%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP041^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n028.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:04 AM UTC 2026
% Result : Theorem 2.60s 1.76s
% Output : Refutation 2.60s
% Verified :
% SZS Type : Refutation
% Derivation depth : 6
% Number of leaves : 2
% Syntax : Number of formulae : 11 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 26 ( 6 equ; 0 cnn)
% Maximal formula atoms : 2 ( 2 avg)
% Number of connectives : 213 ( 5 ~; 0 |; 3 &; 205 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 5 avg)
% Number of types : 24 ( 23 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 153 ( 150 usr; 19 con; 0-4 aty)
% Number of variables : 0 ( 0 ^; 0 !; 0 ?; 0 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
set_Pr1389752855real_c: $tType ).
thf(type_def_6,type,
finite1398487019real_c: $tType ).
thf(type_def_7,type,
real: $tType ).
thf(type_def_9,type,
set_Sum_sum_c_c: $tType ).
thf(type_def_10,type,
formula_a_b_c: $tType ).
thf(type_def_11,type,
hp_a_b_c: $tType ).
thf(type_def_12,type,
denota231621370t_unit: $tType ).
thf(type_def_13,type,
set_Su1783761653um_b_c: $tType ).
thf(type_def_14,type,
oDE_a_c: $tType ).
thf(type_def_15,type,
set_c: $tType ).
thf(type_def_16,type,
produc190496183real_c: $tType ).
thf(type_def_17,type,
filter_real: $tType ).
thf(type_def_18,type,
sum_sum_c_c: $tType ).
thf(type_def_19,type,
sum_sum_c_d: $tType ).
thf(type_def_20,type,
c: $tType ).
thf(type_def_21,type,
produc957004601l_real: $tType ).
thf(type_def_22,type,
produc754691304um_c_c: $tType ).
thf(type_def_23,type,
produc1398711912c_real: $tType ).
thf(type_def_24,type,
produc214722967um_c_c: $tType ).
thf(type_def_25,type,
set_Fi1407883041real_c: $tType ).
thf(type_def_26,type,
set_Sum_sum_c_d: $tType ).
thf(type_def_27,type,
set_real: $tType ).
thf(type_def_28,type,
set_set_Sum_sum_c_c: $tType ).
thf(func_def_0,type,
set_Pr1389752855real_c: $tType ).
thf(func_def_1,type,
produc190496183real_c: $tType ).
thf(func_def_2,type,
produc214722967um_c_c: $tType ).
thf(func_def_3,type,
denota231621370t_unit: $tType ).
thf(func_def_4,type,
produc1398711912c_real: $tType ).
thf(func_def_5,type,
produc754691304um_c_c: $tType ).
thf(func_def_6,type,
set_Su1783761653um_b_c: $tType ).
thf(func_def_7,type,
set_Fi1407883041real_c: $tType ).
thf(func_def_8,type,
set_set_Sum_sum_c_c: $tType ).
thf(func_def_9,type,
finite1398487019real_c: $tType ).
thf(func_def_10,type,
produc957004601l_real: $tType ).
thf(func_def_11,type,
set_Sum_sum_c_d: $tType ).
thf(func_def_12,type,
set_Sum_sum_c_c: $tType ).
thf(func_def_13,type,
formula_a_b_c: $tType ).
thf(func_def_14,type,
hp_a_b_c: $tType ).
thf(func_def_15,type,
sum_sum_c_d: $tType ).
thf(func_def_16,type,
sum_sum_c_c: $tType ).
thf(func_def_17,type,
filter_real: $tType ).
thf(func_def_18,type,
oDE_a_c: $tType ).
thf(func_def_19,type,
set_real: $tType ).
thf(func_def_20,type,
set_c: $tType ).
thf(func_def_21,type,
real: $tType ).
thf(func_def_22,type,
c: $tType ).
thf(func_def_23,type,
bNF_Ca449034301c_real: set_Pr1389752855real_c > ( finite1398487019real_c > real ) > $o ).
thf(func_def_24,type,
bNF_Ca1693016812um_c_c: set_Pr1389752855real_c > ( finite1398487019real_c > set_Sum_sum_c_c ) > $o ).
thf(func_def_25,type,
coinci501627771_a_b_c: formula_a_b_c > $o ).
thf(func_def_26,type,
coinci354700372_a_b_c: hp_a_b_c > denota231621370t_unit > denota231621370t_unit > $o ).
thf(func_def_27,type,
coinci1852866155_a_b_c: hp_a_b_c > $o ).
thf(func_def_28,type,
coinci772602909_a_b_c: hp_a_b_c > denota231621370t_unit > $o ).
thf(func_def_29,type,
denota69106024_a_b_c: denota231621370t_unit > denota231621370t_unit > set_Su1783761653um_b_c > $o ).
thf(func_def_30,type,
denota1275485728_a_b_c: denota231621370t_unit > oDE_a_c > finite1398487019real_c > finite1398487019real_c ).
thf(func_def_31,type,
denota256060419gree_c: finite1398487019real_c > finite1398487019real_c > set_c > $o ).
thf(func_def_32,type,
denota1997846518gree_c: produc190496183real_c > produc190496183real_c > set_Sum_sum_c_c > $o ).
thf(func_def_33,type,
denota968303861_a_b_c: denota231621370t_unit > formula_a_b_c > set_Pr1389752855real_c ).
thf(func_def_34,type,
denota161327353_a_b_c: denota231621370t_unit > oDE_a_c > produc190496183real_c > finite1398487019real_c > produc190496183real_c ).
thf(func_def_35,type,
denota1896987029_a_b_c: denota231621370t_unit > oDE_a_c > finite1398487019real_c > produc190496183real_c ).
thf(func_def_36,type,
denota1293057250_a_b_c: denota231621370t_unit > oDE_a_c > set_Sum_sum_c_c ).
thf(func_def_37,type,
has_ve1904308670real_c: ( real > finite1398487019real_c ) > finite1398487019real_c > filter_real > $o ).
thf(func_def_38,type,
has_ve373335844real_c: ( real > produc190496183real_c ) > produc190496183real_c > filter_real > $o ).
thf(func_def_39,type,
has_ve217091583e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_40,type,
uminus23580315al_c_o: ( finite1398487019real_c > $o ) > finite1398487019real_c > $o ).
thf(func_def_41,type,
uminus32155087al_c_o: ( produc190496183real_c > $o ) > produc190496183real_c > $o ).
thf(func_def_42,type,
uminus88729697_c_c_o: ( sum_sum_c_c > $o ) > sum_sum_c_c > $o ).
thf(func_def_43,type,
uminus916499104_c_d_o: ( sum_sum_c_d > $o ) > sum_sum_c_d > $o ).
thf(func_def_44,type,
uminus_uminus_c_o: ( c > $o ) > c > $o ).
thf(func_def_45,type,
uminus1737280820real_c: finite1398487019real_c > finite1398487019real_c ).
thf(func_def_46,type,
uminus17964590real_c: produc190496183real_c > produc190496183real_c ).
thf(func_def_47,type,
uminus856758960l_real: produc957004601l_real > produc957004601l_real ).
thf(func_def_48,type,
uminus1629073887um_c_c: produc754691304um_c_c > produc754691304um_c_c ).
thf(func_def_49,type,
uminus125610847c_real: produc1398711912c_real > produc1398711912c_real ).
thf(func_def_50,type,
uminus1225118734um_c_c: produc214722967um_c_c > produc214722967um_c_c ).
thf(func_def_51,type,
uminus_uminus_real: real > real ).
thf(func_def_52,type,
uminus102402154real_c: set_Fi1407883041real_c > set_Fi1407883041real_c ).
thf(func_def_53,type,
uminus232257166real_c: set_Pr1389752855real_c > set_Pr1389752855real_c ).
thf(func_def_54,type,
uminus1381786404um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c ).
thf(func_def_55,type,
uminus373867045um_c_d: set_Sum_sum_c_d > set_Sum_sum_c_d ).
thf(func_def_56,type,
uminus_uminus_set_c: set_c > set_c ).
thf(func_def_57,type,
zero_z109254132real_c: finite1398487019real_c ).
thf(func_def_58,type,
zero_z1506780526real_c: produc190496183real_c ).
thf(func_def_59,type,
zero_z659284464l_real: produc957004601l_real ).
thf(func_def_60,type,
zero_zero_real: real ).
thf(func_def_61,type,
the_Pr1290640446real_c: ( produc190496183real_c > $o ) > produc190496183real_c ).
thf(func_def_62,type,
initia1631504802real_c: ( real > finite1398487019real_c ) > ( real > finite1398487019real_c > finite1398487019real_c ) > set_real > set_Fi1407883041real_c > $o ).
thf(func_def_63,type,
sup_su1349021703um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > set_Sum_sum_c_c ).
thf(func_def_64,type,
ord_le1382916304al_c_o: ( finite1398487019real_c > finite1398487019real_c > $o ) > ( finite1398487019real_c > finite1398487019real_c > $o ) > $o ).
thf(func_def_65,type,
ord_le1132241894al_c_o: ( produc190496183real_c > $o ) > ( produc190496183real_c > $o ) > $o ).
thf(func_def_66,type,
ord_le1315653386_c_c_o: ( sum_sum_c_c > $o ) > ( sum_sum_c_c > $o ) > $o ).
