%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP118^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:42 AM UTC 2026
% Result : Theorem 3.18s 1.79s
% Output : Refutation 3.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 6
% Number of leaves : 4
% Syntax : Number of formulae : 16 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 16 ( 15 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 105 ( 5 ~; 0 |; 0 &; 100 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 4 ( 2 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 96 ( 94 usr; 18 con; 0-3 aty)
% Number of variables : 12 ( 0 ^; 12 !; 0 ?; 12 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
real: $tType ).
thf(type_def_6,type,
int: $tType ).
thf(type_def_8,type,
numeral_num1: $tType ).
thf(type_def_9,type,
finite2063899472l_num1: $tType ).
thf(type_def_10,type,
finite1183840848l_num1: $tType ).
thf(type_def_11,type,
n: $tType ).
thf(type_def_12,type,
finite964658038_int_n: $tType ).
thf(type_def_13,type,
finite1489363574real_n: $tType ).
thf(type_def_14,type,
finite463740460l_num1: $tType ).
thf(type_def_15,type,
finite337957458num1_n: $tType ).
thf(type_def_16,type,
finite121157254l_num1: $tType ).
thf(type_def_17,type,
finite318270636al_n_n: $tType ).
thf(type_def_18,type,
set_Fi1257234438l_num1: $tType ).
thf(type_def_19,type,
set_Fi1058188332real_n: $tType ).
thf(type_def_20,type,
set_int: $tType ).
thf(type_def_21,type,
set_real: $tType ).
thf(func_def_0,type,
finite463740460l_num1: $tType ).
thf(func_def_1,type,
finite121157254l_num1: $tType ).
thf(func_def_2,type,
finite337957458num1_n: $tType ).
thf(func_def_3,type,
finite318270636al_n_n: $tType ).
thf(func_def_4,type,
set_Fi1257234438l_num1: $tType ).
thf(func_def_5,type,
finite1183840848l_num1: $tType ).
thf(func_def_6,type,
finite2063899472l_num1: $tType ).
thf(func_def_7,type,
set_Fi1058188332real_n: $tType ).
thf(func_def_8,type,
finite1489363574real_n: $tType ).
thf(func_def_9,type,
finite964658038_int_n: $tType ).
thf(func_def_10,type,
set_real: $tType ).
thf(func_def_11,type,
set_int: $tType ).
thf(func_def_12,type,
numeral_num1: $tType ).
thf(func_def_13,type,
real: $tType ).
thf(func_def_14,type,
int: $tType ).
thf(func_def_15,type,
n: $tType ).
thf(func_def_16,type,
archim1371465213g_real: real > int ).
thf(func_def_17,type,
archim1031974863r_real: real > int ).
thf(func_def_18,type,
finite627713486m1_int: numeral_num1 > int > finite2063899472l_num1 ).
thf(func_def_19,type,
finite15144922381_real: numeral_num1 > real > finite1183840848l_num1 ).
thf(func_def_20,type,
finite1888777460_n_int: n > int > finite964658038_int_n ).
thf(func_def_21,type,
finite703491444n_real: n > real > finite1489363574real_n ).
thf(func_def_22,type,
finite1470207077l_num1: int > finite2063899472l_num1 ).
thf(func_def_23,type,
finite55766923_int_n: int > finite964658038_int_n ).
thf(func_def_24,type,
finite1329045861l_num1: real > finite1183840848l_num1 ).
thf(func_def_25,type,
finite667548811real_n: real > finite1489363574real_n ).
thf(func_def_26,type,
finite1449546019l_num1: finite463740460l_num1 > numeral_num1 > finite1183840848l_num1 ).
thf(func_def_27,type,
finite1639785033num1_n: finite337957458num1_n > n > finite1183840848l_num1 ).
thf(func_def_28,type,
finite1503577981l_num1: finite121157254l_num1 > numeral_num1 > finite1489363574real_n ).
thf(func_def_29,type,
finite625646243al_n_n: finite318270636al_n_n > n > finite1489363574real_n ).
thf(func_def_30,type,
finite1800583751l_num1: finite2063899472l_num1 > numeral_num1 > int ).
thf(func_def_31,type,
finite566214509_int_n: finite964658038_int_n > n > int ).
thf(func_def_32,type,
finite1106245191l_num1: finite1183840848l_num1 > numeral_num1 > real ).
thf(func_def_33,type,
finite772340589real_n: finite1489363574real_n > n > real ).
thf(func_def_34,type,
finite1961675766l_num1: real > finite1183840848l_num1 > finite1183840848l_num1 ).
thf(func_def_35,type,
finite2053577756real_n: real > finite1489363574real_n > finite1489363574real_n ).
thf(func_def_36,type,
abs_ab948591917l_num1: finite1183840848l_num1 > finite1183840848l_num1 ).
thf(func_def_37,type,
abs_ab1599551059real_n: finite1489363574real_n > finite1489363574real_n ).
thf(func_def_38,type,
abs_abs_int: int > int ).
thf(func_def_39,type,
abs_abs_real: real > real ).
thf(func_def_40,type,
minus_779068585l_num1: finite2063899472l_num1 > finite2063899472l_num1 > finite2063899472l_num1 ).
thf(func_def_41,type,
minus_1196255695_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n ).
thf(func_def_42,type,
minus_249161513l_num1: finite1183840848l_num1 > finite1183840848l_num1 > finite1183840848l_num1 ).
thf(func_def_43,type,
minus_1037315151real_n: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ).
thf(func_def_44,type,
minus_minus_int: int > int > int ).
thf(func_def_45,type,
minus_minus_real: real > real > real ).
thf(func_def_46,type,
one_on601734293l_num1: finite2063899472l_num1 ).
thf(func_def_47,type,
one_on1516019131_int_n: finite964658038_int_n ).
thf(func_def_48,type,
one_on1961019669l_num1: finite1183840848l_num1 ).
thf(func_def_49,type,
one_on1253059131real_n: finite1489363574real_n ).
thf(func_def_50,type,
one_one_int: int ).
thf(func_def_51,type,
one_one_Numeral_num1: numeral_num1 ).
thf(func_def_52,type,
one_one_real: real ).
thf(func_def_53,type,
if_Fin1570437642l_num1: $o > finite2063899472l_num1 > finite2063899472l_num1 > finite2063899472l_num1 ).
thf(func_def_54,type,
if_Fin1767949360_int_n: $o > finite964658038_int_n > finite964658038_int_n > finite964658038_int_n ).
thf(func_def_55,type,
if_Fin1668193290l_num1: $o > finite1183840848l_num1 > finite1183840848l_num1 > finite1183840848l_num1 ).
thf(func_def_56,type,
if_Fin127821360real_n: $o > finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ).
thf(func_def_57,type,
ring_11529266783l_num1: set_Fi1257234438l_num1 ).
thf(func_def_58,type,
ring_1723899781real_n: set_Fi1058188332real_n ).
thf(func_def_59,type,
ring_1_Ints_int: set_int ).
thf(func_def_60,type,
ring_1_Ints_real: set_real ).
thf(func_def_61,type,
ring_1154201247l_num1: int > finite1183840848l_num1 ).
thf(func_def_62,type,
ring_11737179077real_n: int > finite1489363574real_n ).
thf(func_def_63,type,
ring_1_of_int_int: int > int ).
thf(func_def_64,type,
ring_1_of_int_real: int > real ).
thf(func_def_65,type,
minkow781993583real_n: finite2063899472l_num1 > finite121157254l_num1 ).
thf(func_def_66,type,
minkow2063319845m1_int: finite2063899472l_num1 > finite2063899472l_num1 ).
thf(func_def_67,type,
minkow19073400691_real: finite2063899472l_num1 > finite1183840848l_num1 ).
thf(func_def_68,type,
minkow371489429real_n: finite964658038_int_n > finite318270636al_n_n ).
thf(func_def_69,type,
minkow1468704075_n_int: finite964658038_int_n > finite964658038_int_n ).
thf(func_def_70,type,
minkow1134813771n_real: finite964658038_int_n > finite1489363574real_n ).
thf(func_def_71,type,
ord_le2035778724l_num1: finite1183840848l_num1 > finite1183840848l_num1 > $o ).
thf(func_def_72,type,
ord_le1017090250real_n: finite1489363574real_n > finite1489363574real_n > $o ).
thf(func_def_73,type,
ord_less_int: int > int > $o ).
thf(func_def_74,type,
ord_less_real: real > real > $o ).
thf(func_def_75,type,
ord_le1202326448l_num1: finite2063899472l_num1 > finite2063899472l_num1 > $o ).
thf(func_def_76,type,
ord_le1908593622_int_n: finite964658038_int_n > finite964658038_int_n > $o ).
thf(func_def_77,type,
ord_le1724070576l_num1: finite1183840848l_num1 > finite1183840848l_num1 > $o ).
thf(func_def_78,type,
ord_le866583254real_n: finite1489363574real_n > finite1489363574real_n > $o ).
thf(func_def_79,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_80,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_81,type,
real_V1974343547l_num1: finite463740460l_num1 > finite463740460l_num1 > real ).
thf(func_def_82,type,
real_V845814485l_num1: finite121157254l_num1 > finite121157254l_num1 > real ).
thf(func_def_83,type,
real_V488409631l_num1: finite1183840848l_num1 > finite1183840848l_num1 > real ).
thf(func_def_84,type,
real_V783965509real_n: finite1489363574real_n > finite1489363574real_n > real ).
thf(func_def_85,type,
real_V1934908667t_real: real > real > real ).
thf(func_def_86,type,
real_V316032763l_num1: finite463740460l_num1 > real ).
thf(func_def_87,type,
real_V1121707553num1_n: finite337957458num1_n > real ).
thf(func_def_88,type,
real_V400292255l_num1: finite1183840848l_num1 > real ).
thf(func_def_89,type,
real_V739724485real_n: finite1489363574real_n > real ).
thf(func_def_90,type,
real_V646646907m_real: real > real ).
thf(func_def_91,type,
collec321817931real_n: ( finite1489363574real_n > $o ) > set_Fi1058188332real_n ).
thf(func_def_92,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_93,type,
member1413569767l_num1: finite1183840848l_num1 > set_Fi1257234438l_num1 > $o ).
thf(func_def_94,type,
member1352538125real_n: finite1489363574real_n > set_Fi1058188332real_n > $o ).
thf(func_def_95,type,
member_int: int > set_int > $o ).
thf(func_def_96,type,
member_real: real > set_real > $o ).
thf(func_def_97,type,
a: set_Fi1058188332real_n ).
thf(func_def_98,type,
i: n ).
thf(func_def_99,type,
m: int ).
thf(func_def_100,type,
x: finite1489363574real_n ).
thf(func_def_101,type,
y: finite1489363574real_n ).
thf(func_def_115,type,
sK0: n > ( n > real > $o ) > real ).
thf(func_def_116,type,
sK1: finite1489363574real_n > ( n > real > $o ) > n ).
thf(func_def_117,type,
sK2: ( n > real > $o ) > finite1489363574real_n ).
thf(func_def_118,type,
sK3: ( n > real > $o ) > n ).
thf(func_def_119,type,
sK4: real > int ).
thf(func_def_120,type,
sK5: int ).
thf(func_def_122,type,
ph7:
!>[X0: $tType] : X0 ).
thf(func_def_123,type,
sK8: finite1489363574real_n > finite1489363574real_n > n ).
thf(f1318,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1317]) ).
thf(f1317,plain,
( ( abs_abs_real @ ( ring_1_of_int_real @ m ) )
!= ( abs_abs_real @ ( ring_1_of_int_real @ m ) ) ),
inference(backward_demodulation,[status(thm)],[f926,f1309]) ).
thf(f1309,plain,
( ( ring_1_of_int_real @ m )
= ( finite772340589real_n @ ( minus_1037315151real_n @ x @ y ) @ i ) ),
inference(superposition,[status(thm)],[f891,f886]) ).
thf(f886,plain,
! [X2: finite1489363574real_n,X0: finite1489363574real_n,X1: n] :
( ( finite772340589real_n @ ( minus_1037315151real_n @ X0 @ X2 ) @ X1 )
= ( minus_minus_real @ ( finite772340589real_n @ X0 @ X1 ) @ ( finite772340589real_n @ X2 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f852]) ).
thf(f852,plain,
! [X0: finite1489363574real_n,X1: n,X2: finite1489363574real_n] :
( ( finite772340589real_n @ ( minus_1037315151real_n @ X0 @ X2 ) @ X1 )
= ( minus_minus_real @ ( finite772340589real_n @ X0 @ X1 ) @ ( finite772340589real_n @ X2 @ X1 ) ) ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,axiom,
! [X1: finite1489363574real_n,X3: n,X2: finite1489363574real_n] :
( ( finite772340589real_n @ ( minus_1037315151real_n @ X1 @ X2 ) @ X3 )
= ( minus_minus_real @ ( finite772340589real_n @ X1 @ X3 ) @ ( finite772340589real_n @ X2 @ X3 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.LSq3s3M90o/DTF2THF_11239.p',fact_6_vector__minus__component) ).
thf(f891,plain,
( ( ring_1_of_int_real @ m )
= ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) ),
inference(cnf_transformation,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( ring_1_of_int_real @ m )
= ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) ),
file('/export/starexec/sandbox/tmp/tmp.LSq3s3M90o/DTF2THF_11239.p',fact_2_m) ).
thf(f926,plain,
( ( abs_abs_real @ ( finite772340589real_n @ ( minus_1037315151real_n @ x @ y ) @ i ) )
!= ( abs_abs_real @ ( ring_1_of_int_real @ m ) ) ),
inference(backward_demodulation,[status(thm)],[f907,f910]) ).
thf(f910,plain,
! [X0: int] :
( ( abs_abs_real @ ( ring_1_of_int_real @ X0 ) )
= ( ring_1_of_int_real @ ( abs_abs_int @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f856]) ).
thf(f856,plain,
! [X0: int] :
( ( abs_abs_real @ ( ring_1_of_int_real @ X0 ) )
= ( ring_1_of_int_real @ ( abs_abs_int @ X0 ) ) ),
inference(rectify,[status(thm)],[f10]) ).
thf(f10,axiom,
! [X1: int] :
( ( ring_1_of_int_real @ ( abs_abs_int @ X1 ) )
= ( abs_abs_real @ ( ring_1_of_int_real @ X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.LSq3s3M90o/DTF2THF_11239.p',fact_9_of__int__abs) ).
thf(f907,plain,
( ( ring_1_of_int_real @ ( abs_abs_int @ m ) )
!= ( abs_abs_real @ ( finite772340589real_n @ ( minus_1037315151real_n @ x @ y ) @ i ) ) ),
inference(cnf_transformation,[status(thm)],[f851]) ).
thf(f851,plain,
( ( ring_1_of_int_real @ ( abs_abs_int @ m ) )
!= ( abs_abs_real @ ( finite772340589real_n @ ( minus_1037315151real_n @ x @ y ) @ i ) ) ),
inference(flattening,[status(thm)],[f365]) ).
thf(f365,negated_conjecture,
( ( ring_1_of_int_real @ ( abs_abs_int @ m ) )
!= ( abs_abs_real @ ( finite772340589real_n @ ( minus_1037315151real_n @ x @ y ) @ i ) ) ),
inference(negated_conjecture,[status(cth)],[f364]) ).
thf(f364,conjecture,
( ( ring_1_of_int_real @ ( abs_abs_int @ m ) )
= ( abs_abs_real @ ( finite772340589real_n @ ( minus_1037315151real_n @ x @ y ) @ i ) ) ),
file('/export/starexec/sandbox/tmp/tmp.LSq3s3M90o/DTF2THF_11239.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP118^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n005.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 08:26:44 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.18/0.45 thf(spec,logic,$$dhol).
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % File : ITP118^1 : TPTP v9.2.1. Released v7.5.0.
% 0.18/0.45 % Domain : Interactive Theorem Proving
% 0.18/0.45 % Problem : Sledgehammer Minkowskis_Theorem problem prob_62__6246984_1
% 0.18/0.45 % Version : Especial.
% 0.18/0.45 % English :
% 0.18/0.45
% 0.18/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.18/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.18/0.45 % Source : [Des21]
% 0.18/0.45 % Names : Minkowskis_Theorem/prob_62__6246984_1 [Des21]
% 0.18/0.45
% 0.18/0.45 % Status : Theorem
% 0.18/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.40 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.18/0.45 % Syntax : Number of formulae : 466 ( 241 unt; 102 typ; 0 def)
% 0.18/0.45 % Number of atoms : 767 ( 316 equ; 0 cnn)
% 0.18/0.45 % Maximal formula atoms : 12 ( 2 avg)
% 0.18/0.45 % Number of connectives : 2505 ( 39 ~; 4 |; 25 &;2191 @)
% 0.18/0.45 % ( 0 <=>; 246 =>; 0 <=; 0 <~>)
% 0.18/0.45 % Maximal formula depth : 18 ( 5 avg)
% 0.18/0.45 % Number of types : 17 ( 16 usr)
% 0.18/0.45 % Number of type conns : 153 ( 153 >; 0 *; 0 +; 0 <<)
% 0.18/0.45 % Number of symbols : 89 ( 86 usr; 18 con; 0-3 aty)
% 0.18/0.45 % Number of variables : 842 ( 83 ^; 740 !; 19 ?; 842 :)
% 0.18/0.45 % SPC : TH0_THM_EQU_NAR
% 0.18/0.45
% 0.18/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:43:54.513
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % Could-be-implicit typings (16)
% 0.18/0.45 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mt__Numeral____Type__Onum1_J,type,
% 0.18/0.45 finite463740460l_num1: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Numeral____Type__Onum1_J,type,
% 0.18/0.45 finite121157254l_num1: $tType ).
% 0.18/0.45
% 0.18/0.45 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mtf__n_J,type,
% 0.18/0.45 finite337957458num1_n: $tType ).
% 0.18/0.45
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% 0.18/0.45 real_V1934908667t_real: real > real > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mt__Numeral____Type__Onum1_J,type,
% 0.18/0.45 real_V316032763l_num1: finite463740460l_num1 > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mtf__n_J,type,
% 0.18/0.45 real_V1121707553num1_n: finite337957458num1_n > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 0.18/0.45 real_V400292255l_num1: finite1183840848l_num1 > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 0.18/0.45 real_V739724485real_n: finite1489363574real_n > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 0.18/0.45 real_V646646907m_real: real > real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 0.18/0.45 collec321817931real_n: ( finite1489363574real_n > $o ) > set_Fi1058188332real_n ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.18/0.45 collect_real: ( real > $o ) > set_real ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 0.18/0.45 member1413569767l_num1: finite1183840848l_num1 > set_Fi1257234438l_num1 > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 0.18/0.45 member1352538125real_n: finite1489363574real_n > set_Fi1058188332real_n > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Int__Oint,type,
% 0.18/0.45 member_int: int > set_int > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_c_member_001t__Real__Oreal,type,
% 0.18/0.45 member_real: real > set_real > $o ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_A,type,
% 0.18/0.45 a: set_Fi1058188332real_n ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_i____,type,
% 0.18/0.45 i: n ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_m____,type,
% 0.18/0.45 m: int ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_x____,type,
% 0.18/0.45 x: finite1489363574real_n ).
% 0.18/0.45
% 0.18/0.45 thf(sy_v_y____,type,
% 0.18/0.45 y: finite1489363574real_n ).
% 0.18/0.45
% 0.18/0.45 % Relevant facts (354)
% 0.18/0.45 thf(fact_0__C1_C_I2_J,axiom,
% 0.18/0.45 member1352538125real_n @ y @ a ).
% 0.18/0.45
% 0.18/0.45 % "1"(2)
% 0.18/0.45 thf(fact_1__C1_C_I1_J,axiom,
% 0.18/0.45 member1352538125real_n @ x @ a ).
% 0.18/0.45
% 0.18/0.45 % "1"(1)
% 0.18/0.45 thf(fact_2_m,axiom,
% 0.18/0.45 ( ( ring_1_of_int_real @ m )
% 0.18/0.45 = ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % m
% 0.18/0.45 thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_Areal__of__int_Am_A_061_Ax_A_E_Ai_A_N_Ay_A_E_Ai_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.18/0.45 ~ ! [M: int] :
% 0.18/0.45 ( ( ring_1_of_int_real @ M )
% 0.18/0.45 != ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % \<open>\<And>thesis. (\<And>m. real_of_int m = x $ i - y $ i \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.18/0.45 thf(fact_4_vector__minus__component,axiom,
% 0.18/0.45 ! [X: finite2063899472l_num1,Y: finite2063899472l_num1,I: numeral_num1] :
% 0.18/0.45 ( ( finite1800583751l_num1 @ ( minus_779068585l_num1 @ X @ Y ) @ I )
% 0.18/0.45 = ( minus_minus_int @ ( finite1800583751l_num1 @ X @ I ) @ ( finite1800583751l_num1 @ Y @ I ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % vector_minus_component
% 0.18/0.45 thf(fact_5_vector__minus__component,axiom,
% 0.18/0.45 ! [X: finite964658038_int_n,Y: finite964658038_int_n,I: n] :
% 0.18/0.45 ( ( finite566214509_int_n @ ( minus_1196255695_int_n @ X @ Y ) @ I )
% 0.18/0.45 = ( minus_minus_int @ ( finite566214509_int_n @ X @ I ) @ ( finite566214509_int_n @ Y @ I ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % vector_minus_component
% 0.18/0.45 thf(fact_6_vector__minus__component,axiom,
% 0.18/0.45 ! [X: finite1489363574real_n,Y: finite1489363574real_n,I: n] :
% 0.18/0.45 ( ( finite772340589real_n @ ( minus_1037315151real_n @ X @ Y ) @ I )
% 0.18/0.45 = ( minus_minus_real @ ( finite772340589real_n @ X @ I ) @ ( finite772340589real_n @ Y @ I ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % vector_minus_component
% 0.18/0.45 thf(fact_7_vector__minus__component,axiom,
% 0.18/0.45 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1,I: numeral_num1] :
% 0.18/0.45 ( ( finite1106245191l_num1 @ ( minus_249161513l_num1 @ X @ Y ) @ I )
% 0.18/0.45 = ( minus_minus_real @ ( finite1106245191l_num1 @ X @ I ) @ ( finite1106245191l_num1 @ Y @ I ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % vector_minus_component
% 0.18/0.45 thf(fact_8_of__int__abs,axiom,
% 0.18/0.45 ! [X: int] :
% 0.18/0.45 ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 0.18/0.45 = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_abs
% 0.18/0.45 thf(fact_9_of__int__abs,axiom,
% 0.18/0.45 ! [X: int] :
% 0.18/0.45 ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 0.18/0.45 = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_abs
% 0.18/0.45 thf(fact_10_of__int__diff,axiom,
% 0.18/0.45 ! [W: int,Z: int] :
% 0.18/0.45 ( ( ring_1154201247l_num1 @ ( minus_minus_int @ W @ Z ) )
% 0.18/0.45 = ( minus_249161513l_num1 @ ( ring_1154201247l_num1 @ W ) @ ( ring_1154201247l_num1 @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_diff
% 0.18/0.45 thf(fact_11_of__int__diff,axiom,
% 0.18/0.45 ! [W: int,Z: int] :
% 0.18/0.45 ( ( ring_11737179077real_n @ ( minus_minus_int @ W @ Z ) )
% 0.18/0.45 = ( minus_1037315151real_n @ ( ring_11737179077real_n @ W ) @ ( ring_11737179077real_n @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_diff
% 0.18/0.45 thf(fact_12_of__int__diff,axiom,
% 0.18/0.45 ! [W: int,Z: int] :
% 0.18/0.45 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 0.18/0.45 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_diff
% 0.18/0.45 thf(fact_13_of__int__diff,axiom,
% 0.18/0.45 ! [W: int,Z: int] :
% 0.18/0.45 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 0.18/0.45 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_diff
% 0.18/0.45 thf(fact_14__092_060open_062x_A_E_Ai_A_N_Ay_A_E_Ai_A_092_060in_062_A_092_060int_062_092_060close_062,axiom,
% 0.18/0.45 member_real @ ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) @ ring_1_Ints_real ).
% 0.18/0.45
% 0.18/0.45 % \<open>x $ i - y $ i \<in> \<int>\<close>
% 0.18/0.45 thf(fact_15_of__int__vec__nth,axiom,
% 0.18/0.45 ! [V: finite2063899472l_num1,N: numeral_num1] :
% 0.18/0.45 ( ( finite1503577981l_num1 @ ( minkow781993583real_n @ V ) @ N )
% 0.18/0.45 = ( ring_11737179077real_n @ ( finite1800583751l_num1 @ V @ N ) ) ) ).
% 0.18/0.45
% 0.18/0.45 % of_int_vec_nth
% 0.18/0.45 thf(fact_16_of__int__vec__nth,axiom,
% 0.18/0.45 ! [V: finite964658038_int_n,N: n] :
% 0.18/0.46 ( ( finite625646243al_n_n @ ( minkow371489429real_n @ V ) @ N )
% 0.18/0.46 = ( ring_11737179077real_n @ ( finite566214509_int_n @ V @ N ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_vec_nth
% 0.18/0.46 thf(fact_17_of__int__vec__nth,axiom,
% 0.18/0.46 ! [V: finite2063899472l_num1,N: numeral_num1] :
% 0.18/0.46 ( ( finite1800583751l_num1 @ ( minkow2063319845m1_int @ V ) @ N )
% 0.18/0.46 = ( ring_1_of_int_int @ ( finite1800583751l_num1 @ V @ N ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_vec_nth
% 0.18/0.46 thf(fact_18_of__int__vec__nth,axiom,
% 0.18/0.46 ! [V: finite964658038_int_n,N: n] :
% 0.18/0.46 ( ( finite566214509_int_n @ ( minkow1468704075_n_int @ V ) @ N )
% 0.18/0.46 = ( ring_1_of_int_int @ ( finite566214509_int_n @ V @ N ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_vec_nth
% 0.18/0.46 thf(fact_19_of__int__vec__nth,axiom,
% 0.18/0.46 ! [V: finite964658038_int_n,N: n] :
% 0.18/0.46 ( ( finite772340589real_n @ ( minkow1134813771n_real @ V ) @ N )
% 0.18/0.46 = ( ring_1_of_int_real @ ( finite566214509_int_n @ V @ N ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_vec_nth
% 0.18/0.46 thf(fact_20_of__int__vec__nth,axiom,
% 0.18/0.46 ! [V: finite2063899472l_num1,N: numeral_num1] :
% 0.18/0.46 ( ( finite1106245191l_num1 @ ( minkow19073400691_real @ V ) @ N )
% 0.18/0.46 = ( ring_1_of_int_real @ ( finite1800583751l_num1 @ V @ N ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_vec_nth
% 0.18/0.46 thf(fact_21_of__int__eq__iff,axiom,
% 0.18/0.46 ! [W: int,Z: int] :
% 0.18/0.46 ( ( ( ring_1_of_int_int @ W )
% 0.18/0.46 = ( ring_1_of_int_int @ Z ) )
% 0.18/0.46 = ( W = Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_eq_iff
% 0.18/0.46 thf(fact_22_of__int__eq__iff,axiom,
% 0.18/0.46 ! [W: int,Z: int] :
% 0.18/0.46 ( ( ( ring_11737179077real_n @ W )
% 0.18/0.46 = ( ring_11737179077real_n @ Z ) )
% 0.18/0.46 = ( W = Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_eq_iff
% 0.18/0.46 thf(fact_23_of__int__eq__iff,axiom,
% 0.18/0.46 ! [W: int,Z: int] :
% 0.18/0.46 ( ( ( ring_1_of_int_real @ W )
% 0.18/0.46 = ( ring_1_of_int_real @ Z ) )
% 0.18/0.46 = ( W = Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_eq_iff
% 0.18/0.46 thf(fact_24_abs__abs,axiom,
% 0.18/0.46 ! [A: int] :
% 0.18/0.46 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.18/0.46 = ( abs_abs_int @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_abs
% 0.18/0.46 thf(fact_25_abs__abs,axiom,
% 0.18/0.46 ! [A: real] :
% 0.18/0.46 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.18/0.46 = ( abs_abs_real @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_abs
% 0.18/0.46 thf(fact_26_abs__idempotent,axiom,
% 0.18/0.46 ! [A: int] :
% 0.18/0.46 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.18/0.46 = ( abs_abs_int @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_idempotent
% 0.18/0.46 thf(fact_27_abs__idempotent,axiom,
% 0.18/0.46 ! [A: real] :
% 0.18/0.46 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.18/0.46 = ( abs_abs_real @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_idempotent
% 0.18/0.46 thf(fact_28_abs__minus__commute,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 0.18/0.46 ( ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ A @ B ) )
% 0.18/0.46 = ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_minus_commute
% 0.18/0.46 thf(fact_29_abs__minus__commute,axiom,
% 0.18/0.46 ! [A: real,B: real] :
% 0.18/0.46 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.18/0.46 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_minus_commute
% 0.18/0.46 thf(fact_30_abs__minus__commute,axiom,
% 0.18/0.46 ! [A: int,B: int] :
% 0.18/0.46 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 0.18/0.46 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_minus_commute
% 0.18/0.46 thf(fact_31_abs__minus__commute,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 0.18/0.46 ( ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ A @ B ) )
% 0.18/0.46 = ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_minus_commute
% 0.18/0.46 thf(fact_32_lambda__skolem,axiom,
% 0.18/0.46 ! [P: n > real > $o] :
% 0.18/0.46 ( ( ! [I2: n] :
% 0.18/0.46 ? [X2: real] : ( P @ I2 @ X2 ) )
% 0.18/0.46 = ( ? [X3: finite1489363574real_n] :
% 0.18/0.46 ! [I2: n] : ( P @ I2 @ ( finite772340589real_n @ X3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % lambda_skolem
% 0.18/0.46 thf(fact_33_lambda__skolem,axiom,
% 0.18/0.46 ! [P: numeral_num1 > real > $o] :
% 0.18/0.46 ( ( ! [I2: numeral_num1] :
% 0.18/0.46 ? [X2: real] : ( P @ I2 @ X2 ) )
% 0.18/0.46 = ( ? [X3: finite1183840848l_num1] :
% 0.18/0.46 ! [I2: numeral_num1] : ( P @ I2 @ ( finite1106245191l_num1 @ X3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % lambda_skolem
% 0.18/0.46 thf(fact_34_lambda__skolem,axiom,
% 0.18/0.46 ! [P: numeral_num1 > int > $o] :
% 0.18/0.46 ( ( ! [I2: numeral_num1] :
% 0.18/0.46 ? [X2: int] : ( P @ I2 @ X2 ) )
% 0.18/0.46 = ( ? [X3: finite2063899472l_num1] :
% 0.18/0.46 ! [I2: numeral_num1] : ( P @ I2 @ ( finite1800583751l_num1 @ X3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % lambda_skolem
% 0.18/0.46 thf(fact_35_lambda__skolem,axiom,
% 0.18/0.46 ! [P: n > int > $o] :
% 0.18/0.46 ( ( ! [I2: n] :
% 0.18/0.46 ? [X2: int] : ( P @ I2 @ X2 ) )
% 0.18/0.46 = ( ? [X3: finite964658038_int_n] :
% 0.18/0.46 ! [I2: n] : ( P @ I2 @ ( finite566214509_int_n @ X3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % lambda_skolem
% 0.18/0.46 thf(fact_36_vec__eq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: finite1489363574real_n,Z2: finite1489363574real_n] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] :
% 0.18/0.46 ! [I2: n] :
% 0.18/0.46 ( ( finite772340589real_n @ X3 @ I2 )
% 0.18/0.46 = ( finite772340589real_n @ Y3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_eq_iff
% 0.18/0.46 thf(fact_37_vec__eq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: finite1183840848l_num1,Z2: finite1183840848l_num1] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] :
% 0.18/0.46 ! [I2: numeral_num1] :
% 0.18/0.46 ( ( finite1106245191l_num1 @ X3 @ I2 )
% 0.18/0.46 = ( finite1106245191l_num1 @ Y3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_eq_iff
% 0.18/0.46 thf(fact_38_vec__eq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: finite2063899472l_num1,Z2: finite2063899472l_num1] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [X3: finite2063899472l_num1,Y3: finite2063899472l_num1] :
% 0.18/0.46 ! [I2: numeral_num1] :
% 0.18/0.46 ( ( finite1800583751l_num1 @ X3 @ I2 )
% 0.18/0.46 = ( finite1800583751l_num1 @ Y3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_eq_iff
% 0.18/0.46 thf(fact_39_vec__eq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: finite964658038_int_n,Z2: finite964658038_int_n] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [X3: finite964658038_int_n,Y3: finite964658038_int_n] :
% 0.18/0.46 ! [I2: n] :
% 0.18/0.46 ( ( finite566214509_int_n @ X3 @ I2 )
% 0.18/0.46 = ( finite566214509_int_n @ Y3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_eq_iff
% 0.18/0.46 thf(fact_40_assms,axiom,
% 0.18/0.46 ! [V: finite1489363574real_n,I: n] :
% 0.18/0.46 ( ( member1352538125real_n @ V @ a )
% 0.18/0.46 => ( member_real @ ( finite772340589real_n @ V @ I ) @ ring_1_Ints_real ) ) ).
