%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP114^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:40 AM UTC 2026
% Result : Theorem 2.95s 1.82s
% Output : Refutation 2.95s
% Verified :
% SZS Type : Refutation
% Derivation depth : 6
% Number of leaves : 2
% Syntax : Number of formulae : 11 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 21 ( 5 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 45 ( 5 ~; 0 |; 0 &; 40 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 3 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 84 ( 81 usr; 10 con; 0-3 aty)
% Number of variables : 4 ( 0 ^; 4 !; 0 ?; 4 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_6,type,
extended_enat: $tType ).
thf(type_def_8,type,
extended_ereal: $tType ).
thf(type_def_9,type,
real: $tType ).
thf(type_def_10,type,
a: $tType ).
thf(type_def_11,type,
num: $tType ).
thf(type_def_12,type,
set_Extended_ereal: $tType ).
thf(func_def_0,type,
set_Extended_ereal: $tType ).
thf(func_def_1,type,
extended_ereal: $tType ).
thf(func_def_2,type,
extended_enat: $tType ).
thf(func_def_3,type,
real: $tType ).
thf(func_def_4,type,
num: $tType ).
thf(func_def_5,type,
nat: $tType ).
thf(func_def_6,type,
a: $tType ).
thf(func_def_7,type,
extended_enat2: nat > extended_enat ).
thf(func_def_8,type,
extend1289208545_ereal: extended_ereal ).
thf(func_def_9,type,
extended_the_enat: extended_enat > nat ).
thf(func_def_10,type,
extended_MInfty: extended_ereal ).
thf(func_def_11,type,
extended_PInfty: extended_ereal ).
thf(func_def_12,type,
extended_ereal2: real > extended_ereal ).
thf(func_def_13,type,
extended_size_ereal: extended_ereal > nat ).
thf(func_def_14,type,
extend1771934483f_enat: extended_enat > extended_ereal ).
thf(func_def_15,type,
extend1716541707_ereal: extended_ereal > real ).
thf(func_def_16,type,
abs_ab1260901297_ereal: extended_ereal > extended_ereal ).
thf(func_def_17,type,
uminus1208298309_ereal: extended_ereal > extended_ereal ).
thf(func_def_18,type,
uminus_uminus_real: real > real ).
thf(func_def_19,type,
zero_z491942557d_enat: extended_enat ).
thf(func_def_20,type,
zero_z163181189_ereal: extended_ereal ).
thf(func_def_21,type,
zero_zero_nat: nat ).
thf(func_def_22,type,
zero_zero_real: real ).
thf(func_def_23,type,
lower_1616484581_ereal: ( a > extended_ereal ) > $o ).
thf(func_def_24,type,
lower_881475195hull_a: ( a > extended_ereal ) > a > extended_ereal ).
thf(func_def_25,type,
numera280919179d_enat: num > extended_enat ).
thf(func_def_26,type,
numera1793320307_ereal: num > extended_ereal ).
thf(func_def_27,type,
numeral_numeral_nat: num > nat ).
thf(func_def_28,type,
numeral_numeral_real: num > real ).
thf(func_def_29,type,
ord_le291126163d_enat: ( $o > extended_enat ) > ( $o > extended_enat ) > $o ).
thf(func_def_30,type,
ord_le637473275_ereal: ( $o > extended_ereal ) > ( $o > extended_ereal ) > $o ).
thf(func_def_31,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(func_def_32,type,
ord_less_eq_o_real: ( $o > real ) > ( $o > real ) > $o ).
thf(func_def_33,type,
ord_le1863327750d_enat: extended_enat > extended_enat > $o ).
thf(func_def_34,type,
ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
thf(func_def_35,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_36,type,
ord_less_eq_num: num > num > $o ).
thf(func_def_37,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_38,type,
order_1628344639d_enat: ( extended_enat > $o ) > extended_enat ).
thf(func_def_39,type,
order_1158471719_ereal: ( extended_ereal > $o ) > extended_ereal ).
thf(func_def_40,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(func_def_41,type,
order_Greatest_real: ( real > $o ) > real ).
thf(func_def_42,type,
order_2047034162d_enat: ( extended_enat > extended_enat ) > $o ).
thf(func_def_43,type,
order_1147259034_ereal: ( extended_enat > extended_ereal ) > $o ).
thf(func_def_44,type,
order_442688004at_nat: ( extended_enat > nat ) > $o ).
thf(func_def_45,type,
order_182315744t_real: ( extended_enat > real ) > $o ).
thf(func_def_46,type,
order_680161354d_enat: ( extended_ereal > extended_enat ) > $o ).
thf(func_def_47,type,
order_1408494002_ereal: ( extended_ereal > extended_ereal ) > $o ).
thf(func_def_48,type,
order_523806444al_nat: ( extended_ereal > nat ) > $o ).
thf(func_def_49,type,
order_1800387528l_real: ( extended_ereal > real ) > $o ).
thf(func_def_50,type,
order_1660553314d_enat: ( nat > extended_enat ) > $o ).
thf(func_def_51,type,
order_736687562_ereal: ( nat > extended_ereal ) > $o ).
thf(func_def_52,type,
order_1631207636at_nat: ( nat > nat ) > $o ).
thf(func_def_53,type,
order_106095024t_real: ( nat > real ) > $o ).
thf(func_def_54,type,
order_2106278841d_enat: ( extended_enat > extended_enat ) > $o ).
thf(func_def_55,type,
order_1180956065_ereal: ( extended_enat > extended_ereal ) > $o ).
thf(func_def_56,type,
order_1622825661at_nat: ( extended_enat > nat ) > $o ).
thf(func_def_57,type,
order_135436057t_real: ( extended_enat > real ) > $o ).
thf(func_def_58,type,
order_713858385d_enat: ( extended_ereal > extended_enat ) > $o ).
thf(func_def_59,type,
order_555877177_ereal: ( extended_ereal > extended_ereal ) > $o ).
thf(func_def_60,type,
order_476926757al_nat: ( extended_ereal > nat ) > $o ).
thf(func_def_61,type,
order_1560271745l_real: ( extended_ereal > real ) > $o ).
thf(func_def_62,type,
order_693207323d_enat: ( nat > extended_enat ) > $o ).
thf(func_def_63,type,
order_689807875_ereal: ( nat > extended_ereal ) > $o ).
thf(func_def_64,type,
order_769474267at_nat: ( nat > nat ) > $o ).
thf(func_def_65,type,
order_1962034751d_enat: ( real > extended_enat ) > $o ).
thf(func_def_66,type,
order_946214183_ereal: ( real > extended_ereal ) > $o ).
thf(func_def_67,type,
divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(func_def_68,type,
member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(func_def_69,type,
f: a > extended_ereal ).
thf(func_def_70,type,
g: a > extended_ereal ).
thf(func_def_71,type,
x: a ).
thf(func_def_85,type,
sP0: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_86,type,
sP1: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_87,type,
sP2: extended_ereal > extended_ereal > extended_ereal > $o ).
thf(func_def_88,type,
sK3: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_89,type,
sK4: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_90,type,
sK5: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_91,type,
sK6: ( extended_ereal > extended_ereal > $o ) > extended_ereal ).
thf(func_def_92,type,
sK7: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_93,type,
sK8: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_94,type,
sK9: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_95,type,
sK10: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_96,type,
sK11: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_97,type,
sK12: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_98,type,
sK13: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_99,type,
sK14: ( extended_ereal > extended_ereal ) > extended_ereal ).
thf(func_def_101,type,
ph16:
!>[X0: $tType] : X0 ).
thf(f1040,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f997,f995]) ).
thf(f995,plain,
! [X0: a] :
( ( ord_le824540014_ereal @ ( g @ X0 ) @ ( f @ X0 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f502]) ).
thf(f502,plain,
! [X0: a] :
( ( ord_le824540014_ereal @ ( g @ X0 ) @ ( f @ X0 ) )
= $true ),
inference(fool_elimination,[status(thm)],[f501]) ).
thf(f501,plain,
! [X0: a] : ( ord_le824540014_ereal @ ( g @ X0 ) @ ( f @ X0 ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: a] : ( ord_le824540014_ereal @ ( g @ X0 ) @ ( f @ X0 ) ),
file('/export/starexec/sandbox2/tmp/tmp.nLU4PliHtq/DTF2THF_7494.p',fact_0_assms) ).
thf(f997,plain,
( ( ord_le824540014_ereal @ ( g @ x ) @ ( f @ x ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f927]) ).
thf(f927,plain,
( ( ord_le824540014_ereal @ ( g @ x ) @ ( f @ x ) )
!= $true ),
inference(flattening,[status(thm)],[f612]) ).
thf(f612,plain,
( ( ord_le824540014_ereal @ ( g @ x ) @ ( f @ x ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f611]) ).
thf(f611,plain,
~ ( ord_le824540014_ereal @ ( g @ x ) @ ( f @ x ) ),
inference(rectify,[status(thm)],[f353]) ).
thf(f353,negated_conjecture,
~ ( ord_le824540014_ereal @ ( g @ x ) @ ( f @ x ) ),
inference(negated_conjecture,[status(cth)],[f352]) ).
thf(f352,conjecture,
ord_le824540014_ereal @ ( g @ x ) @ ( f @ x ),
file('/export/starexec/sandbox2/tmp/tmp.nLU4PliHtq/DTF2THF_7494.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP114^1 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n006.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:26:30 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.32/0.46 ---- Original DTF file ---
% 0.32/0.46 thf(spec,logic,$$dhol).
% 0.32/0.46 %------------------------------------------------------------------------------
% 0.32/0.46 % File : ITP114^1 : TPTP v9.2.1. Released v7.5.0.
% 0.32/0.46 % Domain : Interactive Theorem Proving
% 0.32/0.46 % Problem : Sledgehammer Lower_Semicontinuous problem prob_784__6255430_1
% 0.32/0.46 % Version : Especial.
% 0.32/0.46 % English :
% 0.32/0.46
% 0.32/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.32/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.32/0.46 % Source : [Des21]
% 0.32/0.46 % Names : Lower_Semicontinuous/prob_784__6255430_1 [Des21]
% 0.32/0.46
% 0.32/0.46 % Status : Theorem
% 0.32/0.46 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.46 v8.1.0, 0.55 v7.5.0
% 0.32/0.46 % Syntax : Number of formulae : 424 ( 144 unt; 72 typ; 0 def)
% 0.32/0.46 % Number of atoms : 1154 ( 399 equ; 0 cnn)
% 0.32/0.46 % Maximal formula atoms : 81 ( 3 avg)
% 0.32/0.46 % Number of connectives : 3019 ( 124 ~; 19 |; 29 &;2177 @)
% 0.32/0.46 % ( 0 <=>; 670 =>; 0 <=; 0 <~>)
% 0.32/0.46 % Maximal formula depth : 34 ( 7 avg)
% 0.32/0.46 % Number of types : 8 ( 7 usr)
% 0.32/0.46 % Number of type conns : 271 ( 271 >; 0 *; 0 +; 0 <<)
% 0.32/0.46 % Number of symbols : 68 ( 65 usr; 10 con; 0-2 aty)
% 0.32/0.46 % Number of variables : 999 ( 76 ^; 894 !; 29 ?; 999 :)
% 0.32/0.46 % SPC : TH0_THM_EQU_NAR
% 0.32/0.46
% 0.32/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:42:46.608
% 0.32/0.46 %------------------------------------------------------------------------------
% 0.32/0.46 % Could-be-implicit typings (7)
% 0.32/0.46 thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 0.32/0.46 set_Extended_ereal: $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_n_t__Extended____Real__Oereal,type,
% 0.32/0.46 extended_ereal: $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_n_t__Extended____Nat__Oenat,type,
% 0.32/0.46 extended_enat: $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_n_t__Real__Oreal,type,
% 0.32/0.46 real: $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_n_t__Num__Onum,type,
% 0.32/0.46 num: $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.32/0.46 nat: $tType ).
% 0.32/0.46
% 0.32/0.46 thf(ty_n_tf__a,type,
% 0.32/0.46 a: $tType ).
% 0.32/0.46
% 0.32/0.46 % Explicit typings (65)
% 0.32/0.46 thf(sy_c_Extended__Nat_Oenat,type,
% 0.32/0.46 extended_enat2: nat > extended_enat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
% 0.32/0.46 extend1289208545_ereal: extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Nat_Othe__enat,type,
% 0.32/0.46 extended_the_enat: extended_enat > nat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oereal_OMInfty,type,
% 0.32/0.46 extended_MInfty: extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oereal_OPInfty,type,
% 0.32/0.46 extended_PInfty: extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 0.32/0.46 extended_ereal2: real > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oereal_Osize__ereal,type,
% 0.32/0.46 extended_size_ereal: extended_ereal > nat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oereal__of__enat,type,
% 0.32/0.46 extend1771934483f_enat: extended_enat > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Extended__Real_Oreal__of__ereal,type,
% 0.32/0.46 extend1716541707_ereal: extended_ereal > real ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Oabs__class_Oabs_001t__Extended____Real__Oereal,type,
% 0.32/0.46 abs_ab1260901297_ereal: extended_ereal > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
% 0.32/0.46 uminus1208298309_ereal: extended_ereal > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 0.32/0.46 uminus_uminus_real: real > real ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 zero_z491942557d_enat: extended_enat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
% 0.32/0.46 zero_z163181189_ereal: extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 0.32/0.46 zero_zero_nat: nat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 0.32/0.46 zero_zero_real: real ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc_001tf__a_001t__Extended____Real__Oereal,type,
% 0.32/0.46 lower_1616484581_ereal: ( a > extended_ereal ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__hull_001tf__a,type,
% 0.32/0.46 lower_881475195hull_a: ( a > extended_ereal ) > a > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 numera280919179d_enat: num > extended_enat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Real__Oereal,type,
% 0.32/0.46 numera1793320307_ereal: num > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 0.32/0.46 numeral_numeral_nat: num > nat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 0.32/0.46 numeral_numeral_real: num > real ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Nat__Oenat_J,type,
% 0.32/0.46 ord_le291126163d_enat: ( $o > extended_enat ) > ( $o > extended_enat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Real__Oereal_J,type,
% 0.32/0.46 ord_le637473275_ereal: ( $o > extended_ereal ) > ( $o > extended_ereal ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
% 0.32/0.46 ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Real__Oreal_J,type,
% 0.32/0.46 ord_less_eq_o_real: ( $o > real ) > ( $o > real ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 ord_le1863327750d_enat: extended_enat > extended_enat > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
% 0.32/0.46 ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.32/0.46 ord_less_eq_nat: nat > nat > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
% 0.32/0.46 ord_less_eq_num: num > num > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 0.32/0.46 ord_less_eq_real: real > real > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 order_1628344639d_enat: ( extended_enat > $o ) > extended_enat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Real__Oereal,type,
% 0.32/0.46 order_1158471719_ereal: ( extended_ereal > $o ) > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 0.32/0.46 order_Greatest_nat: ( nat > $o ) > nat ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
% 0.32/0.46 order_Greatest_real: ( real > $o ) > real ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 order_2047034162d_enat: ( extended_enat > extended_enat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Extended____Real__Oereal,type,
% 0.32/0.46 order_1147259034_ereal: ( extended_enat > extended_ereal ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
% 0.32/0.46 order_442688004at_nat: ( extended_enat > nat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
% 0.32/0.46 order_182315744t_real: ( extended_enat > real ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 order_680161354d_enat: ( extended_ereal > extended_enat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
% 0.32/0.46 order_1408494002_ereal: ( extended_ereal > extended_ereal ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 0.32/0.46 order_523806444al_nat: ( extended_ereal > nat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
% 0.32/0.46 order_1800387528l_real: ( extended_ereal > real ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 order_1660553314d_enat: ( nat > extended_enat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
% 0.32/0.46 order_736687562_ereal: ( nat > extended_ereal ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.46 order_1631207636at_nat: ( nat > nat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.32/0.46 order_106095024t_real: ( nat > real ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 order_2106278841d_enat: ( extended_enat > extended_enat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Extended____Real__Oereal,type,
% 0.32/0.46 order_1180956065_ereal: ( extended_enat > extended_ereal ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
% 0.32/0.46 order_1622825661at_nat: ( extended_enat > nat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
% 0.32/0.46 order_135436057t_real: ( extended_enat > real ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 order_713858385d_enat: ( extended_ereal > extended_enat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
% 0.32/0.46 order_555877177_ereal: ( extended_ereal > extended_ereal ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 0.32/0.46 order_476926757al_nat: ( extended_ereal > nat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
% 0.32/0.46 order_1560271745l_real: ( extended_ereal > real ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 order_693207323d_enat: ( nat > extended_enat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
% 0.32/0.46 order_689807875_ereal: ( nat > extended_ereal ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.46 order_769474267at_nat: ( nat > nat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Extended____Nat__Oenat,type,
% 0.32/0.46 order_1962034751d_enat: ( real > extended_enat ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Extended____Real__Oereal,type,
% 0.32/0.46 order_946214183_ereal: ( real > extended_ereal ) > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Real__Oereal,type,
% 0.32/0.46 divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_c_member_001t__Extended____Real__Oereal,type,
% 0.32/0.46 member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
% 0.32/0.46
% 0.32/0.46 thf(sy_v_f,type,
% 0.32/0.46 f: a > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_v_g,type,
% 0.32/0.46 g: a > extended_ereal ).
% 0.32/0.46
% 0.32/0.46 thf(sy_v_x____,type,
% 0.32/0.46 x: a ).
% 0.32/0.46
% 0.32/0.46 % Relevant facts (351)
% 0.32/0.46 thf(fact_0_assms,axiom,
% 0.32/0.46 ! [X: a] : ( ord_le824540014_ereal @ ( g @ X ) @ ( f @ X ) ) ).
% 0.32/0.46
% 0.32/0.46 % assms
% 0.32/0.46 thf(fact_1_calculation,axiom,
% 0.32/0.46 ord_le824540014_ereal @ ( lower_881475195hull_a @ g @ x ) @ ( g @ x ) ).
% 0.32/0.46
% 0.32/0.46 % calculation
% 0.32/0.46 thf(fact_2_order__refl,axiom,
% 0.32/0.46 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ X2 ) ).
% 0.32/0.46
% 0.32/0.46 % order_refl
% 0.32/0.46 thf(fact_3_order__refl,axiom,
% 0.32/0.46 ! [X2: extended_enat] : ( ord_le1863327750d_enat @ X2 @ X2 ) ).
% 0.32/0.46
% 0.32/0.46 % order_refl
% 0.32/0.46 thf(fact_4_order__refl,axiom,
% 0.32/0.46 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.32/0.46
% 0.32/0.46 % order_refl
% 0.32/0.46 thf(fact_5_order__refl,axiom,
% 0.32/0.46 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 0.32/0.46
% 0.32/0.46 % order_refl
% 0.32/0.46 thf(fact_6_ereal__complete__Inf,axiom,
% 0.32/0.46 ! [S: set_Extended_ereal] :
% 0.32/0.46 ? [X3: extended_ereal] :
% 0.32/0.46 ( ! [Xa: extended_ereal] :
% 0.32/0.46 ( ( member1900190071_ereal @ Xa @ S )
% 0.32/0.46 => ( ord_le824540014_ereal @ X3 @ Xa ) )
% 0.32/0.46 & ! [Z: extended_ereal] :
% 0.32/0.46 ( ! [Xa2: extended_ereal] :
% 0.32/0.46 ( ( member1900190071_ereal @ Xa2 @ S )
% 0.32/0.46 => ( ord_le824540014_ereal @ Z @ Xa2 ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ Z @ X3 ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % ereal_complete_Inf
% 0.32/0.46 thf(fact_7_ereal__complete__Sup,axiom,
% 0.32/0.46 ! [S: set_Extended_ereal] :
% 0.32/0.46 ? [X3: extended_ereal] :
% 0.32/0.46 ( ! [Xa: extended_ereal] :
% 0.32/0.46 ( ( member1900190071_ereal @ Xa @ S )
% 0.32/0.46 => ( ord_le824540014_ereal @ Xa @ X3 ) )
% 0.32/0.46 & ! [Z: extended_ereal] :
% 0.32/0.46 ( ! [Xa2: extended_ereal] :
% 0.32/0.46 ( ( member1900190071_ereal @ Xa2 @ S )
% 0.32/0.46 => ( ord_le824540014_ereal @ Xa2 @ Z ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ X3 @ Z ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % ereal_complete_Sup
% 0.32/0.46 thf(fact_8_order__subst1,axiom,
% 0.32/0.46 ! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_9_order__subst1,axiom,
% 0.32/0.46 ! [A: extended_ereal,F: extended_enat > extended_ereal,B: extended_enat,C: extended_enat] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_le1863327750d_enat @ B @ C )
% 0.32/0.46 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_10_order__subst1,axiom,
% 0.32/0.46 ! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.46 => ( ! [X3: nat,Y: nat] :
% 0.32/0.46 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_11_order__subst1,axiom,
% 0.32/0.46 ! [A: extended_ereal,F: real > extended_ereal,B: real,C: real] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.46 => ( ! [X3: real,Y: real] :
% 0.32/0.46 ( ( ord_less_eq_real @ X3 @ Y )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_12_order__subst1,axiom,
% 0.32/0.46 ! [A: extended_enat,F: extended_ereal > extended_enat,B: extended_ereal,C: extended_ereal] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_13_order__subst1,axiom,
% 0.32/0.46 ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_le1863327750d_enat @ B @ C )
% 0.32/0.46 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_14_order__subst1,axiom,
% 0.32/0.46 ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.46 => ( ! [X3: nat,Y: nat] :
% 0.32/0.46 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_15_order__subst1,axiom,
% 0.32/0.46 ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.46 => ( ! [X3: real,Y: real] :
% 0.32/0.46 ( ( ord_less_eq_real @ X3 @ Y )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_16_order__subst1,axiom,
% 0.32/0.46 ! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
% 0.32/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_17_order__subst1,axiom,
% 0.32/0.46 ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
% 0.32/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/0.46 => ( ( ord_le1863327750d_enat @ B @ C )
% 0.32/0.46 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst1
% 0.32/0.46 thf(fact_18_order__subst2,axiom,
% 0.32/0.46 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.46 => ( ( ord_le824540014_ereal @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_19_order__subst2,axiom,
% 0.32/0.46 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_enat,C: extended_enat] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.46 => ( ( ord_le1863327750d_enat @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_20_order__subst2,axiom,
% 0.32/0.46 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_21_order__subst2,axiom,
% 0.32/0.46 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > real,C: real] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.46 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_22_order__subst2,axiom,
% 0.32/0.46 ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_ereal,C: extended_ereal] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.46 => ( ( ord_le824540014_ereal @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_23_order__subst2,axiom,
% 0.32/0.46 ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.46 => ( ( ord_le1863327750d_enat @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_24_order__subst2,axiom,
% 0.32/0.46 ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_25_order__subst2,axiom,
% 0.32/0.46 ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.46 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.46 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.46 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_26_order__subst2,axiom,
% 0.32/0.46 ! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
% 0.32/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.46 => ( ( ord_le824540014_ereal @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: nat,Y: nat] :
% 0.32/0.46 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_27_order__subst2,axiom,
% 0.32/0.46 ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
% 0.32/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.46 => ( ( ord_le1863327750d_enat @ ( F @ B ) @ C )
% 0.32/0.46 => ( ! [X3: nat,Y: nat] :
% 0.32/0.46 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % order_subst2
% 0.32/0.46 thf(fact_28_verit__la__disequality,axiom,
% 0.32/0.46 ! [A: extended_ereal,B: extended_ereal] :
% 0.32/0.46 ( ( A = B )
% 0.32/0.46 | ~ ( ord_le824540014_ereal @ A @ B )
% 0.32/0.46 | ~ ( ord_le824540014_ereal @ B @ A ) ) ).
% 0.32/0.46
% 0.32/0.46 % verit_la_disequality
% 0.32/0.46 thf(fact_29_verit__la__disequality,axiom,
% 0.32/0.46 ! [A: extended_enat,B: extended_enat] :
% 0.32/0.46 ( ( A = B )
% 0.32/0.46 | ~ ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.46 | ~ ( ord_le1863327750d_enat @ B @ A ) ) ).