thf(func_def_67,type,
ord_le2143422793_c_d_o: ( sum_sum_c_d > $o ) > ( sum_sum_c_d > $o ) > $o ).
thf(func_def_68,type,
ord_le132810396r_real: filter_real > filter_real > $o ).
thf(func_def_69,type,
ord_le775706699real_c: finite1398487019real_c > finite1398487019real_c > $o ).
thf(func_def_70,type,
ord_le850691415real_c: produc190496183real_c > produc190496183real_c > $o ).
thf(func_def_71,type,
ord_le1342644953l_real: produc957004601l_real > produc957004601l_real > $o ).
thf(func_def_72,type,
ord_le191210632um_c_c: produc754691304um_c_c > produc754691304um_c_c > $o ).
thf(func_def_73,type,
ord_le835231240c_real: produc1398711912c_real > produc1398711912c_real > $o ).
thf(func_def_74,type,
ord_le684454199um_c_c: produc214722967um_c_c > produc214722967um_c_c > $o ).
thf(func_def_75,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_76,type,
ord_le1327118209real_c: set_Fi1407883041real_c > set_Fi1407883041real_c > $o ).
thf(func_def_77,type,
ord_le977353143real_c: set_Pr1389752855real_c > set_Pr1389752855real_c > $o ).
thf(func_def_78,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_79,type,
ord_le657554289um_c_c: set_set_Sum_sum_c_c > set_set_Sum_sum_c_c > $o ).
thf(func_def_80,type,
ord_le929608341um_b_c: set_Su1783761653um_b_c > set_Su1783761653um_b_c > $o ).
thf(func_def_81,type,
ord_le1772180283um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > $o ).
thf(func_def_82,type,
ord_le764260924um_c_d: set_Sum_sum_c_d > set_Sum_sum_c_d > $o ).
thf(func_def_83,type,
ord_less_eq_set_c: set_c > set_c > $o ).
thf(func_def_84,type,
produc394644079real_c: finite1398487019real_c > finite1398487019real_c > produc190496183real_c ).
thf(func_def_85,type,
produc705216881l_real: real > real > produc957004601l_real ).
thf(func_def_86,type,
produc1186709024um_c_c: real > set_Sum_sum_c_c > produc754691304um_c_c ).
thf(func_def_87,type,
produc1528874528c_real: set_Sum_sum_c_c > real > produc1398711912c_real ).
thf(func_def_88,type,
produc1439061071um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > produc214722967um_c_c ).
thf(func_def_89,type,
collec230941376real_c: ( finite1398487019real_c > $o ) > set_Fi1407883041real_c ).
thf(func_def_90,type,
collec1643251106real_c: ( produc190496183real_c > $o ) > set_Pr1389752855real_c ).
thf(func_def_91,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_92,type,
collec95329456um_c_c: ( set_Sum_sum_c_c > $o ) > set_set_Sum_sum_c_c ).
thf(func_def_93,type,
collect_Sum_sum_c_c: ( sum_sum_c_c > $o ) > set_Sum_sum_c_c ).
thf(func_def_94,type,
collect_Sum_sum_c_d: ( sum_sum_c_d > $o ) > set_Sum_sum_c_d ).
thf(func_def_95,type,
collect_c: ( c > $o ) > set_c ).
thf(func_def_96,type,
image_1467045143um_c_c: ( produc190496183real_c > sum_sum_c_c ) > set_Pr1389752855real_c > set_Sum_sum_c_c ).
thf(func_def_97,type,
image_1475306648um_c_d: ( produc190496183real_c > sum_sum_c_d ) > set_Pr1389752855real_c > set_Sum_sum_c_d ).
thf(func_def_98,type,
image_327677598real_c: ( real > finite1398487019real_c ) > set_real > set_Fi1407883041real_c ).
thf(func_def_99,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(func_def_100,type,
image_1146069259real_c: ( sum_sum_c_c > produc190496183real_c ) > set_Sum_sum_c_c > set_Pr1389752855real_c ).
thf(func_def_101,type,
image_666880337um_c_c: ( sum_sum_c_c > sum_sum_c_c ) > set_Sum_sum_c_c > set_Sum_sum_c_c ).
thf(func_def_102,type,
image_675141842um_c_d: ( sum_sum_c_c > sum_sum_c_d ) > set_Sum_sum_c_c > set_Sum_sum_c_d ).
thf(func_def_103,type,
image_1980705354real_c: ( sum_sum_c_d > produc190496183real_c ) > set_Sum_sum_c_d > set_Pr1389752855real_c ).
thf(func_def_104,type,
image_1558941394um_c_c: ( sum_sum_c_d > sum_sum_c_c ) > set_Sum_sum_c_d > set_Sum_sum_c_c ).
thf(func_def_105,type,
image_1567202899um_c_d: ( sum_sum_c_d > sum_sum_c_d ) > set_Sum_sum_c_d > set_Sum_sum_c_d ).
thf(func_def_106,type,
image_c_Sum_sum_c_c: ( c > sum_sum_c_c ) > set_c > set_Sum_sum_c_c ).
thf(func_def_107,type,
image_c_Sum_sum_c_d: ( c > sum_sum_c_d ) > set_c > set_Sum_sum_c_d ).
thf(func_def_108,type,
set_or385782508real_c: produc190496183real_c > produc190496183real_c > set_Pr1389752855real_c ).
thf(func_def_109,type,
set_or656347191t_real: real > real > set_real ).
thf(func_def_110,type,
set_or935154662um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > set_set_Sum_sum_c_c ).
thf(func_def_111,type,
set_or2003963744real_c: finite1398487019real_c > set_Fi1407883041real_c ).
thf(func_def_112,type,
set_or10334466real_c: produc190496183real_c > set_Pr1389752855real_c ).
thf(func_def_113,type,
set_ord_atLeast_real: real > set_real ).
thf(func_def_114,type,
set_or2028906832um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c ).
thf(func_def_115,type,
set_or1478397028real_c: finite1398487019real_c > set_Fi1407883041real_c ).
thf(func_def_116,type,
set_or248124158real_c: produc190496183real_c > set_Pr1389752855real_c ).
thf(func_def_117,type,
set_ord_atMost_real: real > set_real ).
thf(func_def_118,type,
set_or326480468um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c ).
thf(func_def_119,type,
static_BVO_a_c: oDE_a_c > set_Sum_sum_c_c ).
thf(func_def_120,type,
static_FVF_a_b_c: formula_a_b_c > set_Sum_sum_c_c ).
thf(func_def_121,type,
static_FVO_a_c: oDE_a_c > set_c ).
thf(func_def_122,type,
static_SIGF_a_b_c: formula_a_b_c > set_Su1783761653um_b_c ).
thf(func_def_123,type,
sum_Inl_c_c: c > sum_sum_c_c ).
thf(func_def_124,type,
sum_Inl_c_d: c > sum_sum_c_d ).
thf(func_def_125,type,
sum_Inr_c_c: c > sum_sum_c_c ).
thf(func_def_126,type,
equiv_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c ).
thf(func_def_127,type,
implies_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c ).
thf(func_def_128,type,
oDE_dom_a_c: oDE_a_c > set_c ).
thf(func_def_129,type,
fsafe_a_b_c: formula_a_b_c > $o ).
thf(func_def_130,type,
evolveODE_a_c_b: oDE_a_c > formula_a_b_c > hp_a_b_c ).
thf(func_def_131,type,
hpsafe_a_b_c: hp_a_b_c > $o ).
thf(func_def_132,type,
osafe_a_c: oDE_a_c > $o ).
thf(func_def_133,type,
topolo1830106092n_real: real > set_real > filter_real ).
thf(func_def_134,type,
member1261661570real_c: finite1398487019real_c > set_Fi1407883041real_c > $o ).
thf(func_def_135,type,
member1895684704real_c: produc190496183real_c > set_Pr1389752855real_c > $o ).
thf(func_def_136,type,
member_real: real > set_real > $o ).
thf(func_def_137,type,
member2124950898um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c > $o ).
thf(func_def_138,type,
member_Sum_sum_c_c: sum_sum_c_c > set_Sum_sum_c_c > $o ).
thf(func_def_139,type,
member_Sum_sum_c_d: sum_sum_c_d > set_Sum_sum_c_d > $o ).
thf(func_def_140,type,
member_c: c > set_c > $o ).
thf(func_def_141,type,
i: denota231621370t_unit ).
thf(func_def_142,type,
j: denota231621370t_unit ).
thf(func_def_143,type,
ode: oDE_a_c ).
thf(func_def_144,type,
p: formula_a_b_c ).
thf(func_def_145,type,
v: set_Sum_sum_c_c ).
thf(func_def_146,type,
a: finite1398487019real_c ).
thf(func_def_147,type,
aa: finite1398487019real_c ).
thf(func_def_148,type,
ab: finite1398487019real_c ).
thf(func_def_149,type,
b: finite1398487019real_c ).
thf(func_def_150,type,
ba: finite1398487019real_c ).
thf(func_def_151,type,
bb: finite1398487019real_c ).
thf(func_def_152,type,
s: real ).
thf(func_def_153,type,
sol: real > finite1398487019real_c ).
thf(func_def_154,type,
t: real ).