% 0.18/0.46
% 0.18/0.46 % assms
% 0.18/0.46 thf(fact_41_of__int__vec__eq__iff,axiom,
% 0.18/0.46 ! [A: finite2063899472l_num1,B: finite2063899472l_num1] :
% 0.18/0.46 ( ( ( minkow19073400691_real @ A )
% 0.18/0.46 = ( minkow19073400691_real @ B ) )
% 0.18/0.46 = ( A = B ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_vec_eq_iff
% 0.18/0.46 thf(fact_42_of__int__vec__eq__iff,axiom,
% 0.18/0.46 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 0.18/0.46 ( ( ( minkow1134813771n_real @ A )
% 0.18/0.46 = ( minkow1134813771n_real @ B ) )
% 0.18/0.46 = ( A = B ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_vec_eq_iff
% 0.18/0.46 thf(fact_43_Ints__diff,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 0.18/0.46 ( ( member1352538125real_n @ A @ ring_1723899781real_n )
% 0.18/0.46 => ( ( member1352538125real_n @ B @ ring_1723899781real_n )
% 0.18/0.46 => ( member1352538125real_n @ ( minus_1037315151real_n @ A @ B ) @ ring_1723899781real_n ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_diff
% 0.18/0.46 thf(fact_44_Ints__diff,axiom,
% 0.18/0.46 ! [A: real,B: real] :
% 0.18/0.46 ( ( member_real @ A @ ring_1_Ints_real )
% 0.18/0.46 => ( ( member_real @ B @ ring_1_Ints_real )
% 0.18/0.46 => ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_diff
% 0.18/0.46 thf(fact_45_Ints__diff,axiom,
% 0.18/0.46 ! [A: int,B: int] :
% 0.18/0.46 ( ( member_int @ A @ ring_1_Ints_int )
% 0.18/0.46 => ( ( member_int @ B @ ring_1_Ints_int )
% 0.18/0.46 => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_diff
% 0.18/0.46 thf(fact_46_Ints__diff,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 0.18/0.46 ( ( member1413569767l_num1 @ A @ ring_11529266783l_num1 )
% 0.18/0.46 => ( ( member1413569767l_num1 @ B @ ring_11529266783l_num1 )
% 0.18/0.46 => ( member1413569767l_num1 @ ( minus_249161513l_num1 @ A @ B ) @ ring_11529266783l_num1 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_diff
% 0.18/0.46 thf(fact_47_Ints__abs,axiom,
% 0.18/0.46 ! [A: int] :
% 0.18/0.46 ( ( member_int @ A @ ring_1_Ints_int )
% 0.18/0.46 => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_abs
% 0.18/0.46 thf(fact_48_Ints__abs,axiom,
% 0.18/0.46 ! [A: real] :
% 0.18/0.46 ( ( member_real @ A @ ring_1_Ints_real )
% 0.18/0.46 => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_abs
% 0.18/0.46 thf(fact_49_Ints__of__int,axiom,
% 0.18/0.46 ! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_of_int
% 0.18/0.46 thf(fact_50_Ints__of__int,axiom,
% 0.18/0.46 ! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_of_int
% 0.18/0.46 thf(fact_51_Ints__of__int,axiom,
% 0.18/0.46 ! [Z: int] : ( member1352538125real_n @ ( ring_11737179077real_n @ Z ) @ ring_1723899781real_n ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_of_int
% 0.18/0.46 thf(fact_52_Ints__induct,axiom,
% 0.18/0.46 ! [Q: real,P: real > $o] :
% 0.18/0.46 ( ( member_real @ Q @ ring_1_Ints_real )
% 0.18/0.46 => ( ! [Z3: int] : ( P @ ( ring_1_of_int_real @ Z3 ) )
% 0.18/0.46 => ( P @ Q ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_induct
% 0.18/0.46 thf(fact_53_Ints__induct,axiom,
% 0.18/0.46 ! [Q: int,P: int > $o] :
% 0.18/0.46 ( ( member_int @ Q @ ring_1_Ints_int )
% 0.18/0.46 => ( ! [Z3: int] : ( P @ ( ring_1_of_int_int @ Z3 ) )
% 0.18/0.46 => ( P @ Q ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_induct
% 0.18/0.46 thf(fact_54_Ints__induct,axiom,
% 0.18/0.46 ! [Q: finite1489363574real_n,P: finite1489363574real_n > $o] :
% 0.18/0.46 ( ( member1352538125real_n @ Q @ ring_1723899781real_n )
% 0.18/0.46 => ( ! [Z3: int] : ( P @ ( ring_11737179077real_n @ Z3 ) )
% 0.18/0.46 => ( P @ Q ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_induct
% 0.18/0.46 thf(fact_55_Ints__cases,axiom,
% 0.18/0.46 ! [Q: real] :
% 0.18/0.46 ( ( member_real @ Q @ ring_1_Ints_real )
% 0.18/0.46 => ~ ! [Z3: int] :
% 0.18/0.46 ( Q
% 0.18/0.46 != ( ring_1_of_int_real @ Z3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_cases
% 0.18/0.46 thf(fact_56_Ints__cases,axiom,
% 0.18/0.46 ! [Q: int] :
% 0.18/0.46 ( ( member_int @ Q @ ring_1_Ints_int )
% 0.18/0.46 => ~ ! [Z3: int] :
% 0.18/0.46 ( Q
% 0.18/0.46 != ( ring_1_of_int_int @ Z3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_cases
% 0.18/0.46 thf(fact_57_Ints__cases,axiom,
% 0.18/0.46 ! [Q: finite1489363574real_n] :
% 0.18/0.46 ( ( member1352538125real_n @ Q @ ring_1723899781real_n )
% 0.18/0.46 => ~ ! [Z3: int] :
% 0.18/0.46 ( Q
% 0.18/0.46 != ( ring_11737179077real_n @ Z3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_cases
% 0.18/0.46 thf(fact_58_diff__right__commute,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,C: finite1489363574real_n,B: finite1489363574real_n] :
% 0.18/0.46 ( ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ C ) @ B )
% 0.18/0.46 = ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ B ) @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_right_commute
% 0.18/0.46 thf(fact_59_diff__right__commute,axiom,
% 0.18/0.46 ! [A: real,C: real,B: real] :
% 0.18/0.46 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.18/0.46 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_right_commute
% 0.18/0.46 thf(fact_60_diff__right__commute,axiom,
% 0.18/0.46 ! [A: int,C: int,B: int] :
% 0.18/0.46 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.18/0.46 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_right_commute
% 0.18/0.46 thf(fact_61_diff__right__commute,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,C: finite1183840848l_num1,B: finite1183840848l_num1] :
% 0.18/0.46 ( ( minus_249161513l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ B )
% 0.18/0.46 = ( minus_249161513l_num1 @ ( minus_249161513l_num1 @ A @ B ) @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_right_commute
% 0.18/0.46 thf(fact_62_diff__eq__diff__eq,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n,D: finite1489363574real_n] :
% 0.18/0.46 ( ( ( minus_1037315151real_n @ A @ B )
% 0.18/0.46 = ( minus_1037315151real_n @ C @ D ) )
% 0.18/0.46 => ( ( A = B )
% 0.18/0.46 = ( C = D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_eq
% 0.18/0.46 thf(fact_63_diff__eq__diff__eq,axiom,
% 0.18/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.18/0.46 ( ( ( minus_minus_real @ A @ B )
% 0.18/0.46 = ( minus_minus_real @ C @ D ) )
% 0.18/0.46 => ( ( A = B )
% 0.18/0.46 = ( C = D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_eq
% 0.18/0.46 thf(fact_64_diff__eq__diff__eq,axiom,
% 0.18/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.18/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.18/0.46 = ( minus_minus_int @ C @ D ) )
% 0.18/0.46 => ( ( A = B )
% 0.18/0.46 = ( C = D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_eq
% 0.18/0.46 thf(fact_65_diff__eq__diff__eq,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1,D: finite1183840848l_num1] :
% 0.18/0.46 ( ( ( minus_249161513l_num1 @ A @ B )
% 0.18/0.46 = ( minus_249161513l_num1 @ C @ D ) )
% 0.18/0.46 => ( ( A = B )
% 0.18/0.46 = ( C = D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_eq
% 0.18/0.46 thf(fact_66_vec__nth__inject,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 0.18/0.46 ( ( ( finite772340589real_n @ X )
% 0.18/0.46 = ( finite772340589real_n @ Y ) )
% 0.18/0.46 = ( X = Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_nth_inject
% 0.18/0.46 thf(fact_67_vec__nth__inject,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 0.18/0.46 ( ( ( finite1106245191l_num1 @ X )
% 0.18/0.46 = ( finite1106245191l_num1 @ Y ) )
% 0.18/0.46 = ( X = Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_nth_inject
% 0.18/0.46 thf(fact_68_vec__nth__inject,axiom,
% 0.18/0.46 ! [X: finite2063899472l_num1,Y: finite2063899472l_num1] :
% 0.18/0.46 ( ( ( finite1800583751l_num1 @ X )
% 0.18/0.46 = ( finite1800583751l_num1 @ Y ) )
% 0.18/0.46 = ( X = Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_nth_inject
% 0.18/0.46 thf(fact_69_vec__nth__inject,axiom,
% 0.18/0.46 ! [X: finite964658038_int_n,Y: finite964658038_int_n] :
% 0.18/0.46 ( ( ( finite566214509_int_n @ X )
% 0.18/0.46 = ( finite566214509_int_n @ Y ) )
% 0.18/0.46 = ( X = Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_nth_inject
% 0.18/0.46 thf(fact_70_cond__component,axiom,
% 0.18/0.46 ! [B: $o,X: finite1489363574real_n,Y: finite1489363574real_n,I: n] :
% 0.18/0.46 ( ( B
% 0.18/0.46 => ( ( finite772340589real_n @ ( if_Fin127821360real_n @ B @ X @ Y ) @ I )
% 0.18/0.46 = ( finite772340589real_n @ X @ I ) ) )
% 0.18/0.46 & ( ~ B
% 0.18/0.46 => ( ( finite772340589real_n @ ( if_Fin127821360real_n @ B @ X @ Y ) @ I )
% 0.18/0.46 = ( finite772340589real_n @ Y @ I ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cond_component
% 0.18/0.46 thf(fact_71_cond__component,axiom,
% 0.18/0.46 ! [B: $o,X: finite1183840848l_num1,Y: finite1183840848l_num1,I: numeral_num1] :
% 0.18/0.46 ( ( B
% 0.18/0.46 => ( ( finite1106245191l_num1 @ ( if_Fin1668193290l_num1 @ B @ X @ Y ) @ I )
% 0.18/0.46 = ( finite1106245191l_num1 @ X @ I ) ) )
% 0.18/0.46 & ( ~ B
% 0.18/0.46 => ( ( finite1106245191l_num1 @ ( if_Fin1668193290l_num1 @ B @ X @ Y ) @ I )
% 0.18/0.46 = ( finite1106245191l_num1 @ Y @ I ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cond_component
% 0.18/0.46 thf(fact_72_cond__component,axiom,
% 0.18/0.46 ! [B: $o,X: finite2063899472l_num1,Y: finite2063899472l_num1,I: numeral_num1] :
% 0.18/0.46 ( ( B
% 0.18/0.46 => ( ( finite1800583751l_num1 @ ( if_Fin1570437642l_num1 @ B @ X @ Y ) @ I )
% 0.18/0.46 = ( finite1800583751l_num1 @ X @ I ) ) )
% 0.18/0.46 & ( ~ B
% 0.18/0.46 => ( ( finite1800583751l_num1 @ ( if_Fin1570437642l_num1 @ B @ X @ Y ) @ I )
% 0.18/0.46 = ( finite1800583751l_num1 @ Y @ I ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cond_component
% 0.18/0.46 thf(fact_73_cond__component,axiom,
% 0.18/0.46 ! [B: $o,X: finite964658038_int_n,Y: finite964658038_int_n,I: n] :
% 0.18/0.46 ( ( B
% 0.18/0.46 => ( ( finite566214509_int_n @ ( if_Fin1767949360_int_n @ B @ X @ Y ) @ I )
% 0.18/0.46 = ( finite566214509_int_n @ X @ I ) ) )
% 0.18/0.46 & ( ~ B
% 0.18/0.46 => ( ( finite566214509_int_n @ ( if_Fin1767949360_int_n @ B @ X @ Y ) @ I )
% 0.18/0.46 = ( finite566214509_int_n @ Y @ I ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % cond_component
% 0.18/0.46 thf(fact_74_dist__of__int,axiom,
% 0.18/0.46 ! [M2: int,N: int] :
% 0.18/0.46 ( ( real_V1934908667t_real @ ( ring_1_of_int_real @ M2 ) @ ( ring_1_of_int_real @ N ) )
% 0.18/0.46 = ( ring_1_of_int_real @ ( abs_abs_int @ ( minus_minus_int @ M2 @ N ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_of_int
% 0.18/0.46 thf(fact_75_vec__sub,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( finite667548811real_n @ ( minus_minus_real @ X @ Y ) )
% 0.18/0.46 = ( minus_1037315151real_n @ ( finite667548811real_n @ X ) @ ( finite667548811real_n @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_sub
% 0.18/0.46 thf(fact_76_vec__sub,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( finite1329045861l_num1 @ ( minus_minus_real @ X @ Y ) )
% 0.18/0.46 = ( minus_249161513l_num1 @ ( finite1329045861l_num1 @ X ) @ ( finite1329045861l_num1 @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec_sub
% 0.18/0.46 thf(fact_77_vector__sub__rdistrib,axiom,
% 0.18/0.46 ! [A: real,B: real,X: finite1489363574real_n] :
% 0.18/0.46 ( ( finite2053577756real_n @ ( minus_minus_real @ A @ B ) @ X )
% 0.18/0.46 = ( minus_1037315151real_n @ ( finite2053577756real_n @ A @ X ) @ ( finite2053577756real_n @ B @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vector_sub_rdistrib
% 0.18/0.46 thf(fact_78_vector__sub__rdistrib,axiom,
% 0.18/0.46 ! [A: real,B: real,X: finite1183840848l_num1] :
% 0.18/0.46 ( ( finite1961675766l_num1 @ ( minus_minus_real @ A @ B ) @ X )
% 0.18/0.46 = ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ A @ X ) @ ( finite1961675766l_num1 @ B @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vector_sub_rdistrib
% 0.18/0.46 thf(fact_79__C1_C_I3_J,axiom,
% 0.18/0.46 ord_less_real @ ( real_V783965509real_n @ y @ x ) @ one_one_real ).
% 0.18/0.46
% 0.18/0.46 % "1"(3)
% 0.18/0.46 thf(fact_80_norm__of__int,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( real_V646646907m_real @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.46 = ( abs_abs_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_of_int
% 0.18/0.46 thf(fact_81_ceiling__diff__of__int,axiom,
% 0.18/0.46 ! [X: real,Z: int] :
% 0.18/0.46 ( ( archim1371465213g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 0.18/0.46 = ( minus_minus_int @ ( archim1371465213g_real @ X ) @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_diff_of_int
% 0.18/0.46 thf(fact_82_axis__nth,axiom,
% 0.18/0.46 ! [I: n,X: real] :
% 0.18/0.46 ( ( finite772340589real_n @ ( finite703491444n_real @ I @ X ) @ I )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % axis_nth
% 0.18/0.46 thf(fact_83_axis__nth,axiom,
% 0.18/0.46 ! [I: numeral_num1,X: real] :
% 0.18/0.46 ( ( finite1106245191l_num1 @ ( finite15144922381_real @ I @ X ) @ I )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % axis_nth
% 0.18/0.46 thf(fact_84_axis__nth,axiom,
% 0.18/0.46 ! [I: numeral_num1,X: int] :
% 0.18/0.46 ( ( finite1800583751l_num1 @ ( finite627713486m1_int @ I @ X ) @ I )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % axis_nth
% 0.18/0.46 thf(fact_85_axis__nth,axiom,
% 0.18/0.46 ! [I: n,X: int] :
% 0.18/0.46 ( ( finite566214509_int_n @ ( finite1888777460_n_int @ I @ X ) @ I )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % axis_nth
% 0.18/0.46 thf(fact_86_vec_Oscale__left__diff__distrib,axiom,
% 0.18/0.46 ! [A: real,B: real,X: finite1489363574real_n] :
% 0.18/0.46 ( ( finite2053577756real_n @ ( minus_minus_real @ A @ B ) @ X )
% 0.18/0.46 = ( minus_1037315151real_n @ ( finite2053577756real_n @ A @ X ) @ ( finite2053577756real_n @ B @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec.scale_left_diff_distrib
% 0.18/0.46 thf(fact_87_vec_Oscale__left__diff__distrib,axiom,
% 0.18/0.46 ! [A: real,B: real,X: finite1183840848l_num1] :
% 0.18/0.46 ( ( finite1961675766l_num1 @ ( minus_minus_real @ A @ B ) @ X )
% 0.18/0.46 = ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ A @ X ) @ ( finite1961675766l_num1 @ B @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec.scale_left_diff_distrib
% 0.18/0.46 thf(fact_88_floor__diff__of__int,axiom,
% 0.18/0.46 ! [X: real,Z: int] :
% 0.18/0.46 ( ( archim1031974863r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 0.18/0.46 = ( minus_minus_int @ ( archim1031974863r_real @ X ) @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % floor_diff_of_int
% 0.18/0.46 thf(fact_89_norm__axis__1,axiom,
% 0.18/0.46 ! [M2: numeral_num1] :
% 0.18/0.46 ( ( real_V400292255l_num1 @ ( finite15144922381_real @ M2 @ one_one_real ) )
% 0.18/0.46 = one_one_real ) ).
% 0.18/0.46
% 0.18/0.46 % norm_axis_1
% 0.18/0.46 thf(fact_90_abs__1,axiom,
% 0.18/0.46 ( ( abs_abs_real @ one_one_real )
% 0.18/0.46 = one_one_real ) ).
% 0.18/0.46
% 0.18/0.46 % abs_1
% 0.18/0.46 thf(fact_91_abs__1,axiom,
% 0.18/0.46 ( ( abs_abs_int @ one_one_int )
% 0.18/0.46 = one_one_int ) ).
% 0.18/0.46
% 0.18/0.46 % abs_1
% 0.18/0.46 thf(fact_92_mem__Collect__eq,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,P: finite1489363574real_n > $o] :
% 0.18/0.46 ( ( member1352538125real_n @ A @ ( collec321817931real_n @ P ) )
% 0.18/0.46 = ( P @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % mem_Collect_eq
% 0.18/0.46 thf(fact_93_mem__Collect__eq,axiom,
% 0.18/0.46 ! [A: real,P: real > $o] :
% 0.18/0.46 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.18/0.46 = ( P @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % mem_Collect_eq
% 0.18/0.46 thf(fact_94_Collect__mem__eq,axiom,
% 0.18/0.46 ! [A2: set_Fi1058188332real_n] :
% 0.18/0.46 ( ( collec321817931real_n
% 0.18/0.46 @ ^ [X3: finite1489363574real_n] : ( member1352538125real_n @ X3 @ A2 ) )
% 0.18/0.46 = A2 ) ).
% 0.18/0.46
% 0.18/0.46 % Collect_mem_eq
% 0.18/0.46 thf(fact_95_Collect__mem__eq,axiom,
% 0.18/0.46 ! [A2: set_real] :
% 0.18/0.46 ( ( collect_real
% 0.18/0.46 @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 0.18/0.46 = A2 ) ).
% 0.18/0.46
% 0.18/0.46 % Collect_mem_eq
% 0.18/0.46 thf(fact_96_of__int__less__iff,axiom,
% 0.18/0.46 ! [W: int,Z: int] :
% 0.18/0.46 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.46 = ( ord_less_int @ W @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_less_iff
% 0.18/0.46 thf(fact_97_of__int__less__iff,axiom,
% 0.18/0.46 ! [W: int,Z: int] :
% 0.18/0.46 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.18/0.46 = ( ord_less_int @ W @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_less_iff
% 0.18/0.46 thf(fact_98_one__index,axiom,
% 0.18/0.46 ! [I: n] :
% 0.18/0.46 ( ( finite772340589real_n @ one_on1253059131real_n @ I )
% 0.18/0.46 = one_one_real ) ).
% 0.18/0.46
% 0.18/0.46 % one_index
% 0.18/0.46 thf(fact_99_one__index,axiom,
% 0.18/0.46 ! [I: numeral_num1] :
% 0.18/0.46 ( ( finite1106245191l_num1 @ one_on1961019669l_num1 @ I )
% 0.18/0.46 = one_one_real ) ).
% 0.18/0.46
% 0.18/0.46 % one_index
% 0.18/0.46 thf(fact_100_one__index,axiom,
% 0.18/0.46 ! [I: numeral_num1] :
% 0.18/0.46 ( ( finite1800583751l_num1 @ one_on601734293l_num1 @ I )
% 0.18/0.46 = one_one_int ) ).
% 0.18/0.46
% 0.18/0.46 % one_index
% 0.18/0.46 thf(fact_101_one__index,axiom,
% 0.18/0.46 ! [I: n] :
% 0.18/0.46 ( ( finite566214509_int_n @ one_on1516019131_int_n @ I )
% 0.18/0.46 = one_one_int ) ).
% 0.18/0.46
% 0.18/0.46 % one_index
% 0.18/0.46 thf(fact_102_of__int__eq__1__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ( ring_1_of_int_real @ Z )
% 0.18/0.46 = one_one_real )
% 0.18/0.46 = ( Z = one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_eq_1_iff
% 0.18/0.46 thf(fact_103_of__int__eq__1__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ( ring_1_of_int_int @ Z )
% 0.18/0.46 = one_one_int )
% 0.18/0.46 = ( Z = one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_eq_1_iff
% 0.18/0.46 thf(fact_104_of__int__eq__1__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ( ring_11737179077real_n @ Z )
% 0.18/0.46 = one_on1253059131real_n )
% 0.18/0.46 = ( Z = one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_eq_1_iff
% 0.18/0.46 thf(fact_105_of__int__1,axiom,
% 0.18/0.46 ( ( ring_1_of_int_real @ one_one_int )
% 0.18/0.46 = one_one_real ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_1
% 0.18/0.46 thf(fact_106_of__int__1,axiom,
% 0.18/0.46 ( ( ring_1_of_int_int @ one_one_int )
% 0.18/0.46 = one_one_int ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_1
% 0.18/0.46 thf(fact_107_of__int__1,axiom,
% 0.18/0.46 ( ( ring_11737179077real_n @ one_one_int )
% 0.18/0.46 = one_on1253059131real_n ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_1
% 0.18/0.46 thf(fact_108_abs__norm__cancel,axiom,
% 0.18/0.46 ! [A: real] :
% 0.18/0.46 ( ( abs_abs_real @ ( real_V646646907m_real @ A ) )
% 0.18/0.46 = ( real_V646646907m_real @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_norm_cancel
% 0.18/0.46 thf(fact_109_abs__norm__cancel,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1] :
% 0.18/0.46 ( ( abs_abs_real @ ( real_V400292255l_num1 @ A ) )
% 0.18/0.46 = ( real_V400292255l_num1 @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_norm_cancel
% 0.18/0.46 thf(fact_110_floor__one,axiom,
% 0.18/0.46 ( ( archim1031974863r_real @ one_one_real )
% 0.18/0.46 = one_one_int ) ).
% 0.18/0.46
% 0.18/0.46 % floor_one
% 0.18/0.46 thf(fact_111_ceiling__one,axiom,
% 0.18/0.46 ( ( archim1371465213g_real @ one_one_real )
% 0.18/0.46 = one_one_int ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_one
% 0.18/0.46 thf(fact_112_floor__of__int,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( archim1031974863r_real @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.46 = Z ) ).
% 0.18/0.46
% 0.18/0.46 % floor_of_int
% 0.18/0.46 thf(fact_113_ceiling__of__int,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( archim1371465213g_real @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.46 = Z ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_of_int
% 0.18/0.46 thf(fact_114_vec__component,axiom,
% 0.18/0.46 ! [X: real,I: n] :
% 0.18/0.46 ( ( finite772340589real_n @ ( finite667548811real_n @ X ) @ I )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % vec_component
% 0.18/0.46 thf(fact_115_vec__component,axiom,
% 0.18/0.46 ! [X: real,I: numeral_num1] :
% 0.18/0.46 ( ( finite1106245191l_num1 @ ( finite1329045861l_num1 @ X ) @ I )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % vec_component
% 0.18/0.46 thf(fact_116_vec__component,axiom,
% 0.18/0.46 ! [X: int,I: numeral_num1] :
% 0.18/0.46 ( ( finite1800583751l_num1 @ ( finite1470207077l_num1 @ X ) @ I )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % vec_component
% 0.18/0.46 thf(fact_117_vec__component,axiom,
% 0.18/0.46 ! [X: int,I: n] :
% 0.18/0.46 ( ( finite566214509_int_n @ ( finite55766923_int_n @ X ) @ I )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % vec_component
% 0.18/0.46 thf(fact_118_vector__ssub__ldistrib,axiom,
% 0.18/0.46 ! [C: real,X: finite1489363574real_n,Y: finite1489363574real_n] :
% 0.18/0.46 ( ( finite2053577756real_n @ C @ ( minus_1037315151real_n @ X @ Y ) )
% 0.18/0.46 = ( minus_1037315151real_n @ ( finite2053577756real_n @ C @ X ) @ ( finite2053577756real_n @ C @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vector_ssub_ldistrib
% 0.18/0.46 thf(fact_119_vector__ssub__ldistrib,axiom,
% 0.18/0.46 ! [C: real,X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 0.18/0.46 ( ( finite1961675766l_num1 @ C @ ( minus_249161513l_num1 @ X @ Y ) )
% 0.18/0.46 = ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ C @ X ) @ ( finite1961675766l_num1 @ C @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vector_ssub_ldistrib
% 0.18/0.46 thf(fact_120_of__int__1__less__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.46 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_1_less_iff
% 0.18/0.46 thf(fact_121_of__int__1__less__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.18/0.46 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_1_less_iff
% 0.18/0.46 thf(fact_122_of__int__less__1__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.18/0.46 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_less_1_iff
% 0.18/0.46 thf(fact_123_of__int__less__1__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.18/0.46 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_less_1_iff
% 0.18/0.46 thf(fact_124_norm__one,axiom,
% 0.18/0.46 ( ( real_V646646907m_real @ one_one_real )
% 0.18/0.46 = one_one_real ) ).
% 0.18/0.46
% 0.18/0.46 % norm_one
% 0.18/0.46 thf(fact_125_dist__diff_I2_J,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 0.18/0.46 ( ( real_V783965509real_n @ ( minus_1037315151real_n @ A @ B ) @ A )
% 0.18/0.46 = ( real_V739724485real_n @ B ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_diff(2)
% 0.18/0.46 thf(fact_126_dist__diff_I2_J,axiom,
% 0.18/0.46 ! [A: real,B: real] :
% 0.18/0.46 ( ( real_V1934908667t_real @ ( minus_minus_real @ A @ B ) @ A )
% 0.18/0.46 = ( real_V646646907m_real @ B ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_diff(2)
% 0.18/0.46 thf(fact_127_dist__diff_I2_J,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 0.18/0.46 ( ( real_V488409631l_num1 @ ( minus_249161513l_num1 @ A @ B ) @ A )
% 0.18/0.46 = ( real_V400292255l_num1 @ B ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_diff(2)
% 0.18/0.46 thf(fact_128_dist__diff_I1_J,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 0.18/0.46 ( ( real_V783965509real_n @ A @ ( minus_1037315151real_n @ A @ B ) )
% 0.18/0.46 = ( real_V739724485real_n @ B ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_diff(1)
% 0.18/0.46 thf(fact_129_dist__diff_I1_J,axiom,
% 0.18/0.46 ! [A: real,B: real] :
% 0.18/0.46 ( ( real_V1934908667t_real @ A @ ( minus_minus_real @ A @ B ) )
% 0.18/0.46 = ( real_V646646907m_real @ B ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_diff(1)
% 0.18/0.46 thf(fact_130_dist__diff_I1_J,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 0.18/0.46 ( ( real_V488409631l_num1 @ A @ ( minus_249161513l_num1 @ A @ B ) )
% 0.18/0.46 = ( real_V400292255l_num1 @ B ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_diff(1)
% 0.18/0.46 thf(fact_131_floor__less__one,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( ord_less_int @ ( archim1031974863r_real @ X ) @ one_one_int )
% 0.18/0.46 = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.18/0.46
% 0.18/0.46 % floor_less_one
% 0.18/0.46 thf(fact_132_one__less__ceiling,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( ord_less_int @ one_one_int @ ( archim1371465213g_real @ X ) )
% 0.18/0.46 = ( ord_less_real @ one_one_real @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % one_less_ceiling
% 0.18/0.46 thf(fact_133_floor__diff__one,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( archim1031974863r_real @ ( minus_minus_real @ X @ one_one_real ) )
% 0.18/0.46 = ( minus_minus_int @ ( archim1031974863r_real @ X ) @ one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % floor_diff_one
% 0.18/0.46 thf(fact_134_ceiling__diff__one,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( archim1371465213g_real @ ( minus_minus_real @ X @ one_one_real ) )
% 0.18/0.46 = ( minus_minus_int @ ( archim1371465213g_real @ X ) @ one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_diff_one
% 0.18/0.46 thf(fact_135_one__reorient,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( one_one_real = X )
% 0.18/0.46 = ( X = one_one_real ) ) ).
% 0.18/0.46
% 0.18/0.46 % one_reorient
% 0.18/0.46 thf(fact_136_one__reorient,axiom,
% 0.18/0.46 ! [X: int] :
% 0.18/0.46 ( ( one_one_int = X )
% 0.18/0.46 = ( X = one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % one_reorient
% 0.18/0.46 thf(fact_137_one__reorient,axiom,
% 0.18/0.46 ! [X: numeral_num1] :
% 0.18/0.46 ( ( one_one_Numeral_num1 = X )
% 0.18/0.46 = ( X = one_one_Numeral_num1 ) ) ).
% 0.18/0.46
% 0.18/0.46 % one_reorient
% 0.18/0.46 thf(fact_138_floor__less__iff,axiom,
% 0.18/0.46 ! [X: real,Z: int] :
% 0.18/0.46 ( ( ord_less_int @ ( archim1031974863r_real @ X ) @ Z )
% 0.18/0.46 = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % floor_less_iff
% 0.18/0.46 thf(fact_139_less__ceiling__iff,axiom,
% 0.18/0.46 ! [Z: int,X: real] :
% 0.18/0.46 ( ( ord_less_int @ Z @ ( archim1371465213g_real @ X ) )
% 0.18/0.46 = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_ceiling_iff
% 0.18/0.46 thf(fact_140_dist__norm,axiom,
% 0.18/0.46 ( real_V783965509real_n
% 0.18/0.46 = ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] : ( real_V739724485real_n @ ( minus_1037315151real_n @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_norm
% 0.18/0.46 thf(fact_141_dist__norm,axiom,
% 0.18/0.46 ( real_V1934908667t_real
% 0.18/0.46 = ( ^ [X3: real,Y3: real] : ( real_V646646907m_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_norm
% 0.18/0.46 thf(fact_142_dist__norm,axiom,
% 0.18/0.46 ( real_V488409631l_num1
% 0.18/0.46 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_norm
% 0.18/0.46 thf(fact_143_linorder__neqE__linordered__idom,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( X != Y )
% 0.18/0.46 => ( ~ ( ord_less_real @ X @ Y )
% 0.18/0.46 => ( ord_less_real @ Y @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % linorder_neqE_linordered_idom
% 0.18/0.46 thf(fact_144_linorder__neqE__linordered__idom,axiom,
% 0.18/0.46 ! [X: int,Y: int] :
% 0.18/0.46 ( ( X != Y )
% 0.18/0.46 => ( ~ ( ord_less_int @ X @ Y )
% 0.18/0.46 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % linorder_neqE_linordered_idom
% 0.18/0.46 thf(fact_145_dist__commute__lessI,axiom,
% 0.18/0.46 ! [Y: finite1489363574real_n,X: finite1489363574real_n,E: real] :
% 0.18/0.46 ( ( ord_less_real @ ( real_V783965509real_n @ Y @ X ) @ E )
% 0.18/0.46 => ( ord_less_real @ ( real_V783965509real_n @ X @ Y ) @ E ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_commute_lessI
% 0.18/0.46 thf(fact_146_dist__commute__lessI,axiom,
% 0.18/0.46 ! [Y: real,X: real,E: real] :
% 0.18/0.46 ( ( ord_less_real @ ( real_V1934908667t_real @ Y @ X ) @ E )
% 0.18/0.46 => ( ord_less_real @ ( real_V1934908667t_real @ X @ Y ) @ E ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_commute_lessI
% 0.18/0.46 thf(fact_147_dist__commute__lessI,axiom,
% 0.18/0.46 ! [Y: finite1183840848l_num1,X: finite1183840848l_num1,E: real] :
% 0.18/0.46 ( ( ord_less_real @ ( real_V488409631l_num1 @ Y @ X ) @ E )
% 0.18/0.46 => ( ord_less_real @ ( real_V488409631l_num1 @ X @ Y ) @ E ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_commute_lessI
% 0.18/0.46 thf(fact_148_dist__commute,axiom,
% 0.18/0.46 ( real_V783965509real_n
% 0.18/0.46 = ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] : ( real_V783965509real_n @ Y3 @ X3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_commute
% 0.18/0.46 thf(fact_149_dist__commute,axiom,
% 0.18/0.46 ( real_V1934908667t_real
% 0.18/0.46 = ( ^ [X3: real,Y3: real] : ( real_V1934908667t_real @ Y3 @ X3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_commute
% 0.18/0.46 thf(fact_150_dist__commute,axiom,
% 0.18/0.46 ( real_V488409631l_num1
% 0.18/0.46 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( real_V488409631l_num1 @ Y3 @ X3 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_commute
% 0.18/0.46 thf(fact_151_ceiling__less__cancel,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( ord_less_int @ ( archim1371465213g_real @ X ) @ ( archim1371465213g_real @ Y ) )
% 0.18/0.46 => ( ord_less_real @ X @ Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_less_cancel
% 0.18/0.46 thf(fact_152_floor__less__cancel,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( ord_less_int @ ( archim1031974863r_real @ X ) @ ( archim1031974863r_real @ Y ) )
% 0.18/0.46 => ( ord_less_real @ X @ Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % floor_less_cancel
% 0.18/0.46 thf(fact_153_norm__minus__commute,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 0.18/0.46 ( ( real_V739724485real_n @ ( minus_1037315151real_n @ A @ B ) )
% 0.18/0.46 = ( real_V739724485real_n @ ( minus_1037315151real_n @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_minus_commute
% 0.18/0.46 thf(fact_154_norm__minus__commute,axiom,
% 0.18/0.46 ! [A: real,B: real] :
% 0.18/0.46 ( ( real_V646646907m_real @ ( minus_minus_real @ A @ B ) )
% 0.18/0.46 = ( real_V646646907m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_minus_commute
% 0.18/0.46 thf(fact_155_norm__minus__commute,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 0.18/0.46 ( ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ A @ B ) )
% 0.18/0.46 = ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_minus_commute
% 0.18/0.46 thf(fact_156_real__norm__def,axiom,
% 0.18/0.46 real_V646646907m_real = abs_abs_real ).
% 0.18/0.46
% 0.18/0.46 % real_norm_def
% 0.18/0.46 thf(fact_157_ex__less__of__int,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % ex_less_of_int
% 0.18/0.46 thf(fact_158_ex__of__int__less,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).
