%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP156^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n019.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 08:56:00 AM UTC 2026
% Result : Theorem 2.71s 1.71s
% Output : Refutation 2.71s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 12
% Syntax : Number of formulae : 62 ( 25 unt; 0 typ; 3 def)
% Number of atoms : 163 ( 87 equ; 0 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 364 ( 43 ~; 33 |; 5 &; 272 @)
% ( 5 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 155 ( 152 usr; 26 con; 0-3 aty)
% Number of variables : 25 ( 0 ^; 25 !; 0 ?; 25 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
real: $tType ).
thf(type_def_6,type,
produc957004601l_real: $tType ).
thf(type_def_8,type,
produc443621487t_real: $tType ).
thf(type_def_9,type,
produc1923333543_set_a: $tType ).
thf(type_def_10,type,
product_prod_real_a: $tType ).
thf(type_def_11,type,
produc391212143l_real: $tType ).
thf(type_def_12,type,
produc1232925073real_a: $tType ).
thf(type_def_13,type,
product_prod_a_real: $tType ).
thf(type_def_14,type,
produc1396156303t_real: $tType ).
thf(type_def_15,type,
product_prod_a_set_a: $tType ).
thf(type_def_16,type,
product_prod_a_a: $tType ).
thf(type_def_17,type,
set_Pr147102617l_real: $tType ).
thf(type_def_18,type,
set_Pr1928503567a_real: $tType ).
thf(type_def_19,type,
set_Product_prod_a_a: $tType ).
thf(type_def_20,type,
set_real: $tType ).
thf(type_def_21,type,
set_se1596668135od_a_a: $tType ).
thf(type_def_22,type,
set_set_real: $tType ).
thf(type_def_23,type,
set_set_a: $tType ).
thf(type_def_24,type,
set_a: $tType ).
thf(type_def_25,type,
a: $tType ).
thf(type_def_26,type,
produc578556564l_real: $tType ).
thf(type_def_27,type,
produc1088645164real_a: $tType ).
thf(type_def_28,type,
produc826782210l_real: $tType ).
thf(type_def_29,type,
produc2096821232od_a_a: $tType ).
thf(type_def_30,type,
produc1286132450l_real: $tType ).
thf(type_def_31,type,
produc1921647824od_a_a: $tType ).
thf(type_def_32,type,
set_Pr1741234931a_real: $tType ).
thf(type_def_33,type,
set_Pr1948701895od_a_a: $tType ).
thf(type_def_34,type,
produc1826081171a_real: $tType ).
thf(type_def_35,type,
produc1572603623od_a_a: $tType ).
thf(type_def_36,type,
produc348734722a_real: $tType ).
thf(type_def_37,type,
produc1016684094_a_a_a: $tType ).
thf(func_def_0,type,
set_Pr1741234931a_real: $tType ).
thf(func_def_1,type,
produc1826081171a_real: $tType ).
thf(func_def_2,type,
set_Pr1948701895od_a_a: $tType ).
thf(func_def_3,type,
produc1572603623od_a_a: $tType ).
thf(func_def_4,type,
produc826782210l_real: $tType ).
thf(func_def_5,type,
produc578556564l_real: $tType ).
thf(func_def_6,type,
produc1286132450l_real: $tType ).
thf(func_def_7,type,
produc1088645164real_a: $tType ).
thf(func_def_8,type,
produc2096821232od_a_a: $tType ).
thf(func_def_9,type,
produc348734722a_real: $tType ).
thf(func_def_10,type,
produc1921647824od_a_a: $tType ).
thf(func_def_11,type,
produc1016684094_a_a_a: $tType ).
thf(func_def_12,type,
set_Pr147102617l_real: $tType ).
thf(func_def_13,type,
produc391212143l_real: $tType ).
thf(func_def_14,type,
produc443621487t_real: $tType ).
thf(func_def_15,type,
set_se1596668135od_a_a: $tType ).
thf(func_def_16,type,
set_Pr1928503567a_real: $tType ).
thf(func_def_17,type,
produc1396156303t_real: $tType ).
thf(func_def_18,type,
produc1232925073real_a: $tType ).
thf(func_def_19,type,
produc1923333543_set_a: $tType ).
thf(func_def_20,type,
produc957004601l_real: $tType ).
thf(func_def_21,type,
set_Product_prod_a_a: $tType ).
thf(func_def_22,type,
product_prod_a_set_a: $tType ).
thf(func_def_23,type,
product_prod_a_real: $tType ).
thf(func_def_24,type,
product_prod_real_a: $tType ).
thf(func_def_25,type,
set_set_real: $tType ).
thf(func_def_26,type,
product_prod_a_a: $tType ).
thf(func_def_27,type,
set_set_a: $tType ).
thf(func_def_28,type,
set_real: $tType ).
thf(func_def_29,type,
set_a: $tType ).
thf(func_def_30,type,
real: $tType ).
thf(func_def_31,type,
a: $tType ).
thf(func_def_32,type,
one_one_real: real ).
thf(func_def_33,type,
plus_p77862768l_real: produc957004601l_real > produc957004601l_real > produc957004601l_real ).
thf(func_def_34,type,
plus_p2005278886t_real: produc443621487t_real > produc443621487t_real > produc443621487t_real ).
thf(func_def_35,type,
plus_p125342128_set_a: produc1923333543_set_a > produc1923333543_set_a > produc1923333543_set_a ).
thf(func_def_36,type,
plus_p1243198544real_a: product_prod_real_a > product_prod_real_a > product_prod_real_a ).
thf(func_def_37,type,
plus_p1952869542l_real: produc391212143l_real > produc391212143l_real > produc391212143l_real ).
thf(func_def_38,type,
plus_p1582417306real_a: produc1232925073real_a > produc1232925073real_a > produc1232925073real_a ).
thf(func_def_39,type,
plus_p541014626a_real: product_prod_a_real > product_prod_a_real > product_prod_a_real ).
thf(func_def_40,type,
plus_p1745648536t_real: produc1396156303t_real > produc1396156303t_real > produc1396156303t_real ).
thf(func_def_41,type,
plus_p1265825342_set_a: product_prod_a_set_a > product_prod_a_set_a > product_prod_a_set_a ).
thf(func_def_42,type,
plus_p1505579230od_a_a: product_prod_a_a > product_prod_a_a > product_prod_a_a ).
thf(func_def_43,type,
plus_plus_real: real > real > real ).
thf(func_def_44,type,
plus_p1708760016l_real: set_Pr147102617l_real > set_Pr147102617l_real > set_Pr147102617l_real ).
thf(func_def_45,type,
plus_p130512152a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > set_Pr1928503567a_real ).
thf(func_def_46,type,
plus_p634297534od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_47,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(func_def_48,type,
plus_p1613276318od_a_a: set_se1596668135od_a_a > set_se1596668135od_a_a > set_se1596668135od_a_a ).
thf(func_def_49,type,
plus_p768704801t_real: set_set_real > set_set_real > set_set_real ).
thf(func_def_50,type,
plus_plus_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(func_def_51,type,
plus_plus_set_a: set_a > set_a > set_a ).
thf(func_def_52,type,
plus_plus_a: a > a > a ).
thf(func_def_53,type,
zero_z342251165l_real: produc578556564l_real ).
thf(func_def_54,type,
zero_z1937883107real_a: produc1088645164real_a ).
thf(func_def_55,type,
zero_z590476811l_real: produc826782210l_real ).
thf(func_def_56,type,
zero_z396988537od_a_a: produc2096821232od_a_a ).
thf(func_def_57,type,
zero_z659284464l_real: produc957004601l_real ).
thf(func_def_58,type,
zero_z1407338960real_a: product_prod_real_a ).
thf(func_def_59,type,
zero_z2135370393l_real: produc1286132450l_real ).
thf(func_def_60,type,
zero_z53797895od_a_a: produc1921647824od_a_a ).
thf(func_def_61,type,
zero_z705155042a_real: product_prod_a_real ).
thf(func_def_62,type,
zero_z950819678od_a_a: product_prod_a_a ).
thf(func_def_63,type,
zero_zero_real: real ).
thf(func_def_64,type,
zero_z257140542od_a_a: set_Product_prod_a_a ).
thf(func_def_65,type,
zero_zero_set_real: set_real ).
thf(func_def_66,type,
zero_zero_set_a: set_a ).
thf(func_def_67,type,
zero_zero_a: a ).
thf(func_def_68,type,
ord_less_real: real > real > $o ).
thf(func_def_69,type,
ord_le108336051od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_70,type,
ord_le1342644953l_real: produc957004601l_real > produc957004601l_real > $o ).
thf(func_def_71,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_72,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_73,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_74,type,
prefer1113819806a_real: set_Pr1741234931a_real > $o ).
thf(func_def_75,type,
prefer225425826od_a_a: set_Pr1948701895od_a_a > $o ).
thf(func_def_76,type,
prefer1247792113f_real: set_Pr147102617l_real > $o ).
thf(func_def_77,type,
prefer529818233pref_a: set_Product_prod_a_a > $o ).
thf(func_def_78,type,
produc1175086478l_real: produc957004601l_real > real > produc578556564l_real ).
thf(func_def_79,type,
produc1430099868real_a: produc957004601l_real > a > produc1088645164real_a ).
thf(func_def_80,type,
produc886678603a_real: product_prod_a_real > product_prod_a_real > produc1826081171a_real ).
thf(func_def_81,type,
produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a ).
thf(func_def_82,type,
produc454274172a_real: product_prod_a_a > real > produc348734722a_real ).
thf(func_def_83,type,
produc2061588782_a_a_a: product_prod_a_a > a > produc1016684094_a_a_a ).
thf(func_def_84,type,
produc1926903988l_real: real > produc957004601l_real > produc826782210l_real ).
thf(func_def_85,type,
produc454550562od_a_a: real > product_prod_a_a > produc2096821232od_a_a ).
thf(func_def_86,type,
produc705216881l_real: real > real > produc957004601l_real ).
thf(func_def_87,type,
produc247649703t_real: real > set_real > produc443621487t_real ).
thf(func_def_88,type,
produc1872317017_set_a: real > set_a > produc1923333543_set_a ).
thf(func_def_89,type,
product_Pair_real_a: real > a > product_prod_real_a ).
thf(func_def_90,type,
produc1812461223l_real: set_real > real > produc391212143l_real ).
thf(func_def_91,type,
produc617496131real_a: set_real > a > produc1232925073real_a ).
thf(func_def_92,type,
produc64206290l_real: a > produc957004601l_real > produc1286132450l_real ).
thf(func_def_93,type,
produc1299253312od_a_a: a > product_prod_a_a > produc1921647824od_a_a ).
thf(func_def_94,type,
product_Pair_a_real: a > real > product_prod_a_real ).
thf(func_def_95,type,
produc1391501065t_real: a > set_real > produc1396156303t_real ).
thf(func_def_96,type,
product_Pair_a_set_a: a > set_a > product_prod_a_set_a ).
thf(func_def_97,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_98,type,
real_V368978776od_a_a: real > produc1572603623od_a_a > produc1572603623od_a_a ).
thf(func_def_99,type,
real_V195069393a_real: real > produc348734722a_real > produc348734722a_real ).
thf(func_def_100,type,
real_V1538726831_a_a_a: real > produc1016684094_a_a_a > produc1016684094_a_a_a ).
thf(func_def_101,type,
real_V1943155903od_a_a: real > produc2096821232od_a_a > produc2096821232od_a_a ).
thf(func_def_102,type,
real_V1139189034l_real: real > produc957004601l_real > produc957004601l_real ).
thf(func_def_103,type,
real_V1214981142real_a: real > product_prod_real_a > product_prod_real_a ).
thf(func_def_104,type,
real_V296206913od_a_a: real > produc1921647824od_a_a > produc1921647824od_a_a ).
thf(func_def_105,type,
real_V512797224a_real: real > product_prod_a_real > product_prod_a_real ).
thf(func_def_106,type,
real_V543523736od_a_a: real > product_prod_a_a > product_prod_a_a ).
thf(func_def_107,type,
real_V453051771R_real: real > real > real ).
thf(func_def_108,type,
real_V1035702895aleR_a: real > a > a ).
thf(func_def_109,type,
collec1300223524l_real: ( produc957004601l_real > $o ) > set_Pr147102617l_real ).
thf(func_def_110,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_111,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_112,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_113,type,
member384820540a_real: produc1826081171a_real > set_Pr1741234931a_real > $o ).
thf(func_def_114,type,
member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
thf(func_def_115,type,
member1068169442l_real: produc957004601l_real > set_Pr147102617l_real > $o ).
thf(func_def_116,type,
member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).
thf(func_def_117,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_118,type,
member_real: real > set_real > $o ).
thf(func_def_119,type,
member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
thf(func_def_120,type,
member_set_real: set_real > set_set_real > $o ).
thf(func_def_121,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_122,type,
member_a: a > set_a > $o ).
thf(func_def_123,type,
relation: set_Product_prod_a_a ).
thf(func_def_124,type,
u: real ).
thf(func_def_125,type,
v: real ).
thf(func_def_126,type,
x: a ).
thf(func_def_127,type,
y: a ).
thf(func_def_141,type,
sP0: real > real > real > $o ).
thf(func_def_142,type,
sP1: real > real > real > $o ).
thf(func_def_143,type,
sP2: real > real > real > $o ).
thf(func_def_144,type,
sK3: set_Pr147102617l_real > real ).
thf(func_def_145,type,
sK4: set_Pr147102617l_real > real ).
thf(func_def_146,type,
sK5: set_Pr147102617l_real > real ).
thf(func_def_147,type,
sK6: set_Pr147102617l_real > real ).
thf(func_def_148,type,
sK7: ( product_prod_a_a > $o ) > a ).
thf(func_def_149,type,
sK8: ( product_prod_a_a > $o ) > a ).
thf(func_def_150,type,
sK9: set_Product_prod_a_a > a ).
thf(func_def_151,type,
sK10: set_Product_prod_a_a > a ).
thf(func_def_152,type,
sK11: set_Product_prod_a_a > real ).
thf(func_def_153,type,
sK12: set_Product_prod_a_a > real ).
thf(func_def_154,type,
sK13: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > product_prod_a_a ).
thf(func_def_155,type,
sK14: product_prod_a_a > a ).
thf(func_def_156,type,
sK15: product_prod_a_a > a ).
thf(func_def_157,type,
sK16: set_Product_prod_a_a > a ).
thf(func_def_158,type,
sK17: set_Product_prod_a_a > real ).
thf(func_def_159,type,
sK18: set_Product_prod_a_a > a ).
thf(func_def_160,type,
sK19: set_Product_prod_a_a > real ).
thf(func_def_161,type,
sK20: product_prod_a_a > a ).
thf(func_def_162,type,
sK21: product_prod_a_a > a ).
thf(func_def_163,type,
sK22: ( real > real ) > real ).
thf(func_def_164,type,
sK23: ( real > real ) > real ).
thf(func_def_165,type,
sK24: set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a > product_prod_a_a ).
thf(func_def_166,type,
sK25: set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a > product_prod_a_a ).
thf(func_def_167,type,
sK26: ( product_prod_a_a > $o ) > a ).
thf(func_def_168,type,
sK27: ( product_prod_a_a > $o ) > a ).
thf(func_def_169,type,
sK28: ( real > real ) > real ).
thf(func_def_170,type,
sK29: ( real > real ) > real ).
thf(func_def_171,type,
sK30: set_Pr1741234931a_real > product_prod_a_real ).
thf(func_def_172,type,
sK31: set_Pr1741234931a_real > product_prod_a_real ).
thf(func_def_173,type,
sK32: set_Pr1741234931a_real > real ).
thf(func_def_174,type,
sK33: set_Pr1741234931a_real > real ).
thf(func_def_175,type,
sK34: set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a ).
thf(func_def_176,type,
sK35: real > set_real > set_real > real ).
thf(func_def_177,type,
sK36: real > set_real > set_real > real ).
thf(func_def_178,type,
sK37: set_Pr1948701895od_a_a > product_prod_a_a ).
thf(func_def_179,type,
sK38: set_Pr1948701895od_a_a > product_prod_a_a ).
thf(func_def_180,type,
sK39: set_Pr1948701895od_a_a > real ).
thf(func_def_181,type,
sK40: set_Pr1948701895od_a_a > real ).
thf(func_def_182,type,
sK41: real > real > real ).
thf(func_def_183,type,
sK42: a > set_a > set_a > a ).
thf(func_def_184,type,
sK43: a > set_a > set_a > a ).
thf(func_def_185,type,
sK44: set_Pr147102617l_real > real ).
thf(func_def_186,type,
sK45: set_Pr147102617l_real > real ).
thf(func_def_187,type,
sK46: set_Pr147102617l_real > real ).
thf(func_def_188,type,
sK47: set_Pr147102617l_real > real ).
thf(func_def_189,type,
sK48: ( real > real ) > real ).
thf(func_def_190,type,
sK49: ( real > real ) > real ).
thf(func_def_191,type,
sK50: ( real > real ) > real ).
thf(func_def_192,type,
sK51: ( real > real ) > real ).
thf(func_def_194,type,
ph53:
!>[X0: $tType] : X0 ).
thf(f2440,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1937,f1942,f1976,f2435]) ).
thf(f2435,plain,
( ~ spl52_5
| ~ spl52_7 ),
inference(avatar_split_clause,[status(thm)],[f2434,f1934,f1925]) ).
thf(f1925,definition,
( spl52_5
<=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl52_5])],[avatar_definition]) ).
thf(f1934,definition,
( spl52_7
<=> ( u = one_one_real ) ),
introduced(definition,[new_symbols(naming,[spl52_7])],[avatar_definition]) ).
thf(f2434,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
!= $true )
| ~ spl52_7 ),
inference(forward_demodulation,[status(thm)],[f2433,f1822]) ).
thf(f1822,plain,
! [X0: a] :
( ( plus_plus_a @ X0 @ zero_zero_a )
= X0 ),
inference(equality_resolution,[status(thm)],[f1486]) ).
thf(f1486,plain,
! [X0: a,X1: a] :
( ( ( plus_plus_a @ X0 @ X1 )
= X0 )
| ( zero_zero_a != X1 ) ),
inference(cnf_transformation,[status(thm)],[f1199]) ).
thf(f1199,plain,
! [X0: a,X1: a] :
( ( ( ( plus_plus_a @ X0 @ X1 )
= X0 )
| ( zero_zero_a != X1 ) )
& ( ( zero_zero_a = X1 )
| ( ( plus_plus_a @ X0 @ X1 )
!= X0 ) ) ),
inference(nnf_transformation,[status(thm)],[f524]) ).
thf(f524,plain,
! [X0: a,X1: a] :
( ( ( plus_plus_a @ X0 @ X1 )
= X0 )
<=> ( zero_zero_a = X1 ) ),
inference(fool_elimination,[status(thm)],[f523]) ).
thf(f523,plain,
! [X0: a,X1: a] :
( ( zero_zero_a = X1 )
= ( ( plus_plus_a @ X0 @ X1 )
= X0 ) ),
inference(rectify,[status(thm)],[f82]) ).
thf(f82,axiom,
! [X4: a,X5: a] :
( ( zero_zero_a = X5 )
= ( ( plus_plus_a @ X4 @ X5 )
= X4 ) ),
file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_81_add__cancel__left__right) ).
thf(f2433,plain,
( ( $true
!= ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ x @ zero_zero_a ) @ y ) @ relation ) )
| ~ spl52_7 ),
inference(forward_demodulation,[status(thm)],[f2432,f1731]) ).
thf(f1731,plain,
! [X0: a] :
( zero_zero_a
= ( real_V1035702895aleR_a @ zero_zero_real @ X0 ) ),
inference(cnf_transformation,[status(thm)],[f239]) ).
thf(f239,axiom,
! [X0: a] :
( zero_zero_a
= ( real_V1035702895aleR_a @ zero_zero_real @ X0 ) ),
file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_238_scale__zero__left) ).
thf(f2432,plain,
( ( $true
!= ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ x @ ( real_V1035702895aleR_a @ zero_zero_real @ y ) ) @ y ) @ relation ) )
| ~ spl52_7 ),
inference(forward_demodulation,[status(thm)],[f2431,f1420]) ).
thf(f1420,plain,
! [X0: a] :
( ( real_V1035702895aleR_a @ one_one_real @ X0 )
= X0 ),
inference(cnf_transformation,[status(thm)],[f242]) ).
thf(f242,axiom,
! [X0: a] :
( ( real_V1035702895aleR_a @ one_one_real @ X0 )
= X0 ),
file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_241_scaleR__one) ).
thf(f2431,plain,
( ( $true
!= ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ one_one_real @ x ) @ ( real_V1035702895aleR_a @ zero_zero_real @ y ) ) @ y ) @ relation ) )
| ~ spl52_7 ),
inference(forward_demodulation,[status(thm)],[f2402,f1936]) ).
thf(f1936,plain,
( ( u = one_one_real )
| ~ spl52_7 ),
inference(avatar_component_clause,[status(thm)],[f1934]) ).
thf(f2402,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ zero_zero_real @ y ) ) @ y ) @ relation )
!= $true )
| ~ spl52_7 ),
inference(superposition,[status(thm)],[f1364,f2243]) ).
thf(f2243,plain,
( ( zero_zero_real = v )
| ~ spl52_7 ),
inference(trivial_inequality_removal,[status(thm)],[f2153]) ).
thf(f2153,plain,
( ( zero_zero_real = v )
| ( one_one_real != one_one_real )
| ~ spl52_7 ),
inference(superposition,[status(thm)],[f1377,f1945]) ).
thf(f1945,plain,
( ( one_one_real
= ( plus_plus_real @ one_one_real @ v ) )
| ~ spl52_7 ),
inference(superposition,[status(thm)],[f1338,f1936]) ).
thf(f1338,plain,
( one_one_real
= ( plus_plus_real @ u @ v ) ),
inference(cnf_transformation,[status(thm)],[f10]) ).
thf(f10,axiom,
( one_one_real
= ( plus_plus_real @ u @ v ) ),
file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_9_assms_I5_J) ).
thf(f1377,plain,
! [X0: real,X1: real] :
( ( ( plus_plus_real @ X0 @ X1 )
!= X0 )
| ( zero_zero_real = X1 ) ),
inference(cnf_transformation,[status(thm)],[f1157]) ).
thf(f1157,plain,
! [X0: real,X1: real] :
( ( ( zero_zero_real = X1 )
| ( ( plus_plus_real @ X0 @ X1 )
!= X0 ) )
& ( ( ( plus_plus_real @ X0 @ X1 )
= X0 )
| ( zero_zero_real != X1 ) ) ),
inference(nnf_transformation,[status(thm)],[f619]) ).
thf(f619,plain,
! [X0: real,X1: real] :
( ( zero_zero_real = X1 )
<=> ( ( plus_plus_real @ X0 @ X1 )
= X0 ) ),
inference(fool_elimination,[status(thm)],[f618]) ).
thf(f618,plain,
! [X0: real,X1: real] :
( ( ( plus_plus_real @ X0 @ X1 )
= X0 )
= ( zero_zero_real = X1 ) ),
inference(rectify,[status(thm)],[f73]) ).
thf(f73,axiom,
! [X4: real,X5: real] :
( ( ( plus_plus_real @ X4 @ X5 )
= X4 )
= ( zero_zero_real = X5 ) ),
file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_72_add__cancel__right__right) ).
thf(f1364,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
!= $true ),
inference(cnf_transformation,[status(thm)],[f885]) ).
thf(f885,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
!= $true ),
inference(flattening,[status(thm)],[f710]) ).
thf(f710,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
!= $true ),
inference(fool_elimination,[status(thm)],[f709]) ).
thf(f709,plain,
~ ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ),
inference(rectify,[status(thm)],[f355]) ).
thf(f355,negated_conjecture,
~ ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ),
inference(negated_conjecture,[status(cth)],[f354]) ).
thf(f354,conjecture,
member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation,
file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',conj_0) ).
thf(f1976,plain,
spl52_5,
inference(avatar_contradiction_clause,[status(thm)],[f1975]) ).
thf(f1975,plain,
( $false
| spl52_5 ),
inference(trivial_inequality_removal,[status(thm)],[f1962]) ).
thf(f1962,plain,
( ( $true != $true )
| spl52_5 ),
inference(superposition,[status(thm)],[f1927,f1354]) ).
thf(f1354,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
= $true ),
inference(cnf_transformation,[status(thm)],[f506]) ).
thf(f506,plain,
( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
= $true ),
inference(fool_elimination,[status(thm)],[f505]) ).
thf(f505,plain,
member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation,
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation,
file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_5_assms_I2_J) ).
thf(f1927,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation )
!= $true )
| spl52_5 ),
inference(avatar_component_clause,[status(thm)],[f1925]) ).
thf(f1942,plain,
~ spl52_6,
inference(avatar_split_clause,[status(thm)],[f1903,f1930]) ).
thf(f1930,definition,
( spl52_6
<=> ( u = zero_zero_real ) ),
introduced(definition,[new_symbols(naming,[spl52_6])],[avatar_definition]) ).
thf(f1903,plain,
u != zero_zero_real,
inference(trivial_inequality_removal,[status(thm)],[f1885]) ).
thf(f1885,plain,
( ( u != zero_zero_real )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1364,f1649]) ).
thf(f1649,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
= $true )
| ( u != zero_zero_real ) ),
inference(cnf_transformation,[status(thm)],[f1093]) ).
thf(f1093,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
= $true )
| ( u != zero_zero_real ) ),
inference(ennf_transformation,[status(thm)],[f675]) ).
thf(f675,plain,
( ( u = zero_zero_real )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
= $true ) ),
inference(fool_elimination,[status(thm)],[f674]) ).
thf(f674,plain,
( ( u = zero_zero_real )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,axiom,
( ( u = zero_zero_real )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ),
file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_7_u__0) ).
thf(f1937,plain,
( spl52_6
| spl52_7 ),
inference(avatar_split_clause,[status(thm)],[f1896,f1934,f1930]) ).
thf(f1896,plain,
( ( u = zero_zero_real )
| ( u = one_one_real ) ),
inference(trivial_inequality_removal,[status(thm)],[f1884]) ).
thf(f1884,plain,
( ( u = one_one_real )
| ( u = zero_zero_real )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1364,f1295]) ).
thf(f1295,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
= $true )
| ( u = zero_zero_real )
| ( u = one_one_real ) ),
inference(cnf_transformation,[status(thm)],[f986]) ).
thf(f986,plain,
( ( u = zero_zero_real )
| ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
= $true )
| ( u = one_one_real ) ),
inference(flattening,[status(thm)],[f985]) ).
thf(f985,plain,
( ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
= $true )
| ( u = zero_zero_real )
| ( u = one_one_real ) ),
inference(ennf_transformation,[status(thm)],[f502]) ).
thf(f502,plain,
( ( ( u != zero_zero_real )
& ( u != one_one_real ) )
=> ( ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation )
= $true ) ),
inference(fool_elimination,[status(thm)],[f501]) ).
thf(f501,plain,
( ( ( u != zero_zero_real )
& ( u != one_one_real ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,axiom,
( ( ( u != zero_zero_real )
& ( u != one_one_real ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ),
file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_6__092_060open_062u_A_092_060noteq_062_A0_A_092_060and_062_Au_A_092_060noteq_062_A1_A_092_060longrightarrow_062_Au_A_K_092_060_094sub_062R_Ax_A_L_Av_A_K_092_060_094sub_062R_Ay_A_092_060succeq_062_Ay_092_060close_062) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP156^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n019.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:28:09 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.45 thf(spec,logic,$$dhol).
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % File : ITP156^1 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.45 % Domain : Interactive Theorem Proving
% 0.30/0.45 % Problem : Sledgehammer Preferences problem prob_474__6256716_1
% 0.30/0.45 % Version : Especial.
% 0.30/0.45 % English :
% 0.30/0.45
% 0.30/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.45 % Source : [Des21]
% 0.30/0.45 % Names : Preferences/prob_474__6256716_1 [Des21]
% 0.30/0.45
% 0.30/0.45 % Status : Theorem
% 0.30/0.45 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.27 v7.5.0
% 0.30/0.45 % Syntax : Number of formulae : 482 ( 221 unt; 128 typ; 0 def)
% 0.30/0.45 % Number of atoms : 879 ( 416 equ; 0 cnn)
% 0.30/0.45 % Maximal formula atoms : 12 ( 2 avg)
% 0.30/0.45 % Number of connectives : 2819 ( 33 ~; 15 |; 57 &;2451 @)
% 0.30/0.45 % ( 0 <=>; 263 =>; 0 <=; 0 <~>)
% 0.30/0.45 % Maximal formula depth : 19 ( 6 avg)
% 0.30/0.45 % Number of types : 33 ( 32 usr)
% 0.30/0.45 % Number of type conns : 158 ( 158 >; 0 *; 0 +; 0 <<)
% 0.30/0.45 % Number of symbols : 97 ( 96 usr; 21 con; 0-2 aty)
% 0.30/0.45 % Number of variables : 912 ( 22 ^; 888 !; 2 ?; 912 :)
% 0.30/0.45 % SPC : TH0_THM_EQU_NAR
% 0.30/0.45
% 0.30/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:46:49.616
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % Could-be-implicit typings (32)
% 0.30/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
% 0.30/0.45 set_Pr1741234931a_real: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
% 0.30/0.45 produc1826081171a_real: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 0.30/0.45 set_Pr1948701895od_a_a: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.30/0.45 produc1572603623od_a_a: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.30/0.45 produc826782210l_real: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
% 0.30/0.45 produc578556564l_real: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.30/0.45 produc1286132450l_real: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_Mtf__a_J,type,
% 0.30/0.45 produc1088645164real_a: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.30/0.45 produc2096821232od_a_a: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Real__Oreal_J,type,
% 0.30/0.45 produc348734722a_real: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.30/0.45 produc1921647824od_a_a: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 0.30/0.45 produc1016684094_a_a_a: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 0.30/0.45 set_Pr147102617l_real: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Real__Oreal_J,type,
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% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.30/0.45 real_V368978776od_a_a: real > produc1572603623od_a_a > produc1572603623od_a_a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Real__Oreal_J,type,
% 0.30/0.45 real_V195069393a_real: real > produc348734722a_real > produc348734722a_real ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 0.30/0.45 real_V1538726831_a_a_a: real > produc1016684094_a_a_a > produc1016684094_a_a_a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.30/0.45 real_V1943155903od_a_a: real > produc2096821232od_a_a > produc2096821232od_a_a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.30/0.45 real_V1139189034l_real: real > produc957004601l_real > produc957004601l_real ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__a_J,type,
% 0.30/0.45 real_V1214981142real_a: real > product_prod_real_a > product_prod_real_a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.30/0.45 real_V296206913od_a_a: real > produc1921647824od_a_a > produc1921647824od_a_a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.30/0.45 real_V512797224a_real: real > product_prod_a_real > product_prod_a_real ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.30/0.45 real_V543523736od_a_a: real > product_prod_a_a > product_prod_a_a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 0.30/0.45 real_V453051771R_real: real > real > real ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001tf__a,type,
% 0.30/0.45 real_V1035702895aleR_a: real > a > a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.30/0.45 collec1300223524l_real: ( produc957004601l_real > $o ) > set_Pr147102617l_real ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.30/0.45 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 0.30/0.45 collect_real: ( real > $o ) > set_real ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.30/0.45 collect_a: ( a > $o ) > set_a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
% 0.30/0.45 member384820540a_real: produc1826081171a_real > set_Pr1741234931a_real > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.30/0.45 member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.30/0.45 member1068169442l_real: produc957004601l_real > set_Pr147102617l_real > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 0.30/0.45 member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.30/0.45 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001t__Real__Oreal,type,
% 0.30/0.45 member_real: real > set_real > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 0.30/0.45 member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.30/0.45 member_set_real: set_real > set_set_real > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 0.30/0.45 member_set_a: set_a > set_set_a > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member_001tf__a,type,
% 0.30/0.45 member_a: a > set_a > $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_relation,type,
% 0.30/0.45 relation: set_Product_prod_a_a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_u,type,
% 0.30/0.45 u: real ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_v,type,
% 0.30/0.45 v: real ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_x,type,
% 0.30/0.45 x: a ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v_y,type,
% 0.30/0.45 y: a ).
% 0.30/0.45
% 0.30/0.45 % Relevant facts (353)
% 0.30/0.45 thf(fact_0_assms_I1_J,axiom,
% 0.30/0.45 prefer529818233pref_a @ relation ).
% 0.30/0.45
% 0.30/0.45 % assms(1)
% 0.30/0.45 thf(fact_1_indifferent__imp__weak__pref_I2_J,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.30/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.30/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % indifferent_imp_weak_pref(2)
% 0.30/0.45 thf(fact_2_indifferent__imp__weak__pref_I1_J,axiom,
% 0.30/0.45 ! [X: a,Y: a] :
% 0.30/0.45 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.30/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.30/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % indifferent_imp_weak_pref(1)
% 0.30/0.45 thf(fact_3_indiff__trans,axiom,
% 0.30/0.45 ! [X: a,Y: a,Z: a] :
% 0.30/0.45 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.30/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.30/0.45 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.30/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.30/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.30/0.45 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % indiff_trans
% 0.30/0.45 thf(fact_4_strict__trans,axiom,
% 0.30/0.45 ! [X: a,Y: a,Z: a] :
% 0.30/0.45 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 0.30/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 0.30/0.45 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 0.30/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 0.30/0.45 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 0.30/0.45 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % strict_trans
% 0.30/0.45 thf(fact_5_assms_I2_J,axiom,
% 0.30/0.45 member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation ).
% 0.30/0.45
% 0.30/0.45 % assms(2)
% 0.30/0.45 thf(fact_6__092_060open_062u_A_092_060noteq_062_A0_A_092_060and_062_Au_A_092_060noteq_062_A1_A_092_060longrightarrow_062_Au_A_K_092_060_094sub_062R_Ax_A_L_Av_A_K_092_060_094sub_062R_Ay_A_092_060succeq_062_Ay_092_060close_062,axiom,
% 0.30/0.45 ( ( ( u != zero_zero_real )
% 0.30/0.45 & ( u != one_one_real ) )
% 0.30/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % \<open>u \<noteq> 0 \<and> u \<noteq> 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<succeq> y\<close>
% 0.30/0.45 thf(fact_7_u__0,axiom,
% 0.30/0.45 ( ( u = zero_zero_real )
% 0.30/0.45 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ) ).
% 0.30/0.45
% 0.30/0.45 % u_0
% 0.30/0.45 thf(fact_8_assms_I4_J,axiom,
% 0.30/0.45 ord_less_eq_real @ zero_zero_real @ v ).
% 0.30/0.45
% 0.30/0.45 % assms(4)
% 0.30/0.45 thf(fact_9_assms_I5_J,axiom,
% 0.30/0.45 ( ( plus_plus_real @ u @ v )
% 0.30/0.45 = one_one_real ) ).
% 0.30/0.45
% 0.30/0.45 % assms(5)
% 0.30/0.45 thf(fact_10_assms_I3_J,axiom,
% 0.30/0.45 ord_less_eq_real @ zero_zero_real @ u ).