% 0.32/0.46
% 0.32/0.46 % verit_la_disequality
% 0.32/0.46 thf(fact_30_verit__la__disequality,axiom,
% 0.32/0.46 ! [A: nat,B: nat] :
% 0.32/0.46 ( ( A = B )
% 0.32/0.46 | ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.46 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.32/0.46
% 0.32/0.46 % verit_la_disequality
% 0.32/0.46 thf(fact_31_verit__la__disequality,axiom,
% 0.32/0.46 ! [A: real,B: real] :
% 0.32/0.46 ( ( A = B )
% 0.32/0.46 | ~ ( ord_less_eq_real @ A @ B )
% 0.32/0.46 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 0.32/0.46
% 0.32/0.46 % verit_la_disequality
% 0.32/0.46 thf(fact_32_ord__eq__le__subst,axiom,
% 0.32/0.46 ! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.32/0.46 ( ( A
% 0.32/0.46 = ( F @ B ) )
% 0.32/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % ord_eq_le_subst
% 0.32/0.46 thf(fact_33_ord__eq__le__subst,axiom,
% 0.32/0.46 ! [A: extended_enat,F: extended_ereal > extended_enat,B: extended_ereal,C: extended_ereal] :
% 0.32/0.46 ( ( A
% 0.32/0.46 = ( F @ B ) )
% 0.32/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % ord_eq_le_subst
% 0.32/0.46 thf(fact_34_ord__eq__le__subst,axiom,
% 0.32/0.46 ! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
% 0.32/0.46 ( ( A
% 0.32/0.46 = ( F @ B ) )
% 0.32/0.46 => ( ( ord_le824540014_ereal @ B @ C )
% 0.32/0.46 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.46 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.46 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 % ord_eq_le_subst
% 0.32/0.46 thf(fact_35_ord__eq__le__subst,axiom,
% 0.32/0.46 ! [A: real,F: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
% 0.32/0.47 ( ( A
% 0.32/0.47 = ( F @ B ) )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ B @ C )
% 0.32/0.47 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_36_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: extended_ereal,F: extended_enat > extended_ereal,B: extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( A
% 0.32/0.47 = ( F @ B ) )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ B @ C )
% 0.32/0.47 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_37_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( A
% 0.32/0.47 = ( F @ B ) )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ B @ C )
% 0.32/0.47 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_38_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( A
% 0.32/0.47 = ( F @ B ) )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ B @ C )
% 0.32/0.47 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_39_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( A
% 0.32/0.47 = ( F @ B ) )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ B @ C )
% 0.32/0.47 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_40_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
% 0.32/0.47 ( ( A
% 0.32/0.47 = ( F @ B ) )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.47 => ( ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_41_ord__eq__le__subst,axiom,
% 0.32/0.47 ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
% 0.32/0.47 ( ( A
% 0.32/0.47 = ( F @ B ) )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.47 => ( ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_subst
% 0.32/0.47 thf(fact_42_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_43_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_44_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_45_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > real,C: real] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_46_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_ereal,C: extended_ereal] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_47_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_48_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_49_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_50_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_51_ord__le__eq__subst,axiom,
% 0.32/0.47 ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.47 => ( ( ( F @ B )
% 0.32/0.47 = C )
% 0.32/0.47 => ( ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_subst
% 0.32/0.47 thf(fact_52_lsc__hull__le,axiom,
% 0.32/0.47 ! [F: a > extended_ereal,X2: a] : ( ord_le824540014_ereal @ ( lower_881475195hull_a @ F @ X2 ) @ ( F @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % lsc_hull_le
% 0.32/0.47 thf(fact_53_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [B: extended_ereal,A: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ B @ A )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( A = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_54_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [B: extended_enat,A: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ B @ A )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.47 => ( A = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_55_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [B: nat,A: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.47 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.47 => ( A = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_56_dual__order_Oantisym,axiom,
% 0.32/0.47 ! [B: real,A: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.47 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.47 => ( A = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.antisym
% 0.32/0.47 thf(fact_57_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: extended_ereal,Z2: extended_ereal] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [A2: extended_ereal,B2: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ B2 @ A2 )
% 0.32/0.47 & ( ord_le824540014_ereal @ A2 @ B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47 thf(fact_58_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: extended_enat,Z2: extended_enat] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [A2: extended_enat,B2: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ B2 @ A2 )
% 0.32/0.47 & ( ord_le1863327750d_enat @ A2 @ B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47 thf(fact_59_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.32/0.47 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47 thf(fact_60_dual__order_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [A2: real,B2: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ B2 @ A2 )
% 0.32/0.47 & ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.eq_iff
% 0.32/0.47 thf(fact_61_dual__order_Otrans,axiom,
% 0.32/0.47 ! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ B @ A )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ C @ B )
% 0.32/0.47 => ( ord_le824540014_ereal @ C @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_62_dual__order_Otrans,axiom,
% 0.32/0.47 ! [B: extended_enat,A: extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ B @ A )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ C @ B )
% 0.32/0.47 => ( ord_le1863327750d_enat @ C @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_63_dual__order_Otrans,axiom,
% 0.32/0.47 ! [B: nat,A: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.47 => ( ( ord_less_eq_nat @ C @ B )
% 0.32/0.47 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_64_dual__order_Otrans,axiom,
% 0.32/0.47 ! [B: real,A: real,C: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.47 => ( ( ord_less_eq_real @ C @ B )
% 0.32/0.47 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.trans
% 0.32/0.47 thf(fact_65_linorder__wlog,axiom,
% 0.32/0.47 ! [P: extended_ereal > extended_ereal > $o,A: extended_ereal,B: extended_ereal] :
% 0.32/0.47 ( ! [A3: extended_ereal,B3: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 0.32/0.47 => ( P @ A3 @ B3 ) )
% 0.32/0.47 => ( ! [A3: extended_ereal,B3: extended_ereal] :
% 0.32/0.47 ( ( P @ B3 @ A3 )
% 0.32/0.47 => ( P @ A3 @ B3 ) )
% 0.32/0.47 => ( P @ A @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47 thf(fact_66_linorder__wlog,axiom,
% 0.32/0.47 ! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
% 0.32/0.47 ( ! [A3: extended_enat,B3: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ A3 @ B3 )
% 0.32/0.47 => ( P @ A3 @ B3 ) )
% 0.32/0.47 => ( ! [A3: extended_enat,B3: extended_enat] :
% 0.32/0.47 ( ( P @ B3 @ A3 )
% 0.32/0.47 => ( P @ A3 @ B3 ) )
% 0.32/0.47 => ( P @ A @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47 thf(fact_67_linorder__wlog,axiom,
% 0.32/0.47 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.32/0.47 ( ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.32/0.47 => ( P @ A3 @ B3 ) )
% 0.32/0.47 => ( ! [A3: nat,B3: nat] :
% 0.32/0.47 ( ( P @ B3 @ A3 )
% 0.32/0.47 => ( P @ A3 @ B3 ) )
% 0.32/0.47 => ( P @ A @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47 thf(fact_68_linorder__wlog,axiom,
% 0.32/0.47 ! [P: real > real > $o,A: real,B: real] :
% 0.32/0.47 ( ! [A3: real,B3: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ A3 @ B3 )
% 0.32/0.47 => ( P @ A3 @ B3 ) )
% 0.32/0.47 => ( ! [A3: real,B3: real] :
% 0.32/0.47 ( ( P @ B3 @ A3 )
% 0.32/0.47 => ( P @ A3 @ B3 ) )
% 0.32/0.47 => ( P @ A @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linorder_wlog
% 0.32/0.47 thf(fact_69_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A: extended_ereal] : ( ord_le824540014_ereal @ A @ A ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_70_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A: extended_enat] : ( ord_le1863327750d_enat @ A @ A ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_71_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_72_dual__order_Orefl,axiom,
% 0.32/0.47 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.32/0.47
% 0.32/0.47 % dual_order.refl
% 0.32/0.47 thf(fact_73_order__trans,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal,Z3: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ Y3 @ Z3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ X2 @ Z3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_74_order__trans,axiom,
% 0.32/0.47 ! [X2: extended_enat,Y3: extended_enat,Z3: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ Y3 @ Z3 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ X2 @ Z3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_75_order__trans,axiom,
% 0.32/0.47 ! [X2: nat,Y3: nat,Z3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ Y3 @ Z3 )
% 0.32/0.47 => ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_76_order__trans,axiom,
% 0.32/0.47 ! [X2: real,Y3: real,Z3: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.32/0.47 => ( ( ord_less_eq_real @ Y3 @ Z3 )
% 0.32/0.47 => ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_trans
% 0.32/0.47 thf(fact_77_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ B @ A )
% 0.32/0.47 => ( A = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_78_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A: extended_enat,B: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ B @ A )
% 0.32/0.47 => ( A = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_79_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A: nat,B: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.47 => ( A = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_80_order__class_Oorder_Oantisym,axiom,
% 0.32/0.47 ! [A: real,B: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.47 => ( ( ord_less_eq_real @ B @ A )
% 0.32/0.47 => ( A = B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.antisym
% 0.32/0.47 thf(fact_81_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( ( B = C )
% 0.32/0.47 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_82_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.47 => ( ( B = C )
% 0.32/0.47 => ( ord_le1863327750d_enat @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_83_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A: nat,B: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.47 => ( ( B = C )
% 0.32/0.47 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_84_ord__le__eq__trans,axiom,
% 0.32/0.47 ! [A: real,B: real,C: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.47 => ( ( B = C )
% 0.32/0.47 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_le_eq_trans
% 0.32/0.47 thf(fact_85_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ B @ C )
% 0.32/0.47 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_86_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ B @ C )
% 0.32/0.47 => ( ord_le1863327750d_enat @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_87_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A: nat,B: nat,C: nat] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.47 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_88_ord__eq__le__trans,axiom,
% 0.32/0.47 ! [A: real,B: real,C: real] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.47 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord_eq_le_trans
% 0.32/0.47 thf(fact_89_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: extended_ereal,Z2: extended_ereal] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [A2: extended_ereal,B2: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A2 @ B2 )
% 0.32/0.47 & ( ord_le824540014_ereal @ B2 @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_90_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: extended_enat,Z2: extended_enat] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [A2: extended_enat,B2: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ A2 @ B2 )
% 0.32/0.47 & ( ord_le1863327750d_enat @ B2 @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_91_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [A2: nat,B2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.32/0.47 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_92_order__class_Oorder_Oeq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [A2: real,B2: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ A2 @ B2 )
% 0.32/0.47 & ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order_class.order.eq_iff
% 0.32/0.47 thf(fact_93_antisym__conv,axiom,
% 0.32/0.47 ! [Y3: extended_ereal,X2: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ Y3 @ X2 )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 = ( X2 = Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_94_antisym__conv,axiom,
% 0.32/0.47 ! [Y3: extended_enat,X2: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ Y3 @ X2 )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 = ( X2 = Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_95_antisym__conv,axiom,
% 0.32/0.47 ! [Y3: nat,X2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 = ( X2 = Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_96_antisym__conv,axiom,
% 0.32/0.47 ! [Y3: real,X2: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.32/0.47 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.32/0.47 = ( X2 = Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym_conv
% 0.32/0.47 thf(fact_97_le__cases3,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal,Z3: extended_ereal] :
% 0.32/0.47 ( ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ~ ( ord_le824540014_ereal @ Y3 @ Z3 ) )
% 0.32/0.47 => ( ( ( ord_le824540014_ereal @ Y3 @ X2 )
% 0.32/0.47 => ~ ( ord_le824540014_ereal @ X2 @ Z3 ) )
% 0.32/0.47 => ( ( ( ord_le824540014_ereal @ X2 @ Z3 )
% 0.32/0.47 => ~ ( ord_le824540014_ereal @ Z3 @ Y3 ) )
% 0.32/0.47 => ( ( ( ord_le824540014_ereal @ Z3 @ Y3 )
% 0.32/0.47 => ~ ( ord_le824540014_ereal @ Y3 @ X2 ) )
% 0.32/0.47 => ( ( ( ord_le824540014_ereal @ Y3 @ Z3 )
% 0.32/0.47 => ~ ( ord_le824540014_ereal @ Z3 @ X2 ) )
% 0.32/0.47 => ~ ( ( ord_le824540014_ereal @ Z3 @ X2 )
% 0.32/0.47 => ~ ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_98_le__cases3,axiom,
% 0.32/0.47 ! [X2: extended_enat,Y3: extended_enat,Z3: extended_enat] :
% 0.32/0.47 ( ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ~ ( ord_le1863327750d_enat @ Y3 @ Z3 ) )
% 0.32/0.47 => ( ( ( ord_le1863327750d_enat @ Y3 @ X2 )
% 0.32/0.47 => ~ ( ord_le1863327750d_enat @ X2 @ Z3 ) )
% 0.32/0.47 => ( ( ( ord_le1863327750d_enat @ X2 @ Z3 )
% 0.32/0.47 => ~ ( ord_le1863327750d_enat @ Z3 @ Y3 ) )
% 0.32/0.47 => ( ( ( ord_le1863327750d_enat @ Z3 @ Y3 )
% 0.32/0.47 => ~ ( ord_le1863327750d_enat @ Y3 @ X2 ) )
% 0.32/0.47 => ( ( ( ord_le1863327750d_enat @ Y3 @ Z3 )
% 0.32/0.47 => ~ ( ord_le1863327750d_enat @ Z3 @ X2 ) )
% 0.32/0.47 => ~ ( ( ord_le1863327750d_enat @ Z3 @ X2 )
% 0.32/0.47 => ~ ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_99_le__cases3,axiom,
% 0.32/0.47 ! [X2: nat,Y3: nat,Z3: nat] :
% 0.32/0.47 ( ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_nat @ X2 @ Z3 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
% 0.32/0.47 => ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
% 0.32/0.47 => ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_100_le__cases3,axiom,
% 0.32/0.47 ! [X2: real,Y3: real,Z3: real] :
% 0.32/0.47 ( ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.32/0.47 => ~ ( ord_less_eq_real @ Y3 @ Z3 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.32/0.47 => ~ ( ord_less_eq_real @ X2 @ Z3 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_real @ X2 @ Z3 )
% 0.32/0.47 => ~ ( ord_less_eq_real @ Z3 @ Y3 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_real @ Z3 @ Y3 )
% 0.32/0.47 => ~ ( ord_less_eq_real @ Y3 @ X2 ) )
% 0.32/0.47 => ( ( ( ord_less_eq_real @ Y3 @ Z3 )
% 0.32/0.47 => ~ ( ord_less_eq_real @ Z3 @ X2 ) )
% 0.32/0.47 => ~ ( ( ord_less_eq_real @ Z3 @ X2 )
% 0.32/0.47 => ~ ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases3
% 0.32/0.47 thf(fact_101_order_Otrans,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ B @ C )
% 0.32/0.47 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_102_order_Otrans,axiom,
% 0.32/0.47 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ A @ B )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ B @ C )
% 0.32/0.47 => ( ord_le1863327750d_enat @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_103_order_Otrans,axiom,
% 0.32/0.47 ! [A: nat,B: nat,C: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.47 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.47 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_104_order_Otrans,axiom,
% 0.32/0.47 ! [A: real,B: real,C: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.47 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.47 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % order.trans
% 0.32/0.47 thf(fact_105_le__cases,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ~ ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ Y3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_106_le__cases,axiom,
% 0.32/0.47 ! [X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ~ ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ Y3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_107_le__cases,axiom,
% 0.32/0.47 ! [X2: nat,Y3: nat] :
% 0.32/0.47 ( ~ ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_108_le__cases,axiom,
% 0.32/0.47 ! [X2: real,Y3: real] :
% 0.32/0.47 ( ~ ( ord_less_eq_real @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_cases
% 0.32/0.47 thf(fact_109_eq__refl,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( X2 = Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ X2 @ Y3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_110_eq__refl,axiom,
% 0.32/0.47 ! [X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( X2 = Y3 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_111_eq__refl,axiom,
% 0.32/0.47 ! [X2: nat,Y3: nat] :
% 0.32/0.47 ( ( X2 = Y3 )
% 0.32/0.47 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_112_eq__refl,axiom,
% 0.32/0.47 ! [X2: real,Y3: real] :
% 0.32/0.47 ( ( X2 = Y3 )
% 0.32/0.47 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_refl
% 0.32/0.47 thf(fact_113_linear,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 | ( ord_le824540014_ereal @ Y3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_114_linear,axiom,
% 0.32/0.47 ! [X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 | ( ord_le1863327750d_enat @ Y3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_115_linear,axiom,
% 0.32/0.47 ! [X2: nat,Y3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 | ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_116_linear,axiom,
% 0.32/0.47 ! [X2: real,Y3: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.32/0.47 | ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % linear
% 0.32/0.47 thf(fact_117_antisym,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ Y3 @ X2 )
% 0.32/0.47 => ( X2 = Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_118_antisym,axiom,
% 0.32/0.47 ! [X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ Y3 @ X2 )
% 0.32/0.47 => ( X2 = Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_119_antisym,axiom,
% 0.32/0.47 ! [X2: nat,Y3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.32/0.47 => ( X2 = Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_120_antisym,axiom,
% 0.32/0.47 ! [X2: real,Y3: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.32/0.47 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.32/0.47 => ( X2 = Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antisym
% 0.32/0.47 thf(fact_121_eq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: extended_ereal,Z2: extended_ereal] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [X4: extended_ereal,Y4: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 0.32/0.47 & ( ord_le824540014_ereal @ Y4 @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_122_eq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: extended_enat,Z2: extended_enat] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [X4: extended_enat,Y4: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 0.32/0.47 & ( ord_le1863327750d_enat @ Y4 @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_123_eq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [X4: nat,Y4: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.47 & ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_124_eq__iff,axiom,
% 0.32/0.47 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 0.32/0.47 = ( ^ [X4: real,Y4: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ X4 @ Y4 )
% 0.32/0.47 & ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_iff
% 0.32/0.47 thf(fact_125_lsc__hull__iff__greatest,axiom,
% 0.32/0.47 ! [G: a > extended_ereal,F: a > extended_ereal] :
% 0.32/0.47 ( ( G
% 0.32/0.47 = ( lower_881475195hull_a @ F ) )
% 0.32/0.47 = ( ( lower_1616484581_ereal @ G )
% 0.32/0.47 & ! [X4: a] : ( ord_le824540014_ereal @ ( G @ X4 ) @ ( F @ X4 ) )
% 0.32/0.47 & ! [H: a > extended_ereal] :
% 0.32/0.47 ( ( ( lower_1616484581_ereal @ H )
% 0.32/0.47 & ! [X4: a] : ( ord_le824540014_ereal @ ( H @ X4 ) @ ( F @ X4 ) ) )
% 0.32/0.47 => ! [X4: a] : ( ord_le824540014_ereal @ ( H @ X4 ) @ ( G @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lsc_hull_iff_greatest
% 0.32/0.47 thf(fact_126_lsc__hull__greatest,axiom,
% 0.32/0.47 ! [G: a > extended_ereal,F: a > extended_ereal] :
% 0.32/0.47 ( ( lower_1616484581_ereal @ G )
% 0.32/0.47 => ( ! [X3: a] : ( ord_le824540014_ereal @ ( G @ X3 ) @ ( F @ X3 ) )
% 0.32/0.47 => ! [X: a] : ( ord_le824540014_ereal @ ( G @ X ) @ ( lower_881475195hull_a @ F @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lsc_hull_greatest
% 0.32/0.47 thf(fact_127_Greatest__equality,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,X2: extended_ereal] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 => ( ! [Y: extended_ereal] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ Y @ X2 ) )
% 0.32/0.47 => ( ( order_1158471719_ereal @ P )
% 0.32/0.47 = X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Greatest_equality
% 0.32/0.47 thf(fact_128_Greatest__equality,axiom,
% 0.32/0.47 ! [P: extended_enat > $o,X2: extended_enat] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 => ( ! [Y: extended_enat] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ Y @ X2 ) )
% 0.32/0.47 => ( ( order_1628344639d_enat @ P )
% 0.32/0.47 = X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Greatest_equality
% 0.32/0.47 thf(fact_129_Greatest__equality,axiom,
% 0.32/0.47 ! [P: real > $o,X2: real] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 => ( ! [Y: real] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq_real @ Y @ X2 ) )
% 0.32/0.47 => ( ( order_Greatest_real @ P )
% 0.32/0.47 = X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Greatest_equality
% 0.32/0.47 thf(fact_130_Greatest__equality,axiom,
% 0.32/0.47 ! [P: nat > $o,X2: nat] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 => ( ! [Y: nat] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ Y @ X2 ) )
% 0.32/0.47 => ( ( order_Greatest_nat @ P )
% 0.32/0.47 = X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Greatest_equality
% 0.32/0.47 thf(fact_131_GreatestI2__order,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,X2: extended_ereal,Q: extended_ereal > $o] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 => ( ! [Y: extended_ereal] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ Y @ X2 ) )
% 0.32/0.47 => ( ! [X3: extended_ereal] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 => ( ! [Y5: extended_ereal] :
% 0.32/0.47 ( ( P @ Y5 )
% 0.32/0.47 => ( ord_le824540014_ereal @ Y5 @ X3 ) )
% 0.32/0.47 => ( Q @ X3 ) ) )
% 0.32/0.47 => ( Q @ ( order_1158471719_ereal @ P ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % GreatestI2_order
% 0.32/0.47 thf(fact_132_GreatestI2__order,axiom,
% 0.32/0.47 ! [P: extended_enat > $o,X2: extended_enat,Q: extended_enat > $o] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 => ( ! [Y: extended_enat] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ Y @ X2 ) )
% 0.32/0.47 => ( ! [X3: extended_enat] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 => ( ! [Y5: extended_enat] :
% 0.32/0.47 ( ( P @ Y5 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ Y5 @ X3 ) )
% 0.32/0.47 => ( Q @ X3 ) ) )
% 0.32/0.47 => ( Q @ ( order_1628344639d_enat @ P ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % GreatestI2_order
% 0.32/0.47 thf(fact_133_GreatestI2__order,axiom,
% 0.32/0.47 ! [P: real > $o,X2: real,Q: real > $o] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 => ( ! [Y: real] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq_real @ Y @ X2 ) )
% 0.32/0.47 => ( ! [X3: real] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 => ( ! [Y5: real] :
% 0.32/0.47 ( ( P @ Y5 )
% 0.32/0.47 => ( ord_less_eq_real @ Y5 @ X3 ) )
% 0.32/0.47 => ( Q @ X3 ) ) )
% 0.32/0.47 => ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % GreatestI2_order
% 0.32/0.47 thf(fact_134_GreatestI2__order,axiom,
% 0.32/0.47 ! [P: nat > $o,X2: nat,Q: nat > $o] :
% 0.32/0.47 ( ( P @ X2 )
% 0.32/0.47 => ( ! [Y: nat] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ Y @ X2 ) )
% 0.32/0.47 => ( ! [X3: nat] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 => ( ! [Y5: nat] :
% 0.32/0.47 ( ( P @ Y5 )
% 0.32/0.47 => ( ord_less_eq_nat @ Y5 @ X3 ) )
% 0.32/0.47 => ( Q @ X3 ) ) )
% 0.32/0.47 => ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % GreatestI2_order
% 0.32/0.47 thf(fact_135_le__rel__bool__arg__iff,axiom,
% 0.32/0.47 ( ord_le637473275_ereal
% 0.32/0.47 = ( ^ [X5: $o > extended_ereal,Y6: $o > extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 0.32/0.47 & ( ord_le824540014_ereal @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_rel_bool_arg_iff
% 0.32/0.47 thf(fact_136_le__rel__bool__arg__iff,axiom,
% 0.32/0.47 ( ord_le291126163d_enat
% 0.32/0.47 = ( ^ [X5: $o > extended_enat,Y6: $o > extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 0.32/0.47 & ( ord_le1863327750d_enat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_rel_bool_arg_iff
% 0.32/0.47 thf(fact_137_le__rel__bool__arg__iff,axiom,
% 0.32/0.47 ( ord_less_eq_o_nat
% 0.32/0.47 = ( ^ [X5: $o > nat,Y6: $o > nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 0.32/0.47 & ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_rel_bool_arg_iff
% 0.32/0.47 thf(fact_138_le__rel__bool__arg__iff,axiom,
% 0.32/0.47 ( ord_less_eq_o_real
% 0.32/0.47 = ( ^ [X5: $o > real,Y6: $o > real] :
% 0.32/0.47 ( ( ord_less_eq_real @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 0.32/0.47 & ( ord_less_eq_real @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_rel_bool_arg_iff
% 0.32/0.47 thf(fact_139_lsc__lsc__hull,axiom,
% 0.32/0.47 ! [F: a > extended_ereal] : ( lower_1616484581_ereal @ ( lower_881475195hull_a @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % lsc_lsc_hull
% 0.32/0.47 thf(fact_140_ereal__of__enat__le__iff,axiom,
% 0.32/0.47 ! [M: extended_enat,N: extended_enat] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ ( extend1771934483f_enat @ M ) @ ( extend1771934483f_enat @ N ) )
% 0.32/0.47 = ( ord_le1863327750d_enat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_of_enat_le_iff
% 0.32/0.47 thf(fact_141_antimono__def,axiom,
% 0.32/0.47 ( order_1408494002_ereal
% 0.32/0.47 = ( ^ [F2: extended_ereal > extended_ereal] :
% 0.32/0.47 ! [X4: extended_ereal,Y4: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_142_antimono__def,axiom,
% 0.32/0.47 ( order_680161354d_enat
% 0.32/0.47 = ( ^ [F2: extended_ereal > extended_enat] :
% 0.32/0.47 ! [X4: extended_ereal,Y4: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_143_antimono__def,axiom,
% 0.32/0.47 ( order_523806444al_nat
% 0.32/0.47 = ( ^ [F2: extended_ereal > nat] :
% 0.32/0.47 ! [X4: extended_ereal,Y4: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_144_antimono__def,axiom,
% 0.32/0.47 ( order_1800387528l_real
% 0.32/0.47 = ( ^ [F2: extended_ereal > real] :
% 0.32/0.47 ! [X4: extended_ereal,Y4: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 0.32/0.47 => ( ord_less_eq_real @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_145_antimono__def,axiom,
% 0.32/0.47 ( order_1147259034_ereal
% 0.32/0.47 = ( ^ [F2: extended_enat > extended_ereal] :
% 0.32/0.47 ! [X4: extended_enat,Y4: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_146_antimono__def,axiom,
% 0.32/0.47 ( order_2047034162d_enat
% 0.32/0.47 = ( ^ [F2: extended_enat > extended_enat] :
% 0.32/0.47 ! [X4: extended_enat,Y4: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_147_antimono__def,axiom,
% 0.32/0.47 ( order_442688004at_nat
% 0.32/0.47 = ( ^ [F2: extended_enat > nat] :
% 0.32/0.47 ! [X4: extended_enat,Y4: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_148_antimono__def,axiom,
% 0.32/0.47 ( order_182315744t_real
% 0.32/0.47 = ( ^ [F2: extended_enat > real] :
% 0.32/0.47 ! [X4: extended_enat,Y4: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 0.32/0.47 => ( ord_less_eq_real @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_149_antimono__def,axiom,
% 0.32/0.47 ( order_736687562_ereal
% 0.32/0.47 = ( ^ [F2: nat > extended_ereal] :
% 0.32/0.47 ! [X4: nat,Y4: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_150_antimono__def,axiom,
% 0.32/0.47 ( order_1660553314d_enat
% 0.32/0.47 = ( ^ [F2: nat > extended_enat] :
% 0.32/0.47 ! [X4: nat,Y4: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimono_def
% 0.32/0.47 thf(fact_151_antimonoI,axiom,
% 0.32/0.47 ! [F: extended_ereal > extended_ereal] :
% 0.32/0.47 ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_1408494002_ereal @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_152_antimonoI,axiom,
% 0.32/0.47 ! [F: extended_ereal > extended_enat] :
% 0.32/0.47 ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_680161354d_enat @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_153_antimonoI,axiom,
% 0.32/0.47 ! [F: extended_ereal > nat] :
% 0.32/0.47 ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_523806444al_nat @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_154_antimonoI,axiom,
% 0.32/0.47 ! [F: extended_ereal > real] :
% 0.32/0.47 ( ! [X3: extended_ereal,Y: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_1800387528l_real @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_155_antimonoI,axiom,
% 0.32/0.47 ! [F: extended_enat > extended_ereal] :
% 0.32/0.47 ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_1147259034_ereal @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_156_antimonoI,axiom,
% 0.32/0.47 ! [F: extended_enat > extended_enat] :
% 0.32/0.47 ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_2047034162d_enat @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_157_antimonoI,axiom,
% 0.32/0.47 ! [F: extended_enat > nat] :
% 0.32/0.47 ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_442688004at_nat @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_158_antimonoI,axiom,
% 0.32/0.47 ! [F: extended_enat > real] :
% 0.32/0.47 ( ! [X3: extended_enat,Y: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_182315744t_real @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_159_antimonoI,axiom,
% 0.32/0.47 ! [F: nat > extended_ereal] :
% 0.32/0.47 ( ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_736687562_ereal @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_160_antimonoI,axiom,
% 0.32/0.47 ! [F: nat > extended_enat] :
% 0.32/0.47 ( ! [X3: nat,Y: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ X3 @ Y )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 0.32/0.47 => ( order_1660553314d_enat @ F ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoI
% 0.32/0.47 thf(fact_161_antimonoE,axiom,
% 0.32/0.47 ! [F: extended_ereal > extended_ereal,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_1408494002_ereal @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_162_antimonoE,axiom,
% 0.32/0.47 ! [F: extended_ereal > extended_enat,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_680161354d_enat @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_163_antimonoE,axiom,
% 0.32/0.47 ! [F: extended_ereal > nat,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_523806444al_nat @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_164_antimonoE,axiom,
% 0.32/0.47 ! [F: extended_ereal > real,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_1800387528l_real @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_165_antimonoE,axiom,
% 0.32/0.47 ! [F: extended_enat > extended_ereal,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_1147259034_ereal @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_166_antimonoE,axiom,
% 0.32/0.47 ! [F: extended_enat > extended_enat,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_2047034162d_enat @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_167_antimonoE,axiom,
% 0.32/0.47 ! [F: extended_enat > nat,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_442688004at_nat @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_168_antimonoE,axiom,
% 0.32/0.47 ! [F: extended_enat > real,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_182315744t_real @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_169_antimonoE,axiom,
% 0.32/0.47 ! [F: nat > extended_ereal,X2: nat,Y3: nat] :
% 0.32/0.47 ( ( order_736687562_ereal @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_170_antimonoE,axiom,
% 0.32/0.47 ! [F: nat > extended_enat,X2: nat,Y3: nat] :
% 0.32/0.47 ( ( order_1660553314d_enat @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoE
% 0.32/0.47 thf(fact_171_antimonoD,axiom,
% 0.32/0.47 ! [F: extended_ereal > extended_ereal,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_1408494002_ereal @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_172_antimonoD,axiom,
% 0.32/0.47 ! [F: extended_ereal > extended_enat,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_680161354d_enat @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_173_antimonoD,axiom,
% 0.32/0.47 ! [F: extended_ereal > nat,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_523806444al_nat @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_174_antimonoD,axiom,
% 0.32/0.47 ! [F: extended_ereal > real,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_1800387528l_real @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_175_antimonoD,axiom,
% 0.32/0.47 ! [F: extended_enat > extended_ereal,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_1147259034_ereal @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_176_antimonoD,axiom,
% 0.32/0.47 ! [F: extended_enat > extended_enat,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_2047034162d_enat @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_177_antimonoD,axiom,
% 0.32/0.47 ! [F: extended_enat > nat,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_442688004at_nat @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_178_antimonoD,axiom,
% 0.32/0.47 ! [F: extended_enat > real,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_182315744t_real @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_179_antimonoD,axiom,
% 0.32/0.47 ! [F: nat > extended_ereal,X2: nat,Y3: nat] :
% 0.32/0.47 ( ( order_736687562_ereal @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_180_antimonoD,axiom,
% 0.32/0.47 ! [F: nat > extended_enat,X2: nat,Y3: nat] :
% 0.32/0.47 ( ( order_1660553314d_enat @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % antimonoD
% 0.32/0.47 thf(fact_181_decseqD,axiom,
% 0.32/0.47 ! [F: nat > extended_ereal,I: nat,J: nat] :
% 0.32/0.47 ( ( order_736687562_ereal @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % decseqD
% 0.32/0.47 thf(fact_182_decseqD,axiom,
% 0.32/0.47 ! [F: nat > extended_enat,I: nat,J: nat] :
% 0.32/0.47 ( ( order_1660553314d_enat @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % decseqD
% 0.32/0.47 thf(fact_183_decseqD,axiom,
% 0.32/0.47 ! [F: nat > nat,I: nat,J: nat] :
% 0.32/0.47 ( ( order_1631207636at_nat @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.47 => ( ord_less_eq_nat @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % decseqD
% 0.32/0.47 thf(fact_184_decseqD,axiom,
% 0.32/0.47 ! [F: nat > real,I: nat,J: nat] :
% 0.32/0.47 ( ( order_106095024t_real @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.47 => ( ord_less_eq_real @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % decseqD
% 0.32/0.47 thf(fact_185_decseq__def,axiom,
% 0.32/0.47 ( order_736687562_ereal
% 0.32/0.47 = ( ^ [X5: nat > extended_ereal] :
% 0.32/0.47 ! [M2: nat,N2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % decseq_def
% 0.32/0.47 thf(fact_186_decseq__def,axiom,
% 0.32/0.47 ( order_1660553314d_enat
% 0.32/0.47 = ( ^ [X5: nat > extended_enat] :
% 0.32/0.47 ! [M2: nat,N2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % decseq_def
% 0.32/0.47 thf(fact_187_decseq__def,axiom,
% 0.32/0.47 ( order_1631207636at_nat
% 0.32/0.47 = ( ^ [X5: nat > nat] :
% 0.32/0.47 ! [M2: nat,N2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.32/0.47 => ( ord_less_eq_nat @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % decseq_def
% 0.32/0.47 thf(fact_188_decseq__def,axiom,
% 0.32/0.47 ( order_106095024t_real
% 0.32/0.47 = ( ^ [X5: nat > real] :
% 0.32/0.47 ! [M2: nat,N2: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.32/0.47 => ( ord_less_eq_real @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % decseq_def
% 0.32/0.47 thf(fact_189_ereal__of__enat__nonneg,axiom,
% 0.32/0.47 ! [N: extended_enat] : ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extend1771934483f_enat @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_of_enat_nonneg
% 0.32/0.47 thf(fact_190_ereal__of__enat__ge__zero__cancel__iff,axiom,
% 0.32/0.47 ! [N: extended_enat] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extend1771934483f_enat @ N ) )
% 0.32/0.47 = ( ord_le1863327750d_enat @ zero_z491942557d_enat @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_of_enat_ge_zero_cancel_iff
% 0.32/0.47 thf(fact_191_numeral__le__ereal__of__enat__iff,axiom,
% 0.32/0.47 ! [M: num,N: extended_enat] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ ( numera1793320307_ereal @ M ) @ ( extend1771934483f_enat @ N ) )
% 0.32/0.47 = ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % numeral_le_ereal_of_enat_iff
% 0.32/0.47 thf(fact_192_ereal__infty__less__eq_I1_J,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ extend1289208545_ereal @ X2 )
% 0.32/0.47 = ( X2 = extend1289208545_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_infty_less_eq(1)
% 0.32/0.47 thf(fact_193_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: extended_ereal > extended_ereal,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_555877177_ereal @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_194_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: extended_enat > extended_ereal,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_1180956065_ereal @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_195_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: nat > extended_ereal,X2: nat,Y3: nat] :
% 0.32/0.47 ( ( order_689807875_ereal @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_196_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: real > extended_ereal,X2: real,Y3: real] :
% 0.32/0.47 ( ( order_946214183_ereal @ F )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_197_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: extended_ereal > extended_enat,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_713858385d_enat @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_198_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: extended_enat > extended_enat,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_2106278841d_enat @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_199_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: nat > extended_enat,X2: nat,Y3: nat] :
% 0.32/0.47 ( ( order_693207323d_enat @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_200_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: real > extended_enat,X2: real,Y3: real] :
% 0.32/0.47 ( ( order_1962034751d_enat @ F )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_201_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: extended_ereal > nat,X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( order_476926757al_nat @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_202_strict__mono__less__eq,axiom,
% 0.32/0.47 ! [F: extended_enat > nat,X2: extended_enat,Y3: extended_enat] :
% 0.32/0.47 ( ( order_1622825661at_nat @ F )
% 0.32/0.47 => ( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 0.32/0.47 = ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_less_eq
% 0.32/0.47 thf(fact_203_ereal__of__enat__zero,axiom,
% 0.32/0.47 ( ( extend1771934483f_enat @ zero_z491942557d_enat )
% 0.32/0.47 = zero_z163181189_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_of_enat_zero
% 0.32/0.47 thf(fact_204_strict__mono__eq,axiom,
% 0.32/0.47 ! [F: nat > nat,X2: nat,Y3: nat] :
% 0.32/0.47 ( ( order_769474267at_nat @ F )
% 0.32/0.47 => ( ( ( F @ X2 )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 = ( X2 = Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_eq
% 0.32/0.47 thf(fact_205_Infty__neq__0_I1_J,axiom,
% 0.32/0.47 extend1289208545_ereal != zero_z163181189_ereal ).