% 0.18/0.46
% 0.18/0.46 % ex_of_int_less
% 0.18/0.46 thf(fact_159_norm__bound__component__lt__cart,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,E: real,I: n] :
% 0.18/0.46 ( ( ord_less_real @ ( real_V739724485real_n @ X ) @ E )
% 0.18/0.46 => ( ord_less_real @ ( abs_abs_real @ ( finite772340589real_n @ X @ I ) ) @ E ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_bound_component_lt_cart
% 0.18/0.46 thf(fact_160_norm__bound__component__lt__cart,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,E: real,I: numeral_num1] :
% 0.18/0.46 ( ( ord_less_real @ ( real_V400292255l_num1 @ X ) @ E )
% 0.18/0.46 => ( ord_less_real @ ( abs_abs_real @ ( finite1106245191l_num1 @ X @ I ) ) @ E ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_bound_component_lt_cart
% 0.18/0.46 thf(fact_161_diff__strict__mono,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n,D: finite1489363574real_n,C: finite1489363574real_n] :
% 0.18/0.46 ( ( ord_le1017090250real_n @ A @ B )
% 0.18/0.46 => ( ( ord_le1017090250real_n @ D @ C )
% 0.18/0.46 => ( ord_le1017090250real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_mono
% 0.18/0.46 thf(fact_162_diff__strict__mono,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,D: finite1183840848l_num1,C: finite1183840848l_num1] :
% 0.18/0.46 ( ( ord_le2035778724l_num1 @ A @ B )
% 0.18/0.46 => ( ( ord_le2035778724l_num1 @ D @ C )
% 0.18/0.46 => ( ord_le2035778724l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_mono
% 0.18/0.46 thf(fact_163_diff__strict__mono,axiom,
% 0.18/0.46 ! [A: real,B: real,D: real,C: real] :
% 0.18/0.46 ( ( ord_less_real @ A @ B )
% 0.18/0.46 => ( ( ord_less_real @ D @ C )
% 0.18/0.46 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_mono
% 0.18/0.46 thf(fact_164_diff__strict__mono,axiom,
% 0.18/0.46 ! [A: int,B: int,D: int,C: int] :
% 0.18/0.46 ( ( ord_less_int @ A @ B )
% 0.18/0.46 => ( ( ord_less_int @ D @ C )
% 0.18/0.46 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_mono
% 0.18/0.46 thf(fact_165_diff__eq__diff__less,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n,D: finite1489363574real_n] :
% 0.18/0.46 ( ( ( minus_1037315151real_n @ A @ B )
% 0.18/0.46 = ( minus_1037315151real_n @ C @ D ) )
% 0.18/0.46 => ( ( ord_le1017090250real_n @ A @ B )
% 0.18/0.46 = ( ord_le1017090250real_n @ C @ D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_less
% 0.18/0.46 thf(fact_166_diff__eq__diff__less,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1,D: finite1183840848l_num1] :
% 0.18/0.46 ( ( ( minus_249161513l_num1 @ A @ B )
% 0.18/0.46 = ( minus_249161513l_num1 @ C @ D ) )
% 0.18/0.46 => ( ( ord_le2035778724l_num1 @ A @ B )
% 0.18/0.46 = ( ord_le2035778724l_num1 @ C @ D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_less
% 0.18/0.46 thf(fact_167_diff__eq__diff__less,axiom,
% 0.18/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.18/0.46 ( ( ( minus_minus_real @ A @ B )
% 0.18/0.46 = ( minus_minus_real @ C @ D ) )
% 0.18/0.46 => ( ( ord_less_real @ A @ B )
% 0.18/0.46 = ( ord_less_real @ C @ D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_less
% 0.18/0.46 thf(fact_168_diff__eq__diff__less,axiom,
% 0.18/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.18/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.18/0.46 = ( minus_minus_int @ C @ D ) )
% 0.18/0.46 => ( ( ord_less_int @ A @ B )
% 0.18/0.46 = ( ord_less_int @ C @ D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_less
% 0.18/0.46 thf(fact_169_diff__strict__left__mono,axiom,
% 0.18/0.46 ! [B: finite1489363574real_n,A: finite1489363574real_n,C: finite1489363574real_n] :
% 0.18/0.46 ( ( ord_le1017090250real_n @ B @ A )
% 0.18/0.46 => ( ord_le1017090250real_n @ ( minus_1037315151real_n @ C @ A ) @ ( minus_1037315151real_n @ C @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_left_mono
% 0.18/0.46 thf(fact_170_diff__strict__left__mono,axiom,
% 0.18/0.46 ! [B: finite1183840848l_num1,A: finite1183840848l_num1,C: finite1183840848l_num1] :
% 0.18/0.46 ( ( ord_le2035778724l_num1 @ B @ A )
% 0.18/0.46 => ( ord_le2035778724l_num1 @ ( minus_249161513l_num1 @ C @ A ) @ ( minus_249161513l_num1 @ C @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_left_mono
% 0.18/0.46 thf(fact_171_diff__strict__left__mono,axiom,
% 0.18/0.46 ! [B: real,A: real,C: real] :
% 0.18/0.46 ( ( ord_less_real @ B @ A )
% 0.18/0.46 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_left_mono
% 0.18/0.46 thf(fact_172_diff__strict__left__mono,axiom,
% 0.18/0.46 ! [B: int,A: int,C: int] :
% 0.18/0.46 ( ( ord_less_int @ B @ A )
% 0.18/0.46 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_left_mono
% 0.18/0.46 thf(fact_173_diff__strict__right__mono,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] :
% 0.18/0.46 ( ( ord_le1017090250real_n @ A @ B )
% 0.18/0.46 => ( ord_le1017090250real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_right_mono
% 0.18/0.46 thf(fact_174_diff__strict__right__mono,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1] :
% 0.18/0.46 ( ( ord_le2035778724l_num1 @ A @ B )
% 0.18/0.46 => ( ord_le2035778724l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_right_mono
% 0.18/0.46 thf(fact_175_diff__strict__right__mono,axiom,
% 0.18/0.46 ! [A: real,B: real,C: real] :
% 0.18/0.46 ( ( ord_less_real @ A @ B )
% 0.18/0.46 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_right_mono
% 0.18/0.46 thf(fact_176_diff__strict__right__mono,axiom,
% 0.18/0.46 ! [A: int,B: int,C: int] :
% 0.18/0.46 ( ( ord_less_int @ A @ B )
% 0.18/0.46 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_strict_right_mono
% 0.18/0.46 thf(fact_177_abs__one,axiom,
% 0.18/0.46 ( ( abs_abs_real @ one_one_real )
% 0.18/0.46 = one_one_real ) ).
% 0.18/0.46
% 0.18/0.46 % abs_one
% 0.18/0.46 thf(fact_178_abs__one,axiom,
% 0.18/0.46 ( ( abs_abs_int @ one_one_int )
% 0.18/0.46 = one_one_int ) ).
% 0.18/0.46
% 0.18/0.46 % abs_one
% 0.18/0.46 thf(fact_179_dist__real__def,axiom,
% 0.18/0.46 ( real_V1934908667t_real
% 0.18/0.46 = ( ^ [X3: real,Y3: real] : ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_real_def
% 0.18/0.46 thf(fact_180_Ints__1,axiom,
% 0.18/0.46 member1352538125real_n @ one_on1253059131real_n @ ring_1723899781real_n ).
% 0.18/0.46
% 0.18/0.46 % Ints_1
% 0.18/0.46 thf(fact_181_Ints__1,axiom,
% 0.18/0.46 member_int @ one_one_int @ ring_1_Ints_int ).
% 0.18/0.46
% 0.18/0.46 % Ints_1
% 0.18/0.46 thf(fact_182_Ints__1,axiom,
% 0.18/0.46 member_real @ one_one_real @ ring_1_Ints_real ).
% 0.18/0.46
% 0.18/0.46 % Ints_1
% 0.18/0.46 thf(fact_183_vec_Oscale__right__diff__distrib,axiom,
% 0.18/0.46 ! [A: real,X: finite1489363574real_n,Y: finite1489363574real_n] :
% 0.18/0.46 ( ( finite2053577756real_n @ A @ ( minus_1037315151real_n @ X @ Y ) )
% 0.18/0.46 = ( minus_1037315151real_n @ ( finite2053577756real_n @ A @ X ) @ ( finite2053577756real_n @ A @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec.scale_right_diff_distrib
% 0.18/0.46 thf(fact_184_vec_Oscale__right__diff__distrib,axiom,
% 0.18/0.46 ! [A: real,X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 0.18/0.46 ( ( finite1961675766l_num1 @ A @ ( minus_249161513l_num1 @ X @ Y ) )
% 0.18/0.46 = ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ A @ X ) @ ( finite1961675766l_num1 @ A @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % vec.scale_right_diff_distrib
% 0.18/0.46 thf(fact_185_Ints__eq__abs__less1,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( member_real @ X @ ring_1_Ints_real )
% 0.18/0.46 => ( ( member_real @ Y @ ring_1_Ints_real )
% 0.18/0.46 => ( ( X = Y )
% 0.18/0.46 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_eq_abs_less1
% 0.18/0.46 thf(fact_186_Ints__eq__abs__less1,axiom,
% 0.18/0.46 ! [X: int,Y: int] :
% 0.18/0.46 ( ( member_int @ X @ ring_1_Ints_int )
% 0.18/0.46 => ( ( member_int @ Y @ ring_1_Ints_int )
% 0.18/0.46 => ( ( X = Y )
% 0.18/0.46 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % Ints_eq_abs_less1
% 0.18/0.46 thf(fact_187_of__int__ceiling__cancel,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) )
% 0.18/0.46 = X )
% 0.18/0.46 = ( ? [N2: int] :
% 0.18/0.46 ( X
% 0.18/0.46 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_ceiling_cancel
% 0.18/0.46 thf(fact_188_of__int__floor__cancel,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( ( ring_1_of_int_real @ ( archim1031974863r_real @ X ) )
% 0.18/0.46 = X )
% 0.18/0.46 = ( ? [N2: int] :
% 0.18/0.46 ( X
% 0.18/0.46 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_floor_cancel
% 0.18/0.46 thf(fact_189_real__of__int__floor__gt__diff__one,axiom,
% 0.18/0.46 ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim1031974863r_real @ R ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % real_of_int_floor_gt_diff_one
% 0.18/0.46 thf(fact_190_real__abs__dist,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 0.18/0.46 ( ( abs_abs_real @ ( real_V783965509real_n @ X @ Y ) )
% 0.18/0.46 = ( real_V783965509real_n @ X @ Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % real_abs_dist
% 0.18/0.46 thf(fact_191_real__abs__dist,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( abs_abs_real @ ( real_V1934908667t_real @ X @ Y ) )
% 0.18/0.46 = ( real_V1934908667t_real @ X @ Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % real_abs_dist
% 0.18/0.46 thf(fact_192_real__abs__dist,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 0.18/0.46 ( ( abs_abs_real @ ( real_V488409631l_num1 @ X @ Y ) )
% 0.18/0.46 = ( real_V488409631l_num1 @ X @ Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % real_abs_dist
% 0.18/0.46 thf(fact_193_less__numeral__extra_I4_J,axiom,
% 0.18/0.46 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.18/0.46
% 0.18/0.46 % less_numeral_extra(4)
% 0.18/0.46 thf(fact_194_less__numeral__extra_I4_J,axiom,
% 0.18/0.46 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.18/0.46
% 0.18/0.46 % less_numeral_extra(4)
% 0.18/0.46 thf(fact_195_le__ceiling__iff,axiom,
% 0.18/0.46 ! [Z: int,X: real] :
% 0.18/0.46 ( ( ord_less_eq_int @ Z @ ( archim1371465213g_real @ X ) )
% 0.18/0.46 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_ceiling_iff
% 0.18/0.46 thf(fact_196_ceiling__split,axiom,
% 0.18/0.46 ! [P: int > $o,T: real] :
% 0.18/0.46 ( ( P @ ( archim1371465213g_real @ T ) )
% 0.18/0.46 = ( ! [I2: int] :
% 0.18/0.46 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
% 0.18/0.46 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
% 0.18/0.46 => ( P @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_split
% 0.18/0.46 thf(fact_197_ceiling__eq__iff,axiom,
% 0.18/0.46 ! [X: real,A: int] :
% 0.18/0.46 ( ( ( archim1371465213g_real @ X )
% 0.18/0.46 = A )
% 0.18/0.46 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 0.18/0.46 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_eq_iff
% 0.18/0.46 thf(fact_198_of__int__le__iff,axiom,
% 0.18/0.46 ! [W: int,Z: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.18/0.46 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_le_iff
% 0.18/0.46 thf(fact_199_of__int__le__iff,axiom,
% 0.18/0.46 ! [W: int,Z: int] :
% 0.18/0.46 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.46 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_le_iff
% 0.18/0.46 thf(fact_200_zle__diff1__eq,axiom,
% 0.18/0.46 ! [W: int,Z: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.18/0.46 = ( ord_less_int @ W @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % zle_diff1_eq
% 0.18/0.46 thf(fact_201_of__int__le__1__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.18/0.46 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_le_1_iff
% 0.18/0.46 thf(fact_202_of__int__le__1__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.18/0.46 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_le_1_iff
% 0.18/0.46 thf(fact_203_of__int__1__le__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.18/0.46 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_1_le_iff
% 0.18/0.46 thf(fact_204_of__int__1__le__iff,axiom,
% 0.18/0.46 ! [Z: int] :
% 0.18/0.46 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.46 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_1_le_iff
% 0.18/0.46 thf(fact_205_one__le__floor,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( ord_less_eq_int @ one_one_int @ ( archim1031974863r_real @ X ) )
% 0.18/0.46 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % one_le_floor
% 0.18/0.46 thf(fact_206_ceiling__le__one,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ one_one_int )
% 0.18/0.46 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_le_one
% 0.18/0.46 thf(fact_207_ceiling__mono,axiom,
% 0.18/0.46 ! [Y: real,X: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ Y @ X )
% 0.18/0.46 => ( ord_less_eq_int @ ( archim1371465213g_real @ Y ) @ ( archim1371465213g_real @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_mono
% 0.18/0.46 thf(fact_208_floor__mono,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.18/0.46 => ( ord_less_eq_int @ ( archim1031974863r_real @ X ) @ ( archim1031974863r_real @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % floor_mono
% 0.18/0.46 thf(fact_209_less__eq__real__def,axiom,
% 0.18/0.46 ( ord_less_eq_real
% 0.18/0.46 = ( ^ [X3: real,Y3: real] :
% 0.18/0.46 ( ( ord_less_real @ X3 @ Y3 )
% 0.18/0.46 | ( X3 = Y3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_eq_real_def
% 0.18/0.46 thf(fact_210_le__numeral__extra_I4_J,axiom,
% 0.18/0.46 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.18/0.46
% 0.18/0.46 % le_numeral_extra(4)
% 0.18/0.46 thf(fact_211_le__numeral__extra_I4_J,axiom,
% 0.18/0.46 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.18/0.46
% 0.18/0.46 % le_numeral_extra(4)
% 0.18/0.46 thf(fact_212_less__eq__vec__def,axiom,
% 0.18/0.46 ( ord_le866583254real_n
% 0.18/0.46 = ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] :
% 0.18/0.46 ! [I2: n] : ( ord_less_eq_real @ ( finite772340589real_n @ X3 @ I2 ) @ ( finite772340589real_n @ Y3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_eq_vec_def
% 0.18/0.46 thf(fact_213_less__eq__vec__def,axiom,
% 0.18/0.46 ( ord_le1724070576l_num1
% 0.18/0.46 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] :
% 0.18/0.46 ! [I2: numeral_num1] : ( ord_less_eq_real @ ( finite1106245191l_num1 @ X3 @ I2 ) @ ( finite1106245191l_num1 @ Y3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_eq_vec_def
% 0.18/0.46 thf(fact_214_less__eq__vec__def,axiom,
% 0.18/0.46 ( ord_le1202326448l_num1
% 0.18/0.46 = ( ^ [X3: finite2063899472l_num1,Y3: finite2063899472l_num1] :
% 0.18/0.46 ! [I2: numeral_num1] : ( ord_less_eq_int @ ( finite1800583751l_num1 @ X3 @ I2 ) @ ( finite1800583751l_num1 @ Y3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_eq_vec_def
% 0.18/0.46 thf(fact_215_less__eq__vec__def,axiom,
% 0.18/0.46 ( ord_le1908593622_int_n
% 0.18/0.46 = ( ^ [X3: finite964658038_int_n,Y3: finite964658038_int_n] :
% 0.18/0.46 ! [I2: n] : ( ord_less_eq_int @ ( finite566214509_int_n @ X3 @ I2 ) @ ( finite566214509_int_n @ Y3 @ I2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % less_eq_vec_def
% 0.18/0.46 thf(fact_216_int__less__induct,axiom,
% 0.18/0.46 ! [I: int,K: int,P: int > $o] :
% 0.18/0.46 ( ( ord_less_int @ I @ K )
% 0.18/0.46 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.18/0.46 => ( ! [I3: int] :
% 0.18/0.46 ( ( ord_less_int @ I3 @ K )
% 0.18/0.46 => ( ( P @ I3 )
% 0.18/0.46 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.18/0.46 => ( P @ I ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % int_less_induct
% 0.18/0.46 thf(fact_217_int__le__induct,axiom,
% 0.18/0.46 ! [I: int,K: int,P: int > $o] :
% 0.18/0.46 ( ( ord_less_eq_int @ I @ K )
% 0.18/0.46 => ( ( P @ K )
% 0.18/0.46 => ( ! [I3: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ I3 @ K )
% 0.18/0.46 => ( ( P @ I3 )
% 0.18/0.46 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.18/0.46 => ( P @ I ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % int_le_induct
% 0.18/0.46 thf(fact_218_ceiling__le,axiom,
% 0.18/0.46 ! [X: real,A: int] :
% 0.18/0.46 ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 0.18/0.46 => ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_le
% 0.18/0.46 thf(fact_219_le__floor__iff,axiom,
% 0.18/0.46 ! [Z: int,X: real] :
% 0.18/0.46 ( ( ord_less_eq_int @ Z @ ( archim1031974863r_real @ X ) )
% 0.18/0.46 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_floor_iff
% 0.18/0.46 thf(fact_220_ceiling__le__iff,axiom,
% 0.18/0.46 ! [X: real,Z: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ Z )
% 0.18/0.46 = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_le_iff
% 0.18/0.46 thf(fact_221_diff__eq__diff__less__eq,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n,D: finite1489363574real_n] :
% 0.18/0.46 ( ( ( minus_1037315151real_n @ A @ B )
% 0.18/0.46 = ( minus_1037315151real_n @ C @ D ) )
% 0.18/0.46 => ( ( ord_le866583254real_n @ A @ B )
% 0.18/0.46 = ( ord_le866583254real_n @ C @ D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_less_eq
% 0.18/0.46 thf(fact_222_diff__eq__diff__less__eq,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1,D: finite1183840848l_num1] :
% 0.18/0.46 ( ( ( minus_249161513l_num1 @ A @ B )
% 0.18/0.46 = ( minus_249161513l_num1 @ C @ D ) )
% 0.18/0.46 => ( ( ord_le1724070576l_num1 @ A @ B )
% 0.18/0.46 = ( ord_le1724070576l_num1 @ C @ D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_less_eq
% 0.18/0.46 thf(fact_223_diff__eq__diff__less__eq,axiom,
% 0.18/0.46 ! [A: int,B: int,C: int,D: int] :
% 0.18/0.46 ( ( ( minus_minus_int @ A @ B )
% 0.18/0.46 = ( minus_minus_int @ C @ D ) )
% 0.18/0.46 => ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_less_eq
% 0.18/0.46 thf(fact_224_diff__eq__diff__less__eq,axiom,
% 0.18/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.18/0.46 ( ( ( minus_minus_real @ A @ B )
% 0.18/0.46 = ( minus_minus_real @ C @ D ) )
% 0.18/0.46 => ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_eq_diff_less_eq
% 0.18/0.46 thf(fact_225_diff__right__mono,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] :
% 0.18/0.46 ( ( ord_le866583254real_n @ A @ B )
% 0.18/0.46 => ( ord_le866583254real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_right_mono
% 0.18/0.46 thf(fact_226_diff__right__mono,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1] :
% 0.18/0.46 ( ( ord_le1724070576l_num1 @ A @ B )
% 0.18/0.46 => ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_right_mono
% 0.18/0.46 thf(fact_227_diff__right__mono,axiom,
% 0.18/0.46 ! [A: int,B: int,C: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_right_mono
% 0.18/0.46 thf(fact_228_diff__right__mono,axiom,
% 0.18/0.46 ! [A: real,B: real,C: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_right_mono
% 0.18/0.46 thf(fact_229_diff__left__mono,axiom,
% 0.18/0.46 ! [B: finite1489363574real_n,A: finite1489363574real_n,C: finite1489363574real_n] :
% 0.18/0.46 ( ( ord_le866583254real_n @ B @ A )
% 0.18/0.46 => ( ord_le866583254real_n @ ( minus_1037315151real_n @ C @ A ) @ ( minus_1037315151real_n @ C @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_left_mono
% 0.18/0.46 thf(fact_230_diff__left__mono,axiom,
% 0.18/0.46 ! [B: finite1183840848l_num1,A: finite1183840848l_num1,C: finite1183840848l_num1] :
% 0.18/0.46 ( ( ord_le1724070576l_num1 @ B @ A )
% 0.18/0.46 => ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ C @ A ) @ ( minus_249161513l_num1 @ C @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_left_mono
% 0.18/0.46 thf(fact_231_diff__left__mono,axiom,
% 0.18/0.46 ! [B: int,A: int,C: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.18/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_left_mono
% 0.18/0.46 thf(fact_232_diff__left__mono,axiom,
% 0.18/0.46 ! [B: real,A: real,C: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.18/0.46 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_left_mono
% 0.18/0.46 thf(fact_233_diff__mono,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n,D: finite1489363574real_n,C: finite1489363574real_n] :
% 0.18/0.46 ( ( ord_le866583254real_n @ A @ B )
% 0.18/0.46 => ( ( ord_le866583254real_n @ D @ C )
% 0.18/0.46 => ( ord_le866583254real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_mono
% 0.18/0.46 thf(fact_234_diff__mono,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,D: finite1183840848l_num1,C: finite1183840848l_num1] :
% 0.18/0.46 ( ( ord_le1724070576l_num1 @ A @ B )
% 0.18/0.46 => ( ( ord_le1724070576l_num1 @ D @ C )
% 0.18/0.46 => ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_mono
% 0.18/0.46 thf(fact_235_diff__mono,axiom,
% 0.18/0.46 ! [A: int,B: int,D: int,C: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 => ( ( ord_less_eq_int @ D @ C )
% 0.18/0.46 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_mono
% 0.18/0.46 thf(fact_236_diff__mono,axiom,
% 0.18/0.46 ! [A: real,B: real,D: real,C: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 => ( ( ord_less_eq_real @ D @ C )
% 0.18/0.46 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % diff_mono
% 0.18/0.46 thf(fact_237_abs__ge__self,axiom,
% 0.18/0.46 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_ge_self
% 0.18/0.46 thf(fact_238_abs__ge__self,axiom,
% 0.18/0.46 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_ge_self
% 0.18/0.46 thf(fact_239_abs__le__D1,axiom,
% 0.18/0.46 ! [A: int,B: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.18/0.46 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_le_D1
% 0.18/0.46 thf(fact_240_abs__le__D1,axiom,
% 0.18/0.46 ! [A: real,B: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.18/0.46 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_le_D1
% 0.18/0.46 thf(fact_241_norm__vector__1,axiom,
% 0.18/0.46 ( real_V316032763l_num1
% 0.18/0.46 = ( ^ [X3: finite463740460l_num1] : ( real_V400292255l_num1 @ ( finite1449546019l_num1 @ X3 @ one_one_Numeral_num1 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_vector_1
% 0.18/0.46 thf(fact_242_norm__vector__1,axiom,
% 0.18/0.46 ( real_V400292255l_num1
% 0.18/0.46 = ( ^ [X3: finite1183840848l_num1] : ( real_V646646907m_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_vector_1
% 0.18/0.46 thf(fact_243_ex__le__of__int,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% 0.18/0.46
% 0.18/0.46 % ex_le_of_int
% 0.18/0.46 thf(fact_244_dist__vector__1,axiom,
% 0.18/0.46 ( real_V845814485l_num1
% 0.18/0.46 = ( ^ [X3: finite121157254l_num1,Y3: finite121157254l_num1] : ( real_V783965509real_n @ ( finite1503577981l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1503577981l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_vector_1
% 0.18/0.46 thf(fact_245_dist__vector__1,axiom,
% 0.18/0.46 ( real_V1974343547l_num1
% 0.18/0.46 = ( ^ [X3: finite463740460l_num1,Y3: finite463740460l_num1] : ( real_V488409631l_num1 @ ( finite1449546019l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1449546019l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_vector_1
% 0.18/0.46 thf(fact_246_dist__vector__1,axiom,
% 0.18/0.46 ( real_V488409631l_num1
% 0.18/0.46 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( real_V1934908667t_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1106245191l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_vector_1
% 0.18/0.46 thf(fact_247_norm__real,axiom,
% 0.18/0.46 ( real_V400292255l_num1
% 0.18/0.46 = ( ^ [X3: finite1183840848l_num1] : ( abs_abs_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_real
% 0.18/0.46 thf(fact_248_ceiling__diff__floor__le__1,axiom,
% 0.18/0.46 ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim1371465213g_real @ X ) @ ( archim1031974863r_real @ X ) ) @ one_one_int ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_diff_floor_le_1
% 0.18/0.46 thf(fact_249_ceiling__less__iff,axiom,
% 0.18/0.46 ! [X: real,Z: int] :
% 0.18/0.46 ( ( ord_less_int @ ( archim1371465213g_real @ X ) @ Z )
% 0.18/0.46 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_less_iff
% 0.18/0.46 thf(fact_250_abs__triangle__ineq2__sym,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_le866583254real_n @ ( minus_1037315151real_n @ ( abs_ab1599551059real_n @ A ) @ ( abs_ab1599551059real_n @ B ) ) @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq2_sym
% 0.18/0.46 thf(fact_251_abs__triangle__ineq2__sym,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ ( abs_ab948591917l_num1 @ A ) @ ( abs_ab948591917l_num1 @ B ) ) @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq2_sym
% 0.18/0.46 thf(fact_252_abs__triangle__ineq2__sym,axiom,
% 0.18/0.46 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq2_sym
% 0.18/0.46 thf(fact_253_abs__triangle__ineq2__sym,axiom,
% 0.18/0.46 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq2_sym
% 0.18/0.46 thf(fact_254_abs__triangle__ineq3,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_le866583254real_n @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ ( abs_ab1599551059real_n @ A ) @ ( abs_ab1599551059real_n @ B ) ) ) @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq3
% 0.18/0.46 thf(fact_255_abs__triangle__ineq3,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_le1724070576l_num1 @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ ( abs_ab948591917l_num1 @ A ) @ ( abs_ab948591917l_num1 @ B ) ) ) @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq3
% 0.18/0.46 thf(fact_256_abs__triangle__ineq3,axiom,
% 0.18/0.46 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq3
% 0.18/0.46 thf(fact_257_abs__triangle__ineq3,axiom,
% 0.18/0.46 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq3
% 0.18/0.46 thf(fact_258_abs__triangle__ineq2,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_le866583254real_n @ ( minus_1037315151real_n @ ( abs_ab1599551059real_n @ A ) @ ( abs_ab1599551059real_n @ B ) ) @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq2
% 0.18/0.46 thf(fact_259_abs__triangle__ineq2,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ ( abs_ab948591917l_num1 @ A ) @ ( abs_ab948591917l_num1 @ B ) ) @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq2
% 0.18/0.46 thf(fact_260_abs__triangle__ineq2,axiom,
% 0.18/0.46 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq2
% 0.18/0.46 thf(fact_261_abs__triangle__ineq2,axiom,
% 0.18/0.46 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_triangle_ineq2
% 0.18/0.46 thf(fact_262_norm__le__componentwise__cart,axiom,
% 0.18/0.46 ! [X: finite337957458num1_n,Y: finite1489363574real_n] :
% 0.18/0.46 ( ! [I3: n] : ( ord_less_eq_real @ ( real_V400292255l_num1 @ ( finite1639785033num1_n @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite772340589real_n @ Y @ I3 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ ( real_V1121707553num1_n @ X ) @ ( real_V739724485real_n @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_le_componentwise_cart
% 0.18/0.46 thf(fact_263_norm__le__componentwise__cart,axiom,
% 0.18/0.46 ! [X: finite463740460l_num1,Y: finite1183840848l_num1] :
% 0.18/0.46 ( ! [I3: numeral_num1] : ( ord_less_eq_real @ ( real_V400292255l_num1 @ ( finite1449546019l_num1 @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ Y @ I3 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ ( real_V316032763l_num1 @ X ) @ ( real_V400292255l_num1 @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_le_componentwise_cart
% 0.18/0.46 thf(fact_264_norm__le__componentwise__cart,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,Y: finite337957458num1_n] :
% 0.18/0.46 ( ! [I3: n] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite772340589real_n @ X @ I3 ) ) @ ( real_V400292255l_num1 @ ( finite1639785033num1_n @ Y @ I3 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ ( real_V739724485real_n @ X ) @ ( real_V1121707553num1_n @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_le_componentwise_cart
% 0.18/0.46 thf(fact_265_norm__le__componentwise__cart,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 0.18/0.46 ( ! [I3: n] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite772340589real_n @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite772340589real_n @ Y @ I3 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ ( real_V739724485real_n @ X ) @ ( real_V739724485real_n @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_le_componentwise_cart
% 0.18/0.46 thf(fact_266_norm__le__componentwise__cart,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,Y: finite463740460l_num1] :
% 0.18/0.46 ( ! [I3: numeral_num1] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ X @ I3 ) ) @ ( real_V400292255l_num1 @ ( finite1449546019l_num1 @ Y @ I3 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ ( real_V400292255l_num1 @ X ) @ ( real_V316032763l_num1 @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_le_componentwise_cart
% 0.18/0.46 thf(fact_267_norm__le__componentwise__cart,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 0.18/0.46 ( ! [I3: numeral_num1] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ Y @ I3 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ ( real_V400292255l_num1 @ X ) @ ( real_V400292255l_num1 @ Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_le_componentwise_cart
% 0.18/0.46 thf(fact_268_Finite__Cartesian__Product_Onorm__nth__le,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,I: n] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite772340589real_n @ X @ I ) ) @ ( real_V739724485real_n @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % Finite_Cartesian_Product.norm_nth_le
% 0.18/0.46 thf(fact_269_Finite__Cartesian__Product_Onorm__nth__le,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,I: numeral_num1] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ X @ I ) ) @ ( real_V400292255l_num1 @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % Finite_Cartesian_Product.norm_nth_le
% 0.18/0.46 thf(fact_270_real__of__int__floor__ge__diff__one,axiom,
% 0.18/0.46 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim1031974863r_real @ R ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % real_of_int_floor_ge_diff_one
% 0.18/0.46 thf(fact_271_of__int__floor__le,axiom,
% 0.18/0.46 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim1031974863r_real @ X ) ) @ X ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_floor_le
% 0.18/0.46 thf(fact_272_le__of__int__ceiling,axiom,
% 0.18/0.46 ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_of_int_ceiling
% 0.18/0.46 thf(fact_273_norm__bound__component__le__cart,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,E: real,I: n] :
% 0.18/0.46 ( ( ord_less_eq_real @ ( real_V739724485real_n @ X ) @ E )
% 0.18/0.46 => ( ord_less_eq_real @ ( abs_abs_real @ ( finite772340589real_n @ X @ I ) ) @ E ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_bound_component_le_cart
% 0.18/0.46 thf(fact_274_norm__bound__component__le__cart,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,E: real,I: numeral_num1] :
% 0.18/0.46 ( ( ord_less_eq_real @ ( real_V400292255l_num1 @ X ) @ E )
% 0.18/0.46 => ( ord_less_eq_real @ ( abs_abs_real @ ( finite1106245191l_num1 @ X @ I ) ) @ E ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_bound_component_le_cart
% 0.18/0.46 thf(fact_275_component__le__norm__cart,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,I: n] : ( ord_less_eq_real @ ( abs_abs_real @ ( finite772340589real_n @ X @ I ) ) @ ( real_V739724485real_n @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % component_le_norm_cart
% 0.18/0.46 thf(fact_276_component__le__norm__cart,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,I: numeral_num1] : ( ord_less_eq_real @ ( abs_abs_real @ ( finite1106245191l_num1 @ X @ I ) ) @ ( real_V400292255l_num1 @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % component_le_norm_cart
% 0.18/0.46 thf(fact_277_dist__real,axiom,
% 0.18/0.46 ( real_V488409631l_num1
% 0.18/0.46 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( abs_abs_real @ ( minus_minus_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1106245191l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_real
% 0.18/0.46 thf(fact_278_floor__le__ceiling,axiom,
% 0.18/0.46 ! [X: real] : ( ord_less_eq_int @ ( archim1031974863r_real @ X ) @ ( archim1371465213g_real @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % floor_le_ceiling
% 0.18/0.46 thf(fact_279_of__int__ceiling__diff__one__le,axiom,
% 0.18/0.46 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim1371465213g_real @ R ) ) @ one_one_real ) @ R ) ).
% 0.18/0.46
% 0.18/0.46 % of_int_ceiling_diff_one_le
% 0.18/0.46 thf(fact_280_norm__triangle__ineq2,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V739724485real_n @ A ) @ ( real_V739724485real_n @ B ) ) @ ( real_V739724485real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_triangle_ineq2
% 0.18/0.46 thf(fact_281_norm__triangle__ineq2,axiom,
% 0.18/0.46 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V646646907m_real @ A ) @ ( real_V646646907m_real @ B ) ) @ ( real_V646646907m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_triangle_ineq2
% 0.18/0.46 thf(fact_282_norm__triangle__ineq2,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V400292255l_num1 @ A ) @ ( real_V400292255l_num1 @ B ) ) @ ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_triangle_ineq2
% 0.18/0.46 thf(fact_283_abs__dist__diff__le,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V783965509real_n @ A @ B ) @ ( real_V783965509real_n @ B @ C ) ) ) @ ( real_V783965509real_n @ A @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_dist_diff_le
% 0.18/0.46 thf(fact_284_abs__dist__diff__le,axiom,
% 0.18/0.46 ! [A: real,B: real,C: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1934908667t_real @ A @ B ) @ ( real_V1934908667t_real @ B @ C ) ) ) @ ( real_V1934908667t_real @ A @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_dist_diff_le
% 0.18/0.46 thf(fact_285_abs__dist__diff__le,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V488409631l_num1 @ A @ B ) @ ( real_V488409631l_num1 @ B @ C ) ) ) @ ( real_V488409631l_num1 @ A @ C ) ) ).