thf(func_def_170,type,
sK0: ( set_Sum_sum_c_c > set_Sum_sum_c_c ) > set_Sum_sum_c_c ).
thf(func_def_171,type,
sK1: ( set_Sum_sum_c_c > set_Sum_sum_c_c ) > set_Sum_sum_c_c ).
thf(func_def_172,type,
sK2: produc190496183real_c > finite1398487019real_c ).
thf(func_def_173,type,
sK3: produc190496183real_c > finite1398487019real_c ).
thf(func_def_174,type,
sK4: ( set_Sum_sum_c_c > set_Sum_sum_c_c ) > set_Sum_sum_c_c ).
thf(func_def_175,type,
sK5: ( set_Sum_sum_c_c > set_Sum_sum_c_c ) > set_Sum_sum_c_c ).
thf(func_def_176,type,
sK6: ( produc190496183real_c > $o ) > finite1398487019real_c ).
thf(func_def_177,type,
sK7: ( produc190496183real_c > $o ) > finite1398487019real_c ).
thf(func_def_178,type,
sK8: ( produc190496183real_c > $o ) > finite1398487019real_c ).
thf(func_def_179,type,
sK9: ( produc190496183real_c > $o ) > finite1398487019real_c ).
thf(func_def_180,type,
sK10: ( set_Sum_sum_c_c > set_Sum_sum_c_c ) > set_Sum_sum_c_c ).
thf(func_def_181,type,
sK11: ( set_Sum_sum_c_c > set_Sum_sum_c_c ) > set_Sum_sum_c_c ).
thf(func_def_182,type,
sK12: produc190496183real_c > finite1398487019real_c ).
thf(func_def_183,type,
sK13: produc190496183real_c > finite1398487019real_c ).
thf(func_def_184,type,
sK14: ( set_Sum_sum_c_c > set_Sum_sum_c_c ) > set_Sum_sum_c_c ).
thf(func_def_185,type,
sK15: ( set_Sum_sum_c_c > set_Sum_sum_c_c ) > set_Sum_sum_c_c ).
thf(func_def_186,type,
sK16: produc190496183real_c > finite1398487019real_c > denota231621370t_unit > oDE_a_c > produc190496183real_c ).
thf(func_def_188,type,
ph18:
!>[X0: $tType] : X0 ).
thf(f1148,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1069,f1066]) ).
thf(f1066,plain,
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f901]) ).
thf(f901,plain,
( ( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) )
= $true )
& ( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( produc394644079real_c @ aa @ ba ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ j @ ode ) ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f900]) ).
thf(f900,plain,
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) )
& ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( produc394644079real_c @ aa @ ba ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ j @ ode ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) )
& ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( produc394644079real_c @ aa @ ba ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ j @ ode ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.VZsX1DUqbC/DTF2THF_10391.p',fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062) ).
thf(f1069,plain,
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f964]) ).
thf(f964,plain,
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) )
!= $true ),
inference(flattening,[status(thm)],[f455]) ).
thf(f455,plain,
( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f454]) ).
thf(f454,plain,
~ ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) ),
inference(rectify,[status(thm)],[f354]) ).
thf(f354,negated_conjecture,
~ ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) ),
inference(negated_conjecture,[status(cth)],[f353]) ).
thf(f353,conjecture,
denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ),
file('/export/starexec/sandbox/tmp/tmp.VZsX1DUqbC/DTF2THF_10391.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP041^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n028.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:24:12 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : ITP041^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.46 % Domain : Interactive Theorem Proving
% 0.19/0.46 % Problem : Sledgehammer Coincidence problem prob_905__7236378_1
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Des21]
% 0.19/0.46 % Names : Coincidence/prob_905__7236378_1 [Des21]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.19/0.46 % Syntax : Number of formulae : 508 ( 181 unt; 155 typ; 0 def)
% 0.19/0.46 % Number of atoms : 1012 ( 266 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 12 ( 2 avg)
% 0.19/0.46 % Number of connectives : 2981 ( 51 ~; 13 |; 80 &;2490 @)
% 0.19/0.46 % ( 0 <=>; 347 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 16 ( 6 avg)
% 0.19/0.46 % Number of types : 24 ( 23 usr)
% 0.19/0.46 % Number of type conns : 396 ( 396 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 133 ( 132 usr; 17 con; 0-4 aty)
% 0.19/0.46 % Number of variables : 1051 ( 158 ^; 881 !; 12 ?;1051 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:45:05.406
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % Could-be-implicit typings (23)
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_J,type,
% 0.19/0.46 set_Pr1389752855real_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 0.19/0.46 produc190496183real_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_Mt__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_J,type,
% 0.19/0.46 produc214722967um_c_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__a_Mtf__b_Mtf__c_Mt__Product____Type__Ounit_J,type,
% 0.19/0.46 denota231621370t_unit: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_Mt__Real__Oreal_J,type,
% 0.19/0.46 produc1398711912c_real: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_J,type,
% 0.19/0.46 produc754691304um_c_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Sum____Type__Osum_Itf__b_Mtf__c_J_J_J,type,
% 0.19/0.46 set_Su1783761653um_b_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 0.19/0.46 set_Fi1407883041real_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_J,type,
% 0.19/0.46 set_set_Sum_sum_c_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
% 0.19/0.46 finite1398487019real_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.19/0.46 produc957004601l_real: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__d_J_J,type,
% 0.19/0.46 set_Sum_sum_c_d: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
% 0.19/0.46 set_Sum_sum_c_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Syntax__Oformula_Itf__a_Mtf__b_Mtf__c_J,type,
% 0.19/0.46 formula_a_b_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Syntax__Ohp_Itf__a_Mtf__b_Mtf__c_J,type,
% 0.19/0.46 hp_a_b_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
% 0.19/0.46 sum_sum_c_d: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
% 0.19/0.46 sum_sum_c_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 0.19/0.46 filter_real: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Syntax__OODE_Itf__a_Mtf__c_J,type,
% 0.19/0.46 oDE_a_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 0.19/0.46 set_real: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Set__Oset_Itf__c_J,type,
% 0.19/0.46 set_c: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Real__Oreal,type,
% 0.19/0.46 real: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_tf__c,type,
% 0.19/0.46 c: $tType ).
% 0.19/0.46
% 0.19/0.46 % Explicit typings (132)
% 0.19/0.46 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_001t__Real__Oreal,type,
% 0.19/0.46 bNF_Ca449034301c_real: set_Pr1389752855real_c > ( finite1398487019real_c > real ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
% 0.19/0.46 bNF_Ca1693016812um_c_c: set_Pr1389752855real_c > ( finite1398487019real_c > set_Sum_sum_c_c ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Coincidence__Mirabelle__veeocdlawq_Oids_Ocoincide__fml_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 coinci501627771_a_b_c: formula_a_b_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Coincidence__Mirabelle__veeocdlawq_Oids_Ocoincide__hp_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 coinci354700372_a_b_c: hp_a_b_c > denota231621370t_unit > denota231621370t_unit > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Coincidence__Mirabelle__veeocdlawq_Oids_Ocoincide__hp_H_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 coinci1852866155_a_b_c: hp_a_b_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Coincidence__Mirabelle__veeocdlawq_Oids_Oode__sem__equiv_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 coinci772602909_a_b_c: hp_a_b_c > denota231621370t_unit > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Denotational__Semantics_OIagree_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 denota69106024_a_b_c: denota231621370t_unit > denota231621370t_unit > set_Su1783761653um_b_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Denotational__Semantics_OODE__sem_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 denota1275485728_a_b_c: denota231621370t_unit > oDE_a_c > finite1398487019real_c > finite1398487019real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Denotational__Semantics_OVSagree_001tf__c,type,
% 0.19/0.46 denota256060419gree_c: finite1398487019real_c > finite1398487019real_c > set_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Denotational__Semantics_OVagree_001tf__c,type,
% 0.19/0.46 denota1997846518gree_c: produc190496183real_c > produc190496183real_c > set_Sum_sum_c_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Denotational__Semantics_Ofml__sem_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 denota968303861_a_b_c: denota231621370t_unit > formula_a_b_c > set_Pr1389752855real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Denotational__Semantics_Omk__v_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 denota161327353_a_b_c: denota231621370t_unit > oDE_a_c > produc190496183real_c > finite1398487019real_c > produc190496183real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Denotational__Semantics_Omk__xode_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 denota1896987029_a_b_c: denota231621370t_unit > oDE_a_c > finite1398487019real_c > produc190496183real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Denotational__Semantics_OsemBV_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 denota1293057250_a_b_c: denota231621370t_unit > oDE_a_c > set_Sum_sum_c_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Deriv_Ohas__vector__derivative_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