% 0.30/0.45
% 0.30/0.45 % assms(3)
% 0.30/0.45 thf(fact_11_scaleR__Pair,axiom,
% 0.30/0.45 ! [R: real,A: a,B: product_prod_a_a] :
% 0.30/0.45 ( ( real_V296206913od_a_a @ R @ ( produc1299253312od_a_a @ A @ B ) )
% 0.30/0.45 = ( produc1299253312od_a_a @ ( real_V1035702895aleR_a @ R @ A ) @ ( real_V543523736od_a_a @ R @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scaleR_Pair
% 0.30/0.45 thf(fact_12_scaleR__Pair,axiom,
% 0.30/0.45 ! [R: real,A: a,B: real] :
% 0.30/0.45 ( ( real_V512797224a_real @ R @ ( product_Pair_a_real @ A @ B ) )
% 0.30/0.45 = ( product_Pair_a_real @ ( real_V1035702895aleR_a @ R @ A ) @ ( real_V453051771R_real @ R @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scaleR_Pair
% 0.30/0.45 thf(fact_13_scaleR__Pair,axiom,
% 0.30/0.45 ! [R: real,A: product_prod_a_a,B: a] :
% 0.30/0.45 ( ( real_V1538726831_a_a_a @ R @ ( produc2061588782_a_a_a @ A @ B ) )
% 0.30/0.45 = ( produc2061588782_a_a_a @ ( real_V543523736od_a_a @ R @ A ) @ ( real_V1035702895aleR_a @ R @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scaleR_Pair
% 0.30/0.45 thf(fact_14_scaleR__Pair,axiom,
% 0.30/0.45 ! [R: real,A: product_prod_a_a,B: product_prod_a_a] :
% 0.30/0.45 ( ( real_V368978776od_a_a @ R @ ( produc1474507607od_a_a @ A @ B ) )
% 0.30/0.45 = ( produc1474507607od_a_a @ ( real_V543523736od_a_a @ R @ A ) @ ( real_V543523736od_a_a @ R @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scaleR_Pair
% 0.30/0.45 thf(fact_15_scaleR__Pair,axiom,
% 0.30/0.45 ! [R: real,A: product_prod_a_a,B: real] :
% 0.30/0.45 ( ( real_V195069393a_real @ R @ ( produc454274172a_real @ A @ B ) )
% 0.30/0.45 = ( produc454274172a_real @ ( real_V543523736od_a_a @ R @ A ) @ ( real_V453051771R_real @ R @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scaleR_Pair
% 0.30/0.45 thf(fact_16_scaleR__Pair,axiom,
% 0.30/0.45 ! [R: real,A: real,B: a] :
% 0.30/0.45 ( ( real_V1214981142real_a @ R @ ( product_Pair_real_a @ A @ B ) )
% 0.30/0.45 = ( product_Pair_real_a @ ( real_V453051771R_real @ R @ A ) @ ( real_V1035702895aleR_a @ R @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scaleR_Pair
% 0.30/0.45 thf(fact_17_scaleR__Pair,axiom,
% 0.30/0.45 ! [R: real,A: real,B: product_prod_a_a] :
% 0.30/0.45 ( ( real_V1943155903od_a_a @ R @ ( produc454550562od_a_a @ A @ B ) )
% 0.30/0.45 = ( produc454550562od_a_a @ ( real_V453051771R_real @ R @ A ) @ ( real_V543523736od_a_a @ R @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scaleR_Pair
% 0.30/0.45 thf(fact_18_scaleR__Pair,axiom,
% 0.30/0.45 ! [R: real,A: real,B: real] :
% 0.30/0.45 ( ( real_V1139189034l_real @ R @ ( produc705216881l_real @ A @ B ) )
% 0.30/0.45 = ( produc705216881l_real @ ( real_V453051771R_real @ R @ A ) @ ( real_V453051771R_real @ R @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scaleR_Pair
% 0.30/0.45 thf(fact_19_scaleR__Pair,axiom,
% 0.30/0.45 ! [R: real,A: a,B: a] :
% 0.30/0.45 ( ( real_V543523736od_a_a @ R @ ( product_Pair_a_a @ A @ B ) )
% 0.30/0.45 = ( product_Pair_a_a @ ( real_V1035702895aleR_a @ R @ A ) @ ( real_V1035702895aleR_a @ R @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scaleR_Pair
% 0.30/0.45 thf(fact_20_scale__prod,axiom,
% 0.30/0.45 ! [X: real,A: a,B: product_prod_a_a] :
% 0.30/0.45 ( ( real_V296206913od_a_a @ X @ ( produc1299253312od_a_a @ A @ B ) )
% 0.30/0.45 = ( produc1299253312od_a_a @ ( real_V1035702895aleR_a @ X @ A ) @ ( real_V543523736od_a_a @ X @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scale_prod
% 0.30/0.45 thf(fact_21_scale__prod,axiom,
% 0.30/0.45 ! [X: real,A: a,B: real] :
% 0.30/0.45 ( ( real_V512797224a_real @ X @ ( product_Pair_a_real @ A @ B ) )
% 0.30/0.45 = ( product_Pair_a_real @ ( real_V1035702895aleR_a @ X @ A ) @ ( real_V453051771R_real @ X @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scale_prod
% 0.30/0.45 thf(fact_22_scale__prod,axiom,
% 0.30/0.45 ! [X: real,A: product_prod_a_a,B: a] :
% 0.30/0.45 ( ( real_V1538726831_a_a_a @ X @ ( produc2061588782_a_a_a @ A @ B ) )
% 0.30/0.45 = ( produc2061588782_a_a_a @ ( real_V543523736od_a_a @ X @ A ) @ ( real_V1035702895aleR_a @ X @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scale_prod
% 0.30/0.45 thf(fact_23_scale__prod,axiom,
% 0.30/0.45 ! [X: real,A: product_prod_a_a,B: product_prod_a_a] :
% 0.30/0.45 ( ( real_V368978776od_a_a @ X @ ( produc1474507607od_a_a @ A @ B ) )
% 0.30/0.45 = ( produc1474507607od_a_a @ ( real_V543523736od_a_a @ X @ A ) @ ( real_V543523736od_a_a @ X @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scale_prod
% 0.30/0.45 thf(fact_24_scale__prod,axiom,
% 0.30/0.45 ! [X: real,A: product_prod_a_a,B: real] :
% 0.30/0.45 ( ( real_V195069393a_real @ X @ ( produc454274172a_real @ A @ B ) )
% 0.30/0.45 = ( produc454274172a_real @ ( real_V543523736od_a_a @ X @ A ) @ ( real_V453051771R_real @ X @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scale_prod
% 0.30/0.45 thf(fact_25_scale__prod,axiom,
% 0.30/0.45 ! [X: real,A: real,B: a] :
% 0.30/0.45 ( ( real_V1214981142real_a @ X @ ( product_Pair_real_a @ A @ B ) )
% 0.30/0.45 = ( product_Pair_real_a @ ( real_V453051771R_real @ X @ A ) @ ( real_V1035702895aleR_a @ X @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scale_prod
% 0.30/0.45 thf(fact_26_scale__prod,axiom,
% 0.30/0.45 ! [X: real,A: real,B: product_prod_a_a] :
% 0.30/0.45 ( ( real_V1943155903od_a_a @ X @ ( produc454550562od_a_a @ A @ B ) )
% 0.30/0.45 = ( produc454550562od_a_a @ ( real_V453051771R_real @ X @ A ) @ ( real_V543523736od_a_a @ X @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scale_prod
% 0.30/0.45 thf(fact_27_scale__prod,axiom,
% 0.30/0.45 ! [X: real,A: real,B: real] :
% 0.30/0.45 ( ( real_V1139189034l_real @ X @ ( produc705216881l_real @ A @ B ) )
% 0.30/0.45 = ( produc705216881l_real @ ( real_V453051771R_real @ X @ A ) @ ( real_V453051771R_real @ X @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scale_prod
% 0.30/0.45 thf(fact_28_scale__prod,axiom,
% 0.30/0.45 ! [X: real,A: a,B: a] :
% 0.30/0.45 ( ( real_V543523736od_a_a @ X @ ( product_Pair_a_a @ A @ B ) )
% 0.30/0.45 = ( product_Pair_a_a @ ( real_V1035702895aleR_a @ X @ A ) @ ( real_V1035702895aleR_a @ X @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % scale_prod
% 0.30/0.45 thf(fact_29_add__Pair,axiom,
% 0.30/0.45 ! [A: a,B: a,C: a,D: a] :
% 0.30/0.45 ( ( plus_p1505579230od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Pair_a_a @ C @ D ) )
% 0.30/0.45 = ( product_Pair_a_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_30_add__Pair,axiom,
% 0.30/0.45 ! [A: a,B: real,C: a,D: real] :
% 0.30/0.45 ( ( plus_p541014626a_real @ ( product_Pair_a_real @ A @ B ) @ ( product_Pair_a_real @ C @ D ) )
% 0.30/0.45 = ( product_Pair_a_real @ ( plus_plus_a @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_31_add__Pair,axiom,
% 0.30/0.45 ! [A: real,B: a,C: real,D: a] :
% 0.30/0.45 ( ( plus_p1243198544real_a @ ( product_Pair_real_a @ A @ B ) @ ( product_Pair_real_a @ C @ D ) )
% 0.30/0.45 = ( product_Pair_real_a @ ( plus_plus_real @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_32_add__Pair,axiom,
% 0.30/0.45 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.45 ( ( plus_p77862768l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
% 0.30/0.45 = ( produc705216881l_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_33_add__Pair,axiom,
% 0.30/0.45 ! [A: a,B: set_real,C: a,D: set_real] :
% 0.30/0.45 ( ( plus_p1745648536t_real @ ( produc1391501065t_real @ A @ B ) @ ( produc1391501065t_real @ C @ D ) )
% 0.30/0.45 = ( produc1391501065t_real @ ( plus_plus_a @ A @ C ) @ ( plus_plus_set_real @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_34_add__Pair,axiom,
% 0.30/0.45 ! [A: a,B: set_a,C: a,D: set_a] :
% 0.30/0.45 ( ( plus_p1265825342_set_a @ ( product_Pair_a_set_a @ A @ B ) @ ( product_Pair_a_set_a @ C @ D ) )
% 0.30/0.45 = ( product_Pair_a_set_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_set_a @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_35_add__Pair,axiom,
% 0.30/0.45 ! [A: real,B: set_real,C: real,D: set_real] :
% 0.30/0.45 ( ( plus_p2005278886t_real @ ( produc247649703t_real @ A @ B ) @ ( produc247649703t_real @ C @ D ) )
% 0.30/0.45 = ( produc247649703t_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_set_real @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_36_add__Pair,axiom,
% 0.30/0.45 ! [A: real,B: set_a,C: real,D: set_a] :
% 0.30/0.45 ( ( plus_p125342128_set_a @ ( produc1872317017_set_a @ A @ B ) @ ( produc1872317017_set_a @ C @ D ) )
% 0.30/0.45 = ( produc1872317017_set_a @ ( plus_plus_real @ A @ C ) @ ( plus_plus_set_a @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_37_add__Pair,axiom,
% 0.30/0.45 ! [A: set_real,B: a,C: set_real,D: a] :
% 0.30/0.45 ( ( plus_p1582417306real_a @ ( produc617496131real_a @ A @ B ) @ ( produc617496131real_a @ C @ D ) )
% 0.30/0.45 = ( produc617496131real_a @ ( plus_plus_set_real @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_38_add__Pair,axiom,
% 0.30/0.45 ! [A: set_real,B: real,C: set_real,D: real] :
% 0.30/0.45 ( ( plus_p1952869542l_real @ ( produc1812461223l_real @ A @ B ) @ ( produc1812461223l_real @ C @ D ) )
% 0.30/0.45 = ( produc1812461223l_real @ ( plus_plus_set_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_Pair
% 0.30/0.45 thf(fact_39_prod_Oinject,axiom,
% 0.30/0.45 ! [X1: real,X2: real,Y1: real,Y2: real] :
% 0.30/0.45 ( ( ( produc705216881l_real @ X1 @ X2 )
% 0.30/0.45 = ( produc705216881l_real @ Y1 @ Y2 ) )
% 0.30/0.45 = ( ( X1 = Y1 )
% 0.30/0.45 & ( X2 = Y2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod.inject
% 0.30/0.45 thf(fact_40_prod_Oinject,axiom,
% 0.30/0.45 ! [X1: real,X2: a,Y1: real,Y2: a] :
% 0.30/0.45 ( ( ( product_Pair_real_a @ X1 @ X2 )
% 0.30/0.45 = ( product_Pair_real_a @ Y1 @ Y2 ) )
% 0.30/0.45 = ( ( X1 = Y1 )
% 0.30/0.45 & ( X2 = Y2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod.inject
% 0.30/0.45 thf(fact_41_prod_Oinject,axiom,
% 0.30/0.45 ! [X1: a,X2: real,Y1: a,Y2: real] :
% 0.30/0.45 ( ( ( product_Pair_a_real @ X1 @ X2 )
% 0.30/0.45 = ( product_Pair_a_real @ Y1 @ Y2 ) )
% 0.30/0.45 = ( ( X1 = Y1 )
% 0.30/0.45 & ( X2 = Y2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod.inject
% 0.30/0.45 thf(fact_42_prod_Oinject,axiom,
% 0.30/0.45 ! [X1: a,X2: a,Y1: a,Y2: a] :
% 0.30/0.45 ( ( ( product_Pair_a_a @ X1 @ X2 )
% 0.30/0.45 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 0.30/0.45 = ( ( X1 = Y1 )
% 0.30/0.45 & ( X2 = Y2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod.inject
% 0.30/0.45 thf(fact_43_old_Oprod_Oinject,axiom,
% 0.30/0.45 ! [A: real,B: real,A2: real,B2: real] :
% 0.30/0.45 ( ( ( produc705216881l_real @ A @ B )
% 0.30/0.45 = ( produc705216881l_real @ A2 @ B2 ) )
% 0.30/0.45 = ( ( A = A2 )
% 0.30/0.45 & ( B = B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.inject
% 0.30/0.45 thf(fact_44_old_Oprod_Oinject,axiom,
% 0.30/0.45 ! [A: real,B: a,A2: real,B2: a] :
% 0.30/0.45 ( ( ( product_Pair_real_a @ A @ B )
% 0.30/0.45 = ( product_Pair_real_a @ A2 @ B2 ) )
% 0.30/0.45 = ( ( A = A2 )
% 0.30/0.45 & ( B = B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.inject
% 0.30/0.45 thf(fact_45_old_Oprod_Oinject,axiom,
% 0.30/0.45 ! [A: a,B: real,A2: a,B2: real] :
% 0.30/0.45 ( ( ( product_Pair_a_real @ A @ B )
% 0.30/0.45 = ( product_Pair_a_real @ A2 @ B2 ) )
% 0.30/0.45 = ( ( A = A2 )
% 0.30/0.45 & ( B = B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.inject
% 0.30/0.45 thf(fact_46_old_Oprod_Oinject,axiom,
% 0.30/0.45 ! [A: a,B: a,A2: a,B2: a] :
% 0.30/0.45 ( ( ( product_Pair_a_a @ A @ B )
% 0.30/0.45 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.30/0.45 = ( ( A = A2 )
% 0.30/0.45 & ( B = B2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.inject
% 0.30/0.45 thf(fact_47_set__plus__intro,axiom,
% 0.30/0.45 ! [A: produc957004601l_real,C2: set_Pr147102617l_real,B: produc957004601l_real,D2: set_Pr147102617l_real] :
% 0.30/0.45 ( ( member1068169442l_real @ A @ C2 )
% 0.30/0.45 => ( ( member1068169442l_real @ B @ D2 )
% 0.30/0.45 => ( member1068169442l_real @ ( plus_p77862768l_real @ A @ B ) @ ( plus_p1708760016l_real @ C2 @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_plus_intro
% 0.30/0.45 thf(fact_48_set__plus__intro,axiom,
% 0.30/0.45 ! [A: product_prod_a_real,C2: set_Pr1928503567a_real,B: product_prod_a_real,D2: set_Pr1928503567a_real] :
% 0.30/0.45 ( ( member1103263856a_real @ A @ C2 )
% 0.30/0.45 => ( ( member1103263856a_real @ B @ D2 )
% 0.30/0.45 => ( member1103263856a_real @ ( plus_p541014626a_real @ A @ B ) @ ( plus_p130512152a_real @ C2 @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_plus_intro
% 0.30/0.45 thf(fact_49_set__plus__intro,axiom,
% 0.30/0.45 ! [A: set_Product_prod_a_a,C2: set_se1596668135od_a_a,B: set_Product_prod_a_a,D2: set_se1596668135od_a_a] :
% 0.30/0.45 ( ( member1838126896od_a_a @ A @ C2 )
% 0.30/0.45 => ( ( member1838126896od_a_a @ B @ D2 )
% 0.30/0.45 => ( member1838126896od_a_a @ ( plus_p634297534od_a_a @ A @ B ) @ ( plus_p1613276318od_a_a @ C2 @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_plus_intro
% 0.30/0.45 thf(fact_50_set__plus__intro,axiom,
% 0.30/0.45 ! [A: set_real,C2: set_set_real,B: set_real,D2: set_set_real] :
% 0.30/0.45 ( ( member_set_real @ A @ C2 )
% 0.30/0.45 => ( ( member_set_real @ B @ D2 )
% 0.30/0.45 => ( member_set_real @ ( plus_plus_set_real @ A @ B ) @ ( plus_p768704801t_real @ C2 @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_plus_intro
% 0.30/0.45 thf(fact_51_set__plus__intro,axiom,
% 0.30/0.45 ! [A: set_a,C2: set_set_a,B: set_a,D2: set_set_a] :
% 0.30/0.45 ( ( member_set_a @ A @ C2 )
% 0.30/0.45 => ( ( member_set_a @ B @ D2 )
% 0.30/0.45 => ( member_set_a @ ( plus_plus_set_a @ A @ B ) @ ( plus_plus_set_set_a @ C2 @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_plus_intro
% 0.30/0.45 thf(fact_52_set__plus__intro,axiom,
% 0.30/0.45 ! [A: product_prod_a_a,C2: set_Product_prod_a_a,B: product_prod_a_a,D2: set_Product_prod_a_a] :
% 0.30/0.45 ( ( member449909584od_a_a @ A @ C2 )
% 0.30/0.45 => ( ( member449909584od_a_a @ B @ D2 )
% 0.30/0.45 => ( member449909584od_a_a @ ( plus_p1505579230od_a_a @ A @ B ) @ ( plus_p634297534od_a_a @ C2 @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_plus_intro
% 0.30/0.45 thf(fact_53_set__plus__intro,axiom,
% 0.30/0.45 ! [A: a,C2: set_a,B: a,D2: set_a] :
% 0.30/0.45 ( ( member_a @ A @ C2 )
% 0.30/0.45 => ( ( member_a @ B @ D2 )
% 0.30/0.45 => ( member_a @ ( plus_plus_a @ A @ B ) @ ( plus_plus_set_a @ C2 @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_plus_intro
% 0.30/0.45 thf(fact_54_set__plus__intro,axiom,
% 0.30/0.45 ! [A: real,C2: set_real,B: real,D2: set_real] :
% 0.30/0.45 ( ( member_real @ A @ C2 )
% 0.30/0.45 => ( ( member_real @ B @ D2 )
% 0.30/0.45 => ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % set_plus_intro
% 0.30/0.45 thf(fact_55_add__left__cancel,axiom,
% 0.30/0.45 ! [A: product_prod_a_real,B: product_prod_a_real,C: product_prod_a_real] :
% 0.30/0.45 ( ( ( plus_p541014626a_real @ A @ B )
% 0.30/0.45 = ( plus_p541014626a_real @ A @ C ) )
% 0.30/0.45 = ( B = C ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_left_cancel
% 0.30/0.45 thf(fact_56_add__left__cancel,axiom,
% 0.30/0.45 ! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a] :
% 0.30/0.45 ( ( ( plus_p1505579230od_a_a @ A @ B )
% 0.30/0.45 = ( plus_p1505579230od_a_a @ A @ C ) )
% 0.30/0.45 = ( B = C ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_left_cancel
% 0.30/0.45 thf(fact_57_add__left__cancel,axiom,
% 0.30/0.45 ! [A: a,B: a,C: a] :
% 0.30/0.45 ( ( ( plus_plus_a @ A @ B )
% 0.30/0.45 = ( plus_plus_a @ A @ C ) )
% 0.30/0.45 = ( B = C ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_left_cancel
% 0.30/0.45 thf(fact_58_add__left__cancel,axiom,
% 0.30/0.45 ! [A: real,B: real,C: real] :
% 0.30/0.45 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.45 = ( plus_plus_real @ A @ C ) )
% 0.30/0.45 = ( B = C ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_left_cancel
% 0.30/0.45 thf(fact_59_add__right__cancel,axiom,
% 0.30/0.45 ! [B: product_prod_a_real,A: product_prod_a_real,C: product_prod_a_real] :
% 0.30/0.45 ( ( ( plus_p541014626a_real @ B @ A )
% 0.30/0.45 = ( plus_p541014626a_real @ C @ A ) )
% 0.30/0.45 = ( B = C ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_right_cancel
% 0.30/0.45 thf(fact_60_add__right__cancel,axiom,
% 0.30/0.45 ! [B: product_prod_a_a,A: product_prod_a_a,C: product_prod_a_a] :
% 0.30/0.45 ( ( ( plus_p1505579230od_a_a @ B @ A )
% 0.30/0.45 = ( plus_p1505579230od_a_a @ C @ A ) )
% 0.30/0.45 = ( B = C ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_right_cancel
% 0.30/0.45 thf(fact_61_add__right__cancel,axiom,
% 0.30/0.45 ! [B: a,A: a,C: a] :
% 0.30/0.45 ( ( ( plus_plus_a @ B @ A )
% 0.30/0.45 = ( plus_plus_a @ C @ A ) )
% 0.30/0.45 = ( B = C ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_right_cancel
% 0.30/0.45 thf(fact_62_add__right__cancel,axiom,
% 0.30/0.45 ! [B: real,A: real,C: real] :
% 0.30/0.45 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.45 = ( plus_plus_real @ C @ A ) )
% 0.30/0.45 = ( B = C ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_right_cancel
% 0.30/0.45 thf(fact_63_pth__9_I3_J,axiom,
% 0.30/0.45 ! [C: real,X: real,W: real,D: real,Z: real] :
% 0.30/0.45 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ X ) @ Z ) )
% 0.30/0.45 = ( plus_plus_real @ ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) @ ( plus_plus_real @ W @ Z ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % pth_9(3)
% 0.30/0.45 thf(fact_64_add__le__cancel__right,axiom,
% 0.30/0.45 ! [A: produc957004601l_real,C: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.45 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ C ) )
% 0.30/0.45 = ( ord_le1342644953l_real @ A @ B ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_le_cancel_right
% 0.30/0.45 thf(fact_65_add__le__cancel__right,axiom,
% 0.30/0.45 ! [A: real,C: real,B: real] :
% 0.30/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.45 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_le_cancel_right
% 0.30/0.45 thf(fact_66_add__le__cancel__left,axiom,
% 0.30/0.45 ! [C: produc957004601l_real,A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.45 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ C @ A ) @ ( plus_p77862768l_real @ C @ B ) )
% 0.30/0.45 = ( ord_le1342644953l_real @ A @ B ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_le_cancel_left
% 0.30/0.45 thf(fact_67_add__le__cancel__left,axiom,
% 0.30/0.45 ! [C: real,A: real,B: real] :
% 0.30/0.45 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.45 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_le_cancel_left
% 0.30/0.45 thf(fact_68_add__cancel__right__right,axiom,
% 0.30/0.45 ! [A: product_prod_a_real,B: product_prod_a_real] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_p541014626a_real @ A @ B ) )
% 0.30/0.45 = ( B = zero_z705155042a_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_right
% 0.30/0.45 thf(fact_69_add__cancel__right__right,axiom,
% 0.30/0.45 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_p77862768l_real @ A @ B ) )
% 0.30/0.45 = ( B = zero_z659284464l_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_right
% 0.30/0.45 thf(fact_70_add__cancel__right__right,axiom,
% 0.30/0.45 ! [A: product_prod_a_a,B: product_prod_a_a] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_p1505579230od_a_a @ A @ B ) )
% 0.30/0.45 = ( B = zero_z950819678od_a_a ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_right
% 0.30/0.45 thf(fact_71_add__cancel__right__right,axiom,
% 0.30/0.45 ! [A: a,B: a] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_plus_a @ A @ B ) )
% 0.30/0.45 = ( B = zero_zero_a ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_right
% 0.30/0.45 thf(fact_72_add__cancel__right__right,axiom,
% 0.30/0.45 ! [A: real,B: real] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_plus_real @ A @ B ) )
% 0.30/0.45 = ( B = zero_zero_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_right
% 0.30/0.45 thf(fact_73_add__cancel__right__left,axiom,
% 0.30/0.45 ! [A: product_prod_a_real,B: product_prod_a_real] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_p541014626a_real @ B @ A ) )
% 0.30/0.45 = ( B = zero_z705155042a_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_left
% 0.30/0.45 thf(fact_74_add__cancel__right__left,axiom,
% 0.30/0.45 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_p77862768l_real @ B @ A ) )
% 0.30/0.45 = ( B = zero_z659284464l_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_left
% 0.30/0.45 thf(fact_75_add__cancel__right__left,axiom,
% 0.30/0.45 ! [A: product_prod_a_a,B: product_prod_a_a] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_p1505579230od_a_a @ B @ A ) )
% 0.30/0.45 = ( B = zero_z950819678od_a_a ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_left
% 0.30/0.45 thf(fact_76_add__cancel__right__left,axiom,
% 0.30/0.45 ! [A: a,B: a] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_plus_a @ B @ A ) )
% 0.30/0.45 = ( B = zero_zero_a ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_left
% 0.30/0.45 thf(fact_77_add__cancel__right__left,axiom,
% 0.30/0.45 ! [A: real,B: real] :
% 0.30/0.45 ( ( A
% 0.30/0.45 = ( plus_plus_real @ B @ A ) )
% 0.30/0.45 = ( B = zero_zero_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_right_left
% 0.30/0.45 thf(fact_78_add__cancel__left__right,axiom,
% 0.30/0.45 ! [A: product_prod_a_real,B: product_prod_a_real] :
% 0.30/0.45 ( ( ( plus_p541014626a_real @ A @ B )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_z705155042a_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_right
% 0.30/0.45 thf(fact_79_add__cancel__left__right,axiom,
% 0.30/0.45 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.45 ( ( ( plus_p77862768l_real @ A @ B )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_z659284464l_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_right
% 0.30/0.45 thf(fact_80_add__cancel__left__right,axiom,
% 0.30/0.45 ! [A: product_prod_a_a,B: product_prod_a_a] :
% 0.30/0.45 ( ( ( plus_p1505579230od_a_a @ A @ B )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_z950819678od_a_a ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_right
% 0.30/0.45 thf(fact_81_add__cancel__left__right,axiom,
% 0.30/0.45 ! [A: a,B: a] :
% 0.30/0.45 ( ( ( plus_plus_a @ A @ B )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_zero_a ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_right
% 0.30/0.45 thf(fact_82_add__cancel__left__right,axiom,
% 0.30/0.45 ! [A: real,B: real] :
% 0.30/0.45 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_zero_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_right
% 0.30/0.45 thf(fact_83_add__cancel__left__left,axiom,
% 0.30/0.45 ! [B: product_prod_a_real,A: product_prod_a_real] :
% 0.30/0.45 ( ( ( plus_p541014626a_real @ B @ A )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_z705155042a_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_left
% 0.30/0.45 thf(fact_84_add__cancel__left__left,axiom,
% 0.30/0.45 ! [B: produc957004601l_real,A: produc957004601l_real] :
% 0.30/0.45 ( ( ( plus_p77862768l_real @ B @ A )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_z659284464l_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_left
% 0.30/0.45 thf(fact_85_add__cancel__left__left,axiom,
% 0.30/0.45 ! [B: product_prod_a_a,A: product_prod_a_a] :
% 0.30/0.45 ( ( ( plus_p1505579230od_a_a @ B @ A )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_z950819678od_a_a ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_left
% 0.30/0.45 thf(fact_86_add__cancel__left__left,axiom,
% 0.30/0.45 ! [B: a,A: a] :
% 0.30/0.45 ( ( ( plus_plus_a @ B @ A )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_zero_a ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_left
% 0.30/0.45 thf(fact_87_add__cancel__left__left,axiom,
% 0.30/0.45 ! [B: real,A: real] :
% 0.30/0.45 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.45 = A )
% 0.30/0.45 = ( B = zero_zero_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % add_cancel_left_left
% 0.30/0.45 thf(fact_88_double__zero__sym,axiom,
% 0.30/0.45 ! [A: real] :
% 0.30/0.45 ( ( zero_zero_real
% 0.30/0.45 = ( plus_plus_real @ A @ A ) )
% 0.30/0.45 = ( A = zero_zero_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % double_zero_sym
% 0.30/0.45 thf(fact_89_double__zero,axiom,
% 0.30/0.45 ! [A: real] :
% 0.30/0.45 ( ( ( plus_plus_real @ A @ A )
% 0.30/0.45 = zero_zero_real )
% 0.30/0.45 = ( A = zero_zero_real ) ) ).
% 0.30/0.45
% 0.30/0.45 % double_zero
% 0.30/0.45 thf(fact_90_add_Oright__neutral,axiom,
% 0.30/0.45 ! [A: product_prod_a_real] :
% 0.30/0.45 ( ( plus_p541014626a_real @ A @ zero_z705155042a_real )
% 0.30/0.45 = A ) ).
% 0.30/0.45
% 0.30/0.45 % add.right_neutral
% 0.30/0.45 thf(fact_91_add_Oright__neutral,axiom,
% 0.30/0.45 ! [A: set_Product_prod_a_a] :
% 0.30/0.45 ( ( plus_p634297534od_a_a @ A @ zero_z257140542od_a_a )
% 0.30/0.45 = A ) ).
% 0.30/0.45
% 0.30/0.45 % add.right_neutral
% 0.30/0.45 thf(fact_92_add_Oright__neutral,axiom,
% 0.30/0.45 ! [A: set_real] :
% 0.30/0.45 ( ( plus_plus_set_real @ A @ zero_zero_set_real )
% 0.30/0.45 = A ) ).
