [X0 : filter_nat] : ((filter1531173832_ereal @ (comp_a1112243075al_nat @ f @ x) @ (topolo2140997059_ereal @ a2) @ X0) => (filter1531173832_ereal @ (^[X1 : nat] : (uminus1208298309_ereal @ (comp_a1112243075al_nat @ f @ x @ X1))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ X0))),
% 3.04/1.71 file('/export/starexec/sandbox/tmp/tmp.z9pBe0D4OI/DTF2THF_9897.p',fact_2__092_060open_062_092_060And_062F_O_A_I_If_A_092_060circ_062_Ax_J_A_092_060longlongrightarrow_062_AA_J_AF_A_092_060Longrightarrow_062_A_I_I_092_060lambda_062xa_O_A_N_A_If_A_092_060circ_062_Ax_J_Axa_J_A_092_060longlongrightarrow_062_A_N_AA_J_AF_092_060close_062)).
% 3.04/1.71 thf(f1261,plain,(
% 3.04/1.71 ($true != (filter1531173832_ereal @ (^[Y0 : nat]: (uminus1208298309_ereal @ (f @ (x @ Y0)))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ at_top_nat))),
% 3.04/1.71 inference(cnf_transformation,[status(thm)],[f991])).
% 3.04/1.71 thf(f991,plain,(
% 3.04/1.71 ($true != (filter1531173832_ereal @ (^[Y0 : nat]: (uminus1208298309_ereal @ (f @ (x @ Y0)))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ at_top_nat))),
% 3.04/1.71 inference(flattening,[status(thm)],[f782])).
% 3.04/1.71 thf(f782,plain,(
% 3.04/1.71 ~($true = (filter1531173832_ereal @ (^[Y0 : nat]: (uminus1208298309_ereal @ (f @ (x @ Y0)))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ at_top_nat))),
% 3.04/1.71 inference(fool_elimination,[status(thm)],[f781])).
% 3.04/1.71 thf(f781,plain,(
% 3.04/1.71 ~(filter1531173832_ereal @ (^[X0 : nat] : (uminus1208298309_ereal @ (f @ (x @ X0)))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ at_top_nat)),
% 3.04/1.71 inference(rectify,[status(thm)],[f357])).
% 3.04/1.71 thf(f357,negated_conjecture,(
% 3.04/1.71 ~(filter1531173832_ereal @ (^[X85 : nat] : (uminus1208298309_ereal @ (f @ (x @ X85)))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ at_top_nat)),
% 3.04/1.71 inference(negated_conjecture,[status(cth)],[f356])).
% 3.04/1.71 thf(f356,conjecture,(
% 3.04/1.71 (filter1531173832_ereal @ (^[X85 : nat] : (uminus1208298309_ereal @ (f @ (x @ X85)))) @ (topolo2140997059_ereal @ (uminus1208298309_ereal @ a2)) @ at_top_nat)),
% 3.04/1.71 file('/export/starexec/sandbox/tmp/tmp.z9pBe0D4OI/DTF2THF_9897.p',conj_0)).
% 3.04/1.71 % SZS output end Proof for DTF2THF_9897
% 3.04/1.71 % (10231)------------------------------
% 3.04/1.71 % (10231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.04/1.71 % (10231)Termination reason: Refutation
% 3.04/1.71
% 3.04/1.71 % (10231)Memory used [KB]: 7164
% 3.04/1.71 % (10231)Time elapsed: 0.031 s
% 3.04/1.71 % (10231)Instructions burned: 98 (million)
% 3.04/1.71 % (10231)------------------------------
% 3.04/1.71 % (10231)------------------------------
% 3.04/1.71 % (10225)Success in time 0.066 s
%------------------------------------------------------------------------------