% 0.19/0.46 has_ve1904308670real_c: ( real > finite1398487019real_c ) > finite1398487019real_c > filter_real > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Deriv_Ohas__vector__derivative_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 0.19/0.46 has_ve373335844real_c: ( real > produc190496183real_c ) > produc190496183real_c > filter_real > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Deriv_Ohas__vector__derivative_001t__Real__Oreal,type,
% 0.19/0.46 has_ve217091583e_real: ( real > real ) > real > filter_real > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_M_Eo_J,type,
% 0.19/0.46 uminus23580315al_c_o: ( finite1398487019real_c > $o ) > finite1398487019real_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_M_Eo_J,type,
% 0.19/0.46 uminus32155087al_c_o: ( produc190496183real_c > $o ) > produc190496183real_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Sum____Type__Osum_Itf__c_Mtf__c_J_M_Eo_J,type,
% 0.19/0.46 uminus88729697_c_c_o: ( sum_sum_c_c > $o ) > sum_sum_c_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Sum____Type__Osum_Itf__c_Mtf__d_J_M_Eo_J,type,
% 0.19/0.46 uminus916499104_c_d_o: ( sum_sum_c_d > $o ) > sum_sum_c_d > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__c_M_Eo_J,type,
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% 0.19/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
% 0.19/0.46 set_or935154662um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > set_set_Sum_sum_c_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
% 0.19/0.46 set_or2003963744real_c: finite1398487019real_c > set_Fi1407883041real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 0.19/0.46 set_or10334466real_c: produc190496183real_c > set_Pr1389752855real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
% 0.19/0.46 set_ord_atLeast_real: real > set_real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
% 0.19/0.46 set_or2028906832um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
% 0.19/0.46 set_or1478397028real_c: finite1398487019real_c > set_Fi1407883041real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 0.19/0.46 set_or248124158real_c: produc190496183real_c > set_Pr1389752855real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
% 0.19/0.46 set_ord_atMost_real: real > set_real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
% 0.19/0.46 set_or326480468um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Static__Semantics_OBVO_001tf__a_001tf__c,type,
% 0.19/0.46 static_BVO_a_c: oDE_a_c > set_Sum_sum_c_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Static__Semantics_OFVF_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 static_FVF_a_b_c: formula_a_b_c > set_Sum_sum_c_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Static__Semantics_OFVO_001tf__a_001tf__c,type,
% 0.19/0.46 static_FVO_a_c: oDE_a_c > set_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Static__Semantics_OSIGF_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 static_SIGF_a_b_c: formula_a_b_c > set_Su1783761653um_b_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Sum__Type_OInl_001tf__c_001tf__c,type,
% 0.19/0.46 sum_Inl_c_c: c > sum_sum_c_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Sum__Type_OInl_001tf__c_001tf__d,type,
% 0.19/0.46 sum_Inl_c_d: c > sum_sum_c_d ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Sum__Type_OInr_001tf__c_001tf__c,type,
% 0.19/0.46 sum_Inr_c_c: c > sum_sum_c_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Syntax_OEquiv_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 equiv_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Syntax_OImplies_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 implies_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Syntax_OODE__dom_001tf__a_001tf__c,type,
% 0.19/0.46 oDE_dom_a_c: oDE_a_c > set_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Syntax_Ofsafe_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 fsafe_a_b_c: formula_a_b_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Syntax_Ohp_OEvolveODE_001tf__a_001tf__c_001tf__b,type,
% 0.19/0.46 evolveODE_a_c_b: oDE_a_c > formula_a_b_c > hp_a_b_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Syntax_Ohpsafe_001tf__a_001tf__b_001tf__c,type,
% 0.19/0.46 hpsafe_a_b_c: hp_a_b_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Syntax_Oosafe_001tf__a_001tf__c,type,
% 0.19/0.46 osafe_a_c: oDE_a_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 0.19/0.46 topolo1830106092n_real: real > set_real > filter_real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
% 0.19/0.46 member1261661570real_c: finite1398487019real_c > set_Fi1407883041real_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 0.19/0.46 member1895684704real_c: produc190496183real_c > set_Pr1389752855real_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Real__Oreal,type,
% 0.19/0.46 member_real: real > set_real > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
% 0.19/0.46 member2124950898um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
% 0.19/0.46 member_Sum_sum_c_c: sum_sum_c_c > set_Sum_sum_c_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
% 0.19/0.46 member_Sum_sum_c_d: sum_sum_c_d > set_Sum_sum_c_d > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member_001tf__c,type,
% 0.19/0.46 member_c: c > set_c > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_I____,type,
% 0.19/0.46 i: denota231621370t_unit ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_J____,type,
% 0.19/0.46 j: denota231621370t_unit ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_ODE____,type,
% 0.19/0.46 ode: oDE_a_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_P____,type,
% 0.19/0.46 p: formula_a_b_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_V____,type,
% 0.19/0.46 v: set_Sum_sum_c_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_a____,type,
% 0.19/0.46 a: finite1398487019real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_aa____,type,
% 0.19/0.46 aa: finite1398487019real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_ab____,type,
% 0.19/0.46 ab: finite1398487019real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_b____,type,
% 0.19/0.46 b: finite1398487019real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_ba____,type,
% 0.19/0.46 ba: finite1398487019real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_bb____,type,
% 0.19/0.46 bb: finite1398487019real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_s____,type,
% 0.19/0.46 s: real ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_sol____,type,
% 0.19/0.46 sol: real > finite1398487019real_c ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_t____,type,
% 0.19/0.46 t: real ).
% 0.19/0.46
% 0.19/0.46 % Relevant facts (352)
% 0.19/0.46 thf(fact_0_osafe,axiom,
% 0.19/0.46 osafe_a_c @ ode ).
% 0.19/0.46
% 0.19/0.46 % osafe
% 0.19/0.46 thf(fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062,axiom,
% 0.19/0.46 ( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( produc394644079real_c @ aa @ ba ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ j @ ode ) ) )
% 0.19/0.46 & ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>Vagree (mk_v J ODE (aa, ba) (sol s)) (aa, ba) (- semBV J ODE) \<and> Vagree (mk_v J ODE (aa, ba) (sol s)) (mk_xode J ODE (sol s)) (semBV J ODE)\<close>
% 0.19/0.46 thf(fact_2__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_Asa_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_092_060close_062,axiom,
% 0.19/0.46 ! [S: real] : ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ S ) ) @ ( produc394644079real_c @ aa @ ba ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ j @ ode ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>\<And>sa. Vagree (mk_v J ODE (aa, ba) (sol sa)) (aa, ba) (- semBV J ODE)\<close>
% 0.19/0.46 thf(fact_3_mk__v__agree,axiom,
% 0.19/0.46 ! [I: denota231621370t_unit,ODE: oDE_a_c,Nu: produc190496183real_c,Sol: finite1398487019real_c] :
% 0.19/0.46 ( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ I @ ODE @ Nu @ Sol ) @ Nu @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I @ ODE ) ) )
% 0.19/0.46 & ( denota1997846518gree_c @ ( denota161327353_a_b_c @ I @ ODE @ Nu @ Sol ) @ ( denota1896987029_a_b_c @ I @ ODE @ Sol ) @ ( denota1293057250_a_b_c @ I @ ODE ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mk_v_agree
% 0.19/0.46 thf(fact_4_prod_Oinject,axiom,
% 0.19/0.46 ! [X1: finite1398487019real_c,X2: finite1398487019real_c,Y1: finite1398487019real_c,Y2: finite1398487019real_c] :
% 0.19/0.46 ( ( ( produc394644079real_c @ X1 @ X2 )
% 0.19/0.46 = ( produc394644079real_c @ Y1 @ Y2 ) )
% 0.19/0.46 = ( ( X1 = Y1 )
% 0.19/0.46 & ( X2 = Y2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % prod.inject
% 0.19/0.46 thf(fact_5_old_Oprod_Oinject,axiom,
% 0.19/0.46 ! [A: finite1398487019real_c,B: finite1398487019real_c,A2: finite1398487019real_c,B2: finite1398487019real_c] :
% 0.19/0.46 ( ( ( produc394644079real_c @ A @ B )
% 0.19/0.46 = ( produc394644079real_c @ A2 @ B2 ) )
% 0.19/0.46 = ( ( A = A2 )
% 0.19/0.46 & ( B = B2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % old.prod.inject
% 0.19/0.46 thf(fact_6_mk__v__exists,axiom,
% 0.19/0.46 ! [Nu: produc190496183real_c,I: denota231621370t_unit,ODE: oDE_a_c,Sol: finite1398487019real_c] :
% 0.19/0.46 ? [Omega: produc190496183real_c] :
% 0.19/0.46 ( ( denota1997846518gree_c @ Omega @ Nu @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I @ ODE ) ) )
% 0.19/0.46 & ( denota1997846518gree_c @ Omega @ ( denota1896987029_a_b_c @ I @ ODE @ Sol ) @ ( denota1293057250_a_b_c @ I @ ODE ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mk_v_exists
% 0.19/0.46 thf(fact_7_VA,axiom,
% 0.19/0.46 denota1997846518gree_c @ ( produc394644079real_c @ a @ b ) @ ( produc394644079real_c @ aa @ ba ) @ v ).
% 0.19/0.46
% 0.19/0.46 % VA
% 0.19/0.46 thf(fact_8__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Imk__xode_AI_AODE_A_Isol_Asa_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
% 0.19/0.46 ! [S: real] : ( denota1997846518gree_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ S ) ) @ ( denota1896987029_a_b_c @ i @ ode @ ( sol @ S ) ) @ ( denota1293057250_a_b_c @ i @ ode ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>\<And>sa. Vagree (mk_v I ODE (a, b) (sol sa)) (mk_xode I ODE (sol sa)) (semBV I ODE)\<close>
% 0.19/0.46 thf(fact_9_VAOV,axiom,
% 0.19/0.46 denota1997846518gree_c @ ( produc394644079real_c @ a @ b ) @ ( produc394644079real_c @ aa @ ba ) @ ( static_BVO_a_c @ ode ) ).