% 0.30/0.45
% 0.30/0.45 % add.right_neutral
% 0.30/0.45 thf(fact_93_add_Oright__neutral,axiom,
% 0.30/0.45 ! [A: set_a] :
% 0.30/0.45 ( ( plus_plus_set_a @ A @ zero_zero_set_a )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.right_neutral
% 0.30/0.46 thf(fact_94_add_Oright__neutral,axiom,
% 0.30/0.46 ! [A: produc957004601l_real] :
% 0.30/0.46 ( ( plus_p77862768l_real @ A @ zero_z659284464l_real )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.right_neutral
% 0.30/0.46 thf(fact_95_add_Oright__neutral,axiom,
% 0.30/0.46 ! [A: product_prod_a_a] :
% 0.30/0.46 ( ( plus_p1505579230od_a_a @ A @ zero_z950819678od_a_a )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.right_neutral
% 0.30/0.46 thf(fact_96_add_Oright__neutral,axiom,
% 0.30/0.46 ! [A: a] :
% 0.30/0.46 ( ( plus_plus_a @ A @ zero_zero_a )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.right_neutral
% 0.30/0.46 thf(fact_97_add_Oright__neutral,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.right_neutral
% 0.30/0.46 thf(fact_98_add_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: product_prod_a_real] :
% 0.30/0.46 ( ( plus_p541014626a_real @ zero_z705155042a_real @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_neutral
% 0.30/0.46 thf(fact_99_add_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: set_Product_prod_a_a] :
% 0.30/0.46 ( ( plus_p634297534od_a_a @ zero_z257140542od_a_a @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_neutral
% 0.30/0.46 thf(fact_100_add_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: set_real] :
% 0.30/0.46 ( ( plus_plus_set_real @ zero_zero_set_real @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_neutral
% 0.30/0.46 thf(fact_101_add_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: set_a] :
% 0.30/0.46 ( ( plus_plus_set_a @ zero_zero_set_a @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_neutral
% 0.30/0.46 thf(fact_102_add_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: produc957004601l_real] :
% 0.30/0.46 ( ( plus_p77862768l_real @ zero_z659284464l_real @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_neutral
% 0.30/0.46 thf(fact_103_add_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: product_prod_a_a] :
% 0.30/0.46 ( ( plus_p1505579230od_a_a @ zero_z950819678od_a_a @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_neutral
% 0.30/0.46 thf(fact_104_add_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: a] :
% 0.30/0.46 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_neutral
% 0.30/0.46 thf(fact_105_add_Oleft__neutral,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_neutral
% 0.30/0.46 thf(fact_106_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.46 thf(fact_107_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.46 thf(fact_108_le__add__same__cancel2,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ A @ ( plus_p77862768l_real @ B @ A ) )
% 0.30/0.46 = ( ord_le1342644953l_real @ zero_z659284464l_real @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_add_same_cancel2
% 0.30/0.46 thf(fact_109_le__add__same__cancel2,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/0.46 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_add_same_cancel2
% 0.30/0.46 thf(fact_110_le__add__same__cancel1,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ A @ ( plus_p77862768l_real @ A @ B ) )
% 0.30/0.46 = ( ord_le1342644953l_real @ zero_z659284464l_real @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_add_same_cancel1
% 0.30/0.46 thf(fact_111_le__add__same__cancel1,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.46 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_add_same_cancel1
% 0.30/0.46 thf(fact_112_add__le__same__cancel2,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ B ) @ B )
% 0.30/0.46 = ( ord_le1342644953l_real @ A @ zero_z659284464l_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_same_cancel2
% 0.30/0.46 thf(fact_113_add__le__same__cancel2,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_same_cancel2
% 0.30/0.46 thf(fact_114_add__le__same__cancel1,axiom,
% 0.30/0.46 ! [B: produc957004601l_real,A: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ B @ A ) @ B )
% 0.30/0.46 = ( ord_le1342644953l_real @ A @ zero_z659284464l_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_same_cancel1
% 0.30/0.46 thf(fact_115_add__le__same__cancel1,axiom,
% 0.30/0.46 ! [B: real,A: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.30/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_same_cancel1
% 0.30/0.46 thf(fact_116_one__reorient,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( one_one_real = X )
% 0.30/0.46 = ( X = one_one_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % one_reorient
% 0.30/0.46 thf(fact_117_zero__reorient,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( zero_zero_a = X )
% 0.30/0.46 = ( X = zero_zero_a ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_reorient
% 0.30/0.46 thf(fact_118_zero__reorient,axiom,
% 0.30/0.46 ! [X: produc957004601l_real] :
% 0.30/0.46 ( ( zero_z659284464l_real = X )
% 0.30/0.46 = ( X = zero_z659284464l_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_reorient
% 0.30/0.46 thf(fact_119_zero__reorient,axiom,
% 0.30/0.46 ! [X: product_prod_a_a] :
% 0.30/0.46 ( ( zero_z950819678od_a_a = X )
% 0.30/0.46 = ( X = zero_z950819678od_a_a ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_reorient
% 0.30/0.46 thf(fact_120_zero__reorient,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( zero_zero_real = X )
% 0.30/0.46 = ( X = zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_reorient
% 0.30/0.46 thf(fact_121_add__nonpos__eq__0__iff,axiom,
% 0.30/0.46 ! [X: produc957004601l_real,Y: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ X @ zero_z659284464l_real )
% 0.30/0.46 => ( ( ord_le1342644953l_real @ Y @ zero_z659284464l_real )
% 0.30/0.46 => ( ( ( plus_p77862768l_real @ X @ Y )
% 0.30/0.46 = zero_z659284464l_real )
% 0.30/0.46 = ( ( X = zero_z659284464l_real )
% 0.30/0.46 & ( Y = zero_z659284464l_real ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonpos_eq_0_iff
% 0.30/0.46 thf(fact_122_add__nonpos__eq__0__iff,axiom,
% 0.30/0.46 ! [X: real,Y: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.30/0.46 => ( ( ( plus_plus_real @ X @ Y )
% 0.30/0.46 = zero_zero_real )
% 0.30/0.46 = ( ( X = zero_zero_real )
% 0.30/0.46 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonpos_eq_0_iff
% 0.30/0.46 thf(fact_123_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: real,P: real > $o] :
% 0.30/0.46 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.30/0.46 = ( P @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_124_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: a,P: a > $o] :
% 0.30/0.46 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.30/0.46 = ( P @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_125_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,P: produc957004601l_real > $o] :
% 0.30/0.46 ( ( member1068169442l_real @ A @ ( collec1300223524l_real @ P ) )
% 0.30/0.46 = ( P @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_126_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 0.30/0.46 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 0.30/0.46 = ( P @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_127_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A3: set_real] :
% 0.30/0.46 ( ( collect_real
% 0.30/0.46 @ ^ [X3: real] : ( member_real @ X3 @ A3 ) )
% 0.30/0.46 = A3 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_128_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A3: set_a] :
% 0.30/0.46 ( ( collect_a
% 0.30/0.46 @ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
% 0.30/0.46 = A3 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_129_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A3: set_Pr147102617l_real] :
% 0.30/0.46 ( ( collec1300223524l_real
% 0.30/0.46 @ ^ [X3: produc957004601l_real] : ( member1068169442l_real @ X3 @ A3 ) )
% 0.30/0.46 = A3 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_130_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A3: set_Product_prod_a_a] :
% 0.30/0.46 ( ( collec645855634od_a_a
% 0.30/0.46 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 0.30/0.46 = A3 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_131_Collect__cong,axiom,
% 0.30/0.46 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 0.30/0.46 ( ! [X4: product_prod_a_a] :
% 0.30/0.46 ( ( P @ X4 )
% 0.30/0.46 = ( Q @ X4 ) )
% 0.30/0.46 => ( ( collec645855634od_a_a @ P )
% 0.30/0.46 = ( collec645855634od_a_a @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_132_add__nonneg__eq__0__iff,axiom,
% 0.30/0.46 ! [X: produc957004601l_real,Y: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ zero_z659284464l_real @ X )
% 0.30/0.46 => ( ( ord_le1342644953l_real @ zero_z659284464l_real @ Y )
% 0.30/0.46 => ( ( ( plus_p77862768l_real @ X @ Y )
% 0.30/0.46 = zero_z659284464l_real )
% 0.30/0.46 = ( ( X = zero_z659284464l_real )
% 0.30/0.46 & ( Y = zero_z659284464l_real ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonneg_eq_0_iff
% 0.30/0.46 thf(fact_133_add__nonneg__eq__0__iff,axiom,
% 0.30/0.46 ! [X: real,Y: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/0.46 => ( ( ( plus_plus_real @ X @ Y )
% 0.30/0.46 = zero_zero_real )
% 0.30/0.46 = ( ( X = zero_zero_real )
% 0.30/0.46 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonneg_eq_0_iff
% 0.30/0.46 thf(fact_134_add__nonpos__nonpos,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ A @ zero_z659284464l_real )
% 0.30/0.46 => ( ( ord_le1342644953l_real @ B @ zero_z659284464l_real )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ B ) @ zero_z659284464l_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonpos_nonpos
% 0.30/0.46 thf(fact_135_add__nonpos__nonpos,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonpos_nonpos
% 0.30/0.46 thf(fact_136_add__nonneg__nonneg,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ zero_z659284464l_real @ A )
% 0.30/0.46 => ( ( ord_le1342644953l_real @ zero_z659284464l_real @ B )
% 0.30/0.46 => ( ord_le1342644953l_real @ zero_z659284464l_real @ ( plus_p77862768l_real @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonneg_nonneg
% 0.30/0.46 thf(fact_137_add__nonneg__nonneg,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonneg_nonneg
% 0.30/0.46 thf(fact_138_add__increasing2,axiom,
% 0.30/0.46 ! [C: produc957004601l_real,B: produc957004601l_real,A: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ zero_z659284464l_real @ C )
% 0.30/0.46 => ( ( ord_le1342644953l_real @ B @ A )
% 0.30/0.46 => ( ord_le1342644953l_real @ B @ ( plus_p77862768l_real @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_increasing2
% 0.30/0.46 thf(fact_139_add__increasing2,axiom,
% 0.30/0.46 ! [C: real,B: real,A: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.46 => ( ( ord_less_eq_real @ B @ A )
% 0.30/0.46 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_increasing2
% 0.30/0.46 thf(fact_140_add__decreasing2,axiom,
% 0.30/0.46 ! [C: produc957004601l_real,A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ C @ zero_z659284464l_real )
% 0.30/0.46 => ( ( ord_le1342644953l_real @ A @ B )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_decreasing2
% 0.30/0.46 thf(fact_141_add__decreasing2,axiom,
% 0.30/0.46 ! [C: real,A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_decreasing2
% 0.30/0.46 thf(fact_142_add__increasing,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ zero_z659284464l_real @ A )
% 0.30/0.46 => ( ( ord_le1342644953l_real @ B @ C )
% 0.30/0.46 => ( ord_le1342644953l_real @ B @ ( plus_p77862768l_real @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_increasing
% 0.30/0.46 thf(fact_143_add__increasing,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.46 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_increasing
% 0.30/0.46 thf(fact_144_add__decreasing,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,C: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ A @ zero_z659284464l_real )
% 0.30/0.46 => ( ( ord_le1342644953l_real @ C @ B )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_decreasing
% 0.30/0.46 thf(fact_145_add__decreasing,axiom,
% 0.30/0.46 ! [A: real,C: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ C @ B )
% 0.30/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_decreasing
% 0.30/0.46 thf(fact_146_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z950819678od_a_a
% 0.30/0.46 = ( product_Pair_a_a @ zero_zero_a @ zero_zero_a ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_147_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z659284464l_real
% 0.30/0.46 = ( produc705216881l_real @ zero_zero_real @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_148_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z1407338960real_a
% 0.30/0.46 = ( product_Pair_real_a @ zero_zero_real @ zero_zero_a ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_149_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z705155042a_real
% 0.30/0.46 = ( product_Pair_a_real @ zero_zero_a @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_150_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z590476811l_real
% 0.30/0.46 = ( produc1926903988l_real @ zero_zero_real @ zero_z659284464l_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_151_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z396988537od_a_a
% 0.30/0.46 = ( produc454550562od_a_a @ zero_zero_real @ zero_z950819678od_a_a ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_152_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z2135370393l_real
% 0.30/0.46 = ( produc64206290l_real @ zero_zero_a @ zero_z659284464l_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_153_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z53797895od_a_a
% 0.30/0.46 = ( produc1299253312od_a_a @ zero_zero_a @ zero_z950819678od_a_a ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_154_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z342251165l_real
% 0.30/0.46 = ( produc1175086478l_real @ zero_z659284464l_real @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_155_zero__prod__def,axiom,
% 0.30/0.46 ( zero_z1937883107real_a
% 0.30/0.46 = ( produc1430099868real_a @ zero_z659284464l_real @ zero_zero_a ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_prod_def
% 0.30/0.46 thf(fact_156_add__le__imp__le__right,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,C: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ C ) )
% 0.30/0.46 => ( ord_le1342644953l_real @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_imp_le_right
% 0.30/0.46 thf(fact_157_add__le__imp__le__right,axiom,
% 0.30/0.46 ! [A: real,C: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.46 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_imp_le_right
% 0.30/0.46 thf(fact_158_add__le__imp__le__left,axiom,
% 0.30/0.46 ! [C: produc957004601l_real,A: produc957004601l_real,B: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ C @ A ) @ ( plus_p77862768l_real @ C @ B ) )
% 0.30/0.46 => ( ord_le1342644953l_real @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_imp_le_left
% 0.30/0.46 thf(fact_159_add__le__imp__le__left,axiom,
% 0.30/0.46 ! [C: real,A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.46 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_imp_le_left
% 0.30/0.46 thf(fact_160_add__right__mono,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ A @ B )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_mono
% 0.30/0.46 thf(fact_161_add__right__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_mono
% 0.30/0.46 thf(fact_162_add__left__mono,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ A @ B )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ C @ A ) @ ( plus_p77862768l_real @ C @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_mono
% 0.30/0.46 thf(fact_163_add__left__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_mono
% 0.30/0.46 thf(fact_164_add__mono,axiom,
% 0.30/0.46 ! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real,D: produc957004601l_real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ A @ B )
% 0.30/0.46 => ( ( ord_le1342644953l_real @ C @ D )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono
% 0.30/0.46 thf(fact_165_add__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono
% 0.30/0.46 thf(fact_166_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.46 ! [I: produc957004601l_real,J: produc957004601l_real,K: produc957004601l_real,L: produc957004601l_real] :
% 0.30/0.46 ( ( ( ord_le1342644953l_real @ I @ J )
% 0.30/0.46 & ( ord_le1342644953l_real @ K @ L ) )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ I @ K ) @ ( plus_p77862768l_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(1)
% 0.30/0.46 thf(fact_167_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.30/0.46 ( ( ( ord_less_eq_real @ I @ J )
% 0.30/0.46 & ( ord_less_eq_real @ K @ L ) )
% 0.30/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(1)
% 0.30/0.46 thf(fact_168_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.46 ! [I: produc957004601l_real,J: produc957004601l_real,K: produc957004601l_real,L: produc957004601l_real] :
% 0.30/0.46 ( ( ( I = J )
% 0.30/0.46 & ( ord_le1342644953l_real @ K @ L ) )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ I @ K ) @ ( plus_p77862768l_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(2)
% 0.30/0.46 thf(fact_169_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.30/0.46 ( ( ( I = J )
% 0.30/0.46 & ( ord_less_eq_real @ K @ L ) )
% 0.30/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(2)
% 0.30/0.46 thf(fact_170_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.46 ! [I: produc957004601l_real,J: produc957004601l_real,K: produc957004601l_real,L: produc957004601l_real] :
% 0.30/0.46 ( ( ( ord_le1342644953l_real @ I @ J )
% 0.30/0.46 & ( K = L ) )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ I @ K ) @ ( plus_p77862768l_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(3)
% 0.30/0.46 thf(fact_171_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.30/0.46 ( ( ( ord_less_eq_real @ I @ J )
% 0.30/0.46 & ( K = L ) )
% 0.30/0.46 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(3)
% 0.30/0.46 thf(fact_172_pth__4_I1_J,axiom,
% 0.30/0.46 ! [X: produc957004601l_real] :
% 0.30/0.46 ( ( real_V1139189034l_real @ zero_zero_real @ X )
% 0.30/0.46 = zero_z659284464l_real ) ).
% 0.30/0.46
% 0.30/0.46 % pth_4(1)
% 0.30/0.46 thf(fact_173_pth__4_I1_J,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( real_V453051771R_real @ zero_zero_real @ X )
% 0.30/0.46 = zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % pth_4(1)
% 0.30/0.46 thf(fact_174_convex__ge__imp__conved,axiom,
% 0.30/0.46 ! [Pr: set_Pr1741234931a_real] :
% 0.30/0.46 ( ! [X4: product_prod_a_real,Y3: product_prod_a_real] :
% 0.30/0.46 ( ( member384820540a_real @ ( produc886678603a_real @ X4 @ Y3 ) @ Pr )
% 0.30/0.46 => ! [Alpha: real,Beta: real] :
% 0.30/0.46 ( ( ( ( plus_plus_real @ Alpha @ Beta )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ Alpha )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ Beta ) )
% 0.30/0.46 => ( member384820540a_real @ ( produc886678603a_real @ ( plus_p541014626a_real @ ( real_V512797224a_real @ Alpha @ X4 ) @ ( real_V512797224a_real @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
% 0.30/0.46 => ( prefer1113819806a_real @ Pr ) ) ).
% 0.30/0.46
% 0.30/0.46 % convex_ge_imp_conved
% 0.30/0.46 thf(fact_175_convex__ge__imp__conved,axiom,
% 0.30/0.46 ! [Pr: set_Pr1948701895od_a_a] :
% 0.30/0.46 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 0.30/0.46 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Pr )
% 0.30/0.46 => ! [Alpha: real,Beta: real] :
% 0.30/0.46 ( ( ( ( plus_plus_real @ Alpha @ Beta )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ Alpha )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ Beta ) )
% 0.30/0.46 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( plus_p1505579230od_a_a @ ( real_V543523736od_a_a @ Alpha @ X4 ) @ ( real_V543523736od_a_a @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
% 0.30/0.46 => ( prefer225425826od_a_a @ Pr ) ) ).
% 0.30/0.46
% 0.30/0.46 % convex_ge_imp_conved
% 0.30/0.46 thf(fact_176_convex__ge__imp__conved,axiom,
% 0.30/0.46 ! [Pr: set_Pr147102617l_real] :
% 0.30/0.46 ( ! [X4: real,Y3: real] :
% 0.30/0.46 ( ( member1068169442l_real @ ( produc705216881l_real @ X4 @ Y3 ) @ Pr )
% 0.30/0.46 => ! [Alpha: real,Beta: real] :
% 0.30/0.46 ( ( ( ( plus_plus_real @ Alpha @ Beta )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ Alpha )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ Beta ) )
% 0.30/0.46 => ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ Alpha @ X4 ) @ ( real_V453051771R_real @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
% 0.30/0.46 => ( prefer1247792113f_real @ Pr ) ) ).
% 0.30/0.46
% 0.30/0.46 % convex_ge_imp_conved
% 0.30/0.46 thf(fact_177_convex__ge__imp__conved,axiom,
% 0.30/0.46 ! [Pr: set_Product_prod_a_a] :
% 0.30/0.46 ( ! [X4: a,Y3: a] :
% 0.30/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Pr )
% 0.30/0.46 => ! [Alpha: real,Beta: real] :
% 0.30/0.46 ( ( ( ( plus_plus_real @ Alpha @ Beta )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ Alpha )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ Beta ) )
% 0.30/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ Alpha @ X4 ) @ ( real_V1035702895aleR_a @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
% 0.30/0.46 => ( prefer529818233pref_a @ Pr ) ) ).
% 0.30/0.46
% 0.30/0.46 % convex_ge_imp_conved
% 0.30/0.46 thf(fact_178_add_Ogroup__left__neutral,axiom,
% 0.30/0.46 ! [A: product_prod_a_real] :
% 0.30/0.46 ( ( plus_p541014626a_real @ zero_z705155042a_real @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.group_left_neutral
% 0.30/0.46 thf(fact_179_add_Ogroup__left__neutral,axiom,
% 0.30/0.46 ! [A: produc957004601l_real] :
% 0.30/0.46 ( ( plus_p77862768l_real @ zero_z659284464l_real @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.group_left_neutral
% 0.30/0.46 thf(fact_180_add_Ogroup__left__neutral,axiom,
% 0.30/0.46 ! [A: product_prod_a_a] :
% 0.30/0.46 ( ( plus_p1505579230od_a_a @ zero_z950819678od_a_a @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.group_left_neutral
% 0.30/0.46 thf(fact_181_add_Ogroup__left__neutral,axiom,
% 0.30/0.46 ! [A: a] :
% 0.30/0.46 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.group_left_neutral
% 0.30/0.46 thf(fact_182_add_Ogroup__left__neutral,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.group_left_neutral
% 0.30/0.46 thf(fact_183_add_Ocomm__neutral,axiom,
% 0.30/0.46 ! [A: set_a] :
% 0.30/0.46 ( ( plus_plus_set_a @ A @ zero_zero_set_a )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.comm_neutral
% 0.30/0.46 thf(fact_184_add_Ocomm__neutral,axiom,
% 0.30/0.46 ! [A: produc957004601l_real] :
% 0.30/0.46 ( ( plus_p77862768l_real @ A @ zero_z659284464l_real )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.comm_neutral
% 0.30/0.46 thf(fact_185_add_Ocomm__neutral,axiom,
% 0.30/0.46 ! [A: product_prod_a_a] :
% 0.30/0.46 ( ( plus_p1505579230od_a_a @ A @ zero_z950819678od_a_a )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.comm_neutral
% 0.30/0.46 thf(fact_186_add_Ocomm__neutral,axiom,
% 0.30/0.46 ! [A: a] :
% 0.30/0.46 ( ( plus_plus_a @ A @ zero_zero_a )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.comm_neutral
% 0.30/0.46 thf(fact_187_add_Ocomm__neutral,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % add.comm_neutral
% 0.30/0.46 thf(fact_188_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.46 ! [A: a] :
% 0.30/0.46 ( ( plus_plus_a @ zero_zero_a @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % comm_monoid_add_class.add_0
% 0.30/0.46 thf(fact_189_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.46 = A ) ).
% 0.30/0.46
% 0.30/0.46 % comm_monoid_add_class.add_0
% 0.30/0.46 thf(fact_190_pth__d,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( plus_plus_real @ X @ zero_zero_real )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 % pth_d
% 0.30/0.46 thf(fact_191_pth__7_I1_J,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( plus_plus_real @ zero_zero_real @ X )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 % pth_7(1)
% 0.30/0.46 thf(fact_192_pth__4_I2_J,axiom,
% 0.30/0.46 ! [C: real] :
% 0.30/0.46 ( ( real_V453051771R_real @ C @ zero_zero_real )
% 0.30/0.46 = zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % pth_4(2)
% 0.30/0.46 thf(fact_193_add__right__imp__eq,axiom,
% 0.30/0.46 ! [B: a,A: a,C: a] :
% 0.30/0.46 ( ( ( plus_plus_a @ B @ A )
% 0.30/0.46 = ( plus_plus_a @ C @ A ) )
% 0.30/0.46 => ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_imp_eq
% 0.30/0.46 thf(fact_194_add__right__imp__eq,axiom,
% 0.30/0.46 ! [B: real,A: real,C: real] :
% 0.30/0.46 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.46 = ( plus_plus_real @ C @ A ) )
% 0.30/0.46 => ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_right_imp_eq
% 0.30/0.46 thf(fact_195_add__left__imp__eq,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( ( plus_plus_a @ A @ B )
% 0.30/0.46 = ( plus_plus_a @ A @ C ) )
% 0.30/0.46 => ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_imp_eq
% 0.30/0.46 thf(fact_196_add__left__imp__eq,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.46 = ( plus_plus_real @ A @ C ) )
% 0.30/0.46 => ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_left_imp_eq
% 0.30/0.46 thf(fact_197_add_Oleft__commute,axiom,
% 0.30/0.46 ! [B: a,A: a,C: a] :
% 0.30/0.46 ( ( plus_plus_a @ B @ ( plus_plus_a @ A @ C ) )
% 0.30/0.46 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_commute
% 0.30/0.46 thf(fact_198_add_Oleft__commute,axiom,
% 0.30/0.46 ! [B: real,A: real,C: real] :
% 0.30/0.46 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.30/0.46 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_commute
% 0.30/0.46 thf(fact_199_add_Ocommute,axiom,
% 0.30/0.46 ( plus_plus_a
% 0.30/0.46 = ( ^ [A4: a,B3: a] : ( plus_plus_a @ B3 @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.commute
% 0.30/0.46 thf(fact_200_add_Ocommute,axiom,
% 0.30/0.46 ( plus_plus_real
% 0.30/0.46 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.commute
% 0.30/0.46 thf(fact_201_add_Oright__cancel,axiom,
% 0.30/0.46 ! [B: a,A: a,C: a] :
% 0.30/0.46 ( ( ( plus_plus_a @ B @ A )
% 0.30/0.46 = ( plus_plus_a @ C @ A ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.right_cancel
% 0.30/0.46 thf(fact_202_add_Oright__cancel,axiom,
% 0.30/0.46 ! [B: real,A: real,C: real] :
% 0.30/0.46 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.46 = ( plus_plus_real @ C @ A ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.right_cancel
% 0.30/0.46 thf(fact_203_add_Oleft__cancel,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( ( plus_plus_a @ A @ B )
% 0.30/0.46 = ( plus_plus_a @ A @ C ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_cancel
% 0.30/0.46 thf(fact_204_add_Oleft__cancel,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.46 = ( plus_plus_real @ A @ C ) )
% 0.30/0.46 = ( B = C ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.left_cancel
% 0.30/0.46 thf(fact_205_add_Oassoc,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
% 0.30/0.46 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.assoc
% 0.30/0.46 thf(fact_206_add_Oassoc,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.46 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add.assoc
% 0.30/0.46 thf(fact_207_set__plus__elim,axiom,
% 0.30/0.46 ! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.30/0.46 ( ( member449909584od_a_a @ X @ ( plus_p634297534od_a_a @ A3 @ B4 ) )
% 0.30/0.46 => ~ ! [A5: product_prod_a_a,B5: product_prod_a_a] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( plus_p1505579230od_a_a @ A5 @ B5 ) )
% 0.30/0.46 => ( ( member449909584od_a_a @ A5 @ A3 )
% 0.30/0.46 => ~ ( member449909584od_a_a @ B5 @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_plus_elim
% 0.30/0.46 thf(fact_208_set__plus__elim,axiom,
% 0.30/0.46 ! [X: a,A3: set_a,B4: set_a] :
% 0.30/0.46 ( ( member_a @ X @ ( plus_plus_set_a @ A3 @ B4 ) )
% 0.30/0.46 => ~ ! [A5: a,B5: a] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( plus_plus_a @ A5 @ B5 ) )
% 0.30/0.46 => ( ( member_a @ A5 @ A3 )
% 0.30/0.46 => ~ ( member_a @ B5 @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_plus_elim
% 0.30/0.46 thf(fact_209_set__plus__elim,axiom,
% 0.30/0.46 ! [X: real,A3: set_real,B4: set_real] :
% 0.30/0.46 ( ( member_real @ X @ ( plus_plus_set_real @ A3 @ B4 ) )
% 0.30/0.46 => ~ ! [A5: real,B5: real] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( plus_plus_real @ A5 @ B5 ) )
% 0.30/0.46 => ( ( member_real @ A5 @ A3 )
% 0.30/0.46 => ~ ( member_real @ B5 @ B4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_plus_elim
% 0.30/0.46 thf(fact_210_group__cancel_Oadd2,axiom,
% 0.30/0.46 ! [B4: a,K: a,B: a,A: a] :
% 0.30/0.46 ( ( B4
% 0.30/0.46 = ( plus_plus_a @ K @ B ) )
% 0.30/0.46 => ( ( plus_plus_a @ A @ B4 )
% 0.30/0.46 = ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add2
% 0.30/0.46 thf(fact_211_group__cancel_Oadd2,axiom,
% 0.30/0.46 ! [B4: real,K: real,B: real,A: real] :
% 0.30/0.46 ( ( B4
% 0.30/0.46 = ( plus_plus_real @ K @ B ) )
% 0.30/0.46 => ( ( plus_plus_real @ A @ B4 )
% 0.30/0.46 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add2
% 0.30/0.46 thf(fact_212_group__cancel_Oadd1,axiom,
% 0.30/0.46 ! [A3: a,K: a,A: a,B: a] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( plus_plus_a @ K @ A ) )
% 0.30/0.46 => ( ( plus_plus_a @ A3 @ B )
% 0.30/0.46 = ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add1
% 0.30/0.46 thf(fact_213_group__cancel_Oadd1,axiom,
% 0.30/0.46 ! [A3: real,K: real,A: real,B: real] :
% 0.30/0.46 ( ( A3
% 0.30/0.46 = ( plus_plus_real @ K @ A ) )
% 0.30/0.46 => ( ( plus_plus_real @ A3 @ B )
% 0.30/0.46 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % group_cancel.add1
% 0.30/0.46 thf(fact_214_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.30/0.46 ( ( ( I = J )
% 0.30/0.46 & ( K = L ) )
% 0.30/0.46 => ( ( plus_plus_real @ I @ K )
% 0.30/0.46 = ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_semiring(4)
% 0.30/0.46 thf(fact_215_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.46 ! [A: a,B: a,C: a] :
% 0.30/0.46 ( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
% 0.30/0.46 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.46 thf(fact_216_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.46 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.46 thf(fact_217_old_Oprod_Oinducts,axiom,
% 0.30/0.46 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.30/0.46 ( ! [A5: a,B5: a] : ( P @ ( product_Pair_a_a @ A5 @ B5 ) )
% 0.30/0.46 => ( P @ Prod ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.inducts
% 0.30/0.46 thf(fact_218_old_Oprod_Oexhaust,axiom,
% 0.30/0.46 ! [Y: product_prod_a_a] :
% 0.30/0.46 ~ ! [A5: a,B5: a] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( product_Pair_a_a @ A5 @ B5 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.exhaust
% 0.30/0.46 thf(fact_219_Pair__inject,axiom,
% 0.30/0.46 ! [A: a,B: a,A2: a,B2: a] :
% 0.30/0.46 ( ( ( product_Pair_a_a @ A @ B )
% 0.30/0.46 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.30/0.46 => ~ ( ( A = A2 )
% 0.30/0.46 => ( B != B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pair_inject
% 0.30/0.46 thf(fact_220_prod__cases,axiom,
% 0.30/0.46 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 0.30/0.46 ( ! [A5: a,B5: a] : ( P @ ( product_Pair_a_a @ A5 @ B5 ) )
% 0.30/0.46 => ( P @ P2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_cases
% 0.30/0.46 thf(fact_221_surj__pair,axiom,
% 0.30/0.46 ! [P2: product_prod_a_a] :
% 0.30/0.46 ? [X4: a,Y3: a] :
% 0.30/0.46 ( P2
% 0.30/0.46 = ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_pair
% 0.30/0.46 thf(fact_222_pth__a,axiom,
% 0.30/0.46 ! [X: real,Y: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( real_V453051771R_real @ zero_zero_real @ X ) @ Y )
% 0.30/0.46 = Y ) ).
% 0.30/0.46
% 0.30/0.46 % pth_a
% 0.30/0.46 thf(fact_223_pth__8,axiom,
% 0.30/0.46 ! [C: real,X: real,D: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ X ) )
% 0.30/0.46 = ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_8
% 0.30/0.46 thf(fact_224_pth__6,axiom,
% 0.30/0.46 ! [C: real,X: real,Y: real] :
% 0.30/0.46 ( ( real_V453051771R_real @ C @ ( plus_plus_real @ X @ Y ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ C @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_6
% 0.30/0.46 thf(fact_225_pth__c_I1_J,axiom,
% 0.30/0.46 ! [C: real,X: real,D: real,Y: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ Y ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( real_V453051771R_real @ C @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_c(1)
% 0.30/0.46 thf(fact_226_pth__c_I2_J,axiom,
% 0.30/0.46 ! [C: real,X: real,Z: real,D: real,Y: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) @ ( real_V453051771R_real @ D @ Y ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_c(2)
% 0.30/0.46 thf(fact_227_pth__c_I3_J,axiom,
% 0.30/0.46 ! [C: real,X: real,D: real,Y: real,Z: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_c(3)
% 0.30/0.46 thf(fact_228_pth__c_I4_J,axiom,
% 0.30/0.46 ! [C: real,X: real,W: real,D: real,Y: real,Z: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_c(4)
% 0.30/0.46 thf(fact_229_pth__b_I1_J,axiom,
% 0.30/0.46 ! [C: real,X: real,D: real,Y: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ Y ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_b(1)
% 0.30/0.46 thf(fact_230_pth__b_I2_J,axiom,
% 0.30/0.46 ! [C: real,X: real,Z: real,D: real,Y: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) @ ( real_V453051771R_real @ D @ Y ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ Z @ ( real_V453051771R_real @ D @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_b(2)
% 0.30/0.46 thf(fact_231_pth__b_I3_J,axiom,
% 0.30/0.46 ! [C: real,X: real,D: real,Y: real,Z: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_b(3)
% 0.30/0.46 thf(fact_232_pth__b_I4_J,axiom,
% 0.30/0.46 ! [C: real,X: real,W: real,D: real,Y: real,Z: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ W @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_b(4)
% 0.30/0.46 thf(fact_233_pth__9_I1_J,axiom,
% 0.30/0.46 ! [C: real,X: real,Z: real,D: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) @ ( real_V453051771R_real @ D @ X ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_9(1)
% 0.30/0.46 thf(fact_234_pth__9_I2_J,axiom,
% 0.30/0.46 ! [C: real,X: real,D: real,Z: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ X ) @ Z ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) @ Z ) ) ).
% 0.30/0.46
% 0.30/0.46 % pth_9(2)
% 0.30/0.46 thf(fact_235_scaleR__eq__iff,axiom,
% 0.30/0.46 ! [B: real,U: real,A: real] :
% 0.30/0.46 ( ( ( plus_plus_real @ B @ ( real_V453051771R_real @ U @ A ) )
% 0.30/0.46 = ( plus_plus_real @ A @ ( real_V453051771R_real @ U @ B ) ) )
% 0.30/0.46 = ( ( A = B )
% 0.30/0.46 | ( U = one_one_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_eq_iff
% 0.30/0.46 thf(fact_236_scaleR__eq__iff,axiom,
% 0.30/0.46 ! [B: a,U: real,A: a] :
% 0.30/0.46 ( ( ( plus_plus_a @ B @ ( real_V1035702895aleR_a @ U @ A ) )
% 0.30/0.46 = ( plus_plus_a @ A @ ( real_V1035702895aleR_a @ U @ B ) ) )
% 0.30/0.46 = ( ( A = B )
% 0.30/0.46 | ( U = one_one_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_eq_iff
% 0.30/0.46 thf(fact_237_scale__zero__left,axiom,
% 0.30/0.46 ! [X: real] :
% 0.30/0.46 ( ( real_V453051771R_real @ zero_zero_real @ X )
% 0.30/0.46 = zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % scale_zero_left
% 0.30/0.46 thf(fact_238_scale__zero__left,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( real_V1035702895aleR_a @ zero_zero_real @ X )
% 0.30/0.46 = zero_zero_a ) ).
% 0.30/0.46
% 0.30/0.46 % scale_zero_left
% 0.30/0.46 thf(fact_239_scale__eq__0__iff,axiom,
% 0.30/0.46 ! [A: real,X: real] :
% 0.30/0.46 ( ( ( real_V453051771R_real @ A @ X )
% 0.30/0.46 = zero_zero_real )
% 0.30/0.46 = ( ( A = zero_zero_real )
% 0.30/0.46 | ( X = zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scale_eq_0_iff
% 0.30/0.46 thf(fact_240_scale__eq__0__iff,axiom,
% 0.30/0.46 ! [A: real,X: a] :
% 0.30/0.46 ( ( ( real_V1035702895aleR_a @ A @ X )
% 0.30/0.46 = zero_zero_a )
% 0.30/0.46 = ( ( A = zero_zero_real )
% 0.30/0.46 | ( X = zero_zero_a ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scale_eq_0_iff
% 0.30/0.46 thf(fact_241_scaleR__one,axiom,
% 0.30/0.46 ! [X: a] :
% 0.30/0.46 ( ( real_V1035702895aleR_a @ one_one_real @ X )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_one
% 0.30/0.46 thf(fact_242_scale__cancel__left,axiom,
% 0.30/0.46 ! [A: real,X: a,Y: a] :
% 0.30/0.46 ( ( ( real_V1035702895aleR_a @ A @ X )
% 0.30/0.46 = ( real_V1035702895aleR_a @ A @ Y ) )
% 0.30/0.46 = ( ( X = Y )
% 0.30/0.46 | ( A = zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scale_cancel_left
% 0.30/0.46 thf(fact_243_scaleR__left__le__one__le,axiom,
% 0.30/0.46 ! [X: real,A: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.46 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_left_le_one_le
% 0.30/0.46 thf(fact_244_scale__cancel__right,axiom,
% 0.30/0.46 ! [A: real,X: real,B: real] :
% 0.30/0.46 ( ( ( real_V453051771R_real @ A @ X )
% 0.30/0.46 = ( real_V453051771R_real @ B @ X ) )
% 0.30/0.46 = ( ( A = B )
% 0.30/0.46 | ( X = zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scale_cancel_right
% 0.30/0.46 thf(fact_245_scale__cancel__right,axiom,
% 0.30/0.46 ! [A: real,X: a,B: real] :
% 0.30/0.46 ( ( ( real_V1035702895aleR_a @ A @ X )
% 0.30/0.46 = ( real_V1035702895aleR_a @ B @ X ) )
% 0.30/0.46 = ( ( A = B )
% 0.30/0.46 | ( X = zero_zero_a ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scale_cancel_right
% 0.30/0.46 thf(fact_246_scale__zero__right,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( real_V453051771R_real @ A @ zero_zero_real )
% 0.30/0.46 = zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % scale_zero_right
% 0.30/0.46 thf(fact_247_scale__zero__right,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( real_V1035702895aleR_a @ A @ zero_zero_a )
% 0.30/0.46 = zero_zero_a ) ).