% 0.32/0.47
% 0.32/0.47 % Infty_neq_0(1)
% 0.32/0.47 thf(fact_206_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: extended_ereal > extended_ereal,M: extended_ereal,N: extended_ereal] :
% 0.32/0.47 ( ( order_555877177_ereal @ R )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ M @ N )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_207_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: extended_ereal > extended_enat,M: extended_ereal,N: extended_ereal] :
% 0.32/0.47 ( ( order_713858385d_enat @ R )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ M @ N )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_208_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: extended_ereal > nat,M: extended_ereal,N: extended_ereal] :
% 0.32/0.47 ( ( order_476926757al_nat @ R )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ M @ N )
% 0.32/0.47 => ( ord_less_eq_nat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_209_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: extended_ereal > real,M: extended_ereal,N: extended_ereal] :
% 0.32/0.47 ( ( order_1560271745l_real @ R )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ M @ N )
% 0.32/0.47 => ( ord_less_eq_real @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_210_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: extended_enat > extended_ereal,M: extended_enat,N: extended_enat] :
% 0.32/0.47 ( ( order_1180956065_ereal @ R )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ M @ N )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_211_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: extended_enat > extended_enat,M: extended_enat,N: extended_enat] :
% 0.32/0.47 ( ( order_2106278841d_enat @ R )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ M @ N )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_212_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: extended_enat > nat,M: extended_enat,N: extended_enat] :
% 0.32/0.47 ( ( order_1622825661at_nat @ R )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ M @ N )
% 0.32/0.47 => ( ord_less_eq_nat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_213_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: extended_enat > real,M: extended_enat,N: extended_enat] :
% 0.32/0.47 ( ( order_135436057t_real @ R )
% 0.32/0.47 => ( ( ord_le1863327750d_enat @ M @ N )
% 0.32/0.47 => ( ord_less_eq_real @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_214_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: nat > extended_ereal,M: nat,N: nat] :
% 0.32/0.47 ( ( order_689807875_ereal @ R )
% 0.32/0.47 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_215_strict__mono__leD,axiom,
% 0.32/0.47 ! [R: nat > extended_enat,M: nat,N: nat] :
% 0.32/0.47 ( ( order_693207323d_enat @ R )
% 0.32/0.47 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.47 => ( ord_le1863327750d_enat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_leD
% 0.32/0.47 thf(fact_216_neq__PInf__trans,axiom,
% 0.32/0.47 ! [Y3: extended_ereal,X2: extended_ereal] :
% 0.32/0.47 ( ( Y3 != extend1289208545_ereal )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( X2 != extend1289208545_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % neq_PInf_trans
% 0.32/0.47 thf(fact_217_ereal__infty__less__eq2_I1_J,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( ( A = extend1289208545_ereal )
% 0.32/0.47 => ( B = extend1289208545_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_infty_less_eq2(1)
% 0.32/0.47 thf(fact_218_ereal__less__eq_I1_J,axiom,
% 0.32/0.47 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ extend1289208545_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(1)
% 0.32/0.47 thf(fact_219_numeral__le__iff,axiom,
% 0.32/0.47 ! [M: num,N: num] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ ( numera280919179d_enat @ N ) )
% 0.32/0.47 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % numeral_le_iff
% 0.32/0.47 thf(fact_220_numeral__le__iff,axiom,
% 0.32/0.47 ! [M: num,N: num] :
% 0.32/0.47 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.47 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % numeral_le_iff
% 0.32/0.47 thf(fact_221_numeral__le__iff,axiom,
% 0.32/0.47 ! [M: num,N: num] :
% 0.32/0.47 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.47 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % numeral_le_iff
% 0.32/0.47 thf(fact_222_le__zero__eq,axiom,
% 0.32/0.47 ! [N: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ N @ zero_z491942557d_enat )
% 0.32/0.47 = ( N = zero_z491942557d_enat ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_zero_eq
% 0.32/0.47 thf(fact_223_le__zero__eq,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/0.47 = ( N = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_zero_eq
% 0.32/0.47 thf(fact_224_zero__le__numeral,axiom,
% 0.32/0.47 ! [N: num] : ( ord_le1863327750d_enat @ zero_z491942557d_enat @ ( numera280919179d_enat @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le_numeral
% 0.32/0.47 thf(fact_225_zero__le__numeral,axiom,
% 0.32/0.47 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le_numeral
% 0.32/0.47 thf(fact_226_zero__le__numeral,axiom,
% 0.32/0.47 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le_numeral
% 0.32/0.47 thf(fact_227_not__numeral__le__zero,axiom,
% 0.32/0.47 ! [N: num] :
% 0.32/0.47 ~ ( ord_le1863327750d_enat @ ( numera280919179d_enat @ N ) @ zero_z491942557d_enat ) ).
% 0.32/0.47
% 0.32/0.47 % not_numeral_le_zero
% 0.32/0.47 thf(fact_228_not__numeral__le__zero,axiom,
% 0.32/0.47 ! [N: num] :
% 0.32/0.47 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.32/0.47
% 0.32/0.47 % not_numeral_le_zero
% 0.32/0.47 thf(fact_229_not__numeral__le__zero,axiom,
% 0.32/0.47 ! [N: num] :
% 0.32/0.47 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.32/0.47
% 0.32/0.47 % not_numeral_le_zero
% 0.32/0.47 thf(fact_230_numeral__eq__iff,axiom,
% 0.32/0.47 ! [M: num,N: num] :
% 0.32/0.47 ( ( ( numeral_numeral_real @ M )
% 0.32/0.47 = ( numeral_numeral_real @ N ) )
% 0.32/0.47 = ( M = N ) ) ).
% 0.32/0.47
% 0.32/0.47 % numeral_eq_iff
% 0.32/0.47 thf(fact_231_ile0__eq,axiom,
% 0.32/0.47 ! [N: extended_enat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ N @ zero_z491942557d_enat )
% 0.32/0.47 = ( N = zero_z491942557d_enat ) ) ).
% 0.32/0.47
% 0.32/0.47 % ile0_eq
% 0.32/0.47 thf(fact_232_i0__lb,axiom,
% 0.32/0.47 ! [N: extended_enat] : ( ord_le1863327750d_enat @ zero_z491942557d_enat @ N ) ).
% 0.32/0.47
% 0.32/0.47 % i0_lb
% 0.32/0.47 thf(fact_233_enat__ord__number_I1_J,axiom,
% 0.32/0.47 ! [M: num,N: num] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ ( numera280919179d_enat @ N ) )
% 0.32/0.47 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % enat_ord_number(1)
% 0.32/0.47 thf(fact_234_bot__nat__0_Oextremum,axiom,
% 0.32/0.47 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum
% 0.32/0.47 thf(fact_235_le0,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/0.47
% 0.32/0.47 % le0
% 0.32/0.47 thf(fact_236_numeral__le__enat__iff,axiom,
% 0.32/0.47 ! [M: num,N: nat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ ( extended_enat2 @ N ) )
% 0.32/0.47 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % numeral_le_enat_iff
% 0.32/0.47 thf(fact_237_real__of__ereal__le__0,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq_real @ ( extend1716541707_ereal @ X2 ) @ zero_zero_real )
% 0.32/0.47 = ( ( ord_le824540014_ereal @ X2 @ zero_z163181189_ereal )
% 0.32/0.47 | ( X2 = extend1289208545_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal_le_0
% 0.32/0.47 thf(fact_238_ereal__uminus__zero__iff,axiom,
% 0.32/0.47 ! [A: extended_ereal] :
% 0.32/0.47 ( ( ( uminus1208298309_ereal @ A )
% 0.32/0.47 = zero_z163181189_ereal )
% 0.32/0.47 = ( A = zero_z163181189_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_zero_iff
% 0.32/0.47 thf(fact_239_ereal__uminus__zero,axiom,
% 0.32/0.47 ( ( uminus1208298309_ereal @ zero_z163181189_ereal )
% 0.32/0.47 = zero_z163181189_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_zero
% 0.32/0.47 thf(fact_240_ereal__minus__le__minus,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ ( uminus1208298309_ereal @ B ) )
% 0.32/0.47 = ( ord_le824540014_ereal @ B @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_minus_le_minus
% 0.32/0.47 thf(fact_241_enat_Oinject,axiom,
% 0.32/0.47 ! [Nat: nat,Nat2: nat] :
% 0.32/0.47 ( ( ( extended_enat2 @ Nat )
% 0.32/0.47 = ( extended_enat2 @ Nat2 ) )
% 0.32/0.47 = ( Nat = Nat2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % enat.inject
% 0.32/0.47 thf(fact_242_real__of__ereal,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( extend1716541707_ereal @ ( uminus1208298309_ereal @ X2 ) )
% 0.32/0.47 = ( uminus_uminus_real @ ( extend1716541707_ereal @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal
% 0.32/0.47 thf(fact_243_ereal__infty__less__eq_I2_J,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 = ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_infty_less_eq(2)
% 0.32/0.47 thf(fact_244_ereal__uminus__le__0__iff,axiom,
% 0.32/0.47 ! [A: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ zero_z163181189_ereal )
% 0.32/0.47 = ( ord_le824540014_ereal @ zero_z163181189_ereal @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_le_0_iff
% 0.32/0.47 thf(fact_245_ereal__0__le__uminus__iff,axiom,
% 0.32/0.47 ! [A: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( uminus1208298309_ereal @ A ) )
% 0.32/0.47 = ( ord_le824540014_ereal @ A @ zero_z163181189_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_0_le_uminus_iff
% 0.32/0.47 thf(fact_246_real__of__ereal__0,axiom,
% 0.32/0.47 ( ( extend1716541707_ereal @ zero_z163181189_ereal )
% 0.32/0.47 = zero_zero_real ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal_0
% 0.32/0.47 thf(fact_247_enat__ord__simps_I1_J,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.32/0.47 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % enat_ord_simps(1)
% 0.32/0.47 thf(fact_248_real__of__ereal_Osimps_I3_J,axiom,
% 0.32/0.47 ( ( extend1716541707_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 = zero_zero_real ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal.simps(3)
% 0.32/0.47 thf(fact_249_real__of__ereal__eq__0,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ( extend1716541707_ereal @ X2 )
% 0.32/0.47 = zero_zero_real )
% 0.32/0.47 = ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 | ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 | ( X2 = zero_z163181189_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal_eq_0
% 0.32/0.47 thf(fact_250_real__of__ereal_Osimps_I2_J,axiom,
% 0.32/0.47 ( ( extend1716541707_ereal @ extend1289208545_ereal )
% 0.32/0.47 = zero_zero_real ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal.simps(2)
% 0.32/0.47 thf(fact_251_zero__enat__def,axiom,
% 0.32/0.47 ( zero_z491942557d_enat
% 0.32/0.47 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_enat_def
% 0.32/0.47 thf(fact_252_enat__0__iff_I1_J,axiom,
% 0.32/0.47 ! [X2: nat] :
% 0.32/0.47 ( ( ( extended_enat2 @ X2 )
% 0.32/0.47 = zero_z491942557d_enat )
% 0.32/0.47 = ( X2 = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % enat_0_iff(1)
% 0.32/0.47 thf(fact_253_enat__0__iff_I2_J,axiom,
% 0.32/0.47 ! [X2: nat] :
% 0.32/0.47 ( ( zero_z491942557d_enat
% 0.32/0.47 = ( extended_enat2 @ X2 ) )
% 0.32/0.47 = ( X2 = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % enat_0_iff(2)
% 0.32/0.47 thf(fact_254_MInfty__neq__PInfty_I1_J,axiom,
% 0.32/0.47 ( extend1289208545_ereal
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % MInfty_neq_PInfty(1)
% 0.32/0.47 thf(fact_255_ereal__uminus__le__reorder,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ B )
% 0.32/0.47 = ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ B ) @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_le_reorder
% 0.32/0.47 thf(fact_256_enat__ile,axiom,
% 0.32/0.47 ! [N: extended_enat,M: nat] :
% 0.32/0.47 ( ( ord_le1863327750d_enat @ N @ ( extended_enat2 @ M ) )
% 0.32/0.47 => ? [K: nat] :
% 0.32/0.47 ( N
% 0.32/0.47 = ( extended_enat2 @ K ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % enat_ile
% 0.32/0.47 thf(fact_257_numeral__eq__enat,axiom,
% 0.32/0.47 ( numera280919179d_enat
% 0.32/0.47 = ( ^ [K2: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % numeral_eq_enat
% 0.32/0.47 thf(fact_258_real__of__ereal__pos,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
% 0.32/0.47 => ( ord_less_eq_real @ zero_zero_real @ ( extend1716541707_ereal @ X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal_pos
% 0.32/0.47 thf(fact_259_Infty__neq__0_I3_J,axiom,
% 0.32/0.47 ( ( uminus1208298309_ereal @ extend1289208545_ereal )
% 0.32/0.47 != zero_z163181189_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % Infty_neq_0(3)
% 0.32/0.47 thf(fact_260_ereal__infty__less__eq2_I2_J,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ B )
% 0.32/0.47 => ( ( B
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( A
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_infty_less_eq2(2)
% 0.32/0.47 thf(fact_261_ereal__less__eq_I2_J,axiom,
% 0.32/0.47 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X2 ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(2)
% 0.32/0.47 thf(fact_262_real__of__ereal__positive__mono,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ( Y3 != extend1289208545_ereal )
% 0.32/0.47 => ( ord_less_eq_real @ ( extend1716541707_ereal @ X2 ) @ ( extend1716541707_ereal @ Y3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal_positive_mono
% 0.32/0.47 thf(fact_263_not__MInfty__nonneg,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
% 0.32/0.47 => ( X2
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_MInfty_nonneg
% 0.32/0.47 thf(fact_264_le__refl,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.32/0.47
% 0.32/0.47 % le_refl
% 0.32/0.47 thf(fact_265_le__trans,axiom,
% 0.32/0.47 ! [I: nat,J: nat,K3: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.47 => ( ( ord_less_eq_nat @ J @ K3 )
% 0.32/0.47 => ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_trans
% 0.32/0.47 thf(fact_266_eq__imp__le,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( M = N )
% 0.32/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % eq_imp_le
% 0.32/0.47 thf(fact_267_le__antisym,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.47 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.47 => ( M = N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_antisym
% 0.32/0.47 thf(fact_268_GreatestI__nat,axiom,
% 0.32/0.47 ! [P: nat > $o,K3: nat,B: nat] :
% 0.32/0.47 ( ( P @ K3 )
% 0.32/0.47 => ( ! [Y: nat] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ Y @ B ) )
% 0.32/0.47 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % GreatestI_nat
% 0.32/0.47 thf(fact_269_nat__le__linear,axiom,
% 0.32/0.47 ! [M: nat,N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.47 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_le_linear
% 0.32/0.47 thf(fact_270_Greatest__le__nat,axiom,
% 0.32/0.47 ! [P: nat > $o,K3: nat,B: nat] :
% 0.32/0.47 ( ( P @ K3 )
% 0.32/0.47 => ( ! [Y: nat] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ Y @ B ) )
% 0.32/0.47 => ( ord_less_eq_nat @ K3 @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Greatest_le_nat
% 0.32/0.47 thf(fact_271_GreatestI__ex__nat,axiom,
% 0.32/0.47 ! [P: nat > $o,B: nat] :
% 0.32/0.47 ( ? [X_1: nat] : ( P @ X_1 )
% 0.32/0.47 => ( ! [Y: nat] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ Y @ B ) )
% 0.32/0.47 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % GreatestI_ex_nat
% 0.32/0.47 thf(fact_272_Nat_Oex__has__greatest__nat,axiom,
% 0.32/0.47 ! [P: nat > $o,K3: nat,B: nat] :
% 0.32/0.47 ( ( P @ K3 )
% 0.32/0.47 => ( ! [Y: nat] :
% 0.32/0.47 ( ( P @ Y )
% 0.32/0.47 => ( ord_less_eq_nat @ Y @ B ) )
% 0.32/0.47 => ? [X3: nat] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 & ! [Y5: nat] :
% 0.32/0.47 ( ( P @ Y5 )
% 0.32/0.47 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Nat.ex_has_greatest_nat
% 0.32/0.47 thf(fact_273_strict__mono__imp__increasing,axiom,
% 0.32/0.47 ! [F: nat > nat,N: nat] :
% 0.32/0.47 ( ( order_769474267at_nat @ F )
% 0.32/0.47 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % strict_mono_imp_increasing
% 0.32/0.47 thf(fact_274_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.32/0.47 ! [A: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.47 => ( A = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum_uniqueI
% 0.32/0.47 thf(fact_275_bot__nat__0_Oextremum__unique,axiom,
% 0.32/0.47 ! [A: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.47 = ( A = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % bot_nat_0.extremum_unique
% 0.32/0.47 thf(fact_276_le__0__eq,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/0.47 = ( N = zero_zero_nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_0_eq
% 0.32/0.47 thf(fact_277_less__eq__nat_Osimps_I1_J,axiom,
% 0.32/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/0.47
% 0.32/0.47 % less_eq_nat.simps(1)
% 0.32/0.47 thf(fact_278_the__enat_Osimps,axiom,
% 0.32/0.47 ! [N: nat] :
% 0.32/0.47 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 0.32/0.47 = N ) ).