% 0.19/0.46
% 0.19/0.46 % VAOV
% 0.19/0.46 thf(fact_10_agree__comm,axiom,
% 0.19/0.46 ! [A3: produc190496183real_c,B3: produc190496183real_c,V: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( denota1997846518gree_c @ A3 @ B3 @ V )
% 0.19/0.46 => ( denota1997846518gree_c @ B3 @ A3 @ V ) ) ).
% 0.19/0.46
% 0.19/0.46 % agree_comm
% 0.19/0.46 thf(fact_11_agree__refl,axiom,
% 0.19/0.46 ! [Nu: produc190496183real_c,A3: set_Sum_sum_c_c] : ( denota1997846518gree_c @ Nu @ Nu @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % agree_refl
% 0.19/0.46 thf(fact_12__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Ia_M_Ab_J_A_I_N_AsemBV_AI_AODE_J_092_060close_062,axiom,
% 0.19/0.46 ! [S: real] : ( denota1997846518gree_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ S ) ) @ ( produc394644079real_c @ a @ b ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ i @ ode ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>\<And>sa. Vagree (mk_v I ODE (a, b) (sol sa)) (a, b) (- semBV I ODE)\<close>
% 0.19/0.46 thf(fact_13_surj__pair,axiom,
% 0.19/0.46 ! [P: produc190496183real_c] :
% 0.19/0.46 ? [X: finite1398487019real_c,Y: finite1398487019real_c] :
% 0.19/0.46 ( P
% 0.19/0.46 = ( produc394644079real_c @ X @ Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % surj_pair
% 0.19/0.46 thf(fact_14_prod__cases,axiom,
% 0.19/0.46 ! [P2: produc190496183real_c > $o,P: produc190496183real_c] :
% 0.19/0.46 ( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P2 @ ( produc394644079real_c @ A4 @ B4 ) )
% 0.19/0.46 => ( P2 @ P ) ) ).
% 0.19/0.46
% 0.19/0.46 % prod_cases
% 0.19/0.46 thf(fact_15_OVsub,axiom,
% 0.19/0.46 ord_le1772180283um_c_c @ ( static_BVO_a_c @ ode ) @ v ).
% 0.19/0.46
% 0.19/0.46 % OVsub
% 0.19/0.46 thf(fact_16_veq,axiom,
% 0.19/0.46 ( ( produc394644079real_c @ ab @ bb )
% 0.19/0.46 = ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ t ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % veq
% 0.19/0.46 thf(fact_17_old_Oprod_Oinducts,axiom,
% 0.19/0.46 ! [P2: produc190496183real_c > $o,Prod: produc190496183real_c] :
% 0.19/0.46 ( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P2 @ ( produc394644079real_c @ A4 @ B4 ) )
% 0.19/0.46 => ( P2 @ Prod ) ) ).
% 0.19/0.46
% 0.19/0.46 % old.prod.inducts
% 0.19/0.46 thf(fact_18_old_Oprod_Oexhaust,axiom,
% 0.19/0.46 ! [Y3: produc190496183real_c] :
% 0.19/0.46 ~ ! [A4: finite1398487019real_c,B4: finite1398487019real_c] :
% 0.19/0.46 ( Y3
% 0.19/0.46 != ( produc394644079real_c @ A4 @ B4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % old.prod.exhaust
% 0.19/0.46 thf(fact_19_Pair__inject,axiom,
% 0.19/0.46 ! [A: finite1398487019real_c,B: finite1398487019real_c,A2: finite1398487019real_c,B2: finite1398487019real_c] :
% 0.19/0.46 ( ( ( produc394644079real_c @ A @ B )
% 0.19/0.46 = ( produc394644079real_c @ A2 @ B2 ) )
% 0.19/0.46 => ~ ( ( A = A2 )
% 0.19/0.46 => ( B != B2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_inject
% 0.19/0.46 thf(fact_20_OVsub_H_H,axiom,
% 0.19/0.46 ord_le1772180283um_c_c @ ( denota1293057250_a_b_c @ i @ ode ) @ v ).
% 0.19/0.46
% 0.19/0.46 % OVsub''
% 0.19/0.46 thf(fact_21_semBVsub,axiom,
% 0.19/0.46 ord_le1772180283um_c_c @ ( denota1293057250_a_b_c @ i @ ode ) @ ( static_BVO_a_c @ ode ) ).
% 0.19/0.46
% 0.19/0.46 % semBVsub
% 0.19/0.46 thf(fact_22_uminus__Pair,axiom,
% 0.19/0.46 ! [A: finite1398487019real_c,B: finite1398487019real_c] :
% 0.19/0.46 ( ( uminus17964590real_c @ ( produc394644079real_c @ A @ B ) )
% 0.19/0.46 = ( produc394644079real_c @ ( uminus1737280820real_c @ A ) @ ( uminus1737280820real_c @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_Pair
% 0.19/0.46 thf(fact_23_uminus__Pair,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( uminus1225118734um_c_c @ ( produc1439061071um_c_c @ A @ B ) )
% 0.19/0.46 = ( produc1439061071um_c_c @ ( uminus1381786404um_c_c @ A ) @ ( uminus1381786404um_c_c @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_Pair
% 0.19/0.46 thf(fact_24_uminus__Pair,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: real] :
% 0.19/0.46 ( ( uminus125610847c_real @ ( produc1528874528c_real @ A @ B ) )
% 0.19/0.46 = ( produc1528874528c_real @ ( uminus1381786404um_c_c @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_Pair
% 0.19/0.46 thf(fact_25_uminus__Pair,axiom,
% 0.19/0.46 ! [A: real,B: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( uminus1629073887um_c_c @ ( produc1186709024um_c_c @ A @ B ) )
% 0.19/0.46 = ( produc1186709024um_c_c @ ( uminus_uminus_real @ A ) @ ( uminus1381786404um_c_c @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_Pair
% 0.19/0.46 thf(fact_26_uminus__Pair,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( uminus856758960l_real @ ( produc705216881l_real @ A @ B ) )
% 0.19/0.46 = ( produc705216881l_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_Pair
% 0.19/0.46 thf(fact_27_ComplI,axiom,
% 0.19/0.46 ! [C: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 0.19/0.46 ( ~ ( member1895684704real_c @ C @ A3 )
% 0.19/0.46 => ( member1895684704real_c @ C @ ( uminus232257166real_c @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ComplI
% 0.19/0.46 thf(fact_28_ComplI,axiom,
% 0.19/0.46 ! [C: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 0.19/0.46 ( ~ ( member_Sum_sum_c_d @ C @ A3 )
% 0.19/0.46 => ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ComplI
% 0.19/0.46 thf(fact_29_ComplI,axiom,
% 0.19/0.46 ! [C: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ~ ( member_Sum_sum_c_c @ C @ A3 )
% 0.19/0.46 => ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ComplI
% 0.19/0.46 thf(fact_30_Compl__iff,axiom,
% 0.19/0.46 ! [C: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ C @ ( uminus232257166real_c @ A3 ) )
% 0.19/0.46 = ( ~ ( member1895684704real_c @ C @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_iff
% 0.19/0.46 thf(fact_31_Compl__iff,axiom,
% 0.19/0.46 ! [C: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 0.19/0.46 ( ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A3 ) )
% 0.19/0.46 = ( ~ ( member_Sum_sum_c_d @ C @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_iff
% 0.19/0.46 thf(fact_32_Compl__iff,axiom,
% 0.19/0.46 ! [C: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A3 ) )
% 0.19/0.46 = ( ~ ( member_Sum_sum_c_c @ C @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_iff
% 0.19/0.46 thf(fact_33_Compl__eq__Compl__iff,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ( uminus1381786404um_c_c @ A3 )
% 0.19/0.46 = ( uminus1381786404um_c_c @ B3 ) )
% 0.19/0.46 = ( A3 = B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_eq_Compl_iff
% 0.19/0.46 thf(fact_34_verit__minus__simplify_I4_J,axiom,
% 0.19/0.46 ! [B: real] :
% 0.19/0.46 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % verit_minus_simplify(4)
% 0.19/0.46 thf(fact_35_add_Oinverse__inverse,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add.inverse_inverse
% 0.19/0.46 thf(fact_36_subset__antisym,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ B3 @ A3 )
% 0.19/0.46 => ( A3 = B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_antisym
% 0.19/0.46 thf(fact_37_subsetI,axiom,
% 0.19/0.46 ! [A3: set_Pr1389752855real_c,B3: set_Pr1389752855real_c] :
% 0.19/0.46 ( ! [X: produc190496183real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ X @ A3 )
% 0.19/0.46 => ( member1895684704real_c @ X @ B3 ) )
% 0.19/0.46 => ( ord_le977353143real_c @ A3 @ B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % subsetI
% 0.19/0.46 thf(fact_38_subsetI,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 0.19/0.46 ( ! [X: sum_sum_c_d] :
% 0.19/0.46 ( ( member_Sum_sum_c_d @ X @ A3 )
% 0.19/0.46 => ( member_Sum_sum_c_d @ X @ B3 ) )
% 0.19/0.46 => ( ord_le764260924um_c_d @ A3 @ B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % subsetI
% 0.19/0.46 thf(fact_39_subsetI,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ! [X: sum_sum_c_c] :
% 0.19/0.46 ( ( member_Sum_sum_c_c @ X @ A3 )
% 0.19/0.46 => ( member_Sum_sum_c_c @ X @ B3 ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A3 @ B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % subsetI
% 0.19/0.46 thf(fact_40_compl__eq__compl__iff,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ( uminus1381786404um_c_c @ X3 )
% 0.19/0.46 = ( uminus1381786404um_c_c @ Y3 ) )
% 0.19/0.46 = ( X3 = Y3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % compl_eq_compl_iff
% 0.19/0.46 thf(fact_41_double__compl,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( uminus1381786404um_c_c @ ( uminus1381786404um_c_c @ X3 ) )
% 0.19/0.46 = X3 ) ).