% 0.18/0.46
% 0.18/0.46 % abs_dist_diff_le
% 0.18/0.46 thf(fact_286_metric__eq__thm,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,S: set_Fi1058188332real_n,Y: finite1489363574real_n] :
% 0.18/0.46 ( ( member1352538125real_n @ X @ S )
% 0.18/0.46 => ( ( member1352538125real_n @ Y @ S )
% 0.18/0.46 => ( ( X = Y )
% 0.18/0.46 = ( ! [X3: finite1489363574real_n] :
% 0.18/0.46 ( ( member1352538125real_n @ X3 @ S )
% 0.18/0.46 => ( ( real_V783965509real_n @ X @ X3 )
% 0.18/0.46 = ( real_V783965509real_n @ Y @ X3 ) ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % metric_eq_thm
% 0.18/0.46 thf(fact_287_metric__eq__thm,axiom,
% 0.18/0.46 ! [X: real,S: set_real,Y: real] :
% 0.18/0.46 ( ( member_real @ X @ S )
% 0.18/0.46 => ( ( member_real @ Y @ S )
% 0.18/0.46 => ( ( X = Y )
% 0.18/0.46 = ( ! [X3: real] :
% 0.18/0.46 ( ( member_real @ X3 @ S )
% 0.18/0.46 => ( ( real_V1934908667t_real @ X @ X3 )
% 0.18/0.46 = ( real_V1934908667t_real @ Y @ X3 ) ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % metric_eq_thm
% 0.18/0.46 thf(fact_288_metric__eq__thm,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,S: set_Fi1257234438l_num1,Y: finite1183840848l_num1] :
% 0.18/0.46 ( ( member1413569767l_num1 @ X @ S )
% 0.18/0.46 => ( ( member1413569767l_num1 @ Y @ S )
% 0.18/0.46 => ( ( X = Y )
% 0.18/0.46 = ( ! [X3: finite1183840848l_num1] :
% 0.18/0.46 ( ( member1413569767l_num1 @ X3 @ S )
% 0.18/0.46 => ( ( real_V488409631l_num1 @ X @ X3 )
% 0.18/0.46 = ( real_V488409631l_num1 @ Y @ X3 ) ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % metric_eq_thm
% 0.18/0.46 thf(fact_289_dist__vec__nth__le,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,I: n,Y: finite1489363574real_n] : ( ord_less_eq_real @ ( real_V1934908667t_real @ ( finite772340589real_n @ X @ I ) @ ( finite772340589real_n @ Y @ I ) ) @ ( real_V783965509real_n @ X @ Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_vec_nth_le
% 0.18/0.46 thf(fact_290_dist__vec__nth__le,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,I: numeral_num1,Y: finite1183840848l_num1] : ( ord_less_eq_real @ ( real_V1934908667t_real @ ( finite1106245191l_num1 @ X @ I ) @ ( finite1106245191l_num1 @ Y @ I ) ) @ ( real_V488409631l_num1 @ X @ Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % dist_vec_nth_le
% 0.18/0.46 thf(fact_291_norm__triangle__ineq3,axiom,
% 0.18/0.46 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V739724485real_n @ A ) @ ( real_V739724485real_n @ B ) ) ) @ ( real_V739724485real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_triangle_ineq3
% 0.18/0.46 thf(fact_292_norm__triangle__ineq3,axiom,
% 0.18/0.46 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V646646907m_real @ A ) @ ( real_V646646907m_real @ B ) ) ) @ ( real_V646646907m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_triangle_ineq3
% 0.18/0.46 thf(fact_293_norm__triangle__ineq3,axiom,
% 0.18/0.46 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V400292255l_num1 @ A ) @ ( real_V400292255l_num1 @ B ) ) ) @ ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % norm_triangle_ineq3
% 0.18/0.46 thf(fact_294_ceiling__correct,axiom,
% 0.18/0.46 ! [X: real] :
% 0.18/0.46 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) @ one_one_real ) @ X )
% 0.18/0.46 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_correct
% 0.18/0.46 thf(fact_295_ceiling__unique,axiom,
% 0.18/0.46 ! [Z: int,X: real] :
% 0.18/0.46 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
% 0.18/0.46 => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
% 0.18/0.46 => ( ( archim1371465213g_real @ X )
% 0.18/0.46 = Z ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ceiling_unique
% 0.18/0.46 thf(fact_296_order__refl,axiom,
% 0.18/0.46 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.18/0.46
% 0.18/0.46 % order_refl
% 0.18/0.46 thf(fact_297_order__refl,axiom,
% 0.18/0.46 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 0.18/0.46
% 0.18/0.46 % order_refl
% 0.18/0.46 thf(fact_298_complete__interval,axiom,
% 0.18/0.46 ! [A: int,B: int,P: int > $o] :
% 0.18/0.46 ( ( ord_less_int @ A @ B )
% 0.18/0.46 => ( ( P @ A )
% 0.18/0.46 => ( ~ ( P @ B )
% 0.18/0.46 => ? [C2: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ A @ C2 )
% 0.18/0.46 & ( ord_less_eq_int @ C2 @ B )
% 0.18/0.46 & ! [X4: int] :
% 0.18/0.46 ( ( ( ord_less_eq_int @ A @ X4 )
% 0.18/0.46 & ( ord_less_int @ X4 @ C2 ) )
% 0.18/0.46 => ( P @ X4 ) )
% 0.18/0.46 & ! [D2: int] :
% 0.18/0.46 ( ! [X5: int] :
% 0.18/0.46 ( ( ( ord_less_eq_int @ A @ X5 )
% 0.18/0.46 & ( ord_less_int @ X5 @ D2 ) )
% 0.18/0.46 => ( P @ X5 ) )
% 0.18/0.46 => ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % complete_interval
% 0.18/0.46 thf(fact_299_complete__interval,axiom,
% 0.18/0.46 ! [A: real,B: real,P: real > $o] :
% 0.18/0.46 ( ( ord_less_real @ A @ B )
% 0.18/0.46 => ( ( P @ A )
% 0.18/0.46 => ( ~ ( P @ B )
% 0.18/0.46 => ? [C2: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ A @ C2 )
% 0.18/0.46 & ( ord_less_eq_real @ C2 @ B )
% 0.18/0.46 & ! [X4: real] :
% 0.18/0.46 ( ( ( ord_less_eq_real @ A @ X4 )
% 0.18/0.46 & ( ord_less_real @ X4 @ C2 ) )
% 0.18/0.46 => ( P @ X4 ) )
% 0.18/0.46 & ! [D2: real] :
% 0.18/0.46 ( ! [X5: real] :
% 0.18/0.46 ( ( ( ord_less_eq_real @ A @ X5 )
% 0.18/0.46 & ( ord_less_real @ X5 @ D2 ) )
% 0.18/0.46 => ( P @ X5 ) )
% 0.18/0.46 => ( ord_less_eq_real @ D2 @ C2 ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % complete_interval
% 0.18/0.46 thf(fact_300_eucl__less__le__not__le,axiom,
% 0.18/0.46 ( ord_less_real
% 0.18/0.46 = ( ^ [X3: real,Y3: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.18/0.46 & ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eucl_less_le_not_le
% 0.18/0.46 thf(fact_301_order_Onot__eq__order__implies__strict,axiom,
% 0.18/0.46 ! [A: int,B: int] :
% 0.18/0.46 ( ( A != B )
% 0.18/0.46 => ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 => ( ord_less_int @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order.not_eq_order_implies_strict
% 0.18/0.46 thf(fact_302_order_Onot__eq__order__implies__strict,axiom,
% 0.18/0.46 ! [A: real,B: real] :
% 0.18/0.46 ( ( A != B )
% 0.18/0.46 => ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 => ( ord_less_real @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order.not_eq_order_implies_strict
% 0.18/0.46 thf(fact_303_dual__order_Ostrict__implies__order,axiom,
% 0.18/0.46 ! [B: int,A: int] :
% 0.18/0.46 ( ( ord_less_int @ B @ A )
% 0.18/0.46 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.strict_implies_order
% 0.18/0.46 thf(fact_304_dual__order_Ostrict__implies__order,axiom,
% 0.18/0.46 ! [B: real,A: real] :
% 0.18/0.46 ( ( ord_less_real @ B @ A )
% 0.18/0.46 => ( ord_less_eq_real @ B @ A ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.strict_implies_order
% 0.18/0.46 thf(fact_305_vector__one__nth,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1] :
% 0.18/0.46 ( ( finite1329045861l_num1 @ ( finite1106245191l_num1 @ X @ one_one_Numeral_num1 ) )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % vector_one_nth
% 0.18/0.46 thf(fact_306_vector__one__nth,axiom,
% 0.18/0.46 ! [X: finite2063899472l_num1] :
% 0.18/0.46 ( ( finite1470207077l_num1 @ ( finite1800583751l_num1 @ X @ one_one_Numeral_num1 ) )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % vector_one_nth
% 0.18/0.46 thf(fact_307_complete__real,axiom,
% 0.18/0.46 ! [S2: set_real] :
% 0.18/0.46 ( ? [X4: real] : ( member_real @ X4 @ S2 )
% 0.18/0.46 => ( ? [Z4: real] :
% 0.18/0.46 ! [X5: real] :
% 0.18/0.46 ( ( member_real @ X5 @ S2 )
% 0.18/0.46 => ( ord_less_eq_real @ X5 @ Z4 ) )
% 0.18/0.46 => ? [Y4: real] :
% 0.18/0.46 ( ! [X4: real] :
% 0.18/0.46 ( ( member_real @ X4 @ S2 )
% 0.18/0.46 => ( ord_less_eq_real @ X4 @ Y4 ) )
% 0.18/0.46 & ! [Z4: real] :
% 0.18/0.46 ( ! [X5: real] :
% 0.18/0.46 ( ( member_real @ X5 @ S2 )
% 0.18/0.46 => ( ord_less_eq_real @ X5 @ Z4 ) )
% 0.18/0.46 => ( ord_less_eq_real @ Y4 @ Z4 ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % complete_real
% 0.18/0.46 thf(fact_308_dual__order_Oantisym,axiom,
% 0.18/0.46 ! [B: int,A: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.18/0.46 => ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 => ( A = B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.antisym
% 0.18/0.46 thf(fact_309_dual__order_Oantisym,axiom,
% 0.18/0.46 ! [B: real,A: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.18/0.46 => ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 => ( A = B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.antisym
% 0.18/0.46 thf(fact_310_dual__order_Oeq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [A3: int,B2: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ B2 @ A3 )
% 0.18/0.46 & ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.eq_iff
% 0.18/0.46 thf(fact_311_dual__order_Oeq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [A3: real,B2: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ B2 @ A3 )
% 0.18/0.46 & ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.eq_iff
% 0.18/0.46 thf(fact_312_dual__order_Otrans,axiom,
% 0.18/0.46 ! [B: int,A: int,C: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ B @ A )
% 0.18/0.46 => ( ( ord_less_eq_int @ C @ B )
% 0.18/0.46 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.trans
% 0.18/0.46 thf(fact_313_dual__order_Otrans,axiom,
% 0.18/0.46 ! [B: real,A: real,C: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.18/0.46 => ( ( ord_less_eq_real @ C @ B )
% 0.18/0.46 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.trans
% 0.18/0.46 thf(fact_314_linorder__wlog,axiom,
% 0.18/0.46 ! [P: int > int > $o,A: int,B: int] :
% 0.18/0.46 ( ! [A4: int,B3: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ A4 @ B3 )
% 0.18/0.46 => ( P @ A4 @ B3 ) )
% 0.18/0.46 => ( ! [A4: int,B3: int] :
% 0.18/0.46 ( ( P @ B3 @ A4 )
% 0.18/0.46 => ( P @ A4 @ B3 ) )
% 0.18/0.46 => ( P @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % linorder_wlog
% 0.18/0.46 thf(fact_315_linorder__wlog,axiom,
% 0.18/0.46 ! [P: real > real > $o,A: real,B: real] :
% 0.18/0.46 ( ! [A4: real,B3: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ A4 @ B3 )
% 0.18/0.46 => ( P @ A4 @ B3 ) )
% 0.18/0.46 => ( ! [A4: real,B3: real] :
% 0.18/0.46 ( ( P @ B3 @ A4 )
% 0.18/0.46 => ( P @ A4 @ B3 ) )
% 0.18/0.46 => ( P @ A @ B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % linorder_wlog
% 0.18/0.46 thf(fact_316_dual__order_Orefl,axiom,
% 0.18/0.46 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.refl
% 0.18/0.46 thf(fact_317_dual__order_Orefl,axiom,
% 0.18/0.46 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.18/0.46
% 0.18/0.46 % dual_order.refl
% 0.18/0.46 thf(fact_318_order__trans,axiom,
% 0.18/0.46 ! [X: int,Y: int,Z: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ X @ Y )
% 0.18/0.46 => ( ( ord_less_eq_int @ Y @ Z )
% 0.18/0.46 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_trans
% 0.18/0.46 thf(fact_319_order__trans,axiom,
% 0.18/0.46 ! [X: real,Y: real,Z: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.18/0.46 => ( ( ord_less_eq_real @ Y @ Z )
% 0.18/0.46 => ( ord_less_eq_real @ X @ Z ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_trans
% 0.18/0.46 thf(fact_320_order__class_Oorder_Oantisym,axiom,
% 0.18/0.46 ! [A: int,B: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 => ( ( ord_less_eq_int @ B @ A )
% 0.18/0.46 => ( A = B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.antisym
% 0.18/0.46 thf(fact_321_order__class_Oorder_Oantisym,axiom,
% 0.18/0.46 ! [A: real,B: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 => ( ( ord_less_eq_real @ B @ A )
% 0.18/0.46 => ( A = B ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.antisym
% 0.18/0.46 thf(fact_322_ord__le__eq__trans,axiom,
% 0.18/0.46 ! [A: int,B: int,C: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 => ( ( B = C )
% 0.18/0.46 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_trans
% 0.18/0.46 thf(fact_323_ord__le__eq__trans,axiom,
% 0.18/0.46 ! [A: real,B: real,C: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 => ( ( B = C )
% 0.18/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_trans
% 0.18/0.46 thf(fact_324_ord__eq__le__trans,axiom,
% 0.18/0.46 ! [A: int,B: int,C: int] :
% 0.18/0.46 ( ( A = B )
% 0.18/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.46 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_trans
% 0.18/0.46 thf(fact_325_ord__eq__le__trans,axiom,
% 0.18/0.46 ! [A: real,B: real,C: real] :
% 0.18/0.46 ( ( A = B )
% 0.18/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_trans
% 0.18/0.46 thf(fact_326_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [A3: int,B2: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.18/0.46 & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.eq_iff
% 0.18/0.46 thf(fact_327_order__class_Oorder_Oeq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [A3: real,B2: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ A3 @ B2 )
% 0.18/0.46 & ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order_class.order.eq_iff
% 0.18/0.46 thf(fact_328_antisym__conv,axiom,
% 0.18/0.46 ! [Y: int,X: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ Y @ X )
% 0.18/0.46 => ( ( ord_less_eq_int @ X @ Y )
% 0.18/0.46 = ( X = Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym_conv
% 0.18/0.46 thf(fact_329_antisym__conv,axiom,
% 0.18/0.46 ! [Y: real,X: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ Y @ X )
% 0.18/0.46 => ( ( ord_less_eq_real @ X @ Y )
% 0.18/0.46 = ( X = Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym_conv
% 0.18/0.46 thf(fact_330_le__cases3,axiom,
% 0.18/0.46 ! [X: int,Y: int,Z: int] :
% 0.18/0.46 ( ( ( ord_less_eq_int @ X @ Y )
% 0.18/0.46 => ~ ( ord_less_eq_int @ Y @ Z ) )
% 0.18/0.46 => ( ( ( ord_less_eq_int @ Y @ X )
% 0.18/0.46 => ~ ( ord_less_eq_int @ X @ Z ) )
% 0.18/0.46 => ( ( ( ord_less_eq_int @ X @ Z )
% 0.18/0.46 => ~ ( ord_less_eq_int @ Z @ Y ) )
% 0.18/0.46 => ( ( ( ord_less_eq_int @ Z @ Y )
% 0.18/0.46 => ~ ( ord_less_eq_int @ Y @ X ) )
% 0.18/0.46 => ( ( ( ord_less_eq_int @ Y @ Z )
% 0.18/0.46 => ~ ( ord_less_eq_int @ Z @ X ) )
% 0.18/0.46 => ~ ( ( ord_less_eq_int @ Z @ X )
% 0.18/0.46 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_cases3
% 0.18/0.46 thf(fact_331_le__cases3,axiom,
% 0.18/0.46 ! [X: real,Y: real,Z: real] :
% 0.18/0.46 ( ( ( ord_less_eq_real @ X @ Y )
% 0.18/0.46 => ~ ( ord_less_eq_real @ Y @ Z ) )
% 0.18/0.46 => ( ( ( ord_less_eq_real @ Y @ X )
% 0.18/0.46 => ~ ( ord_less_eq_real @ X @ Z ) )
% 0.18/0.46 => ( ( ( ord_less_eq_real @ X @ Z )
% 0.18/0.46 => ~ ( ord_less_eq_real @ Z @ Y ) )
% 0.18/0.46 => ( ( ( ord_less_eq_real @ Z @ Y )
% 0.18/0.46 => ~ ( ord_less_eq_real @ Y @ X ) )
% 0.18/0.46 => ( ( ( ord_less_eq_real @ Y @ Z )
% 0.18/0.46 => ~ ( ord_less_eq_real @ Z @ X ) )
% 0.18/0.46 => ~ ( ( ord_less_eq_real @ Z @ X )
% 0.18/0.46 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_cases3
% 0.18/0.46 thf(fact_332_order_Otrans,axiom,
% 0.18/0.46 ! [A: int,B: int,C: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.46 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order.trans
% 0.18/0.46 thf(fact_333_order_Otrans,axiom,
% 0.18/0.46 ! [A: real,B: real,C: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % order.trans
% 0.18/0.46 thf(fact_334_le__cases,axiom,
% 0.18/0.46 ! [X: int,Y: int] :
% 0.18/0.46 ( ~ ( ord_less_eq_int @ X @ Y )
% 0.18/0.46 => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_cases
% 0.18/0.46 thf(fact_335_le__cases,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ~ ( ord_less_eq_real @ X @ Y )
% 0.18/0.46 => ( ord_less_eq_real @ Y @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % le_cases
% 0.18/0.46 thf(fact_336_eq__refl,axiom,
% 0.18/0.46 ! [X: int,Y: int] :
% 0.18/0.46 ( ( X = Y )
% 0.18/0.46 => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_refl
% 0.18/0.46 thf(fact_337_eq__refl,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( X = Y )
% 0.18/0.46 => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_refl
% 0.18/0.46 thf(fact_338_linear,axiom,
% 0.18/0.46 ! [X: int,Y: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ X @ Y )
% 0.18/0.46 | ( ord_less_eq_int @ Y @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % linear
% 0.18/0.46 thf(fact_339_linear,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.18/0.46 | ( ord_less_eq_real @ Y @ X ) ) ).
% 0.18/0.46
% 0.18/0.46 % linear
% 0.18/0.46 thf(fact_340_antisym,axiom,
% 0.18/0.46 ! [X: int,Y: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ X @ Y )
% 0.18/0.46 => ( ( ord_less_eq_int @ Y @ X )
% 0.18/0.46 => ( X = Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym
% 0.18/0.46 thf(fact_341_antisym,axiom,
% 0.18/0.46 ! [X: real,Y: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.18/0.46 => ( ( ord_less_eq_real @ Y @ X )
% 0.18/0.46 => ( X = Y ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % antisym
% 0.18/0.46 thf(fact_342_eq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [X3: int,Y3: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.18/0.46 & ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_iff
% 0.18/0.46 thf(fact_343_eq__iff,axiom,
% 0.18/0.46 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 0.18/0.46 = ( ^ [X3: real,Y3: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.18/0.46 & ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % eq_iff
% 0.18/0.46 thf(fact_344_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A: int,B: int,F: int > int,C: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 => ( ( ( F @ B )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X5: int,Y4: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.18/0.46 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_345_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A: int,B: int,F: int > real,C: real] :
% 0.18/0.46 ( ( ord_less_eq_int @ A @ B )
% 0.18/0.46 => ( ( ( F @ B )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X5: int,Y4: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.18/0.46 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_346_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A: real,B: real,F: real > int,C: int] :
% 0.18/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 => ( ( ( F @ B )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X5: real,Y4: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X5 @ Y4 )
% 0.18/0.46 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_347_ord__le__eq__subst,axiom,
% 0.18/0.46 ! [A: real,B: real,F: real > real,C: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.18/0.46 => ( ( ( F @ B )
% 0.18/0.46 = C )
% 0.18/0.46 => ( ! [X5: real,Y4: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X5 @ Y4 )
% 0.18/0.46 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_le_eq_subst
% 0.18/0.46 thf(fact_348_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A: int,F: int > int,B: int,C: int] :
% 0.18/0.46 ( ( A
% 0.18/0.46 = ( F @ B ) )
% 0.18/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.46 => ( ! [X5: int,Y4: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.18/0.46 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_349_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A: real,F: int > real,B: int,C: int] :
% 0.18/0.46 ( ( A
% 0.18/0.46 = ( F @ B ) )
% 0.18/0.46 => ( ( ord_less_eq_int @ B @ C )
% 0.18/0.46 => ( ! [X5: int,Y4: int] :
% 0.18/0.46 ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.18/0.46 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_350_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A: int,F: real > int,B: real,C: real] :
% 0.18/0.46 ( ( A
% 0.18/0.46 = ( F @ B ) )
% 0.18/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.46 => ( ! [X5: real,Y4: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X5 @ Y4 )
% 0.18/0.46 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_351_ord__eq__le__subst,axiom,
% 0.18/0.46 ! [A: real,F: real > real,B: real,C: real] :
% 0.18/0.46 ( ( A
% 0.18/0.46 = ( F @ B ) )
% 0.18/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.18/0.46 => ( ! [X5: real,Y4: real] :
% 0.18/0.46 ( ( ord_less_eq_real @ X5 @ Y4 )
% 0.18/0.46 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.18/0.46 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ord_eq_le_subst
% 0.18/0.46 thf(fact_352_forall__1,axiom,
% 0.18/0.46 ( ( ^ [P2: numeral_num1 > $o] :
% 0.18/0.46 ! [X2: numeral_num1] : ( P2 @ X2 ) )
% 0.18/0.46 = ( ^ [P3: numeral_num1 > $o] : ( P3 @ one_one_Numeral_num1 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % forall_1
% 0.18/0.46 thf(fact_353_ex__1,axiom,
% 0.18/0.46 ( ( ^ [P2: numeral_num1 > $o] :
% 0.18/0.46 ? [X2: numeral_num1] : ( P2 @ X2 ) )
% 0.18/0.46 = ( ^ [P3: numeral_num1 > $o] : ( P3 @ one_one_Numeral_num1 ) ) ) ).
% 0.18/0.46
% 0.18/0.46 % ex_1
% 0.18/0.46
% 0.18/0.46 % Helper facts (9)
% 0.18/0.46 thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_T,axiom,
% 0.18/0.46 ! [X: finite964658038_int_n,Y: finite964658038_int_n] :
% 0.18/0.46 ( ( if_Fin1767949360_int_n @ $false @ X @ Y )
% 0.18/0.46 = Y ) ).
% 0.18/0.46
% 0.18/0.46 thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_T,axiom,
% 0.18/0.46 ! [X: finite964658038_int_n,Y: finite964658038_int_n] :
% 0.18/0.46 ( ( if_Fin1767949360_int_n @ $true @ X @ Y )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_T,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 0.18/0.46 ( ( if_Fin127821360real_n @ $false @ X @ Y )
% 0.18/0.46 = Y ) ).
% 0.18/0.46
% 0.18/0.46 thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_T,axiom,
% 0.18/0.46 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 0.18/0.46 ( ( if_Fin127821360real_n @ $true @ X @ Y )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J_T,axiom,
% 0.18/0.46 ! [X: finite2063899472l_num1,Y: finite2063899472l_num1] :
% 0.18/0.46 ( ( if_Fin1570437642l_num1 @ $false @ X @ Y )
% 0.18/0.46 = Y ) ).
% 0.18/0.46
% 0.18/0.46 thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J_T,axiom,
% 0.18/0.46 ! [X: finite2063899472l_num1,Y: finite2063899472l_num1] :
% 0.18/0.46 ( ( if_Fin1570437642l_num1 @ $true @ X @ Y )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 thf(help_If_3_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_T,axiom,
% 0.18/0.46 ! [P: $o] :
% 0.18/0.46 ( ( P = $true )
% 0.18/0.46 | ( P = $false ) ) ).
% 0.18/0.46
% 0.18/0.46 thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_T,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 0.18/0.46 ( ( if_Fin1668193290l_num1 @ $false @ X @ Y )
% 0.18/0.46 = Y ) ).
% 0.18/0.46
% 0.18/0.46 thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_T,axiom,
% 0.18/0.46 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 0.18/0.46 ( ( if_Fin1668193290l_num1 @ $true @ X @ Y )
% 0.18/0.46 = X ) ).
% 0.18/0.46
% 0.18/0.46 % Conjectures (1)
% 0.18/0.46 thf(conj_0,conjecture,
% 0.18/0.46 ( ( ring_1_of_int_real @ ( abs_abs_int @ m ) )
% 0.18/0.46 = ( abs_abs_real @ ( finite772340589real_n @ ( minus_1037315151real_n @ x @ y ) @ i ) ) ) ).
% 0.18/0.46
% 0.18/0.46 %------------------------------------------------------------------------------
% 0.18/0.46 ------------------------
% 2.88/1.58 ---- Embedded in THF ---
% 2.88/1.58 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.88/1.58 %%% Generated on Tue Feb 24 08:26:45 EST 2026
% 2.88/1.58 %%% using '$$dhol' embedding, version 1.3.0.
% 2.88/1.58 %%% Logic specification used:
% 2.88/1.58 %%% thf(spec, logic, $$dhol).
% 2.88/1.58
% 2.88/1.58 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.LSq3s3M90o/DTF2DTF11239.p
% See solution above
% 2.88/1.58 ------------------------
% 3.16/1.61 ---- Cleaned THF ---
% 3.16/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 finite463740460l_num1: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 finite121157254l_num1: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mtf__n_J,type,
% 3.16/1.61 finite337957458num1_n: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mtf__n_J,type,
% 3.16/1.61 finite318270636al_n_n: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_J,type,
% 3.16/1.61 set_Fi1257234438l_num1: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 finite1183840848l_num1: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 finite2063899472l_num1: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_J,type,
% 3.16/1.61 set_Fi1058188332real_n: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 finite1489363574real_n: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 3.16/1.61 finite964658038_int_n: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 3.16/1.61 set_real: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 3.16/1.61 set_int: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Numeral____Type__Onum1,type,
% 3.16/1.61 numeral_num1: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Real__Oreal,type,
% 3.16/1.61 real: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_t__Int__Oint,type,
% 3.16/1.61 int: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(ty_n_tf__n,type,
% 3.16/1.61 n: $tType ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 3.16/1.61 archim1371465213g_real: real > int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
% 3.16/1.61 archim1031974863r_real: real > int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Oaxis_001t__Numeral____Type__Onum1_001t__Int__Oint,type,
% 3.16/1.61 finite627713486m1_int: numeral_num1 > int > finite2063899472l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Oaxis_001t__Numeral____Type__Onum1_001t__Real__Oreal,type,
% 3.16/1.61 finite15144922381_real: numeral_num1 > real > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Oaxis_001tf__n_001t__Int__Oint,type,
% 3.16/1.61 finite1888777460_n_int: n > int > finite964658038_int_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Oaxis_001tf__n_001t__Real__Oreal,type,
% 3.16/1.61 finite703491444n_real: n > real > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_001t__Int__Oint_001t__Numeral____Type__Onum1,type,
% 3.16/1.61 finite1470207077l_num1: int > finite2063899472l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_001t__Int__Oint_001tf__n,type,
% 3.16/1.61 finite55766923_int_n: int > finite964658038_int_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_001t__Real__Oreal_001t__Numeral____Type__Onum1,type,
% 3.16/1.61 finite1329045861l_num1: real > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_001t__Real__Oreal_001tf__n,type,
% 3.16/1.61 finite667548811real_n: real > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_001t__Numeral____Type__Onum1,type,
% 3.16/1.61 finite1449546019l_num1: finite463740460l_num1 > numeral_num1 > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_001tf__n,type,
% 3.16/1.61 finite1639785033num1_n: finite337957458num1_n > n > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_001t__Numeral____Type__Onum1,type,
% 3.16/1.61 finite1503577981l_num1: finite121157254l_num1 > numeral_num1 > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_001tf__n,type,
% 3.16/1.61 finite625646243al_n_n: finite318270636al_n_n > n > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth_001t__Int__Oint_001t__Numeral____Type__Onum1,type,
% 3.16/1.61 finite1800583751l_num1: finite2063899472l_num1 > numeral_num1 > int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth_001t__Int__Oint_001tf__n,type,
% 3.16/1.61 finite566214509_int_n: finite964658038_int_n > n > int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth_001t__Real__Oreal_001t__Numeral____Type__Onum1,type,
% 3.16/1.61 finite1106245191l_num1: finite1183840848l_num1 > numeral_num1 > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovec_Ovec__nth_001t__Real__Oreal_001tf__n,type,
% 3.16/1.61 finite772340589real_n: finite1489363574real_n > n > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovector__scalar__mult_001t__Real__Oreal_001t__Numeral____Type__Onum1,type,
% 3.16/1.61 finite1961675766l_num1: real > finite1183840848l_num1 > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Finite__Cartesian__Product_Ovector__scalar__mult_001t__Real__Oreal_001tf__n,type,
% 3.16/1.61 finite2053577756real_n: real > finite1489363574real_n > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oabs__class_Oabs_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 abs_ab948591917l_num1: finite1183840848l_num1 > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oabs__class_Oabs_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 abs_ab1599551059real_n: finite1489363574real_n > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 3.16/1.61 abs_abs_int: int > int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 3.16/1.61 abs_abs_real: real > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 minus_779068585l_num1: finite2063899472l_num1 > finite2063899472l_num1 > finite2063899472l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 3.16/1.61 minus_1196255695_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 minus_249161513l_num1: finite1183840848l_num1 > finite1183840848l_num1 > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 minus_1037315151real_n: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 3.16/1.61 minus_minus_int: int > int > int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 3.16/1.61 minus_minus_real: real > real > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oone__class_Oone_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 one_on601734293l_num1: finite2063899472l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oone__class_Oone_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 3.16/1.61 one_on1516019131_int_n: finite964658038_int_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oone__class_Oone_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 one_on1961019669l_num1: finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oone__class_Oone_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 one_on1253059131real_n: finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
% 3.16/1.61 one_one_int: int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oone__class_Oone_001t__Numeral____Type__Onum1,type,
% 3.16/1.61 one_one_Numeral_num1: numeral_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 3.16/1.61 one_one_real: real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 if_Fin1570437642l_num1: $o > finite2063899472l_num1 > finite2063899472l_num1 > finite2063899472l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 3.16/1.61 if_Fin1767949360_int_n: $o > finite964658038_int_n > finite964658038_int_n > finite964658038_int_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 if_Fin1668193290l_num1: $o > finite1183840848l_num1 > finite1183840848l_num1 > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 if_Fin127821360real_n: $o > finite1489363574real_n > finite1489363574real_n > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Int_Oring__1__class_OInts_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 ring_11529266783l_num1: set_Fi1257234438l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Int_Oring__1__class_OInts_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 ring_1723899781real_n: set_Fi1058188332real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
% 3.16/1.61 ring_1_Ints_int: set_int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 3.16/1.61 ring_1_Ints_real: set_real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 ring_1154201247l_num1: int > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 ring_11737179077real_n: int > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 3.16/1.61 ring_1_of_int_int: int > int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 3.16/1.61 ring_1_of_int_real: int > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001t__Numeral____Type__Onum1_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 minkow781993583real_n: finite2063899472l_num1 > finite121157254l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001t__Numeral____Type__Onum1_001t__Int__Oint,type,
% 3.16/1.61 minkow2063319845m1_int: finite2063899472l_num1 > finite2063899472l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001t__Numeral____Type__Onum1_001t__Real__Oreal,type,
% 3.16/1.61 minkow19073400691_real: finite2063899472l_num1 > finite1183840848l_num1 ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001tf__n_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 minkow371489429real_n: finite964658038_int_n > finite318270636al_n_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001tf__n_001t__Int__Oint,type,
% 3.16/1.61 minkow1468704075_n_int: finite964658038_int_n > finite964658038_int_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Minkowskis__Theorem__Mirabelle__uzuvqgwfeb_Oof__int__vec_001tf__n_001t__Real__Oreal,type,
% 3.16/1.61 minkow1134813771n_real: finite964658038_int_n > finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 ord_le2035778724l_num1: finite1183840848l_num1 > finite1183840848l_num1 > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 ord_le1017090250real_n: finite1489363574real_n > finite1489363574real_n > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 3.16/1.61 ord_less_int: int > int > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 3.16/1.61 ord_less_real: real > real > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 ord_le1202326448l_num1: finite2063899472l_num1 > finite2063899472l_num1 > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J,type,
% 3.16/1.61 ord_le1908593622_int_n: finite964658038_int_n > finite964658038_int_n > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 ord_le1724070576l_num1: finite1183840848l_num1 > finite1183840848l_num1 > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 ord_le866583254real_n: finite1489363574real_n > finite1489363574real_n > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 3.16/1.61 ord_less_eq_int: int > int > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 3.16/1.61 ord_less_eq_real: real > real > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 real_V1974343547l_num1: finite463740460l_num1 > finite463740460l_num1 > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 real_V845814485l_num1: finite121157254l_num1 > finite121157254l_num1 > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 real_V488409631l_num1: finite1183840848l_num1 > finite1183840848l_num1 > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 real_V783965509real_n: finite1489363574real_n > finite1489363574real_n > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 3.16/1.61 real_V1934908667t_real: real > real > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 real_V316032763l_num1: finite463740460l_num1 > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_Mtf__n_J,type,
% 3.16/1.61 real_V1121707553num1_n: finite337957458num1_n > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 real_V400292255l_num1: finite1183840848l_num1 > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 real_V739724485real_n: finite1489363574real_n > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 3.16/1.61 real_V646646907m_real: real > real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Set_OCollect_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 collec321817931real_n: ( finite1489363574real_n > $o ) > set_Fi1058188332real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 3.16/1.61 collect_real: ( real > $o ) > set_real ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J,type,
% 3.16/1.61 member1413569767l_num1: finite1183840848l_num1 > set_Fi1257234438l_num1 > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J,type,
% 3.16/1.61 member1352538125real_n: finite1489363574real_n > set_Fi1058188332real_n > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_member_001t__Int__Oint,type,
% 3.16/1.61 member_int: int > set_int > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_c_member_001t__Real__Oreal,type,
% 3.16/1.61 member_real: real > set_real > $o ).