% 0.30/0.46
% 0.30/0.46 % scale_zero_right
% 0.30/0.46 thf(fact_248_split__scaleR__neg__le,axiom,
% 0.30/0.46 ! [A: real,X: real] :
% 0.30/0.46 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 & ( ord_less_eq_real @ X @ zero_zero_real ) )
% 0.30/0.46 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_scaleR_neg_le
% 0.30/0.46 thf(fact_249_split__scaleR__pos__le,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.30/0.46 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.46 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.30/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_scaleR_pos_le
% 0.30/0.46 thf(fact_250_set__plus__mono2__b,axiom,
% 0.30/0.46 ! [C2: set_Product_prod_a_a,D2: set_Product_prod_a_a,E: set_Product_prod_a_a,F: set_Product_prod_a_a,X: product_prod_a_a] :
% 0.30/0.46 ( ( ord_le1824328871od_a_a @ C2 @ D2 )
% 0.30/0.46 => ( ( ord_le1824328871od_a_a @ E @ F )
% 0.30/0.46 => ( ( member449909584od_a_a @ X @ ( plus_p634297534od_a_a @ C2 @ E ) )
% 0.30/0.46 => ( member449909584od_a_a @ X @ ( plus_p634297534od_a_a @ D2 @ F ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_plus_mono2_b
% 0.30/0.46 thf(fact_251_set__zero__plus2,axiom,
% 0.30/0.46 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.30/0.46 ( ( member449909584od_a_a @ zero_z950819678od_a_a @ A3 )
% 0.30/0.46 => ( ord_le1824328871od_a_a @ B4 @ ( plus_p634297534od_a_a @ A3 @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_zero_plus2
% 0.30/0.46 thf(fact_252_set__zero__plus2,axiom,
% 0.30/0.46 ! [A3: set_real,B4: set_real] :
% 0.30/0.46 ( ( member_real @ zero_zero_real @ A3 )
% 0.30/0.46 => ( ord_less_eq_set_real @ B4 @ ( plus_plus_set_real @ A3 @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_zero_plus2
% 0.30/0.46 thf(fact_253_scale__left__commute,axiom,
% 0.30/0.46 ! [A: real,B: real,X: a] :
% 0.30/0.46 ( ( real_V1035702895aleR_a @ A @ ( real_V1035702895aleR_a @ B @ X ) )
% 0.30/0.46 = ( real_V1035702895aleR_a @ B @ ( real_V1035702895aleR_a @ A @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scale_left_commute
% 0.30/0.46 thf(fact_254_scale__right__imp__eq,axiom,
% 0.30/0.46 ! [X: real,A: real,B: real] :
% 0.30/0.46 ( ( X != zero_zero_real )
% 0.30/0.46 => ( ( ( real_V453051771R_real @ A @ X )
% 0.30/0.46 = ( real_V453051771R_real @ B @ X ) )
% 0.30/0.46 => ( A = B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scale_right_imp_eq
% 0.30/0.46 thf(fact_255_scale__right__imp__eq,axiom,
% 0.30/0.46 ! [X: a,A: real,B: real] :
% 0.30/0.46 ( ( X != zero_zero_a )
% 0.30/0.46 => ( ( ( real_V1035702895aleR_a @ A @ X )
% 0.30/0.46 = ( real_V1035702895aleR_a @ B @ X ) )
% 0.30/0.46 => ( A = B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scale_right_imp_eq
% 0.30/0.46 thf(fact_256_scaleR__add__right,axiom,
% 0.30/0.46 ! [A: real,X: real,Y: real] :
% 0.30/0.46 ( ( real_V453051771R_real @ A @ ( plus_plus_real @ X @ Y ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ A @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_add_right
% 0.30/0.46 thf(fact_257_scaleR__add__right,axiom,
% 0.30/0.46 ! [A: real,X: a,Y: a] :
% 0.30/0.46 ( ( real_V1035702895aleR_a @ A @ ( plus_plus_a @ X @ Y ) )
% 0.30/0.46 = ( plus_plus_a @ ( real_V1035702895aleR_a @ A @ X ) @ ( real_V1035702895aleR_a @ A @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_add_right
% 0.30/0.46 thf(fact_258_scale__left__imp__eq,axiom,
% 0.30/0.46 ! [A: real,X: a,Y: a] :
% 0.30/0.46 ( ( A != zero_zero_real )
% 0.30/0.46 => ( ( ( real_V1035702895aleR_a @ A @ X )
% 0.30/0.46 = ( real_V1035702895aleR_a @ A @ Y ) )
% 0.30/0.46 => ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scale_left_imp_eq
% 0.30/0.46 thf(fact_259_scaleR__left_Oadd,axiom,
% 0.30/0.46 ! [X: real,Y: real,Xa: real] :
% 0.30/0.46 ( ( real_V453051771R_real @ ( plus_plus_real @ X @ Y ) @ Xa )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ X @ Xa ) @ ( real_V453051771R_real @ Y @ Xa ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_left.add
% 0.30/0.46 thf(fact_260_scaleR__left_Oadd,axiom,
% 0.30/0.46 ! [X: real,Y: real,Xa: a] :
% 0.30/0.46 ( ( real_V1035702895aleR_a @ ( plus_plus_real @ X @ Y ) @ Xa )
% 0.30/0.46 = ( plus_plus_a @ ( real_V1035702895aleR_a @ X @ Xa ) @ ( real_V1035702895aleR_a @ Y @ Xa ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_left.add
% 0.30/0.46 thf(fact_261_scaleR__add__left,axiom,
% 0.30/0.46 ! [A: real,B: real,X: real] :
% 0.30/0.46 ( ( real_V453051771R_real @ ( plus_plus_real @ A @ B ) @ X )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ B @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_add_left
% 0.30/0.46 thf(fact_262_scaleR__add__left,axiom,
% 0.30/0.46 ! [A: real,B: real,X: a] :
% 0.30/0.46 ( ( real_V1035702895aleR_a @ ( plus_plus_real @ A @ B ) @ X )
% 0.30/0.46 = ( plus_plus_a @ ( real_V1035702895aleR_a @ A @ X ) @ ( real_V1035702895aleR_a @ B @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_add_left
% 0.30/0.46 thf(fact_263_scaleR__right__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,X: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ B @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_right_mono
% 0.30/0.46 thf(fact_264_scaleR__right__mono__neg,axiom,
% 0.30/0.46 ! [B: real,A: real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ C ) @ ( real_V453051771R_real @ B @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_right_mono_neg
% 0.30/0.46 thf(fact_265_scaleR__left__mono,axiom,
% 0.30/0.46 ! [X: real,Y: real,A: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ A @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_left_mono
% 0.30/0.46 thf(fact_266_scaleR__left__mono__neg,axiom,
% 0.30/0.46 ! [B: real,A: real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_left_mono_neg
% 0.30/0.46 thf(fact_267_scaleR__mono_H,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ C ) @ ( real_V453051771R_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_mono'
% 0.30/0.46 thf(fact_268_scaleR__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,X: real,Y: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ B @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_mono
% 0.30/0.46 thf(fact_269_scaleR__nonpos__nonpos,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_nonpos_nonpos
% 0.30/0.46 thf(fact_270_scaleR__nonpos__nonneg,axiom,
% 0.30/0.46 ! [A: real,X: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_nonpos_nonneg
% 0.30/0.46 thf(fact_271_scaleR__nonneg__nonpos,axiom,
% 0.30/0.46 ! [A: real,X: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.46 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_nonneg_nonpos
% 0.30/0.46 thf(fact_272_scaleR__nonneg__nonneg,axiom,
% 0.30/0.46 ! [A: real,X: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.46 => ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_nonneg_nonneg
% 0.30/0.46 thf(fact_273_Pair__le,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.46 ( ( ord_le1342644953l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
% 0.30/0.46 = ( ( ord_less_eq_real @ A @ C )
% 0.30/0.46 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pair_le
% 0.30/0.46 thf(fact_274_not__one__le__zero,axiom,
% 0.30/0.46 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % not_one_le_zero
% 0.30/0.46 thf(fact_275_zero__le__one,axiom,
% 0.30/0.46 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.30/0.46
% 0.30/0.46 % zero_le_one
% 0.30/0.46 thf(fact_276_zero__neq__one,axiom,
% 0.30/0.46 zero_zero_real != one_one_real ).
% 0.30/0.46
% 0.30/0.46 % zero_neq_one
% 0.30/0.46 thf(fact_277_Pair__mono,axiom,
% 0.30/0.46 ! [X: real,X5: real,Y: real,Y4: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X @ X5 )
% 0.30/0.46 => ( ( ord_less_eq_real @ Y @ Y4 )
% 0.30/0.46 => ( ord_le1342644953l_real @ ( produc705216881l_real @ X @ Y ) @ ( produc705216881l_real @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Pair_mono
% 0.30/0.46 thf(fact_278_weak__convexI,axiom,
% 0.30/0.46 ! [Pr: set_Pr147102617l_real] :
% 0.30/0.46 ( ! [X4: real,Y3: real,Alpha: real,Beta: real] :
% 0.30/0.46 ( ( member1068169442l_real @ ( produc705216881l_real @ X4 @ Y3 ) @ Pr )
% 0.30/0.46 => ( ( ( plus_plus_real @ Alpha @ Beta )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ Alpha )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ Beta )
% 0.30/0.46 => ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ Alpha @ X4 ) @ ( real_V453051771R_real @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) ) ) )
% 0.30/0.46 => ( prefer1247792113f_real @ Pr ) ) ).
% 0.30/0.46
% 0.30/0.46 % weak_convexI
% 0.30/0.46 thf(fact_279_weak__convexI,axiom,
% 0.30/0.46 ! [Pr: set_Product_prod_a_a] :
% 0.30/0.46 ( ! [X4: a,Y3: a,Alpha: real,Beta: real] :
% 0.30/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Pr )
% 0.30/0.46 => ( ( ( plus_plus_real @ Alpha @ Beta )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ Alpha )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ Beta )
% 0.30/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ Alpha @ X4 ) @ ( real_V1035702895aleR_a @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) ) ) )
% 0.30/0.46 => ( prefer529818233pref_a @ Pr ) ) ).
% 0.30/0.46
% 0.30/0.46 % weak_convexI
% 0.30/0.46 thf(fact_280_subsetI,axiom,
% 0.30/0.46 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.30/0.46 ( ! [X4: product_prod_a_a] :
% 0.30/0.46 ( ( member449909584od_a_a @ X4 @ A3 )
% 0.30/0.46 => ( member449909584od_a_a @ X4 @ B4 ) )
% 0.30/0.46 => ( ord_le1824328871od_a_a @ A3 @ B4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetI
% 0.30/0.46 thf(fact_281_add__less__cancel__right,axiom,
% 0.30/0.46 ! [A: real,C: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.46 = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_less_cancel_right
% 0.30/0.46 thf(fact_282_add__less__cancel__left,axiom,
% 0.30/0.46 ! [C: real,A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.46 = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_less_cancel_left
% 0.30/0.46 thf(fact_283_add__less__same__cancel1,axiom,
% 0.30/0.46 ! [B: real,A: real] :
% 0.30/0.46 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.30/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_less_same_cancel1
% 0.30/0.46 thf(fact_284_add__less__same__cancel2,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_less_same_cancel2
% 0.30/0.46 thf(fact_285_less__add__same__cancel1,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.46 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_add_same_cancel1
% 0.30/0.46 thf(fact_286_less__add__same__cancel2,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/0.46 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_add_same_cancel2
% 0.30/0.46 thf(fact_287_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/0.46 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.46 thf(fact_288_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.46 ! [A: real] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/0.46 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.46 thf(fact_289_add__less__imp__less__right,axiom,
% 0.30/0.46 ! [A: real,C: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.46 => ( ord_less_real @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_less_imp_less_right
% 0.30/0.46 thf(fact_290_add__less__imp__less__left,axiom,
% 0.30/0.46 ! [C: real,A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.46 => ( ord_less_real @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_less_imp_less_left
% 0.30/0.46 thf(fact_291_add__strict__right__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ B )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_strict_right_mono
% 0.30/0.46 thf(fact_292_add__strict__left__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ B )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_strict_left_mono
% 0.30/0.46 thf(fact_293_add__strict__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ B )
% 0.30/0.46 => ( ( ord_less_real @ C @ D )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_strict_mono
% 0.30/0.46 thf(fact_294_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.30/0.46 ( ( ( ord_less_real @ I @ J )
% 0.30/0.46 & ( K = L ) )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_field(1)
% 0.30/0.46 thf(fact_295_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.30/0.46 ( ( ( I = J )
% 0.30/0.46 & ( ord_less_real @ K @ L ) )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_field(2)
% 0.30/0.46 thf(fact_296_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.30/0.46 ( ( ( ord_less_real @ I @ J )
% 0.30/0.46 & ( ord_less_real @ K @ L ) )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_field(5)
% 0.30/0.46 thf(fact_297_add__less__le__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ B )
% 0.30/0.46 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_less_le_mono
% 0.30/0.46 thf(fact_298_add__le__less__mono,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ( ord_less_real @ C @ D )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_le_less_mono
% 0.30/0.46 thf(fact_299_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.30/0.46 ( ( ( ord_less_real @ I @ J )
% 0.30/0.46 & ( ord_less_eq_real @ K @ L ) )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_field(3)
% 0.30/0.46 thf(fact_300_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.46 ! [I: real,J: real,K: real,L: real] :
% 0.30/0.46 ( ( ( ord_less_eq_real @ I @ J )
% 0.30/0.46 & ( ord_less_real @ K @ L ) )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono_thms_linordered_field(4)
% 0.30/0.46 thf(fact_301_add__neg__neg,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_neg_neg
% 0.30/0.46 thf(fact_302_add__pos__pos,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.46 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_pos_pos
% 0.30/0.46 thf(fact_303_pos__add__strict,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_real @ B @ C )
% 0.30/0.46 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pos_add_strict
% 0.30/0.46 thf(fact_304_add__less__zeroD,axiom,
% 0.30/0.46 ! [X: real,Y: real] :
% 0.30/0.46 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.30/0.46 | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_less_zeroD
% 0.30/0.46 thf(fact_305_zero__less__one,axiom,
% 0.30/0.46 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.30/0.46
% 0.30/0.46 % zero_less_one
% 0.30/0.46 thf(fact_306_not__one__less__zero,axiom,
% 0.30/0.46 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % not_one_less_zero
% 0.30/0.46 thf(fact_307_add__mono1,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ B )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_mono1
% 0.30/0.46 thf(fact_308_less__add__one,axiom,
% 0.30/0.46 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.30/0.46
% 0.30/0.46 % less_add_one
% 0.30/0.46 thf(fact_309_add__strict__increasing2,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_real @ B @ C )
% 0.30/0.46 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_strict_increasing2
% 0.30/0.46 thf(fact_310_add__strict__increasing,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.46 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_strict_increasing
% 0.30/0.46 thf(fact_311_add__pos__nonneg,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.46 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_pos_nonneg
% 0.30/0.46 thf(fact_312_add__nonpos__neg,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonpos_neg
% 0.30/0.46 thf(fact_313_add__nonneg__pos,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.46 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_nonneg_pos
% 0.30/0.46 thf(fact_314_add__neg__nonpos,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.46 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_neg_nonpos
% 0.30/0.46 thf(fact_315_zero__less__two,axiom,
% 0.30/0.46 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.30/0.46
% 0.30/0.46 % zero_less_two
% 0.30/0.46 thf(fact_316_scaleR__le__cancel__left__pos,axiom,
% 0.30/0.46 ! [C: real,A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.46 => ( ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) )
% 0.30/0.46 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_le_cancel_left_pos
% 0.30/0.46 thf(fact_317_scaleR__le__cancel__left__neg,axiom,
% 0.30/0.46 ! [C: real,A: real,B: real] :
% 0.30/0.46 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.46 => ( ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) )
% 0.30/0.46 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_le_cancel_left_neg
% 0.30/0.46 thf(fact_318_scaleR__le__cancel__left,axiom,
% 0.30/0.46 ! [C: real,A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) )
% 0.30/0.46 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.46 => ( ord_less_eq_real @ A @ B ) )
% 0.30/0.46 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.46 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_le_cancel_left
% 0.30/0.46 thf(fact_319_subset__iff,axiom,
% 0.30/0.46 ( ord_le1824328871od_a_a
% 0.30/0.46 = ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
% 0.30/0.46 ! [T: product_prod_a_a] :
% 0.30/0.46 ( ( member449909584od_a_a @ T @ A6 )
% 0.30/0.46 => ( member449909584od_a_a @ T @ B6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_iff
% 0.30/0.46 thf(fact_320_subset__eq,axiom,
% 0.30/0.46 ( ord_le1824328871od_a_a
% 0.30/0.46 = ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
% 0.30/0.46 ! [X3: product_prod_a_a] :
% 0.30/0.46 ( ( member449909584od_a_a @ X3 @ A6 )
% 0.30/0.46 => ( member449909584od_a_a @ X3 @ B6 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_eq
% 0.30/0.46 thf(fact_321_subsetD,axiom,
% 0.30/0.46 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
% 0.30/0.46 ( ( ord_le1824328871od_a_a @ A3 @ B4 )
% 0.30/0.46 => ( ( member449909584od_a_a @ C @ A3 )
% 0.30/0.46 => ( member449909584od_a_a @ C @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetD
% 0.30/0.46 thf(fact_322_in__mono,axiom,
% 0.30/0.46 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,X: product_prod_a_a] :
% 0.30/0.46 ( ( ord_le1824328871od_a_a @ A3 @ B4 )
% 0.30/0.46 => ( ( member449909584od_a_a @ X @ A3 )
% 0.30/0.46 => ( member449909584od_a_a @ X @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_mono
% 0.30/0.46 thf(fact_323_zero__le__scaleR__iff,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ B ) )
% 0.30/0.46 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.30/0.46 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.46 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.30/0.46 | ( A = zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % zero_le_scaleR_iff
% 0.30/0.46 thf(fact_324_scaleR__le__0__iff,axiom,
% 0.30/0.46 ! [A: real,B: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ ( real_V453051771R_real @ A @ B ) @ zero_zero_real )
% 0.30/0.46 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.46 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.30/0.46 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.46 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.30/0.46 | ( A = zero_zero_real ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % scaleR_le_0_iff
% 0.30/0.46 thf(fact_325_weak__convexD,axiom,
% 0.30/0.46 ! [Pr: set_Pr147102617l_real,X: real,Y: real,U: real,V: real] :
% 0.30/0.46 ( ( prefer1247792113f_real @ Pr )
% 0.30/0.46 => ( ( member1068169442l_real @ ( produc705216881l_real @ X @ Y ) @ Pr )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ U )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ V )
% 0.30/0.46 => ( ( ( plus_plus_real @ U @ V )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 => ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ U @ X ) @ ( real_V453051771R_real @ V @ Y ) ) @ Y ) @ Pr ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % weak_convexD
% 0.30/0.46 thf(fact_326_weak__convexD,axiom,
% 0.30/0.46 ! [Pr: set_Product_prod_a_a,X: a,Y: a,U: real,V: real] :
% 0.30/0.46 ( ( prefer529818233pref_a @ Pr )
% 0.30/0.46 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ U )
% 0.30/0.46 => ( ( ord_less_real @ zero_zero_real @ V )
% 0.30/0.46 => ( ( ( plus_plus_real @ U @ V )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ U @ X ) @ ( real_V1035702895aleR_a @ V @ Y ) ) @ Y ) @ Pr ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % weak_convexD
% 0.30/0.46 thf(fact_327_weak__convex__pref__def,axiom,
% 0.30/0.46 ( prefer1247792113f_real
% 0.30/0.46 = ( ^ [Pr2: set_Pr147102617l_real] :
% 0.30/0.46 ! [X3: real,Y5: real] :
% 0.30/0.46 ( ( member1068169442l_real @ ( produc705216881l_real @ X3 @ Y5 ) @ Pr2 )
% 0.30/0.46 => ! [Alpha2: real,Beta2: real] :
% 0.30/0.46 ( ( ( ( plus_plus_real @ Alpha2 @ Beta2 )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 & ( ord_less_real @ zero_zero_real @ Alpha2 )
% 0.30/0.46 & ( ord_less_real @ zero_zero_real @ Beta2 ) )
% 0.30/0.46 => ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ Alpha2 @ X3 ) @ ( real_V453051771R_real @ Beta2 @ Y5 ) ) @ Y5 ) @ Pr2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % weak_convex_pref_def
% 0.30/0.46 thf(fact_328_weak__convex__pref__def,axiom,
% 0.30/0.46 ( prefer529818233pref_a
% 0.30/0.46 = ( ^ [Pr2: set_Product_prod_a_a] :
% 0.30/0.46 ! [X3: a,Y5: a] :
% 0.30/0.46 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y5 ) @ Pr2 )
% 0.30/0.46 => ! [Alpha2: real,Beta2: real] :
% 0.30/0.46 ( ( ( ( plus_plus_real @ Alpha2 @ Beta2 )
% 0.30/0.46 = one_one_real )
% 0.30/0.46 & ( ord_less_real @ zero_zero_real @ Alpha2 )
% 0.30/0.46 & ( ord_less_real @ zero_zero_real @ Beta2 ) )
% 0.30/0.46 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ Alpha2 @ X3 ) @ ( real_V1035702895aleR_a @ Beta2 @ Y5 ) ) @ Y5 ) @ Pr2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % weak_convex_pref_def
% 0.30/0.46 thf(fact_329_field__le__epsilon,axiom,
% 0.30/0.46 ! [X: real,Y: real] :
% 0.30/0.46 ( ! [E2: real] :
% 0.30/0.46 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.46 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
% 0.30/0.46 => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % field_le_epsilon
% 0.30/0.46 thf(fact_330_less__numeral__extra_I1_J,axiom,
% 0.30/0.46 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.30/0.46
% 0.30/0.46 % less_numeral_extra(1)
% 0.30/0.46 thf(fact_331_psubsetD,axiom,
% 0.30/0.46 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
% 0.30/0.46 ( ( ord_le108336051od_a_a @ A3 @ B4 )
% 0.30/0.46 => ( ( member449909584od_a_a @ C @ A3 )
% 0.30/0.46 => ( member449909584od_a_a @ C @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubsetD
% 0.30/0.46 thf(fact_332_is__num__normalize_I1_J,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.46 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_num_normalize(1)
% 0.30/0.46 thf(fact_333_le__numeral__extra_I3_J,axiom,
% 0.30/0.46 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.30/0.46
% 0.30/0.46 % le_numeral_extra(3)
% 0.30/0.46 thf(fact_334_less__numeral__extra_I3_J,axiom,
% 0.30/0.46 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.30/0.46
% 0.30/0.46 % less_numeral_extra(3)
% 0.30/0.46 thf(fact_335_le__numeral__extra_I4_J,axiom,
% 0.30/0.46 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.30/0.46
% 0.30/0.46 % le_numeral_extra(4)
% 0.30/0.46 thf(fact_336_less__numeral__extra_I4_J,axiom,
% 0.30/0.46 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.30/0.46
% 0.30/0.46 % less_numeral_extra(4)
% 0.30/0.46 thf(fact_337_order__refl,axiom,
% 0.30/0.46 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 0.30/0.46
% 0.30/0.46 % order_refl
% 0.30/0.46 thf(fact_338_add__scaleR__degen,axiom,
% 0.30/0.46 ! [U: real,B: real,V: real,A: real] :
% 0.30/0.46 ( ( ( plus_plus_real @ ( real_V453051771R_real @ U @ B ) @ ( real_V453051771R_real @ V @ A ) )
% 0.30/0.46 = ( plus_plus_real @ ( real_V453051771R_real @ U @ A ) @ ( real_V453051771R_real @ V @ B ) ) )
% 0.30/0.46 => ( ( U != V )
% 0.30/0.46 => ( A = B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_scaleR_degen
% 0.30/0.46 thf(fact_339_add__scaleR__degen,axiom,
% 0.30/0.46 ! [U: real,B: a,V: real,A: a] :
% 0.30/0.46 ( ( ( plus_plus_a @ ( real_V1035702895aleR_a @ U @ B ) @ ( real_V1035702895aleR_a @ V @ A ) )
% 0.30/0.46 = ( plus_plus_a @ ( real_V1035702895aleR_a @ U @ A ) @ ( real_V1035702895aleR_a @ V @ B ) ) )
% 0.30/0.46 => ( ( U != V )
% 0.30/0.46 => ( A = B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % add_scaleR_degen
% 0.30/0.46 thf(fact_340_order__subst1,axiom,
% 0.30/0.46 ! [A: real,F2: real > real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
% 0.30/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.46 => ( ! [X4: real,Y3: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.30/0.46 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst1
% 0.30/0.46 thf(fact_341_order__subst2,axiom,
% 0.30/0.46 ! [A: real,B: real,F2: real > real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
% 0.30/0.46 => ( ! [X4: real,Y3: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.30/0.46 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_subst2
% 0.30/0.46 thf(fact_342_ord__eq__le__subst,axiom,
% 0.30/0.46 ! [A: real,F2: real > real,B: real,C: real] :
% 0.30/0.46 ( ( A
% 0.30/0.46 = ( F2 @ B ) )
% 0.30/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.46 => ( ! [X4: real,Y3: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.30/0.46 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_eq_le_subst
% 0.30/0.46 thf(fact_343_ord__le__eq__subst,axiom,
% 0.30/0.46 ! [A: real,B: real,F2: real > real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ( ( F2 @ B )
% 0.30/0.46 = C )
% 0.30/0.46 => ( ! [X4: real,Y3: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X4 @ Y3 )
% 0.30/0.46 => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 0.30/0.46 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ord_le_eq_subst
% 0.30/0.46 thf(fact_344_eq__iff,axiom,
% 0.30/0.46 ( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
% 0.30/0.46 = ( ^ [X3: real,Y5: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X3 @ Y5 )
% 0.30/0.46 & ( ord_less_eq_real @ Y5 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_iff
% 0.30/0.46 thf(fact_345_antisym,axiom,
% 0.30/0.46 ! [X: real,Y: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.46 => ( ( ord_less_eq_real @ Y @ X )
% 0.30/0.46 => ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym
% 0.30/0.46 thf(fact_346_linear,axiom,
% 0.30/0.46 ! [X: real,Y: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.46 | ( ord_less_eq_real @ Y @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % linear
% 0.30/0.46 thf(fact_347_eq__refl,axiom,
% 0.30/0.46 ! [X: real,Y: real] :
% 0.30/0.46 ( ( X = Y )
% 0.30/0.46 => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % eq_refl
% 0.30/0.46 thf(fact_348_le__cases,axiom,
% 0.30/0.46 ! [X: real,Y: real] :
% 0.30/0.46 ( ~ ( ord_less_eq_real @ X @ Y )
% 0.30/0.46 => ( ord_less_eq_real @ Y @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases
% 0.30/0.46 thf(fact_349_order_Otrans,axiom,
% 0.30/0.46 ! [A: real,B: real,C: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order.trans
% 0.30/0.46 thf(fact_350_le__cases3,axiom,
% 0.30/0.46 ! [X: real,Y: real,Z: real] :
% 0.30/0.46 ( ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.46 => ~ ( ord_less_eq_real @ Y @ Z ) )
% 0.30/0.46 => ( ( ( ord_less_eq_real @ Y @ X )
% 0.30/0.46 => ~ ( ord_less_eq_real @ X @ Z ) )
% 0.30/0.46 => ( ( ( ord_less_eq_real @ X @ Z )
% 0.30/0.46 => ~ ( ord_less_eq_real @ Z @ Y ) )
% 0.30/0.46 => ( ( ( ord_less_eq_real @ Z @ Y )
% 0.30/0.46 => ~ ( ord_less_eq_real @ Y @ X ) )
% 0.30/0.46 => ( ( ( ord_less_eq_real @ Y @ Z )
% 0.30/0.46 => ~ ( ord_less_eq_real @ Z @ X ) )
% 0.30/0.46 => ~ ( ( ord_less_eq_real @ Z @ X )
% 0.30/0.46 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % le_cases3
% 0.30/0.46 thf(fact_351_antisym__conv,axiom,
% 0.30/0.46 ! [Y: real,X: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ Y @ X )
% 0.30/0.46 => ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.46 = ( X = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % antisym_conv
% 0.30/0.46 thf(fact_352_order__class_Oorder_Oeq__iff,axiom,
% 0.30/0.46 ( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
% 0.30/0.46 = ( ^ [A4: real,B3: real] :
% 0.30/0.46 ( ( ord_less_eq_real @ A4 @ B3 )
% 0.30/0.46 & ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % order_class.order.eq_iff
% 0.30/0.46
% 0.30/0.46 % Conjectures (1)
% 0.30/0.46 thf(conj_0,conjecture,
% 0.30/0.46 member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.39/1.49 ---- Embedded in THF ---
% 2.39/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.39/1.49 %%% Generated on Tue Feb 24 08:28:10 EST 2026
% 2.39/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.39/1.49 %%% Logic specification used:
% 2.39/1.49 %%% thf(spec, logic, $$dhol).
% 2.39/1.49
% 2.39/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2DTF13924.p
% See solution above
% 2.39/1.49 ------------------------
% 2.66/1.54 ---- Cleaned THF ---
% 2.66/1.54 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J_J,type,
% 2.66/1.54 set_Pr1741234931a_real: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
% 2.66/1.54 produc1826081171a_real: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
% 2.66/1.54 set_Pr1948701895od_a_a: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.66/1.54 produc1572603623od_a_a: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 2.66/1.54 produc826782210l_real: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
% 2.66/1.54 produc578556564l_real: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 2.66/1.54 produc1286132450l_real: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_Mtf__a_J,type,
% 2.66/1.54 produc1088645164real_a: $tType ).
% 2.66/1.54
% 2.66/1.54 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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% 2.66/1.54 thf(sy_c_Product__Type_OPair_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.66/1.54 produc1299253312od_a_a: a > product_prod_a_a > produc1921647824od_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Product__Type_OPair_001tf__a_001t__Real__Oreal,type,
% 2.66/1.54 product_Pair_a_real: a > real > product_prod_a_real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Product__Type_OPair_001tf__a_001t__Set__Oset_It__Real__Oreal_J,type,
% 2.66/1.54 produc1391501065t_real: a > set_real > produc1396156303t_real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Product__Type_OPair_001tf__a_001t__Set__Oset_Itf__a_J,type,
% 2.66/1.54 product_Pair_a_set_a: a > set_a > product_prod_a_set_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 2.66/1.54 product_Pair_a_a: a > a > product_prod_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.66/1.54 real_V368978776od_a_a: real > produc1572603623od_a_a > produc1572603623od_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Real__Oreal_J,type,
% 2.66/1.54 real_V195069393a_real: real > produc348734722a_real > produc348734722a_real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 2.66/1.54 real_V1538726831_a_a_a: real > produc1016684094_a_a_a > produc1016684094_a_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.66/1.54 real_V1943155903od_a_a: real > produc2096821232od_a_a > produc2096821232od_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 2.66/1.54 real_V1139189034l_real: real > produc957004601l_real > produc957004601l_real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__a_J,type,
% 2.66/1.54 real_V1214981142real_a: real > product_prod_real_a > product_prod_real_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.66/1.54 real_V296206913od_a_a: real > produc1921647824od_a_a > produc1921647824od_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 2.66/1.54 real_V512797224a_real: real > product_prod_a_real > product_prod_a_real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.66/1.54 real_V543523736od_a_a: real > product_prod_a_a > product_prod_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 2.66/1.54 real_V453051771R_real: real > real > real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001tf__a,type,
% 2.66/1.54 real_V1035702895aleR_a: real > a > a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 2.66/1.54 collec1300223524l_real: ( produc957004601l_real > $o ) > set_Pr147102617l_real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.66/1.54 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 2.66/1.54 collect_real: ( real > $o ) > set_real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.66/1.54 collect_a: ( a > $o ) > set_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J_J,type,
% 2.66/1.54 member384820540a_real: produc1826081171a_real > set_Pr1741234931a_real > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.66/1.54 member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 2.66/1.54 member1068169442l_real: produc957004601l_real > set_Pr147102617l_real > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
% 2.66/1.54 member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 2.66/1.54 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001t__Real__Oreal,type,
% 2.66/1.54 member_real: real > set_real > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 2.66/1.54 member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
% 2.66/1.54 member_set_real: set_real > set_set_real > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 2.66/1.54 member_set_a: set_a > set_set_a > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_c_member_001tf__a,type,
% 2.66/1.54 member_a: a > set_a > $o ).
% 2.66/1.54
% 2.66/1.54 thf(sy_v_relation,type,
% 2.66/1.54 relation: set_Product_prod_a_a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_v_u,type,
% 2.66/1.54 u: real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_v_v,type,
% 2.66/1.54 v: real ).
% 2.66/1.54
% 2.66/1.54 thf(sy_v_x,type,
% 2.66/1.54 x: a ).
% 2.66/1.54
% 2.66/1.54 thf(sy_v_y,type,
% 2.66/1.54 y: a ).
% 2.66/1.54
% 2.66/1.54 thf(fact_0_assms_I1_J,axiom,
% 2.66/1.54 prefer529818233pref_a @ relation ).
% 2.66/1.54
% 2.66/1.54 thf(fact_1_indifferent__imp__weak__pref_I2_J,axiom,
% 2.66/1.54 ! [X: a,Y: a] :
% 2.66/1.54 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.66/1.54 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.66/1.54 => ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_2_indifferent__imp__weak__pref_I1_J,axiom,
% 2.66/1.54 ! [X: a,Y: a] :
% 2.66/1.54 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.66/1.54 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.66/1.54 => ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_3_indiff__trans,axiom,
% 2.66/1.54 ! [X: a,Y: a,Z: a] :
% 2.66/1.54 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.66/1.54 & ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.66/1.54 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.66/1.54 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 2.66/1.54 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 2.66/1.54 & ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 2.66/1.54
% 2.66/1.54 thf(fact_4_strict__trans,axiom,
% 2.66/1.54 ! [X: a,Y: a,Z: a] :
% 2.66/1.54 ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ relation )
% 2.66/1.54 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ X ) @ relation ) )
% 2.66/1.55 => ( ( ( member449909584od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ relation )
% 2.66/1.55 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ relation ) )
% 2.66/1.55 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Z ) @ relation )
% 2.66/1.55 & ~ ( member449909584od_a_a @ ( product_Pair_a_a @ Z @ X ) @ relation ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_5_assms_I2_J,axiom,
% 2.66/1.55 member449909584od_a_a @ ( product_Pair_a_a @ x @ y ) @ relation ).
% 2.66/1.55
% 2.66/1.55 thf(fact_6__092_060open_062u_A_092_060noteq_062_A0_A_092_060and_062_Au_A_092_060noteq_062_A1_A_092_060longrightarrow_062_Au_A_K_092_060_094sub_062R_Ax_A_L_Av_A_K_092_060_094sub_062R_Ay_A_092_060succeq_062_Ay_092_060close_062,axiom,
% 2.66/1.55 ( ( ( u != zero_zero_real )
% 2.66/1.55 & ( u != one_one_real ) )
% 2.66/1.55 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_7_u__0,axiom,
% 2.66/1.55 ( ( u = zero_zero_real )
% 2.66/1.55 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_8_assms_I4_J,axiom,
% 2.66/1.55 ord_less_eq_real @ zero_zero_real @ v ).
% 2.66/1.55
% 2.66/1.55 thf(fact_9_assms_I5_J,axiom,
% 2.66/1.55 ( ( plus_plus_real @ u @ v )
% 2.66/1.55 = one_one_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_10_assms_I3_J,axiom,
% 2.66/1.55 ord_less_eq_real @ zero_zero_real @ u ).