% 0.19/0.46
% 0.19/0.46 % double_compl
% 0.19/0.46 thf(fact_42_neg__equal__iff__equal,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = ( uminus_uminus_real @ B ) )
% 0.19/0.46 = ( A = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_iff_equal
% 0.19/0.46 thf(fact_43_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: produc190496183real_c,P2: produc190496183real_c > $o] :
% 0.19/0.46 ( ( member1895684704real_c @ A @ ( collec1643251106real_c @ P2 ) )
% 0.19/0.46 = ( P2 @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_44_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 0.19/0.46 ( ( member_Sum_sum_c_c @ A @ ( collect_Sum_sum_c_c @ P2 ) )
% 0.19/0.46 = ( P2 @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 0.19/0.46 ( ( member_Sum_sum_c_d @ A @ ( collect_Sum_sum_c_d @ P2 ) )
% 0.19/0.46 = ( P2 @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_46_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: finite1398487019real_c,P2: finite1398487019real_c > $o] :
% 0.19/0.46 ( ( member1261661570real_c @ A @ ( collec230941376real_c @ P2 ) )
% 0.19/0.46 = ( P2 @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_47_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: c,P2: c > $o] :
% 0.19/0.46 ( ( member_c @ A @ ( collect_c @ P2 ) )
% 0.19/0.46 = ( P2 @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_48_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A3: set_Pr1389752855real_c] :
% 0.19/0.46 ( ( collec1643251106real_c
% 0.19/0.46 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A3 ) )
% 0.19/0.46 = A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_49_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( collect_Sum_sum_c_c
% 0.19/0.46 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ A3 ) )
% 0.19/0.46 = A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_50_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_d] :
% 0.19/0.46 ( ( collect_Sum_sum_c_d
% 0.19/0.46 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ A3 ) )
% 0.19/0.46 = A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_51_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A3: set_Fi1407883041real_c] :
% 0.19/0.46 ( ( collec230941376real_c
% 0.19/0.46 @ ^ [X4: finite1398487019real_c] : ( member1261661570real_c @ X4 @ A3 ) )
% 0.19/0.46 = A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_52_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A3: set_c] :
% 0.19/0.46 ( ( collect_c
% 0.19/0.46 @ ^ [X4: c] : ( member_c @ X4 @ A3 ) )
% 0.19/0.46 = A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_53_Collect__cong,axiom,
% 0.19/0.46 ! [P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
% 0.19/0.46 ( ! [X: finite1398487019real_c] :
% 0.19/0.46 ( ( P2 @ X )
% 0.19/0.46 = ( Q @ X ) )
% 0.19/0.46 => ( ( collec230941376real_c @ P2 )
% 0.19/0.46 = ( collec230941376real_c @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_cong
% 0.19/0.46 thf(fact_54_Collect__cong,axiom,
% 0.19/0.46 ! [P2: c > $o,Q: c > $o] :
% 0.19/0.46 ( ! [X: c] :
% 0.19/0.46 ( ( P2 @ X )
% 0.19/0.46 = ( Q @ X ) )
% 0.19/0.46 => ( ( collect_c @ P2 )
% 0.19/0.46 = ( collect_c @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_cong
% 0.19/0.46 thf(fact_55_compl__le__compl__iff,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ X3 ) @ ( uminus1381786404um_c_c @ Y3 ) )
% 0.19/0.46 = ( ord_le1772180283um_c_c @ Y3 @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % compl_le_compl_iff
% 0.19/0.46 thf(fact_56_neg__le__iff__le,axiom,
% 0.19/0.46 ! [B: real,A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_le_iff_le
% 0.19/0.46 thf(fact_57_Compl__subset__Compl__iff,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ A3 ) @ ( uminus1381786404um_c_c @ B3 ) )
% 0.19/0.46 = ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_subset_Compl_iff
% 0.19/0.46 thf(fact_58_Compl__anti__mono,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ B3 ) @ ( uminus1381786404um_c_c @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_anti_mono
% 0.19/0.46 thf(fact_59_verit__la__disequality,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 | ~ ( ord_less_eq_real @ A @ B )
% 0.19/0.46 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_la_disequality
% 0.19/0.46 thf(fact_60_Collect__mono__iff,axiom,
% 0.19/0.46 ! [P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
% 0.19/0.46 ( ( ord_le1327118209real_c @ ( collec230941376real_c @ P2 ) @ ( collec230941376real_c @ Q ) )
% 0.19/0.46 = ( ! [X4: finite1398487019real_c] :
% 0.19/0.46 ( ( P2 @ X4 )
% 0.19/0.46 => ( Q @ X4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mono_iff
% 0.19/0.46 thf(fact_61_Collect__mono__iff,axiom,
% 0.19/0.46 ! [P2: c > $o,Q: c > $o] :
% 0.19/0.46 ( ( ord_less_eq_set_c @ ( collect_c @ P2 ) @ ( collect_c @ Q ) )
% 0.19/0.46 = ( ! [X4: c] :
% 0.19/0.46 ( ( P2 @ X4 )
% 0.19/0.46 => ( Q @ X4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mono_iff
% 0.19/0.46 thf(fact_62_Collect__mono__iff,axiom,
% 0.19/0.46 ! [P2: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ ( collect_Sum_sum_c_c @ P2 ) @ ( collect_Sum_sum_c_c @ Q ) )
% 0.19/0.46 = ( ! [X4: sum_sum_c_c] :
% 0.19/0.46 ( ( P2 @ X4 )
% 0.19/0.46 => ( Q @ X4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mono_iff
% 0.19/0.46 thf(fact_63_set__eq__subset,axiom,
% 0.19/0.46 ( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A5 @ B5 )
% 0.19/0.46 & ( ord_le1772180283um_c_c @ B5 @ A5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % set_eq_subset
% 0.19/0.46 thf(fact_64_subset__trans,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,C2: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ B3 @ C2 )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A3 @ C2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_trans
% 0.19/0.46 thf(fact_65_Collect__mono,axiom,
% 0.19/0.46 ! [P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
% 0.19/0.46 ( ! [X: finite1398487019real_c] :
% 0.19/0.46 ( ( P2 @ X )
% 0.19/0.46 => ( Q @ X ) )
% 0.19/0.46 => ( ord_le1327118209real_c @ ( collec230941376real_c @ P2 ) @ ( collec230941376real_c @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mono
% 0.19/0.46 thf(fact_66_Collect__mono,axiom,
% 0.19/0.46 ! [P2: c > $o,Q: c > $o] :
% 0.19/0.46 ( ! [X: c] :
% 0.19/0.46 ( ( P2 @ X )
% 0.19/0.46 => ( Q @ X ) )
% 0.19/0.46 => ( ord_less_eq_set_c @ ( collect_c @ P2 ) @ ( collect_c @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mono
% 0.19/0.46 thf(fact_67_Collect__mono,axiom,
% 0.19/0.46 ! [P2: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
% 0.19/0.46 ( ! [X: sum_sum_c_c] :
% 0.19/0.46 ( ( P2 @ X )
% 0.19/0.46 => ( Q @ X ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( collect_Sum_sum_c_c @ P2 ) @ ( collect_Sum_sum_c_c @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mono
% 0.19/0.46 thf(fact_68_subset__refl,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A3 @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % subset_refl
% 0.19/0.46 thf(fact_69_subset__iff,axiom,
% 0.19/0.46 ( ord_le977353143real_c
% 0.19/0.46 = ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
% 0.19/0.46 ! [T: produc190496183real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ T @ A5 )
% 0.19/0.46 => ( member1895684704real_c @ T @ B5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_iff
% 0.19/0.46 thf(fact_70_subset__iff,axiom,
% 0.19/0.46 ( ord_le764260924um_c_d
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
% 0.19/0.46 ! [T: sum_sum_c_d] :
% 0.19/0.46 ( ( member_Sum_sum_c_d @ T @ A5 )
% 0.19/0.46 => ( member_Sum_sum_c_d @ T @ B5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_iff
% 0.19/0.46 thf(fact_71_subset__iff,axiom,
% 0.19/0.46 ( ord_le1772180283um_c_c
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
% 0.19/0.46 ! [T: sum_sum_c_c] :
% 0.19/0.46 ( ( member_Sum_sum_c_c @ T @ A5 )
% 0.19/0.46 => ( member_Sum_sum_c_c @ T @ B5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_iff
% 0.19/0.46 thf(fact_72_equalityD2,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( A3 = B3 )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % equalityD2
% 0.19/0.46 thf(fact_73_equalityD1,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( A3 = B3 )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A3 @ B3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % equalityD1
% 0.19/0.46 thf(fact_74_subset__eq,axiom,
% 0.19/0.46 ( ord_le977353143real_c
% 0.19/0.46 = ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
% 0.19/0.46 ! [X4: produc190496183real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ X4 @ A5 )
% 0.19/0.46 => ( member1895684704real_c @ X4 @ B5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_eq
% 0.19/0.46 thf(fact_75_subset__eq,axiom,
% 0.19/0.46 ( ord_le764260924um_c_d
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
% 0.19/0.46 ! [X4: sum_sum_c_d] :
% 0.19/0.46 ( ( member_Sum_sum_c_d @ X4 @ A5 )
% 0.19/0.46 => ( member_Sum_sum_c_d @ X4 @ B5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_eq
% 0.19/0.46 thf(fact_76_subset__eq,axiom,
% 0.19/0.46 ( ord_le1772180283um_c_c
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
% 0.19/0.46 ! [X4: sum_sum_c_c] :
% 0.19/0.46 ( ( member_Sum_sum_c_c @ X4 @ A5 )
% 0.19/0.46 => ( member_Sum_sum_c_c @ X4 @ B5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subset_eq
% 0.19/0.46 thf(fact_77_equalityE,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( A3 = B3 )
% 0.19/0.46 => ~ ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 0.19/0.46 => ~ ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % equalityE
% 0.19/0.46 thf(fact_78_subsetD,axiom,
% 0.19/0.46 ! [A3: set_Pr1389752855real_c,B3: set_Pr1389752855real_c,C: produc190496183real_c] :
% 0.19/0.46 ( ( ord_le977353143real_c @ A3 @ B3 )