% 0.32/0.47
% 0.32/0.47 % the_enat.simps
% 0.32/0.47 thf(fact_279_ereal__uminus__uminus,axiom,
% 0.32/0.47 ! [A: extended_ereal] :
% 0.32/0.47 ( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A ) )
% 0.32/0.47 = A ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_uminus
% 0.32/0.47 thf(fact_280_ereal__uminus__eq__iff,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal] :
% 0.32/0.47 ( ( ( uminus1208298309_ereal @ A )
% 0.32/0.47 = ( uminus1208298309_ereal @ B ) )
% 0.32/0.47 = ( A = B ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_eq_iff
% 0.32/0.47 thf(fact_281_ereal__uminus__eq__reorder,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal] :
% 0.32/0.47 ( ( ( uminus1208298309_ereal @ A )
% 0.32/0.47 = B )
% 0.32/0.47 = ( A
% 0.32/0.47 = ( uminus1208298309_ereal @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_eq_reorder
% 0.32/0.47 thf(fact_282_real__eq__0__iff__le__ge__0,axiom,
% 0.32/0.47 ! [X2: real] :
% 0.32/0.47 ( ( X2 = zero_zero_real )
% 0.32/0.47 = ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.32/0.47 & ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_eq_0_iff_le_ge_0
% 0.32/0.47 thf(fact_283_ereal__less__eq_I5_J,axiom,
% 0.32/0.47 ! [R: real] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extended_ereal2 @ R ) )
% 0.32/0.47 = ( ord_less_eq_real @ zero_zero_real @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(5)
% 0.32/0.47 thf(fact_284_ereal__less__eq_I4_J,axiom,
% 0.32/0.47 ! [R: real] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R ) @ zero_z163181189_ereal )
% 0.32/0.47 = ( ord_less_eq_real @ R @ zero_zero_real ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(4)
% 0.32/0.47 thf(fact_285_MInfty__eq__minfinity,axiom,
% 0.32/0.47 ( extended_MInfty
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % MInfty_eq_minfinity
% 0.32/0.47 thf(fact_286_ereal__cong,axiom,
% 0.32/0.47 ! [X2: real,Y3: real] :
% 0.32/0.47 ( ( X2 = Y3 )
% 0.32/0.47 => ( ( extended_ereal2 @ X2 )
% 0.32/0.47 = ( extended_ereal2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_cong
% 0.32/0.47 thf(fact_287_ereal_Oinject,axiom,
% 0.32/0.47 ! [X1: real,Y1: real] :
% 0.32/0.47 ( ( ( extended_ereal2 @ X1 )
% 0.32/0.47 = ( extended_ereal2 @ Y1 ) )
% 0.32/0.47 = ( X1 = Y1 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.inject
% 0.32/0.47 thf(fact_288_numeral__eq__ereal,axiom,
% 0.32/0.47 ( numera1793320307_ereal
% 0.32/0.47 = ( ^ [W: num] : ( extended_ereal2 @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % numeral_eq_ereal
% 0.32/0.47 thf(fact_289_ereal__eq__0_I1_J,axiom,
% 0.32/0.47 ! [R: real] :
% 0.32/0.47 ( ( ( extended_ereal2 @ R )
% 0.32/0.47 = zero_z163181189_ereal )
% 0.32/0.47 = ( R = zero_zero_real ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_eq_0(1)
% 0.32/0.47 thf(fact_290_ereal__eq__0_I2_J,axiom,
% 0.32/0.47 ! [R: real] :
% 0.32/0.47 ( ( zero_z163181189_ereal
% 0.32/0.47 = ( extended_ereal2 @ R ) )
% 0.32/0.47 = ( R = zero_zero_real ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_eq_0(2)
% 0.32/0.47 thf(fact_291_ereal__less__eq_I3_J,axiom,
% 0.32/0.47 ! [R: real,P2: real] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
% 0.32/0.47 = ( ord_less_eq_real @ R @ P2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_less_eq(3)
% 0.32/0.47 thf(fact_292_ereal_Odistinct_I3_J,axiom,
% 0.32/0.47 ! [X1: real] :
% 0.32/0.47 ( ( extended_ereal2 @ X1 )
% 0.32/0.47 != extended_MInfty ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.distinct(3)
% 0.32/0.47 thf(fact_293_real__of__ereal_Osimps_I1_J,axiom,
% 0.32/0.47 ! [R: real] :
% 0.32/0.47 ( ( extend1716541707_ereal @ ( extended_ereal2 @ R ) )
% 0.32/0.47 = R ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal.simps(1)
% 0.32/0.47 thf(fact_294_ereal__le__real,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ! [Z4: real] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
% 0.32/0.47 => ( ord_le824540014_ereal @ Y3 @ ( extended_ereal2 @ Z4 ) ) )
% 0.32/0.47 => ( ord_le824540014_ereal @ Y3 @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_le_real
% 0.32/0.47 thf(fact_295_PInfty__neq__ereal_I1_J,axiom,
% 0.32/0.47 ! [R: real] :
% 0.32/0.47 ( ( extended_ereal2 @ R )
% 0.32/0.47 != extend1289208545_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % PInfty_neq_ereal(1)
% 0.32/0.47 thf(fact_296_zero__ereal__def,axiom,
% 0.32/0.47 ( zero_z163181189_ereal
% 0.32/0.47 = ( extended_ereal2 @ zero_zero_real ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_ereal_def
% 0.32/0.47 thf(fact_297_le__ereal__le,axiom,
% 0.32/0.47 ! [A: extended_ereal,X2: real,Y3: real] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ X2 ) )
% 0.32/0.47 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ Y3 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % le_ereal_le
% 0.32/0.47 thf(fact_298_ereal__le__le,axiom,
% 0.32/0.47 ! [Y3: real,A: extended_ereal,X2: real] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ Y3 ) @ A )
% 0.32/0.47 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_le_le
% 0.32/0.47 thf(fact_299_real__of__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.32/0.47 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( P @ extend1289208545_ereal )
% 0.32/0.47 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( P @ A0 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal.induct
% 0.32/0.47 thf(fact_300_real__of__ereal_Ocases,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ! [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 != ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( X2 != extend1289208545_ereal )
% 0.32/0.47 => ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal.cases
% 0.32/0.47 thf(fact_301_times__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.32/0.47 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 0.32/0.47 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ extend1289208545_ereal )
% 0.32/0.47 => ( ! [R2: real] : ( P @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( P @ extend1289208545_ereal @ extend1289208545_ereal )
% 0.32/0.47 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 0.32/0.47 => ( ( P @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % times_ereal.induct
% 0.32/0.47 thf(fact_302_plus__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.32/0.47 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 0.32/0.47 => ( ! [X_12: extended_ereal] : ( P @ extend1289208545_ereal @ X_12 )
% 0.32/0.47 => ( ! [A3: extended_ereal] : ( P @ A3 @ extend1289208545_ereal )
% 0.32/0.47 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ! [P3: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P3 ) )
% 0.32/0.47 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % plus_ereal.induct
% 0.32/0.47 thf(fact_303_less__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 0.32/0.47 ( ! [X3: real,Y: real] : ( P @ ( extended_ereal2 @ X3 ) @ ( extended_ereal2 @ Y ) )
% 0.32/0.47 => ( ! [X_12: extended_ereal] : ( P @ extend1289208545_ereal @ X_12 )
% 0.32/0.47 => ( ! [A3: extended_ereal] : ( P @ A3 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) @ extend1289208545_ereal )
% 0.32/0.47 => ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 0.32/0.47 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_ereal.induct
% 0.32/0.47 thf(fact_304_abs__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.32/0.47 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ( P @ extend1289208545_ereal )
% 0.32/0.47 => ( P @ A0 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal.induct
% 0.32/0.47 thf(fact_305_ereal__all__split,axiom,
% 0.32/0.47 ( ( ^ [P4: extended_ereal > $o] :
% 0.32/0.47 ! [X6: extended_ereal] : ( P4 @ X6 ) )
% 0.32/0.47 = ( ^ [P5: extended_ereal > $o] :
% 0.32/0.47 ( ( P5 @ extend1289208545_ereal )
% 0.32/0.47 & ! [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
% 0.32/0.47 & ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_all_split
% 0.32/0.47 thf(fact_306_abs__ereal_Ocases,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ! [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 != ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( X2
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( X2 = extend1289208545_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal.cases
% 0.32/0.47 thf(fact_307_ereal__ex__split,axiom,
% 0.32/0.47 ( ( ^ [P4: extended_ereal > $o] :
% 0.32/0.47 ? [X6: extended_ereal] : ( P4 @ X6 ) )
% 0.32/0.47 = ( ^ [P5: extended_ereal > $o] :
% 0.32/0.47 ( ( P5 @ extend1289208545_ereal )
% 0.32/0.47 | ? [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
% 0.32/0.47 | ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_ex_split
% 0.32/0.47 thf(fact_308_ereal3__cases,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 0.32/0.47 ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ? [Ra: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ Ra ) )
% 0.32/0.47 => ! [Rb: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Rb ) ) ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ? [Ra: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ Ra ) )
% 0.32/0.47 => ( Xb != extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ? [Ra: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ Ra ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( Xa3 = extend1289208545_ereal )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( Xa3 = extend1289208545_ereal )
% 0.32/0.47 => ( Xb != extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( Xa3 = extend1289208545_ereal )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( Xb != extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( ? [R2: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( ? [R2: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( Xb != extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( ? [R2: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( ( Xa3 = extend1289208545_ereal )
% 0.32/0.47 => ! [R2: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ R2 ) ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( ( Xa3 = extend1289208545_ereal )
% 0.32/0.47 => ( Xb != extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( ( Xa3 = extend1289208545_ereal )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ! [R2: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ R2 ) ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( Xb != extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ? [R2: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ? [R2: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( Xb != extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ? [R2: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ( Xa3 = extend1289208545_ereal )
% 0.32/0.47 => ! [R2: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ R2 ) ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ( Xa3 = extend1289208545_ereal )
% 0.32/0.47 => ( Xb != extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ( Xa3 = extend1289208545_ereal )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ! [R2: real] :
% 0.32/0.47 ( Xb
% 0.32/0.47 != ( extended_ereal2 @ R2 ) ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( Xb != extend1289208545_ereal ) ) )
% 0.32/0.47 => ~ ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ( Xa3
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( Xb
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal3_cases
% 0.32/0.47 thf(fact_309_ereal2__cases,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Xa3: extended_ereal] :
% 0.32/0.47 ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ! [Ra: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 != ( extended_ereal2 @ Ra ) ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( Xa3 != extend1289208545_ereal ) )
% 0.32/0.47 => ( ( ? [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( Xa3
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ! [R2: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 != ( extended_ereal2 @ R2 ) ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( Xa3 != extend1289208545_ereal ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( Xa3
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ! [R2: real] :
% 0.32/0.47 ( Xa3
% 0.32/0.47 != ( extended_ereal2 @ R2 ) ) )
% 0.32/0.47 => ( ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( Xa3 != extend1289208545_ereal ) )
% 0.32/0.47 => ~ ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( Xa3
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal2_cases
% 0.32/0.47 thf(fact_310_ereal__cases,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ! [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 != ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( X2 != extend1289208545_ereal )
% 0.32/0.47 => ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_cases
% 0.32/0.47 thf(fact_311_MInfty__neq__ereal_I1_J,axiom,
% 0.32/0.47 ! [R: real] :
% 0.32/0.47 ( ( extended_ereal2 @ R )
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % MInfty_neq_ereal(1)
% 0.32/0.47 thf(fact_312_ereal__top,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ! [B4: real] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B4 ) @ X2 )
% 0.32/0.47 => ( X2 = extend1289208545_ereal ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_top
% 0.32/0.47 thf(fact_313_uminus__ereal_Osimps_I1_J,axiom,
% 0.32/0.47 ! [R: real] :
% 0.32/0.47 ( ( uminus1208298309_ereal @ ( extended_ereal2 @ R ) )
% 0.32/0.47 = ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.simps(1)
% 0.32/0.47 thf(fact_314_ereal__bot,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ! [B4: real] : ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ B4 ) )
% 0.32/0.47 => ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_bot
% 0.32/0.47 thf(fact_315_real__of__ereal_Oelims,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: real] :
% 0.32/0.47 ( ( ( extend1716541707_ereal @ X2 )
% 0.32/0.47 = Y3 )
% 0.32/0.47 => ( ! [R2: real] :
% 0.32/0.47 ( ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( Y3 != R2 ) )
% 0.32/0.47 => ( ( ( X2 = extend1289208545_ereal )
% 0.32/0.47 => ( Y3 != zero_zero_real ) )
% 0.32/0.47 => ~ ( ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( Y3 != zero_zero_real ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_of_ereal.elims
% 0.32/0.47 thf(fact_316_ereal_Osize__gen_I3_J,axiom,
% 0.32/0.47 ( ( extended_size_ereal @ extended_MInfty )
% 0.32/0.47 = zero_zero_nat ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.size_gen(3)
% 0.32/0.47 thf(fact_317_ereal_Osize__gen_I1_J,axiom,
% 0.32/0.47 ! [X1: real] :
% 0.32/0.47 ( ( extended_size_ereal @ ( extended_ereal2 @ X1 ) )
% 0.32/0.47 = zero_zero_nat ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.size_gen(1)
% 0.32/0.47 thf(fact_318_uminus__ereal_Oelims,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( ( uminus1208298309_ereal @ X2 )
% 0.32/0.47 = Y3 )
% 0.32/0.47 => ( ! [R2: real] :
% 0.32/0.47 ( ( X2
% 0.32/0.47 = ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( Y3
% 0.32/0.47 != ( extended_ereal2 @ ( uminus_uminus_real @ R2 ) ) ) )
% 0.32/0.47 => ( ( ( X2 = extended_PInfty )
% 0.32/0.47 => ( Y3 != extended_MInfty ) )
% 0.32/0.47 => ~ ( ( X2 = extended_MInfty )
% 0.32/0.47 => ( Y3 != extended_PInfty ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.elims
% 0.32/0.47 thf(fact_319_ereal_Odistinct_I1_J,axiom,
% 0.32/0.47 ! [X1: real] :
% 0.32/0.47 ( ( extended_ereal2 @ X1 )
% 0.32/0.47 != extended_PInfty ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.distinct(1)
% 0.32/0.47 thf(fact_320_ereal_Osize__gen_I2_J,axiom,
% 0.32/0.47 ( ( extended_size_ereal @ extended_PInfty )
% 0.32/0.47 = zero_zero_nat ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.size_gen(2)
% 0.32/0.47 thf(fact_321_infinity__ereal__def,axiom,
% 0.32/0.47 extend1289208545_ereal = extended_PInfty ).
% 0.32/0.47
% 0.32/0.47 % infinity_ereal_def
% 0.32/0.47 thf(fact_322_ereal_Odistinct_I5_J,axiom,
% 0.32/0.47 extended_PInfty != extended_MInfty ).
% 0.32/0.47
% 0.32/0.47 % ereal.distinct(5)
% 0.32/0.47 thf(fact_323_uminus__ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 0.32/0.47 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( P @ extended_PInfty )
% 0.32/0.47 => ( ( P @ extended_MInfty )
% 0.32/0.47 => ( P @ A0 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.induct
% 0.32/0.47 thf(fact_324_uminus__ereal_Ocases,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ! [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 != ( extended_ereal2 @ R2 ) )
% 0.32/0.47 => ( ( X2 != extended_PInfty )
% 0.32/0.47 => ( X2 = extended_MInfty ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.cases
% 0.32/0.47 thf(fact_325_ereal_Oexhaust,axiom,
% 0.32/0.47 ! [Y3: extended_ereal] :
% 0.32/0.47 ( ! [X12: real] :
% 0.32/0.47 ( Y3
% 0.32/0.47 != ( extended_ereal2 @ X12 ) )
% 0.32/0.47 => ( ( Y3 != extended_PInfty )
% 0.32/0.47 => ( Y3 = extended_MInfty ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.exhaust
% 0.32/0.47 thf(fact_326_ereal_Oinduct,axiom,
% 0.32/0.47 ! [P: extended_ereal > $o,Ereal: extended_ereal] :
% 0.32/0.47 ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) )
% 0.32/0.47 => ( ( P @ extended_PInfty )
% 0.32/0.47 => ( ( P @ extended_MInfty )
% 0.32/0.47 => ( P @ Ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal.induct
% 0.32/0.47 thf(fact_327_uminus__ereal_Osimps_I2_J,axiom,
% 0.32/0.47 ( ( uminus1208298309_ereal @ extended_PInfty )
% 0.32/0.47 = extended_MInfty ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.simps(2)
% 0.32/0.47 thf(fact_328_uminus__ereal_Osimps_I3_J,axiom,
% 0.32/0.47 ( ( uminus1208298309_ereal @ extended_MInfty )
% 0.32/0.47 = extended_PInfty ) ).
% 0.32/0.47
% 0.32/0.47 % uminus_ereal.simps(3)
% 0.32/0.47 thf(fact_329_ereal__divide__ereal,axiom,
% 0.32/0.47 ! [R: real] :
% 0.32/0.47 ( ( ( ord_less_eq_real @ zero_zero_real @ R )
% 0.32/0.47 => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R ) )
% 0.32/0.47 = extend1289208545_ereal ) )
% 0.32/0.47 & ( ~ ( ord_less_eq_real @ zero_zero_real @ R )
% 0.32/0.47 => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R ) )
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide_ereal
% 0.32/0.47 thf(fact_330_ereal__le__real__iff,axiom,
% 0.32/0.47 ! [X2: real,Y3: extended_ereal] :
% 0.32/0.47 ( ( ord_less_eq_real @ X2 @ ( extend1716541707_ereal @ Y3 ) )
% 0.32/0.47 = ( ( ( ( abs_ab1260901297_ereal @ Y3 )
% 0.32/0.47 != extend1289208545_ereal )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( extended_ereal2 @ X2 ) @ Y3 ) )
% 0.32/0.47 & ( ( ( abs_ab1260901297_ereal @ Y3 )
% 0.32/0.47 = extend1289208545_ereal )
% 0.32/0.47 => ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_le_real_iff
% 0.32/0.47 thf(fact_331_abs__ereal__uminus,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( abs_ab1260901297_ereal @ ( uminus1208298309_ereal @ X2 ) )
% 0.32/0.47 = ( abs_ab1260901297_ereal @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal_uminus
% 0.32/0.47 thf(fact_332_abs__ereal__zero,axiom,
% 0.32/0.47 ( ( abs_ab1260901297_ereal @ zero_z163181189_ereal )
% 0.32/0.47 = zero_z163181189_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal_zero
% 0.32/0.47 thf(fact_333_ereal__uminus__divide,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( divide595620860_ereal @ ( uminus1208298309_ereal @ X2 ) @ Y3 )
% 0.32/0.47 = ( uminus1208298309_ereal @ ( divide595620860_ereal @ X2 @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_uminus_divide
% 0.32/0.47 thf(fact_334_ereal__divide__zero__left,axiom,
% 0.32/0.47 ! [A: extended_ereal] :
% 0.32/0.47 ( ( divide595620860_ereal @ zero_z163181189_ereal @ A )
% 0.32/0.47 = zero_z163181189_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide_zero_left
% 0.32/0.47 thf(fact_335_abs__ereal__ge0,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
% 0.32/0.47 => ( ( abs_ab1260901297_ereal @ X2 )
% 0.32/0.47 = X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal_ge0
% 0.32/0.47 thf(fact_336_ereal__divide__Infty_I1_J,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( divide595620860_ereal @ X2 @ extend1289208545_ereal )
% 0.32/0.47 = zero_z163181189_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide_Infty(1)
% 0.32/0.47 thf(fact_337_ereal__divide__Infty_I2_J,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( divide595620860_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 = zero_z163181189_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide_Infty(2)
% 0.32/0.47 thf(fact_338_not__infty__ereal,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ( abs_ab1260901297_ereal @ X2 )
% 0.32/0.47 != extend1289208545_ereal )
% 0.32/0.47 = ( ? [X7: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 = ( extended_ereal2 @ X7 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % not_infty_ereal
% 0.32/0.47 thf(fact_339_abs__neq__infinity__cases,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ( abs_ab1260901297_ereal @ X2 )
% 0.32/0.47 != extend1289208545_ereal )
% 0.32/0.47 => ~ ! [R2: real] :
% 0.32/0.47 ( X2
% 0.32/0.47 != ( extended_ereal2 @ R2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % abs_neq_infinity_cases
% 0.32/0.47 thf(fact_340_abs__ereal_Osimps_I2_J,axiom,
% 0.32/0.47 ( ( abs_ab1260901297_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 = extend1289208545_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal.simps(2)
% 0.32/0.47 thf(fact_341_ereal__infinity__cases,axiom,
% 0.32/0.47 ! [A: extended_ereal] :
% 0.32/0.47 ( ( A != extend1289208545_ereal )
% 0.32/0.47 => ( ( A
% 0.32/0.47 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 0.32/0.47 => ( ( abs_ab1260901297_ereal @ A )
% 0.32/0.47 != extend1289208545_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_infinity_cases
% 0.32/0.47 thf(fact_342_abs__eq__infinity__cases,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ( abs_ab1260901297_ereal @ X2 )
% 0.32/0.47 = extend1289208545_ereal )
% 0.32/0.47 => ( ( X2 != extend1289208545_ereal )
% 0.32/0.47 => ( X2
% 0.32/0.47 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % abs_eq_infinity_cases
% 0.32/0.47 thf(fact_343_ereal__abs__leI,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ X2 ) @ Y3 )
% 0.32/0.47 => ( ord_le824540014_ereal @ ( abs_ab1260901297_ereal @ X2 ) @ Y3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_abs_leI
% 0.32/0.47 thf(fact_344_abs__ereal__pos,axiom,
% 0.32/0.47 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( abs_ab1260901297_ereal @ X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal_pos
% 0.32/0.47 thf(fact_345_zero__le__divide__ereal,axiom,
% 0.32/0.47 ! [A: extended_ereal,B: extended_ereal] :
% 0.32/0.47 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ A )
% 0.32/0.47 => ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ B )
% 0.32/0.47 => ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( divide595620860_ereal @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % zero_le_divide_ereal
% 0.32/0.47 thf(fact_346_ereal__divide__eq__0__iff,axiom,
% 0.32/0.47 ! [X2: extended_ereal,Y3: extended_ereal] :
% 0.32/0.47 ( ( ( divide595620860_ereal @ X2 @ Y3 )
% 0.32/0.47 = zero_z163181189_ereal )
% 0.32/0.47 = ( ( X2 = zero_z163181189_ereal )
% 0.32/0.47 | ( ( abs_ab1260901297_ereal @ Y3 )
% 0.32/0.47 = extend1289208545_ereal ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_divide_eq_0_iff
% 0.32/0.47 thf(fact_347_abs__ereal_Osimps_I3_J,axiom,
% 0.32/0.47 ( ( abs_ab1260901297_ereal @ extend1289208545_ereal )
% 0.32/0.47 = extend1289208545_ereal ) ).
% 0.32/0.47
% 0.32/0.47 % abs_ereal.simps(3)
% 0.32/0.47 thf(fact_348_ereal__real_H,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ( abs_ab1260901297_ereal @ X2 )
% 0.32/0.47 != extend1289208545_ereal )
% 0.32/0.47 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 0.32/0.47 = X2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_real'
% 0.32/0.47 thf(fact_349_ereal__real,axiom,
% 0.32/0.47 ! [X2: extended_ereal] :
% 0.32/0.47 ( ( ( ( abs_ab1260901297_ereal @ X2 )
% 0.32/0.47 = extend1289208545_ereal )
% 0.32/0.47 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 0.32/0.47 = zero_z163181189_ereal ) )
% 0.32/0.47 & ( ( ( abs_ab1260901297_ereal @ X2 )
% 0.32/0.47 != extend1289208545_ereal )
% 0.32/0.47 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 0.32/0.47 = X2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ereal_real
% 0.32/0.47 thf(fact_350_real__le__ereal__iff,axiom,
% 0.32/0.47 ! [Y3: extended_ereal,X2: real] :
% 0.32/0.47 ( ( ord_less_eq_real @ ( extend1716541707_ereal @ Y3 ) @ X2 )
% 0.32/0.47 = ( ( ( ( abs_ab1260901297_ereal @ Y3 )
% 0.32/0.47 != extend1289208545_ereal )
% 0.32/0.47 => ( ord_le824540014_ereal @ Y3 @ ( extended_ereal2 @ X2 ) ) )
% 0.32/0.47 & ( ( ( abs_ab1260901297_ereal @ Y3 )
% 0.32/0.47 = extend1289208545_ereal )
% 0.32/0.47 => ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % real_le_ereal_iff
% 0.32/0.47
% 0.32/0.47 % Conjectures (1)
% 0.32/0.47 thf(conj_0,conjecture,
% 0.32/0.47 ord_le824540014_ereal @ ( g @ x ) @ ( f @ x ) ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 2.87/1.71 ---- Embedded in THF ---
% 2.87/1.71 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.87/1.71 %%% Generated on Tue Feb 24 08:26:31 EST 2026
% 2.87/1.71 %%% using '$$dhol' embedding, version 1.3.0.
% 2.87/1.71 %%% Logic specification used:
% 2.87/1.71 %%% thf(spec, logic, $$dhol).
% 2.87/1.71
% 2.87/1.71 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.nLU4PliHtq/DTF2DTF7494.p
% See solution above
% 2.87/1.71 ------------------------
% 2.87/1.76 ---- Cleaned THF ---
% 2.87/1.76 thf(ty_n_t__Set__Oset_It__Extended____Real__Oereal_J,type,
% 2.87/1.76 set_Extended_ereal: $tType ).
% 2.87/1.76
% 2.87/1.76 thf(ty_n_t__Extended____Real__Oereal,type,
% 2.87/1.76 extended_ereal: $tType ).
% 2.87/1.76
% 2.87/1.76 thf(ty_n_t__Extended____Nat__Oenat,type,
% 2.87/1.76 extended_enat: $tType ).
% 2.87/1.76
% 2.87/1.76 thf(ty_n_t__Real__Oreal,type,
% 2.87/1.76 real: $tType ).
% 2.87/1.76
% 2.87/1.76 thf(ty_n_t__Num__Onum,type,
% 2.87/1.76 num: $tType ).
% 2.87/1.76
% 2.87/1.76 thf(ty_n_t__Nat__Onat,type,
% 2.87/1.76 nat: $tType ).
% 2.87/1.76
% 2.87/1.76 thf(ty_n_tf__a,type,
% 2.87/1.76 a: $tType ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Extended__Nat_Oenat,type,
% 2.87/1.76 extended_enat2: nat > extended_enat ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Real__Oereal,type,
% 2.87/1.76 extend1289208545_ereal: extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Extended__Nat_Othe__enat,type,
% 2.87/1.76 extended_the_enat: extended_enat > nat ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Extended__Real_Oereal_OMInfty,type,
% 2.87/1.76 extended_MInfty: extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Extended__Real_Oereal_OPInfty,type,
% 2.87/1.76 extended_PInfty: extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Extended__Real_Oereal_Oereal,type,
% 2.87/1.76 extended_ereal2: real > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Extended__Real_Oereal_Osize__ereal,type,
% 2.87/1.76 extended_size_ereal: extended_ereal > nat ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Extended__Real_Oereal__of__enat,type,
% 2.87/1.76 extend1771934483f_enat: extended_enat > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Extended__Real_Oreal__of__ereal,type,
% 2.87/1.76 extend1716541707_ereal: extended_ereal > real ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Groups_Oabs__class_Oabs_001t__Extended____Real__Oereal,type,
% 2.87/1.76 abs_ab1260901297_ereal: extended_ereal > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Extended____Real__Oereal,type,
% 2.87/1.76 uminus1208298309_ereal: extended_ereal > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
% 2.87/1.76 uminus_uminus_real: real > real ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 zero_z491942557d_enat: extended_enat ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
% 2.87/1.76 zero_z163181189_ereal: extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.87/1.76 zero_zero_nat: nat ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
% 2.87/1.76 zero_zero_real: real ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc_001tf__a_001t__Extended____Real__Oereal,type,
% 2.87/1.76 lower_1616484581_ereal: ( a > extended_ereal ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Lower__Semicontinuous__Mirabelle__mxyexokbxt_Olsc__hull_001tf__a,type,
% 2.87/1.76 lower_881475195hull_a: ( a > extended_ereal ) > a > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 numera280919179d_enat: num > extended_enat ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Real__Oereal,type,
% 2.87/1.76 numera1793320307_ereal: num > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 2.87/1.76 numeral_numeral_nat: num > nat ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 2.87/1.76 numeral_numeral_real: num > real ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Nat__Oenat_J,type,
% 2.87/1.76 ord_le291126163d_enat: ( $o > extended_enat ) > ( $o > extended_enat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Real__Oereal_J,type,
% 2.87/1.76 ord_le637473275_ereal: ( $o > extended_ereal ) > ( $o > extended_ereal ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
% 2.87/1.76 ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Real__Oreal_J,type,
% 2.87/1.76 ord_less_eq_o_real: ( $o > real ) > ( $o > real ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 ord_le1863327750d_enat: extended_enat > extended_enat > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
% 2.87/1.76 ord_le824540014_ereal: extended_ereal > extended_ereal > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.87/1.76 ord_less_eq_nat: nat > nat > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
% 2.87/1.76 ord_less_eq_num: num > num > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 2.87/1.76 ord_less_eq_real: real > real > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 order_1628344639d_enat: ( extended_enat > $o ) > extended_enat ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Real__Oereal,type,
% 2.87/1.76 order_1158471719_ereal: ( extended_ereal > $o ) > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 2.87/1.76 order_Greatest_nat: ( nat > $o ) > nat ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
% 2.87/1.76 order_Greatest_real: ( real > $o ) > real ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 order_2047034162d_enat: ( extended_enat > extended_enat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Extended____Real__Oereal,type,
% 2.87/1.76 order_1147259034_ereal: ( extended_enat > extended_ereal ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
% 2.87/1.76 order_442688004at_nat: ( extended_enat > nat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
% 2.87/1.76 order_182315744t_real: ( extended_enat > real ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 order_680161354d_enat: ( extended_ereal > extended_enat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
% 2.87/1.76 order_1408494002_ereal: ( extended_ereal > extended_ereal ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 2.87/1.76 order_523806444al_nat: ( extended_ereal > nat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
% 2.87/1.76 order_1800387528l_real: ( extended_ereal > real ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 order_1660553314d_enat: ( nat > extended_enat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
% 2.87/1.76 order_736687562_ereal: ( nat > extended_ereal ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.87/1.76 order_1631207636at_nat: ( nat > nat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
% 2.87/1.76 order_106095024t_real: ( nat > real ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 order_2106278841d_enat: ( extended_enat > extended_enat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Extended____Real__Oereal,type,
% 2.87/1.76 order_1180956065_ereal: ( extended_enat > extended_ereal ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
% 2.87/1.76 order_1622825661at_nat: ( extended_enat > nat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
% 2.87/1.76 order_135436057t_real: ( extended_enat > real ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 order_713858385d_enat: ( extended_ereal > extended_enat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
% 2.87/1.76 order_555877177_ereal: ( extended_ereal > extended_ereal ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
% 2.87/1.76 order_476926757al_nat: ( extended_ereal > nat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Extended____Real__Oereal_001t__Real__Oreal,type,
% 2.87/1.76 order_1560271745l_real: ( extended_ereal > real ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 order_693207323d_enat: ( nat > extended_enat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
% 2.87/1.76 order_689807875_ereal: ( nat > extended_ereal ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.87/1.76 order_769474267at_nat: ( nat > nat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Extended____Nat__Oenat,type,
% 2.87/1.76 order_1962034751d_enat: ( real > extended_enat ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Extended____Real__Oereal,type,
% 2.87/1.76 order_946214183_ereal: ( real > extended_ereal ) > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Real__Oereal,type,
% 2.87/1.76 divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_c_member_001t__Extended____Real__Oereal,type,
% 2.87/1.76 member1900190071_ereal: extended_ereal > set_Extended_ereal > $o ).