% 2.66/1.55
% 2.66/1.55 thf(fact_11_scaleR__Pair,axiom,
% 2.66/1.55 ! [R: real,A: a,B: product_prod_a_a] :
% 2.66/1.55 ( ( real_V296206913od_a_a @ R @ ( produc1299253312od_a_a @ A @ B ) )
% 2.66/1.55 = ( produc1299253312od_a_a @ ( real_V1035702895aleR_a @ R @ A ) @ ( real_V543523736od_a_a @ R @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_12_scaleR__Pair,axiom,
% 2.66/1.55 ! [R: real,A: a,B: real] :
% 2.66/1.55 ( ( real_V512797224a_real @ R @ ( product_Pair_a_real @ A @ B ) )
% 2.66/1.55 = ( product_Pair_a_real @ ( real_V1035702895aleR_a @ R @ A ) @ ( real_V453051771R_real @ R @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_13_scaleR__Pair,axiom,
% 2.66/1.55 ! [R: real,A: product_prod_a_a,B: a] :
% 2.66/1.55 ( ( real_V1538726831_a_a_a @ R @ ( produc2061588782_a_a_a @ A @ B ) )
% 2.66/1.55 = ( produc2061588782_a_a_a @ ( real_V543523736od_a_a @ R @ A ) @ ( real_V1035702895aleR_a @ R @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_14_scaleR__Pair,axiom,
% 2.66/1.55 ! [R: real,A: product_prod_a_a,B: product_prod_a_a] :
% 2.66/1.55 ( ( real_V368978776od_a_a @ R @ ( produc1474507607od_a_a @ A @ B ) )
% 2.66/1.55 = ( produc1474507607od_a_a @ ( real_V543523736od_a_a @ R @ A ) @ ( real_V543523736od_a_a @ R @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_15_scaleR__Pair,axiom,
% 2.66/1.55 ! [R: real,A: product_prod_a_a,B: real] :
% 2.66/1.55 ( ( real_V195069393a_real @ R @ ( produc454274172a_real @ A @ B ) )
% 2.66/1.55 = ( produc454274172a_real @ ( real_V543523736od_a_a @ R @ A ) @ ( real_V453051771R_real @ R @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_16_scaleR__Pair,axiom,
% 2.66/1.55 ! [R: real,A: real,B: a] :
% 2.66/1.55 ( ( real_V1214981142real_a @ R @ ( product_Pair_real_a @ A @ B ) )
% 2.66/1.55 = ( product_Pair_real_a @ ( real_V453051771R_real @ R @ A ) @ ( real_V1035702895aleR_a @ R @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_17_scaleR__Pair,axiom,
% 2.66/1.55 ! [R: real,A: real,B: product_prod_a_a] :
% 2.66/1.55 ( ( real_V1943155903od_a_a @ R @ ( produc454550562od_a_a @ A @ B ) )
% 2.66/1.55 = ( produc454550562od_a_a @ ( real_V453051771R_real @ R @ A ) @ ( real_V543523736od_a_a @ R @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_18_scaleR__Pair,axiom,
% 2.66/1.55 ! [R: real,A: real,B: real] :
% 2.66/1.55 ( ( real_V1139189034l_real @ R @ ( produc705216881l_real @ A @ B ) )
% 2.66/1.55 = ( produc705216881l_real @ ( real_V453051771R_real @ R @ A ) @ ( real_V453051771R_real @ R @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_19_scaleR__Pair,axiom,
% 2.66/1.55 ! [R: real,A: a,B: a] :
% 2.66/1.55 ( ( real_V543523736od_a_a @ R @ ( product_Pair_a_a @ A @ B ) )
% 2.66/1.55 = ( product_Pair_a_a @ ( real_V1035702895aleR_a @ R @ A ) @ ( real_V1035702895aleR_a @ R @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_20_scale__prod,axiom,
% 2.66/1.55 ! [X: real,A: a,B: product_prod_a_a] :
% 2.66/1.55 ( ( real_V296206913od_a_a @ X @ ( produc1299253312od_a_a @ A @ B ) )
% 2.66/1.55 = ( produc1299253312od_a_a @ ( real_V1035702895aleR_a @ X @ A ) @ ( real_V543523736od_a_a @ X @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_21_scale__prod,axiom,
% 2.66/1.55 ! [X: real,A: a,B: real] :
% 2.66/1.55 ( ( real_V512797224a_real @ X @ ( product_Pair_a_real @ A @ B ) )
% 2.66/1.55 = ( product_Pair_a_real @ ( real_V1035702895aleR_a @ X @ A ) @ ( real_V453051771R_real @ X @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_22_scale__prod,axiom,
% 2.66/1.55 ! [X: real,A: product_prod_a_a,B: a] :
% 2.66/1.55 ( ( real_V1538726831_a_a_a @ X @ ( produc2061588782_a_a_a @ A @ B ) )
% 2.66/1.55 = ( produc2061588782_a_a_a @ ( real_V543523736od_a_a @ X @ A ) @ ( real_V1035702895aleR_a @ X @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_23_scale__prod,axiom,
% 2.66/1.55 ! [X: real,A: product_prod_a_a,B: product_prod_a_a] :
% 2.66/1.55 ( ( real_V368978776od_a_a @ X @ ( produc1474507607od_a_a @ A @ B ) )
% 2.66/1.55 = ( produc1474507607od_a_a @ ( real_V543523736od_a_a @ X @ A ) @ ( real_V543523736od_a_a @ X @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_24_scale__prod,axiom,
% 2.66/1.55 ! [X: real,A: product_prod_a_a,B: real] :
% 2.66/1.55 ( ( real_V195069393a_real @ X @ ( produc454274172a_real @ A @ B ) )
% 2.66/1.55 = ( produc454274172a_real @ ( real_V543523736od_a_a @ X @ A ) @ ( real_V453051771R_real @ X @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_25_scale__prod,axiom,
% 2.66/1.55 ! [X: real,A: real,B: a] :
% 2.66/1.55 ( ( real_V1214981142real_a @ X @ ( product_Pair_real_a @ A @ B ) )
% 2.66/1.55 = ( product_Pair_real_a @ ( real_V453051771R_real @ X @ A ) @ ( real_V1035702895aleR_a @ X @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_26_scale__prod,axiom,
% 2.66/1.55 ! [X: real,A: real,B: product_prod_a_a] :
% 2.66/1.55 ( ( real_V1943155903od_a_a @ X @ ( produc454550562od_a_a @ A @ B ) )
% 2.66/1.55 = ( produc454550562od_a_a @ ( real_V453051771R_real @ X @ A ) @ ( real_V543523736od_a_a @ X @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_27_scale__prod,axiom,
% 2.66/1.55 ! [X: real,A: real,B: real] :
% 2.66/1.55 ( ( real_V1139189034l_real @ X @ ( produc705216881l_real @ A @ B ) )
% 2.66/1.55 = ( produc705216881l_real @ ( real_V453051771R_real @ X @ A ) @ ( real_V453051771R_real @ X @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_28_scale__prod,axiom,
% 2.66/1.55 ! [X: real,A: a,B: a] :
% 2.66/1.55 ( ( real_V543523736od_a_a @ X @ ( product_Pair_a_a @ A @ B ) )
% 2.66/1.55 = ( product_Pair_a_a @ ( real_V1035702895aleR_a @ X @ A ) @ ( real_V1035702895aleR_a @ X @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_29_add__Pair,axiom,
% 2.66/1.55 ! [A: a,B: a,C: a,D: a] :
% 2.66/1.55 ( ( plus_p1505579230od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( product_Pair_a_a @ C @ D ) )
% 2.66/1.55 = ( product_Pair_a_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_30_add__Pair,axiom,
% 2.66/1.55 ! [A: a,B: real,C: a,D: real] :
% 2.66/1.55 ( ( plus_p541014626a_real @ ( product_Pair_a_real @ A @ B ) @ ( product_Pair_a_real @ C @ D ) )
% 2.66/1.55 = ( product_Pair_a_real @ ( plus_plus_a @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_31_add__Pair,axiom,
% 2.66/1.55 ! [A: real,B: a,C: real,D: a] :
% 2.66/1.55 ( ( plus_p1243198544real_a @ ( product_Pair_real_a @ A @ B ) @ ( product_Pair_real_a @ C @ D ) )
% 2.66/1.55 = ( product_Pair_real_a @ ( plus_plus_real @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_32_add__Pair,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real,D: real] :
% 2.66/1.55 ( ( plus_p77862768l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
% 2.66/1.55 = ( produc705216881l_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_33_add__Pair,axiom,
% 2.66/1.55 ! [A: a,B: set_real,C: a,D: set_real] :
% 2.66/1.55 ( ( plus_p1745648536t_real @ ( produc1391501065t_real @ A @ B ) @ ( produc1391501065t_real @ C @ D ) )
% 2.66/1.55 = ( produc1391501065t_real @ ( plus_plus_a @ A @ C ) @ ( plus_plus_set_real @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_34_add__Pair,axiom,
% 2.66/1.55 ! [A: a,B: set_a,C: a,D: set_a] :
% 2.66/1.55 ( ( plus_p1265825342_set_a @ ( product_Pair_a_set_a @ A @ B ) @ ( product_Pair_a_set_a @ C @ D ) )
% 2.66/1.55 = ( product_Pair_a_set_a @ ( plus_plus_a @ A @ C ) @ ( plus_plus_set_a @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_35_add__Pair,axiom,
% 2.66/1.55 ! [A: real,B: set_real,C: real,D: set_real] :
% 2.66/1.55 ( ( plus_p2005278886t_real @ ( produc247649703t_real @ A @ B ) @ ( produc247649703t_real @ C @ D ) )
% 2.66/1.55 = ( produc247649703t_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_set_real @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_36_add__Pair,axiom,
% 2.66/1.55 ! [A: real,B: set_a,C: real,D: set_a] :
% 2.66/1.55 ( ( plus_p125342128_set_a @ ( produc1872317017_set_a @ A @ B ) @ ( produc1872317017_set_a @ C @ D ) )
% 2.66/1.55 = ( produc1872317017_set_a @ ( plus_plus_real @ A @ C ) @ ( plus_plus_set_a @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_37_add__Pair,axiom,
% 2.66/1.55 ! [A: set_real,B: a,C: set_real,D: a] :
% 2.66/1.55 ( ( plus_p1582417306real_a @ ( produc617496131real_a @ A @ B ) @ ( produc617496131real_a @ C @ D ) )
% 2.66/1.55 = ( produc617496131real_a @ ( plus_plus_set_real @ A @ C ) @ ( plus_plus_a @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_38_add__Pair,axiom,
% 2.66/1.55 ! [A: set_real,B: real,C: set_real,D: real] :
% 2.66/1.55 ( ( plus_p1952869542l_real @ ( produc1812461223l_real @ A @ B ) @ ( produc1812461223l_real @ C @ D ) )
% 2.66/1.55 = ( produc1812461223l_real @ ( plus_plus_set_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_39_prod_Oinject,axiom,
% 2.66/1.55 ! [X1: real,X2: real,Y1: real,Y2: real] :
% 2.66/1.55 ( ( ( produc705216881l_real @ X1 @ X2 )
% 2.66/1.55 = ( produc705216881l_real @ Y1 @ Y2 ) )
% 2.66/1.55 = ( ( X1 = Y1 )
% 2.66/1.55 & ( X2 = Y2 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_40_prod_Oinject,axiom,
% 2.66/1.55 ! [X1: real,X2: a,Y1: real,Y2: a] :
% 2.66/1.55 ( ( ( product_Pair_real_a @ X1 @ X2 )
% 2.66/1.55 = ( product_Pair_real_a @ Y1 @ Y2 ) )
% 2.66/1.55 = ( ( X1 = Y1 )
% 2.66/1.55 & ( X2 = Y2 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_41_prod_Oinject,axiom,
% 2.66/1.55 ! [X1: a,X2: real,Y1: a,Y2: real] :
% 2.66/1.55 ( ( ( product_Pair_a_real @ X1 @ X2 )
% 2.66/1.55 = ( product_Pair_a_real @ Y1 @ Y2 ) )
% 2.66/1.55 = ( ( X1 = Y1 )
% 2.66/1.55 & ( X2 = Y2 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_42_prod_Oinject,axiom,
% 2.66/1.55 ! [X1: a,X2: a,Y1: a,Y2: a] :
% 2.66/1.55 ( ( ( product_Pair_a_a @ X1 @ X2 )
% 2.66/1.55 = ( product_Pair_a_a @ Y1 @ Y2 ) )
% 2.66/1.55 = ( ( X1 = Y1 )
% 2.66/1.55 & ( X2 = Y2 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_43_old_Oprod_Oinject,axiom,
% 2.66/1.55 ! [A: real,B: real,A2: real,B2: real] :
% 2.66/1.55 ( ( ( produc705216881l_real @ A @ B )
% 2.66/1.55 = ( produc705216881l_real @ A2 @ B2 ) )
% 2.66/1.55 = ( ( A = A2 )
% 2.66/1.55 & ( B = B2 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_44_old_Oprod_Oinject,axiom,
% 2.66/1.55 ! [A: real,B: a,A2: real,B2: a] :
% 2.66/1.55 ( ( ( product_Pair_real_a @ A @ B )
% 2.66/1.55 = ( product_Pair_real_a @ A2 @ B2 ) )
% 2.66/1.55 = ( ( A = A2 )
% 2.66/1.55 & ( B = B2 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_45_old_Oprod_Oinject,axiom,
% 2.66/1.55 ! [A: a,B: real,A2: a,B2: real] :
% 2.66/1.55 ( ( ( product_Pair_a_real @ A @ B )
% 2.66/1.55 = ( product_Pair_a_real @ A2 @ B2 ) )
% 2.66/1.55 = ( ( A = A2 )
% 2.66/1.55 & ( B = B2 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_46_old_Oprod_Oinject,axiom,
% 2.66/1.55 ! [A: a,B: a,A2: a,B2: a] :
% 2.66/1.55 ( ( ( product_Pair_a_a @ A @ B )
% 2.66/1.55 = ( product_Pair_a_a @ A2 @ B2 ) )
% 2.66/1.55 = ( ( A = A2 )
% 2.66/1.55 & ( B = B2 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_47_set__plus__intro,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,C2: set_Pr147102617l_real,B: produc957004601l_real,D2: set_Pr147102617l_real] :
% 2.66/1.55 ( ( member1068169442l_real @ A @ C2 )
% 2.66/1.55 => ( ( member1068169442l_real @ B @ D2 )
% 2.66/1.55 => ( member1068169442l_real @ ( plus_p77862768l_real @ A @ B ) @ ( plus_p1708760016l_real @ C2 @ D2 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_48_set__plus__intro,axiom,
% 2.66/1.55 ! [A: product_prod_a_real,C2: set_Pr1928503567a_real,B: product_prod_a_real,D2: set_Pr1928503567a_real] :
% 2.66/1.55 ( ( member1103263856a_real @ A @ C2 )
% 2.66/1.55 => ( ( member1103263856a_real @ B @ D2 )
% 2.66/1.55 => ( member1103263856a_real @ ( plus_p541014626a_real @ A @ B ) @ ( plus_p130512152a_real @ C2 @ D2 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_49_set__plus__intro,axiom,
% 2.66/1.55 ! [A: set_Product_prod_a_a,C2: set_se1596668135od_a_a,B: set_Product_prod_a_a,D2: set_se1596668135od_a_a] :
% 2.66/1.55 ( ( member1838126896od_a_a @ A @ C2 )
% 2.66/1.55 => ( ( member1838126896od_a_a @ B @ D2 )
% 2.66/1.55 => ( member1838126896od_a_a @ ( plus_p634297534od_a_a @ A @ B ) @ ( plus_p1613276318od_a_a @ C2 @ D2 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_50_set__plus__intro,axiom,
% 2.66/1.55 ! [A: set_real,C2: set_set_real,B: set_real,D2: set_set_real] :
% 2.66/1.55 ( ( member_set_real @ A @ C2 )
% 2.66/1.55 => ( ( member_set_real @ B @ D2 )
% 2.66/1.55 => ( member_set_real @ ( plus_plus_set_real @ A @ B ) @ ( plus_p768704801t_real @ C2 @ D2 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_51_set__plus__intro,axiom,
% 2.66/1.55 ! [A: set_a,C2: set_set_a,B: set_a,D2: set_set_a] :
% 2.66/1.55 ( ( member_set_a @ A @ C2 )
% 2.66/1.55 => ( ( member_set_a @ B @ D2 )
% 2.66/1.55 => ( member_set_a @ ( plus_plus_set_a @ A @ B ) @ ( plus_plus_set_set_a @ C2 @ D2 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_52_set__plus__intro,axiom,
% 2.66/1.55 ! [A: product_prod_a_a,C2: set_Product_prod_a_a,B: product_prod_a_a,D2: set_Product_prod_a_a] :
% 2.66/1.55 ( ( member449909584od_a_a @ A @ C2 )
% 2.66/1.55 => ( ( member449909584od_a_a @ B @ D2 )
% 2.66/1.55 => ( member449909584od_a_a @ ( plus_p1505579230od_a_a @ A @ B ) @ ( plus_p634297534od_a_a @ C2 @ D2 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_53_set__plus__intro,axiom,
% 2.66/1.55 ! [A: a,C2: set_a,B: a,D2: set_a] :
% 2.66/1.55 ( ( member_a @ A @ C2 )
% 2.66/1.55 => ( ( member_a @ B @ D2 )
% 2.66/1.55 => ( member_a @ ( plus_plus_a @ A @ B ) @ ( plus_plus_set_a @ C2 @ D2 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_54_set__plus__intro,axiom,
% 2.66/1.55 ! [A: real,C2: set_real,B: real,D2: set_real] :
% 2.66/1.55 ( ( member_real @ A @ C2 )
% 2.66/1.55 => ( ( member_real @ B @ D2 )
% 2.66/1.55 => ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D2 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_55_add__left__cancel,axiom,
% 2.66/1.55 ! [A: product_prod_a_real,B: product_prod_a_real,C: product_prod_a_real] :
% 2.66/1.55 ( ( ( plus_p541014626a_real @ A @ B )
% 2.66/1.55 = ( plus_p541014626a_real @ A @ C ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_56_add__left__cancel,axiom,
% 2.66/1.55 ! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a] :
% 2.66/1.55 ( ( ( plus_p1505579230od_a_a @ A @ B )
% 2.66/1.55 = ( plus_p1505579230od_a_a @ A @ C ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_57_add__left__cancel,axiom,
% 2.66/1.55 ! [A: a,B: a,C: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ A @ B )
% 2.66/1.55 = ( plus_plus_a @ A @ C ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_58_add__left__cancel,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ A @ B )
% 2.66/1.55 = ( plus_plus_real @ A @ C ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_59_add__right__cancel,axiom,
% 2.66/1.55 ! [B: product_prod_a_real,A: product_prod_a_real,C: product_prod_a_real] :
% 2.66/1.55 ( ( ( plus_p541014626a_real @ B @ A )
% 2.66/1.55 = ( plus_p541014626a_real @ C @ A ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_60_add__right__cancel,axiom,
% 2.66/1.55 ! [B: product_prod_a_a,A: product_prod_a_a,C: product_prod_a_a] :
% 2.66/1.55 ( ( ( plus_p1505579230od_a_a @ B @ A )
% 2.66/1.55 = ( plus_p1505579230od_a_a @ C @ A ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_61_add__right__cancel,axiom,
% 2.66/1.55 ! [B: a,A: a,C: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ B @ A )
% 2.66/1.55 = ( plus_plus_a @ C @ A ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_62_add__right__cancel,axiom,
% 2.66/1.55 ! [B: real,A: real,C: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ B @ A )
% 2.66/1.55 = ( plus_plus_real @ C @ A ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_63_pth__9_I3_J,axiom,
% 2.66/1.55 ! [C: real,X: real,W: real,D: real,Z: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ X ) @ Z ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) @ ( plus_plus_real @ W @ Z ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_64_add__le__cancel__right,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,C: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ C ) )
% 2.66/1.55 = ( ord_le1342644953l_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_65_add__le__cancel__right,axiom,
% 2.66/1.55 ! [A: real,C: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.66/1.55 = ( ord_less_eq_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_66_add__le__cancel__left,axiom,
% 2.66/1.55 ! [C: produc957004601l_real,A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ C @ A ) @ ( plus_p77862768l_real @ C @ B ) )
% 2.66/1.55 = ( ord_le1342644953l_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_67_add__le__cancel__left,axiom,
% 2.66/1.55 ! [C: real,A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.66/1.55 = ( ord_less_eq_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_68_add__cancel__right__right,axiom,
% 2.66/1.55 ! [A: product_prod_a_real,B: product_prod_a_real] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_p541014626a_real @ A @ B ) )
% 2.66/1.55 = ( B = zero_z705155042a_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_69_add__cancel__right__right,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_p77862768l_real @ A @ B ) )
% 2.66/1.55 = ( B = zero_z659284464l_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_70_add__cancel__right__right,axiom,
% 2.66/1.55 ! [A: product_prod_a_a,B: product_prod_a_a] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_p1505579230od_a_a @ A @ B ) )
% 2.66/1.55 = ( B = zero_z950819678od_a_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_71_add__cancel__right__right,axiom,
% 2.66/1.55 ! [A: a,B: a] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_plus_a @ A @ B ) )
% 2.66/1.55 = ( B = zero_zero_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_72_add__cancel__right__right,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_plus_real @ A @ B ) )
% 2.66/1.55 = ( B = zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_73_add__cancel__right__left,axiom,
% 2.66/1.55 ! [A: product_prod_a_real,B: product_prod_a_real] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_p541014626a_real @ B @ A ) )
% 2.66/1.55 = ( B = zero_z705155042a_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_74_add__cancel__right__left,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_p77862768l_real @ B @ A ) )
% 2.66/1.55 = ( B = zero_z659284464l_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_75_add__cancel__right__left,axiom,
% 2.66/1.55 ! [A: product_prod_a_a,B: product_prod_a_a] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_p1505579230od_a_a @ B @ A ) )
% 2.66/1.55 = ( B = zero_z950819678od_a_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_76_add__cancel__right__left,axiom,
% 2.66/1.55 ! [A: a,B: a] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_plus_a @ B @ A ) )
% 2.66/1.55 = ( B = zero_zero_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_77_add__cancel__right__left,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( plus_plus_real @ B @ A ) )
% 2.66/1.55 = ( B = zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_78_add__cancel__left__right,axiom,
% 2.66/1.55 ! [A: product_prod_a_real,B: product_prod_a_real] :
% 2.66/1.55 ( ( ( plus_p541014626a_real @ A @ B )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_z705155042a_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_79_add__cancel__left__right,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ( plus_p77862768l_real @ A @ B )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_z659284464l_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_80_add__cancel__left__right,axiom,
% 2.66/1.55 ! [A: product_prod_a_a,B: product_prod_a_a] :
% 2.66/1.55 ( ( ( plus_p1505579230od_a_a @ A @ B )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_z950819678od_a_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_81_add__cancel__left__right,axiom,
% 2.66/1.55 ! [A: a,B: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ A @ B )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_zero_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_82_add__cancel__left__right,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ A @ B )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_83_add__cancel__left__left,axiom,
% 2.66/1.55 ! [B: product_prod_a_real,A: product_prod_a_real] :
% 2.66/1.55 ( ( ( plus_p541014626a_real @ B @ A )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_z705155042a_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_84_add__cancel__left__left,axiom,
% 2.66/1.55 ! [B: produc957004601l_real,A: produc957004601l_real] :
% 2.66/1.55 ( ( ( plus_p77862768l_real @ B @ A )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_z659284464l_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_85_add__cancel__left__left,axiom,
% 2.66/1.55 ! [B: product_prod_a_a,A: product_prod_a_a] :
% 2.66/1.55 ( ( ( plus_p1505579230od_a_a @ B @ A )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_z950819678od_a_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_86_add__cancel__left__left,axiom,
% 2.66/1.55 ! [B: a,A: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ B @ A )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_zero_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_87_add__cancel__left__left,axiom,
% 2.66/1.55 ! [B: real,A: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ B @ A )
% 2.66/1.55 = A )
% 2.66/1.55 = ( B = zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_88_double__zero__sym,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( zero_zero_real
% 2.66/1.55 = ( plus_plus_real @ A @ A ) )
% 2.66/1.55 = ( A = zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_89_double__zero,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ A @ A )
% 2.66/1.55 = zero_zero_real )
% 2.66/1.55 = ( A = zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_90_add_Oright__neutral,axiom,
% 2.66/1.55 ! [A: product_prod_a_real] :
% 2.66/1.55 ( ( plus_p541014626a_real @ A @ zero_z705155042a_real )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_91_add_Oright__neutral,axiom,
% 2.66/1.55 ! [A: set_Product_prod_a_a] :
% 2.66/1.55 ( ( plus_p634297534od_a_a @ A @ zero_z257140542od_a_a )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_92_add_Oright__neutral,axiom,
% 2.66/1.55 ! [A: set_real] :
% 2.66/1.55 ( ( plus_plus_set_real @ A @ zero_zero_set_real )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_93_add_Oright__neutral,axiom,
% 2.66/1.55 ! [A: set_a] :
% 2.66/1.55 ( ( plus_plus_set_a @ A @ zero_zero_set_a )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_94_add_Oright__neutral,axiom,
% 2.66/1.55 ! [A: produc957004601l_real] :
% 2.66/1.55 ( ( plus_p77862768l_real @ A @ zero_z659284464l_real )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_95_add_Oright__neutral,axiom,
% 2.66/1.55 ! [A: product_prod_a_a] :
% 2.66/1.55 ( ( plus_p1505579230od_a_a @ A @ zero_z950819678od_a_a )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_96_add_Oright__neutral,axiom,
% 2.66/1.55 ! [A: a] :
% 2.66/1.55 ( ( plus_plus_a @ A @ zero_zero_a )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_97_add_Oright__neutral,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( plus_plus_real @ A @ zero_zero_real )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_98_add_Oleft__neutral,axiom,
% 2.66/1.55 ! [A: product_prod_a_real] :
% 2.66/1.55 ( ( plus_p541014626a_real @ zero_z705155042a_real @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_99_add_Oleft__neutral,axiom,
% 2.66/1.55 ! [A: set_Product_prod_a_a] :
% 2.66/1.55 ( ( plus_p634297534od_a_a @ zero_z257140542od_a_a @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_100_add_Oleft__neutral,axiom,
% 2.66/1.55 ! [A: set_real] :
% 2.66/1.55 ( ( plus_plus_set_real @ zero_zero_set_real @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_101_add_Oleft__neutral,axiom,
% 2.66/1.55 ! [A: set_a] :
% 2.66/1.55 ( ( plus_plus_set_a @ zero_zero_set_a @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_102_add_Oleft__neutral,axiom,
% 2.66/1.55 ! [A: produc957004601l_real] :
% 2.66/1.55 ( ( plus_p77862768l_real @ zero_z659284464l_real @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_103_add_Oleft__neutral,axiom,
% 2.66/1.55 ! [A: product_prod_a_a] :
% 2.66/1.55 ( ( plus_p1505579230od_a_a @ zero_z950819678od_a_a @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_104_add_Oleft__neutral,axiom,
% 2.66/1.55 ! [A: a] :
% 2.66/1.55 ( ( plus_plus_a @ zero_zero_a @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_105_add_Oleft__neutral,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( plus_plus_real @ zero_zero_real @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_106_zero__le__double__add__iff__zero__le__single__add,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 2.66/1.55 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_107_double__add__le__zero__iff__single__add__le__zero,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 2.66/1.55 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_108_le__add__same__cancel2,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ A @ ( plus_p77862768l_real @ B @ A ) )
% 2.66/1.55 = ( ord_le1342644953l_real @ zero_z659284464l_real @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_109_le__add__same__cancel2,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 2.66/1.55 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_110_le__add__same__cancel1,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ A @ ( plus_p77862768l_real @ A @ B ) )
% 2.66/1.55 = ( ord_le1342644953l_real @ zero_z659284464l_real @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_111_le__add__same__cancel1,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 2.66/1.55 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_112_add__le__same__cancel2,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ B ) @ B )
% 2.66/1.55 = ( ord_le1342644953l_real @ A @ zero_z659284464l_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_113_add__le__same__cancel2,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 2.66/1.55 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_114_add__le__same__cancel1,axiom,
% 2.66/1.55 ! [B: produc957004601l_real,A: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ B @ A ) @ B )
% 2.66/1.55 = ( ord_le1342644953l_real @ A @ zero_z659284464l_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_115_add__le__same__cancel1,axiom,
% 2.66/1.55 ! [B: real,A: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 2.66/1.55 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_116_one__reorient,axiom,
% 2.66/1.55 ! [X: real] :
% 2.66/1.55 ( ( one_one_real = X )
% 2.66/1.55 = ( X = one_one_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_117_zero__reorient,axiom,
% 2.66/1.55 ! [X: a] :
% 2.66/1.55 ( ( zero_zero_a = X )
% 2.66/1.55 = ( X = zero_zero_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_118_zero__reorient,axiom,
% 2.66/1.55 ! [X: produc957004601l_real] :
% 2.66/1.55 ( ( zero_z659284464l_real = X )
% 2.66/1.55 = ( X = zero_z659284464l_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_119_zero__reorient,axiom,
% 2.66/1.55 ! [X: product_prod_a_a] :
% 2.66/1.55 ( ( zero_z950819678od_a_a = X )
% 2.66/1.55 = ( X = zero_z950819678od_a_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_120_zero__reorient,axiom,
% 2.66/1.55 ! [X: real] :
% 2.66/1.55 ( ( zero_zero_real = X )
% 2.66/1.55 = ( X = zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_121_add__nonpos__eq__0__iff,axiom,
% 2.66/1.55 ! [X: produc957004601l_real,Y: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ X @ zero_z659284464l_real )
% 2.66/1.55 => ( ( ord_le1342644953l_real @ Y @ zero_z659284464l_real )
% 2.66/1.55 => ( ( ( plus_p77862768l_real @ X @ Y )
% 2.66/1.55 = zero_z659284464l_real )
% 2.66/1.55 = ( ( X = zero_z659284464l_real )
% 2.66/1.55 & ( Y = zero_z659284464l_real ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_122_add__nonpos__eq__0__iff,axiom,
% 2.66/1.55 ! [X: real,Y: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 2.66/1.55 => ( ( ( plus_plus_real @ X @ Y )
% 2.66/1.55 = zero_zero_real )
% 2.66/1.55 = ( ( X = zero_zero_real )
% 2.66/1.55 & ( Y = zero_zero_real ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_123_mem__Collect__eq,axiom,
% 2.66/1.55 ! [A: real,P: real > $o] :
% 2.66/1.55 ( ( member_real @ A @ ( collect_real @ P ) )
% 2.66/1.55 = ( P @ A ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_124_mem__Collect__eq,axiom,
% 2.66/1.55 ! [A: a,P: a > $o] :
% 2.66/1.55 ( ( member_a @ A @ ( collect_a @ P ) )
% 2.66/1.55 = ( P @ A ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_125_mem__Collect__eq,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,P: produc957004601l_real > $o] :
% 2.66/1.55 ( ( member1068169442l_real @ A @ ( collec1300223524l_real @ P ) )
% 2.66/1.55 = ( P @ A ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_126_mem__Collect__eq,axiom,
% 2.66/1.55 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 2.66/1.55 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 2.66/1.55 = ( P @ A ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_127_Collect__mem__eq,axiom,
% 2.66/1.55 ! [A3: set_real] :
% 2.66/1.55 ( ( collect_real
% 2.66/1.55 @ ^ [X3: real] : ( member_real @ X3 @ A3 ) )
% 2.66/1.55 = A3 ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_128_Collect__mem__eq,axiom,
% 2.66/1.55 ! [A3: set_a] :
% 2.66/1.55 ( ( collect_a
% 2.66/1.55 @ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
% 2.66/1.55 = A3 ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_129_Collect__mem__eq,axiom,
% 2.66/1.55 ! [A3: set_Pr147102617l_real] :
% 2.66/1.55 ( ( collec1300223524l_real
% 2.66/1.55 @ ^ [X3: produc957004601l_real] : ( member1068169442l_real @ X3 @ A3 ) )
% 2.66/1.55 = A3 ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_130_Collect__mem__eq,axiom,
% 2.66/1.55 ! [A3: set_Product_prod_a_a] :
% 2.66/1.55 ( ( collec645855634od_a_a
% 2.66/1.55 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 2.66/1.55 = A3 ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_131_Collect__cong,axiom,
% 2.66/1.55 ! [P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
% 2.66/1.55 ( ! [X4: product_prod_a_a] :
% 2.66/1.55 ( ( P @ X4 )
% 2.66/1.55 = ( Q @ X4 ) )
% 2.66/1.55 => ( ( collec645855634od_a_a @ P )
% 2.66/1.55 = ( collec645855634od_a_a @ Q ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_132_add__nonneg__eq__0__iff,axiom,
% 2.66/1.55 ! [X: produc957004601l_real,Y: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ zero_z659284464l_real @ X )
% 2.66/1.55 => ( ( ord_le1342644953l_real @ zero_z659284464l_real @ Y )
% 2.66/1.55 => ( ( ( plus_p77862768l_real @ X @ Y )
% 2.66/1.55 = zero_z659284464l_real )
% 2.66/1.55 = ( ( X = zero_z659284464l_real )
% 2.66/1.55 & ( Y = zero_z659284464l_real ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_133_add__nonneg__eq__0__iff,axiom,
% 2.66/1.55 ! [X: real,Y: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 2.66/1.55 => ( ( ( plus_plus_real @ X @ Y )
% 2.66/1.55 = zero_zero_real )
% 2.66/1.55 = ( ( X = zero_zero_real )
% 2.66/1.55 & ( Y = zero_zero_real ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_134_add__nonpos__nonpos,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ A @ zero_z659284464l_real )
% 2.66/1.55 => ( ( ord_le1342644953l_real @ B @ zero_z659284464l_real )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ B ) @ zero_z659284464l_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_135_add__nonpos__nonpos,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 2.66/1.55 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_136_add__nonneg__nonneg,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ zero_z659284464l_real @ A )
% 2.66/1.55 => ( ( ord_le1342644953l_real @ zero_z659284464l_real @ B )
% 2.66/1.55 => ( ord_le1342644953l_real @ zero_z659284464l_real @ ( plus_p77862768l_real @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_137_add__nonneg__nonneg,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.66/1.55 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_138_add__increasing2,axiom,
% 2.66/1.55 ! [C: produc957004601l_real,B: produc957004601l_real,A: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ zero_z659284464l_real @ C )
% 2.66/1.55 => ( ( ord_le1342644953l_real @ B @ A )
% 2.66/1.55 => ( ord_le1342644953l_real @ B @ ( plus_p77862768l_real @ A @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_139_add__increasing2,axiom,
% 2.66/1.55 ! [C: real,B: real,A: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.66/1.55 => ( ( ord_less_eq_real @ B @ A )
% 2.66/1.55 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_140_add__decreasing2,axiom,
% 2.66/1.55 ! [C: produc957004601l_real,A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ C @ zero_z659284464l_real )
% 2.66/1.55 => ( ( ord_le1342644953l_real @ A @ B )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_141_add__decreasing2,axiom,
% 2.66/1.55 ! [C: real,A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_142_add__increasing,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ zero_z659284464l_real @ A )
% 2.66/1.55 => ( ( ord_le1342644953l_real @ B @ C )
% 2.66/1.55 => ( ord_le1342644953l_real @ B @ ( plus_p77862768l_real @ A @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_143_add__increasing,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_eq_real @ B @ C )
% 2.66/1.55 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_144_add__decreasing,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,C: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ A @ zero_z659284464l_real )
% 2.66/1.55 => ( ( ord_le1342644953l_real @ C @ B )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_145_add__decreasing,axiom,
% 2.66/1.55 ! [A: real,C: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_eq_real @ C @ B )
% 2.66/1.55 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_146_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z950819678od_a_a
% 2.66/1.55 = ( product_Pair_a_a @ zero_zero_a @ zero_zero_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_147_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z659284464l_real
% 2.66/1.55 = ( produc705216881l_real @ zero_zero_real @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_148_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z1407338960real_a
% 2.66/1.55 = ( product_Pair_real_a @ zero_zero_real @ zero_zero_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_149_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z705155042a_real
% 2.66/1.55 = ( product_Pair_a_real @ zero_zero_a @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_150_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z590476811l_real
% 2.66/1.55 = ( produc1926903988l_real @ zero_zero_real @ zero_z659284464l_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_151_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z396988537od_a_a
% 2.66/1.55 = ( produc454550562od_a_a @ zero_zero_real @ zero_z950819678od_a_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_152_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z2135370393l_real
% 2.66/1.55 = ( produc64206290l_real @ zero_zero_a @ zero_z659284464l_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_153_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z53797895od_a_a
% 2.66/1.55 = ( produc1299253312od_a_a @ zero_zero_a @ zero_z950819678od_a_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_154_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z342251165l_real
% 2.66/1.55 = ( produc1175086478l_real @ zero_z659284464l_real @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_155_zero__prod__def,axiom,
% 2.66/1.55 ( zero_z1937883107real_a
% 2.66/1.55 = ( produc1430099868real_a @ zero_z659284464l_real @ zero_zero_a ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_156_add__le__imp__le__right,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,C: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ C ) )
% 2.66/1.55 => ( ord_le1342644953l_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_157_add__le__imp__le__right,axiom,
% 2.66/1.55 ! [A: real,C: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.66/1.55 => ( ord_less_eq_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_158_add__le__imp__le__left,axiom,