% 0.19/0.46 => ( ( member1895684704real_c @ C @ A3 )
% 0.19/0.46 => ( member1895684704real_c @ C @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subsetD
% 0.19/0.46 thf(fact_79_subsetD,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_d,B3: set_Sum_sum_c_d,C: sum_sum_c_d] :
% 0.19/0.46 ( ( ord_le764260924um_c_d @ A3 @ B3 )
% 0.19/0.46 => ( ( member_Sum_sum_c_d @ C @ A3 )
% 0.19/0.46 => ( member_Sum_sum_c_d @ C @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subsetD
% 0.19/0.46 thf(fact_80_subsetD,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,C: sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 0.19/0.46 => ( ( member_Sum_sum_c_c @ C @ A3 )
% 0.19/0.46 => ( member_Sum_sum_c_c @ C @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % subsetD
% 0.19/0.46 thf(fact_81_in__mono,axiom,
% 0.19/0.46 ! [A3: set_Pr1389752855real_c,B3: set_Pr1389752855real_c,X3: produc190496183real_c] :
% 0.19/0.46 ( ( ord_le977353143real_c @ A3 @ B3 )
% 0.19/0.46 => ( ( member1895684704real_c @ X3 @ A3 )
% 0.19/0.46 => ( member1895684704real_c @ X3 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % in_mono
% 0.19/0.46 thf(fact_82_in__mono,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_d,B3: set_Sum_sum_c_d,X3: sum_sum_c_d] :
% 0.19/0.46 ( ( ord_le764260924um_c_d @ A3 @ B3 )
% 0.19/0.46 => ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 0.19/0.46 => ( member_Sum_sum_c_d @ X3 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % in_mono
% 0.19/0.46 thf(fact_83_in__mono,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,X3: sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 0.19/0.46 => ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 0.19/0.46 => ( member_Sum_sum_c_c @ X3 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % in_mono
% 0.19/0.46 thf(fact_84_le__imp__neg__le,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_imp_neg_le
% 0.19/0.46 thf(fact_85_minus__le__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.19/0.46 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_le_iff
% 0.19/0.46 thf(fact_86_le__minus__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.19/0.46 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_minus_iff
% 0.19/0.46 thf(fact_87_compl__le__swap2,axiom,
% 0.19/0.46 ! [Y3: set_Sum_sum_c_c,X3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ Y3 ) @ X3 )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ X3 ) @ Y3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % compl_le_swap2
% 0.19/0.46 thf(fact_88_compl__le__swap1,axiom,
% 0.19/0.46 ! [Y3: set_Sum_sum_c_c,X3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ Y3 @ ( uminus1381786404um_c_c @ X3 ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ X3 @ ( uminus1381786404um_c_c @ Y3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % compl_le_swap1
% 0.19/0.46 thf(fact_89_compl__mono,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X3 @ Y3 )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ Y3 ) @ ( uminus1381786404um_c_c @ X3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % compl_mono
% 0.19/0.46 thf(fact_90_agree__sub,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,Nu: produc190496183real_c,Omega2: produc190496183real_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 0.19/0.46 => ( ( denota1997846518gree_c @ Nu @ Omega2 @ B3 )
% 0.19/0.46 => ( denota1997846518gree_c @ Nu @ Omega2 @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % agree_sub
% 0.19/0.46 thf(fact_91_agree__supset,axiom,
% 0.19/0.46 ! [B3: set_Sum_sum_c_c,A3: set_Sum_sum_c_c,Nu: produc190496183real_c,Nu2: produc190496183real_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ B3 @ A3 )
% 0.19/0.46 => ( ( denota1997846518gree_c @ Nu @ Nu2 @ A3 )
% 0.19/0.46 => ( denota1997846518gree_c @ Nu @ Nu2 @ B3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % agree_supset
% 0.19/0.46 thf(fact_92_minus__equation__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = B )
% 0.19/0.46 = ( ( uminus_uminus_real @ B )
% 0.19/0.46 = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_equation_iff
% 0.19/0.46 thf(fact_93_equation__minus__iff,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( uminus_uminus_real @ B ) )
% 0.19/0.46 = ( B
% 0.19/0.46 = ( uminus_uminus_real @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % equation_minus_iff
% 0.19/0.46 thf(fact_94_verit__negate__coefficient_I3_J,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( uminus_uminus_real @ A )
% 0.19/0.46 = ( uminus_uminus_real @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % verit_negate_coefficient(3)
% 0.19/0.46 thf(fact_95_double__complement,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( uminus1381786404um_c_c @ ( uminus1381786404um_c_c @ A3 ) )
% 0.19/0.46 = A3 ) ).
% 0.19/0.46
% 0.19/0.46 % double_complement
% 0.19/0.46 thf(fact_96_ComplD,axiom,
% 0.19/0.46 ! [C: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ C @ ( uminus232257166real_c @ A3 ) )
% 0.19/0.46 => ~ ( member1895684704real_c @ C @ A3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % ComplD
% 0.19/0.46 thf(fact_97_ComplD,axiom,
% 0.19/0.46 ! [C: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 0.19/0.46 ( ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A3 ) )
% 0.19/0.46 => ~ ( member_Sum_sum_c_d @ C @ A3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % ComplD
% 0.19/0.46 thf(fact_98_ComplD,axiom,
% 0.19/0.46 ! [C: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A3 ) )
% 0.19/0.46 => ~ ( member_Sum_sum_c_c @ C @ A3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % ComplD
% 0.19/0.46 thf(fact_99_Pair__le,axiom,
% 0.19/0.46 ! [A: finite1398487019real_c,B: finite1398487019real_c,C: finite1398487019real_c,D: finite1398487019real_c] :
% 0.19/0.46 ( ( ord_le850691415real_c @ ( produc394644079real_c @ A @ B ) @ ( produc394644079real_c @ C @ D ) )
% 0.19/0.46 = ( ( ord_le775706699real_c @ A @ C )
% 0.19/0.46 & ( ord_le775706699real_c @ B @ D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_le
% 0.19/0.46 thf(fact_100_Pair__le,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,D: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le684454199um_c_c @ ( produc1439061071um_c_c @ A @ B ) @ ( produc1439061071um_c_c @ C @ D ) )
% 0.19/0.46 = ( ( ord_le1772180283um_c_c @ A @ C )
% 0.19/0.46 & ( ord_le1772180283um_c_c @ B @ D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_le
% 0.19/0.46 thf(fact_101_Pair__le,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: real,C: set_Sum_sum_c_c,D: real] :
% 0.19/0.46 ( ( ord_le835231240c_real @ ( produc1528874528c_real @ A @ B ) @ ( produc1528874528c_real @ C @ D ) )
% 0.19/0.46 = ( ( ord_le1772180283um_c_c @ A @ C )
% 0.19/0.46 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_le
% 0.19/0.46 thf(fact_102_Pair__le,axiom,
% 0.19/0.46 ! [A: real,B: set_Sum_sum_c_c,C: real,D: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le191210632um_c_c @ ( produc1186709024um_c_c @ A @ B ) @ ( produc1186709024um_c_c @ C @ D ) )
% 0.19/0.46 = ( ( ord_less_eq_real @ A @ C )
% 0.19/0.46 & ( ord_le1772180283um_c_c @ B @ D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_le
% 0.19/0.46 thf(fact_103_Pair__le,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.19/0.46 ( ( ord_le1342644953l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
% 0.19/0.46 = ( ( ord_less_eq_real @ A @ C )
% 0.19/0.46 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_le
% 0.19/0.46 thf(fact_104_order__refl,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X3 @ X3 ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_105_order__refl,axiom,
% 0.19/0.46 ! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_106_Pair__mono,axiom,
% 0.19/0.46 ! [X3: finite1398487019real_c,X5: finite1398487019real_c,Y3: finite1398487019real_c,Y5: finite1398487019real_c] :
% 0.19/0.46 ( ( ord_le775706699real_c @ X3 @ X5 )
% 0.19/0.46 => ( ( ord_le775706699real_c @ Y3 @ Y5 )
% 0.19/0.46 => ( ord_le850691415real_c @ ( produc394644079real_c @ X3 @ Y3 ) @ ( produc394644079real_c @ X5 @ Y5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_mono
% 0.19/0.46 thf(fact_107_Pair__mono,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c,X5: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X3 @ X5 )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ Y3 @ Y5 )
% 0.19/0.46 => ( ord_le684454199um_c_c @ ( produc1439061071um_c_c @ X3 @ Y3 ) @ ( produc1439061071um_c_c @ X5 @ Y5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_mono
% 0.19/0.46 thf(fact_108_Pair__mono,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c,X5: set_Sum_sum_c_c,Y3: real,Y5: real] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X3 @ X5 )
% 0.19/0.46 => ( ( ord_less_eq_real @ Y3 @ Y5 )
% 0.19/0.46 => ( ord_le835231240c_real @ ( produc1528874528c_real @ X3 @ Y3 ) @ ( produc1528874528c_real @ X5 @ Y5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_mono
% 0.19/0.46 thf(fact_109_Pair__mono,axiom,
% 0.19/0.46 ! [X3: real,X5: real,Y3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_less_eq_real @ X3 @ X5 )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ Y3 @ Y5 )
% 0.19/0.46 => ( ord_le191210632um_c_c @ ( produc1186709024um_c_c @ X3 @ Y3 ) @ ( produc1186709024um_c_c @ X5 @ Y5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_mono
% 0.19/0.46 thf(fact_110_Pair__mono,axiom,
% 0.19/0.46 ! [X3: real,X5: real,Y3: real,Y5: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X3 @ X5 )
% 0.19/0.46 => ( ( ord_less_eq_real @ Y3 @ Y5 )
% 0.19/0.46 => ( ord_le1342644953l_real @ ( produc705216881l_real @ X3 @ Y3 ) @ ( produc705216881l_real @ X5 @ Y5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Pair_mono
% 0.19/0.46 thf(fact_111_Fsub,axiom,
% 0.19/0.46 ord_le1772180283um_c_c @ ( static_FVF_a_b_c @ p ) @ v ).