% 3.16/1.61
% 3.16/1.61 thf(sy_v_A,type,
% 3.16/1.61 a: set_Fi1058188332real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_v_i____,type,
% 3.16/1.61 i: n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_v_m____,type,
% 3.16/1.61 m: int ).
% 3.16/1.61
% 3.16/1.61 thf(sy_v_x____,type,
% 3.16/1.61 x: finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(sy_v_y____,type,
% 3.16/1.61 y: finite1489363574real_n ).
% 3.16/1.61
% 3.16/1.61 thf(fact_0__C1_C_I2_J,axiom,
% 3.16/1.61 member1352538125real_n @ y @ a ).
% 3.16/1.61
% 3.16/1.61 thf(fact_1__C1_C_I1_J,axiom,
% 3.16/1.61 member1352538125real_n @ x @ a ).
% 3.16/1.61
% 3.16/1.61 thf(fact_2_m,axiom,
% 3.16/1.61 ( ( ring_1_of_int_real @ m )
% 3.16/1.61 = ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_Areal__of__int_Am_A_061_Ax_A_E_Ai_A_N_Ay_A_E_Ai_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 3.16/1.61 ~ ! [M: int] :
% 3.16/1.61 ( ( ring_1_of_int_real @ M )
% 3.16/1.61 != ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_4_vector__minus__component,axiom,
% 3.16/1.61 ! [X: finite2063899472l_num1,Y: finite2063899472l_num1,I: numeral_num1] :
% 3.16/1.61 ( ( finite1800583751l_num1 @ ( minus_779068585l_num1 @ X @ Y ) @ I )
% 3.16/1.61 = ( minus_minus_int @ ( finite1800583751l_num1 @ X @ I ) @ ( finite1800583751l_num1 @ Y @ I ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_5_vector__minus__component,axiom,
% 3.16/1.61 ! [X: finite964658038_int_n,Y: finite964658038_int_n,I: n] :
% 3.16/1.61 ( ( finite566214509_int_n @ ( minus_1196255695_int_n @ X @ Y ) @ I )
% 3.16/1.61 = ( minus_minus_int @ ( finite566214509_int_n @ X @ I ) @ ( finite566214509_int_n @ Y @ I ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_6_vector__minus__component,axiom,
% 3.16/1.61 ! [X: finite1489363574real_n,Y: finite1489363574real_n,I: n] :
% 3.16/1.61 ( ( finite772340589real_n @ ( minus_1037315151real_n @ X @ Y ) @ I )
% 3.16/1.61 = ( minus_minus_real @ ( finite772340589real_n @ X @ I ) @ ( finite772340589real_n @ Y @ I ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_7_vector__minus__component,axiom,
% 3.16/1.61 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1,I: numeral_num1] :
% 3.16/1.61 ( ( finite1106245191l_num1 @ ( minus_249161513l_num1 @ X @ Y ) @ I )
% 3.16/1.61 = ( minus_minus_real @ ( finite1106245191l_num1 @ X @ I ) @ ( finite1106245191l_num1 @ Y @ I ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_8_of__int__abs,axiom,
% 3.16/1.61 ! [X: int] :
% 3.16/1.61 ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
% 3.16/1.61 = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_9_of__int__abs,axiom,
% 3.16/1.61 ! [X: int] :
% 3.16/1.61 ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
% 3.16/1.61 = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_10_of__int__diff,axiom,
% 3.16/1.61 ! [W: int,Z: int] :
% 3.16/1.61 ( ( ring_1154201247l_num1 @ ( minus_minus_int @ W @ Z ) )
% 3.16/1.61 = ( minus_249161513l_num1 @ ( ring_1154201247l_num1 @ W ) @ ( ring_1154201247l_num1 @ Z ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_11_of__int__diff,axiom,
% 3.16/1.61 ! [W: int,Z: int] :
% 3.16/1.61 ( ( ring_11737179077real_n @ ( minus_minus_int @ W @ Z ) )
% 3.16/1.61 = ( minus_1037315151real_n @ ( ring_11737179077real_n @ W ) @ ( ring_11737179077real_n @ Z ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_12_of__int__diff,axiom,
% 3.16/1.61 ! [W: int,Z: int] :
% 3.16/1.61 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 3.16/1.61 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_13_of__int__diff,axiom,
% 3.16/1.61 ! [W: int,Z: int] :
% 3.16/1.61 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 3.16/1.61 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_14__092_060open_062x_A_E_Ai_A_N_Ay_A_E_Ai_A_092_060in_062_A_092_060int_062_092_060close_062,axiom,
% 3.16/1.61 member_real @ ( minus_minus_real @ ( finite772340589real_n @ x @ i ) @ ( finite772340589real_n @ y @ i ) ) @ ring_1_Ints_real ).
% 3.16/1.61
% 3.16/1.61 thf(fact_15_of__int__vec__nth,axiom,
% 3.16/1.61 ! [V: finite2063899472l_num1,N: numeral_num1] :
% 3.16/1.61 ( ( finite1503577981l_num1 @ ( minkow781993583real_n @ V ) @ N )
% 3.16/1.61 = ( ring_11737179077real_n @ ( finite1800583751l_num1 @ V @ N ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_16_of__int__vec__nth,axiom,
% 3.16/1.61 ! [V: finite964658038_int_n,N: n] :
% 3.16/1.61 ( ( finite625646243al_n_n @ ( minkow371489429real_n @ V ) @ N )
% 3.16/1.61 = ( ring_11737179077real_n @ ( finite566214509_int_n @ V @ N ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_17_of__int__vec__nth,axiom,
% 3.16/1.61 ! [V: finite2063899472l_num1,N: numeral_num1] :
% 3.16/1.61 ( ( finite1800583751l_num1 @ ( minkow2063319845m1_int @ V ) @ N )
% 3.16/1.61 = ( ring_1_of_int_int @ ( finite1800583751l_num1 @ V @ N ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_18_of__int__vec__nth,axiom,
% 3.16/1.61 ! [V: finite964658038_int_n,N: n] :
% 3.16/1.61 ( ( finite566214509_int_n @ ( minkow1468704075_n_int @ V ) @ N )
% 3.16/1.61 = ( ring_1_of_int_int @ ( finite566214509_int_n @ V @ N ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_19_of__int__vec__nth,axiom,
% 3.16/1.61 ! [V: finite964658038_int_n,N: n] :
% 3.16/1.61 ( ( finite772340589real_n @ ( minkow1134813771n_real @ V ) @ N )
% 3.16/1.61 = ( ring_1_of_int_real @ ( finite566214509_int_n @ V @ N ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_20_of__int__vec__nth,axiom,
% 3.16/1.61 ! [V: finite2063899472l_num1,N: numeral_num1] :
% 3.16/1.61 ( ( finite1106245191l_num1 @ ( minkow19073400691_real @ V ) @ N )
% 3.16/1.61 = ( ring_1_of_int_real @ ( finite1800583751l_num1 @ V @ N ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_21_of__int__eq__iff,axiom,
% 3.16/1.61 ! [W: int,Z: int] :
% 3.16/1.61 ( ( ( ring_1_of_int_int @ W )
% 3.16/1.61 = ( ring_1_of_int_int @ Z ) )
% 3.16/1.61 = ( W = Z ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_22_of__int__eq__iff,axiom,
% 3.16/1.61 ! [W: int,Z: int] :
% 3.16/1.61 ( ( ( ring_11737179077real_n @ W )
% 3.16/1.61 = ( ring_11737179077real_n @ Z ) )
% 3.16/1.61 = ( W = Z ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_23_of__int__eq__iff,axiom,
% 3.16/1.61 ! [W: int,Z: int] :
% 3.16/1.61 ( ( ( ring_1_of_int_real @ W )
% 3.16/1.61 = ( ring_1_of_int_real @ Z ) )
% 3.16/1.61 = ( W = Z ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_24_abs__abs,axiom,
% 3.16/1.61 ! [A: int] :
% 3.16/1.61 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 3.16/1.61 = ( abs_abs_int @ A ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_25_abs__abs,axiom,
% 3.16/1.61 ! [A: real] :
% 3.16/1.61 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 3.16/1.61 = ( abs_abs_real @ A ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_26_abs__idempotent,axiom,
% 3.16/1.61 ! [A: int] :
% 3.16/1.61 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 3.16/1.61 = ( abs_abs_int @ A ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_27_abs__idempotent,axiom,
% 3.16/1.61 ! [A: real] :
% 3.16/1.61 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 3.16/1.61 = ( abs_abs_real @ A ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_28_abs__minus__commute,axiom,
% 3.16/1.61 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 3.16/1.61 ( ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ A @ B ) )
% 3.16/1.61 = ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ B @ A ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_29_abs__minus__commute,axiom,
% 3.16/1.61 ! [A: real,B: real] :
% 3.16/1.61 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 3.16/1.61 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_30_abs__minus__commute,axiom,
% 3.16/1.61 ! [A: int,B: int] :
% 3.16/1.61 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 3.16/1.61 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_31_abs__minus__commute,axiom,
% 3.16/1.61 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 3.16/1.61 ( ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ A @ B ) )
% 3.16/1.61 = ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ B @ A ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_32_lambda__skolem,axiom,
% 3.16/1.61 ! [P: n > real > $o] :
% 3.16/1.61 ( ( ! [I2: n] :
% 3.16/1.61 ? [X2: real] : ( P @ I2 @ X2 ) )
% 3.16/1.61 = ( ? [X3: finite1489363574real_n] :
% 3.16/1.61 ! [I2: n] : ( P @ I2 @ ( finite772340589real_n @ X3 @ I2 ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_33_lambda__skolem,axiom,
% 3.16/1.61 ! [P: numeral_num1 > real > $o] :
% 3.16/1.61 ( ( ! [I2: numeral_num1] :
% 3.16/1.61 ? [X2: real] : ( P @ I2 @ X2 ) )
% 3.16/1.61 = ( ? [X3: finite1183840848l_num1] :
% 3.16/1.61 ! [I2: numeral_num1] : ( P @ I2 @ ( finite1106245191l_num1 @ X3 @ I2 ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_34_lambda__skolem,axiom,
% 3.16/1.61 ! [P: numeral_num1 > int > $o] :
% 3.16/1.61 ( ( ! [I2: numeral_num1] :
% 3.16/1.61 ? [X2: int] : ( P @ I2 @ X2 ) )
% 3.16/1.61 = ( ? [X3: finite2063899472l_num1] :
% 3.16/1.61 ! [I2: numeral_num1] : ( P @ I2 @ ( finite1800583751l_num1 @ X3 @ I2 ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_35_lambda__skolem,axiom,
% 3.16/1.61 ! [P: n > int > $o] :
% 3.16/1.61 ( ( ! [I2: n] :
% 3.16/1.61 ? [X2: int] : ( P @ I2 @ X2 ) )
% 3.16/1.61 = ( ? [X3: finite964658038_int_n] :
% 3.16/1.61 ! [I2: n] : ( P @ I2 @ ( finite566214509_int_n @ X3 @ I2 ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_36_vec__eq__iff,axiom,
% 3.16/1.61 ( ( ^ [Y2: finite1489363574real_n,Z2: finite1489363574real_n] : ( Y2 = Z2 ) )
% 3.16/1.61 = ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] :
% 3.16/1.61 ! [I2: n] :
% 3.16/1.61 ( ( finite772340589real_n @ X3 @ I2 )
% 3.16/1.61 = ( finite772340589real_n @ Y3 @ I2 ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_37_vec__eq__iff,axiom,
% 3.16/1.61 ( ( ^ [Y2: finite1183840848l_num1,Z2: finite1183840848l_num1] : ( Y2 = Z2 ) )
% 3.16/1.61 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] :
% 3.16/1.61 ! [I2: numeral_num1] :
% 3.16/1.61 ( ( finite1106245191l_num1 @ X3 @ I2 )
% 3.16/1.61 = ( finite1106245191l_num1 @ Y3 @ I2 ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_38_vec__eq__iff,axiom,
% 3.16/1.61 ( ( ^ [Y2: finite2063899472l_num1,Z2: finite2063899472l_num1] : ( Y2 = Z2 ) )
% 3.16/1.61 = ( ^ [X3: finite2063899472l_num1,Y3: finite2063899472l_num1] :
% 3.16/1.61 ! [I2: numeral_num1] :
% 3.16/1.61 ( ( finite1800583751l_num1 @ X3 @ I2 )
% 3.16/1.61 = ( finite1800583751l_num1 @ Y3 @ I2 ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_39_vec__eq__iff,axiom,
% 3.16/1.61 ( ( ^ [Y2: finite964658038_int_n,Z2: finite964658038_int_n] : ( Y2 = Z2 ) )
% 3.16/1.61 = ( ^ [X3: finite964658038_int_n,Y3: finite964658038_int_n] :
% 3.16/1.61 ! [I2: n] :
% 3.16/1.61 ( ( finite566214509_int_n @ X3 @ I2 )
% 3.16/1.61 = ( finite566214509_int_n @ Y3 @ I2 ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_40_assms,axiom,
% 3.16/1.61 ! [V: finite1489363574real_n,I: n] :
% 3.16/1.61 ( ( member1352538125real_n @ V @ a )
% 3.16/1.61 => ( member_real @ ( finite772340589real_n @ V @ I ) @ ring_1_Ints_real ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_41_of__int__vec__eq__iff,axiom,
% 3.16/1.61 ! [A: finite2063899472l_num1,B: finite2063899472l_num1] :
% 3.16/1.61 ( ( ( minkow19073400691_real @ A )
% 3.16/1.61 = ( minkow19073400691_real @ B ) )
% 3.16/1.61 = ( A = B ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_42_of__int__vec__eq__iff,axiom,
% 3.16/1.61 ! [A: finite964658038_int_n,B: finite964658038_int_n] :
% 3.16/1.61 ( ( ( minkow1134813771n_real @ A )
% 3.16/1.61 = ( minkow1134813771n_real @ B ) )
% 3.16/1.61 = ( A = B ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_43_Ints__diff,axiom,
% 3.16/1.61 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 3.16/1.61 ( ( member1352538125real_n @ A @ ring_1723899781real_n )
% 3.16/1.61 => ( ( member1352538125real_n @ B @ ring_1723899781real_n )
% 3.16/1.61 => ( member1352538125real_n @ ( minus_1037315151real_n @ A @ B ) @ ring_1723899781real_n ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_44_Ints__diff,axiom,
% 3.16/1.61 ! [A: real,B: real] :
% 3.16/1.61 ( ( member_real @ A @ ring_1_Ints_real )
% 3.16/1.61 => ( ( member_real @ B @ ring_1_Ints_real )
% 3.16/1.61 => ( member_real @ ( minus_minus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_45_Ints__diff,axiom,
% 3.16/1.61 ! [A: int,B: int] :
% 3.16/1.61 ( ( member_int @ A @ ring_1_Ints_int )
% 3.16/1.61 => ( ( member_int @ B @ ring_1_Ints_int )
% 3.16/1.61 => ( member_int @ ( minus_minus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_46_Ints__diff,axiom,
% 3.16/1.61 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 3.16/1.61 ( ( member1413569767l_num1 @ A @ ring_11529266783l_num1 )
% 3.16/1.61 => ( ( member1413569767l_num1 @ B @ ring_11529266783l_num1 )
% 3.16/1.61 => ( member1413569767l_num1 @ ( minus_249161513l_num1 @ A @ B ) @ ring_11529266783l_num1 ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_47_Ints__abs,axiom,
% 3.16/1.61 ! [A: int] :
% 3.16/1.61 ( ( member_int @ A @ ring_1_Ints_int )
% 3.16/1.61 => ( member_int @ ( abs_abs_int @ A ) @ ring_1_Ints_int ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_48_Ints__abs,axiom,
% 3.16/1.61 ! [A: real] :
% 3.16/1.61 ( ( member_real @ A @ ring_1_Ints_real )
% 3.16/1.61 => ( member_real @ ( abs_abs_real @ A ) @ ring_1_Ints_real ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_49_Ints__of__int,axiom,
% 3.16/1.61 ! [Z: int] : ( member_real @ ( ring_1_of_int_real @ Z ) @ ring_1_Ints_real ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_50_Ints__of__int,axiom,
% 3.16/1.61 ! [Z: int] : ( member_int @ ( ring_1_of_int_int @ Z ) @ ring_1_Ints_int ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_51_Ints__of__int,axiom,
% 3.16/1.61 ! [Z: int] : ( member1352538125real_n @ ( ring_11737179077real_n @ Z ) @ ring_1723899781real_n ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_52_Ints__induct,axiom,
% 3.16/1.61 ! [Q: real,P: real > $o] :
% 3.16/1.61 ( ( member_real @ Q @ ring_1_Ints_real )
% 3.16/1.61 => ( ! [Z3: int] : ( P @ ( ring_1_of_int_real @ Z3 ) )
% 3.16/1.61 => ( P @ Q ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_53_Ints__induct,axiom,
% 3.16/1.61 ! [Q: int,P: int > $o] :
% 3.16/1.61 ( ( member_int @ Q @ ring_1_Ints_int )
% 3.16/1.61 => ( ! [Z3: int] : ( P @ ( ring_1_of_int_int @ Z3 ) )
% 3.16/1.61 => ( P @ Q ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_54_Ints__induct,axiom,
% 3.16/1.61 ! [Q: finite1489363574real_n,P: finite1489363574real_n > $o] :
% 3.16/1.61 ( ( member1352538125real_n @ Q @ ring_1723899781real_n )
% 3.16/1.61 => ( ! [Z3: int] : ( P @ ( ring_11737179077real_n @ Z3 ) )
% 3.16/1.61 => ( P @ Q ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_55_Ints__cases,axiom,
% 3.16/1.61 ! [Q: real] :
% 3.16/1.61 ( ( member_real @ Q @ ring_1_Ints_real )
% 3.16/1.61 => ~ ! [Z3: int] :
% 3.16/1.61 ( Q
% 3.16/1.61 != ( ring_1_of_int_real @ Z3 ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_56_Ints__cases,axiom,
% 3.16/1.61 ! [Q: int] :
% 3.16/1.61 ( ( member_int @ Q @ ring_1_Ints_int )
% 3.16/1.61 => ~ ! [Z3: int] :
% 3.16/1.61 ( Q
% 3.16/1.61 != ( ring_1_of_int_int @ Z3 ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_57_Ints__cases,axiom,
% 3.16/1.61 ! [Q: finite1489363574real_n] :
% 3.16/1.61 ( ( member1352538125real_n @ Q @ ring_1723899781real_n )
% 3.16/1.61 => ~ ! [Z3: int] :
% 3.16/1.61 ( Q
% 3.16/1.61 != ( ring_11737179077real_n @ Z3 ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_58_diff__right__commute,axiom,
% 3.16/1.61 ! [A: finite1489363574real_n,C: finite1489363574real_n,B: finite1489363574real_n] :
% 3.16/1.61 ( ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ C ) @ B )
% 3.16/1.61 = ( minus_1037315151real_n @ ( minus_1037315151real_n @ A @ B ) @ C ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_59_diff__right__commute,axiom,
% 3.16/1.61 ! [A: real,C: real,B: real] :
% 3.16/1.61 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 3.16/1.61 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_60_diff__right__commute,axiom,
% 3.16/1.61 ! [A: int,C: int,B: int] :
% 3.16/1.61 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 3.16/1.61 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_61_diff__right__commute,axiom,
% 3.16/1.61 ! [A: finite1183840848l_num1,C: finite1183840848l_num1,B: finite1183840848l_num1] :
% 3.16/1.61 ( ( minus_249161513l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ B )
% 3.16/1.61 = ( minus_249161513l_num1 @ ( minus_249161513l_num1 @ A @ B ) @ C ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_62_diff__eq__diff__eq,axiom,
% 3.16/1.61 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n,D: finite1489363574real_n] :
% 3.16/1.61 ( ( ( minus_1037315151real_n @ A @ B )
% 3.16/1.61 = ( minus_1037315151real_n @ C @ D ) )
% 3.16/1.61 => ( ( A = B )
% 3.16/1.61 = ( C = D ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_63_diff__eq__diff__eq,axiom,
% 3.16/1.61 ! [A: real,B: real,C: real,D: real] :
% 3.16/1.61 ( ( ( minus_minus_real @ A @ B )
% 3.16/1.61 = ( minus_minus_real @ C @ D ) )
% 3.16/1.61 => ( ( A = B )
% 3.16/1.61 = ( C = D ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_64_diff__eq__diff__eq,axiom,
% 3.16/1.61 ! [A: int,B: int,C: int,D: int] :
% 3.16/1.61 ( ( ( minus_minus_int @ A @ B )
% 3.16/1.61 = ( minus_minus_int @ C @ D ) )
% 3.16/1.61 => ( ( A = B )
% 3.16/1.61 = ( C = D ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_65_diff__eq__diff__eq,axiom,
% 3.16/1.61 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1,D: finite1183840848l_num1] :
% 3.16/1.61 ( ( ( minus_249161513l_num1 @ A @ B )
% 3.16/1.61 = ( minus_249161513l_num1 @ C @ D ) )
% 3.16/1.61 => ( ( A = B )
% 3.16/1.61 = ( C = D ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_66_vec__nth__inject,axiom,
% 3.16/1.61 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 3.16/1.61 ( ( ( finite772340589real_n @ X )
% 3.16/1.61 = ( finite772340589real_n @ Y ) )
% 3.16/1.61 = ( X = Y ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_67_vec__nth__inject,axiom,
% 3.16/1.61 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 3.16/1.61 ( ( ( finite1106245191l_num1 @ X )
% 3.16/1.61 = ( finite1106245191l_num1 @ Y ) )
% 3.16/1.61 = ( X = Y ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_68_vec__nth__inject,axiom,
% 3.16/1.61 ! [X: finite2063899472l_num1,Y: finite2063899472l_num1] :
% 3.16/1.61 ( ( ( finite1800583751l_num1 @ X )
% 3.16/1.61 = ( finite1800583751l_num1 @ Y ) )
% 3.16/1.61 = ( X = Y ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_69_vec__nth__inject,axiom,
% 3.16/1.61 ! [X: finite964658038_int_n,Y: finite964658038_int_n] :
% 3.16/1.61 ( ( ( finite566214509_int_n @ X )
% 3.16/1.61 = ( finite566214509_int_n @ Y ) )
% 3.16/1.61 = ( X = Y ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_70_cond__component,axiom,
% 3.16/1.61 ! [B: $o,X: finite1489363574real_n,Y: finite1489363574real_n,I: n] :
% 3.16/1.61 ( ( B
% 3.16/1.61 => ( ( finite772340589real_n @ ( if_Fin127821360real_n @ B @ X @ Y ) @ I )
% 3.16/1.61 = ( finite772340589real_n @ X @ I ) ) )
% 3.16/1.61 & ( ~ B
% 3.16/1.61 => ( ( finite772340589real_n @ ( if_Fin127821360real_n @ B @ X @ Y ) @ I )
% 3.16/1.61 = ( finite772340589real_n @ Y @ I ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_71_cond__component,axiom,
% 3.16/1.61 ! [B: $o,X: finite1183840848l_num1,Y: finite1183840848l_num1,I: numeral_num1] :
% 3.16/1.61 ( ( B
% 3.16/1.61 => ( ( finite1106245191l_num1 @ ( if_Fin1668193290l_num1 @ B @ X @ Y ) @ I )
% 3.16/1.61 = ( finite1106245191l_num1 @ X @ I ) ) )
% 3.16/1.61 & ( ~ B
% 3.16/1.61 => ( ( finite1106245191l_num1 @ ( if_Fin1668193290l_num1 @ B @ X @ Y ) @ I )
% 3.16/1.61 = ( finite1106245191l_num1 @ Y @ I ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_72_cond__component,axiom,
% 3.16/1.61 ! [B: $o,X: finite2063899472l_num1,Y: finite2063899472l_num1,I: numeral_num1] :
% 3.16/1.61 ( ( B
% 3.16/1.61 => ( ( finite1800583751l_num1 @ ( if_Fin1570437642l_num1 @ B @ X @ Y ) @ I )
% 3.16/1.61 = ( finite1800583751l_num1 @ X @ I ) ) )
% 3.16/1.61 & ( ~ B
% 3.16/1.61 => ( ( finite1800583751l_num1 @ ( if_Fin1570437642l_num1 @ B @ X @ Y ) @ I )
% 3.16/1.61 = ( finite1800583751l_num1 @ Y @ I ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_73_cond__component,axiom,
% 3.16/1.61 ! [B: $o,X: finite964658038_int_n,Y: finite964658038_int_n,I: n] :
% 3.16/1.61 ( ( B
% 3.16/1.61 => ( ( finite566214509_int_n @ ( if_Fin1767949360_int_n @ B @ X @ Y ) @ I )
% 3.16/1.61 = ( finite566214509_int_n @ X @ I ) ) )
% 3.16/1.61 & ( ~ B
% 3.16/1.61 => ( ( finite566214509_int_n @ ( if_Fin1767949360_int_n @ B @ X @ Y ) @ I )
% 3.16/1.61 = ( finite566214509_int_n @ Y @ I ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_74_dist__of__int,axiom,
% 3.16/1.61 ! [M2: int,N: int] :
% 3.16/1.61 ( ( real_V1934908667t_real @ ( ring_1_of_int_real @ M2 ) @ ( ring_1_of_int_real @ N ) )
% 3.16/1.61 = ( ring_1_of_int_real @ ( abs_abs_int @ ( minus_minus_int @ M2 @ N ) ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_75_vec__sub,axiom,
% 3.16/1.61 ! [X: real,Y: real] :
% 3.16/1.61 ( ( finite667548811real_n @ ( minus_minus_real @ X @ Y ) )
% 3.16/1.61 = ( minus_1037315151real_n @ ( finite667548811real_n @ X ) @ ( finite667548811real_n @ Y ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_76_vec__sub,axiom,
% 3.16/1.61 ! [X: real,Y: real] :
% 3.16/1.61 ( ( finite1329045861l_num1 @ ( minus_minus_real @ X @ Y ) )
% 3.16/1.61 = ( minus_249161513l_num1 @ ( finite1329045861l_num1 @ X ) @ ( finite1329045861l_num1 @ Y ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_77_vector__sub__rdistrib,axiom,
% 3.16/1.61 ! [A: real,B: real,X: finite1489363574real_n] :
% 3.16/1.61 ( ( finite2053577756real_n @ ( minus_minus_real @ A @ B ) @ X )
% 3.16/1.61 = ( minus_1037315151real_n @ ( finite2053577756real_n @ A @ X ) @ ( finite2053577756real_n @ B @ X ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_78_vector__sub__rdistrib,axiom,
% 3.16/1.61 ! [A: real,B: real,X: finite1183840848l_num1] :
% 3.16/1.61 ( ( finite1961675766l_num1 @ ( minus_minus_real @ A @ B ) @ X )
% 3.16/1.61 = ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ A @ X ) @ ( finite1961675766l_num1 @ B @ X ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_79__C1_C_I3_J,axiom,
% 3.16/1.61 ord_less_real @ ( real_V783965509real_n @ y @ x ) @ one_one_real ).
% 3.16/1.61
% 3.16/1.61 thf(fact_80_norm__of__int,axiom,
% 3.16/1.61 ! [Z: int] :
% 3.16/1.61 ( ( real_V646646907m_real @ ( ring_1_of_int_real @ Z ) )
% 3.16/1.61 = ( abs_abs_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_81_ceiling__diff__of__int,axiom,
% 3.16/1.61 ! [X: real,Z: int] :
% 3.16/1.61 ( ( archim1371465213g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 3.16/1.61 = ( minus_minus_int @ ( archim1371465213g_real @ X ) @ Z ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_82_axis__nth,axiom,
% 3.16/1.61 ! [I: n,X: real] :
% 3.16/1.61 ( ( finite772340589real_n @ ( finite703491444n_real @ I @ X ) @ I )
% 3.16/1.61 = X ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_83_axis__nth,axiom,
% 3.16/1.61 ! [I: numeral_num1,X: real] :
% 3.16/1.61 ( ( finite1106245191l_num1 @ ( finite15144922381_real @ I @ X ) @ I )
% 3.16/1.61 = X ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_84_axis__nth,axiom,
% 3.16/1.61 ! [I: numeral_num1,X: int] :
% 3.16/1.61 ( ( finite1800583751l_num1 @ ( finite627713486m1_int @ I @ X ) @ I )
% 3.16/1.61 = X ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_85_axis__nth,axiom,
% 3.16/1.61 ! [I: n,X: int] :
% 3.16/1.61 ( ( finite566214509_int_n @ ( finite1888777460_n_int @ I @ X ) @ I )
% 3.16/1.61 = X ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_86_vec_Oscale__left__diff__distrib,axiom,
% 3.16/1.61 ! [A: real,B: real,X: finite1489363574real_n] :
% 3.16/1.61 ( ( finite2053577756real_n @ ( minus_minus_real @ A @ B ) @ X )
% 3.16/1.61 = ( minus_1037315151real_n @ ( finite2053577756real_n @ A @ X ) @ ( finite2053577756real_n @ B @ X ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_87_vec_Oscale__left__diff__distrib,axiom,
% 3.16/1.61 ! [A: real,B: real,X: finite1183840848l_num1] :
% 3.16/1.61 ( ( finite1961675766l_num1 @ ( minus_minus_real @ A @ B ) @ X )
% 3.16/1.61 = ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ A @ X ) @ ( finite1961675766l_num1 @ B @ X ) ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_88_floor__diff__of__int,axiom,
% 3.16/1.61 ! [X: real,Z: int] :
% 3.16/1.61 ( ( archim1031974863r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
% 3.16/1.61 = ( minus_minus_int @ ( archim1031974863r_real @ X ) @ Z ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_89_norm__axis__1,axiom,
% 3.16/1.61 ! [M2: numeral_num1] :
% 3.16/1.61 ( ( real_V400292255l_num1 @ ( finite15144922381_real @ M2 @ one_one_real ) )
% 3.16/1.61 = one_one_real ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_90_abs__1,axiom,
% 3.16/1.61 ( ( abs_abs_real @ one_one_real )
% 3.16/1.61 = one_one_real ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_91_abs__1,axiom,
% 3.16/1.61 ( ( abs_abs_int @ one_one_int )
% 3.16/1.61 = one_one_int ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_92_mem__Collect__eq,axiom,
% 3.16/1.61 ! [A: finite1489363574real_n,P: finite1489363574real_n > $o] :
% 3.16/1.61 ( ( member1352538125real_n @ A @ ( collec321817931real_n @ P ) )
% 3.16/1.61 = ( P @ A ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_93_mem__Collect__eq,axiom,
% 3.16/1.61 ! [A: real,P: real > $o] :
% 3.16/1.61 ( ( member_real @ A @ ( collect_real @ P ) )
% 3.16/1.61 = ( P @ A ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_94_Collect__mem__eq,axiom,
% 3.16/1.61 ! [A2: set_Fi1058188332real_n] :
% 3.16/1.61 ( ( collec321817931real_n
% 3.16/1.61 @ ^ [X3: finite1489363574real_n] : ( member1352538125real_n @ X3 @ A2 ) )
% 3.16/1.61 = A2 ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_95_Collect__mem__eq,axiom,
% 3.16/1.61 ! [A2: set_real] :
% 3.16/1.61 ( ( collect_real
% 3.16/1.61 @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
% 3.16/1.61 = A2 ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_96_of__int__less__iff,axiom,
% 3.16/1.61 ! [W: int,Z: int] :
% 3.16/1.61 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 3.16/1.61 = ( ord_less_int @ W @ Z ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_97_of__int__less__iff,axiom,
% 3.16/1.61 ! [W: int,Z: int] :
% 3.16/1.61 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 3.16/1.61 = ( ord_less_int @ W @ Z ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_98_one__index,axiom,
% 3.16/1.61 ! [I: n] :
% 3.16/1.61 ( ( finite772340589real_n @ one_on1253059131real_n @ I )
% 3.16/1.61 = one_one_real ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_99_one__index,axiom,
% 3.16/1.61 ! [I: numeral_num1] :
% 3.16/1.61 ( ( finite1106245191l_num1 @ one_on1961019669l_num1 @ I )
% 3.16/1.61 = one_one_real ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_100_one__index,axiom,
% 3.16/1.61 ! [I: numeral_num1] :
% 3.16/1.61 ( ( finite1800583751l_num1 @ one_on601734293l_num1 @ I )
% 3.16/1.61 = one_one_int ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_101_one__index,axiom,
% 3.16/1.61 ! [I: n] :
% 3.16/1.61 ( ( finite566214509_int_n @ one_on1516019131_int_n @ I )
% 3.16/1.61 = one_one_int ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_102_of__int__eq__1__iff,axiom,
% 3.16/1.61 ! [Z: int] :
% 3.16/1.61 ( ( ( ring_1_of_int_real @ Z )
% 3.16/1.61 = one_one_real )
% 3.16/1.61 = ( Z = one_one_int ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_103_of__int__eq__1__iff,axiom,
% 3.16/1.61 ! [Z: int] :
% 3.16/1.61 ( ( ( ring_1_of_int_int @ Z )
% 3.16/1.61 = one_one_int )
% 3.16/1.61 = ( Z = one_one_int ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_104_of__int__eq__1__iff,axiom,
% 3.16/1.61 ! [Z: int] :
% 3.16/1.61 ( ( ( ring_11737179077real_n @ Z )
% 3.16/1.61 = one_on1253059131real_n )
% 3.16/1.61 = ( Z = one_one_int ) ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_105_of__int__1,axiom,
% 3.16/1.61 ( ( ring_1_of_int_real @ one_one_int )
% 3.16/1.61 = one_one_real ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_106_of__int__1,axiom,
% 3.16/1.61 ( ( ring_1_of_int_int @ one_one_int )
% 3.16/1.61 = one_one_int ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_107_of__int__1,axiom,
% 3.16/1.61 ( ( ring_11737179077real_n @ one_one_int )
% 3.16/1.61 = one_on1253059131real_n ) ).