% 2.66/1.55 ! [C: produc957004601l_real,A: produc957004601l_real,B: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ ( plus_p77862768l_real @ C @ A ) @ ( plus_p77862768l_real @ C @ B ) )
% 2.66/1.55 => ( ord_le1342644953l_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_159_add__le__imp__le__left,axiom,
% 2.66/1.55 ! [C: real,A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.66/1.55 => ( ord_less_eq_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_160_add__right__mono,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ A @ B )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_161_add__right__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_162_add__left__mono,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ A @ B )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ C @ A ) @ ( plus_p77862768l_real @ C @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_163_add__left__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_164_add__mono,axiom,
% 2.66/1.55 ! [A: produc957004601l_real,B: produc957004601l_real,C: produc957004601l_real,D: produc957004601l_real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ A @ B )
% 2.66/1.55 => ( ( ord_le1342644953l_real @ C @ D )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ A @ C ) @ ( plus_p77862768l_real @ B @ D ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_165_add__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real,D: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ( ord_less_eq_real @ C @ D )
% 2.66/1.55 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_166_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.66/1.55 ! [I: produc957004601l_real,J: produc957004601l_real,K: produc957004601l_real,L: produc957004601l_real] :
% 2.66/1.55 ( ( ( ord_le1342644953l_real @ I @ J )
% 2.66/1.55 & ( ord_le1342644953l_real @ K @ L ) )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ I @ K ) @ ( plus_p77862768l_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_167_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.66/1.55 ! [I: real,J: real,K: real,L: real] :
% 2.66/1.55 ( ( ( ord_less_eq_real @ I @ J )
% 2.66/1.55 & ( ord_less_eq_real @ K @ L ) )
% 2.66/1.55 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_168_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.66/1.55 ! [I: produc957004601l_real,J: produc957004601l_real,K: produc957004601l_real,L: produc957004601l_real] :
% 2.66/1.55 ( ( ( I = J )
% 2.66/1.55 & ( ord_le1342644953l_real @ K @ L ) )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ I @ K ) @ ( plus_p77862768l_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_169_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.66/1.55 ! [I: real,J: real,K: real,L: real] :
% 2.66/1.55 ( ( ( I = J )
% 2.66/1.55 & ( ord_less_eq_real @ K @ L ) )
% 2.66/1.55 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_170_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.66/1.55 ! [I: produc957004601l_real,J: produc957004601l_real,K: produc957004601l_real,L: produc957004601l_real] :
% 2.66/1.55 ( ( ( ord_le1342644953l_real @ I @ J )
% 2.66/1.55 & ( K = L ) )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( plus_p77862768l_real @ I @ K ) @ ( plus_p77862768l_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_171_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.66/1.55 ! [I: real,J: real,K: real,L: real] :
% 2.66/1.55 ( ( ( ord_less_eq_real @ I @ J )
% 2.66/1.55 & ( K = L ) )
% 2.66/1.55 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_172_pth__4_I1_J,axiom,
% 2.66/1.55 ! [X: produc957004601l_real] :
% 2.66/1.55 ( ( real_V1139189034l_real @ zero_zero_real @ X )
% 2.66/1.55 = zero_z659284464l_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_173_pth__4_I1_J,axiom,
% 2.66/1.55 ! [X: real] :
% 2.66/1.55 ( ( real_V453051771R_real @ zero_zero_real @ X )
% 2.66/1.55 = zero_zero_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_174_convex__ge__imp__conved,axiom,
% 2.66/1.55 ! [Pr: set_Pr1741234931a_real] :
% 2.66/1.55 ( ! [X4: product_prod_a_real,Y3: product_prod_a_real] :
% 2.66/1.55 ( ( member384820540a_real @ ( produc886678603a_real @ X4 @ Y3 ) @ Pr )
% 2.66/1.55 => ! [Alpha: real,Beta: real] :
% 2.66/1.55 ( ( ( ( plus_plus_real @ Alpha @ Beta )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ Alpha )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ Beta ) )
% 2.66/1.55 => ( member384820540a_real @ ( produc886678603a_real @ ( plus_p541014626a_real @ ( real_V512797224a_real @ Alpha @ X4 ) @ ( real_V512797224a_real @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
% 2.66/1.55 => ( prefer1113819806a_real @ Pr ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_175_convex__ge__imp__conved,axiom,
% 2.66/1.55 ! [Pr: set_Pr1948701895od_a_a] :
% 2.66/1.55 ( ! [X4: product_prod_a_a,Y3: product_prod_a_a] :
% 2.66/1.55 ( ( member2057358096od_a_a @ ( produc1474507607od_a_a @ X4 @ Y3 ) @ Pr )
% 2.66/1.55 => ! [Alpha: real,Beta: real] :
% 2.66/1.55 ( ( ( ( plus_plus_real @ Alpha @ Beta )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ Alpha )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ Beta ) )
% 2.66/1.55 => ( member2057358096od_a_a @ ( produc1474507607od_a_a @ ( plus_p1505579230od_a_a @ ( real_V543523736od_a_a @ Alpha @ X4 ) @ ( real_V543523736od_a_a @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
% 2.66/1.55 => ( prefer225425826od_a_a @ Pr ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_176_convex__ge__imp__conved,axiom,
% 2.66/1.55 ! [Pr: set_Pr147102617l_real] :
% 2.66/1.55 ( ! [X4: real,Y3: real] :
% 2.66/1.55 ( ( member1068169442l_real @ ( produc705216881l_real @ X4 @ Y3 ) @ Pr )
% 2.66/1.55 => ! [Alpha: real,Beta: real] :
% 2.66/1.55 ( ( ( ( plus_plus_real @ Alpha @ Beta )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ Alpha )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ Beta ) )
% 2.66/1.55 => ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ Alpha @ X4 ) @ ( real_V453051771R_real @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
% 2.66/1.55 => ( prefer1247792113f_real @ Pr ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_177_convex__ge__imp__conved,axiom,
% 2.66/1.55 ! [Pr: set_Product_prod_a_a] :
% 2.66/1.55 ( ! [X4: a,Y3: a] :
% 2.66/1.55 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Pr )
% 2.66/1.55 => ! [Alpha: real,Beta: real] :
% 2.66/1.55 ( ( ( ( plus_plus_real @ Alpha @ Beta )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ Alpha )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ Beta ) )
% 2.66/1.55 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ Alpha @ X4 ) @ ( real_V1035702895aleR_a @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) )
% 2.66/1.55 => ( prefer529818233pref_a @ Pr ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_178_add_Ogroup__left__neutral,axiom,
% 2.66/1.55 ! [A: product_prod_a_real] :
% 2.66/1.55 ( ( plus_p541014626a_real @ zero_z705155042a_real @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_179_add_Ogroup__left__neutral,axiom,
% 2.66/1.55 ! [A: produc957004601l_real] :
% 2.66/1.55 ( ( plus_p77862768l_real @ zero_z659284464l_real @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_180_add_Ogroup__left__neutral,axiom,
% 2.66/1.55 ! [A: product_prod_a_a] :
% 2.66/1.55 ( ( plus_p1505579230od_a_a @ zero_z950819678od_a_a @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_181_add_Ogroup__left__neutral,axiom,
% 2.66/1.55 ! [A: a] :
% 2.66/1.55 ( ( plus_plus_a @ zero_zero_a @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_182_add_Ogroup__left__neutral,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( plus_plus_real @ zero_zero_real @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_183_add_Ocomm__neutral,axiom,
% 2.66/1.55 ! [A: set_a] :
% 2.66/1.55 ( ( plus_plus_set_a @ A @ zero_zero_set_a )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_184_add_Ocomm__neutral,axiom,
% 2.66/1.55 ! [A: produc957004601l_real] :
% 2.66/1.55 ( ( plus_p77862768l_real @ A @ zero_z659284464l_real )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_185_add_Ocomm__neutral,axiom,
% 2.66/1.55 ! [A: product_prod_a_a] :
% 2.66/1.55 ( ( plus_p1505579230od_a_a @ A @ zero_z950819678od_a_a )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_186_add_Ocomm__neutral,axiom,
% 2.66/1.55 ! [A: a] :
% 2.66/1.55 ( ( plus_plus_a @ A @ zero_zero_a )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_187_add_Ocomm__neutral,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( plus_plus_real @ A @ zero_zero_real )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_188_comm__monoid__add__class_Oadd__0,axiom,
% 2.66/1.55 ! [A: a] :
% 2.66/1.55 ( ( plus_plus_a @ zero_zero_a @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_189_comm__monoid__add__class_Oadd__0,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( plus_plus_real @ zero_zero_real @ A )
% 2.66/1.55 = A ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_190_pth__d,axiom,
% 2.66/1.55 ! [X: real] :
% 2.66/1.55 ( ( plus_plus_real @ X @ zero_zero_real )
% 2.66/1.55 = X ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_191_pth__7_I1_J,axiom,
% 2.66/1.55 ! [X: real] :
% 2.66/1.55 ( ( plus_plus_real @ zero_zero_real @ X )
% 2.66/1.55 = X ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_192_pth__4_I2_J,axiom,
% 2.66/1.55 ! [C: real] :
% 2.66/1.55 ( ( real_V453051771R_real @ C @ zero_zero_real )
% 2.66/1.55 = zero_zero_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_193_add__right__imp__eq,axiom,
% 2.66/1.55 ! [B: a,A: a,C: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ B @ A )
% 2.66/1.55 = ( plus_plus_a @ C @ A ) )
% 2.66/1.55 => ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_194_add__right__imp__eq,axiom,
% 2.66/1.55 ! [B: real,A: real,C: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ B @ A )
% 2.66/1.55 = ( plus_plus_real @ C @ A ) )
% 2.66/1.55 => ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_195_add__left__imp__eq,axiom,
% 2.66/1.55 ! [A: a,B: a,C: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ A @ B )
% 2.66/1.55 = ( plus_plus_a @ A @ C ) )
% 2.66/1.55 => ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_196_add__left__imp__eq,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ A @ B )
% 2.66/1.55 = ( plus_plus_real @ A @ C ) )
% 2.66/1.55 => ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_197_add_Oleft__commute,axiom,
% 2.66/1.55 ! [B: a,A: a,C: a] :
% 2.66/1.55 ( ( plus_plus_a @ B @ ( plus_plus_a @ A @ C ) )
% 2.66/1.55 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_198_add_Oleft__commute,axiom,
% 2.66/1.55 ! [B: real,A: real,C: real] :
% 2.66/1.55 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 2.66/1.55 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_199_add_Ocommute,axiom,
% 2.66/1.55 ( plus_plus_a
% 2.66/1.55 = ( ^ [A4: a,B3: a] : ( plus_plus_a @ B3 @ A4 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_200_add_Ocommute,axiom,
% 2.66/1.55 ( plus_plus_real
% 2.66/1.55 = ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_201_add_Oright__cancel,axiom,
% 2.66/1.55 ! [B: a,A: a,C: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ B @ A )
% 2.66/1.55 = ( plus_plus_a @ C @ A ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_202_add_Oright__cancel,axiom,
% 2.66/1.55 ! [B: real,A: real,C: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ B @ A )
% 2.66/1.55 = ( plus_plus_real @ C @ A ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_203_add_Oleft__cancel,axiom,
% 2.66/1.55 ! [A: a,B: a,C: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ A @ B )
% 2.66/1.55 = ( plus_plus_a @ A @ C ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_204_add_Oleft__cancel,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ A @ B )
% 2.66/1.55 = ( plus_plus_real @ A @ C ) )
% 2.66/1.55 = ( B = C ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_205_add_Oassoc,axiom,
% 2.66/1.55 ! [A: a,B: a,C: a] :
% 2.66/1.55 ( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
% 2.66/1.55 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_206_add_Oassoc,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.66/1.55 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_207_set__plus__elim,axiom,
% 2.66/1.55 ! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 2.66/1.55 ( ( member449909584od_a_a @ X @ ( plus_p634297534od_a_a @ A3 @ B4 ) )
% 2.66/1.55 => ~ ! [A5: product_prod_a_a,B5: product_prod_a_a] :
% 2.66/1.55 ( ( X
% 2.66/1.55 = ( plus_p1505579230od_a_a @ A5 @ B5 ) )
% 2.66/1.55 => ( ( member449909584od_a_a @ A5 @ A3 )
% 2.66/1.55 => ~ ( member449909584od_a_a @ B5 @ B4 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_208_set__plus__elim,axiom,
% 2.66/1.55 ! [X: a,A3: set_a,B4: set_a] :
% 2.66/1.55 ( ( member_a @ X @ ( plus_plus_set_a @ A3 @ B4 ) )
% 2.66/1.55 => ~ ! [A5: a,B5: a] :
% 2.66/1.55 ( ( X
% 2.66/1.55 = ( plus_plus_a @ A5 @ B5 ) )
% 2.66/1.55 => ( ( member_a @ A5 @ A3 )
% 2.66/1.55 => ~ ( member_a @ B5 @ B4 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_209_set__plus__elim,axiom,
% 2.66/1.55 ! [X: real,A3: set_real,B4: set_real] :
% 2.66/1.55 ( ( member_real @ X @ ( plus_plus_set_real @ A3 @ B4 ) )
% 2.66/1.55 => ~ ! [A5: real,B5: real] :
% 2.66/1.55 ( ( X
% 2.66/1.55 = ( plus_plus_real @ A5 @ B5 ) )
% 2.66/1.55 => ( ( member_real @ A5 @ A3 )
% 2.66/1.55 => ~ ( member_real @ B5 @ B4 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_210_group__cancel_Oadd2,axiom,
% 2.66/1.55 ! [B4: a,K: a,B: a,A: a] :
% 2.66/1.55 ( ( B4
% 2.66/1.55 = ( plus_plus_a @ K @ B ) )
% 2.66/1.55 => ( ( plus_plus_a @ A @ B4 )
% 2.66/1.55 = ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_211_group__cancel_Oadd2,axiom,
% 2.66/1.55 ! [B4: real,K: real,B: real,A: real] :
% 2.66/1.55 ( ( B4
% 2.66/1.55 = ( plus_plus_real @ K @ B ) )
% 2.66/1.55 => ( ( plus_plus_real @ A @ B4 )
% 2.66/1.55 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_212_group__cancel_Oadd1,axiom,
% 2.66/1.55 ! [A3: a,K: a,A: a,B: a] :
% 2.66/1.55 ( ( A3
% 2.66/1.55 = ( plus_plus_a @ K @ A ) )
% 2.66/1.55 => ( ( plus_plus_a @ A3 @ B )
% 2.66/1.55 = ( plus_plus_a @ K @ ( plus_plus_a @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_213_group__cancel_Oadd1,axiom,
% 2.66/1.55 ! [A3: real,K: real,A: real,B: real] :
% 2.66/1.55 ( ( A3
% 2.66/1.55 = ( plus_plus_real @ K @ A ) )
% 2.66/1.55 => ( ( plus_plus_real @ A3 @ B )
% 2.66/1.55 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_214_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.66/1.55 ! [I: real,J: real,K: real,L: real] :
% 2.66/1.55 ( ( ( I = J )
% 2.66/1.55 & ( K = L ) )
% 2.66/1.55 => ( ( plus_plus_real @ I @ K )
% 2.66/1.55 = ( plus_plus_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_215_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.66/1.55 ! [A: a,B: a,C: a] :
% 2.66/1.55 ( ( plus_plus_a @ ( plus_plus_a @ A @ B ) @ C )
% 2.66/1.55 = ( plus_plus_a @ A @ ( plus_plus_a @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_216_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.66/1.55 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_217_old_Oprod_Oinducts,axiom,
% 2.66/1.55 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 2.66/1.55 ( ! [A5: a,B5: a] : ( P @ ( product_Pair_a_a @ A5 @ B5 ) )
% 2.66/1.55 => ( P @ Prod ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_218_old_Oprod_Oexhaust,axiom,
% 2.66/1.55 ! [Y: product_prod_a_a] :
% 2.66/1.55 ~ ! [A5: a,B5: a] :
% 2.66/1.55 ( Y
% 2.66/1.55 != ( product_Pair_a_a @ A5 @ B5 ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_219_Pair__inject,axiom,
% 2.66/1.55 ! [A: a,B: a,A2: a,B2: a] :
% 2.66/1.55 ( ( ( product_Pair_a_a @ A @ B )
% 2.66/1.55 = ( product_Pair_a_a @ A2 @ B2 ) )
% 2.66/1.55 => ~ ( ( A = A2 )
% 2.66/1.55 => ( B != B2 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_220_prod__cases,axiom,
% 2.66/1.55 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 2.66/1.55 ( ! [A5: a,B5: a] : ( P @ ( product_Pair_a_a @ A5 @ B5 ) )
% 2.66/1.55 => ( P @ P2 ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_221_surj__pair,axiom,
% 2.66/1.55 ! [P2: product_prod_a_a] :
% 2.66/1.55 ? [X4: a,Y3: a] :
% 2.66/1.55 ( P2
% 2.66/1.55 = ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_222_pth__a,axiom,
% 2.66/1.55 ! [X: real,Y: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( real_V453051771R_real @ zero_zero_real @ X ) @ Y )
% 2.66/1.55 = Y ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_223_pth__8,axiom,
% 2.66/1.55 ! [C: real,X: real,D: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ X ) )
% 2.66/1.55 = ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_224_pth__6,axiom,
% 2.66/1.55 ! [C: real,X: real,Y: real] :
% 2.66/1.55 ( ( real_V453051771R_real @ C @ ( plus_plus_real @ X @ Y ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ C @ Y ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_225_pth__c_I1_J,axiom,
% 2.66/1.55 ! [C: real,X: real,D: real,Y: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ Y ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( real_V453051771R_real @ C @ X ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_226_pth__c_I2_J,axiom,
% 2.66/1.55 ! [C: real,X: real,Z: real,D: real,Y: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) @ ( real_V453051771R_real @ D @ Y ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_227_pth__c_I3_J,axiom,
% 2.66/1.55 ! [C: real,X: real,D: real,Y: real,Z: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_228_pth__c_I4_J,axiom,
% 2.66/1.55 ! [C: real,X: real,W: real,D: real,Y: real,Z: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ Z ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_229_pth__b_I1_J,axiom,
% 2.66/1.55 ! [C: real,X: real,D: real,Y: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ Y ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( real_V453051771R_real @ D @ Y ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_230_pth__b_I2_J,axiom,
% 2.66/1.55 ! [C: real,X: real,Z: real,D: real,Y: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) @ ( real_V453051771R_real @ D @ Y ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ Z @ ( real_V453051771R_real @ D @ Y ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_231_pth__b_I3_J,axiom,
% 2.66/1.55 ! [C: real,X: real,D: real,Y: real,Z: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_232_pth__b_I4_J,axiom,
% 2.66/1.55 ! [C: real,X: real,W: real,D: real,Y: real,Z: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ W @ ( plus_plus_real @ ( real_V453051771R_real @ D @ Y ) @ Z ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_233_pth__9_I1_J,axiom,
% 2.66/1.55 ! [C: real,X: real,Z: real,D: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ Z ) @ ( real_V453051771R_real @ D @ X ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) @ Z ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_234_pth__9_I2_J,axiom,
% 2.66/1.55 ! [C: real,X: real,D: real,Z: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( real_V453051771R_real @ C @ X ) @ ( plus_plus_real @ ( real_V453051771R_real @ D @ X ) @ Z ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ ( plus_plus_real @ C @ D ) @ X ) @ Z ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_235_scaleR__eq__iff,axiom,
% 2.66/1.55 ! [B: real,U: real,A: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ B @ ( real_V453051771R_real @ U @ A ) )
% 2.66/1.55 = ( plus_plus_real @ A @ ( real_V453051771R_real @ U @ B ) ) )
% 2.66/1.55 = ( ( A = B )
% 2.66/1.55 | ( U = one_one_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_236_scaleR__eq__iff,axiom,
% 2.66/1.55 ! [B: a,U: real,A: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ B @ ( real_V1035702895aleR_a @ U @ A ) )
% 2.66/1.55 = ( plus_plus_a @ A @ ( real_V1035702895aleR_a @ U @ B ) ) )
% 2.66/1.55 = ( ( A = B )
% 2.66/1.55 | ( U = one_one_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_237_scale__zero__left,axiom,
% 2.66/1.55 ! [X: real] :
% 2.66/1.55 ( ( real_V453051771R_real @ zero_zero_real @ X )
% 2.66/1.55 = zero_zero_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_238_scale__zero__left,axiom,
% 2.66/1.55 ! [X: a] :
% 2.66/1.55 ( ( real_V1035702895aleR_a @ zero_zero_real @ X )
% 2.66/1.55 = zero_zero_a ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_239_scale__eq__0__iff,axiom,
% 2.66/1.55 ! [A: real,X: real] :
% 2.66/1.55 ( ( ( real_V453051771R_real @ A @ X )
% 2.66/1.55 = zero_zero_real )
% 2.66/1.55 = ( ( A = zero_zero_real )
% 2.66/1.55 | ( X = zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_240_scale__eq__0__iff,axiom,
% 2.66/1.55 ! [A: real,X: a] :
% 2.66/1.55 ( ( ( real_V1035702895aleR_a @ A @ X )
% 2.66/1.55 = zero_zero_a )
% 2.66/1.55 = ( ( A = zero_zero_real )
% 2.66/1.55 | ( X = zero_zero_a ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_241_scaleR__one,axiom,
% 2.66/1.55 ! [X: a] :
% 2.66/1.55 ( ( real_V1035702895aleR_a @ one_one_real @ X )
% 2.66/1.55 = X ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_242_scale__cancel__left,axiom,
% 2.66/1.55 ! [A: real,X: a,Y: a] :
% 2.66/1.55 ( ( ( real_V1035702895aleR_a @ A @ X )
% 2.66/1.55 = ( real_V1035702895aleR_a @ A @ Y ) )
% 2.66/1.55 = ( ( X = Y )
% 2.66/1.55 | ( A = zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_243_scaleR__left__le__one__le,axiom,
% 2.66/1.55 ! [X: real,A: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.66/1.55 => ( ( ord_less_eq_real @ A @ one_one_real )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ X ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_244_scale__cancel__right,axiom,
% 2.66/1.55 ! [A: real,X: real,B: real] :
% 2.66/1.55 ( ( ( real_V453051771R_real @ A @ X )
% 2.66/1.55 = ( real_V453051771R_real @ B @ X ) )
% 2.66/1.55 = ( ( A = B )
% 2.66/1.55 | ( X = zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_245_scale__cancel__right,axiom,
% 2.66/1.55 ! [A: real,X: a,B: real] :
% 2.66/1.55 ( ( ( real_V1035702895aleR_a @ A @ X )
% 2.66/1.55 = ( real_V1035702895aleR_a @ B @ X ) )
% 2.66/1.55 = ( ( A = B )
% 2.66/1.55 | ( X = zero_zero_a ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_246_scale__zero__right,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( real_V453051771R_real @ A @ zero_zero_real )
% 2.66/1.55 = zero_zero_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_247_scale__zero__right,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( real_V1035702895aleR_a @ A @ zero_zero_a )
% 2.66/1.55 = zero_zero_a ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_248_split__scaleR__neg__le,axiom,
% 2.66/1.55 ! [A: real,X: real] :
% 2.66/1.55 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 & ( ord_less_eq_real @ X @ zero_zero_real ) )
% 2.66/1.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_249_split__scaleR__pos__le,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 2.66/1.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.66/1.55 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 2.66/1.55 => ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_250_set__plus__mono2__b,axiom,
% 2.66/1.55 ! [C2: set_Product_prod_a_a,D2: set_Product_prod_a_a,E: set_Product_prod_a_a,F: set_Product_prod_a_a,X: product_prod_a_a] :
% 2.66/1.55 ( ( ord_le1824328871od_a_a @ C2 @ D2 )
% 2.66/1.55 => ( ( ord_le1824328871od_a_a @ E @ F )
% 2.66/1.55 => ( ( member449909584od_a_a @ X @ ( plus_p634297534od_a_a @ C2 @ E ) )
% 2.66/1.55 => ( member449909584od_a_a @ X @ ( plus_p634297534od_a_a @ D2 @ F ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_251_set__zero__plus2,axiom,
% 2.66/1.55 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 2.66/1.55 ( ( member449909584od_a_a @ zero_z950819678od_a_a @ A3 )
% 2.66/1.55 => ( ord_le1824328871od_a_a @ B4 @ ( plus_p634297534od_a_a @ A3 @ B4 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_252_set__zero__plus2,axiom,
% 2.66/1.55 ! [A3: set_real,B4: set_real] :
% 2.66/1.55 ( ( member_real @ zero_zero_real @ A3 )
% 2.66/1.55 => ( ord_less_eq_set_real @ B4 @ ( plus_plus_set_real @ A3 @ B4 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_253_scale__left__commute,axiom,
% 2.66/1.55 ! [A: real,B: real,X: a] :
% 2.66/1.55 ( ( real_V1035702895aleR_a @ A @ ( real_V1035702895aleR_a @ B @ X ) )
% 2.66/1.55 = ( real_V1035702895aleR_a @ B @ ( real_V1035702895aleR_a @ A @ X ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_254_scale__right__imp__eq,axiom,
% 2.66/1.55 ! [X: real,A: real,B: real] :
% 2.66/1.55 ( ( X != zero_zero_real )
% 2.66/1.55 => ( ( ( real_V453051771R_real @ A @ X )
% 2.66/1.55 = ( real_V453051771R_real @ B @ X ) )
% 2.66/1.55 => ( A = B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_255_scale__right__imp__eq,axiom,
% 2.66/1.55 ! [X: a,A: real,B: real] :
% 2.66/1.55 ( ( X != zero_zero_a )
% 2.66/1.55 => ( ( ( real_V1035702895aleR_a @ A @ X )
% 2.66/1.55 = ( real_V1035702895aleR_a @ B @ X ) )
% 2.66/1.55 => ( A = B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_256_scaleR__add__right,axiom,
% 2.66/1.55 ! [A: real,X: real,Y: real] :
% 2.66/1.55 ( ( real_V453051771R_real @ A @ ( plus_plus_real @ X @ Y ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ A @ Y ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_257_scaleR__add__right,axiom,
% 2.66/1.55 ! [A: real,X: a,Y: a] :
% 2.66/1.55 ( ( real_V1035702895aleR_a @ A @ ( plus_plus_a @ X @ Y ) )
% 2.66/1.55 = ( plus_plus_a @ ( real_V1035702895aleR_a @ A @ X ) @ ( real_V1035702895aleR_a @ A @ Y ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_258_scale__left__imp__eq,axiom,
% 2.66/1.55 ! [A: real,X: a,Y: a] :
% 2.66/1.55 ( ( A != zero_zero_real )
% 2.66/1.55 => ( ( ( real_V1035702895aleR_a @ A @ X )
% 2.66/1.55 = ( real_V1035702895aleR_a @ A @ Y ) )
% 2.66/1.55 => ( X = Y ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_259_scaleR__left_Oadd,axiom,
% 2.66/1.55 ! [X: real,Y: real,Xa: real] :
% 2.66/1.55 ( ( real_V453051771R_real @ ( plus_plus_real @ X @ Y ) @ Xa )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ X @ Xa ) @ ( real_V453051771R_real @ Y @ Xa ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_260_scaleR__left_Oadd,axiom,
% 2.66/1.55 ! [X: real,Y: real,Xa: a] :
% 2.66/1.55 ( ( real_V1035702895aleR_a @ ( plus_plus_real @ X @ Y ) @ Xa )
% 2.66/1.55 = ( plus_plus_a @ ( real_V1035702895aleR_a @ X @ Xa ) @ ( real_V1035702895aleR_a @ Y @ Xa ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_261_scaleR__add__left,axiom,
% 2.66/1.55 ! [A: real,B: real,X: real] :
% 2.66/1.55 ( ( real_V453051771R_real @ ( plus_plus_real @ A @ B ) @ X )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ B @ X ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_262_scaleR__add__left,axiom,
% 2.66/1.55 ! [A: real,B: real,X: a] :
% 2.66/1.55 ( ( real_V1035702895aleR_a @ ( plus_plus_real @ A @ B ) @ X )
% 2.66/1.55 = ( plus_plus_a @ ( real_V1035702895aleR_a @ A @ X ) @ ( real_V1035702895aleR_a @ B @ X ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_263_scaleR__right__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,X: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ B @ X ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_264_scaleR__right__mono__neg,axiom,
% 2.66/1.55 ! [B: real,A: real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ B @ A )
% 2.66/1.55 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ C ) @ ( real_V453051771R_real @ B @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_265_scaleR__left__mono,axiom,
% 2.66/1.55 ! [X: real,Y: real,A: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X @ Y )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ A @ Y ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_266_scaleR__left__mono__neg,axiom,
% 2.66/1.55 ! [B: real,A: real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ B @ A )
% 2.66/1.55 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_267_scaleR__mono_H,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real,D: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ( ord_less_eq_real @ C @ D )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ C ) @ ( real_V453051771R_real @ B @ D ) ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_268_scaleR__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,X: real,Y: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ( ord_less_eq_real @ X @ Y )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ ( real_V453051771R_real @ B @ Y ) ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_269_scaleR__nonpos__nonpos,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 2.66/1.55 => ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_270_scaleR__nonpos__nonneg,axiom,
% 2.66/1.55 ! [A: real,X: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_271_scaleR__nonneg__nonpos,axiom,
% 2.66/1.55 ! [A: real,X: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 2.66/1.55 => ( ord_less_eq_real @ ( real_V453051771R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_272_scaleR__nonneg__nonneg,axiom,
% 2.66/1.55 ! [A: real,X: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 2.66/1.55 => ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ X ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_273_Pair__le,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real,D: real] :
% 2.66/1.55 ( ( ord_le1342644953l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
% 2.66/1.55 = ( ( ord_less_eq_real @ A @ C )
% 2.66/1.55 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_274_not__one__le__zero,axiom,
% 2.66/1.55 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_275_zero__le__one,axiom,
% 2.66/1.55 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 2.66/1.55
% 2.66/1.55 thf(fact_276_zero__neq__one,axiom,
% 2.66/1.55 zero_zero_real != one_one_real ).
% 2.66/1.55
% 2.66/1.55 thf(fact_277_Pair__mono,axiom,
% 2.66/1.55 ! [X: real,X5: real,Y: real,Y4: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X @ X5 )
% 2.66/1.55 => ( ( ord_less_eq_real @ Y @ Y4 )
% 2.66/1.55 => ( ord_le1342644953l_real @ ( produc705216881l_real @ X @ Y ) @ ( produc705216881l_real @ X5 @ Y4 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_278_weak__convexI,axiom,
% 2.66/1.55 ! [Pr: set_Pr147102617l_real] :
% 2.66/1.55 ( ! [X4: real,Y3: real,Alpha: real,Beta: real] :
% 2.66/1.55 ( ( member1068169442l_real @ ( produc705216881l_real @ X4 @ Y3 ) @ Pr )
% 2.66/1.55 => ( ( ( plus_plus_real @ Alpha @ Beta )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ Alpha )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ Beta )
% 2.66/1.55 => ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ Alpha @ X4 ) @ ( real_V453051771R_real @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) ) ) )
% 2.66/1.55 => ( prefer1247792113f_real @ Pr ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_279_weak__convexI,axiom,
% 2.66/1.55 ! [Pr: set_Product_prod_a_a] :
% 2.66/1.55 ( ! [X4: a,Y3: a,Alpha: real,Beta: real] :
% 2.66/1.55 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Pr )
% 2.66/1.55 => ( ( ( plus_plus_real @ Alpha @ Beta )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ Alpha )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ Beta )
% 2.66/1.55 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ Alpha @ X4 ) @ ( real_V1035702895aleR_a @ Beta @ Y3 ) ) @ Y3 ) @ Pr ) ) ) ) )
% 2.66/1.55 => ( prefer529818233pref_a @ Pr ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_280_subsetI,axiom,
% 2.66/1.55 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 2.66/1.55 ( ! [X4: product_prod_a_a] :
% 2.66/1.55 ( ( member449909584od_a_a @ X4 @ A3 )
% 2.66/1.55 => ( member449909584od_a_a @ X4 @ B4 ) )
% 2.66/1.55 => ( ord_le1824328871od_a_a @ A3 @ B4 ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_281_add__less__cancel__right,axiom,
% 2.66/1.55 ! [A: real,C: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.66/1.55 = ( ord_less_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_282_add__less__cancel__left,axiom,
% 2.66/1.55 ! [C: real,A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.66/1.55 = ( ord_less_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_283_add__less__same__cancel1,axiom,
% 2.66/1.55 ! [B: real,A: real] :
% 2.66/1.55 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 2.66/1.55 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_284_add__less__same__cancel2,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 2.66/1.55 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_285_less__add__same__cancel1,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 2.66/1.55 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_286_less__add__same__cancel2,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 2.66/1.55 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_287_double__add__less__zero__iff__single__add__less__zero,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 2.66/1.55 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_288_zero__less__double__add__iff__zero__less__single__add,axiom,
% 2.66/1.55 ! [A: real] :
% 2.66/1.55 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 2.66/1.55 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_289_add__less__imp__less__right,axiom,
% 2.66/1.55 ! [A: real,C: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 2.66/1.55 => ( ord_less_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_290_add__less__imp__less__left,axiom,
% 2.66/1.55 ! [C: real,A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 2.66/1.55 => ( ord_less_real @ A @ B ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_291_add__strict__right__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_real @ A @ B )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_292_add__strict__left__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_real @ A @ B )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_293_add__strict__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real,D: real] :
% 2.66/1.55 ( ( ord_less_real @ A @ B )
% 2.66/1.55 => ( ( ord_less_real @ C @ D )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_294_add__mono__thms__linordered__field_I1_J,axiom,
% 2.66/1.55 ! [I: real,J: real,K: real,L: real] :
% 2.66/1.55 ( ( ( ord_less_real @ I @ J )
% 2.66/1.55 & ( K = L ) )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_295_add__mono__thms__linordered__field_I2_J,axiom,
% 2.66/1.55 ! [I: real,J: real,K: real,L: real] :
% 2.66/1.55 ( ( ( I = J )
% 2.66/1.55 & ( ord_less_real @ K @ L ) )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_296_add__mono__thms__linordered__field_I5_J,axiom,
% 2.66/1.55 ! [I: real,J: real,K: real,L: real] :
% 2.66/1.55 ( ( ( ord_less_real @ I @ J )
% 2.66/1.55 & ( ord_less_real @ K @ L ) )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_297_add__less__le__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real,D: real] :
% 2.66/1.55 ( ( ord_less_real @ A @ B )
% 2.66/1.55 => ( ( ord_less_eq_real @ C @ D )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_298_add__le__less__mono,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real,D: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ( ord_less_real @ C @ D )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_299_add__mono__thms__linordered__field_I3_J,axiom,
% 2.66/1.55 ! [I: real,J: real,K: real,L: real] :
% 2.66/1.55 ( ( ( ord_less_real @ I @ J )
% 2.66/1.55 & ( ord_less_eq_real @ K @ L ) )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_300_add__mono__thms__linordered__field_I4_J,axiom,
% 2.66/1.55 ! [I: real,J: real,K: real,L: real] :
% 2.66/1.55 ( ( ( ord_less_eq_real @ I @ J )
% 2.66/1.55 & ( ord_less_real @ K @ L ) )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_301_add__neg__neg,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ A @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_real @ B @ zero_zero_real )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_302_add__pos__pos,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ B )
% 2.66/1.55 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_303_pos__add__strict,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_real @ B @ C )
% 2.66/1.55 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_304_add__less__zeroD,axiom,
% 2.66/1.55 ! [X: real,Y: real] :
% 2.66/1.55 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_real @ X @ zero_zero_real )
% 2.66/1.55 | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_305_zero__less__one,axiom,
% 2.66/1.55 ord_less_real @ zero_zero_real @ one_one_real ).