% 2.87/1.76
% 2.87/1.76 thf(sy_v_f,type,
% 2.87/1.76 f: a > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_v_g,type,
% 2.87/1.76 g: a > extended_ereal ).
% 2.87/1.76
% 2.87/1.76 thf(sy_v_x____,type,
% 2.87/1.76 x: a ).
% 2.87/1.76
% 2.87/1.76 thf(fact_0_assms,axiom,
% 2.87/1.76 ! [X: a] : ( ord_le824540014_ereal @ ( g @ X ) @ ( f @ X ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_1_calculation,axiom,
% 2.87/1.76 ord_le824540014_ereal @ ( lower_881475195hull_a @ g @ x ) @ ( g @ x ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_2_order__refl,axiom,
% 2.87/1.76 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ X2 ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_3_order__refl,axiom,
% 2.87/1.76 ! [X2: extended_enat] : ( ord_le1863327750d_enat @ X2 @ X2 ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_4_order__refl,axiom,
% 2.87/1.76 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_5_order__refl,axiom,
% 2.87/1.76 ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_6_ereal__complete__Inf,axiom,
% 2.87/1.76 ! [S: set_Extended_ereal] :
% 2.87/1.76 ? [X3: extended_ereal] :
% 2.87/1.76 ( ! [Xa: extended_ereal] :
% 2.87/1.76 ( ( member1900190071_ereal @ Xa @ S )
% 2.87/1.76 => ( ord_le824540014_ereal @ X3 @ Xa ) )
% 2.87/1.76 & ! [Z: extended_ereal] :
% 2.87/1.76 ( ! [Xa2: extended_ereal] :
% 2.87/1.76 ( ( member1900190071_ereal @ Xa2 @ S )
% 2.87/1.76 => ( ord_le824540014_ereal @ Z @ Xa2 ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ Z @ X3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_7_ereal__complete__Sup,axiom,
% 2.87/1.76 ! [S: set_Extended_ereal] :
% 2.87/1.76 ? [X3: extended_ereal] :
% 2.87/1.76 ( ! [Xa: extended_ereal] :
% 2.87/1.76 ( ( member1900190071_ereal @ Xa @ S )
% 2.87/1.76 => ( ord_le824540014_ereal @ Xa @ X3 ) )
% 2.87/1.76 & ! [Z: extended_ereal] :
% 2.87/1.76 ( ! [Xa2: extended_ereal] :
% 2.87/1.76 ( ( member1900190071_ereal @ Xa2 @ S )
% 2.87/1.76 => ( ord_le824540014_ereal @ Xa2 @ Z ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ X3 @ Z ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_8_order__subst1,axiom,
% 2.87/1.76 ! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_9_order__subst1,axiom,
% 2.87/1.76 ! [A: extended_ereal,F: extended_enat > extended_ereal,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_10_order__subst1,axiom,
% 2.87/1.76 ! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_less_eq_nat @ B @ C )
% 2.87/1.76 => ( ! [X3: nat,Y: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_11_order__subst1,axiom,
% 2.87/1.76 ! [A: extended_ereal,F: real > extended_ereal,B: real,C: real] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_less_eq_real @ B @ C )
% 2.87/1.76 => ( ! [X3: real,Y: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_12_order__subst1,axiom,
% 2.87/1.76 ! [A: extended_enat,F: extended_ereal > extended_enat,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_13_order__subst1,axiom,
% 2.87/1.76 ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_14_order__subst1,axiom,
% 2.87/1.76 ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_less_eq_nat @ B @ C )
% 2.87/1.76 => ( ! [X3: nat,Y: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_15_order__subst1,axiom,
% 2.87/1.76 ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_less_eq_real @ B @ C )
% 2.87/1.76 => ( ! [X3: real,Y: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_16_order__subst1,axiom,
% 2.87/1.76 ! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_17_order__subst1,axiom,
% 2.87/1.76 ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_18_order__subst2,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_19_order__subst2,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_20_order__subst2,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_21_order__subst2,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > real,C: real] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_22_order__subst2,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_23_order__subst2,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_24_order__subst2,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_25_order__subst2,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_26_order__subst2,axiom,
% 2.87/1.76 ! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A @ B )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: nat,Y: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_27_order__subst2,axiom,
% 2.87/1.76 ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A @ B )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ ( F @ B ) @ C )
% 2.87/1.76 => ( ! [X3: nat,Y: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_28_verit__la__disequality,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal] :
% 2.87/1.76 ( ( A = B )
% 2.87/1.76 | ~ ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 | ~ ( ord_le824540014_ereal @ B @ A ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_29_verit__la__disequality,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat] :
% 2.87/1.76 ( ( A = B )
% 2.87/1.76 | ~ ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 | ~ ( ord_le1863327750d_enat @ B @ A ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_30_verit__la__disequality,axiom,
% 2.87/1.76 ! [A: nat,B: nat] :
% 2.87/1.76 ( ( A = B )
% 2.87/1.76 | ~ ( ord_less_eq_nat @ A @ B )
% 2.87/1.76 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_31_verit__la__disequality,axiom,
% 2.87/1.76 ! [A: real,B: real] :
% 2.87/1.76 ( ( A = B )
% 2.87/1.76 | ~ ( ord_less_eq_real @ A @ B )
% 2.87/1.76 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_32_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: extended_ereal,F: extended_ereal > extended_ereal,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_33_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: extended_enat,F: extended_ereal > extended_enat,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_34_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_35_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: real,F: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_36_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: extended_ereal,F: extended_enat > extended_ereal,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_37_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_38_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_39_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_40_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_less_eq_nat @ B @ C )
% 2.87/1.76 => ( ! [X3: nat,Y: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_41_ord__eq__le__subst,axiom,
% 2.87/1.76 ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
% 2.87/1.76 ( ( A
% 2.87/1.76 = ( F @ B ) )
% 2.87/1.76 => ( ( ord_less_eq_nat @ B @ C )
% 2.87/1.76 => ( ! [X3: nat,Y: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ ( F @ C ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_42_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_43_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_44_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_45_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,F: extended_ereal > real,C: real] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_46_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_47_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_48_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_49_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_50_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: nat,Y: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le824540014_ereal @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_51_ord__le__eq__subst,axiom,
% 2.87/1.76 ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A @ B )
% 2.87/1.76 => ( ( ( F @ B )
% 2.87/1.76 = C )
% 2.87/1.76 => ( ! [X3: nat,Y: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ X3 ) @ ( F @ Y ) ) )
% 2.87/1.76 => ( ord_le1863327750d_enat @ ( F @ A ) @ C ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_52_lsc__hull__le,axiom,
% 2.87/1.76 ! [F: a > extended_ereal,X2: a] : ( ord_le824540014_ereal @ ( lower_881475195hull_a @ F @ X2 ) @ ( F @ X2 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_53_dual__order_Oantisym,axiom,
% 2.87/1.76 ! [B: extended_ereal,A: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ B @ A )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( A = B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_54_dual__order_Oantisym,axiom,
% 2.87/1.76 ! [B: extended_enat,A: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ B @ A )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( A = B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_55_dual__order_Oantisym,axiom,
% 2.87/1.76 ! [B: nat,A: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ B @ A )
% 2.87/1.76 => ( ( ord_less_eq_nat @ A @ B )
% 2.87/1.76 => ( A = B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_56_dual__order_Oantisym,axiom,
% 2.87/1.76 ! [B: real,A: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ B @ A )
% 2.87/1.76 => ( ( ord_less_eq_real @ A @ B )
% 2.87/1.76 => ( A = B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_57_dual__order_Oeq__iff,axiom,
% 2.87/1.76 ( ( ^ [Y2: extended_ereal,Z2: extended_ereal] : ( Y2 = Z2 ) )
% 2.87/1.76 = ( ^ [A2: extended_ereal,B2: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ B2 @ A2 )
% 2.87/1.76 & ( ord_le824540014_ereal @ A2 @ B2 ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_58_dual__order_Oeq__iff,axiom,
% 2.87/1.76 ( ( ^ [Y2: extended_enat,Z2: extended_enat] : ( Y2 = Z2 ) )
% 2.87/1.76 = ( ^ [A2: extended_enat,B2: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ B2 @ A2 )
% 2.87/1.76 & ( ord_le1863327750d_enat @ A2 @ B2 ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_59_dual__order_Oeq__iff,axiom,
% 2.87/1.76 ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
% 2.87/1.76 = ( ^ [A2: nat,B2: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.87/1.76 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_60_dual__order_Oeq__iff,axiom,
% 2.87/1.76 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 2.87/1.76 = ( ^ [A2: real,B2: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ B2 @ A2 )
% 2.87/1.76 & ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_61_dual__order_Otrans,axiom,
% 2.87/1.76 ! [B: extended_ereal,A: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ B @ A )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ C @ B )
% 2.87/1.76 => ( ord_le824540014_ereal @ C @ A ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_62_dual__order_Otrans,axiom,
% 2.87/1.76 ! [B: extended_enat,A: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ B @ A )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ C @ B )
% 2.87/1.76 => ( ord_le1863327750d_enat @ C @ A ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_63_dual__order_Otrans,axiom,
% 2.87/1.76 ! [B: nat,A: nat,C: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ B @ A )
% 2.87/1.76 => ( ( ord_less_eq_nat @ C @ B )
% 2.87/1.76 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_64_dual__order_Otrans,axiom,
% 2.87/1.76 ! [B: real,A: real,C: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ B @ A )
% 2.87/1.76 => ( ( ord_less_eq_real @ C @ B )
% 2.87/1.76 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_65_linorder__wlog,axiom,
% 2.87/1.76 ! [P: extended_ereal > extended_ereal > $o,A: extended_ereal,B: extended_ereal] :
% 2.87/1.76 ( ! [A3: extended_ereal,B3: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A3 @ B3 )
% 2.87/1.76 => ( P @ A3 @ B3 ) )
% 2.87/1.76 => ( ! [A3: extended_ereal,B3: extended_ereal] :
% 2.87/1.76 ( ( P @ B3 @ A3 )
% 2.87/1.76 => ( P @ A3 @ B3 ) )
% 2.87/1.76 => ( P @ A @ B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_66_linorder__wlog,axiom,
% 2.87/1.76 ! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
% 2.87/1.76 ( ! [A3: extended_enat,B3: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A3 @ B3 )
% 2.87/1.76 => ( P @ A3 @ B3 ) )
% 2.87/1.76 => ( ! [A3: extended_enat,B3: extended_enat] :
% 2.87/1.76 ( ( P @ B3 @ A3 )
% 2.87/1.76 => ( P @ A3 @ B3 ) )
% 2.87/1.76 => ( P @ A @ B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_67_linorder__wlog,axiom,
% 2.87/1.76 ! [P: nat > nat > $o,A: nat,B: nat] :
% 2.87/1.76 ( ! [A3: nat,B3: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A3 @ B3 )
% 2.87/1.76 => ( P @ A3 @ B3 ) )
% 2.87/1.76 => ( ! [A3: nat,B3: nat] :
% 2.87/1.76 ( ( P @ B3 @ A3 )
% 2.87/1.76 => ( P @ A3 @ B3 ) )
% 2.87/1.76 => ( P @ A @ B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_68_linorder__wlog,axiom,
% 2.87/1.76 ! [P: real > real > $o,A: real,B: real] :
% 2.87/1.76 ( ! [A3: real,B3: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ A3 @ B3 )
% 2.87/1.76 => ( P @ A3 @ B3 ) )
% 2.87/1.76 => ( ! [A3: real,B3: real] :
% 2.87/1.76 ( ( P @ B3 @ A3 )
% 2.87/1.76 => ( P @ A3 @ B3 ) )
% 2.87/1.76 => ( P @ A @ B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_69_dual__order_Orefl,axiom,
% 2.87/1.76 ! [A: extended_ereal] : ( ord_le824540014_ereal @ A @ A ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_70_dual__order_Orefl,axiom,
% 2.87/1.76 ! [A: extended_enat] : ( ord_le1863327750d_enat @ A @ A ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_71_dual__order_Orefl,axiom,
% 2.87/1.76 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_72_dual__order_Orefl,axiom,
% 2.87/1.76 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_73_order__trans,axiom,
% 2.87/1.76 ! [X2: extended_ereal,Y3: extended_ereal,Z3: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ Y3 @ Z3 )
% 2.87/1.76 => ( ord_le824540014_ereal @ X2 @ Z3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_74_order__trans,axiom,
% 2.87/1.76 ! [X2: extended_enat,Y3: extended_enat,Z3: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ Y3 @ Z3 )
% 2.87/1.76 => ( ord_le1863327750d_enat @ X2 @ Z3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_75_order__trans,axiom,
% 2.87/1.76 ! [X2: nat,Y3: nat,Z3: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.76 => ( ( ord_less_eq_nat @ Y3 @ Z3 )
% 2.87/1.76 => ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_76_order__trans,axiom,
% 2.87/1.76 ! [X2: real,Y3: real,Z3: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ X2 @ Y3 )
% 2.87/1.76 => ( ( ord_less_eq_real @ Y3 @ Z3 )
% 2.87/1.76 => ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_77_order__class_Oorder_Oantisym,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ A )
% 2.87/1.76 => ( A = B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_78_order__class_Oorder_Oantisym,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ A )
% 2.87/1.76 => ( A = B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_79_order__class_Oorder_Oantisym,axiom,
% 2.87/1.76 ! [A: nat,B: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A @ B )
% 2.87/1.76 => ( ( ord_less_eq_nat @ B @ A )
% 2.87/1.76 => ( A = B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_80_order__class_Oorder_Oantisym,axiom,
% 2.87/1.76 ! [A: real,B: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ A @ B )
% 2.87/1.76 => ( ( ord_less_eq_real @ B @ A )
% 2.87/1.76 => ( A = B ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_81_ord__le__eq__trans,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( B = C )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_82_ord__le__eq__trans,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( B = C )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_83_ord__le__eq__trans,axiom,
% 2.87/1.76 ! [A: nat,B: nat,C: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A @ B )
% 2.87/1.76 => ( ( B = C )
% 2.87/1.76 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_84_ord__le__eq__trans,axiom,
% 2.87/1.76 ! [A: real,B: real,C: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ A @ B )
% 2.87/1.76 => ( ( B = C )
% 2.87/1.76 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_85_ord__eq__le__trans,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( A = B )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ C )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_86_ord__eq__le__trans,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( A = B )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ C )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_87_ord__eq__le__trans,axiom,
% 2.87/1.76 ! [A: nat,B: nat,C: nat] :
% 2.87/1.76 ( ( A = B )
% 2.87/1.76 => ( ( ord_less_eq_nat @ B @ C )
% 2.87/1.76 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_88_ord__eq__le__trans,axiom,
% 2.87/1.76 ! [A: real,B: real,C: real] :
% 2.87/1.76 ( ( A = B )
% 2.87/1.76 => ( ( ord_less_eq_real @ B @ C )
% 2.87/1.76 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_89_order__class_Oorder_Oeq__iff,axiom,
% 2.87/1.76 ( ( ^ [Y2: extended_ereal,Z2: extended_ereal] : ( Y2 = Z2 ) )
% 2.87/1.76 = ( ^ [A2: extended_ereal,B2: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A2 @ B2 )
% 2.87/1.76 & ( ord_le824540014_ereal @ B2 @ A2 ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_90_order__class_Oorder_Oeq__iff,axiom,
% 2.87/1.76 ( ( ^ [Y2: extended_enat,Z2: extended_enat] : ( Y2 = Z2 ) )
% 2.87/1.76 = ( ^ [A2: extended_enat,B2: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A2 @ B2 )
% 2.87/1.76 & ( ord_le1863327750d_enat @ B2 @ A2 ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_91_order__class_Oorder_Oeq__iff,axiom,
% 2.87/1.76 ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
% 2.87/1.76 = ( ^ [A2: nat,B2: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.87/1.76 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_92_order__class_Oorder_Oeq__iff,axiom,
% 2.87/1.76 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 2.87/1.76 = ( ^ [A2: real,B2: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ A2 @ B2 )
% 2.87/1.76 & ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_93_antisym__conv,axiom,
% 2.87/1.76 ! [Y3: extended_ereal,X2: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ Y3 @ X2 )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.76 = ( X2 = Y3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_94_antisym__conv,axiom,
% 2.87/1.76 ! [Y3: extended_enat,X2: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ Y3 @ X2 )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.76 = ( X2 = Y3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_95_antisym__conv,axiom,
% 2.87/1.76 ! [Y3: nat,X2: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 2.87/1.76 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.76 = ( X2 = Y3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_96_antisym__conv,axiom,
% 2.87/1.76 ! [Y3: real,X2: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ Y3 @ X2 )
% 2.87/1.76 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 2.87/1.76 = ( X2 = Y3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_97_le__cases3,axiom,
% 2.87/1.76 ! [X2: extended_ereal,Y3: extended_ereal,Z3: extended_ereal] :
% 2.87/1.76 ( ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.76 => ~ ( ord_le824540014_ereal @ Y3 @ Z3 ) )
% 2.87/1.76 => ( ( ( ord_le824540014_ereal @ Y3 @ X2 )
% 2.87/1.76 => ~ ( ord_le824540014_ereal @ X2 @ Z3 ) )
% 2.87/1.76 => ( ( ( ord_le824540014_ereal @ X2 @ Z3 )
% 2.87/1.76 => ~ ( ord_le824540014_ereal @ Z3 @ Y3 ) )
% 2.87/1.76 => ( ( ( ord_le824540014_ereal @ Z3 @ Y3 )
% 2.87/1.76 => ~ ( ord_le824540014_ereal @ Y3 @ X2 ) )
% 2.87/1.76 => ( ( ( ord_le824540014_ereal @ Y3 @ Z3 )
% 2.87/1.76 => ~ ( ord_le824540014_ereal @ Z3 @ X2 ) )
% 2.87/1.76 => ~ ( ( ord_le824540014_ereal @ Z3 @ X2 )
% 2.87/1.76 => ~ ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_98_le__cases3,axiom,
% 2.87/1.76 ! [X2: extended_enat,Y3: extended_enat,Z3: extended_enat] :
% 2.87/1.76 ( ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.76 => ~ ( ord_le1863327750d_enat @ Y3 @ Z3 ) )
% 2.87/1.76 => ( ( ( ord_le1863327750d_enat @ Y3 @ X2 )
% 2.87/1.76 => ~ ( ord_le1863327750d_enat @ X2 @ Z3 ) )
% 2.87/1.76 => ( ( ( ord_le1863327750d_enat @ X2 @ Z3 )
% 2.87/1.76 => ~ ( ord_le1863327750d_enat @ Z3 @ Y3 ) )
% 2.87/1.76 => ( ( ( ord_le1863327750d_enat @ Z3 @ Y3 )
% 2.87/1.76 => ~ ( ord_le1863327750d_enat @ Y3 @ X2 ) )
% 2.87/1.76 => ( ( ( ord_le1863327750d_enat @ Y3 @ Z3 )
% 2.87/1.76 => ~ ( ord_le1863327750d_enat @ Z3 @ X2 ) )
% 2.87/1.76 => ~ ( ( ord_le1863327750d_enat @ Z3 @ X2 )
% 2.87/1.76 => ~ ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_99_le__cases3,axiom,
% 2.87/1.76 ! [X2: nat,Y3: nat,Z3: nat] :
% 2.87/1.76 ( ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.76 => ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
% 2.87/1.76 => ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 2.87/1.76 => ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
% 2.87/1.76 => ( ( ( ord_less_eq_nat @ X2 @ Z3 )
% 2.87/1.76 => ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
% 2.87/1.76 => ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
% 2.87/1.76 => ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
% 2.87/1.76 => ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
% 2.87/1.76 => ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
% 2.87/1.76 => ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
% 2.87/1.76 => ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_100_le__cases3,axiom,
% 2.87/1.76 ! [X2: real,Y3: real,Z3: real] :
% 2.87/1.76 ( ( ( ord_less_eq_real @ X2 @ Y3 )
% 2.87/1.76 => ~ ( ord_less_eq_real @ Y3 @ Z3 ) )
% 2.87/1.76 => ( ( ( ord_less_eq_real @ Y3 @ X2 )
% 2.87/1.76 => ~ ( ord_less_eq_real @ X2 @ Z3 ) )
% 2.87/1.76 => ( ( ( ord_less_eq_real @ X2 @ Z3 )
% 2.87/1.76 => ~ ( ord_less_eq_real @ Z3 @ Y3 ) )
% 2.87/1.76 => ( ( ( ord_less_eq_real @ Z3 @ Y3 )
% 2.87/1.76 => ~ ( ord_less_eq_real @ Y3 @ X2 ) )
% 2.87/1.76 => ( ( ( ord_less_eq_real @ Y3 @ Z3 )
% 2.87/1.76 => ~ ( ord_less_eq_real @ Z3 @ X2 ) )
% 2.87/1.76 => ~ ( ( ord_less_eq_real @ Z3 @ X2 )
% 2.87/1.76 => ~ ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_101_order_Otrans,axiom,
% 2.87/1.76 ! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ B @ C )
% 2.87/1.76 => ( ord_le824540014_ereal @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_102_order_Otrans,axiom,
% 2.87/1.76 ! [A: extended_enat,B: extended_enat,C: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ A @ B )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ B @ C )
% 2.87/1.76 => ( ord_le1863327750d_enat @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_103_order_Otrans,axiom,
% 2.87/1.76 ! [A: nat,B: nat,C: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ A @ B )
% 2.87/1.76 => ( ( ord_less_eq_nat @ B @ C )
% 2.87/1.76 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_104_order_Otrans,axiom,
% 2.87/1.76 ! [A: real,B: real,C: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ A @ B )
% 2.87/1.76 => ( ( ord_less_eq_real @ B @ C )
% 2.87/1.76 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_105_le__cases,axiom,
% 2.87/1.76 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.76 ( ~ ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.76 => ( ord_le824540014_ereal @ Y3 @ X2 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_106_le__cases,axiom,
% 2.87/1.76 ! [X2: extended_enat,Y3: extended_enat] :
% 2.87/1.76 ( ~ ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.76 => ( ord_le1863327750d_enat @ Y3 @ X2 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_107_le__cases,axiom,
% 2.87/1.76 ! [X2: nat,Y3: nat] :
% 2.87/1.76 ( ~ ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.76 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_108_le__cases,axiom,
% 2.87/1.76 ! [X2: real,Y3: real] :
% 2.87/1.76 ( ~ ( ord_less_eq_real @ X2 @ Y3 )
% 2.87/1.76 => ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_109_eq__refl,axiom,
% 2.87/1.76 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.76 ( ( X2 = Y3 )
% 2.87/1.76 => ( ord_le824540014_ereal @ X2 @ Y3 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_110_eq__refl,axiom,
% 2.87/1.76 ! [X2: extended_enat,Y3: extended_enat] :
% 2.87/1.76 ( ( X2 = Y3 )
% 2.87/1.76 => ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_111_eq__refl,axiom,
% 2.87/1.76 ! [X2: nat,Y3: nat] :
% 2.87/1.76 ( ( X2 = Y3 )
% 2.87/1.76 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_112_eq__refl,axiom,
% 2.87/1.76 ! [X2: real,Y3: real] :
% 2.87/1.76 ( ( X2 = Y3 )
% 2.87/1.76 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_113_linear,axiom,
% 2.87/1.76 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.76 | ( ord_le824540014_ereal @ Y3 @ X2 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_114_linear,axiom,
% 2.87/1.76 ! [X2: extended_enat,Y3: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.76 | ( ord_le1863327750d_enat @ Y3 @ X2 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_115_linear,axiom,
% 2.87/1.76 ! [X2: nat,Y3: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.76 | ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_116_linear,axiom,
% 2.87/1.76 ! [X2: real,Y3: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ X2 @ Y3 )
% 2.87/1.76 | ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_117_antisym,axiom,
% 2.87/1.76 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.76 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.76 => ( ( ord_le824540014_ereal @ Y3 @ X2 )
% 2.87/1.76 => ( X2 = Y3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_118_antisym,axiom,
% 2.87/1.76 ! [X2: extended_enat,Y3: extended_enat] :
% 2.87/1.76 ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.76 => ( ( ord_le1863327750d_enat @ Y3 @ X2 )
% 2.87/1.76 => ( X2 = Y3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_119_antisym,axiom,
% 2.87/1.76 ! [X2: nat,Y3: nat] :
% 2.87/1.76 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.76 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 2.87/1.76 => ( X2 = Y3 ) ) ) ).