% 3.16/1.61
% 3.16/1.61 thf(fact_108_abs__norm__cancel,axiom,
% 3.16/1.61 ! [A: real] :
% 3.16/1.61 ( ( abs_abs_real @ ( real_V646646907m_real @ A ) )
% 3.16/1.61 = ( real_V646646907m_real @ A ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_109_abs__norm__cancel,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1] :
% 3.16/1.62 ( ( abs_abs_real @ ( real_V400292255l_num1 @ A ) )
% 3.16/1.62 = ( real_V400292255l_num1 @ A ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_110_floor__one,axiom,
% 3.16/1.62 ( ( archim1031974863r_real @ one_one_real )
% 3.16/1.62 = one_one_int ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_111_ceiling__one,axiom,
% 3.16/1.62 ( ( archim1371465213g_real @ one_one_real )
% 3.16/1.62 = one_one_int ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_112_floor__of__int,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( archim1031974863r_real @ ( ring_1_of_int_real @ Z ) )
% 3.16/1.62 = Z ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_113_ceiling__of__int,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( archim1371465213g_real @ ( ring_1_of_int_real @ Z ) )
% 3.16/1.62 = Z ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_114_vec__component,axiom,
% 3.16/1.62 ! [X: real,I: n] :
% 3.16/1.62 ( ( finite772340589real_n @ ( finite667548811real_n @ X ) @ I )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_115_vec__component,axiom,
% 3.16/1.62 ! [X: real,I: numeral_num1] :
% 3.16/1.62 ( ( finite1106245191l_num1 @ ( finite1329045861l_num1 @ X ) @ I )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_116_vec__component,axiom,
% 3.16/1.62 ! [X: int,I: numeral_num1] :
% 3.16/1.62 ( ( finite1800583751l_num1 @ ( finite1470207077l_num1 @ X ) @ I )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_117_vec__component,axiom,
% 3.16/1.62 ! [X: int,I: n] :
% 3.16/1.62 ( ( finite566214509_int_n @ ( finite55766923_int_n @ X ) @ I )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_118_vector__ssub__ldistrib,axiom,
% 3.16/1.62 ! [C: real,X: finite1489363574real_n,Y: finite1489363574real_n] :
% 3.16/1.62 ( ( finite2053577756real_n @ C @ ( minus_1037315151real_n @ X @ Y ) )
% 3.16/1.62 = ( minus_1037315151real_n @ ( finite2053577756real_n @ C @ X ) @ ( finite2053577756real_n @ C @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_119_vector__ssub__ldistrib,axiom,
% 3.16/1.62 ! [C: real,X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 3.16/1.62 ( ( finite1961675766l_num1 @ C @ ( minus_249161513l_num1 @ X @ Y ) )
% 3.16/1.62 = ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ C @ X ) @ ( finite1961675766l_num1 @ C @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_120_of__int__1__less__iff,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 3.16/1.62 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_121_of__int__1__less__iff,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 3.16/1.62 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_122_of__int__less__1__iff,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 3.16/1.62 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_123_of__int__less__1__iff,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 3.16/1.62 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_124_norm__one,axiom,
% 3.16/1.62 ( ( real_V646646907m_real @ one_one_real )
% 3.16/1.62 = one_one_real ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_125_dist__diff_I2_J,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 3.16/1.62 ( ( real_V783965509real_n @ ( minus_1037315151real_n @ A @ B ) @ A )
% 3.16/1.62 = ( real_V739724485real_n @ B ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_126_dist__diff_I2_J,axiom,
% 3.16/1.62 ! [A: real,B: real] :
% 3.16/1.62 ( ( real_V1934908667t_real @ ( minus_minus_real @ A @ B ) @ A )
% 3.16/1.62 = ( real_V646646907m_real @ B ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_127_dist__diff_I2_J,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 3.16/1.62 ( ( real_V488409631l_num1 @ ( minus_249161513l_num1 @ A @ B ) @ A )
% 3.16/1.62 = ( real_V400292255l_num1 @ B ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_128_dist__diff_I1_J,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 3.16/1.62 ( ( real_V783965509real_n @ A @ ( minus_1037315151real_n @ A @ B ) )
% 3.16/1.62 = ( real_V739724485real_n @ B ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_129_dist__diff_I1_J,axiom,
% 3.16/1.62 ! [A: real,B: real] :
% 3.16/1.62 ( ( real_V1934908667t_real @ A @ ( minus_minus_real @ A @ B ) )
% 3.16/1.62 = ( real_V646646907m_real @ B ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_130_dist__diff_I1_J,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 3.16/1.62 ( ( real_V488409631l_num1 @ A @ ( minus_249161513l_num1 @ A @ B ) )
% 3.16/1.62 = ( real_V400292255l_num1 @ B ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_131_floor__less__one,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( ord_less_int @ ( archim1031974863r_real @ X ) @ one_one_int )
% 3.16/1.62 = ( ord_less_real @ X @ one_one_real ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_132_one__less__ceiling,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( ord_less_int @ one_one_int @ ( archim1371465213g_real @ X ) )
% 3.16/1.62 = ( ord_less_real @ one_one_real @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_133_floor__diff__one,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( archim1031974863r_real @ ( minus_minus_real @ X @ one_one_real ) )
% 3.16/1.62 = ( minus_minus_int @ ( archim1031974863r_real @ X ) @ one_one_int ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_134_ceiling__diff__one,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( archim1371465213g_real @ ( minus_minus_real @ X @ one_one_real ) )
% 3.16/1.62 = ( minus_minus_int @ ( archim1371465213g_real @ X ) @ one_one_int ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_135_one__reorient,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( one_one_real = X )
% 3.16/1.62 = ( X = one_one_real ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_136_one__reorient,axiom,
% 3.16/1.62 ! [X: int] :
% 3.16/1.62 ( ( one_one_int = X )
% 3.16/1.62 = ( X = one_one_int ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_137_one__reorient,axiom,
% 3.16/1.62 ! [X: numeral_num1] :
% 3.16/1.62 ( ( one_one_Numeral_num1 = X )
% 3.16/1.62 = ( X = one_one_Numeral_num1 ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_138_floor__less__iff,axiom,
% 3.16/1.62 ! [X: real,Z: int] :
% 3.16/1.62 ( ( ord_less_int @ ( archim1031974863r_real @ X ) @ Z )
% 3.16/1.62 = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_139_less__ceiling__iff,axiom,
% 3.16/1.62 ! [Z: int,X: real] :
% 3.16/1.62 ( ( ord_less_int @ Z @ ( archim1371465213g_real @ X ) )
% 3.16/1.62 = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_140_dist__norm,axiom,
% 3.16/1.62 ( real_V783965509real_n
% 3.16/1.62 = ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] : ( real_V739724485real_n @ ( minus_1037315151real_n @ X3 @ Y3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_141_dist__norm,axiom,
% 3.16/1.62 ( real_V1934908667t_real
% 3.16/1.62 = ( ^ [X3: real,Y3: real] : ( real_V646646907m_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_142_dist__norm,axiom,
% 3.16/1.62 ( real_V488409631l_num1
% 3.16/1.62 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ X3 @ Y3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_143_linorder__neqE__linordered__idom,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ( X != Y )
% 3.16/1.62 => ( ~ ( ord_less_real @ X @ Y )
% 3.16/1.62 => ( ord_less_real @ Y @ X ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_144_linorder__neqE__linordered__idom,axiom,
% 3.16/1.62 ! [X: int,Y: int] :
% 3.16/1.62 ( ( X != Y )
% 3.16/1.62 => ( ~ ( ord_less_int @ X @ Y )
% 3.16/1.62 => ( ord_less_int @ Y @ X ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_145_dist__commute__lessI,axiom,
% 3.16/1.62 ! [Y: finite1489363574real_n,X: finite1489363574real_n,E: real] :
% 3.16/1.62 ( ( ord_less_real @ ( real_V783965509real_n @ Y @ X ) @ E )
% 3.16/1.62 => ( ord_less_real @ ( real_V783965509real_n @ X @ Y ) @ E ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_146_dist__commute__lessI,axiom,
% 3.16/1.62 ! [Y: real,X: real,E: real] :
% 3.16/1.62 ( ( ord_less_real @ ( real_V1934908667t_real @ Y @ X ) @ E )
% 3.16/1.62 => ( ord_less_real @ ( real_V1934908667t_real @ X @ Y ) @ E ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_147_dist__commute__lessI,axiom,
% 3.16/1.62 ! [Y: finite1183840848l_num1,X: finite1183840848l_num1,E: real] :
% 3.16/1.62 ( ( ord_less_real @ ( real_V488409631l_num1 @ Y @ X ) @ E )
% 3.16/1.62 => ( ord_less_real @ ( real_V488409631l_num1 @ X @ Y ) @ E ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_148_dist__commute,axiom,
% 3.16/1.62 ( real_V783965509real_n
% 3.16/1.62 = ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] : ( real_V783965509real_n @ Y3 @ X3 ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_149_dist__commute,axiom,
% 3.16/1.62 ( real_V1934908667t_real
% 3.16/1.62 = ( ^ [X3: real,Y3: real] : ( real_V1934908667t_real @ Y3 @ X3 ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_150_dist__commute,axiom,
% 3.16/1.62 ( real_V488409631l_num1
% 3.16/1.62 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( real_V488409631l_num1 @ Y3 @ X3 ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_151_ceiling__less__cancel,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ( ord_less_int @ ( archim1371465213g_real @ X ) @ ( archim1371465213g_real @ Y ) )
% 3.16/1.62 => ( ord_less_real @ X @ Y ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_152_floor__less__cancel,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ( ord_less_int @ ( archim1031974863r_real @ X ) @ ( archim1031974863r_real @ Y ) )
% 3.16/1.62 => ( ord_less_real @ X @ Y ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_153_norm__minus__commute,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n] :
% 3.16/1.62 ( ( real_V739724485real_n @ ( minus_1037315151real_n @ A @ B ) )
% 3.16/1.62 = ( real_V739724485real_n @ ( minus_1037315151real_n @ B @ A ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_154_norm__minus__commute,axiom,
% 3.16/1.62 ! [A: real,B: real] :
% 3.16/1.62 ( ( real_V646646907m_real @ ( minus_minus_real @ A @ B ) )
% 3.16/1.62 = ( real_V646646907m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_155_norm__minus__commute,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] :
% 3.16/1.62 ( ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ A @ B ) )
% 3.16/1.62 = ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ B @ A ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_156_real__norm__def,axiom,
% 3.16/1.62 real_V646646907m_real = abs_abs_real ).
% 3.16/1.62
% 3.16/1.62 thf(fact_157_ex__less__of__int,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_158_ex__of__int__less,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_159_norm__bound__component__lt__cart,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,E: real,I: n] :
% 3.16/1.62 ( ( ord_less_real @ ( real_V739724485real_n @ X ) @ E )
% 3.16/1.62 => ( ord_less_real @ ( abs_abs_real @ ( finite772340589real_n @ X @ I ) ) @ E ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_160_norm__bound__component__lt__cart,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,E: real,I: numeral_num1] :
% 3.16/1.62 ( ( ord_less_real @ ( real_V400292255l_num1 @ X ) @ E )
% 3.16/1.62 => ( ord_less_real @ ( abs_abs_real @ ( finite1106245191l_num1 @ X @ I ) ) @ E ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_161_diff__strict__mono,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n,D: finite1489363574real_n,C: finite1489363574real_n] :
% 3.16/1.62 ( ( ord_le1017090250real_n @ A @ B )
% 3.16/1.62 => ( ( ord_le1017090250real_n @ D @ C )
% 3.16/1.62 => ( ord_le1017090250real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ D ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_162_diff__strict__mono,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,D: finite1183840848l_num1,C: finite1183840848l_num1] :
% 3.16/1.62 ( ( ord_le2035778724l_num1 @ A @ B )
% 3.16/1.62 => ( ( ord_le2035778724l_num1 @ D @ C )
% 3.16/1.62 => ( ord_le2035778724l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ D ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_163_diff__strict__mono,axiom,
% 3.16/1.62 ! [A: real,B: real,D: real,C: real] :
% 3.16/1.62 ( ( ord_less_real @ A @ B )
% 3.16/1.62 => ( ( ord_less_real @ D @ C )
% 3.16/1.62 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_164_diff__strict__mono,axiom,
% 3.16/1.62 ! [A: int,B: int,D: int,C: int] :
% 3.16/1.62 ( ( ord_less_int @ A @ B )
% 3.16/1.62 => ( ( ord_less_int @ D @ C )
% 3.16/1.62 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_165_diff__eq__diff__less,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n,D: finite1489363574real_n] :
% 3.16/1.62 ( ( ( minus_1037315151real_n @ A @ B )
% 3.16/1.62 = ( minus_1037315151real_n @ C @ D ) )
% 3.16/1.62 => ( ( ord_le1017090250real_n @ A @ B )
% 3.16/1.62 = ( ord_le1017090250real_n @ C @ D ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_166_diff__eq__diff__less,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1,D: finite1183840848l_num1] :
% 3.16/1.62 ( ( ( minus_249161513l_num1 @ A @ B )
% 3.16/1.62 = ( minus_249161513l_num1 @ C @ D ) )
% 3.16/1.62 => ( ( ord_le2035778724l_num1 @ A @ B )
% 3.16/1.62 = ( ord_le2035778724l_num1 @ C @ D ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_167_diff__eq__diff__less,axiom,
% 3.16/1.62 ! [A: real,B: real,C: real,D: real] :
% 3.16/1.62 ( ( ( minus_minus_real @ A @ B )
% 3.16/1.62 = ( minus_minus_real @ C @ D ) )
% 3.16/1.62 => ( ( ord_less_real @ A @ B )
% 3.16/1.62 = ( ord_less_real @ C @ D ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_168_diff__eq__diff__less,axiom,
% 3.16/1.62 ! [A: int,B: int,C: int,D: int] :
% 3.16/1.62 ( ( ( minus_minus_int @ A @ B )
% 3.16/1.62 = ( minus_minus_int @ C @ D ) )
% 3.16/1.62 => ( ( ord_less_int @ A @ B )
% 3.16/1.62 = ( ord_less_int @ C @ D ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_169_diff__strict__left__mono,axiom,
% 3.16/1.62 ! [B: finite1489363574real_n,A: finite1489363574real_n,C: finite1489363574real_n] :
% 3.16/1.62 ( ( ord_le1017090250real_n @ B @ A )
% 3.16/1.62 => ( ord_le1017090250real_n @ ( minus_1037315151real_n @ C @ A ) @ ( minus_1037315151real_n @ C @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_170_diff__strict__left__mono,axiom,
% 3.16/1.62 ! [B: finite1183840848l_num1,A: finite1183840848l_num1,C: finite1183840848l_num1] :
% 3.16/1.62 ( ( ord_le2035778724l_num1 @ B @ A )
% 3.16/1.62 => ( ord_le2035778724l_num1 @ ( minus_249161513l_num1 @ C @ A ) @ ( minus_249161513l_num1 @ C @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_171_diff__strict__left__mono,axiom,
% 3.16/1.62 ! [B: real,A: real,C: real] :
% 3.16/1.62 ( ( ord_less_real @ B @ A )
% 3.16/1.62 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_172_diff__strict__left__mono,axiom,
% 3.16/1.62 ! [B: int,A: int,C: int] :
% 3.16/1.62 ( ( ord_less_int @ B @ A )
% 3.16/1.62 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_173_diff__strict__right__mono,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] :
% 3.16/1.62 ( ( ord_le1017090250real_n @ A @ B )
% 3.16/1.62 => ( ord_le1017090250real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_174_diff__strict__right__mono,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1] :
% 3.16/1.62 ( ( ord_le2035778724l_num1 @ A @ B )
% 3.16/1.62 => ( ord_le2035778724l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_175_diff__strict__right__mono,axiom,
% 3.16/1.62 ! [A: real,B: real,C: real] :
% 3.16/1.62 ( ( ord_less_real @ A @ B )
% 3.16/1.62 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_176_diff__strict__right__mono,axiom,
% 3.16/1.62 ! [A: int,B: int,C: int] :
% 3.16/1.62 ( ( ord_less_int @ A @ B )
% 3.16/1.62 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_177_abs__one,axiom,
% 3.16/1.62 ( ( abs_abs_real @ one_one_real )
% 3.16/1.62 = one_one_real ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_178_abs__one,axiom,
% 3.16/1.62 ( ( abs_abs_int @ one_one_int )
% 3.16/1.62 = one_one_int ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_179_dist__real__def,axiom,
% 3.16/1.62 ( real_V1934908667t_real
% 3.16/1.62 = ( ^ [X3: real,Y3: real] : ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_180_Ints__1,axiom,
% 3.16/1.62 member1352538125real_n @ one_on1253059131real_n @ ring_1723899781real_n ).
% 3.16/1.62
% 3.16/1.62 thf(fact_181_Ints__1,axiom,
% 3.16/1.62 member_int @ one_one_int @ ring_1_Ints_int ).
% 3.16/1.62
% 3.16/1.62 thf(fact_182_Ints__1,axiom,
% 3.16/1.62 member_real @ one_one_real @ ring_1_Ints_real ).
% 3.16/1.62
% 3.16/1.62 thf(fact_183_vec_Oscale__right__diff__distrib,axiom,
% 3.16/1.62 ! [A: real,X: finite1489363574real_n,Y: finite1489363574real_n] :
% 3.16/1.62 ( ( finite2053577756real_n @ A @ ( minus_1037315151real_n @ X @ Y ) )
% 3.16/1.62 = ( minus_1037315151real_n @ ( finite2053577756real_n @ A @ X ) @ ( finite2053577756real_n @ A @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_184_vec_Oscale__right__diff__distrib,axiom,
% 3.16/1.62 ! [A: real,X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 3.16/1.62 ( ( finite1961675766l_num1 @ A @ ( minus_249161513l_num1 @ X @ Y ) )
% 3.16/1.62 = ( minus_249161513l_num1 @ ( finite1961675766l_num1 @ A @ X ) @ ( finite1961675766l_num1 @ A @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_185_Ints__eq__abs__less1,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ( member_real @ X @ ring_1_Ints_real )
% 3.16/1.62 => ( ( member_real @ Y @ ring_1_Ints_real )
% 3.16/1.62 => ( ( X = Y )
% 3.16/1.62 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) @ one_one_real ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_186_Ints__eq__abs__less1,axiom,
% 3.16/1.62 ! [X: int,Y: int] :
% 3.16/1.62 ( ( member_int @ X @ ring_1_Ints_int )
% 3.16/1.62 => ( ( member_int @ Y @ ring_1_Ints_int )
% 3.16/1.62 => ( ( X = Y )
% 3.16/1.62 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_187_of__int__ceiling__cancel,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) )
% 3.16/1.62 = X )
% 3.16/1.62 = ( ? [N2: int] :
% 3.16/1.62 ( X
% 3.16/1.62 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_188_of__int__floor__cancel,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( ( ring_1_of_int_real @ ( archim1031974863r_real @ X ) )
% 3.16/1.62 = X )
% 3.16/1.62 = ( ? [N2: int] :
% 3.16/1.62 ( X
% 3.16/1.62 = ( ring_1_of_int_real @ N2 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_189_real__of__int__floor__gt__diff__one,axiom,
% 3.16/1.62 ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim1031974863r_real @ R ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_190_real__abs__dist,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 3.16/1.62 ( ( abs_abs_real @ ( real_V783965509real_n @ X @ Y ) )
% 3.16/1.62 = ( real_V783965509real_n @ X @ Y ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_191_real__abs__dist,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ( abs_abs_real @ ( real_V1934908667t_real @ X @ Y ) )
% 3.16/1.62 = ( real_V1934908667t_real @ X @ Y ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_192_real__abs__dist,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 3.16/1.62 ( ( abs_abs_real @ ( real_V488409631l_num1 @ X @ Y ) )
% 3.16/1.62 = ( real_V488409631l_num1 @ X @ Y ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_193_less__numeral__extra_I4_J,axiom,
% 3.16/1.62 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_194_less__numeral__extra_I4_J,axiom,
% 3.16/1.62 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_195_le__ceiling__iff,axiom,
% 3.16/1.62 ! [Z: int,X: real] :
% 3.16/1.62 ( ( ord_less_eq_int @ Z @ ( archim1371465213g_real @ X ) )
% 3.16/1.62 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_196_ceiling__split,axiom,
% 3.16/1.62 ! [P: int > $o,T: real] :
% 3.16/1.62 ( ( P @ ( archim1371465213g_real @ T ) )
% 3.16/1.62 = ( ! [I2: int] :
% 3.16/1.62 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
% 3.16/1.62 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
% 3.16/1.62 => ( P @ I2 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_197_ceiling__eq__iff,axiom,
% 3.16/1.62 ! [X: real,A: int] :
% 3.16/1.62 ( ( ( archim1371465213g_real @ X )
% 3.16/1.62 = A )
% 3.16/1.62 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
% 3.16/1.62 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_198_of__int__le__iff,axiom,
% 3.16/1.62 ! [W: int,Z: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 3.16/1.62 = ( ord_less_eq_int @ W @ Z ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_199_of__int__le__iff,axiom,
% 3.16/1.62 ! [W: int,Z: int] :
% 3.16/1.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 3.16/1.62 = ( ord_less_eq_int @ W @ Z ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_200_zle__diff1__eq,axiom,
% 3.16/1.62 ! [W: int,Z: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 3.16/1.62 = ( ord_less_int @ W @ Z ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_201_of__int__le__1__iff,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 3.16/1.62 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_202_of__int__le__1__iff,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 3.16/1.62 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_203_of__int__1__le__iff,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 3.16/1.62 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_204_of__int__1__le__iff,axiom,
% 3.16/1.62 ! [Z: int] :
% 3.16/1.62 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 3.16/1.62 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_205_one__le__floor,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( ord_less_eq_int @ one_one_int @ ( archim1031974863r_real @ X ) )
% 3.16/1.62 = ( ord_less_eq_real @ one_one_real @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_206_ceiling__le__one,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ one_one_int )
% 3.16/1.62 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_207_ceiling__mono,axiom,
% 3.16/1.62 ! [Y: real,X: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ Y @ X )
% 3.16/1.62 => ( ord_less_eq_int @ ( archim1371465213g_real @ Y ) @ ( archim1371465213g_real @ X ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_208_floor__mono,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X @ Y )
% 3.16/1.62 => ( ord_less_eq_int @ ( archim1031974863r_real @ X ) @ ( archim1031974863r_real @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_209_less__eq__real__def,axiom,
% 3.16/1.62 ( ord_less_eq_real
% 3.16/1.62 = ( ^ [X3: real,Y3: real] :
% 3.16/1.62 ( ( ord_less_real @ X3 @ Y3 )
% 3.16/1.62 | ( X3 = Y3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_210_le__numeral__extra_I4_J,axiom,
% 3.16/1.62 ord_less_eq_int @ one_one_int @ one_one_int ).
% 3.16/1.62
% 3.16/1.62 thf(fact_211_le__numeral__extra_I4_J,axiom,
% 3.16/1.62 ord_less_eq_real @ one_one_real @ one_one_real ).
% 3.16/1.62
% 3.16/1.62 thf(fact_212_less__eq__vec__def,axiom,
% 3.16/1.62 ( ord_le866583254real_n
% 3.16/1.62 = ( ^ [X3: finite1489363574real_n,Y3: finite1489363574real_n] :
% 3.16/1.62 ! [I2: n] : ( ord_less_eq_real @ ( finite772340589real_n @ X3 @ I2 ) @ ( finite772340589real_n @ Y3 @ I2 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_213_less__eq__vec__def,axiom,
% 3.16/1.62 ( ord_le1724070576l_num1
% 3.16/1.62 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] :
% 3.16/1.62 ! [I2: numeral_num1] : ( ord_less_eq_real @ ( finite1106245191l_num1 @ X3 @ I2 ) @ ( finite1106245191l_num1 @ Y3 @ I2 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_214_less__eq__vec__def,axiom,
% 3.16/1.62 ( ord_le1202326448l_num1
% 3.16/1.62 = ( ^ [X3: finite2063899472l_num1,Y3: finite2063899472l_num1] :
% 3.16/1.62 ! [I2: numeral_num1] : ( ord_less_eq_int @ ( finite1800583751l_num1 @ X3 @ I2 ) @ ( finite1800583751l_num1 @ Y3 @ I2 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_215_less__eq__vec__def,axiom,
% 3.16/1.62 ( ord_le1908593622_int_n
% 3.16/1.62 = ( ^ [X3: finite964658038_int_n,Y3: finite964658038_int_n] :
% 3.16/1.62 ! [I2: n] : ( ord_less_eq_int @ ( finite566214509_int_n @ X3 @ I2 ) @ ( finite566214509_int_n @ Y3 @ I2 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_216_int__less__induct,axiom,
% 3.16/1.62 ! [I: int,K: int,P: int > $o] :
% 3.16/1.62 ( ( ord_less_int @ I @ K )
% 3.16/1.62 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 3.16/1.62 => ( ! [I3: int] :
% 3.16/1.62 ( ( ord_less_int @ I3 @ K )
% 3.16/1.62 => ( ( P @ I3 )
% 3.16/1.62 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 3.16/1.62 => ( P @ I ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_217_int__le__induct,axiom,
% 3.16/1.62 ! [I: int,K: int,P: int > $o] :
% 3.16/1.62 ( ( ord_less_eq_int @ I @ K )
% 3.16/1.62 => ( ( P @ K )
% 3.16/1.62 => ( ! [I3: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ I3 @ K )
% 3.16/1.62 => ( ( P @ I3 )
% 3.16/1.62 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 3.16/1.62 => ( P @ I ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_218_ceiling__le,axiom,
% 3.16/1.62 ! [X: real,A: int] :
% 3.16/1.62 ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
% 3.16/1.62 => ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ A ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_219_le__floor__iff,axiom,
% 3.16/1.62 ! [Z: int,X: real] :
% 3.16/1.62 ( ( ord_less_eq_int @ Z @ ( archim1031974863r_real @ X ) )
% 3.16/1.62 = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_220_ceiling__le__iff,axiom,
% 3.16/1.62 ! [X: real,Z: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ ( archim1371465213g_real @ X ) @ Z )
% 3.16/1.62 = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_221_diff__eq__diff__less__eq,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n,D: finite1489363574real_n] :
% 3.16/1.62 ( ( ( minus_1037315151real_n @ A @ B )
% 3.16/1.62 = ( minus_1037315151real_n @ C @ D ) )
% 3.16/1.62 => ( ( ord_le866583254real_n @ A @ B )
% 3.16/1.62 = ( ord_le866583254real_n @ C @ D ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_222_diff__eq__diff__less__eq,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1,D: finite1183840848l_num1] :
% 3.16/1.62 ( ( ( minus_249161513l_num1 @ A @ B )
% 3.16/1.62 = ( minus_249161513l_num1 @ C @ D ) )
% 3.16/1.62 => ( ( ord_le1724070576l_num1 @ A @ B )
% 3.16/1.62 = ( ord_le1724070576l_num1 @ C @ D ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_223_diff__eq__diff__less__eq,axiom,
% 3.16/1.62 ! [A: int,B: int,C: int,D: int] :
% 3.16/1.62 ( ( ( minus_minus_int @ A @ B )
% 3.16/1.62 = ( minus_minus_int @ C @ D ) )
% 3.16/1.62 => ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_224_diff__eq__diff__less__eq,axiom,
% 3.16/1.62 ! [A: real,B: real,C: real,D: real] :
% 3.16/1.62 ( ( ( minus_minus_real @ A @ B )
% 3.16/1.62 = ( minus_minus_real @ C @ D ) )
% 3.16/1.62 => ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_225_diff__right__mono,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] :
% 3.16/1.62 ( ( ord_le866583254real_n @ A @ B )
% 3.16/1.62 => ( ord_le866583254real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_226_diff__right__mono,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1] :
% 3.16/1.62 ( ( ord_le1724070576l_num1 @ A @ B )
% 3.16/1.62 => ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_227_diff__right__mono,axiom,
% 3.16/1.62 ! [A: int,B: int,C: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_228_diff__right__mono,axiom,
% 3.16/1.62 ! [A: real,B: real,C: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_229_diff__left__mono,axiom,
% 3.16/1.62 ! [B: finite1489363574real_n,A: finite1489363574real_n,C: finite1489363574real_n] :
% 3.16/1.62 ( ( ord_le866583254real_n @ B @ A )
% 3.16/1.62 => ( ord_le866583254real_n @ ( minus_1037315151real_n @ C @ A ) @ ( minus_1037315151real_n @ C @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_230_diff__left__mono,axiom,
% 3.16/1.62 ! [B: finite1183840848l_num1,A: finite1183840848l_num1,C: finite1183840848l_num1] :
% 3.16/1.62 ( ( ord_le1724070576l_num1 @ B @ A )
% 3.16/1.62 => ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ C @ A ) @ ( minus_249161513l_num1 @ C @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_231_diff__left__mono,axiom,
% 3.16/1.62 ! [B: int,A: int,C: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ B @ A )
% 3.16/1.62 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_232_diff__left__mono,axiom,
% 3.16/1.62 ! [B: real,A: real,C: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ B @ A )
% 3.16/1.62 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_233_diff__mono,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n,D: finite1489363574real_n,C: finite1489363574real_n] :
% 3.16/1.62 ( ( ord_le866583254real_n @ A @ B )
% 3.16/1.62 => ( ( ord_le866583254real_n @ D @ C )
% 3.16/1.62 => ( ord_le866583254real_n @ ( minus_1037315151real_n @ A @ C ) @ ( minus_1037315151real_n @ B @ D ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_234_diff__mono,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,D: finite1183840848l_num1,C: finite1183840848l_num1] :
% 3.16/1.62 ( ( ord_le1724070576l_num1 @ A @ B )
% 3.16/1.62 => ( ( ord_le1724070576l_num1 @ D @ C )
% 3.16/1.62 => ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ A @ C ) @ ( minus_249161513l_num1 @ B @ D ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_235_diff__mono,axiom,
% 3.16/1.62 ! [A: int,B: int,D: int,C: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 => ( ( ord_less_eq_int @ D @ C )
% 3.16/1.62 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_236_diff__mono,axiom,
% 3.16/1.62 ! [A: real,B: real,D: real,C: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 => ( ( ord_less_eq_real @ D @ C )
% 3.16/1.62 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_237_abs__ge__self,axiom,
% 3.16/1.62 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_238_abs__ge__self,axiom,
% 3.16/1.62 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_239_abs__le__D1,axiom,
% 3.16/1.62 ! [A: int,B: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 3.16/1.62 => ( ord_less_eq_int @ A @ B ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_240_abs__le__D1,axiom,
% 3.16/1.62 ! [A: real,B: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 3.16/1.62 => ( ord_less_eq_real @ A @ B ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_241_norm__vector__1,axiom,
% 3.16/1.62 ( real_V316032763l_num1
% 3.16/1.62 = ( ^ [X3: finite463740460l_num1] : ( real_V400292255l_num1 @ ( finite1449546019l_num1 @ X3 @ one_one_Numeral_num1 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_242_norm__vector__1,axiom,
% 3.16/1.62 ( real_V400292255l_num1
% 3.16/1.62 = ( ^ [X3: finite1183840848l_num1] : ( real_V646646907m_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_243_ex__le__of__int,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_244_dist__vector__1,axiom,
% 3.16/1.62 ( real_V845814485l_num1
% 3.16/1.62 = ( ^ [X3: finite121157254l_num1,Y3: finite121157254l_num1] : ( real_V783965509real_n @ ( finite1503577981l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1503577981l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_245_dist__vector__1,axiom,
% 3.16/1.62 ( real_V1974343547l_num1
% 3.16/1.62 = ( ^ [X3: finite463740460l_num1,Y3: finite463740460l_num1] : ( real_V488409631l_num1 @ ( finite1449546019l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1449546019l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_246_dist__vector__1,axiom,
% 3.16/1.62 ( real_V488409631l_num1
% 3.16/1.62 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( real_V1934908667t_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1106245191l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_247_norm__real,axiom,
% 3.16/1.62 ( real_V400292255l_num1
% 3.16/1.62 = ( ^ [X3: finite1183840848l_num1] : ( abs_abs_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_248_ceiling__diff__floor__le__1,axiom,
% 3.16/1.62 ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim1371465213g_real @ X ) @ ( archim1031974863r_real @ X ) ) @ one_one_int ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_249_ceiling__less__iff,axiom,
% 3.16/1.62 ! [X: real,Z: int] :
% 3.16/1.62 ( ( ord_less_int @ ( archim1371465213g_real @ X ) @ Z )
% 3.16/1.62 = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_250_abs__triangle__ineq2__sym,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_le866583254real_n @ ( minus_1037315151real_n @ ( abs_ab1599551059real_n @ A ) @ ( abs_ab1599551059real_n @ B ) ) @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ B @ A ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_251_abs__triangle__ineq2__sym,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ ( abs_ab948591917l_num1 @ A ) @ ( abs_ab948591917l_num1 @ B ) ) @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ B @ A ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_252_abs__triangle__ineq2__sym,axiom,
% 3.16/1.62 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_253_abs__triangle__ineq2__sym,axiom,
% 3.16/1.62 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_254_abs__triangle__ineq3,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_le866583254real_n @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ ( abs_ab1599551059real_n @ A ) @ ( abs_ab1599551059real_n @ B ) ) ) @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_255_abs__triangle__ineq3,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_le1724070576l_num1 @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ ( abs_ab948591917l_num1 @ A ) @ ( abs_ab948591917l_num1 @ B ) ) ) @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_256_abs__triangle__ineq3,axiom,
% 3.16/1.62 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_257_abs__triangle__ineq3,axiom,