% 2.66/1.55
% 2.66/1.55 thf(fact_306_not__one__less__zero,axiom,
% 2.66/1.55 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_307_add__mono1,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ A @ B )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_308_less__add__one,axiom,
% 2.66/1.55 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_309_add__strict__increasing2,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_real @ B @ C )
% 2.66/1.55 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_310_add__strict__increasing,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_eq_real @ B @ C )
% 2.66/1.55 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_311_add__pos__nonneg,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 2.66/1.55 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_312_add__nonpos__neg,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_real @ B @ zero_zero_real )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_313_add__nonneg__pos,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ B )
% 2.66/1.55 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_314_add__neg__nonpos,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ A @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 2.66/1.55 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_315_zero__less__two,axiom,
% 2.66/1.55 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_316_scaleR__le__cancel__left__pos,axiom,
% 2.66/1.55 ! [C: real,A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ zero_zero_real @ C )
% 2.66/1.55 => ( ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) )
% 2.66/1.55 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_317_scaleR__le__cancel__left__neg,axiom,
% 2.66/1.55 ! [C: real,A: real,B: real] :
% 2.66/1.55 ( ( ord_less_real @ C @ zero_zero_real )
% 2.66/1.55 => ( ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) )
% 2.66/1.55 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_318_scaleR__le__cancel__left,axiom,
% 2.66/1.55 ! [C: real,A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ ( real_V453051771R_real @ C @ A ) @ ( real_V453051771R_real @ C @ B ) )
% 2.66/1.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 2.66/1.55 => ( ord_less_eq_real @ A @ B ) )
% 2.66/1.55 & ( ( ord_less_real @ C @ zero_zero_real )
% 2.66/1.55 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_319_subset__iff,axiom,
% 2.66/1.55 ( ord_le1824328871od_a_a
% 2.66/1.55 = ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
% 2.66/1.55 ! [T: product_prod_a_a] :
% 2.66/1.55 ( ( member449909584od_a_a @ T @ A6 )
% 2.66/1.55 => ( member449909584od_a_a @ T @ B6 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_320_subset__eq,axiom,
% 2.66/1.55 ( ord_le1824328871od_a_a
% 2.66/1.55 = ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
% 2.66/1.55 ! [X3: product_prod_a_a] :
% 2.66/1.55 ( ( member449909584od_a_a @ X3 @ A6 )
% 2.66/1.55 => ( member449909584od_a_a @ X3 @ B6 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_321_subsetD,axiom,
% 2.66/1.55 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
% 2.66/1.55 ( ( ord_le1824328871od_a_a @ A3 @ B4 )
% 2.66/1.55 => ( ( member449909584od_a_a @ C @ A3 )
% 2.66/1.55 => ( member449909584od_a_a @ C @ B4 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_322_in__mono,axiom,
% 2.66/1.55 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,X: product_prod_a_a] :
% 2.66/1.55 ( ( ord_le1824328871od_a_a @ A3 @ B4 )
% 2.66/1.55 => ( ( member449909584od_a_a @ X @ A3 )
% 2.66/1.55 => ( member449909584od_a_a @ X @ B4 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_323_zero__le__scaleR__iff,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ zero_zero_real @ ( real_V453051771R_real @ A @ B ) )
% 2.66/1.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 2.66/1.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 2.66/1.55 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 2.66/1.55 | ( A = zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_324_scaleR__le__0__iff,axiom,
% 2.66/1.55 ! [A: real,B: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ ( real_V453051771R_real @ A @ B ) @ zero_zero_real )
% 2.66/1.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 2.66/1.55 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 2.66/1.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 2.66/1.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 2.66/1.55 | ( A = zero_zero_real ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_325_weak__convexD,axiom,
% 2.66/1.55 ! [Pr: set_Pr147102617l_real,X: real,Y: real,U: real,V: real] :
% 2.66/1.55 ( ( prefer1247792113f_real @ Pr )
% 2.66/1.55 => ( ( member1068169442l_real @ ( produc705216881l_real @ X @ Y ) @ Pr )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ U )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ V )
% 2.66/1.55 => ( ( ( plus_plus_real @ U @ V )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 => ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ U @ X ) @ ( real_V453051771R_real @ V @ Y ) ) @ Y ) @ Pr ) ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_326_weak__convexD,axiom,
% 2.66/1.55 ! [Pr: set_Product_prod_a_a,X: a,Y: a,U: real,V: real] :
% 2.66/1.55 ( ( prefer529818233pref_a @ Pr )
% 2.66/1.55 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ X @ Y ) @ Pr )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ U )
% 2.66/1.55 => ( ( ord_less_real @ zero_zero_real @ V )
% 2.66/1.55 => ( ( ( plus_plus_real @ U @ V )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ U @ X ) @ ( real_V1035702895aleR_a @ V @ Y ) ) @ Y ) @ Pr ) ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_327_weak__convex__pref__def,axiom,
% 2.66/1.55 ( prefer1247792113f_real
% 2.66/1.55 = ( ^ [Pr2: set_Pr147102617l_real] :
% 2.66/1.55 ! [X3: real,Y5: real] :
% 2.66/1.55 ( ( member1068169442l_real @ ( produc705216881l_real @ X3 @ Y5 ) @ Pr2 )
% 2.66/1.55 => ! [Alpha2: real,Beta2: real] :
% 2.66/1.55 ( ( ( ( plus_plus_real @ Alpha2 @ Beta2 )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 & ( ord_less_real @ zero_zero_real @ Alpha2 )
% 2.66/1.55 & ( ord_less_real @ zero_zero_real @ Beta2 ) )
% 2.66/1.55 => ( member1068169442l_real @ ( produc705216881l_real @ ( plus_plus_real @ ( real_V453051771R_real @ Alpha2 @ X3 ) @ ( real_V453051771R_real @ Beta2 @ Y5 ) ) @ Y5 ) @ Pr2 ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_328_weak__convex__pref__def,axiom,
% 2.66/1.55 ( prefer529818233pref_a
% 2.66/1.55 = ( ^ [Pr2: set_Product_prod_a_a] :
% 2.66/1.55 ! [X3: a,Y5: a] :
% 2.66/1.55 ( ( member449909584od_a_a @ ( product_Pair_a_a @ X3 @ Y5 ) @ Pr2 )
% 2.66/1.55 => ! [Alpha2: real,Beta2: real] :
% 2.66/1.55 ( ( ( ( plus_plus_real @ Alpha2 @ Beta2 )
% 2.66/1.55 = one_one_real )
% 2.66/1.55 & ( ord_less_real @ zero_zero_real @ Alpha2 )
% 2.66/1.55 & ( ord_less_real @ zero_zero_real @ Beta2 ) )
% 2.66/1.55 => ( member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ Alpha2 @ X3 ) @ ( real_V1035702895aleR_a @ Beta2 @ Y5 ) ) @ Y5 ) @ Pr2 ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_329_field__le__epsilon,axiom,
% 2.66/1.55 ! [X: real,Y: real] :
% 2.66/1.55 ( ! [E2: real] :
% 2.66/1.55 ( ( ord_less_real @ zero_zero_real @ E2 )
% 2.66/1.55 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
% 2.66/1.55 => ( ord_less_eq_real @ X @ Y ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_330_less__numeral__extra_I1_J,axiom,
% 2.66/1.55 ord_less_real @ zero_zero_real @ one_one_real ).
% 2.66/1.55
% 2.66/1.55 thf(fact_331_psubsetD,axiom,
% 2.66/1.55 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
% 2.66/1.55 ( ( ord_le108336051od_a_a @ A3 @ B4 )
% 2.66/1.55 => ( ( member449909584od_a_a @ C @ A3 )
% 2.66/1.55 => ( member449909584od_a_a @ C @ B4 ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_332_is__num__normalize_I1_J,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 2.66/1.55 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_333_le__numeral__extra_I3_J,axiom,
% 2.66/1.55 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 2.66/1.55
% 2.66/1.55 thf(fact_334_less__numeral__extra_I3_J,axiom,
% 2.66/1.55 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_335_le__numeral__extra_I4_J,axiom,
% 2.66/1.55 ord_less_eq_real @ one_one_real @ one_one_real ).
% 2.66/1.55
% 2.66/1.55 thf(fact_336_less__numeral__extra_I4_J,axiom,
% 2.66/1.55 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_337_order__refl,axiom,
% 2.66/1.55 ! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_338_add__scaleR__degen,axiom,
% 2.66/1.55 ! [U: real,B: real,V: real,A: real] :
% 2.66/1.55 ( ( ( plus_plus_real @ ( real_V453051771R_real @ U @ B ) @ ( real_V453051771R_real @ V @ A ) )
% 2.66/1.55 = ( plus_plus_real @ ( real_V453051771R_real @ U @ A ) @ ( real_V453051771R_real @ V @ B ) ) )
% 2.66/1.55 => ( ( U != V )
% 2.66/1.55 => ( A = B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_339_add__scaleR__degen,axiom,
% 2.66/1.55 ! [U: real,B: a,V: real,A: a] :
% 2.66/1.55 ( ( ( plus_plus_a @ ( real_V1035702895aleR_a @ U @ B ) @ ( real_V1035702895aleR_a @ V @ A ) )
% 2.66/1.55 = ( plus_plus_a @ ( real_V1035702895aleR_a @ U @ A ) @ ( real_V1035702895aleR_a @ V @ B ) ) )
% 2.66/1.55 => ( ( U != V )
% 2.66/1.55 => ( A = B ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_340_order__subst1,axiom,
% 2.66/1.55 ! [A: real,F2: real > real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
% 2.66/1.55 => ( ( ord_less_eq_real @ B @ C )
% 2.66/1.55 => ( ! [X4: real,Y3: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X4 @ Y3 )
% 2.66/1.55 => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 2.66/1.55 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_341_order__subst2,axiom,
% 2.66/1.55 ! [A: real,B: real,F2: real > real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ( ord_less_eq_real @ ( F2 @ B ) @ C )
% 2.66/1.55 => ( ! [X4: real,Y3: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X4 @ Y3 )
% 2.66/1.55 => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 2.66/1.55 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_342_ord__eq__le__subst,axiom,
% 2.66/1.55 ! [A: real,F2: real > real,B: real,C: real] :
% 2.66/1.55 ( ( A
% 2.66/1.55 = ( F2 @ B ) )
% 2.66/1.55 => ( ( ord_less_eq_real @ B @ C )
% 2.66/1.55 => ( ! [X4: real,Y3: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X4 @ Y3 )
% 2.66/1.55 => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 2.66/1.55 => ( ord_less_eq_real @ A @ ( F2 @ C ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_343_ord__le__eq__subst,axiom,
% 2.66/1.55 ! [A: real,B: real,F2: real > real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ( ( F2 @ B )
% 2.66/1.55 = C )
% 2.66/1.55 => ( ! [X4: real,Y3: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X4 @ Y3 )
% 2.66/1.55 => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
% 2.66/1.55 => ( ord_less_eq_real @ ( F2 @ A ) @ C ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_344_eq__iff,axiom,
% 2.66/1.55 ( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
% 2.66/1.55 = ( ^ [X3: real,Y5: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X3 @ Y5 )
% 2.66/1.55 & ( ord_less_eq_real @ Y5 @ X3 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_345_antisym,axiom,
% 2.66/1.55 ! [X: real,Y: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X @ Y )
% 2.66/1.55 => ( ( ord_less_eq_real @ Y @ X )
% 2.66/1.55 => ( X = Y ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_346_linear,axiom,
% 2.66/1.55 ! [X: real,Y: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ X @ Y )
% 2.66/1.55 | ( ord_less_eq_real @ Y @ X ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_347_eq__refl,axiom,
% 2.66/1.55 ! [X: real,Y: real] :
% 2.66/1.55 ( ( X = Y )
% 2.66/1.55 => ( ord_less_eq_real @ X @ Y ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_348_le__cases,axiom,
% 2.66/1.55 ! [X: real,Y: real] :
% 2.66/1.55 ( ~ ( ord_less_eq_real @ X @ Y )
% 2.66/1.55 => ( ord_less_eq_real @ Y @ X ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_349_order_Otrans,axiom,
% 2.66/1.55 ! [A: real,B: real,C: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A @ B )
% 2.66/1.55 => ( ( ord_less_eq_real @ B @ C )
% 2.66/1.55 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_350_le__cases3,axiom,
% 2.66/1.55 ! [X: real,Y: real,Z: real] :
% 2.66/1.55 ( ( ( ord_less_eq_real @ X @ Y )
% 2.66/1.55 => ~ ( ord_less_eq_real @ Y @ Z ) )
% 2.66/1.55 => ( ( ( ord_less_eq_real @ Y @ X )
% 2.66/1.55 => ~ ( ord_less_eq_real @ X @ Z ) )
% 2.66/1.55 => ( ( ( ord_less_eq_real @ X @ Z )
% 2.66/1.55 => ~ ( ord_less_eq_real @ Z @ Y ) )
% 2.66/1.55 => ( ( ( ord_less_eq_real @ Z @ Y )
% 2.66/1.55 => ~ ( ord_less_eq_real @ Y @ X ) )
% 2.66/1.55 => ( ( ( ord_less_eq_real @ Y @ Z )
% 2.66/1.55 => ~ ( ord_less_eq_real @ Z @ X ) )
% 2.66/1.55 => ~ ( ( ord_less_eq_real @ Z @ X )
% 2.66/1.55 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_351_antisym__conv,axiom,
% 2.66/1.55 ! [Y: real,X: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ Y @ X )
% 2.66/1.55 => ( ( ord_less_eq_real @ X @ Y )
% 2.66/1.55 = ( X = Y ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(fact_352_order__class_Oorder_Oeq__iff,axiom,
% 2.66/1.55 ( ( ^ [Y6: real,Z2: real] : ( Y6 = Z2 ) )
% 2.66/1.55 = ( ^ [A4: real,B3: real] :
% 2.66/1.55 ( ( ord_less_eq_real @ A4 @ B3 )
% 2.66/1.55 & ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% 2.66/1.55
% 2.66/1.55 thf(conj_0,conjecture,
% 2.66/1.55 member449909584od_a_a @ ( product_Pair_a_a @ ( plus_plus_a @ ( real_V1035702895aleR_a @ u @ x ) @ ( real_V1035702895aleR_a @ v @ y ) ) @ y ) @ relation ).
% 2.66/1.55 ------------------------
% 2.66/1.58 % (14143)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/183Mi)
% 2.66/1.58 % (14147)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_13924 for (2999ds/2Mi)
% 2.66/1.58 % (14147)Instruction limit reached!
% 2.66/1.58 % (14147)------------------------------
% 2.66/1.58 % (14147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.58 % (14147)Termination reason: Unknown
% 2.66/1.58 % (14147)Termination phase: shuffling
% 2.66/1.58
% 2.66/1.58 % (14147)Memory used [KB]: 1407
% 2.66/1.58 % (14147)Time elapsed: 0.003 s
% 2.66/1.58 % (14147)Instructions burned: 4 (million)
% 2.66/1.58 % (14147)------------------------------
% 2.66/1.58 % (14147)------------------------------
% 2.66/1.59 % (14144)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_13924 for (2999ds/4Mi)
% 2.66/1.59 % (14146)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/2Mi)
% 2.66/1.59 % (14145)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_13924 for (2999ds/27Mi)
% 2.66/1.59 % (14148)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_13924 for (2999ds/275Mi)
% 2.66/1.59 % (14149)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/18Mi)
% 2.66/1.59 % (14146)Instruction limit reached!
% 2.66/1.59 % (14146)------------------------------
% 2.66/1.59 % (14146)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.59 % (14146)Termination reason: Unknown
% 2.66/1.59 % (14146)Termination phase: shuffling
% 2.66/1.59
% 2.66/1.59 % (14146)Memory used [KB]: 1407
% 2.66/1.59 % (14146)Time elapsed: 0.004 s
% 2.66/1.59 % (14146)Instructions burned: 2 (million)
% 2.66/1.59 % (14146)------------------------------
% 2.66/1.59 % (14146)------------------------------
% 2.66/1.59 % (14144)Instruction limit reached!
% 2.66/1.59 % (14144)------------------------------
% 2.66/1.59 % (14144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.59 % (14144)Termination reason: Unknown
% 2.66/1.59 % (14144)Termination phase: shuffling
% 2.66/1.59
% 2.66/1.59 % (14144)Memory used [KB]: 1407
% 2.66/1.59 % (14144)Time elapsed: 0.005 s
% 2.66/1.59 % (14144)Instructions burned: 5 (million)
% 2.66/1.59 % (14144)------------------------------
% 2.66/1.59 % (14144)------------------------------
% 2.66/1.59 % (14150)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/3Mi)
% 2.66/1.59 % (14150)Instruction limit reached!
% 2.66/1.59 % (14150)------------------------------
% 2.66/1.59 % (14150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.59 % (14150)Termination reason: Unknown
% 2.66/1.59 % (14150)Termination phase: shuffling
% 2.66/1.59
% 2.66/1.59 % (14150)Memory used [KB]: 1407
% 2.66/1.59 % (14150)Time elapsed: 0.003 s
% 2.66/1.59 % (14150)Instructions burned: 5 (million)
% 2.66/1.59 % (14150)------------------------------
% 2.66/1.59 % (14150)------------------------------
% 2.71/1.60 % (14149)Instruction limit reached!
% 2.71/1.60 % (14149)------------------------------
% 2.71/1.60 % (14149)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (14149)Termination reason: Unknown
% 2.71/1.60 % (14149)Termination phase: shuffling
% 2.71/1.60
% 2.71/1.60 % (14149)Memory used [KB]: 1791
% 2.71/1.60 % (14149)Time elapsed: 0.014 s
% 2.71/1.60 % (14149)Instructions burned: 18 (million)
% 2.71/1.60 % (14149)------------------------------
% 2.71/1.60 % (14149)------------------------------
% 2.71/1.60 % (14153)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_13924 for (2999ds/3Mi)
% 2.71/1.60 % (14153)Instruction limit reached!
% 2.71/1.60 % (14153)------------------------------
% 2.71/1.60 % (14153)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (14153)Termination reason: Unknown
% 2.71/1.60 % (14153)Termination phase: shuffling
% 2.71/1.60
% 2.71/1.60 % (14153)Memory used [KB]: 1407
% 2.71/1.60 % (14153)Time elapsed: 0.002 s
% 2.71/1.60 % (14153)Instructions burned: 3 (million)
% 2.71/1.60 % (14153)------------------------------
% 2.71/1.60 % (14153)------------------------------
% 2.71/1.61 % (14145)Instruction limit reached!
% 2.71/1.61 % (14145)------------------------------
% 2.71/1.61 % (14145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (14145)Termination reason: Unknown
% 2.71/1.61 % (14145)Termination phase: Property scanning
% 2.71/1.61
% 2.71/1.61 % (14145)Memory used [KB]: 1791
% 2.71/1.61 % (14145)Time elapsed: 0.018 s
% 2.71/1.61 % (14145)Instructions burned: 28 (million)
% 2.71/1.61 % (14145)------------------------------
% 2.71/1.61 % (14145)------------------------------
% 2.71/1.61 % (14152)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_13924 for (2999ds/15Mi)
% 2.71/1.61 % (14151)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_13924 for (2999ds/37Mi)
% 2.71/1.61 % (14152)Instruction limit reached!
% 2.71/1.61 % (14152)------------------------------
% 2.71/1.61 % (14152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (14152)Termination reason: Unknown
% 2.71/1.61 % (14152)Termination phase: shuffling
% 2.71/1.61
% 2.71/1.61 % (14152)Memory used [KB]: 1663
% 2.71/1.61 % (14152)Time elapsed: 0.008 s
% 2.71/1.61 % (14152)Instructions burned: 15 (million)
% 2.71/1.61 % (14152)------------------------------
% 2.71/1.61 % (14152)------------------------------
% 2.71/1.61 % (14155)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/7Mi)
% 2.71/1.62 % (14155)Instruction limit reached!
% 2.71/1.62 % (14155)------------------------------
% 2.71/1.62 % (14155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.62 % (14155)Termination reason: Unknown
% 2.71/1.62 % (14155)Termination phase: shuffling
% 2.71/1.62
% 2.71/1.62 % (14155)Memory used [KB]: 1535
% 2.71/1.62 % (14155)Time elapsed: 0.004 s
% 2.71/1.62 % (14155)Instructions burned: 8 (million)
% 2.71/1.62 % (14155)------------------------------
% 2.71/1.62 % (14155)------------------------------
% 2.71/1.62 % (14154)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_13924 for (2999ds/1041Mi)
% 2.71/1.62 % (14157)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_13924 for (2999ds/3Mi)
% 2.71/1.62 % (14156)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/16Mi)
% 2.71/1.62 % (14157)Instruction limit reached!
% 2.71/1.62 % (14157)------------------------------
% 2.71/1.62 % (14157)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.62 % (14157)Termination reason: Unknown
% 2.71/1.62 % (14157)Termination phase: shuffling
% 2.71/1.62
% 2.71/1.62 % (14157)Memory used [KB]: 1407
% 2.71/1.62 % (14157)Time elapsed: 0.003 s
% 2.71/1.62 % (14157)Instructions burned: 3 (million)
% 2.71/1.62 % (14157)------------------------------
% 2.71/1.62 % (14157)------------------------------
% 2.71/1.63 % (14158)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_13924 for (2999ds/3Mi)
% 2.71/1.63 % (14158)Instruction limit reached!
% 2.71/1.63 % (14158)------------------------------
% 2.71/1.63 % (14158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (14158)Termination reason: Unknown
% 2.71/1.63 % (14158)Termination phase: shuffling
% 2.71/1.63
% 2.71/1.63 % (14158)Memory used [KB]: 1407
% 2.71/1.63 % (14158)Time elapsed: 0.002 s
% 2.71/1.63 % (14158)Instructions burned: 3 (million)
% 2.71/1.63 % (14158)------------------------------
% 2.71/1.63 % (14158)------------------------------
% 2.71/1.63 % (14151)Instruction limit reached!
% 2.71/1.63 % (14151)------------------------------
% 2.71/1.63 % (14151)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (14151)Termination reason: Unknown
% 2.71/1.63 % (14151)Termination phase: shuffling
% 2.71/1.63
% 2.71/1.63 % (14151)Memory used [KB]: 2174
% 2.71/1.63 % (14151)Time elapsed: 0.021 s
% 2.71/1.63 % (14151)Instructions burned: 38 (million)
% 2.71/1.63 % (14151)------------------------------
% 2.71/1.63 % (14151)------------------------------
% 2.71/1.63 % (14156)Instruction limit reached!
% 2.71/1.63 % (14156)------------------------------
% 2.71/1.63 % (14156)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (14156)Termination reason: Unknown
% 2.71/1.63 % (14156)Termination phase: shuffling
% 2.71/1.63
% 2.71/1.63 % (14156)Memory used [KB]: 1791
% 2.71/1.63 % (14156)Time elapsed: 0.012 s
% 2.71/1.63 % (14156)Instructions burned: 16 (million)
% 2.71/1.63 % (14156)------------------------------
% 2.71/1.63 % (14156)------------------------------
% 2.71/1.63 % (14143)Instruction limit reached!
% 2.71/1.63 % (14143)------------------------------
% 2.71/1.63 % (14143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.63 % (14143)Termination reason: Unknown
% 2.71/1.63 % (14143)Termination phase: Saturation
% 2.71/1.63
% 2.71/1.63 % (14143)Memory used [KB]: 7164
% 2.71/1.63 % (14143)Time elapsed: 0.055 s
% 2.71/1.63 % (14143)Instructions burned: 183 (million)
% 2.71/1.63 % (14143)------------------------------
% 2.71/1.63 % (14143)------------------------------
% 2.71/1.63 % (14159)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/7Mi)
% 2.71/1.64 % (14160)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/3Mi)
% 2.71/1.64 % (14160)Instruction limit reached!
% 2.71/1.64 % (14160)------------------------------
% 2.71/1.64 % (14160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.64 % (14160)Termination reason: Unknown
% 2.71/1.64 % (14160)Termination phase: shuffling
% 2.71/1.64
% 2.71/1.64 % (14160)Memory used [KB]: 1407
% 2.71/1.64 % (14160)Time elapsed: 0.002 s
% 2.71/1.64 % (14160)Instructions burned: 3 (million)
% 2.71/1.64 % (14160)------------------------------
% 2.71/1.64 % (14160)------------------------------
% 2.71/1.64 % (14159)Instruction limit reached!
% 2.71/1.64 % (14159)------------------------------
% 2.71/1.64 % (14159)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.64 % (14159)Termination reason: Unknown
% 2.71/1.64 % (14159)Termination phase: shuffling
% 2.71/1.64
% 2.71/1.64 % (14159)Memory used [KB]: 1535
% 2.71/1.64 % (14159)Time elapsed: 0.005 s
% 2.71/1.64 % (14159)Instructions burned: 9 (million)
% 2.71/1.64 % (14159)------------------------------
% 2.71/1.64 % (14159)------------------------------
% 2.71/1.64 % (14163)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/710Mi)
% 2.71/1.65 % (14161)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/4Mi)
% 2.71/1.65 % (14164)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/6Mi)
% 2.71/1.65 % (14165)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_13924 for (2999ds/902Mi)
% 2.71/1.65 % (14164)Instruction limit reached!
% 2.71/1.65 % (14164)------------------------------
% 2.71/1.65 % (14164)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.65 % (14164)Termination reason: Unknown
% 2.71/1.65 % (14164)Termination phase: shuffling
% 2.71/1.65
% 2.71/1.65 % (14164)Memory used [KB]: 1535
% 2.71/1.65 % (14164)Time elapsed: 0.003 s
% 2.71/1.65 % (14164)Instructions burned: 6 (million)
% 2.71/1.65 % (14164)------------------------------
% 2.71/1.65 % (14164)------------------------------
% 2.71/1.65 % (14161)Instruction limit reached!
% 2.71/1.65 % (14161)------------------------------
% 2.71/1.65 % (14161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.65 % (14161)Termination reason: Unknown
% 2.71/1.65 % (14161)Termination phase: shuffling
% 2.71/1.65
% 2.71/1.65 % (14161)Memory used [KB]: 1407
% 2.71/1.65 % (14161)Time elapsed: 0.005 s
% 2.71/1.65 % (14161)Instructions burned: 5 (million)
% 2.71/1.65 % (14161)------------------------------
% 2.71/1.65 % (14161)------------------------------
% 2.71/1.65 % (14162)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/18Mi)
% 2.71/1.66 % (14166)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_13924 for (2999ds/21Mi)
% 2.71/1.66 % (14162)Instruction limit reached!
% 2.71/1.66 % (14162)------------------------------
% 2.71/1.66 % (14162)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.66 % (14162)Termination reason: Unknown
% 2.71/1.66 % (14162)Termination phase: shuffling
% 2.71/1.66
% 2.71/1.66 % (14162)Memory used [KB]: 1663
% 2.71/1.66 % (14162)Time elapsed: 0.013 s
% 2.71/1.66 % (14162)Instructions burned: 18 (million)
% 2.71/1.66 % (14162)------------------------------
% 2.71/1.66 % (14162)------------------------------
% 2.71/1.66 % (14167)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13924 for (2999ds/5Mi)
% 2.71/1.66 % (14166)Instruction limit reached!
% 2.71/1.66 % (14166)------------------------------
% 2.71/1.66 % (14166)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.66 % (14166)Termination reason: Unknown
% 2.71/1.66 % (14166)Termination phase: shuffling
% 2.71/1.66
% 2.71/1.66 % (14166)Memory used [KB]: 1791
% 2.71/1.66 % (14166)Time elapsed: 0.008 s
% 2.71/1.66 % (14166)Instructions burned: 22 (million)
% 2.71/1.66 % (14166)------------------------------
% 2.71/1.66 % (14166)------------------------------
% 2.71/1.67 % (14167)Instruction limit reached!
% 2.71/1.67 % (14167)------------------------------
% 2.71/1.67 % (14167)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.67 % (14167)Termination reason: Unknown
% 2.71/1.67 % (14167)Termination phase: shuffling
% 2.71/1.67
% 2.71/1.67 % (14167)Memory used [KB]: 1535
% 2.71/1.67 % (14167)Time elapsed: 0.004 s
% 2.71/1.67 % (14167)Instructions burned: 7 (million)
% 2.71/1.67 % (14167)------------------------------
% 2.71/1.67 % (14167)------------------------------
% 2.71/1.67 % (14169)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13924 for (2998ds/377Mi)
% 2.71/1.68 % (14170)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_13924 for (2998ds/779Mi)
% 2.71/1.68 % (14168)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_13924 for (2998ds/6Mi)
% 2.71/1.68 % (14168)Instruction limit reached!
% 2.71/1.68 % (14168)------------------------------
% 2.71/1.68 % (14168)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.68 % (14168)Termination reason: Unknown
% 2.71/1.68 % (14168)Termination phase: shuffling
% 2.71/1.68
% 2.71/1.68 % (14168)Memory used [KB]: 1535
% 2.71/1.68 % (14168)Time elapsed: 0.006 s
% 2.71/1.68 % (14168)Instructions burned: 7 (million)
% 2.71/1.68 % (14168)------------------------------
% 2.71/1.68 % (14168)------------------------------
% 2.71/1.70 % (14171)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_13924 for (2998ds/19Mi)
% 2.71/1.70 % (14148)Instruction limit reached!
% 2.71/1.70 % (14148)------------------------------
% 2.71/1.70 % (14148)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.70 % (14148)Termination reason: Unknown
% 2.71/1.70 % (14148)Termination phase: Saturation
% 2.71/1.70
% 2.71/1.70 % (14148)Memory used [KB]: 6780
% 2.71/1.70 % (14148)Time elapsed: 0.116 s
% 2.71/1.70 % (14148)Instructions burned: 276 (million)
% 2.71/1.70 % (14148)------------------------------
% 2.71/1.70 % (14148)------------------------------
% 2.71/1.71 % (14171)Instruction limit reached!
% 2.71/1.71 % (14171)------------------------------
% 2.71/1.71 % (14171)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.71 % (14171)Termination reason: Unknown
% 2.71/1.71 % (14171)Termination phase: shuffling
% 2.71/1.71
% 2.71/1.71 % (14171)Memory used [KB]: 1791
% 2.71/1.71 % (14171)Time elapsed: 0.008 s
% 2.71/1.71 % (14171)Instructions burned: 19 (million)
% 2.71/1.71 % (14171)------------------------------
% 2.71/1.71 % (14171)------------------------------
% 2.71/1.71 % (14165)First to succeed.
% 2.71/1.71 % (14172)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13924 for (2998ds/879Mi)
% 2.71/1.71 % (14165)Refutation found. Thanks to Tanya!
% 2.71/1.71 % SZS status Theorem for DTF2THF_13924
% 2.71/1.71 % SZS output start Proof for DTF2THF_13924
% 2.71/1.71 thf(type_def_5, type, real: $tType).
% 2.71/1.71 thf(type_def_6, type, produc957004601l_real: $tType).
% 2.71/1.71 thf(type_def_8, type, produc443621487t_real: $tType).
% 2.71/1.71 thf(type_def_9, type, produc1923333543_set_a: $tType).
% 2.71/1.71 thf(type_def_10, type, product_prod_real_a: $tType).
% 2.71/1.71 thf(type_def_11, type, produc391212143l_real: $tType).
% 2.71/1.71 thf(type_def_12, type, produc1232925073real_a: $tType).
% 2.71/1.71 thf(type_def_13, type, product_prod_a_real: $tType).
% 2.71/1.71 thf(type_def_14, type, produc1396156303t_real: $tType).
% 2.71/1.71 thf(type_def_15, type, product_prod_a_set_a: $tType).
% 2.71/1.71 thf(type_def_16, type, product_prod_a_a: $tType).
% 2.71/1.71 thf(type_def_17, type, set_Pr147102617l_real: $tType).
% 2.71/1.71 thf(type_def_18, type, set_Pr1928503567a_real: $tType).
% 2.71/1.71 thf(type_def_19, type, set_Product_prod_a_a: $tType).
% 2.71/1.71 thf(type_def_20, type, set_real: $tType).
% 2.71/1.71 thf(type_def_21, type, set_se1596668135od_a_a: $tType).
% 2.71/1.71 thf(type_def_22, type, set_set_real: $tType).
% 2.71/1.71 thf(type_def_23, type, set_set_a: $tType).
% 2.71/1.71 thf(type_def_24, type, set_a: $tType).
% 2.71/1.71 thf(type_def_25, type, a: $tType).
% 2.71/1.71 thf(type_def_26, type, produc578556564l_real: $tType).
% 2.71/1.71 thf(type_def_27, type, produc1088645164real_a: $tType).
% 2.71/1.71 thf(type_def_28, type, produc826782210l_real: $tType).
% 2.71/1.71 thf(type_def_29, type, produc2096821232od_a_a: $tType).