% 2.87/1.76
% 2.87/1.76 thf(fact_120_antisym,axiom,
% 2.87/1.76 ! [X2: real,Y3: real] :
% 2.87/1.76 ( ( ord_less_eq_real @ X2 @ Y3 )
% 2.87/1.77 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 2.87/1.77 => ( X2 = Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_121_eq__iff,axiom,
% 2.87/1.77 ( ( ^ [Y2: extended_ereal,Z2: extended_ereal] : ( Y2 = Z2 ) )
% 2.87/1.77 = ( ^ [X4: extended_ereal,Y4: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 2.87/1.77 & ( ord_le824540014_ereal @ Y4 @ X4 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_122_eq__iff,axiom,
% 2.87/1.77 ( ( ^ [Y2: extended_enat,Z2: extended_enat] : ( Y2 = Z2 ) )
% 2.87/1.77 = ( ^ [X4: extended_enat,Y4: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 2.87/1.77 & ( ord_le1863327750d_enat @ Y4 @ X4 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_123_eq__iff,axiom,
% 2.87/1.77 ( ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 ) )
% 2.87/1.77 = ( ^ [X4: nat,Y4: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 2.87/1.77 & ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_124_eq__iff,axiom,
% 2.87/1.77 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 2.87/1.77 = ( ^ [X4: real,Y4: real] :
% 2.87/1.77 ( ( ord_less_eq_real @ X4 @ Y4 )
% 2.87/1.77 & ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_125_lsc__hull__iff__greatest,axiom,
% 2.87/1.77 ! [G: a > extended_ereal,F: a > extended_ereal] :
% 2.87/1.77 ( ( G
% 2.87/1.77 = ( lower_881475195hull_a @ F ) )
% 2.87/1.77 = ( ( lower_1616484581_ereal @ G )
% 2.87/1.77 & ! [X4: a] : ( ord_le824540014_ereal @ ( G @ X4 ) @ ( F @ X4 ) )
% 2.87/1.77 & ! [H: a > extended_ereal] :
% 2.87/1.77 ( ( ( lower_1616484581_ereal @ H )
% 2.87/1.77 & ! [X4: a] : ( ord_le824540014_ereal @ ( H @ X4 ) @ ( F @ X4 ) ) )
% 2.87/1.77 => ! [X4: a] : ( ord_le824540014_ereal @ ( H @ X4 ) @ ( G @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_126_lsc__hull__greatest,axiom,
% 2.87/1.77 ! [G: a > extended_ereal,F: a > extended_ereal] :
% 2.87/1.77 ( ( lower_1616484581_ereal @ G )
% 2.87/1.77 => ( ! [X3: a] : ( ord_le824540014_ereal @ ( G @ X3 ) @ ( F @ X3 ) )
% 2.87/1.77 => ! [X: a] : ( ord_le824540014_ereal @ ( G @ X ) @ ( lower_881475195hull_a @ F @ X ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_127_Greatest__equality,axiom,
% 2.87/1.77 ! [P: extended_ereal > $o,X2: extended_ereal] :
% 2.87/1.77 ( ( P @ X2 )
% 2.87/1.77 => ( ! [Y: extended_ereal] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_le824540014_ereal @ Y @ X2 ) )
% 2.87/1.77 => ( ( order_1158471719_ereal @ P )
% 2.87/1.77 = X2 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_128_Greatest__equality,axiom,
% 2.87/1.77 ! [P: extended_enat > $o,X2: extended_enat] :
% 2.87/1.77 ( ( P @ X2 )
% 2.87/1.77 => ( ! [Y: extended_enat] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_le1863327750d_enat @ Y @ X2 ) )
% 2.87/1.77 => ( ( order_1628344639d_enat @ P )
% 2.87/1.77 = X2 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_129_Greatest__equality,axiom,
% 2.87/1.77 ! [P: real > $o,X2: real] :
% 2.87/1.77 ( ( P @ X2 )
% 2.87/1.77 => ( ! [Y: real] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_less_eq_real @ Y @ X2 ) )
% 2.87/1.77 => ( ( order_Greatest_real @ P )
% 2.87/1.77 = X2 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_130_Greatest__equality,axiom,
% 2.87/1.77 ! [P: nat > $o,X2: nat] :
% 2.87/1.77 ( ( P @ X2 )
% 2.87/1.77 => ( ! [Y: nat] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_less_eq_nat @ Y @ X2 ) )
% 2.87/1.77 => ( ( order_Greatest_nat @ P )
% 2.87/1.77 = X2 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_131_GreatestI2__order,axiom,
% 2.87/1.77 ! [P: extended_ereal > $o,X2: extended_ereal,Q: extended_ereal > $o] :
% 2.87/1.77 ( ( P @ X2 )
% 2.87/1.77 => ( ! [Y: extended_ereal] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_le824540014_ereal @ Y @ X2 ) )
% 2.87/1.77 => ( ! [X3: extended_ereal] :
% 2.87/1.77 ( ( P @ X3 )
% 2.87/1.77 => ( ! [Y5: extended_ereal] :
% 2.87/1.77 ( ( P @ Y5 )
% 2.87/1.77 => ( ord_le824540014_ereal @ Y5 @ X3 ) )
% 2.87/1.77 => ( Q @ X3 ) ) )
% 2.87/1.77 => ( Q @ ( order_1158471719_ereal @ P ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_132_GreatestI2__order,axiom,
% 2.87/1.77 ! [P: extended_enat > $o,X2: extended_enat,Q: extended_enat > $o] :
% 2.87/1.77 ( ( P @ X2 )
% 2.87/1.77 => ( ! [Y: extended_enat] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_le1863327750d_enat @ Y @ X2 ) )
% 2.87/1.77 => ( ! [X3: extended_enat] :
% 2.87/1.77 ( ( P @ X3 )
% 2.87/1.77 => ( ! [Y5: extended_enat] :
% 2.87/1.77 ( ( P @ Y5 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ Y5 @ X3 ) )
% 2.87/1.77 => ( Q @ X3 ) ) )
% 2.87/1.77 => ( Q @ ( order_1628344639d_enat @ P ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_133_GreatestI2__order,axiom,
% 2.87/1.77 ! [P: real > $o,X2: real,Q: real > $o] :
% 2.87/1.77 ( ( P @ X2 )
% 2.87/1.77 => ( ! [Y: real] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_less_eq_real @ Y @ X2 ) )
% 2.87/1.77 => ( ! [X3: real] :
% 2.87/1.77 ( ( P @ X3 )
% 2.87/1.77 => ( ! [Y5: real] :
% 2.87/1.77 ( ( P @ Y5 )
% 2.87/1.77 => ( ord_less_eq_real @ Y5 @ X3 ) )
% 2.87/1.77 => ( Q @ X3 ) ) )
% 2.87/1.77 => ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_134_GreatestI2__order,axiom,
% 2.87/1.77 ! [P: nat > $o,X2: nat,Q: nat > $o] :
% 2.87/1.77 ( ( P @ X2 )
% 2.87/1.77 => ( ! [Y: nat] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_less_eq_nat @ Y @ X2 ) )
% 2.87/1.77 => ( ! [X3: nat] :
% 2.87/1.77 ( ( P @ X3 )
% 2.87/1.77 => ( ! [Y5: nat] :
% 2.87/1.77 ( ( P @ Y5 )
% 2.87/1.77 => ( ord_less_eq_nat @ Y5 @ X3 ) )
% 2.87/1.77 => ( Q @ X3 ) ) )
% 2.87/1.77 => ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_135_le__rel__bool__arg__iff,axiom,
% 2.87/1.77 ( ord_le637473275_ereal
% 2.87/1.77 = ( ^ [X5: $o > extended_ereal,Y6: $o > extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 2.87/1.77 & ( ord_le824540014_ereal @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_136_le__rel__bool__arg__iff,axiom,
% 2.87/1.77 ( ord_le291126163d_enat
% 2.87/1.77 = ( ^ [X5: $o > extended_enat,Y6: $o > extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 2.87/1.77 & ( ord_le1863327750d_enat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_137_le__rel__bool__arg__iff,axiom,
% 2.87/1.77 ( ord_less_eq_o_nat
% 2.87/1.77 = ( ^ [X5: $o > nat,Y6: $o > nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 2.87/1.77 & ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_138_le__rel__bool__arg__iff,axiom,
% 2.87/1.77 ( ord_less_eq_o_real
% 2.87/1.77 = ( ^ [X5: $o > real,Y6: $o > real] :
% 2.87/1.77 ( ( ord_less_eq_real @ ( X5 @ $false ) @ ( Y6 @ $false ) )
% 2.87/1.77 & ( ord_less_eq_real @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_139_lsc__lsc__hull,axiom,
% 2.87/1.77 ! [F: a > extended_ereal] : ( lower_1616484581_ereal @ ( lower_881475195hull_a @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_140_ereal__of__enat__le__iff,axiom,
% 2.87/1.77 ! [M: extended_enat,N: extended_enat] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ ( extend1771934483f_enat @ M ) @ ( extend1771934483f_enat @ N ) )
% 2.87/1.77 = ( ord_le1863327750d_enat @ M @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_141_antimono__def,axiom,
% 2.87/1.77 ( order_1408494002_ereal
% 2.87/1.77 = ( ^ [F2: extended_ereal > extended_ereal] :
% 2.87/1.77 ! [X4: extended_ereal,Y4: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_142_antimono__def,axiom,
% 2.87/1.77 ( order_680161354d_enat
% 2.87/1.77 = ( ^ [F2: extended_ereal > extended_enat] :
% 2.87/1.77 ! [X4: extended_ereal,Y4: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_143_antimono__def,axiom,
% 2.87/1.77 ( order_523806444al_nat
% 2.87/1.77 = ( ^ [F2: extended_ereal > nat] :
% 2.87/1.77 ! [X4: extended_ereal,Y4: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 2.87/1.77 => ( ord_less_eq_nat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_144_antimono__def,axiom,
% 2.87/1.77 ( order_1800387528l_real
% 2.87/1.77 = ( ^ [F2: extended_ereal > real] :
% 2.87/1.77 ! [X4: extended_ereal,Y4: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X4 @ Y4 )
% 2.87/1.77 => ( ord_less_eq_real @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_145_antimono__def,axiom,
% 2.87/1.77 ( order_1147259034_ereal
% 2.87/1.77 = ( ^ [F2: extended_enat > extended_ereal] :
% 2.87/1.77 ! [X4: extended_enat,Y4: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_146_antimono__def,axiom,
% 2.87/1.77 ( order_2047034162d_enat
% 2.87/1.77 = ( ^ [F2: extended_enat > extended_enat] :
% 2.87/1.77 ! [X4: extended_enat,Y4: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_147_antimono__def,axiom,
% 2.87/1.77 ( order_442688004at_nat
% 2.87/1.77 = ( ^ [F2: extended_enat > nat] :
% 2.87/1.77 ! [X4: extended_enat,Y4: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 2.87/1.77 => ( ord_less_eq_nat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_148_antimono__def,axiom,
% 2.87/1.77 ( order_182315744t_real
% 2.87/1.77 = ( ^ [F2: extended_enat > real] :
% 2.87/1.77 ! [X4: extended_enat,Y4: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ X4 @ Y4 )
% 2.87/1.77 => ( ord_less_eq_real @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_149_antimono__def,axiom,
% 2.87/1.77 ( order_736687562_ereal
% 2.87/1.77 = ( ^ [F2: nat > extended_ereal] :
% 2.87/1.77 ! [X4: nat,Y4: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_150_antimono__def,axiom,
% 2.87/1.77 ( order_1660553314d_enat
% 2.87/1.77 = ( ^ [F2: nat > extended_enat] :
% 2.87/1.77 ! [X4: nat,Y4: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_151_antimonoI,axiom,
% 2.87/1.77 ! [F: extended_ereal > extended_ereal] :
% 2.87/1.77 ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_1408494002_ereal @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_152_antimonoI,axiom,
% 2.87/1.77 ! [F: extended_ereal > extended_enat] :
% 2.87/1.77 ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_680161354d_enat @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_153_antimonoI,axiom,
% 2.87/1.77 ! [F: extended_ereal > nat] :
% 2.87/1.77 ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.77 => ( ord_less_eq_nat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_523806444al_nat @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_154_antimonoI,axiom,
% 2.87/1.77 ! [F: extended_ereal > real] :
% 2.87/1.77 ( ! [X3: extended_ereal,Y: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X3 @ Y )
% 2.87/1.77 => ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_1800387528l_real @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_155_antimonoI,axiom,
% 2.87/1.77 ! [F: extended_enat > extended_ereal] :
% 2.87/1.77 ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_1147259034_ereal @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_156_antimonoI,axiom,
% 2.87/1.77 ! [F: extended_enat > extended_enat] :
% 2.87/1.77 ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_2047034162d_enat @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_157_antimonoI,axiom,
% 2.87/1.77 ! [F: extended_enat > nat] :
% 2.87/1.77 ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.77 => ( ord_less_eq_nat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_442688004at_nat @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_158_antimonoI,axiom,
% 2.87/1.77 ! [F: extended_enat > real] :
% 2.87/1.77 ( ! [X3: extended_enat,Y: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ X3 @ Y )
% 2.87/1.77 => ( ord_less_eq_real @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_182315744t_real @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_159_antimonoI,axiom,
% 2.87/1.77 ! [F: nat > extended_ereal] :
% 2.87/1.77 ( ! [X3: nat,Y: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_736687562_ereal @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_160_antimonoI,axiom,
% 2.87/1.77 ! [F: nat > extended_enat] :
% 2.87/1.77 ( ! [X3: nat,Y: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ X3 @ Y )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ Y ) @ ( F @ X3 ) ) )
% 2.87/1.77 => ( order_1660553314d_enat @ F ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_161_antimonoE,axiom,
% 2.87/1.77 ! [F: extended_ereal > extended_ereal,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_1408494002_ereal @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_162_antimonoE,axiom,
% 2.87/1.77 ! [F: extended_ereal > extended_enat,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_680161354d_enat @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_163_antimonoE,axiom,
% 2.87/1.77 ! [F: extended_ereal > nat,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_523806444al_nat @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_164_antimonoE,axiom,
% 2.87/1.77 ! [F: extended_ereal > real,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_1800387528l_real @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_165_antimonoE,axiom,
% 2.87/1.77 ! [F: extended_enat > extended_ereal,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_1147259034_ereal @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_166_antimonoE,axiom,
% 2.87/1.77 ! [F: extended_enat > extended_enat,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_2047034162d_enat @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_167_antimonoE,axiom,
% 2.87/1.77 ! [F: extended_enat > nat,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_442688004at_nat @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_168_antimonoE,axiom,
% 2.87/1.77 ! [F: extended_enat > real,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_182315744t_real @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_169_antimonoE,axiom,
% 2.87/1.77 ! [F: nat > extended_ereal,X2: nat,Y3: nat] :
% 2.87/1.77 ( ( order_736687562_ereal @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_170_antimonoE,axiom,
% 2.87/1.77 ! [F: nat > extended_enat,X2: nat,Y3: nat] :
% 2.87/1.77 ( ( order_1660553314d_enat @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_171_antimonoD,axiom,
% 2.87/1.77 ! [F: extended_ereal > extended_ereal,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_1408494002_ereal @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_172_antimonoD,axiom,
% 2.87/1.77 ! [F: extended_ereal > extended_enat,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_680161354d_enat @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_173_antimonoD,axiom,
% 2.87/1.77 ! [F: extended_ereal > nat,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_523806444al_nat @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_174_antimonoD,axiom,
% 2.87/1.77 ! [F: extended_ereal > real,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_1800387528l_real @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_175_antimonoD,axiom,
% 2.87/1.77 ! [F: extended_enat > extended_ereal,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_1147259034_ereal @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_176_antimonoD,axiom,
% 2.87/1.77 ! [F: extended_enat > extended_enat,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_2047034162d_enat @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_177_antimonoD,axiom,
% 2.87/1.77 ! [F: extended_enat > nat,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_442688004at_nat @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_less_eq_nat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_178_antimonoD,axiom,
% 2.87/1.77 ! [F: extended_enat > real,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_182315744t_real @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_179_antimonoD,axiom,
% 2.87/1.77 ! [F: nat > extended_ereal,X2: nat,Y3: nat] :
% 2.87/1.77 ( ( order_736687562_ereal @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_180_antimonoD,axiom,
% 2.87/1.77 ! [F: nat > extended_enat,X2: nat,Y3: nat] :
% 2.87/1.77 ( ( order_1660553314d_enat @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_181_decseqD,axiom,
% 2.87/1.77 ! [F: nat > extended_ereal,I: nat,J: nat] :
% 2.87/1.77 ( ( order_736687562_ereal @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ I @ J )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_182_decseqD,axiom,
% 2.87/1.77 ! [F: nat > extended_enat,I: nat,J: nat] :
% 2.87/1.77 ( ( order_1660553314d_enat @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ I @ J )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_183_decseqD,axiom,
% 2.87/1.77 ! [F: nat > nat,I: nat,J: nat] :
% 2.87/1.77 ( ( order_1631207636at_nat @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ I @ J )
% 2.87/1.77 => ( ord_less_eq_nat @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_184_decseqD,axiom,
% 2.87/1.77 ! [F: nat > real,I: nat,J: nat] :
% 2.87/1.77 ( ( order_106095024t_real @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ I @ J )
% 2.87/1.77 => ( ord_less_eq_real @ ( F @ J ) @ ( F @ I ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_185_decseq__def,axiom,
% 2.87/1.77 ( order_736687562_ereal
% 2.87/1.77 = ( ^ [X5: nat > extended_ereal] :
% 2.87/1.77 ! [M2: nat,N2: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_186_decseq__def,axiom,
% 2.87/1.77 ( order_1660553314d_enat
% 2.87/1.77 = ( ^ [X5: nat > extended_enat] :
% 2.87/1.77 ! [M2: nat,N2: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_187_decseq__def,axiom,
% 2.87/1.77 ( order_1631207636at_nat
% 2.87/1.77 = ( ^ [X5: nat > nat] :
% 2.87/1.77 ! [M2: nat,N2: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.87/1.77 => ( ord_less_eq_nat @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_188_decseq__def,axiom,
% 2.87/1.77 ( order_106095024t_real
% 2.87/1.77 = ( ^ [X5: nat > real] :
% 2.87/1.77 ! [M2: nat,N2: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.87/1.77 => ( ord_less_eq_real @ ( X5 @ N2 ) @ ( X5 @ M2 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_189_ereal__of__enat__nonneg,axiom,
% 2.87/1.77 ! [N: extended_enat] : ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extend1771934483f_enat @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_190_ereal__of__enat__ge__zero__cancel__iff,axiom,
% 2.87/1.77 ! [N: extended_enat] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extend1771934483f_enat @ N ) )
% 2.87/1.77 = ( ord_le1863327750d_enat @ zero_z491942557d_enat @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_191_numeral__le__ereal__of__enat__iff,axiom,
% 2.87/1.77 ! [M: num,N: extended_enat] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ ( numera1793320307_ereal @ M ) @ ( extend1771934483f_enat @ N ) )
% 2.87/1.77 = ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_192_ereal__infty__less__eq_I1_J,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ extend1289208545_ereal @ X2 )
% 2.87/1.77 = ( X2 = extend1289208545_ereal ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_193_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: extended_ereal > extended_ereal,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_555877177_ereal @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_194_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: extended_enat > extended_ereal,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_1180956065_ereal @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_195_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: nat > extended_ereal,X2: nat,Y3: nat] :
% 2.87/1.77 ( ( order_689807875_ereal @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_196_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: real > extended_ereal,X2: real,Y3: real] :
% 2.87/1.77 ( ( order_946214183_ereal @ F )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_197_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: extended_ereal > extended_enat,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_713858385d_enat @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_198_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: extended_enat > extended_enat,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_2106278841d_enat @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_199_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: nat > extended_enat,X2: nat,Y3: nat] :
% 2.87/1.77 ( ( order_693207323d_enat @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_200_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: real > extended_enat,X2: real,Y3: real] :
% 2.87/1.77 ( ( order_1962034751d_enat @ F )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_201_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: extended_ereal > nat,X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( order_476926757al_nat @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_le824540014_ereal @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_202_strict__mono__less__eq,axiom,
% 2.87/1.77 ! [F: extended_enat > nat,X2: extended_enat,Y3: extended_enat] :
% 2.87/1.77 ( ( order_1622825661at_nat @ F )
% 2.87/1.77 => ( ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
% 2.87/1.77 = ( ord_le1863327750d_enat @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_203_ereal__of__enat__zero,axiom,
% 2.87/1.77 ( ( extend1771934483f_enat @ zero_z491942557d_enat )
% 2.87/1.77 = zero_z163181189_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_204_strict__mono__eq,axiom,
% 2.87/1.77 ! [F: nat > nat,X2: nat,Y3: nat] :
% 2.87/1.77 ( ( order_769474267at_nat @ F )
% 2.87/1.77 => ( ( ( F @ X2 )
% 2.87/1.77 = ( F @ Y3 ) )
% 2.87/1.77 = ( X2 = Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_205_Infty__neq__0_I1_J,axiom,
% 2.87/1.77 extend1289208545_ereal != zero_z163181189_ereal ).