% 3.16/1.62 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_258_abs__triangle__ineq2,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_le866583254real_n @ ( minus_1037315151real_n @ ( abs_ab1599551059real_n @ A ) @ ( abs_ab1599551059real_n @ B ) ) @ ( abs_ab1599551059real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_259_abs__triangle__ineq2,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_le1724070576l_num1 @ ( minus_249161513l_num1 @ ( abs_ab948591917l_num1 @ A ) @ ( abs_ab948591917l_num1 @ B ) ) @ ( abs_ab948591917l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_260_abs__triangle__ineq2,axiom,
% 3.16/1.62 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_261_abs__triangle__ineq2,axiom,
% 3.16/1.62 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_262_norm__le__componentwise__cart,axiom,
% 3.16/1.62 ! [X: finite337957458num1_n,Y: finite1489363574real_n] :
% 3.16/1.62 ( ! [I3: n] : ( ord_less_eq_real @ ( real_V400292255l_num1 @ ( finite1639785033num1_n @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite772340589real_n @ Y @ I3 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ ( real_V1121707553num1_n @ X ) @ ( real_V739724485real_n @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_263_norm__le__componentwise__cart,axiom,
% 3.16/1.62 ! [X: finite463740460l_num1,Y: finite1183840848l_num1] :
% 3.16/1.62 ( ! [I3: numeral_num1] : ( ord_less_eq_real @ ( real_V400292255l_num1 @ ( finite1449546019l_num1 @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ Y @ I3 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ ( real_V316032763l_num1 @ X ) @ ( real_V400292255l_num1 @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_264_norm__le__componentwise__cart,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,Y: finite337957458num1_n] :
% 3.16/1.62 ( ! [I3: n] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite772340589real_n @ X @ I3 ) ) @ ( real_V400292255l_num1 @ ( finite1639785033num1_n @ Y @ I3 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ ( real_V739724485real_n @ X ) @ ( real_V1121707553num1_n @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_265_norm__le__componentwise__cart,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 3.16/1.62 ( ! [I3: n] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite772340589real_n @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite772340589real_n @ Y @ I3 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ ( real_V739724485real_n @ X ) @ ( real_V739724485real_n @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_266_norm__le__componentwise__cart,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,Y: finite463740460l_num1] :
% 3.16/1.62 ( ! [I3: numeral_num1] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ X @ I3 ) ) @ ( real_V400292255l_num1 @ ( finite1449546019l_num1 @ Y @ I3 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ ( real_V400292255l_num1 @ X ) @ ( real_V316032763l_num1 @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_267_norm__le__componentwise__cart,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 3.16/1.62 ( ! [I3: numeral_num1] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ X @ I3 ) ) @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ Y @ I3 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ ( real_V400292255l_num1 @ X ) @ ( real_V400292255l_num1 @ Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_268_Finite__Cartesian__Product_Onorm__nth__le,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,I: n] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite772340589real_n @ X @ I ) ) @ ( real_V739724485real_n @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_269_Finite__Cartesian__Product_Onorm__nth__le,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,I: numeral_num1] : ( ord_less_eq_real @ ( real_V646646907m_real @ ( finite1106245191l_num1 @ X @ I ) ) @ ( real_V400292255l_num1 @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_270_real__of__int__floor__ge__diff__one,axiom,
% 3.16/1.62 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim1031974863r_real @ R ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_271_of__int__floor__le,axiom,
% 3.16/1.62 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim1031974863r_real @ X ) ) @ X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_272_le__of__int__ceiling,axiom,
% 3.16/1.62 ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_273_norm__bound__component__le__cart,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,E: real,I: n] :
% 3.16/1.62 ( ( ord_less_eq_real @ ( real_V739724485real_n @ X ) @ E )
% 3.16/1.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( finite772340589real_n @ X @ I ) ) @ E ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_274_norm__bound__component__le__cart,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,E: real,I: numeral_num1] :
% 3.16/1.62 ( ( ord_less_eq_real @ ( real_V400292255l_num1 @ X ) @ E )
% 3.16/1.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( finite1106245191l_num1 @ X @ I ) ) @ E ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_275_component__le__norm__cart,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,I: n] : ( ord_less_eq_real @ ( abs_abs_real @ ( finite772340589real_n @ X @ I ) ) @ ( real_V739724485real_n @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_276_component__le__norm__cart,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,I: numeral_num1] : ( ord_less_eq_real @ ( abs_abs_real @ ( finite1106245191l_num1 @ X @ I ) ) @ ( real_V400292255l_num1 @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_277_dist__real,axiom,
% 3.16/1.62 ( real_V488409631l_num1
% 3.16/1.62 = ( ^ [X3: finite1183840848l_num1,Y3: finite1183840848l_num1] : ( abs_abs_real @ ( minus_minus_real @ ( finite1106245191l_num1 @ X3 @ one_one_Numeral_num1 ) @ ( finite1106245191l_num1 @ Y3 @ one_one_Numeral_num1 ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_278_floor__le__ceiling,axiom,
% 3.16/1.62 ! [X: real] : ( ord_less_eq_int @ ( archim1031974863r_real @ X ) @ ( archim1371465213g_real @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_279_of__int__ceiling__diff__one__le,axiom,
% 3.16/1.62 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim1371465213g_real @ R ) ) @ one_one_real ) @ R ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_280_norm__triangle__ineq2,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V739724485real_n @ A ) @ ( real_V739724485real_n @ B ) ) @ ( real_V739724485real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_281_norm__triangle__ineq2,axiom,
% 3.16/1.62 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V646646907m_real @ A ) @ ( real_V646646907m_real @ B ) ) @ ( real_V646646907m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_282_norm__triangle__ineq2,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V400292255l_num1 @ A ) @ ( real_V400292255l_num1 @ B ) ) @ ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_283_abs__dist__diff__le,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n,C: finite1489363574real_n] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V783965509real_n @ A @ B ) @ ( real_V783965509real_n @ B @ C ) ) ) @ ( real_V783965509real_n @ A @ C ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_284_abs__dist__diff__le,axiom,
% 3.16/1.62 ! [A: real,B: real,C: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1934908667t_real @ A @ B ) @ ( real_V1934908667t_real @ B @ C ) ) ) @ ( real_V1934908667t_real @ A @ C ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_285_abs__dist__diff__le,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1,C: finite1183840848l_num1] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V488409631l_num1 @ A @ B ) @ ( real_V488409631l_num1 @ B @ C ) ) ) @ ( real_V488409631l_num1 @ A @ C ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_286_metric__eq__thm,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,S: set_Fi1058188332real_n,Y: finite1489363574real_n] :
% 3.16/1.62 ( ( member1352538125real_n @ X @ S )
% 3.16/1.62 => ( ( member1352538125real_n @ Y @ S )
% 3.16/1.62 => ( ( X = Y )
% 3.16/1.62 = ( ! [X3: finite1489363574real_n] :
% 3.16/1.62 ( ( member1352538125real_n @ X3 @ S )
% 3.16/1.62 => ( ( real_V783965509real_n @ X @ X3 )
% 3.16/1.62 = ( real_V783965509real_n @ Y @ X3 ) ) ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_287_metric__eq__thm,axiom,
% 3.16/1.62 ! [X: real,S: set_real,Y: real] :
% 3.16/1.62 ( ( member_real @ X @ S )
% 3.16/1.62 => ( ( member_real @ Y @ S )
% 3.16/1.62 => ( ( X = Y )
% 3.16/1.62 = ( ! [X3: real] :
% 3.16/1.62 ( ( member_real @ X3 @ S )
% 3.16/1.62 => ( ( real_V1934908667t_real @ X @ X3 )
% 3.16/1.62 = ( real_V1934908667t_real @ Y @ X3 ) ) ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_288_metric__eq__thm,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,S: set_Fi1257234438l_num1,Y: finite1183840848l_num1] :
% 3.16/1.62 ( ( member1413569767l_num1 @ X @ S )
% 3.16/1.62 => ( ( member1413569767l_num1 @ Y @ S )
% 3.16/1.62 => ( ( X = Y )
% 3.16/1.62 = ( ! [X3: finite1183840848l_num1] :
% 3.16/1.62 ( ( member1413569767l_num1 @ X3 @ S )
% 3.16/1.62 => ( ( real_V488409631l_num1 @ X @ X3 )
% 3.16/1.62 = ( real_V488409631l_num1 @ Y @ X3 ) ) ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_289_dist__vec__nth__le,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,I: n,Y: finite1489363574real_n] : ( ord_less_eq_real @ ( real_V1934908667t_real @ ( finite772340589real_n @ X @ I ) @ ( finite772340589real_n @ Y @ I ) ) @ ( real_V783965509real_n @ X @ Y ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_290_dist__vec__nth__le,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,I: numeral_num1,Y: finite1183840848l_num1] : ( ord_less_eq_real @ ( real_V1934908667t_real @ ( finite1106245191l_num1 @ X @ I ) @ ( finite1106245191l_num1 @ Y @ I ) ) @ ( real_V488409631l_num1 @ X @ Y ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_291_norm__triangle__ineq3,axiom,
% 3.16/1.62 ! [A: finite1489363574real_n,B: finite1489363574real_n] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V739724485real_n @ A ) @ ( real_V739724485real_n @ B ) ) ) @ ( real_V739724485real_n @ ( minus_1037315151real_n @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_292_norm__triangle__ineq3,axiom,
% 3.16/1.62 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V646646907m_real @ A ) @ ( real_V646646907m_real @ B ) ) ) @ ( real_V646646907m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_293_norm__triangle__ineq3,axiom,
% 3.16/1.62 ! [A: finite1183840848l_num1,B: finite1183840848l_num1] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V400292255l_num1 @ A ) @ ( real_V400292255l_num1 @ B ) ) ) @ ( real_V400292255l_num1 @ ( minus_249161513l_num1 @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_294_ceiling__correct,axiom,
% 3.16/1.62 ! [X: real] :
% 3.16/1.62 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) @ one_one_real ) @ X )
% 3.16/1.62 & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim1371465213g_real @ X ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_295_ceiling__unique,axiom,
% 3.16/1.62 ! [Z: int,X: real] :
% 3.16/1.62 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
% 3.16/1.62 => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
% 3.16/1.62 => ( ( archim1371465213g_real @ X )
% 3.16/1.62 = Z ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_296_order__refl,axiom,
% 3.16/1.62 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_297_order__refl,axiom,
% 3.16/1.62 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_298_complete__interval,axiom,
% 3.16/1.62 ! [A: int,B: int,P: int > $o] :
% 3.16/1.62 ( ( ord_less_int @ A @ B )
% 3.16/1.62 => ( ( P @ A )
% 3.16/1.62 => ( ~ ( P @ B )
% 3.16/1.62 => ? [C2: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ A @ C2 )
% 3.16/1.62 & ( ord_less_eq_int @ C2 @ B )
% 3.16/1.62 & ! [X4: int] :
% 3.16/1.62 ( ( ( ord_less_eq_int @ A @ X4 )
% 3.16/1.62 & ( ord_less_int @ X4 @ C2 ) )
% 3.16/1.62 => ( P @ X4 ) )
% 3.16/1.62 & ! [D2: int] :
% 3.16/1.62 ( ! [X5: int] :
% 3.16/1.62 ( ( ( ord_less_eq_int @ A @ X5 )
% 3.16/1.62 & ( ord_less_int @ X5 @ D2 ) )
% 3.16/1.62 => ( P @ X5 ) )
% 3.16/1.62 => ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_299_complete__interval,axiom,
% 3.16/1.62 ! [A: real,B: real,P: real > $o] :
% 3.16/1.62 ( ( ord_less_real @ A @ B )
% 3.16/1.62 => ( ( P @ A )
% 3.16/1.62 => ( ~ ( P @ B )
% 3.16/1.62 => ? [C2: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ A @ C2 )
% 3.16/1.62 & ( ord_less_eq_real @ C2 @ B )
% 3.16/1.62 & ! [X4: real] :
% 3.16/1.62 ( ( ( ord_less_eq_real @ A @ X4 )
% 3.16/1.62 & ( ord_less_real @ X4 @ C2 ) )
% 3.16/1.62 => ( P @ X4 ) )
% 3.16/1.62 & ! [D2: real] :
% 3.16/1.62 ( ! [X5: real] :
% 3.16/1.62 ( ( ( ord_less_eq_real @ A @ X5 )
% 3.16/1.62 & ( ord_less_real @ X5 @ D2 ) )
% 3.16/1.62 => ( P @ X5 ) )
% 3.16/1.62 => ( ord_less_eq_real @ D2 @ C2 ) ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_300_eucl__less__le__not__le,axiom,
% 3.16/1.62 ( ord_less_real
% 3.16/1.62 = ( ^ [X3: real,Y3: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X3 @ Y3 )
% 3.16/1.62 & ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_301_order_Onot__eq__order__implies__strict,axiom,
% 3.16/1.62 ! [A: int,B: int] :
% 3.16/1.62 ( ( A != B )
% 3.16/1.62 => ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 => ( ord_less_int @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_302_order_Onot__eq__order__implies__strict,axiom,
% 3.16/1.62 ! [A: real,B: real] :
% 3.16/1.62 ( ( A != B )
% 3.16/1.62 => ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 => ( ord_less_real @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_303_dual__order_Ostrict__implies__order,axiom,
% 3.16/1.62 ! [B: int,A: int] :
% 3.16/1.62 ( ( ord_less_int @ B @ A )
% 3.16/1.62 => ( ord_less_eq_int @ B @ A ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_304_dual__order_Ostrict__implies__order,axiom,
% 3.16/1.62 ! [B: real,A: real] :
% 3.16/1.62 ( ( ord_less_real @ B @ A )
% 3.16/1.62 => ( ord_less_eq_real @ B @ A ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_305_vector__one__nth,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1] :
% 3.16/1.62 ( ( finite1329045861l_num1 @ ( finite1106245191l_num1 @ X @ one_one_Numeral_num1 ) )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_306_vector__one__nth,axiom,
% 3.16/1.62 ! [X: finite2063899472l_num1] :
% 3.16/1.62 ( ( finite1470207077l_num1 @ ( finite1800583751l_num1 @ X @ one_one_Numeral_num1 ) )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_307_complete__real,axiom,
% 3.16/1.62 ! [S2: set_real] :
% 3.16/1.62 ( ? [X4: real] : ( member_real @ X4 @ S2 )
% 3.16/1.62 => ( ? [Z4: real] :
% 3.16/1.62 ! [X5: real] :
% 3.16/1.62 ( ( member_real @ X5 @ S2 )
% 3.16/1.62 => ( ord_less_eq_real @ X5 @ Z4 ) )
% 3.16/1.62 => ? [Y4: real] :
% 3.16/1.62 ( ! [X4: real] :
% 3.16/1.62 ( ( member_real @ X4 @ S2 )
% 3.16/1.62 => ( ord_less_eq_real @ X4 @ Y4 ) )
% 3.16/1.62 & ! [Z4: real] :
% 3.16/1.62 ( ! [X5: real] :
% 3.16/1.62 ( ( member_real @ X5 @ S2 )
% 3.16/1.62 => ( ord_less_eq_real @ X5 @ Z4 ) )
% 3.16/1.62 => ( ord_less_eq_real @ Y4 @ Z4 ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_308_dual__order_Oantisym,axiom,
% 3.16/1.62 ! [B: int,A: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ B @ A )
% 3.16/1.62 => ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 => ( A = B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_309_dual__order_Oantisym,axiom,
% 3.16/1.62 ! [B: real,A: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ B @ A )
% 3.16/1.62 => ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 => ( A = B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_310_dual__order_Oeq__iff,axiom,
% 3.16/1.62 ( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
% 3.16/1.62 = ( ^ [A3: int,B2: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ B2 @ A3 )
% 3.16/1.62 & ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_311_dual__order_Oeq__iff,axiom,
% 3.16/1.62 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 3.16/1.62 = ( ^ [A3: real,B2: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ B2 @ A3 )
% 3.16/1.62 & ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_312_dual__order_Otrans,axiom,
% 3.16/1.62 ! [B: int,A: int,C: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ B @ A )
% 3.16/1.62 => ( ( ord_less_eq_int @ C @ B )
% 3.16/1.62 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_313_dual__order_Otrans,axiom,
% 3.16/1.62 ! [B: real,A: real,C: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ B @ A )
% 3.16/1.62 => ( ( ord_less_eq_real @ C @ B )
% 3.16/1.62 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_314_linorder__wlog,axiom,
% 3.16/1.62 ! [P: int > int > $o,A: int,B: int] :
% 3.16/1.62 ( ! [A4: int,B3: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ A4 @ B3 )
% 3.16/1.62 => ( P @ A4 @ B3 ) )
% 3.16/1.62 => ( ! [A4: int,B3: int] :
% 3.16/1.62 ( ( P @ B3 @ A4 )
% 3.16/1.62 => ( P @ A4 @ B3 ) )
% 3.16/1.62 => ( P @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_315_linorder__wlog,axiom,
% 3.16/1.62 ! [P: real > real > $o,A: real,B: real] :
% 3.16/1.62 ( ! [A4: real,B3: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ A4 @ B3 )
% 3.16/1.62 => ( P @ A4 @ B3 ) )
% 3.16/1.62 => ( ! [A4: real,B3: real] :
% 3.16/1.62 ( ( P @ B3 @ A4 )
% 3.16/1.62 => ( P @ A4 @ B3 ) )
% 3.16/1.62 => ( P @ A @ B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_316_dual__order_Orefl,axiom,
% 3.16/1.62 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_317_dual__order_Orefl,axiom,
% 3.16/1.62 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_318_order__trans,axiom,
% 3.16/1.62 ! [X: int,Y: int,Z: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ X @ Y )
% 3.16/1.62 => ( ( ord_less_eq_int @ Y @ Z )
% 3.16/1.62 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_319_order__trans,axiom,
% 3.16/1.62 ! [X: real,Y: real,Z: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X @ Y )
% 3.16/1.62 => ( ( ord_less_eq_real @ Y @ Z )
% 3.16/1.62 => ( ord_less_eq_real @ X @ Z ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_320_order__class_Oorder_Oantisym,axiom,
% 3.16/1.62 ! [A: int,B: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 => ( ( ord_less_eq_int @ B @ A )
% 3.16/1.62 => ( A = B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_321_order__class_Oorder_Oantisym,axiom,
% 3.16/1.62 ! [A: real,B: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 => ( ( ord_less_eq_real @ B @ A )
% 3.16/1.62 => ( A = B ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_322_ord__le__eq__trans,axiom,
% 3.16/1.62 ! [A: int,B: int,C: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 => ( ( B = C )
% 3.16/1.62 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_323_ord__le__eq__trans,axiom,
% 3.16/1.62 ! [A: real,B: real,C: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 => ( ( B = C )
% 3.16/1.62 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_324_ord__eq__le__trans,axiom,
% 3.16/1.62 ! [A: int,B: int,C: int] :
% 3.16/1.62 ( ( A = B )
% 3.16/1.62 => ( ( ord_less_eq_int @ B @ C )
% 3.16/1.62 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_325_ord__eq__le__trans,axiom,
% 3.16/1.62 ! [A: real,B: real,C: real] :
% 3.16/1.62 ( ( A = B )
% 3.16/1.62 => ( ( ord_less_eq_real @ B @ C )
% 3.16/1.62 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_326_order__class_Oorder_Oeq__iff,axiom,
% 3.16/1.62 ( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
% 3.16/1.62 = ( ^ [A3: int,B2: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ A3 @ B2 )
% 3.16/1.62 & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_327_order__class_Oorder_Oeq__iff,axiom,
% 3.16/1.62 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 3.16/1.62 = ( ^ [A3: real,B2: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ A3 @ B2 )
% 3.16/1.62 & ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_328_antisym__conv,axiom,
% 3.16/1.62 ! [Y: int,X: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ Y @ X )
% 3.16/1.62 => ( ( ord_less_eq_int @ X @ Y )
% 3.16/1.62 = ( X = Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_329_antisym__conv,axiom,
% 3.16/1.62 ! [Y: real,X: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ Y @ X )
% 3.16/1.62 => ( ( ord_less_eq_real @ X @ Y )
% 3.16/1.62 = ( X = Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_330_le__cases3,axiom,
% 3.16/1.62 ! [X: int,Y: int,Z: int] :
% 3.16/1.62 ( ( ( ord_less_eq_int @ X @ Y )
% 3.16/1.62 => ~ ( ord_less_eq_int @ Y @ Z ) )
% 3.16/1.62 => ( ( ( ord_less_eq_int @ Y @ X )
% 3.16/1.62 => ~ ( ord_less_eq_int @ X @ Z ) )
% 3.16/1.62 => ( ( ( ord_less_eq_int @ X @ Z )
% 3.16/1.62 => ~ ( ord_less_eq_int @ Z @ Y ) )
% 3.16/1.62 => ( ( ( ord_less_eq_int @ Z @ Y )
% 3.16/1.62 => ~ ( ord_less_eq_int @ Y @ X ) )
% 3.16/1.62 => ( ( ( ord_less_eq_int @ Y @ Z )
% 3.16/1.62 => ~ ( ord_less_eq_int @ Z @ X ) )
% 3.16/1.62 => ~ ( ( ord_less_eq_int @ Z @ X )
% 3.16/1.62 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_331_le__cases3,axiom,
% 3.16/1.62 ! [X: real,Y: real,Z: real] :
% 3.16/1.62 ( ( ( ord_less_eq_real @ X @ Y )
% 3.16/1.62 => ~ ( ord_less_eq_real @ Y @ Z ) )
% 3.16/1.62 => ( ( ( ord_less_eq_real @ Y @ X )
% 3.16/1.62 => ~ ( ord_less_eq_real @ X @ Z ) )
% 3.16/1.62 => ( ( ( ord_less_eq_real @ X @ Z )
% 3.16/1.62 => ~ ( ord_less_eq_real @ Z @ Y ) )
% 3.16/1.62 => ( ( ( ord_less_eq_real @ Z @ Y )
% 3.16/1.62 => ~ ( ord_less_eq_real @ Y @ X ) )
% 3.16/1.62 => ( ( ( ord_less_eq_real @ Y @ Z )
% 3.16/1.62 => ~ ( ord_less_eq_real @ Z @ X ) )
% 3.16/1.62 => ~ ( ( ord_less_eq_real @ Z @ X )
% 3.16/1.62 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_332_order_Otrans,axiom,
% 3.16/1.62 ! [A: int,B: int,C: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 => ( ( ord_less_eq_int @ B @ C )
% 3.16/1.62 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_333_order_Otrans,axiom,
% 3.16/1.62 ! [A: real,B: real,C: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 => ( ( ord_less_eq_real @ B @ C )
% 3.16/1.62 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_334_le__cases,axiom,
% 3.16/1.62 ! [X: int,Y: int] :
% 3.16/1.62 ( ~ ( ord_less_eq_int @ X @ Y )
% 3.16/1.62 => ( ord_less_eq_int @ Y @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_335_le__cases,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ~ ( ord_less_eq_real @ X @ Y )
% 3.16/1.62 => ( ord_less_eq_real @ Y @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_336_eq__refl,axiom,
% 3.16/1.62 ! [X: int,Y: int] :
% 3.16/1.62 ( ( X = Y )
% 3.16/1.62 => ( ord_less_eq_int @ X @ Y ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_337_eq__refl,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ( X = Y )
% 3.16/1.62 => ( ord_less_eq_real @ X @ Y ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_338_linear,axiom,
% 3.16/1.62 ! [X: int,Y: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ X @ Y )
% 3.16/1.62 | ( ord_less_eq_int @ Y @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_339_linear,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X @ Y )
% 3.16/1.62 | ( ord_less_eq_real @ Y @ X ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_340_antisym,axiom,
% 3.16/1.62 ! [X: int,Y: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ X @ Y )
% 3.16/1.62 => ( ( ord_less_eq_int @ Y @ X )
% 3.16/1.62 => ( X = Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_341_antisym,axiom,
% 3.16/1.62 ! [X: real,Y: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X @ Y )
% 3.16/1.62 => ( ( ord_less_eq_real @ Y @ X )
% 3.16/1.62 => ( X = Y ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_342_eq__iff,axiom,
% 3.16/1.62 ( ( ^ [Y2: int,Z2: int] : ( Y2 = Z2 ) )
% 3.16/1.62 = ( ^ [X3: int,Y3: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ X3 @ Y3 )
% 3.16/1.62 & ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_343_eq__iff,axiom,
% 3.16/1.62 ( ( ^ [Y2: real,Z2: real] : ( Y2 = Z2 ) )
% 3.16/1.62 = ( ^ [X3: real,Y3: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X3 @ Y3 )
% 3.16/1.62 & ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_344_ord__le__eq__subst,axiom,
% 3.16/1.62 ! [A: int,B: int,F: int > int,C: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 => ( ( ( F @ B )
% 3.16/1.62 = C )
% 3.16/1.62 => ( ! [X5: int,Y4: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ X5 @ Y4 )
% 3.16/1.62 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 3.16/1.62 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_345_ord__le__eq__subst,axiom,
% 3.16/1.62 ! [A: int,B: int,F: int > real,C: real] :
% 3.16/1.62 ( ( ord_less_eq_int @ A @ B )
% 3.16/1.62 => ( ( ( F @ B )
% 3.16/1.62 = C )
% 3.16/1.62 => ( ! [X5: int,Y4: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ X5 @ Y4 )
% 3.16/1.62 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_346_ord__le__eq__subst,axiom,
% 3.16/1.62 ! [A: real,B: real,F: real > int,C: int] :
% 3.16/1.62 ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 => ( ( ( F @ B )
% 3.16/1.62 = C )
% 3.16/1.62 => ( ! [X5: real,Y4: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X5 @ Y4 )
% 3.16/1.62 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 3.16/1.62 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_347_ord__le__eq__subst,axiom,
% 3.16/1.62 ! [A: real,B: real,F: real > real,C: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ A @ B )
% 3.16/1.62 => ( ( ( F @ B )
% 3.16/1.62 = C )
% 3.16/1.62 => ( ! [X5: real,Y4: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X5 @ Y4 )
% 3.16/1.62 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_348_ord__eq__le__subst,axiom,
% 3.16/1.62 ! [A: int,F: int > int,B: int,C: int] :
% 3.16/1.62 ( ( A
% 3.16/1.62 = ( F @ B ) )
% 3.16/1.62 => ( ( ord_less_eq_int @ B @ C )
% 3.16/1.62 => ( ! [X5: int,Y4: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ X5 @ Y4 )
% 3.16/1.62 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 3.16/1.62 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_349_ord__eq__le__subst,axiom,
% 3.16/1.62 ! [A: real,F: int > real,B: int,C: int] :
% 3.16/1.62 ( ( A
% 3.16/1.62 = ( F @ B ) )
% 3.16/1.62 => ( ( ord_less_eq_int @ B @ C )
% 3.16/1.62 => ( ! [X5: int,Y4: int] :
% 3.16/1.62 ( ( ord_less_eq_int @ X5 @ Y4 )
% 3.16/1.62 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_350_ord__eq__le__subst,axiom,
% 3.16/1.62 ! [A: int,F: real > int,B: real,C: real] :
% 3.16/1.62 ( ( A
% 3.16/1.62 = ( F @ B ) )
% 3.16/1.62 => ( ( ord_less_eq_real @ B @ C )
% 3.16/1.62 => ( ! [X5: real,Y4: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X5 @ Y4 )
% 3.16/1.62 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 3.16/1.62 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_351_ord__eq__le__subst,axiom,
% 3.16/1.62 ! [A: real,F: real > real,B: real,C: real] :
% 3.16/1.62 ( ( A
% 3.16/1.62 = ( F @ B ) )
% 3.16/1.62 => ( ( ord_less_eq_real @ B @ C )
% 3.16/1.62 => ( ! [X5: real,Y4: real] :
% 3.16/1.62 ( ( ord_less_eq_real @ X5 @ Y4 )
% 3.16/1.62 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 3.16/1.62 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_352_forall__1,axiom,
% 3.16/1.62 ( ( ^ [P2: numeral_num1 > $o] :
% 3.16/1.62 ! [X2: numeral_num1] : ( P2 @ X2 ) )
% 3.16/1.62 = ( ^ [P3: numeral_num1 > $o] : ( P3 @ one_one_Numeral_num1 ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(fact_353_ex__1,axiom,
% 3.16/1.62 ( ( ^ [P2: numeral_num1 > $o] :
% 3.16/1.62 ? [X2: numeral_num1] : ( P2 @ X2 ) )
% 3.16/1.62 = ( ^ [P3: numeral_num1 > $o] : ( P3 @ one_one_Numeral_num1 ) ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_T,axiom,
% 3.16/1.62 ! [X: finite964658038_int_n,Y: finite964658038_int_n] :
% 3.16/1.62 ( ( if_Fin1767949360_int_n @ $false @ X @ Y )
% 3.16/1.62 = Y ) ).
% 3.16/1.62
% 3.16/1.62 thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mtf__n_J_T,axiom,
% 3.16/1.62 ! [X: finite964658038_int_n,Y: finite964658038_int_n] :
% 3.16/1.62 ( ( if_Fin1767949360_int_n @ $true @ X @ Y )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_T,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 3.16/1.62 ( ( if_Fin127821360real_n @ $false @ X @ Y )
% 3.16/1.62 = Y ) ).
% 3.16/1.62
% 3.16/1.62 thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__n_J_T,axiom,
% 3.16/1.62 ! [X: finite1489363574real_n,Y: finite1489363574real_n] :
% 3.16/1.62 ( ( if_Fin127821360real_n @ $true @ X @ Y )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J_T,axiom,
% 3.16/1.62 ! [X: finite2063899472l_num1,Y: finite2063899472l_num1] :
% 3.16/1.62 ( ( if_Fin1570437642l_num1 @ $false @ X @ Y )
% 3.16/1.62 = Y ) ).
% 3.16/1.62
% 3.16/1.62 thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Int__Oint_Mt__Numeral____Type__Onum1_J_T,axiom,
% 3.16/1.62 ! [X: finite2063899472l_num1,Y: finite2063899472l_num1] :
% 3.16/1.62 ( ( if_Fin1570437642l_num1 @ $true @ X @ Y )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(help_If_3_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_T,axiom,
% 3.16/1.62 ! [P: $o] :
% 3.16/1.62 ( ( P = $true )
% 3.16/1.62 | ( P = $false ) ) ).
% 3.16/1.62
% 3.16/1.62 thf(help_If_2_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_T,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 3.16/1.62 ( ( if_Fin1668193290l_num1 @ $false @ X @ Y )
% 3.16/1.62 = Y ) ).
% 3.16/1.62
% 3.16/1.62 thf(help_If_1_1_If_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mt__Numeral____Type__Onum1_J_T,axiom,
% 3.16/1.62 ! [X: finite1183840848l_num1,Y: finite1183840848l_num1] :
% 3.16/1.62 ( ( if_Fin1668193290l_num1 @ $true @ X @ Y )
% 3.16/1.62 = X ) ).
% 3.16/1.62
% 3.16/1.62 thf(conj_0,conjecture,
% 3.16/1.62 ( ( ring_1_of_int_real @ ( abs_abs_int @ m ) )
% 3.16/1.62 = ( abs_abs_real @ ( finite772340589real_n @ ( minus_1037315151real_n @ x @ y ) @ i ) ) ) ).
% 3.16/1.62 ------------------------
% 3.16/1.64 % (11449)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/183Mi)
% 3.16/1.65 % (11450)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_11239 for (2999ds/4Mi)
% 3.16/1.65 % (11451)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_11239 for (2999ds/27Mi)
% 3.16/1.65 % (11452)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/2Mi)
% 3.16/1.65 % (11452)Instruction limit reached!
% 3.16/1.65 % (11452)------------------------------
% 3.16/1.65 % (11452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.65 % (11452)Termination reason: Unknown
% 3.16/1.65 % (11452)Termination phase: shuffling
% 3.16/1.65
% 3.16/1.65 % (11452)Memory used [KB]: 1407
% 3.16/1.65 % (11452)Time elapsed: 0.003 s
% 3.16/1.65 % (11452)Instructions burned: 2 (million)
% 3.16/1.65 % (11452)------------------------------
% 3.16/1.65 % (11452)------------------------------
% 3.16/1.65 % (11455)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/18Mi)
% 3.16/1.65 % (11453)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_11239 for (2999ds/2Mi)
% 3.16/1.65 % (11450)Instruction limit reached!
% 3.16/1.65 % (11450)------------------------------
% 3.16/1.65 % (11450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.65 % (11450)Termination reason: Unknown
% 3.16/1.65 % (11450)Termination phase: shuffling
% 3.16/1.65
% 3.16/1.65 % (11450)Memory used [KB]: 1535
% 3.16/1.65 % (11450)Time elapsed: 0.004 s
% 3.16/1.65 % (11450)Instructions burned: 5 (million)
% 3.16/1.65 % (11450)------------------------------
% 3.16/1.65 % (11450)------------------------------
% 3.16/1.65 % (11453)Instruction limit reached!
% 3.16/1.65 % (11453)------------------------------
% 3.16/1.65 % (11453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.65 % (11453)Termination reason: Unknown
% 3.16/1.65 % (11453)Termination phase: shuffling
% 3.16/1.65
% 3.16/1.65 % (11453)Memory used [KB]: 1407
% 3.16/1.65 % (11453)Time elapsed: 0.003 s
% 3.16/1.65 % (11453)Instructions burned: 4 (million)
% 3.16/1.65 % (11453)------------------------------
% 3.16/1.65 % (11453)------------------------------
% 3.16/1.66 % (11455)Instruction limit reached!
% 3.16/1.66 % (11455)------------------------------
% 3.16/1.66 % (11455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.66 % (11455)Termination reason: Unknown
% 3.16/1.66 % (11455)Termination phase: Property scanning
% 3.16/1.66
% 3.16/1.66 % (11455)Memory used [KB]: 1791
% 3.16/1.66 % (11455)Time elapsed: 0.012 s
% 3.16/1.66 % (11455)Instructions burned: 20 (million)
% 3.16/1.66 % (11455)------------------------------
% 3.16/1.66 % (11455)------------------------------
% 3.16/1.66 % (11451)Instruction limit reached!
% 3.16/1.66 % (11451)------------------------------
% 3.16/1.66 % (11451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.66 % (11451)Termination reason: Unknown
% 3.16/1.66 % (11451)Termination phase: Property scanning
% 3.16/1.66
% 3.16/1.66 % (11451)Memory used [KB]: 1791
% 3.16/1.66 % (11451)Time elapsed: 0.015 s
% 3.16/1.66 % (11451)Instructions burned: 28 (million)
% 3.16/1.66 % (11451)------------------------------
% 3.16/1.66 % (11451)------------------------------
% 3.16/1.66 % (11454)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_11239 for (2999ds/275Mi)
% 3.16/1.66 % (11461)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_11239 for (2999ds/15Mi)
% 3.16/1.66 % (11460)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_11239 for (2999ds/37Mi)
% 3.16/1.67 % (11459)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/3Mi)
% 3.16/1.67 % (11459)Instruction limit reached!
% 3.16/1.67 % (11459)------------------------------
% 3.16/1.67 % (11459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.67 % (11459)Termination reason: Unknown
% 3.16/1.67 % (11459)Termination phase: shuffling
% 3.16/1.67
% 3.16/1.67 % (11459)Memory used [KB]: 1407
% 3.16/1.67 % (11459)Time elapsed: 0.004 s
% 3.16/1.67 % (11459)Instructions burned: 3 (million)
% 3.16/1.67 % (11459)------------------------------
% 3.16/1.67 % (11459)------------------------------
% 3.16/1.67 % (11463)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_11239 for (2999ds/3Mi)
% 3.16/1.67 % (11463)Instruction limit reached!