% 2.71/1.71 thf(type_def_30, type, produc1286132450l_real: $tType).
% 2.71/1.71 thf(type_def_31, type, produc1921647824od_a_a: $tType).
% 2.71/1.71 thf(type_def_32, type, set_Pr1741234931a_real: $tType).
% 2.71/1.71 thf(type_def_33, type, set_Pr1948701895od_a_a: $tType).
% 2.71/1.71 thf(type_def_34, type, produc1826081171a_real: $tType).
% 2.71/1.71 thf(type_def_35, type, produc1572603623od_a_a: $tType).
% 2.71/1.71 thf(type_def_36, type, produc348734722a_real: $tType).
% 2.71/1.71 thf(type_def_37, type, produc1016684094_a_a_a: $tType).
% 2.71/1.71 thf(func_def_0, type, set_Pr1741234931a_real: $tType).
% 2.71/1.71 thf(func_def_1, type, produc1826081171a_real: $tType).
% 2.71/1.71 thf(func_def_2, type, set_Pr1948701895od_a_a: $tType).
% 2.71/1.71 thf(func_def_3, type, produc1572603623od_a_a: $tType).
% 2.71/1.71 thf(func_def_4, type, produc826782210l_real: $tType).
% 2.71/1.71 thf(func_def_5, type, produc578556564l_real: $tType).
% 2.71/1.71 thf(func_def_6, type, produc1286132450l_real: $tType).
% 2.71/1.71 thf(func_def_7, type, produc1088645164real_a: $tType).
% 2.71/1.71 thf(func_def_8, type, produc2096821232od_a_a: $tType).
% 2.71/1.71 thf(func_def_9, type, produc348734722a_real: $tType).
% 2.71/1.71 thf(func_def_10, type, produc1921647824od_a_a: $tType).
% 2.71/1.71 thf(func_def_11, type, produc1016684094_a_a_a: $tType).
% 2.71/1.71 thf(func_def_12, type, set_Pr147102617l_real: $tType).
% 2.71/1.71 thf(func_def_13, type, produc391212143l_real: $tType).
% 2.71/1.71 thf(func_def_14, type, produc443621487t_real: $tType).
% 2.71/1.71 thf(func_def_15, type, set_se1596668135od_a_a: $tType).
% 2.71/1.71 thf(func_def_16, type, set_Pr1928503567a_real: $tType).
% 2.71/1.71 thf(func_def_17, type, produc1396156303t_real: $tType).
% 2.71/1.71 thf(func_def_18, type, produc1232925073real_a: $tType).
% 2.71/1.71 thf(func_def_19, type, produc1923333543_set_a: $tType).
% 2.71/1.71 thf(func_def_20, type, produc957004601l_real: $tType).
% 2.71/1.71 thf(func_def_21, type, set_Product_prod_a_a: $tType).
% 2.71/1.71 thf(func_def_22, type, product_prod_a_set_a: $tType).
% 2.71/1.71 thf(func_def_23, type, product_prod_a_real: $tType).
% 2.71/1.71 thf(func_def_24, type, product_prod_real_a: $tType).
% 2.71/1.71 thf(func_def_25, type, set_set_real: $tType).
% 2.71/1.71 thf(func_def_26, type, product_prod_a_a: $tType).
% 2.71/1.71 thf(func_def_27, type, set_set_a: $tType).
% 2.71/1.71 thf(func_def_28, type, set_real: $tType).
% 2.71/1.71 thf(func_def_29, type, set_a: $tType).
% 2.71/1.71 thf(func_def_30, type, real: $tType).
% 2.71/1.71 thf(func_def_31, type, a: $tType).
% 2.71/1.71 thf(func_def_32, type, one_one_real: real).
% 2.71/1.71 thf(func_def_33, type, plus_p77862768l_real: produc957004601l_real > produc957004601l_real > produc957004601l_real).
% 2.71/1.71 thf(func_def_34, type, plus_p2005278886t_real: produc443621487t_real > produc443621487t_real > produc443621487t_real).
% 2.71/1.71 thf(func_def_35, type, plus_p125342128_set_a: produc1923333543_set_a > produc1923333543_set_a > produc1923333543_set_a).
% 2.71/1.71 thf(func_def_36, type, plus_p1243198544real_a: product_prod_real_a > product_prod_real_a > product_prod_real_a).
% 2.71/1.71 thf(func_def_37, type, plus_p1952869542l_real: produc391212143l_real > produc391212143l_real > produc391212143l_real).
% 2.71/1.71 thf(func_def_38, type, plus_p1582417306real_a: produc1232925073real_a > produc1232925073real_a > produc1232925073real_a).
% 2.71/1.71 thf(func_def_39, type, plus_p541014626a_real: product_prod_a_real > product_prod_a_real > product_prod_a_real).
% 2.71/1.71 thf(func_def_40, type, plus_p1745648536t_real: produc1396156303t_real > produc1396156303t_real > produc1396156303t_real).
% 2.71/1.71 thf(func_def_41, type, plus_p1265825342_set_a: product_prod_a_set_a > product_prod_a_set_a > product_prod_a_set_a).
% 2.71/1.71 thf(func_def_42, type, plus_p1505579230od_a_a: product_prod_a_a > product_prod_a_a > product_prod_a_a).
% 2.71/1.71 thf(func_def_43, type, plus_plus_real: real > real > real).
% 2.71/1.71 thf(func_def_44, type, plus_p1708760016l_real: set_Pr147102617l_real > set_Pr147102617l_real > set_Pr147102617l_real).
% 2.71/1.71 thf(func_def_45, type, plus_p130512152a_real: set_Pr1928503567a_real > set_Pr1928503567a_real > set_Pr1928503567a_real).
% 2.71/1.71 thf(func_def_46, type, plus_p634297534od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a).
% 2.71/1.71 thf(func_def_47, type, plus_plus_set_real: set_real > set_real > set_real).
% 2.71/1.71 thf(func_def_48, type, plus_p1613276318od_a_a: set_se1596668135od_a_a > set_se1596668135od_a_a > set_se1596668135od_a_a).
% 2.71/1.71 thf(func_def_49, type, plus_p768704801t_real: set_set_real > set_set_real > set_set_real).
% 2.71/1.71 thf(func_def_50, type, plus_plus_set_set_a: set_set_a > set_set_a > set_set_a).
% 2.71/1.71 thf(func_def_51, type, plus_plus_set_a: set_a > set_a > set_a).
% 2.71/1.71 thf(func_def_52, type, plus_plus_a: a > a > a).
% 2.71/1.71 thf(func_def_53, type, zero_z342251165l_real: produc578556564l_real).
% 2.71/1.71 thf(func_def_54, type, zero_z1937883107real_a: produc1088645164real_a).
% 2.71/1.71 thf(func_def_55, type, zero_z590476811l_real: produc826782210l_real).
% 2.71/1.71 thf(func_def_56, type, zero_z396988537od_a_a: produc2096821232od_a_a).
% 2.71/1.71 thf(func_def_57, type, zero_z659284464l_real: produc957004601l_real).
% 2.71/1.71 thf(func_def_58, type, zero_z1407338960real_a: product_prod_real_a).
% 2.71/1.71 thf(func_def_59, type, zero_z2135370393l_real: produc1286132450l_real).
% 2.71/1.71 thf(func_def_60, type, zero_z53797895od_a_a: produc1921647824od_a_a).
% 2.71/1.71 thf(func_def_61, type, zero_z705155042a_real: product_prod_a_real).
% 2.71/1.71 thf(func_def_62, type, zero_z950819678od_a_a: product_prod_a_a).
% 2.71/1.71 thf(func_def_63, type, zero_zero_real: real).
% 2.71/1.71 thf(func_def_64, type, zero_z257140542od_a_a: set_Product_prod_a_a).
% 2.71/1.71 thf(func_def_65, type, zero_zero_set_real: set_real).
% 2.71/1.71 thf(func_def_66, type, zero_zero_set_a: set_a).
% 2.71/1.71 thf(func_def_67, type, zero_zero_a: a).
% 2.71/1.71 thf(func_def_68, type, ord_less_real: real > real > $o).
% 2.71/1.71 thf(func_def_69, type, ord_le108336051od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 2.71/1.71 thf(func_def_70, type, ord_le1342644953l_real: produc957004601l_real > produc957004601l_real > $o).
% 2.71/1.71 thf(func_def_71, type, ord_less_eq_real: real > real > $o).
% 2.71/1.71 thf(func_def_72, type, ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 2.71/1.71 thf(func_def_73, type, ord_less_eq_set_real: set_real > set_real > $o).
% 2.71/1.71 thf(func_def_74, type, prefer1113819806a_real: set_Pr1741234931a_real > $o).
% 2.71/1.71 thf(func_def_75, type, prefer225425826od_a_a: set_Pr1948701895od_a_a > $o).
% 2.71/1.71 thf(func_def_76, type, prefer1247792113f_real: set_Pr147102617l_real > $o).
% 2.71/1.71 thf(func_def_77, type, prefer529818233pref_a: set_Product_prod_a_a > $o).
% 2.71/1.71 thf(func_def_78, type, produc1175086478l_real: produc957004601l_real > real > produc578556564l_real).
% 2.71/1.71 thf(func_def_79, type, produc1430099868real_a: produc957004601l_real > a > produc1088645164real_a).
% 2.71/1.71 thf(func_def_80, type, produc886678603a_real: product_prod_a_real > product_prod_a_real > produc1826081171a_real).
% 2.71/1.71 thf(func_def_81, type, produc1474507607od_a_a: product_prod_a_a > product_prod_a_a > produc1572603623od_a_a).
% 2.71/1.71 thf(func_def_82, type, produc454274172a_real: product_prod_a_a > real > produc348734722a_real).
% 2.71/1.71 thf(func_def_83, type, produc2061588782_a_a_a: product_prod_a_a > a > produc1016684094_a_a_a).
% 2.71/1.71 thf(func_def_84, type, produc1926903988l_real: real > produc957004601l_real > produc826782210l_real).
% 2.71/1.71 thf(func_def_85, type, produc454550562od_a_a: real > product_prod_a_a > produc2096821232od_a_a).
% 2.71/1.71 thf(func_def_86, type, produc705216881l_real: real > real > produc957004601l_real).
% 2.71/1.71 thf(func_def_87, type, produc247649703t_real: real > set_real > produc443621487t_real).
% 2.71/1.71 thf(func_def_88, type, produc1872317017_set_a: real > set_a > produc1923333543_set_a).
% 2.71/1.71 thf(func_def_89, type, product_Pair_real_a: real > a > product_prod_real_a).
% 2.71/1.71 thf(func_def_90, type, produc1812461223l_real: set_real > real > produc391212143l_real).
% 2.71/1.71 thf(func_def_91, type, produc617496131real_a: set_real > a > produc1232925073real_a).
% 2.71/1.71 thf(func_def_92, type, produc64206290l_real: a > produc957004601l_real > produc1286132450l_real).
% 2.71/1.71 thf(func_def_93, type, produc1299253312od_a_a: a > product_prod_a_a > produc1921647824od_a_a).
% 2.71/1.71 thf(func_def_94, type, product_Pair_a_real: a > real > product_prod_a_real).
% 2.71/1.71 thf(func_def_95, type, produc1391501065t_real: a > set_real > produc1396156303t_real).
% 2.71/1.71 thf(func_def_96, type, product_Pair_a_set_a: a > set_a > product_prod_a_set_a).
% 2.71/1.71 thf(func_def_97, type, product_Pair_a_a: a > a > product_prod_a_a).
% 2.71/1.71 thf(func_def_98, type, real_V368978776od_a_a: real > produc1572603623od_a_a > produc1572603623od_a_a).
% 2.71/1.71 thf(func_def_99, type, real_V195069393a_real: real > produc348734722a_real > produc348734722a_real).
% 2.71/1.71 thf(func_def_100, type, real_V1538726831_a_a_a: real > produc1016684094_a_a_a > produc1016684094_a_a_a).
% 2.71/1.71 thf(func_def_101, type, real_V1943155903od_a_a: real > produc2096821232od_a_a > produc2096821232od_a_a).
% 2.71/1.71 thf(func_def_102, type, real_V1139189034l_real: real > produc957004601l_real > produc957004601l_real).
% 2.71/1.71 thf(func_def_103, type, real_V1214981142real_a: real > product_prod_real_a > product_prod_real_a).
% 2.71/1.71 thf(func_def_104, type, real_V296206913od_a_a: real > produc1921647824od_a_a > produc1921647824od_a_a).
% 2.71/1.71 thf(func_def_105, type, real_V512797224a_real: real > product_prod_a_real > product_prod_a_real).
% 2.71/1.71 thf(func_def_106, type, real_V543523736od_a_a: real > product_prod_a_a > product_prod_a_a).
% 2.71/1.71 thf(func_def_107, type, real_V453051771R_real: real > real > real).
% 2.71/1.71 thf(func_def_108, type, real_V1035702895aleR_a: real > a > a).
% 2.71/1.71 thf(func_def_109, type, collec1300223524l_real: (produc957004601l_real > $o) > set_Pr147102617l_real).
% 2.71/1.71 thf(func_def_110, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 2.71/1.71 thf(func_def_111, type, collect_real: (real > $o) > set_real).
% 2.71/1.71 thf(func_def_112, type, collect_a: (a > $o) > set_a).
% 2.71/1.71 thf(func_def_113, type, member384820540a_real: produc1826081171a_real > set_Pr1741234931a_real > $o).
% 2.71/1.71 thf(func_def_114, type, member2057358096od_a_a: produc1572603623od_a_a > set_Pr1948701895od_a_a > $o).
% 2.71/1.71 thf(func_def_115, type, member1068169442l_real: produc957004601l_real > set_Pr147102617l_real > $o).
% 2.71/1.71 thf(func_def_116, type, member1103263856a_real: product_prod_a_real > set_Pr1928503567a_real > $o).
% 2.71/1.71 thf(func_def_117, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 2.71/1.71 thf(func_def_118, type, member_real: real > set_real > $o).
% 2.71/1.71 thf(func_def_119, type, member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o).
% 2.71/1.71 thf(func_def_120, type, member_set_real: set_real > set_set_real > $o).
% 2.71/1.71 thf(func_def_121, type, member_set_a: set_a > set_set_a > $o).
% 2.71/1.71 thf(func_def_122, type, member_a: a > set_a > $o).
% 2.71/1.71 thf(func_def_123, type, relation: set_Product_prod_a_a).
% 2.71/1.71 thf(func_def_124, type, u: real).
% 2.71/1.71 thf(func_def_125, type, v: real).
% 2.71/1.71 thf(func_def_126, type, x: a).
% 2.71/1.71 thf(func_def_127, type, y: a).
% 2.71/1.71 thf(func_def_141, type, sP0: real > real > real > $o).
% 2.71/1.71 thf(func_def_142, type, sP1: real > real > real > $o).
% 2.71/1.71 thf(func_def_143, type, sP2: real > real > real > $o).
% 2.71/1.71 thf(func_def_144, type, sK3: set_Pr147102617l_real > real).
% 2.71/1.71 thf(func_def_145, type, sK4: set_Pr147102617l_real > real).
% 2.71/1.71 thf(func_def_146, type, sK5: set_Pr147102617l_real > real).
% 2.71/1.71 thf(func_def_147, type, sK6: set_Pr147102617l_real > real).
% 2.71/1.71 thf(func_def_148, type, sK7: (product_prod_a_a > $o) > a).
% 2.71/1.71 thf(func_def_149, type, sK8: (product_prod_a_a > $o) > a).
% 2.71/1.71 thf(func_def_150, type, sK9: set_Product_prod_a_a > a).
% 2.71/1.71 thf(func_def_151, type, sK10: set_Product_prod_a_a > a).
% 2.71/1.71 thf(func_def_152, type, sK11: set_Product_prod_a_a > real).
% 2.71/1.71 thf(func_def_153, type, sK12: set_Product_prod_a_a > real).
% 2.71/1.71 thf(func_def_154, type, sK13: (product_prod_a_a > $o) > (product_prod_a_a > $o) > product_prod_a_a).
% 2.71/1.71 thf(func_def_155, type, sK14: product_prod_a_a > a).
% 2.71/1.71 thf(func_def_156, type, sK15: product_prod_a_a > a).
% 2.71/1.71 thf(func_def_157, type, sK16: set_Product_prod_a_a > a).
% 2.71/1.71 thf(func_def_158, type, sK17: set_Product_prod_a_a > real).
% 2.71/1.71 thf(func_def_159, type, sK18: set_Product_prod_a_a > a).
% 2.71/1.71 thf(func_def_160, type, sK19: set_Product_prod_a_a > real).
% 2.71/1.71 thf(func_def_161, type, sK20: product_prod_a_a > a).
% 2.71/1.71 thf(func_def_162, type, sK21: product_prod_a_a > a).
% 2.71/1.71 thf(func_def_163, type, sK22: (real > real) > real).
% 2.71/1.71 thf(func_def_164, type, sK23: (real > real) > real).
% 2.71/1.71 thf(func_def_165, type, sK24: set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a > product_prod_a_a).
% 2.71/1.71 thf(func_def_166, type, sK25: set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a > product_prod_a_a).
% 2.71/1.71 thf(func_def_167, type, sK26: (product_prod_a_a > $o) > a).
% 2.71/1.71 thf(func_def_168, type, sK27: (product_prod_a_a > $o) > a).
% 2.71/1.71 thf(func_def_169, type, sK28: (real > real) > real).
% 2.71/1.71 thf(func_def_170, type, sK29: (real > real) > real).
% 2.71/1.71 thf(func_def_171, type, sK30: set_Pr1741234931a_real > product_prod_a_real).
% 2.71/1.71 thf(func_def_172, type, sK31: set_Pr1741234931a_real > product_prod_a_real).
% 2.71/1.71 thf(func_def_173, type, sK32: set_Pr1741234931a_real > real).
% 2.71/1.71 thf(func_def_174, type, sK33: set_Pr1741234931a_real > real).
% 2.71/1.71 thf(func_def_175, type, sK34: set_Product_prod_a_a > set_Product_prod_a_a > product_prod_a_a).
% 2.71/1.71 thf(func_def_176, type, sK35: real > set_real > set_real > real).
% 2.71/1.71 thf(func_def_177, type, sK36: real > set_real > set_real > real).
% 2.71/1.71 thf(func_def_178, type, sK37: set_Pr1948701895od_a_a > product_prod_a_a).
% 2.71/1.71 thf(func_def_179, type, sK38: set_Pr1948701895od_a_a > product_prod_a_a).
% 2.71/1.71 thf(func_def_180, type, sK39: set_Pr1948701895od_a_a > real).
% 2.71/1.71 thf(func_def_181, type, sK40: set_Pr1948701895od_a_a > real).
% 2.71/1.71 thf(func_def_182, type, sK41: real > real > real).
% 2.71/1.71 thf(func_def_183, type, sK42: a > set_a > set_a > a).
% 2.71/1.71 thf(func_def_184, type, sK43: a > set_a > set_a > a).
% 2.71/1.71 thf(func_def_185, type, sK44: set_Pr147102617l_real > real).
% 2.71/1.71 thf(func_def_186, type, sK45: set_Pr147102617l_real > real).
% 2.71/1.71 thf(func_def_187, type, sK46: set_Pr147102617l_real > real).
% 2.71/1.71 thf(func_def_188, type, sK47: set_Pr147102617l_real > real).
% 2.71/1.71 thf(func_def_189, type, sK48: (real > real) > real).
% 2.71/1.71 thf(func_def_190, type, sK49: (real > real) > real).
% 2.71/1.71 thf(func_def_191, type, sK50: (real > real) > real).
% 2.71/1.71 thf(func_def_192, type, sK51: (real > real) > real).
% 2.71/1.71 thf(func_def_194, type, ph53: !>[X0: $tType]:(X0)).
% 2.71/1.71 thf(f2440,plain,(
% 2.71/1.71 $false),
% 2.71/1.71 inference(avatar_sat_refutation,[status(thm)],[f1937,f1942,f1976,f2435])).
% 2.71/1.71 thf(f2435,plain,(
% 2.71/1.71 ~spl52_5 | ~spl52_7),
% 2.71/1.71 inference(avatar_split_clause,[status(thm)],[f2434,f1934,f1925])).
% 2.71/1.71 thf(f1925,definition,(
% 2.71/1.71 spl52_5 <=> ((member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation) = $true)),
% 2.71/1.71 introduced(definition,[new_symbols(naming,[spl52_5])],[avatar_definition])).
% 2.71/1.71 thf(f1934,definition,(
% 2.71/1.71 spl52_7 <=> (u = one_one_real)),
% 2.71/1.71 introduced(definition,[new_symbols(naming,[spl52_7])],[avatar_definition])).
% 2.71/1.71 thf(f2434,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation) != $true) | ~spl52_7),
% 2.71/1.71 inference(forward_demodulation,[status(thm)],[f2433,f1822])).
% 2.71/1.71 thf(f1822,plain,(
% 2.71/1.71 ( ! [X0 : a] : (((plus_plus_a @ X0 @ zero_zero_a) = X0)) )),
% 2.71/1.71 inference(equality_resolution,[status(thm)],[f1486])).
% 2.71/1.71 thf(f1486,plain,(
% 2.71/1.71 ( ! [X0 : a,X1 : a] : (((plus_plus_a @ X0 @ X1) = X0) | (zero_zero_a != X1)) )),
% 2.71/1.71 inference(cnf_transformation,[status(thm)],[f1199])).
% 2.71/1.71 thf(f1199,plain,(
% 2.71/1.71 ! [X0 : a,X1 : a] : ((((plus_plus_a @ X0 @ X1) = X0) | (zero_zero_a != X1)) & ((zero_zero_a = X1) | ((plus_plus_a @ X0 @ X1) != X0)))),
% 2.71/1.71 inference(nnf_transformation,[status(thm)],[f524])).
% 2.71/1.71 thf(f524,plain,(
% 2.71/1.71 ! [X0 : a,X1 : a] : (((plus_plus_a @ X0 @ X1) = X0) <=> (zero_zero_a = X1))),
% 2.71/1.71 inference(fool_elimination,[status(thm)],[f523])).
% 2.71/1.71 thf(f523,plain,(
% 2.71/1.71 ! [X0 : a,X1 : a] : ((zero_zero_a = X1) = ((plus_plus_a @ X0 @ X1) = X0))),
% 2.71/1.71 inference(rectify,[status(thm)],[f82])).
% 2.71/1.71 thf(f82,axiom,(
% 2.71/1.71 ! [X4 : a,X5 : a] : ((zero_zero_a = X5) = ((plus_plus_a @ X4 @ X5) = X4))),
% 2.71/1.71 file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_81_add__cancel__left__right)).
% 2.71/1.71 thf(f2433,plain,(
% 2.71/1.71 ($true != (member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ x @ zero_zero_a) @ y) @ relation)) | ~spl52_7),
% 2.71/1.71 inference(forward_demodulation,[status(thm)],[f2432,f1731])).
% 2.71/1.71 thf(f1731,plain,(
% 2.71/1.71 ( ! [X0 : a] : ((zero_zero_a = (real_V1035702895aleR_a @ zero_zero_real @ X0))) )),
% 2.71/1.71 inference(cnf_transformation,[status(thm)],[f239])).
% 2.71/1.71 thf(f239,axiom,(
% 2.71/1.71 ! [X0 : a] : (zero_zero_a = (real_V1035702895aleR_a @ zero_zero_real @ X0))),
% 2.71/1.71 file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_238_scale__zero__left)).
% 2.71/1.71 thf(f2432,plain,(
% 2.71/1.71 ($true != (member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ x @ (real_V1035702895aleR_a @ zero_zero_real @ y)) @ y) @ relation)) | ~spl52_7),
% 2.71/1.71 inference(forward_demodulation,[status(thm)],[f2431,f1420])).
% 2.71/1.71 thf(f1420,plain,(
% 2.71/1.71 ( ! [X0 : a] : (((real_V1035702895aleR_a @ one_one_real @ X0) = X0)) )),
% 2.71/1.71 inference(cnf_transformation,[status(thm)],[f242])).
% 2.71/1.71 thf(f242,axiom,(
% 2.71/1.71 ! [X0 : a] : ((real_V1035702895aleR_a @ one_one_real @ X0) = X0)),
% 2.71/1.71 file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_241_scaleR__one)).
% 2.71/1.71 thf(f2431,plain,(
% 2.71/1.71 ($true != (member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ one_one_real @ x) @ (real_V1035702895aleR_a @ zero_zero_real @ y)) @ y) @ relation)) | ~spl52_7),
% 2.71/1.71 inference(forward_demodulation,[status(thm)],[f2402,f1936])).
% 2.71/1.71 thf(f1936,plain,(
% 2.71/1.71 (u = one_one_real) | ~spl52_7),
% 2.71/1.71 inference(avatar_component_clause,[status(thm)],[f1934])).
% 2.71/1.71 thf(f2402,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ zero_zero_real @ y)) @ y) @ relation) != $true) | ~spl52_7),
% 2.71/1.71 inference(superposition,[status(thm)],[f1364,f2243])).
% 2.71/1.71 thf(f2243,plain,(
% 2.71/1.71 (zero_zero_real = v) | ~spl52_7),
% 2.71/1.71 inference(trivial_inequality_removal,[status(thm)],[f2153])).
% 2.71/1.71 thf(f2153,plain,(
% 2.71/1.71 (zero_zero_real = v) | (one_one_real != one_one_real) | ~spl52_7),
% 2.71/1.71 inference(superposition,[status(thm)],[f1377,f1945])).
% 2.71/1.71 thf(f1945,plain,(
% 2.71/1.71 (one_one_real = (plus_plus_real @ one_one_real @ v)) | ~spl52_7),
% 2.71/1.71 inference(superposition,[status(thm)],[f1338,f1936])).
% 2.71/1.71 thf(f1338,plain,(
% 2.71/1.71 (one_one_real = (plus_plus_real @ u @ v))),
% 2.71/1.71 inference(cnf_transformation,[status(thm)],[f10])).
% 2.71/1.71 thf(f10,axiom,(
% 2.71/1.71 (one_one_real = (plus_plus_real @ u @ v))),
% 2.71/1.71 file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_9_assms_I5_J)).
% 2.71/1.71 thf(f1377,plain,(
% 2.71/1.71 ( ! [X0 : real,X1 : real] : (((plus_plus_real @ X0 @ X1) != X0) | (zero_zero_real = X1)) )),
% 2.71/1.71 inference(cnf_transformation,[status(thm)],[f1157])).
% 2.71/1.71 thf(f1157,plain,(
% 2.71/1.71 ! [X0 : real,X1 : real] : (((zero_zero_real = X1) | ((plus_plus_real @ X0 @ X1) != X0)) & (((plus_plus_real @ X0 @ X1) = X0) | (zero_zero_real != X1)))),
% 2.71/1.71 inference(nnf_transformation,[status(thm)],[f619])).
% 2.71/1.71 thf(f619,plain,(
% 2.71/1.71 ! [X0 : real,X1 : real] : ((zero_zero_real = X1) <=> ((plus_plus_real @ X0 @ X1) = X0))),
% 2.71/1.71 inference(fool_elimination,[status(thm)],[f618])).
% 2.71/1.71 thf(f618,plain,(
% 2.71/1.71 ! [X0 : real,X1 : real] : (((plus_plus_real @ X0 @ X1) = X0) = (zero_zero_real = X1))),
% 2.71/1.71 inference(rectify,[status(thm)],[f73])).
% 2.71/1.71 thf(f73,axiom,(
% 2.71/1.71 ! [X4 : real,X5 : real] : (((plus_plus_real @ X4 @ X5) = X4) = (zero_zero_real = X5))),
% 2.71/1.71 file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_72_add__cancel__right__right)).
% 2.71/1.71 thf(f1364,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) != $true)),
% 2.71/1.71 inference(cnf_transformation,[status(thm)],[f885])).
% 2.71/1.71 thf(f885,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) != $true)),
% 2.71/1.71 inference(flattening,[status(thm)],[f710])).
% 2.71/1.71 thf(f710,plain,(
% 2.71/1.71 ~((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) = $true)),
% 2.71/1.71 inference(fool_elimination,[status(thm)],[f709])).
% 2.71/1.71 thf(f709,plain,(
% 2.71/1.71 ~(member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation)),
% 2.71/1.71 inference(rectify,[status(thm)],[f355])).
% 2.71/1.71 thf(f355,negated_conjecture,(
% 2.71/1.71 ~(member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation)),
% 2.71/1.71 inference(negated_conjecture,[status(cth)],[f354])).
% 2.71/1.71 thf(f354,conjecture,(
% 2.71/1.71 (member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation)),
% 2.71/1.71 file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',conj_0)).
% 2.71/1.71 thf(f1976,plain,(
% 2.71/1.71 spl52_5),
% 2.71/1.71 inference(avatar_contradiction_clause,[status(thm)],[f1975])).
% 2.71/1.71 thf(f1975,plain,(
% 2.71/1.71 $false | spl52_5),
% 2.71/1.71 inference(trivial_inequality_removal,[status(thm)],[f1962])).
% 2.71/1.71 thf(f1962,plain,(
% 2.71/1.71 ($true != $true) | spl52_5),
% 2.71/1.71 inference(superposition,[status(thm)],[f1927,f1354])).
% 2.71/1.71 thf(f1354,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation) = $true)),
% 2.71/1.71 inference(cnf_transformation,[status(thm)],[f506])).
% 2.71/1.71 thf(f506,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation) = $true)),
% 2.71/1.71 inference(fool_elimination,[status(thm)],[f505])).
% 2.71/1.71 thf(f505,plain,(
% 2.71/1.71 (member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation)),
% 2.71/1.71 inference(rectify,[status(thm)],[f6])).
% 2.71/1.71 thf(f6,axiom,(
% 2.71/1.71 (member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation)),
% 2.71/1.71 file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_5_assms_I2_J)).
% 2.71/1.71 thf(f1927,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ x @ y) @ relation) != $true) | spl52_5),
% 2.71/1.71 inference(avatar_component_clause,[status(thm)],[f1925])).
% 2.71/1.71 thf(f1942,plain,(
% 2.71/1.71 ~spl52_6),
% 2.71/1.71 inference(avatar_split_clause,[status(thm)],[f1903,f1930])).
% 2.71/1.71 thf(f1930,definition,(
% 2.71/1.71 spl52_6 <=> (u = zero_zero_real)),
% 2.71/1.71 introduced(definition,[new_symbols(naming,[spl52_6])],[avatar_definition])).
% 2.71/1.71 thf(f1903,plain,(
% 2.71/1.71 (u != zero_zero_real)),
% 2.71/1.71 inference(trivial_inequality_removal,[status(thm)],[f1885])).
% 2.71/1.71 thf(f1885,plain,(
% 2.71/1.71 (u != zero_zero_real) | ($true != $true)),
% 2.71/1.71 inference(superposition,[status(thm)],[f1364,f1649])).
% 2.71/1.71 thf(f1649,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) = $true) | (u != zero_zero_real)),
% 2.71/1.71 inference(cnf_transformation,[status(thm)],[f1093])).
% 2.71/1.71 thf(f1093,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) = $true) | (u != zero_zero_real)),
% 2.71/1.71 inference(ennf_transformation,[status(thm)],[f675])).
% 2.71/1.71 thf(f675,plain,(
% 2.71/1.71 (u = zero_zero_real) => ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) = $true)),
% 2.71/1.71 inference(fool_elimination,[status(thm)],[f674])).
% 2.71/1.71 thf(f674,plain,(
% 2.71/1.71 (u = zero_zero_real) => (member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation)),
% 2.71/1.71 inference(rectify,[status(thm)],[f8])).
% 2.71/1.71 thf(f8,axiom,(
% 2.71/1.71 (u = zero_zero_real) => (member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation)),
% 2.71/1.71 file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_7_u__0)).
% 2.71/1.71 thf(f1937,plain,(
% 2.71/1.71 spl52_6 | spl52_7),
% 2.71/1.71 inference(avatar_split_clause,[status(thm)],[f1896,f1934,f1930])).
% 2.71/1.71 thf(f1896,plain,(
% 2.71/1.71 (u = zero_zero_real) | (u = one_one_real)),
% 2.71/1.71 inference(trivial_inequality_removal,[status(thm)],[f1884])).
% 2.71/1.71 thf(f1884,plain,(
% 2.71/1.71 (u = one_one_real) | (u = zero_zero_real) | ($true != $true)),
% 2.71/1.71 inference(superposition,[status(thm)],[f1364,f1295])).
% 2.71/1.71 thf(f1295,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) = $true) | (u = zero_zero_real) | (u = one_one_real)),
% 2.71/1.71 inference(cnf_transformation,[status(thm)],[f986])).
% 2.71/1.71 thf(f986,plain,(
% 2.71/1.71 (u = zero_zero_real) | ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) = $true) | (u = one_one_real)),
% 2.71/1.71 inference(flattening,[status(thm)],[f985])).
% 2.71/1.71 thf(f985,plain,(
% 2.71/1.71 ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) = $true) | ((u = zero_zero_real) | (u = one_one_real))),
% 2.71/1.71 inference(ennf_transformation,[status(thm)],[f502])).
% 2.71/1.71 thf(f502,plain,(
% 2.71/1.71 ((u != zero_zero_real) & (u != one_one_real)) => ((member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation) = $true)),
% 2.71/1.71 inference(fool_elimination,[status(thm)],[f501])).
% 2.71/1.71 thf(f501,plain,(
% 2.71/1.71 ((u != zero_zero_real) & (u != one_one_real)) => (member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation)),
% 2.71/1.71 inference(rectify,[status(thm)],[f7])).
% 2.71/1.71 thf(f7,axiom,(
% 2.71/1.71 ((u != zero_zero_real) & (u != one_one_real)) => (member449909584od_a_a @ (product_Pair_a_a @ (plus_plus_a @ (real_V1035702895aleR_a @ u @ x) @ (real_V1035702895aleR_a @ v @ y)) @ y) @ relation)),
% 2.71/1.71 file('/export/starexec/sandbox2/tmp/tmp.AT2sPpftd8/DTF2THF_13924.p',fact_6__092_060open_062u_A_092_060noteq_062_A0_A_092_060and_062_Au_A_092_060noteq_062_A1_A_092_060longrightarrow_062_Au_A_K_092_060_094sub_062R_Ax_A_L_Av_A_K_092_060_094sub_062R_Ay_A_092_060succeq_062_Ay_092_060close_062)).
% 2.71/1.71 % SZS output end Proof for DTF2THF_13924
% 2.71/1.71 % (14165)------------------------------
% 2.71/1.71 % (14165)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.71 % (14165)Termination reason: Refutation
% 2.71/1.71
% 2.71/1.71 % (14165)Memory used [KB]: 7675
% 2.71/1.71 % (14165)Time elapsed: 0.066 s
% 2.71/1.71 % (14165)Instructions burned: 194 (million)
% 2.71/1.71 % (14165)------------------------------
% 2.71/1.71 % (14165)------------------------------
% 2.71/1.71 % (14142)Success in time 0.149 s
%------------------------------------------------------------------------------