% 2.87/1.77
% 2.87/1.77 thf(fact_206_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: extended_ereal > extended_ereal,M: extended_ereal,N: extended_ereal] :
% 2.87/1.77 ( ( order_555877177_ereal @ R )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ M @ N )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_207_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: extended_ereal > extended_enat,M: extended_ereal,N: extended_ereal] :
% 2.87/1.77 ( ( order_713858385d_enat @ R )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ M @ N )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_208_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: extended_ereal > nat,M: extended_ereal,N: extended_ereal] :
% 2.87/1.77 ( ( order_476926757al_nat @ R )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ M @ N )
% 2.87/1.77 => ( ord_less_eq_nat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_209_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: extended_ereal > real,M: extended_ereal,N: extended_ereal] :
% 2.87/1.77 ( ( order_1560271745l_real @ R )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ M @ N )
% 2.87/1.77 => ( ord_less_eq_real @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_210_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: extended_enat > extended_ereal,M: extended_enat,N: extended_enat] :
% 2.87/1.77 ( ( order_1180956065_ereal @ R )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ M @ N )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_211_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: extended_enat > extended_enat,M: extended_enat,N: extended_enat] :
% 2.87/1.77 ( ( order_2106278841d_enat @ R )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ M @ N )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_212_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: extended_enat > nat,M: extended_enat,N: extended_enat] :
% 2.87/1.77 ( ( order_1622825661at_nat @ R )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ M @ N )
% 2.87/1.77 => ( ord_less_eq_nat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_213_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: extended_enat > real,M: extended_enat,N: extended_enat] :
% 2.87/1.77 ( ( order_135436057t_real @ R )
% 2.87/1.77 => ( ( ord_le1863327750d_enat @ M @ N )
% 2.87/1.77 => ( ord_less_eq_real @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_214_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: nat > extended_ereal,M: nat,N: nat] :
% 2.87/1.77 ( ( order_689807875_ereal @ R )
% 2.87/1.77 => ( ( ord_less_eq_nat @ M @ N )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_215_strict__mono__leD,axiom,
% 2.87/1.77 ! [R: nat > extended_enat,M: nat,N: nat] :
% 2.87/1.77 ( ( order_693207323d_enat @ R )
% 2.87/1.77 => ( ( ord_less_eq_nat @ M @ N )
% 2.87/1.77 => ( ord_le1863327750d_enat @ ( R @ M ) @ ( R @ N ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_216_neq__PInf__trans,axiom,
% 2.87/1.77 ! [Y3: extended_ereal,X2: extended_ereal] :
% 2.87/1.77 ( ( Y3 != extend1289208545_ereal )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( X2 != extend1289208545_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_217_ereal__infty__less__eq2_I1_J,axiom,
% 2.87/1.77 ! [A: extended_ereal,B: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.77 => ( ( A = extend1289208545_ereal )
% 2.87/1.77 => ( B = extend1289208545_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_218_ereal__less__eq_I1_J,axiom,
% 2.87/1.77 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ X2 @ extend1289208545_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_219_numeral__le__iff,axiom,
% 2.87/1.77 ! [M: num,N: num] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ ( numera280919179d_enat @ N ) )
% 2.87/1.77 = ( ord_less_eq_num @ M @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_220_numeral__le__iff,axiom,
% 2.87/1.77 ! [M: num,N: num] :
% 2.87/1.77 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 2.87/1.77 = ( ord_less_eq_num @ M @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_221_numeral__le__iff,axiom,
% 2.87/1.77 ! [M: num,N: num] :
% 2.87/1.77 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 2.87/1.77 = ( ord_less_eq_num @ M @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_222_le__zero__eq,axiom,
% 2.87/1.77 ! [N: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ N @ zero_z491942557d_enat )
% 2.87/1.77 = ( N = zero_z491942557d_enat ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_223_le__zero__eq,axiom,
% 2.87/1.77 ! [N: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.87/1.77 = ( N = zero_zero_nat ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_224_zero__le__numeral,axiom,
% 2.87/1.77 ! [N: num] : ( ord_le1863327750d_enat @ zero_z491942557d_enat @ ( numera280919179d_enat @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_225_zero__le__numeral,axiom,
% 2.87/1.77 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_226_zero__le__numeral,axiom,
% 2.87/1.77 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_227_not__numeral__le__zero,axiom,
% 2.87/1.77 ! [N: num] :
% 2.87/1.77 ~ ( ord_le1863327750d_enat @ ( numera280919179d_enat @ N ) @ zero_z491942557d_enat ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_228_not__numeral__le__zero,axiom,
% 2.87/1.77 ! [N: num] :
% 2.87/1.77 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_229_not__numeral__le__zero,axiom,
% 2.87/1.77 ! [N: num] :
% 2.87/1.77 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_230_numeral__eq__iff,axiom,
% 2.87/1.77 ! [M: num,N: num] :
% 2.87/1.77 ( ( ( numeral_numeral_real @ M )
% 2.87/1.77 = ( numeral_numeral_real @ N ) )
% 2.87/1.77 = ( M = N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_231_ile0__eq,axiom,
% 2.87/1.77 ! [N: extended_enat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ N @ zero_z491942557d_enat )
% 2.87/1.77 = ( N = zero_z491942557d_enat ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_232_i0__lb,axiom,
% 2.87/1.77 ! [N: extended_enat] : ( ord_le1863327750d_enat @ zero_z491942557d_enat @ N ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_233_enat__ord__number_I1_J,axiom,
% 2.87/1.77 ! [M: num,N: num] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ ( numera280919179d_enat @ N ) )
% 2.87/1.77 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_234_bot__nat__0_Oextremum,axiom,
% 2.87/1.77 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_235_le0,axiom,
% 2.87/1.77 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_236_numeral__le__enat__iff,axiom,
% 2.87/1.77 ! [M: num,N: nat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ ( numera280919179d_enat @ M ) @ ( extended_enat2 @ N ) )
% 2.87/1.77 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_237_real__of__ereal__le__0,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ord_less_eq_real @ ( extend1716541707_ereal @ X2 ) @ zero_zero_real )
% 2.87/1.77 = ( ( ord_le824540014_ereal @ X2 @ zero_z163181189_ereal )
% 2.87/1.77 | ( X2 = extend1289208545_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_238_ereal__uminus__zero__iff,axiom,
% 2.87/1.77 ! [A: extended_ereal] :
% 2.87/1.77 ( ( ( uminus1208298309_ereal @ A )
% 2.87/1.77 = zero_z163181189_ereal )
% 2.87/1.77 = ( A = zero_z163181189_ereal ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_239_ereal__uminus__zero,axiom,
% 2.87/1.77 ( ( uminus1208298309_ereal @ zero_z163181189_ereal )
% 2.87/1.77 = zero_z163181189_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_240_ereal__minus__le__minus,axiom,
% 2.87/1.77 ! [A: extended_ereal,B: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ ( uminus1208298309_ereal @ B ) )
% 2.87/1.77 = ( ord_le824540014_ereal @ B @ A ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_241_enat_Oinject,axiom,
% 2.87/1.77 ! [Nat: nat,Nat2: nat] :
% 2.87/1.77 ( ( ( extended_enat2 @ Nat )
% 2.87/1.77 = ( extended_enat2 @ Nat2 ) )
% 2.87/1.77 = ( Nat = Nat2 ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_242_real__of__ereal,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( extend1716541707_ereal @ ( uminus1208298309_ereal @ X2 ) )
% 2.87/1.77 = ( uminus_uminus_real @ ( extend1716541707_ereal @ X2 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_243_ereal__infty__less__eq_I2_J,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 = ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_244_ereal__uminus__le__0__iff,axiom,
% 2.87/1.77 ! [A: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ zero_z163181189_ereal )
% 2.87/1.77 = ( ord_le824540014_ereal @ zero_z163181189_ereal @ A ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_245_ereal__0__le__uminus__iff,axiom,
% 2.87/1.77 ! [A: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( uminus1208298309_ereal @ A ) )
% 2.87/1.77 = ( ord_le824540014_ereal @ A @ zero_z163181189_ereal ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_246_real__of__ereal__0,axiom,
% 2.87/1.77 ( ( extend1716541707_ereal @ zero_z163181189_ereal )
% 2.87/1.77 = zero_zero_real ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_247_enat__ord__simps_I1_J,axiom,
% 2.87/1.77 ! [M: nat,N: nat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 2.87/1.77 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_248_real__of__ereal_Osimps_I3_J,axiom,
% 2.87/1.77 ( ( extend1716541707_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 = zero_zero_real ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_249_real__of__ereal__eq__0,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ( extend1716541707_ereal @ X2 )
% 2.87/1.77 = zero_zero_real )
% 2.87/1.77 = ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 | ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 | ( X2 = zero_z163181189_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_250_real__of__ereal_Osimps_I2_J,axiom,
% 2.87/1.77 ( ( extend1716541707_ereal @ extend1289208545_ereal )
% 2.87/1.77 = zero_zero_real ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_251_zero__enat__def,axiom,
% 2.87/1.77 ( zero_z491942557d_enat
% 2.87/1.77 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_252_enat__0__iff_I1_J,axiom,
% 2.87/1.77 ! [X2: nat] :
% 2.87/1.77 ( ( ( extended_enat2 @ X2 )
% 2.87/1.77 = zero_z491942557d_enat )
% 2.87/1.77 = ( X2 = zero_zero_nat ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_253_enat__0__iff_I2_J,axiom,
% 2.87/1.77 ! [X2: nat] :
% 2.87/1.77 ( ( zero_z491942557d_enat
% 2.87/1.77 = ( extended_enat2 @ X2 ) )
% 2.87/1.77 = ( X2 = zero_zero_nat ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_254_MInfty__neq__PInfty_I1_J,axiom,
% 2.87/1.77 ( extend1289208545_ereal
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_255_ereal__uminus__le__reorder,axiom,
% 2.87/1.77 ! [A: extended_ereal,B: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ A ) @ B )
% 2.87/1.77 = ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ B ) @ A ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_256_enat__ile,axiom,
% 2.87/1.77 ! [N: extended_enat,M: nat] :
% 2.87/1.77 ( ( ord_le1863327750d_enat @ N @ ( extended_enat2 @ M ) )
% 2.87/1.77 => ? [K: nat] :
% 2.87/1.77 ( N
% 2.87/1.77 = ( extended_enat2 @ K ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_257_numeral__eq__enat,axiom,
% 2.87/1.77 ( numera280919179d_enat
% 2.87/1.77 = ( ^ [K2: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_258_real__of__ereal__pos,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
% 2.87/1.77 => ( ord_less_eq_real @ zero_zero_real @ ( extend1716541707_ereal @ X2 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_259_Infty__neq__0_I3_J,axiom,
% 2.87/1.77 ( ( uminus1208298309_ereal @ extend1289208545_ereal )
% 2.87/1.77 != zero_z163181189_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_260_ereal__infty__less__eq2_I2_J,axiom,
% 2.87/1.77 ! [A: extended_ereal,B: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ A @ B )
% 2.87/1.77 => ( ( B
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( A
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_261_ereal__less__eq_I2_J,axiom,
% 2.87/1.77 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ X2 ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_262_real__of__ereal__positive__mono,axiom,
% 2.87/1.77 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ( Y3 != extend1289208545_ereal )
% 2.87/1.77 => ( ord_less_eq_real @ ( extend1716541707_ereal @ X2 ) @ ( extend1716541707_ereal @ Y3 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_263_not__MInfty__nonneg,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
% 2.87/1.77 => ( X2
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_264_le__refl,axiom,
% 2.87/1.77 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_265_le__trans,axiom,
% 2.87/1.77 ! [I: nat,J: nat,K3: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ I @ J )
% 2.87/1.77 => ( ( ord_less_eq_nat @ J @ K3 )
% 2.87/1.77 => ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_266_eq__imp__le,axiom,
% 2.87/1.77 ! [M: nat,N: nat] :
% 2.87/1.77 ( ( M = N )
% 2.87/1.77 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_267_le__antisym,axiom,
% 2.87/1.77 ! [M: nat,N: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ M @ N )
% 2.87/1.77 => ( ( ord_less_eq_nat @ N @ M )
% 2.87/1.77 => ( M = N ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_268_GreatestI__nat,axiom,
% 2.87/1.77 ! [P: nat > $o,K3: nat,B: nat] :
% 2.87/1.77 ( ( P @ K3 )
% 2.87/1.77 => ( ! [Y: nat] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_less_eq_nat @ Y @ B ) )
% 2.87/1.77 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_269_nat__le__linear,axiom,
% 2.87/1.77 ! [M: nat,N: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ M @ N )
% 2.87/1.77 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_270_Greatest__le__nat,axiom,
% 2.87/1.77 ! [P: nat > $o,K3: nat,B: nat] :
% 2.87/1.77 ( ( P @ K3 )
% 2.87/1.77 => ( ! [Y: nat] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_less_eq_nat @ Y @ B ) )
% 2.87/1.77 => ( ord_less_eq_nat @ K3 @ ( order_Greatest_nat @ P ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_271_GreatestI__ex__nat,axiom,
% 2.87/1.77 ! [P: nat > $o,B: nat] :
% 2.87/1.77 ( ? [X_1: nat] : ( P @ X_1 )
% 2.87/1.77 => ( ! [Y: nat] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_less_eq_nat @ Y @ B ) )
% 2.87/1.77 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_272_Nat_Oex__has__greatest__nat,axiom,
% 2.87/1.77 ! [P: nat > $o,K3: nat,B: nat] :
% 2.87/1.77 ( ( P @ K3 )
% 2.87/1.77 => ( ! [Y: nat] :
% 2.87/1.77 ( ( P @ Y )
% 2.87/1.77 => ( ord_less_eq_nat @ Y @ B ) )
% 2.87/1.77 => ? [X3: nat] :
% 2.87/1.77 ( ( P @ X3 )
% 2.87/1.77 & ! [Y5: nat] :
% 2.87/1.77 ( ( P @ Y5 )
% 2.87/1.77 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_273_strict__mono__imp__increasing,axiom,
% 2.87/1.77 ! [F: nat > nat,N: nat] :
% 2.87/1.77 ( ( order_769474267at_nat @ F )
% 2.87/1.77 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_274_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.87/1.77 ! [A: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.87/1.77 => ( A = zero_zero_nat ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_275_bot__nat__0_Oextremum__unique,axiom,
% 2.87/1.77 ! [A: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.87/1.77 = ( A = zero_zero_nat ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_276_le__0__eq,axiom,
% 2.87/1.77 ! [N: nat] :
% 2.87/1.77 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.87/1.77 = ( N = zero_zero_nat ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_277_less__eq__nat_Osimps_I1_J,axiom,
% 2.87/1.77 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_278_the__enat_Osimps,axiom,
% 2.87/1.77 ! [N: nat] :
% 2.87/1.77 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 2.87/1.77 = N ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_279_ereal__uminus__uminus,axiom,
% 2.87/1.77 ! [A: extended_ereal] :
% 2.87/1.77 ( ( uminus1208298309_ereal @ ( uminus1208298309_ereal @ A ) )
% 2.87/1.77 = A ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_280_ereal__uminus__eq__iff,axiom,
% 2.87/1.77 ! [A: extended_ereal,B: extended_ereal] :
% 2.87/1.77 ( ( ( uminus1208298309_ereal @ A )
% 2.87/1.77 = ( uminus1208298309_ereal @ B ) )
% 2.87/1.77 = ( A = B ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_281_ereal__uminus__eq__reorder,axiom,
% 2.87/1.77 ! [A: extended_ereal,B: extended_ereal] :
% 2.87/1.77 ( ( ( uminus1208298309_ereal @ A )
% 2.87/1.77 = B )
% 2.87/1.77 = ( A
% 2.87/1.77 = ( uminus1208298309_ereal @ B ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_282_real__eq__0__iff__le__ge__0,axiom,
% 2.87/1.77 ! [X2: real] :
% 2.87/1.77 ( ( X2 = zero_zero_real )
% 2.87/1.77 = ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 2.87/1.77 & ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_283_ereal__less__eq_I5_J,axiom,
% 2.87/1.77 ! [R: real] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( extended_ereal2 @ R ) )
% 2.87/1.77 = ( ord_less_eq_real @ zero_zero_real @ R ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_284_ereal__less__eq_I4_J,axiom,
% 2.87/1.77 ! [R: real] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R ) @ zero_z163181189_ereal )
% 2.87/1.77 = ( ord_less_eq_real @ R @ zero_zero_real ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_285_MInfty__eq__minfinity,axiom,
% 2.87/1.77 ( extended_MInfty
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_286_ereal__cong,axiom,
% 2.87/1.77 ! [X2: real,Y3: real] :
% 2.87/1.77 ( ( X2 = Y3 )
% 2.87/1.77 => ( ( extended_ereal2 @ X2 )
% 2.87/1.77 = ( extended_ereal2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_287_ereal_Oinject,axiom,
% 2.87/1.77 ! [X1: real,Y1: real] :
% 2.87/1.77 ( ( ( extended_ereal2 @ X1 )
% 2.87/1.77 = ( extended_ereal2 @ Y1 ) )
% 2.87/1.77 = ( X1 = Y1 ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_288_numeral__eq__ereal,axiom,
% 2.87/1.77 ( numera1793320307_ereal
% 2.87/1.77 = ( ^ [W: num] : ( extended_ereal2 @ ( numeral_numeral_real @ W ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_289_ereal__eq__0_I1_J,axiom,
% 2.87/1.77 ! [R: real] :
% 2.87/1.77 ( ( ( extended_ereal2 @ R )
% 2.87/1.77 = zero_z163181189_ereal )
% 2.87/1.77 = ( R = zero_zero_real ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_290_ereal__eq__0_I2_J,axiom,
% 2.87/1.77 ! [R: real] :
% 2.87/1.77 ( ( zero_z163181189_ereal
% 2.87/1.77 = ( extended_ereal2 @ R ) )
% 2.87/1.77 = ( R = zero_zero_real ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_291_ereal__less__eq_I3_J,axiom,
% 2.87/1.77 ! [R: real,P2: real] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P2 ) )
% 2.87/1.77 = ( ord_less_eq_real @ R @ P2 ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_292_ereal_Odistinct_I3_J,axiom,
% 2.87/1.77 ! [X1: real] :
% 2.87/1.77 ( ( extended_ereal2 @ X1 )
% 2.87/1.77 != extended_MInfty ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_293_real__of__ereal_Osimps_I1_J,axiom,
% 2.87/1.77 ! [R: real] :
% 2.87/1.77 ( ( extend1716541707_ereal @ ( extended_ereal2 @ R ) )
% 2.87/1.77 = R ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_294_ereal__le__real,axiom,
% 2.87/1.77 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ! [Z4: real] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ Z4 ) )
% 2.87/1.77 => ( ord_le824540014_ereal @ Y3 @ ( extended_ereal2 @ Z4 ) ) )
% 2.87/1.77 => ( ord_le824540014_ereal @ Y3 @ X2 ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_295_PInfty__neq__ereal_I1_J,axiom,
% 2.87/1.77 ! [R: real] :
% 2.87/1.77 ( ( extended_ereal2 @ R )
% 2.87/1.77 != extend1289208545_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_296_zero__ereal__def,axiom,
% 2.87/1.77 ( zero_z163181189_ereal
% 2.87/1.77 = ( extended_ereal2 @ zero_zero_real ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_297_le__ereal__le,axiom,
% 2.87/1.77 ! [A: extended_ereal,X2: real,Y3: real] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ X2 ) )
% 2.87/1.77 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le824540014_ereal @ A @ ( extended_ereal2 @ Y3 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_298_ereal__le__le,axiom,
% 2.87/1.77 ! [Y3: real,A: extended_ereal,X2: real] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ ( extended_ereal2 @ Y3 ) @ A )
% 2.87/1.77 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_299_real__of__ereal_Oinduct,axiom,
% 2.87/1.77 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.87/1.77 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( P @ extend1289208545_ereal )
% 2.87/1.77 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( P @ A0 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_300_real__of__ereal_Ocases,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ! [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 != ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( X2 != extend1289208545_ereal )
% 2.87/1.77 => ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_301_times__ereal_Oinduct,axiom,
% 2.87/1.77 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.87/1.77 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 2.87/1.77 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ extend1289208545_ereal )
% 2.87/1.77 => ( ! [R2: real] : ( P @ extend1289208545_ereal @ ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( P @ extend1289208545_ereal @ extend1289208545_ereal )
% 2.87/1.77 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 2.87/1.77 => ( ( P @ extend1289208545_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_302_plus__ereal_Oinduct,axiom,
% 2.87/1.77 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.87/1.77 ( ! [R2: real,P3: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( extended_ereal2 @ P3 ) )
% 2.87/1.77 => ( ! [X_12: extended_ereal] : ( P @ extend1289208545_ereal @ X_12 )
% 2.87/1.77 => ( ! [A3: extended_ereal] : ( P @ A3 @ extend1289208545_ereal )
% 2.87/1.77 => ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ! [P3: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ P3 ) )
% 2.87/1.77 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_303_less__ereal_Oinduct,axiom,
% 2.87/1.77 ! [P: extended_ereal > extended_ereal > $o,A0: extended_ereal,A1: extended_ereal] :
% 2.87/1.77 ( ! [X3: real,Y: real] : ( P @ ( extended_ereal2 @ X3 ) @ ( extended_ereal2 @ Y ) )
% 2.87/1.77 => ( ! [X_12: extended_ereal] : ( P @ extend1289208545_ereal @ X_12 )
% 2.87/1.77 => ( ! [A3: extended_ereal] : ( P @ A3 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) @ extend1289208545_ereal )
% 2.87/1.77 => ( ! [R2: real] : ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) @ extend1289208545_ereal )
% 2.87/1.77 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_304_abs__ereal_Oinduct,axiom,
% 2.87/1.77 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.87/1.77 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( P @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ( P @ extend1289208545_ereal )
% 2.87/1.77 => ( P @ A0 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_305_ereal__all__split,axiom,
% 2.87/1.77 ( ( ^ [P4: extended_ereal > $o] :
% 2.87/1.77 ! [X6: extended_ereal] : ( P4 @ X6 ) )
% 2.87/1.77 = ( ^ [P5: extended_ereal > $o] :
% 2.87/1.77 ( ( P5 @ extend1289208545_ereal )
% 2.87/1.77 & ! [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
% 2.87/1.77 & ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_306_abs__ereal_Ocases,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ! [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 != ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( X2
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( X2 = extend1289208545_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_307_ereal__ex__split,axiom,
% 2.87/1.77 ( ( ^ [P4: extended_ereal > $o] :
% 2.87/1.77 ? [X6: extended_ereal] : ( P4 @ X6 ) )
% 2.87/1.77 = ( ^ [P5: extended_ereal > $o] :
% 2.87/1.77 ( ( P5 @ extend1289208545_ereal )
% 2.87/1.77 | ? [X4: real] : ( P5 @ ( extended_ereal2 @ X4 ) )
% 2.87/1.77 | ( P5 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_308_ereal3__cases,axiom,
% 2.87/1.77 ! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
% 2.87/1.77 ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ? [Ra: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 = ( extended_ereal2 @ Ra ) )
% 2.87/1.77 => ! [Rb: real] :
% 2.87/1.77 ( Xb
% 2.87/1.77 != ( extended_ereal2 @ Rb ) ) ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ? [Ra: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 = ( extended_ereal2 @ Ra ) )
% 2.87/1.77 => ( Xb != extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ? [Ra: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 = ( extended_ereal2 @ Ra ) )
% 2.87/1.77 => ( Xb
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( Xa3 = extend1289208545_ereal )
% 2.87/1.77 => ! [Ra: real] :
% 2.87/1.77 ( Xb
% 2.87/1.77 != ( extended_ereal2 @ Ra ) ) ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( Xa3 = extend1289208545_ereal )
% 2.87/1.77 => ( Xb != extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( Xa3 = extend1289208545_ereal )
% 2.87/1.77 => ( Xb
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( Xa3
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ! [Ra: real] :
% 2.87/1.77 ( Xb
% 2.87/1.77 != ( extended_ereal2 @ Ra ) ) ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( Xa3
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( Xb != extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( Xa3
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( Xb
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( ? [R2: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ! [Ra: real] :
% 2.87/1.77 ( Xb
% 2.87/1.77 != ( extended_ereal2 @ Ra ) ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( ? [R2: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( Xb != extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( ? [R2: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( Xb
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( ( Xa3 = extend1289208545_ereal )
% 2.87/1.77 => ! [R2: real] :
% 2.87/1.77 ( Xb
% 2.87/1.77 != ( extended_ereal2 @ R2 ) ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( ( Xa3 = extend1289208545_ereal )
% 2.87/1.77 => ( Xb != extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( ( Xa3 = extend1289208545_ereal )
% 2.87/1.77 => ( Xb
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( ( Xa3
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ! [R2: real] :
% 2.87/1.77 ( Xb
% 2.87/1.77 != ( extended_ereal2 @ R2 ) ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( ( Xa3
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( Xb != extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( ( Xa3
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( Xb
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ? [R2: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ! [Ra: real] :
% 2.87/1.77 ( Xb
% 2.87/1.77 != ( extended_ereal2 @ Ra ) ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ? [R2: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( Xb != extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ? [R2: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( Xb
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ( Xa3 = extend1289208545_ereal )
% 2.87/1.77 => ! [R2: real] :
% 2.87/1.77 ( Xb
% 2.87/1.77 != ( extended_ereal2 @ R2 ) ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ( Xa3 = extend1289208545_ereal )
% 2.87/1.77 => ( Xb != extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ( Xa3 = extend1289208545_ereal )
% 2.87/1.77 => ( Xb
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ( Xa3
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ! [R2: real] :
% 2.87/1.77 ( Xb
% 2.87/1.77 != ( extended_ereal2 @ R2 ) ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ( Xa3
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( Xb != extend1289208545_ereal ) ) )
% 2.87/1.77 => ~ ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ( Xa3
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( Xb
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_309_ereal2__cases,axiom,
% 2.87/1.77 ! [X2: extended_ereal,Xa3: extended_ereal] :
% 2.87/1.77 ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ! [Ra: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 != ( extended_ereal2 @ Ra ) ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( Xa3 != extend1289208545_ereal ) )
% 2.87/1.77 => ( ( ? [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( Xa3
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ! [R2: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 != ( extended_ereal2 @ R2 ) ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( Xa3 != extend1289208545_ereal ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( Xa3
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ! [R2: real] :
% 2.87/1.77 ( Xa3
% 2.87/1.77 != ( extended_ereal2 @ R2 ) ) )
% 2.87/1.77 => ( ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( Xa3 != extend1289208545_ereal ) )
% 2.87/1.77 => ~ ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( Xa3
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_310_ereal__cases,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ! [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 != ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( X2 != extend1289208545_ereal )
% 2.87/1.77 => ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_311_MInfty__neq__ereal_I1_J,axiom,
% 2.87/1.77 ! [R: real] :
% 2.87/1.77 ( ( extended_ereal2 @ R )
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_312_ereal__top,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ! [B4: real] : ( ord_le824540014_ereal @ ( extended_ereal2 @ B4 ) @ X2 )
% 2.87/1.77 => ( X2 = extend1289208545_ereal ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_313_uminus__ereal_Osimps_I1_J,axiom,
% 2.87/1.77 ! [R: real] :
% 2.87/1.77 ( ( uminus1208298309_ereal @ ( extended_ereal2 @ R ) )
% 2.87/1.77 = ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_314_ereal__bot,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ! [B4: real] : ( ord_le824540014_ereal @ X2 @ ( extended_ereal2 @ B4 ) )
% 2.87/1.77 => ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_315_real__of__ereal_Oelims,axiom,
% 2.87/1.77 ! [X2: extended_ereal,Y3: real] :
% 2.87/1.77 ( ( ( extend1716541707_ereal @ X2 )
% 2.87/1.77 = Y3 )
% 2.87/1.77 => ( ! [R2: real] :
% 2.87/1.77 ( ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( Y3 != R2 ) )
% 2.87/1.77 => ( ( ( X2 = extend1289208545_ereal )
% 2.87/1.77 => ( Y3 != zero_zero_real ) )
% 2.87/1.77 => ~ ( ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( Y3 != zero_zero_real ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_316_ereal_Osize__gen_I3_J,axiom,
% 2.87/1.77 ( ( extended_size_ereal @ extended_MInfty )
% 2.87/1.77 = zero_zero_nat ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_317_ereal_Osize__gen_I1_J,axiom,
% 2.87/1.77 ! [X1: real] :
% 2.87/1.77 ( ( extended_size_ereal @ ( extended_ereal2 @ X1 ) )
% 2.87/1.77 = zero_zero_nat ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_318_uminus__ereal_Oelims,axiom,
% 2.87/1.77 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( ( uminus1208298309_ereal @ X2 )
% 2.87/1.77 = Y3 )
% 2.87/1.77 => ( ! [R2: real] :
% 2.87/1.77 ( ( X2
% 2.87/1.77 = ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( Y3
% 2.87/1.77 != ( extended_ereal2 @ ( uminus_uminus_real @ R2 ) ) ) )
% 2.87/1.77 => ( ( ( X2 = extended_PInfty )
% 2.87/1.77 => ( Y3 != extended_MInfty ) )
% 2.87/1.77 => ~ ( ( X2 = extended_MInfty )
% 2.87/1.77 => ( Y3 != extended_PInfty ) ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_319_ereal_Odistinct_I1_J,axiom,
% 2.87/1.77 ! [X1: real] :
% 2.87/1.77 ( ( extended_ereal2 @ X1 )
% 2.87/1.77 != extended_PInfty ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_320_ereal_Osize__gen_I2_J,axiom,
% 2.87/1.77 ( ( extended_size_ereal @ extended_PInfty )
% 2.87/1.77 = zero_zero_nat ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_321_infinity__ereal__def,axiom,
% 2.87/1.77 extend1289208545_ereal = extended_PInfty ).
% 2.87/1.77
% 2.87/1.77 thf(fact_322_ereal_Odistinct_I5_J,axiom,
% 2.87/1.77 extended_PInfty != extended_MInfty ).