% 3.16/1.67 % (11463)------------------------------
% 3.16/1.67 % (11463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.67 % (11463)Termination reason: Unknown
% 3.16/1.67 % (11463)Termination phase: shuffling
% 3.16/1.67
% 3.16/1.67 % (11463)Memory used [KB]: 1407
% 3.16/1.67 % (11463)Time elapsed: 0.003 s
% 3.16/1.67 % (11463)Instructions burned: 3 (million)
% 3.16/1.67 % (11463)------------------------------
% 3.16/1.67 % (11463)------------------------------
% 3.16/1.67 % (11461)Instruction limit reached!
% 3.16/1.67 % (11461)------------------------------
% 3.16/1.67 % (11461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.67 % (11461)Termination reason: Unknown
% 3.16/1.67 % (11461)Termination phase: shuffling
% 3.16/1.67
% 3.16/1.67 % (11461)Memory used [KB]: 1791
% 3.16/1.67 % (11461)Time elapsed: 0.009 s
% 3.16/1.67 % (11461)Instructions burned: 16 (million)
% 3.16/1.67 % (11461)------------------------------
% 3.16/1.67 % (11461)------------------------------
% 3.16/1.68 % (11464)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_11239 for (2999ds/1041Mi)
% 3.16/1.68 % (11460)Instruction limit reached!
% 3.16/1.68 % (11460)------------------------------
% 3.16/1.68 % (11460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.68 % (11460)Termination reason: Unknown
% 3.16/1.68 % (11460)Termination phase: shuffling
% 3.16/1.68
% 3.16/1.68 % (11460)Memory used [KB]: 2174
% 3.16/1.68 % (11460)Time elapsed: 0.014 s
% 3.16/1.68 % (11460)Instructions burned: 40 (million)
% 3.16/1.68 % (11460)------------------------------
% 3.16/1.68 % (11460)------------------------------
% 3.16/1.68 % (11467)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/7Mi)
% 3.16/1.69 % (11469)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/16Mi)
% 3.16/1.69 % (11467)Instruction limit reached!
% 3.16/1.69 % (11467)------------------------------
% 3.16/1.69 % (11467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.69 % (11467)Termination reason: Unknown
% 3.16/1.69 % (11467)Termination phase: shuffling
% 3.16/1.69
% 3.16/1.69 % (11467)Memory used [KB]: 1535
% 3.16/1.69 % (11467)Time elapsed: 0.005 s
% 3.16/1.69 % (11467)Instructions burned: 7 (million)
% 3.16/1.69 % (11467)------------------------------
% 3.16/1.69 % (11467)------------------------------
% 3.16/1.69 % (11470)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_11239 for (2999ds/3Mi)
% 3.16/1.69 % (11470)Instruction limit reached!
% 3.16/1.69 % (11470)------------------------------
% 3.16/1.69 % (11470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.69 % (11470)Termination reason: Unknown
% 3.16/1.69 % (11470)Termination phase: shuffling
% 3.16/1.69
% 3.16/1.69 % (11470)Memory used [KB]: 1407
% 3.16/1.69 % (11470)Time elapsed: 0.004 s
% 3.16/1.69 % (11470)Instructions burned: 3 (million)
% 3.16/1.69 % (11470)------------------------------
% 3.16/1.69 % (11470)------------------------------
% 3.16/1.69 % (11469)Instruction limit reached!
% 3.16/1.69 % (11469)------------------------------
% 3.16/1.69 % (11469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.69 % (11469)Termination reason: Unknown
% 3.16/1.69 % (11469)Termination phase: shuffling
% 3.16/1.69
% 3.16/1.69 % (11469)Memory used [KB]: 1791
% 3.16/1.69 % (11469)Time elapsed: 0.009 s
% 3.16/1.69 % (11469)Instructions burned: 16 (million)
% 3.16/1.69 % (11469)------------------------------
% 3.16/1.69 % (11469)------------------------------
% 3.16/1.69 % (11472)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_11239 for (2999ds/3Mi)
% 3.16/1.70 % (11472)Instruction limit reached!
% 3.16/1.70 % (11472)------------------------------
% 3.16/1.70 % (11472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.70 % (11472)Termination reason: Unknown
% 3.16/1.70 % (11472)Termination phase: shuffling
% 3.16/1.70
% 3.16/1.70 % (11472)Memory used [KB]: 1535
% 3.16/1.70 % (11472)Time elapsed: 0.004 s
% 3.16/1.70 % (11472)Instructions burned: 5 (million)
% 3.16/1.70 % (11472)------------------------------
% 3.16/1.70 % (11472)------------------------------
% 3.18/1.70 % (11474)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/7Mi)
% 3.18/1.70 % (11474)Instruction limit reached!
% 3.18/1.70 % (11474)------------------------------
% 3.18/1.70 % (11474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.70 % (11474)Termination reason: Unknown
% 3.18/1.70 % (11474)Termination phase: shuffling
% 3.18/1.70
% 3.18/1.70 % (11474)Memory used [KB]: 1535
% 3.18/1.70 % (11474)Time elapsed: 0.004 s
% 3.18/1.70 % (11474)Instructions burned: 7 (million)
% 3.18/1.70 % (11474)------------------------------
% 3.18/1.70 % (11474)------------------------------
% 3.18/1.71 % (11454)Refutation not found, incomplete strategy
% 3.18/1.71 % (11454)------------------------------
% 3.18/1.71 % (11454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.71 % (11454)Termination reason: Refutation not found, incomplete strategy
% 3.18/1.71
% 3.18/1.71
% 3.18/1.71 % (11454)Memory used [KB]: 6780
% 3.18/1.71 % (11454)Time elapsed: 0.047 s
% 3.18/1.71 % (11454)Instructions burned: 78 (million)
% 3.18/1.71 % (11454)------------------------------
% 3.18/1.71 % (11454)------------------------------
% 3.18/1.71 % (11477)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/3Mi)
% 3.18/1.71 % (11477)Instruction limit reached!
% 3.18/1.71 % (11477)------------------------------
% 3.18/1.71 % (11477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.71 % (11477)Termination reason: Unknown
% 3.18/1.71 % (11477)Termination phase: shuffling
% 3.18/1.71
% 3.18/1.71 % (11477)Memory used [KB]: 1535
% 3.18/1.71 % (11477)Time elapsed: 0.003 s
% 3.18/1.71 % (11477)Instructions burned: 5 (million)
% 3.18/1.71 % (11477)------------------------------
% 3.18/1.71 % (11477)------------------------------
% 3.18/1.71 % (11449)Instruction limit reached!
% 3.18/1.71 % (11449)------------------------------
% 3.18/1.71 % (11449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.71 % (11449)Termination reason: Unknown
% 3.18/1.71 % (11449)Termination phase: Saturation
% 3.18/1.71
% 3.18/1.71 % (11449)Memory used [KB]: 7547
% 3.18/1.71 % (11449)Time elapsed: 0.069 s
% 3.18/1.71 % (11449)Instructions burned: 185 (million)
% 3.18/1.71 % (11449)------------------------------
% 3.18/1.71 % (11449)------------------------------
% 3.18/1.71 % (11481)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/18Mi)
% 3.18/1.71 % (11480)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/4Mi)
% 3.18/1.71 % (11482)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/710Mi)
% 3.18/1.71 % (11480)Instruction limit reached!
% 3.18/1.71 % (11480)------------------------------
% 3.18/1.71 % (11480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.71 % (11480)Termination reason: Unknown
% 3.18/1.71 % (11480)Termination phase: shuffling
% 3.18/1.71
% 3.18/1.71 % (11480)Memory used [KB]: 1535
% 3.18/1.71 % (11480)Time elapsed: 0.003 s
% 3.18/1.71 % (11480)Instructions burned: 6 (million)
% 3.18/1.71 % (11480)------------------------------
% 3.18/1.71 % (11480)------------------------------
% 3.18/1.72 % (11481)Instruction limit reached!
% 3.18/1.72 % (11481)------------------------------
% 3.18/1.72 % (11481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.72 % (11481)Termination reason: Unknown
% 3.18/1.72 % (11481)Termination phase: shuffling
% 3.18/1.72
% 3.18/1.72 % (11481)Memory used [KB]: 1791
% 3.18/1.72 % (11481)Time elapsed: 0.007 s
% 3.18/1.72 % (11481)Instructions burned: 18 (million)
% 3.18/1.72 % (11481)------------------------------
% 3.18/1.72 % (11481)------------------------------
% 3.18/1.72 % (11485)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/6Mi)
% 3.18/1.72 % (11486)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/902Mi)
% 3.18/1.72 % (11487)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/21Mi)
% 3.18/1.72 % (11485)Instruction limit reached!
% 3.18/1.72 % (11485)------------------------------
% 3.18/1.72 % (11485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.72 % (11485)Termination reason: Unknown
% 3.18/1.72 % (11485)Termination phase: shuffling
% 3.18/1.72
% 3.18/1.72 % (11485)Memory used [KB]: 1535
% 3.18/1.72 % (11485)Time elapsed: 0.004 s
% 3.18/1.72 % (11485)Instructions burned: 6 (million)
% 3.18/1.72 % (11485)------------------------------
% 3.18/1.72 % (11485)------------------------------
% 3.18/1.72 % (11488)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/5Mi)
% 3.18/1.72 % (11488)Instruction limit reached!
% 3.18/1.72 % (11488)------------------------------
% 3.18/1.72 % (11488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.72 % (11488)Termination reason: Unknown
% 3.18/1.72 % (11488)Termination phase: shuffling
% 3.18/1.72
% 3.18/1.72 % (11488)Memory used [KB]: 1535
% 3.18/1.72 % (11488)Time elapsed: 0.003 s
% 3.18/1.72 % (11488)Instructions burned: 6 (million)
% 3.18/1.72 % (11488)------------------------------
% 3.18/1.72 % (11488)------------------------------
% 3.18/1.72 % (11489)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/6Mi)
% 3.18/1.72 % (11487)Instruction limit reached!
% 3.18/1.72 % (11487)------------------------------
% 3.18/1.72 % (11487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.72 % (11487)Termination reason: Unknown
% 3.18/1.72 % (11487)Termination phase: shuffling
% 3.18/1.72
% 3.18/1.72 % (11487)Memory used [KB]: 1791
% 3.18/1.72 % (11487)Time elapsed: 0.008 s
% 3.18/1.72 % (11487)Instructions burned: 21 (million)
% 3.18/1.72 % (11487)------------------------------
% 3.18/1.72 % (11487)------------------------------
% 3.18/1.73 % (11489)Instruction limit reached!
% 3.18/1.73 % (11489)------------------------------
% 3.18/1.73 % (11489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.73 % (11489)Termination reason: Unknown
% 3.18/1.73 % (11489)Termination phase: shuffling
% 3.18/1.73
% 3.18/1.73 % (11489)Memory used [KB]: 1535
% 3.18/1.73 % (11489)Time elapsed: 0.003 s
% 3.18/1.73 % (11489)Instructions burned: 6 (million)
% 3.18/1.73 % (11489)------------------------------
% 3.18/1.73 % (11489)------------------------------
% 3.18/1.73 % (11490)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/377Mi)
% 3.18/1.73 % (11491)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/779Mi)
% 3.18/1.73 % (11492)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/19Mi)
% 3.18/1.74 % (11493)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_11239 for (2999ds/879Mi)
% 3.18/1.74 % (11492)Instruction limit reached!
% 3.18/1.74 % (11492)------------------------------
% 3.18/1.74 % (11492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.74 % (11492)Termination reason: Unknown
% 3.18/1.74 % (11492)Termination phase: shuffling
% 3.18/1.74
% 3.18/1.74 % (11492)Memory used [KB]: 1918
% 3.18/1.74 % (11492)Time elapsed: 0.007 s
% 3.18/1.74 % (11492)Instructions burned: 20 (million)
% 3.18/1.74 % (11492)------------------------------
% 3.18/1.74 % (11492)------------------------------
% 3.18/1.75 % (11494)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_11239 for (2998ds/17Mi)
% 3.18/1.75 % (11486)Refutation not found, incomplete strategy
% 3.18/1.75 % (11486)------------------------------
% 3.18/1.75 % (11486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.75 % (11486)Termination reason: Refutation not found, incomplete strategy
% 3.18/1.75
% 3.18/1.75
% 3.18/1.75 % (11486)Memory used [KB]: 7419
% 3.18/1.75 % (11486)Time elapsed: 0.035 s
% 3.18/1.75 % (11486)Instructions burned: 115 (million)
% 3.18/1.75 % (11486)------------------------------
% 3.18/1.75 % (11486)------------------------------
% 3.18/1.76 % (11494)Instruction limit reached!
% 3.18/1.76 % (11494)------------------------------
% 3.18/1.76 % (11494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.76 % (11494)Termination reason: Unknown
% 3.18/1.76 % (11494)Termination phase: shuffling
% 3.18/1.76
% 3.18/1.76 % (11494)Memory used [KB]: 1791
% 3.18/1.76 % (11494)Time elapsed: 0.007 s
% 3.18/1.76 % (11494)Instructions burned: 19 (million)
% 3.18/1.76 % (11494)------------------------------
% 3.18/1.76 % (11494)------------------------------
% 3.18/1.76 % (11495)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_11239 for (2998ds/3Mi)
% 3.18/1.76 % (11495)Instruction limit reached!
% 3.18/1.76 % (11495)------------------------------
% 3.18/1.76 % (11495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.76 % (11495)Termination reason: Unknown
% 3.18/1.76 % (11495)Termination phase: shuffling
% 3.18/1.76
% 3.18/1.76 % (11495)Memory used [KB]: 1535
% 3.18/1.76 % (11495)Time elapsed: 0.003 s
% 3.18/1.76 % (11495)Instructions burned: 5 (million)
% 3.18/1.76 % (11495)------------------------------
% 3.18/1.76 % (11495)------------------------------
% 3.18/1.77 % (11496)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_11239 for (2998ds/30Mi)
% 3.18/1.77 % (11497)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_11239 for (2998ds/127Mi)
% 3.18/1.77 % (11496)Instruction limit reached!
% 3.18/1.77 % (11496)------------------------------
% 3.18/1.77 % (11496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.77 % (11496)Termination reason: Unknown
% 3.18/1.77 % (11496)Termination phase: Property scanning
% 3.18/1.77
% 3.18/1.77 % (11496)Memory used [KB]: 1918
% 3.18/1.77 % (11496)Time elapsed: 0.011 s
% 3.18/1.77 % (11496)Instructions burned: 30 (million)
% 3.18/1.77 % (11496)------------------------------
% 3.18/1.77 % (11496)------------------------------
% 3.18/1.78 % (11498)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_11239 for (2998ds/100Mi)
% 3.18/1.79 % (11482)First to succeed.
% 3.18/1.79 % (11482)Refutation found. Thanks to Tanya!
% 3.18/1.79 % SZS status Theorem for DTF2THF_11239
% 3.18/1.79 % SZS output start Proof for DTF2THF_11239
% 3.18/1.79 thf(type_def_5, type, real: $tType).
% 3.18/1.79 thf(type_def_6, type, int: $tType).
% 3.18/1.79 thf(type_def_8, type, numeral_num1: $tType).
% 3.18/1.79 thf(type_def_9, type, finite2063899472l_num1: $tType).
% 3.18/1.79 thf(type_def_10, type, finite1183840848l_num1: $tType).
% 3.18/1.79 thf(type_def_11, type, n: $tType).
% 3.18/1.79 thf(type_def_12, type, finite964658038_int_n: $tType).
% 3.18/1.79 thf(type_def_13, type, finite1489363574real_n: $tType).
% 3.18/1.79 thf(type_def_14, type, finite463740460l_num1: $tType).
% 3.18/1.79 thf(type_def_15, type, finite337957458num1_n: $tType).
% 3.18/1.79 thf(type_def_16, type, finite121157254l_num1: $tType).
% 3.18/1.79 thf(type_def_17, type, finite318270636al_n_n: $tType).
% 3.18/1.79 thf(type_def_18, type, set_Fi1257234438l_num1: $tType).
% 3.18/1.79 thf(type_def_19, type, set_Fi1058188332real_n: $tType).
% 3.18/1.79 thf(type_def_20, type, set_int: $tType).
% 3.18/1.79 thf(type_def_21, type, set_real: $tType).
% 3.18/1.79 thf(func_def_0, type, finite463740460l_num1: $tType).
% 3.18/1.79 thf(func_def_1, type, finite121157254l_num1: $tType).
% 3.18/1.79 thf(func_def_2, type, finite337957458num1_n: $tType).
% 3.18/1.79 thf(func_def_3, type, finite318270636al_n_n: $tType).
% 3.18/1.79 thf(func_def_4, type, set_Fi1257234438l_num1: $tType).
% 3.18/1.79 thf(func_def_5, type, finite1183840848l_num1: $tType).
% 3.18/1.79 thf(func_def_6, type, finite2063899472l_num1: $tType).
% 3.18/1.79 thf(func_def_7, type, set_Fi1058188332real_n: $tType).
% 3.18/1.79 thf(func_def_8, type, finite1489363574real_n: $tType).
% 3.18/1.79 thf(func_def_9, type, finite964658038_int_n: $tType).
% 3.18/1.79 thf(func_def_10, type, set_real: $tType).
% 3.18/1.79 thf(func_def_11, type, set_int: $tType).
% 3.18/1.79 thf(func_def_12, type, numeral_num1: $tType).
% 3.18/1.79 thf(func_def_13, type, real: $tType).
% 3.18/1.79 thf(func_def_14, type, int: $tType).
% 3.18/1.79 thf(func_def_15, type, n: $tType).
% 3.18/1.79 thf(func_def_16, type, archim1371465213g_real: real > int).
% 3.18/1.79 thf(func_def_17, type, archim1031974863r_real: real > int).
% 3.18/1.79 thf(func_def_18, type, finite627713486m1_int: numeral_num1 > int > finite2063899472l_num1).
% 3.18/1.79 thf(func_def_19, type, finite15144922381_real: numeral_num1 > real > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_20, type, finite1888777460_n_int: n > int > finite964658038_int_n).
% 3.18/1.79 thf(func_def_21, type, finite703491444n_real: n > real > finite1489363574real_n).
% 3.18/1.79 thf(func_def_22, type, finite1470207077l_num1: int > finite2063899472l_num1).
% 3.18/1.79 thf(func_def_23, type, finite55766923_int_n: int > finite964658038_int_n).
% 3.18/1.79 thf(func_def_24, type, finite1329045861l_num1: real > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_25, type, finite667548811real_n: real > finite1489363574real_n).
% 3.18/1.79 thf(func_def_26, type, finite1449546019l_num1: finite463740460l_num1 > numeral_num1 > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_27, type, finite1639785033num1_n: finite337957458num1_n > n > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_28, type, finite1503577981l_num1: finite121157254l_num1 > numeral_num1 > finite1489363574real_n).
% 3.18/1.79 thf(func_def_29, type, finite625646243al_n_n: finite318270636al_n_n > n > finite1489363574real_n).
% 3.18/1.79 thf(func_def_30, type, finite1800583751l_num1: finite2063899472l_num1 > numeral_num1 > int).
% 3.18/1.79 thf(func_def_31, type, finite566214509_int_n: finite964658038_int_n > n > int).
% 3.18/1.79 thf(func_def_32, type, finite1106245191l_num1: finite1183840848l_num1 > numeral_num1 > real).
% 3.18/1.79 thf(func_def_33, type, finite772340589real_n: finite1489363574real_n > n > real).
% 3.18/1.79 thf(func_def_34, type, finite1961675766l_num1: real > finite1183840848l_num1 > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_35, type, finite2053577756real_n: real > finite1489363574real_n > finite1489363574real_n).
% 3.18/1.79 thf(func_def_36, type, abs_ab948591917l_num1: finite1183840848l_num1 > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_37, type, abs_ab1599551059real_n: finite1489363574real_n > finite1489363574real_n).
% 3.18/1.79 thf(func_def_38, type, abs_abs_int: int > int).
% 3.18/1.79 thf(func_def_39, type, abs_abs_real: real > real).
% 3.18/1.79 thf(func_def_40, type, minus_779068585l_num1: finite2063899472l_num1 > finite2063899472l_num1 > finite2063899472l_num1).
% 3.18/1.79 thf(func_def_41, type, minus_1196255695_int_n: finite964658038_int_n > finite964658038_int_n > finite964658038_int_n).
% 3.18/1.79 thf(func_def_42, type, minus_249161513l_num1: finite1183840848l_num1 > finite1183840848l_num1 > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_43, type, minus_1037315151real_n: finite1489363574real_n > finite1489363574real_n > finite1489363574real_n).
% 3.18/1.79 thf(func_def_44, type, minus_minus_int: int > int > int).
% 3.18/1.79 thf(func_def_45, type, minus_minus_real: real > real > real).
% 3.18/1.79 thf(func_def_46, type, one_on601734293l_num1: finite2063899472l_num1).
% 3.18/1.79 thf(func_def_47, type, one_on1516019131_int_n: finite964658038_int_n).
% 3.18/1.79 thf(func_def_48, type, one_on1961019669l_num1: finite1183840848l_num1).
% 3.18/1.79 thf(func_def_49, type, one_on1253059131real_n: finite1489363574real_n).
% 3.18/1.79 thf(func_def_50, type, one_one_int: int).
% 3.18/1.79 thf(func_def_51, type, one_one_Numeral_num1: numeral_num1).
% 3.18/1.79 thf(func_def_52, type, one_one_real: real).
% 3.18/1.79 thf(func_def_53, type, if_Fin1570437642l_num1: $o > finite2063899472l_num1 > finite2063899472l_num1 > finite2063899472l_num1).
% 3.18/1.79 thf(func_def_54, type, if_Fin1767949360_int_n: $o > finite964658038_int_n > finite964658038_int_n > finite964658038_int_n).
% 3.18/1.79 thf(func_def_55, type, if_Fin1668193290l_num1: $o > finite1183840848l_num1 > finite1183840848l_num1 > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_56, type, if_Fin127821360real_n: $o > finite1489363574real_n > finite1489363574real_n > finite1489363574real_n).
% 3.18/1.79 thf(func_def_57, type, ring_11529266783l_num1: set_Fi1257234438l_num1).
% 3.18/1.79 thf(func_def_58, type, ring_1723899781real_n: set_Fi1058188332real_n).
% 3.18/1.79 thf(func_def_59, type, ring_1_Ints_int: set_int).
% 3.18/1.79 thf(func_def_60, type, ring_1_Ints_real: set_real).
% 3.18/1.79 thf(func_def_61, type, ring_1154201247l_num1: int > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_62, type, ring_11737179077real_n: int > finite1489363574real_n).
% 3.18/1.79 thf(func_def_63, type, ring_1_of_int_int: int > int).
% 3.18/1.79 thf(func_def_64, type, ring_1_of_int_real: int > real).
% 3.18/1.79 thf(func_def_65, type, minkow781993583real_n: finite2063899472l_num1 > finite121157254l_num1).
% 3.18/1.79 thf(func_def_66, type, minkow2063319845m1_int: finite2063899472l_num1 > finite2063899472l_num1).
% 3.18/1.79 thf(func_def_67, type, minkow19073400691_real: finite2063899472l_num1 > finite1183840848l_num1).
% 3.18/1.79 thf(func_def_68, type, minkow371489429real_n: finite964658038_int_n > finite318270636al_n_n).
% 3.18/1.79 thf(func_def_69, type, minkow1468704075_n_int: finite964658038_int_n > finite964658038_int_n).
% 3.18/1.79 thf(func_def_70, type, minkow1134813771n_real: finite964658038_int_n > finite1489363574real_n).
% 3.18/1.79 thf(func_def_71, type, ord_le2035778724l_num1: finite1183840848l_num1 > finite1183840848l_num1 > $o).
% 3.18/1.79 thf(func_def_72, type, ord_le1017090250real_n: finite1489363574real_n > finite1489363574real_n > $o).
% 3.18/1.79 thf(func_def_73, type, ord_less_int: int > int > $o).
% 3.18/1.79 thf(func_def_74, type, ord_less_real: real > real > $o).
% 3.18/1.79 thf(func_def_75, type, ord_le1202326448l_num1: finite2063899472l_num1 > finite2063899472l_num1 > $o).
% 3.18/1.79 thf(func_def_76, type, ord_le1908593622_int_n: finite964658038_int_n > finite964658038_int_n > $o).
% 3.18/1.79 thf(func_def_77, type, ord_le1724070576l_num1: finite1183840848l_num1 > finite1183840848l_num1 > $o).
% 3.18/1.79 thf(func_def_78, type, ord_le866583254real_n: finite1489363574real_n > finite1489363574real_n > $o).
% 3.18/1.79 thf(func_def_79, type, ord_less_eq_int: int > int > $o).
% 3.18/1.79 thf(func_def_80, type, ord_less_eq_real: real > real > $o).
% 3.18/1.79 thf(func_def_81, type, real_V1974343547l_num1: finite463740460l_num1 > finite463740460l_num1 > real).
% 3.18/1.79 thf(func_def_82, type, real_V845814485l_num1: finite121157254l_num1 > finite121157254l_num1 > real).
% 3.18/1.79 thf(func_def_83, type, real_V488409631l_num1: finite1183840848l_num1 > finite1183840848l_num1 > real).
% 3.18/1.79 thf(func_def_84, type, real_V783965509real_n: finite1489363574real_n > finite1489363574real_n > real).
% 3.18/1.79 thf(func_def_85, type, real_V1934908667t_real: real > real > real).
% 3.18/1.79 thf(func_def_86, type, real_V316032763l_num1: finite463740460l_num1 > real).
% 3.18/1.79 thf(func_def_87, type, real_V1121707553num1_n: finite337957458num1_n > real).
% 3.18/1.79 thf(func_def_88, type, real_V400292255l_num1: finite1183840848l_num1 > real).
% 3.18/1.79 thf(func_def_89, type, real_V739724485real_n: finite1489363574real_n > real).
% 3.18/1.79 thf(func_def_90, type, real_V646646907m_real: real > real).
% 3.18/1.79 thf(func_def_91, type, collec321817931real_n: (finite1489363574real_n > $o) > set_Fi1058188332real_n).
% 3.18/1.79 thf(func_def_92, type, collect_real: (real > $o) > set_real).
% 3.18/1.79 thf(func_def_93, type, member1413569767l_num1: finite1183840848l_num1 > set_Fi1257234438l_num1 > $o).
% 3.18/1.79 thf(func_def_94, type, member1352538125real_n: finite1489363574real_n > set_Fi1058188332real_n > $o).
% 3.18/1.79 thf(func_def_95, type, member_int: int > set_int > $o).
% 3.18/1.79 thf(func_def_96, type, member_real: real > set_real > $o).
% 3.18/1.79 thf(func_def_97, type, a: set_Fi1058188332real_n).
% 3.18/1.79 thf(func_def_98, type, i: n).
% 3.18/1.79 thf(func_def_99, type, m: int).
% 3.18/1.79 thf(func_def_100, type, x: finite1489363574real_n).
% 3.18/1.79 thf(func_def_101, type, y: finite1489363574real_n).
% 3.18/1.79 thf(func_def_115, type, sK0: n > (n > real > $o) > real).
% 3.18/1.79 thf(func_def_116, type, sK1: finite1489363574real_n > (n > real > $o) > n).
% 3.18/1.79 thf(func_def_117, type, sK2: (n > real > $o) > finite1489363574real_n).
% 3.18/1.79 thf(func_def_118, type, sK3: (n > real > $o) > n).
% 3.18/1.79 thf(func_def_119, type, sK4: real > int).
% 3.18/1.79 thf(func_def_120, type, sK5: int).
% 3.18/1.79 thf(func_def_122, type, ph7: !>[X0: $tType]:(X0)).
% 3.18/1.79 thf(func_def_123, type, sK8: finite1489363574real_n > finite1489363574real_n > n).
% 3.18/1.79 thf(f1318,plain,(
% 3.18/1.79 $false),
% 3.18/1.79 inference(trivial_inequality_removal,[status(thm)],[f1317])).
% 3.18/1.79 thf(f1317,plain,(
% 3.18/1.79 ((abs_abs_real @ (ring_1_of_int_real @ m)) != (abs_abs_real @ (ring_1_of_int_real @ m)))),
% 3.18/1.79 inference(backward_demodulation,[status(thm)],[f926,f1309])).
% 3.18/1.79 thf(f1309,plain,(
% 3.18/1.79 ((ring_1_of_int_real @ m) = (finite772340589real_n @ (minus_1037315151real_n @ x @ y) @ i))),
% 3.18/1.79 inference(superposition,[status(thm)],[f891,f886])).
% 3.18/1.79 thf(f886,plain,(
% 3.18/1.79 ( ! [X2 : finite1489363574real_n,X0 : finite1489363574real_n,X1 : n] : (((finite772340589real_n @ (minus_1037315151real_n @ X0 @ X2) @ X1) = (minus_minus_real @ (finite772340589real_n @ X0 @ X1) @ (finite772340589real_n @ X2 @ X1)))) )),
% 3.18/1.79 inference(cnf_transformation,[status(thm)],[f852])).
% 3.18/1.79 thf(f852,plain,(
% 3.18/1.79 ! [X0 : finite1489363574real_n,X1 : n,X2 : finite1489363574real_n] : ((finite772340589real_n @ (minus_1037315151real_n @ X0 @ X2) @ X1) = (minus_minus_real @ (finite772340589real_n @ X0 @ X1) @ (finite772340589real_n @ X2 @ X1)))),
% 3.18/1.79 inference(rectify,[status(thm)],[f7])).
% 3.18/1.79 thf(f7,axiom,(
% 3.18/1.79 ! [X1 : finite1489363574real_n,X3 : n,X2 : finite1489363574real_n] : ((finite772340589real_n @ (minus_1037315151real_n @ X1 @ X2) @ X3) = (minus_minus_real @ (finite772340589real_n @ X1 @ X3) @ (finite772340589real_n @ X2 @ X3)))),
% 3.18/1.79 file('/export/starexec/sandbox/tmp/tmp.LSq3s3M90o/DTF2THF_11239.p',fact_6_vector__minus__component)).
% 3.18/1.79 thf(f891,plain,(
% 3.18/1.79 ((ring_1_of_int_real @ m) = (minus_minus_real @ (finite772340589real_n @ x @ i) @ (finite772340589real_n @ y @ i)))),
% 3.18/1.79 inference(cnf_transformation,[status(thm)],[f3])).
% 3.18/1.79 thf(f3,axiom,(
% 3.18/1.79 ((ring_1_of_int_real @ m) = (minus_minus_real @ (finite772340589real_n @ x @ i) @ (finite772340589real_n @ y @ i)))),
% 3.18/1.79 file('/export/starexec/sandbox/tmp/tmp.LSq3s3M90o/DTF2THF_11239.p',fact_2_m)).
% 3.18/1.79 thf(f926,plain,(
% 3.18/1.79 ((abs_abs_real @ (finite772340589real_n @ (minus_1037315151real_n @ x @ y) @ i)) != (abs_abs_real @ (ring_1_of_int_real @ m)))),
% 3.18/1.79 inference(backward_demodulation,[status(thm)],[f907,f910])).
% 3.18/1.79 thf(f910,plain,(
% 3.18/1.79 ( ! [X0 : int] : (((abs_abs_real @ (ring_1_of_int_real @ X0)) = (ring_1_of_int_real @ (abs_abs_int @ X0)))) )),
% 3.18/1.79 inference(cnf_transformation,[status(thm)],[f856])).
% 3.18/1.79 thf(f856,plain,(
% 3.18/1.79 ! [X0 : int] : ((abs_abs_real @ (ring_1_of_int_real @ X0)) = (ring_1_of_int_real @ (abs_abs_int @ X0)))),
% 3.18/1.79 inference(rectify,[status(thm)],[f10])).
% 3.18/1.79 thf(f10,axiom,(
% 3.18/1.79 ! [X1 : int] : ((ring_1_of_int_real @ (abs_abs_int @ X1)) = (abs_abs_real @ (ring_1_of_int_real @ X1)))),
% 3.18/1.79 file('/export/starexec/sandbox/tmp/tmp.LSq3s3M90o/DTF2THF_11239.p',fact_9_of__int__abs)).
% 3.18/1.79 thf(f907,plain,(
% 3.18/1.79 ((ring_1_of_int_real @ (abs_abs_int @ m)) != (abs_abs_real @ (finite772340589real_n @ (minus_1037315151real_n @ x @ y) @ i)))),
% 3.18/1.79 inference(cnf_transformation,[status(thm)],[f851])).
% 3.18/1.79 thf(f851,plain,(
% 3.18/1.79 ((ring_1_of_int_real @ (abs_abs_int @ m)) != (abs_abs_real @ (finite772340589real_n @ (minus_1037315151real_n @ x @ y) @ i)))),
% 3.18/1.79 inference(flattening,[status(thm)],[f365])).
% 3.18/1.79 thf(f365,negated_conjecture,(
% 3.18/1.79 ~((ring_1_of_int_real @ (abs_abs_int @ m)) = (abs_abs_real @ (finite772340589real_n @ (minus_1037315151real_n @ x @ y) @ i)))),
% 3.18/1.79 inference(negated_conjecture,[status(cth)],[f364])).
% 3.18/1.79 thf(f364,conjecture,(
% 3.18/1.79 ((ring_1_of_int_real @ (abs_abs_int @ m)) = (abs_abs_real @ (finite772340589real_n @ (minus_1037315151real_n @ x @ y) @ i)))),
% 3.18/1.79 file('/export/starexec/sandbox/tmp/tmp.LSq3s3M90o/DTF2THF_11239.p',conj_0)).
% 3.18/1.79 % SZS output end Proof for DTF2THF_11239
% 3.18/1.79 % (11482)------------------------------
% 3.18/1.79 % (11482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.18/1.79 % (11482)Termination reason: Refutation
% 3.18/1.79
% 3.18/1.79 % (11482)Memory used [KB]: 7291
% 3.18/1.79 % (11482)Time elapsed: 0.082 s
% 3.18/1.79 % (11482)Instructions burned: 232 (million)
% 3.18/1.79 % (11482)------------------------------
% 3.18/1.79 % (11482)------------------------------
% 3.18/1.79 % (11444)Success in time 0.178 s
%------------------------------------------------------------------------------