% 2.87/1.77
% 2.87/1.77 thf(fact_323_uminus__ereal_Oinduct,axiom,
% 2.87/1.77 ! [P: extended_ereal > $o,A0: extended_ereal] :
% 2.87/1.77 ( ! [R2: real] : ( P @ ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( P @ extended_PInfty )
% 2.87/1.77 => ( ( P @ extended_MInfty )
% 2.87/1.77 => ( P @ A0 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_324_uminus__ereal_Ocases,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ! [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 != ( extended_ereal2 @ R2 ) )
% 2.87/1.77 => ( ( X2 != extended_PInfty )
% 2.87/1.77 => ( X2 = extended_MInfty ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_325_ereal_Oexhaust,axiom,
% 2.87/1.77 ! [Y3: extended_ereal] :
% 2.87/1.77 ( ! [X12: real] :
% 2.87/1.77 ( Y3
% 2.87/1.77 != ( extended_ereal2 @ X12 ) )
% 2.87/1.77 => ( ( Y3 != extended_PInfty )
% 2.87/1.77 => ( Y3 = extended_MInfty ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_326_ereal_Oinduct,axiom,
% 2.87/1.77 ! [P: extended_ereal > $o,Ereal: extended_ereal] :
% 2.87/1.77 ( ! [X3: real] : ( P @ ( extended_ereal2 @ X3 ) )
% 2.87/1.77 => ( ( P @ extended_PInfty )
% 2.87/1.77 => ( ( P @ extended_MInfty )
% 2.87/1.77 => ( P @ Ereal ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_327_uminus__ereal_Osimps_I2_J,axiom,
% 2.87/1.77 ( ( uminus1208298309_ereal @ extended_PInfty )
% 2.87/1.77 = extended_MInfty ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_328_uminus__ereal_Osimps_I3_J,axiom,
% 2.87/1.77 ( ( uminus1208298309_ereal @ extended_MInfty )
% 2.87/1.77 = extended_PInfty ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_329_ereal__divide__ereal,axiom,
% 2.87/1.77 ! [R: real] :
% 2.87/1.77 ( ( ( ord_less_eq_real @ zero_zero_real @ R )
% 2.87/1.77 => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R ) )
% 2.87/1.77 = extend1289208545_ereal ) )
% 2.87/1.77 & ( ~ ( ord_less_eq_real @ zero_zero_real @ R )
% 2.87/1.77 => ( ( divide595620860_ereal @ extend1289208545_ereal @ ( extended_ereal2 @ R ) )
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_330_ereal__le__real__iff,axiom,
% 2.87/1.77 ! [X2: real,Y3: extended_ereal] :
% 2.87/1.77 ( ( ord_less_eq_real @ X2 @ ( extend1716541707_ereal @ Y3 ) )
% 2.87/1.77 = ( ( ( ( abs_ab1260901297_ereal @ Y3 )
% 2.87/1.77 != extend1289208545_ereal )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( extended_ereal2 @ X2 ) @ Y3 ) )
% 2.87/1.77 & ( ( ( abs_ab1260901297_ereal @ Y3 )
% 2.87/1.77 = extend1289208545_ereal )
% 2.87/1.77 => ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_331_abs__ereal__uminus,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( abs_ab1260901297_ereal @ ( uminus1208298309_ereal @ X2 ) )
% 2.87/1.77 = ( abs_ab1260901297_ereal @ X2 ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_332_abs__ereal__zero,axiom,
% 2.87/1.77 ( ( abs_ab1260901297_ereal @ zero_z163181189_ereal )
% 2.87/1.77 = zero_z163181189_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_333_ereal__uminus__divide,axiom,
% 2.87/1.77 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( divide595620860_ereal @ ( uminus1208298309_ereal @ X2 ) @ Y3 )
% 2.87/1.77 = ( uminus1208298309_ereal @ ( divide595620860_ereal @ X2 @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_334_ereal__divide__zero__left,axiom,
% 2.87/1.77 ! [A: extended_ereal] :
% 2.87/1.77 ( ( divide595620860_ereal @ zero_z163181189_ereal @ A )
% 2.87/1.77 = zero_z163181189_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_335_abs__ereal__ge0,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ X2 )
% 2.87/1.77 => ( ( abs_ab1260901297_ereal @ X2 )
% 2.87/1.77 = X2 ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_336_ereal__divide__Infty_I1_J,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( divide595620860_ereal @ X2 @ extend1289208545_ereal )
% 2.87/1.77 = zero_z163181189_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_337_ereal__divide__Infty_I2_J,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( divide595620860_ereal @ X2 @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 = zero_z163181189_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_338_not__infty__ereal,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ( abs_ab1260901297_ereal @ X2 )
% 2.87/1.77 != extend1289208545_ereal )
% 2.87/1.77 = ( ? [X7: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 = ( extended_ereal2 @ X7 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_339_abs__neq__infinity__cases,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ( abs_ab1260901297_ereal @ X2 )
% 2.87/1.77 != extend1289208545_ereal )
% 2.87/1.77 => ~ ! [R2: real] :
% 2.87/1.77 ( X2
% 2.87/1.77 != ( extended_ereal2 @ R2 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_340_abs__ereal_Osimps_I2_J,axiom,
% 2.87/1.77 ( ( abs_ab1260901297_ereal @ ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 = extend1289208545_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_341_ereal__infinity__cases,axiom,
% 2.87/1.77 ! [A: extended_ereal] :
% 2.87/1.77 ( ( A != extend1289208545_ereal )
% 2.87/1.77 => ( ( A
% 2.87/1.77 != ( uminus1208298309_ereal @ extend1289208545_ereal ) )
% 2.87/1.77 => ( ( abs_ab1260901297_ereal @ A )
% 2.87/1.77 != extend1289208545_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_342_abs__eq__infinity__cases,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ( abs_ab1260901297_ereal @ X2 )
% 2.87/1.77 = extend1289208545_ereal )
% 2.87/1.77 => ( ( X2 != extend1289208545_ereal )
% 2.87/1.77 => ( X2
% 2.87/1.77 = ( uminus1208298309_ereal @ extend1289208545_ereal ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_343_ereal__abs__leI,axiom,
% 2.87/1.77 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ X2 @ Y3 )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ ( uminus1208298309_ereal @ X2 ) @ Y3 )
% 2.87/1.77 => ( ord_le824540014_ereal @ ( abs_ab1260901297_ereal @ X2 ) @ Y3 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_344_abs__ereal__pos,axiom,
% 2.87/1.77 ! [X2: extended_ereal] : ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( abs_ab1260901297_ereal @ X2 ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_345_zero__le__divide__ereal,axiom,
% 2.87/1.77 ! [A: extended_ereal,B: extended_ereal] :
% 2.87/1.77 ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ A )
% 2.87/1.77 => ( ( ord_le824540014_ereal @ zero_z163181189_ereal @ B )
% 2.87/1.77 => ( ord_le824540014_ereal @ zero_z163181189_ereal @ ( divide595620860_ereal @ A @ B ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_346_ereal__divide__eq__0__iff,axiom,
% 2.87/1.77 ! [X2: extended_ereal,Y3: extended_ereal] :
% 2.87/1.77 ( ( ( divide595620860_ereal @ X2 @ Y3 )
% 2.87/1.77 = zero_z163181189_ereal )
% 2.87/1.77 = ( ( X2 = zero_z163181189_ereal )
% 2.87/1.77 | ( ( abs_ab1260901297_ereal @ Y3 )
% 2.87/1.77 = extend1289208545_ereal ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_347_abs__ereal_Osimps_I3_J,axiom,
% 2.87/1.77 ( ( abs_ab1260901297_ereal @ extend1289208545_ereal )
% 2.87/1.77 = extend1289208545_ereal ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_348_ereal__real_H,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ( abs_ab1260901297_ereal @ X2 )
% 2.87/1.77 != extend1289208545_ereal )
% 2.87/1.77 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 2.87/1.77 = X2 ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_349_ereal__real,axiom,
% 2.87/1.77 ! [X2: extended_ereal] :
% 2.87/1.77 ( ( ( ( abs_ab1260901297_ereal @ X2 )
% 2.87/1.77 = extend1289208545_ereal )
% 2.87/1.77 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 2.87/1.77 = zero_z163181189_ereal ) )
% 2.87/1.77 & ( ( ( abs_ab1260901297_ereal @ X2 )
% 2.87/1.77 != extend1289208545_ereal )
% 2.87/1.77 => ( ( extended_ereal2 @ ( extend1716541707_ereal @ X2 ) )
% 2.87/1.77 = X2 ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(fact_350_real__le__ereal__iff,axiom,
% 2.87/1.77 ! [Y3: extended_ereal,X2: real] :
% 2.87/1.77 ( ( ord_less_eq_real @ ( extend1716541707_ereal @ Y3 ) @ X2 )
% 2.87/1.77 = ( ( ( ( abs_ab1260901297_ereal @ Y3 )
% 2.87/1.77 != extend1289208545_ereal )
% 2.87/1.77 => ( ord_le824540014_ereal @ Y3 @ ( extended_ereal2 @ X2 ) ) )
% 2.87/1.77 & ( ( ( abs_ab1260901297_ereal @ Y3 )
% 2.87/1.77 = extend1289208545_ereal )
% 2.87/1.77 => ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ) ) ).
% 2.87/1.77
% 2.87/1.77 thf(conj_0,conjecture,
% 2.87/1.77 ord_le824540014_ereal @ ( g @ x ) @ ( f @ x ) ).
% 2.87/1.77 ------------------------
% 2.87/1.79 % (7674)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_7494 for (2999ds/27Mi)
% 2.87/1.79 % (7675)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/2Mi)
% 2.87/1.79 % (7676)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_7494 for (2999ds/2Mi)
% 2.87/1.79 % (7679)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/18Mi)
% 2.87/1.79 % (7675)Instruction limit reached!
% 2.87/1.79 % (7675)------------------------------
% 2.87/1.79 % (7675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.79 % (7675)Termination reason: Unknown
% 2.87/1.79 % (7675)Termination phase: shuffling
% 2.87/1.79
% 2.87/1.79 % (7675)Memory used [KB]: 1407
% 2.87/1.79 % (7675)Time elapsed: 0.002 s
% 2.87/1.79 % (7675)Instructions burned: 2 (million)
% 2.87/1.79 % (7675)------------------------------
% 2.87/1.79 % (7675)------------------------------
% 2.87/1.79 % (7676)Instruction limit reached!
% 2.87/1.79 % (7676)------------------------------
% 2.87/1.79 % (7676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.79 % (7676)Termination reason: Unknown
% 2.87/1.79 % (7676)Termination phase: shuffling
% 2.87/1.79
% 2.87/1.79 % (7676)Memory used [KB]: 1407
% 2.87/1.79 % (7676)Time elapsed: 0.002 s
% 2.87/1.79 % (7676)Instructions burned: 2 (million)
% 2.87/1.79 % (7676)------------------------------
% 2.87/1.79 % (7676)------------------------------
% 2.87/1.79 % (7679)Instruction limit reached!
% 2.87/1.79 % (7679)------------------------------
% 2.87/1.79 % (7679)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.79 % (7679)Termination reason: Unknown
% 2.87/1.79 % (7679)Termination phase: shuffling
% 2.87/1.79
% 2.87/1.79 % (7679)Memory used [KB]: 1791
% 2.87/1.79 % (7679)Time elapsed: 0.007 s
% 2.87/1.79 % (7679)Instructions burned: 18 (million)
% 2.87/1.79 % (7679)------------------------------
% 2.87/1.79 % (7679)------------------------------
% 2.87/1.80 % (7672)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/183Mi)
% 2.87/1.80 % (7673)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_7494 for (2999ds/4Mi)
% 2.87/1.80 % (7678)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_7494 for (2999ds/275Mi)
% 2.95/1.80 % (7674)Instruction limit reached!
% 2.95/1.80 % (7674)------------------------------
% 2.95/1.80 % (7674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.80 % (7674)Termination reason: Unknown
% 2.95/1.80 % (7674)Termination phase: Property scanning
% 2.95/1.80
% 2.95/1.80 % (7674)Memory used [KB]: 1918
% 2.95/1.80 % (7674)Time elapsed: 0.010 s
% 2.95/1.80 % (7674)Instructions burned: 27 (million)
% 2.95/1.80 % (7674)------------------------------
% 2.95/1.80 % (7674)------------------------------
% 2.95/1.80 % (7687)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_7494 for (2999ds/37Mi)
% 2.95/1.80 % (7673)Instruction limit reached!
% 2.95/1.80 % (7673)------------------------------
% 2.95/1.80 % (7673)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.80 % (7673)Termination reason: Unknown
% 2.95/1.80 % (7673)Termination phase: shuffling
% 2.95/1.80
% 2.95/1.80 % (7673)Memory used [KB]: 1407
% 2.95/1.80 % (7673)Time elapsed: 0.005 s
% 2.95/1.80 % (7673)Instructions burned: 5 (million)
% 2.95/1.80 % (7673)------------------------------
% 2.95/1.80 % (7673)------------------------------
% 2.95/1.80 % (7686)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/3Mi)
% 2.95/1.80 % (7686)Instruction limit reached!
% 2.95/1.80 % (7686)------------------------------
% 2.95/1.80 % (7686)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.80 % (7686)Termination reason: Unknown
% 2.95/1.80 % (7686)Termination phase: shuffling
% 2.95/1.80
% 2.95/1.80 % (7686)Memory used [KB]: 1407
% 2.95/1.80 % (7686)Time elapsed: 0.002 s
% 2.95/1.80 % (7686)Instructions burned: 4 (million)
% 2.95/1.80 % (7686)------------------------------
% 2.95/1.80 % (7686)------------------------------
% 2.95/1.80 % (7688)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_7494 for (2999ds/15Mi)
% 2.95/1.81 % (7689)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_7494 for (2999ds/3Mi)
% 2.95/1.81 % (7689)Instruction limit reached!
% 2.95/1.81 % (7689)------------------------------
% 2.95/1.81 % (7689)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.81 % (7689)Termination reason: Unknown
% 2.95/1.81 % (7689)Termination phase: shuffling
% 2.95/1.81
% 2.95/1.81 % (7689)Memory used [KB]: 1535
% 2.95/1.81 % (7689)Time elapsed: 0.003 s
% 2.95/1.81 % (7689)Instructions burned: 5 (million)
% 2.95/1.81 % (7689)------------------------------
% 2.95/1.81 % (7689)------------------------------
% 2.95/1.81 % (7687)Instruction limit reached!
% 2.95/1.81 % (7687)------------------------------
% 2.95/1.81 % (7687)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.81 % (7687)Termination reason: Unknown
% 2.95/1.81 % (7687)Termination phase: shuffling
% 2.95/1.81
% 2.95/1.81 % (7687)Memory used [KB]: 2174
% 2.95/1.81 % (7687)Time elapsed: 0.012 s
% 2.95/1.81 % (7687)Instructions burned: 39 (million)
% 2.95/1.81 % (7687)------------------------------
% 2.95/1.81 % (7687)------------------------------
% 2.95/1.81 % (7688)Instruction limit reached!
% 2.95/1.81 % (7688)------------------------------
% 2.95/1.81 % (7688)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.81 % (7688)Termination reason: Unknown
% 2.95/1.81 % (7688)Termination phase: shuffling
% 2.95/1.81
% 2.95/1.81 % (7688)Memory used [KB]: 1791
% 2.95/1.81 % (7688)Time elapsed: 0.007 s
% 2.95/1.81 % (7688)Instructions burned: 16 (million)
% 2.95/1.81 % (7688)------------------------------
% 2.95/1.81 % (7688)------------------------------
% 2.95/1.81 % (7691)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/7Mi)
% 2.95/1.81 % (7691)Instruction limit reached!
% 2.95/1.81 % (7691)------------------------------
% 2.95/1.81 % (7691)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.81 % (7691)Termination reason: Unknown
% 2.95/1.81 % (7691)Termination phase: shuffling
% 2.95/1.81
% 2.95/1.81 % (7691)Memory used [KB]: 1535
% 2.95/1.81 % (7691)Time elapsed: 0.004 s
% 2.95/1.81 % (7691)Instructions burned: 8 (million)
% 2.95/1.81 % (7691)------------------------------
% 2.95/1.81 % (7691)------------------------------
% 2.95/1.81 % (7690)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/1041Mi)
% 2.95/1.82 % (7694)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/16Mi)
% 2.95/1.82 % (7698)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/3Mi)
% 2.95/1.82 % (7699)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/3Mi)
% 2.95/1.82 % (7698)Instruction limit reached!
% 2.95/1.82 % (7698)------------------------------
% 2.95/1.82 % (7698)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.82 % (7698)Termination reason: Unknown
% 2.95/1.82 % (7698)Termination phase: shuffling
% 2.95/1.82
% 2.95/1.82 % (7698)Memory used [KB]: 1407
% 2.95/1.82 % (7698)Time elapsed: 0.002 s
% 2.95/1.82 % (7698)Instructions burned: 3 (million)
% 2.95/1.82 % (7698)------------------------------
% 2.95/1.82 % (7698)------------------------------
% 2.95/1.82 % (7699)Instruction limit reached!
% 2.95/1.82 % (7699)------------------------------
% 2.95/1.82 % (7699)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.82 % (7699)Termination reason: Unknown
% 2.95/1.82 % (7699)Termination phase: shuffling
% 2.95/1.82
% 2.95/1.82 % (7699)Memory used [KB]: 1407
% 2.95/1.82 % (7699)Time elapsed: 0.002 s
% 2.95/1.82 % (7699)Instructions burned: 3 (million)
% 2.95/1.82 % (7699)------------------------------
% 2.95/1.82 % (7699)------------------------------
% 2.95/1.82 % (7703)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_7494 for (2999ds/7Mi)
% 2.95/1.82 % (7694)Instruction limit reached!
% 2.95/1.82 % (7694)------------------------------
% 2.95/1.82 % (7694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.82 % (7694)Termination reason: Unknown
% 2.95/1.82 % (7694)Termination phase: shuffling
% 2.95/1.82
% 2.95/1.82 % (7694)Memory used [KB]: 1791
% 2.95/1.82 % (7694)Time elapsed: 0.006 s
% 2.95/1.82 % (7694)Instructions burned: 16 (million)
% 2.95/1.82 % (7694)------------------------------
% 2.95/1.82 % (7694)------------------------------
% 2.95/1.82 % (7678)First to succeed.
% 2.95/1.82 % (7678)Refutation found. Thanks to Tanya!
% 2.95/1.82 % SZS status Theorem for DTF2THF_7494
% 2.95/1.82 % SZS output start Proof for DTF2THF_7494
% 2.95/1.82 thf(type_def_5, type, nat: $tType).
% 2.95/1.82 thf(type_def_6, type, extended_enat: $tType).
% 2.95/1.82 thf(type_def_8, type, extended_ereal: $tType).
% 2.95/1.82 thf(type_def_9, type, real: $tType).
% 2.95/1.82 thf(type_def_10, type, a: $tType).
% 2.95/1.82 thf(type_def_11, type, num: $tType).
% 2.95/1.82 thf(type_def_12, type, set_Extended_ereal: $tType).
% 2.95/1.82 thf(func_def_0, type, set_Extended_ereal: $tType).
% 2.95/1.82 thf(func_def_1, type, extended_ereal: $tType).
% 2.95/1.82 thf(func_def_2, type, extended_enat: $tType).
% 2.95/1.82 thf(func_def_3, type, real: $tType).
% 2.95/1.82 thf(func_def_4, type, num: $tType).
% 2.95/1.82 thf(func_def_5, type, nat: $tType).
% 2.95/1.82 thf(func_def_6, type, a: $tType).
% 2.95/1.82 thf(func_def_7, type, extended_enat2: nat > extended_enat).
% 2.95/1.82 thf(func_def_8, type, extend1289208545_ereal: extended_ereal).
% 2.95/1.82 thf(func_def_9, type, extended_the_enat: extended_enat > nat).
% 2.95/1.82 thf(func_def_10, type, extended_MInfty: extended_ereal).
% 2.95/1.82 thf(func_def_11, type, extended_PInfty: extended_ereal).
% 2.95/1.82 thf(func_def_12, type, extended_ereal2: real > extended_ereal).
% 2.95/1.82 thf(func_def_13, type, extended_size_ereal: extended_ereal > nat).
% 2.95/1.82 thf(func_def_14, type, extend1771934483f_enat: extended_enat > extended_ereal).
% 2.95/1.82 thf(func_def_15, type, extend1716541707_ereal: extended_ereal > real).
% 2.95/1.82 thf(func_def_16, type, abs_ab1260901297_ereal: extended_ereal > extended_ereal).
% 2.95/1.82 thf(func_def_17, type, uminus1208298309_ereal: extended_ereal > extended_ereal).
% 2.95/1.82 thf(func_def_18, type, uminus_uminus_real: real > real).
% 2.95/1.82 thf(func_def_19, type, zero_z491942557d_enat: extended_enat).
% 2.95/1.82 thf(func_def_20, type, zero_z163181189_ereal: extended_ereal).
% 2.95/1.82 thf(func_def_21, type, zero_zero_nat: nat).
% 2.95/1.82 thf(func_def_22, type, zero_zero_real: real).
% 2.95/1.82 thf(func_def_23, type, lower_1616484581_ereal: (a > extended_ereal) > $o).
% 2.95/1.82 thf(func_def_24, type, lower_881475195hull_a: (a > extended_ereal) > a > extended_ereal).
% 2.95/1.82 thf(func_def_25, type, numera280919179d_enat: num > extended_enat).
% 2.95/1.82 thf(func_def_26, type, numera1793320307_ereal: num > extended_ereal).
% 2.95/1.82 thf(func_def_27, type, numeral_numeral_nat: num > nat).
% 2.95/1.82 thf(func_def_28, type, numeral_numeral_real: num > real).
% 2.95/1.82 thf(func_def_29, type, ord_le291126163d_enat: ($o > extended_enat) > ($o > extended_enat) > $o).
% 2.95/1.82 thf(func_def_30, type, ord_le637473275_ereal: ($o > extended_ereal) > ($o > extended_ereal) > $o).
% 2.95/1.82 thf(func_def_31, type, ord_less_eq_o_nat: ($o > nat) > ($o > nat) > $o).
% 2.95/1.82 thf(func_def_32, type, ord_less_eq_o_real: ($o > real) > ($o > real) > $o).
% 2.95/1.82 thf(func_def_33, type, ord_le1863327750d_enat: extended_enat > extended_enat > $o).
% 2.95/1.82 thf(func_def_34, type, ord_le824540014_ereal: extended_ereal > extended_ereal > $o).
% 2.95/1.82 thf(func_def_35, type, ord_less_eq_nat: nat > nat > $o).
% 2.95/1.82 thf(func_def_36, type, ord_less_eq_num: num > num > $o).
% 2.95/1.82 thf(func_def_37, type, ord_less_eq_real: real > real > $o).
% 2.95/1.82 thf(func_def_38, type, order_1628344639d_enat: (extended_enat > $o) > extended_enat).
% 2.95/1.82 thf(func_def_39, type, order_1158471719_ereal: (extended_ereal > $o) > extended_ereal).
% 2.95/1.82 thf(func_def_40, type, order_Greatest_nat: (nat > $o) > nat).
% 2.95/1.82 thf(func_def_41, type, order_Greatest_real: (real > $o) > real).
% 2.95/1.82 thf(func_def_42, type, order_2047034162d_enat: (extended_enat > extended_enat) > $o).
% 2.95/1.82 thf(func_def_43, type, order_1147259034_ereal: (extended_enat > extended_ereal) > $o).
% 2.95/1.82 thf(func_def_44, type, order_442688004at_nat: (extended_enat > nat) > $o).
% 2.95/1.82 thf(func_def_45, type, order_182315744t_real: (extended_enat > real) > $o).
% 2.95/1.82 thf(func_def_46, type, order_680161354d_enat: (extended_ereal > extended_enat) > $o).
% 2.95/1.82 thf(func_def_47, type, order_1408494002_ereal: (extended_ereal > extended_ereal) > $o).
% 2.95/1.82 thf(func_def_48, type, order_523806444al_nat: (extended_ereal > nat) > $o).
% 2.95/1.82 thf(func_def_49, type, order_1800387528l_real: (extended_ereal > real) > $o).
% 2.95/1.82 thf(func_def_50, type, order_1660553314d_enat: (nat > extended_enat) > $o).
% 2.95/1.82 thf(func_def_51, type, order_736687562_ereal: (nat > extended_ereal) > $o).
% 2.95/1.82 thf(func_def_52, type, order_1631207636at_nat: (nat > nat) > $o).
% 2.95/1.82 thf(func_def_53, type, order_106095024t_real: (nat > real) > $o).
% 2.95/1.82 thf(func_def_54, type, order_2106278841d_enat: (extended_enat > extended_enat) > $o).
% 2.95/1.82 thf(func_def_55, type, order_1180956065_ereal: (extended_enat > extended_ereal) > $o).
% 2.95/1.82 thf(func_def_56, type, order_1622825661at_nat: (extended_enat > nat) > $o).
% 2.95/1.82 thf(func_def_57, type, order_135436057t_real: (extended_enat > real) > $o).
% 2.95/1.82 thf(func_def_58, type, order_713858385d_enat: (extended_ereal > extended_enat) > $o).
% 2.95/1.82 thf(func_def_59, type, order_555877177_ereal: (extended_ereal > extended_ereal) > $o).
% 2.95/1.82 thf(func_def_60, type, order_476926757al_nat: (extended_ereal > nat) > $o).
% 2.95/1.82 thf(func_def_61, type, order_1560271745l_real: (extended_ereal > real) > $o).
% 2.95/1.82 thf(func_def_62, type, order_693207323d_enat: (nat > extended_enat) > $o).
% 2.95/1.82 thf(func_def_63, type, order_689807875_ereal: (nat > extended_ereal) > $o).
% 2.95/1.82 thf(func_def_64, type, order_769474267at_nat: (nat > nat) > $o).
% 2.95/1.82 thf(func_def_65, type, order_1962034751d_enat: (real > extended_enat) > $o).
% 2.95/1.82 thf(func_def_66, type, order_946214183_ereal: (real > extended_ereal) > $o).
% 2.95/1.82 thf(func_def_67, type, divide595620860_ereal: extended_ereal > extended_ereal > extended_ereal).
% 2.95/1.82 thf(func_def_68, type, member1900190071_ereal: extended_ereal > set_Extended_ereal > $o).
% 2.95/1.82 thf(func_def_69, type, f: a > extended_ereal).
% 2.95/1.82 thf(func_def_70, type, g: a > extended_ereal).
% 2.95/1.82 thf(func_def_71, type, x: a).
% 2.95/1.82 thf(func_def_85, type, sP0: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.95/1.82 thf(func_def_86, type, sP1: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.95/1.82 thf(func_def_87, type, sP2: extended_ereal > extended_ereal > extended_ereal > $o).
% 2.95/1.82 thf(func_def_88, type, sK3: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.95/1.82 thf(func_def_89, type, sK4: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.95/1.82 thf(func_def_90, type, sK5: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.95/1.82 thf(func_def_91, type, sK6: (extended_ereal > extended_ereal > $o) > extended_ereal).
% 2.95/1.82 thf(func_def_92, type, sK7: (extended_ereal > extended_ereal) > extended_ereal).
% 2.95/1.82 thf(func_def_93, type, sK8: (extended_ereal > extended_ereal) > extended_ereal).
% 2.95/1.82 thf(func_def_94, type, sK9: (extended_ereal > extended_ereal) > extended_ereal).
% 2.95/1.82 thf(func_def_95, type, sK10: (extended_ereal > extended_ereal) > extended_ereal).
% 2.95/1.82 thf(func_def_96, type, sK11: (extended_ereal > extended_ereal) > extended_ereal).
% 2.95/1.82 thf(func_def_97, type, sK12: (extended_ereal > extended_ereal) > extended_ereal).
% 2.95/1.82 thf(func_def_98, type, sK13: (extended_ereal > extended_ereal) > extended_ereal).
% 2.95/1.82 thf(func_def_99, type, sK14: (extended_ereal > extended_ereal) > extended_ereal).
% 2.95/1.82 thf(func_def_101, type, ph16: !>[X0: $tType]:(X0)).
% 2.95/1.82 thf(f1040,plain,(
% 2.95/1.82 $false),
% 2.95/1.82 inference(subsumption_resolution,[status(thm)],[f997,f995])).
% 2.95/1.82 thf(f995,plain,(
% 2.95/1.82 ( ! [X0 : a] : (((ord_le824540014_ereal @ (g @ X0) @ (f @ X0)) = $true)) )),
% 2.95/1.82 inference(cnf_transformation,[status(thm)],[f502])).
% 2.95/1.82 thf(f502,plain,(
% 2.95/1.82 ! [X0 : a] : ((ord_le824540014_ereal @ (g @ X0) @ (f @ X0)) = $true)),
% 2.95/1.82 inference(fool_elimination,[status(thm)],[f501])).
% 2.95/1.82 thf(f501,plain,(
% 2.95/1.82 ! [X0 : a] : (ord_le824540014_ereal @ (g @ X0) @ (f @ X0))),
% 2.95/1.82 inference(rectify,[status(thm)],[f1])).
% 2.95/1.82 thf(f1,axiom,(
% 2.95/1.82 ! [X0 : a] : (ord_le824540014_ereal @ (g @ X0) @ (f @ X0))),
% 2.95/1.82 file('/export/starexec/sandbox2/tmp/tmp.nLU4PliHtq/DTF2THF_7494.p',fact_0_assms)).
% 2.95/1.82 thf(f997,plain,(
% 2.95/1.82 ((ord_le824540014_ereal @ (g @ x) @ (f @ x)) != $true)),
% 2.95/1.82 inference(cnf_transformation,[status(thm)],[f927])).
% 2.95/1.82 thf(f927,plain,(
% 2.95/1.82 ((ord_le824540014_ereal @ (g @ x) @ (f @ x)) != $true)),
% 2.95/1.82 inference(flattening,[status(thm)],[f612])).
% 2.95/1.82 thf(f612,plain,(
% 2.95/1.82 ~((ord_le824540014_ereal @ (g @ x) @ (f @ x)) = $true)),
% 2.95/1.82 inference(fool_elimination,[status(thm)],[f611])).
% 2.95/1.82 thf(f611,plain,(
% 2.95/1.82 ~(ord_le824540014_ereal @ (g @ x) @ (f @ x))),
% 2.95/1.82 inference(rectify,[status(thm)],[f353])).
% 2.95/1.82 thf(f353,negated_conjecture,(
% 2.95/1.82 ~(ord_le824540014_ereal @ (g @ x) @ (f @ x))),
% 2.95/1.82 inference(negated_conjecture,[status(cth)],[f352])).
% 2.95/1.82 thf(f352,conjecture,(
% 2.95/1.82 (ord_le824540014_ereal @ (g @ x) @ (f @ x))),
% 2.95/1.82 file('/export/starexec/sandbox2/tmp/tmp.nLU4PliHtq/DTF2THF_7494.p',conj_0)).
% 2.95/1.82 % SZS output end Proof for DTF2THF_7494
% 2.95/1.82 % (7678)------------------------------
% 2.95/1.82 % (7678)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.82 % (7678)Termination reason: Refutation
% 2.95/1.82
% 2.95/1.82 % (7678)Memory used [KB]: 6524
% 2.95/1.82 % (7678)Time elapsed: 0.029 s
% 2.95/1.82 % (7678)Instructions burned: 51 (million)
% 2.95/1.82 % (7678)------------------------------
% 2.95/1.82 % (7678)------------------------------
% 2.95/1.82 % (7666)Success in time 0.05 s
%------------------------------------------------------------------------------