% 0.19/0.46
% 0.19/0.46 % Fsub
% 0.19/0.46 thf(fact_112_solSem,axiom,
% 0.19/0.46 ! [X3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.19/0.46 => ( ( ord_less_eq_real @ X3 @ t )
% 0.19/0.46 => ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ X3 ) ) @ ( denota968303861_a_b_c @ i @ p ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % solSem
% 0.19/0.46 thf(fact_113_relChain__def,axiom,
% 0.19/0.46 ( bNF_Ca1693016812um_c_c
% 0.19/0.46 = ( ^ [R: set_Pr1389752855real_c,As: finite1398487019real_c > set_Sum_sum_c_c] :
% 0.19/0.46 ! [I2: finite1398487019real_c,J: finite1398487019real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ ( produc394644079real_c @ I2 @ J ) @ R )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % relChain_def
% 0.19/0.46 thf(fact_114_relChain__def,axiom,
% 0.19/0.46 ( bNF_Ca449034301c_real
% 0.19/0.46 = ( ^ [R: set_Pr1389752855real_c,As: finite1398487019real_c > real] :
% 0.19/0.46 ! [I2: finite1398487019real_c,J: finite1398487019real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ ( produc394644079real_c @ I2 @ J ) @ R )
% 0.19/0.46 => ( ord_less_eq_real @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % relChain_def
% 0.19/0.46 thf(fact_115_hpsafe__Evolve_OIH,axiom,
% 0.19/0.46 coinci501627771_a_b_c @ p ).
% 0.19/0.46
% 0.19/0.46 % hpsafe_Evolve.IH
% 0.19/0.46 thf(fact_116_t,axiom,
% 0.19/0.46 ord_less_eq_real @ zero_zero_real @ t ).
% 0.19/0.46
% 0.19/0.46 % t
% 0.19/0.46 thf(fact_117_add_Oinverse__neutral,axiom,
% 0.19/0.46 ( ( uminus_uminus_real @ zero_zero_real )
% 0.19/0.46 = zero_zero_real ) ).
% 0.19/0.46
% 0.19/0.46 % add.inverse_neutral
% 0.19/0.46 thf(fact_118_neg__0__equal__iff__equal,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( zero_zero_real
% 0.19/0.46 = ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( zero_zero_real = A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_0_equal_iff_equal
% 0.19/0.46 thf(fact_119_neg__equal__0__iff__equal,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = zero_zero_real )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_0_iff_equal
% 0.19/0.46 thf(fact_120_equal__neg__zero,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % equal_neg_zero
% 0.19/0.46 thf(fact_121_neg__equal__zero,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ( uminus_uminus_real @ A )
% 0.19/0.46 = A )
% 0.19/0.46 = ( A = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_equal_zero
% 0.19/0.46 thf(fact_122_IH_H,axiom,
% 0.19/0.46 ! [A6: finite1398487019real_c,B6: finite1398487019real_c,Aa: finite1398487019real_c,Ba: finite1398487019real_c] :
% 0.19/0.46 ( ( denota1997846518gree_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( produc394644079real_c @ Aa @ Ba ) @ ( static_FVF_a_b_c @ p ) )
% 0.19/0.46 => ( ( member1895684704real_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( denota968303861_a_b_c @ i @ p ) )
% 0.19/0.46 = ( member1895684704real_c @ ( produc394644079real_c @ Aa @ Ba ) @ ( denota968303861_a_b_c @ j @ p ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % IH'
% 0.19/0.46 thf(fact_123_neg__less__eq__nonneg,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.19/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_less_eq_nonneg
% 0.19/0.46 thf(fact_124_less__eq__neg__nonpos,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_eq_neg_nonpos
% 0.19/0.46 thf(fact_125_neg__le__0__iff__le,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.19/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_le_0_iff_le
% 0.19/0.46 thf(fact_126_neg__0__le__iff__le,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % neg_0_le_iff_le
% 0.19/0.46 thf(fact_127_fsafe,axiom,
% 0.19/0.46 fsafe_a_b_c @ p ).
% 0.19/0.46
% 0.19/0.46 % fsafe
% 0.19/0.46 thf(fact_128_zero__prod__def,axiom,
% 0.19/0.46 ( zero_z1506780526real_c
% 0.19/0.46 = ( produc394644079real_c @ zero_z109254132real_c @ zero_z109254132real_c ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_prod_def
% 0.19/0.46 thf(fact_129_zero__prod__def,axiom,
% 0.19/0.46 ( zero_z659284464l_real
% 0.19/0.46 = ( produc705216881l_real @ zero_zero_real @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_prod_def
% 0.19/0.46 thf(fact_130_zero__reorient,axiom,
% 0.19/0.46 ! [X3: real] :
% 0.19/0.46 ( ( zero_zero_real = X3 )
% 0.19/0.46 = ( X3 = zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_reorient
% 0.19/0.46 thf(fact_131_iff__to__impl,axiom,
% 0.19/0.46 ! [Nu: produc190496183real_c,I: denota231621370t_unit,A3: formula_a_b_c,B3: formula_a_b_c] :
% 0.19/0.46 ( ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
% 0.19/0.46 = ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) )
% 0.19/0.46 = ( ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
% 0.19/0.46 => ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) )
% 0.19/0.46 & ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) )
% 0.19/0.46 => ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % iff_to_impl
% 0.19/0.46 thf(fact_132_order__subst1,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ B @ C )
% 0.19/0.46 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X @ Y )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_133_order__subst1,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,F: real > set_Sum_sum_c_c,B: real,C: real] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ! [X: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_134_order__subst1,axiom,
% 0.19/0.46 ! [A: real,F: set_Sum_sum_c_c > real,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ B @ C )
% 0.19/0.46 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_135_order__subst1,axiom,
% 0.19/0.46 ! [A: real,F: real > real,B: real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ! [X: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_136_order__subst2,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A @ B )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X @ Y )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_137_order__subst2,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > real,C: real] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_138_order__subst2,axiom,
% 0.19/0.46 ! [A: real,B: real,F: real > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_139_order__subst2,axiom,
% 0.19/0.46 ! [A: real,B: real,F: real > real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_140_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ B @ C )
% 0.19/0.46 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X @ Y )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_141_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: real,F: set_Sum_sum_c_c > real,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ B @ C )
% 0.19/0.46 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_142_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,F: real > set_Sum_sum_c_c,B: real,C: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ! [X: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_143_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: real,F: real > real,B: real,C: real] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ! [X: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_144_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X @ Y )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_145_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > real,C: real] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_146_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: real,B: real,F: real > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_147_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: real,B: real,F: real > real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X: real,Y: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 0.19/0.46 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_148_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [X4: set_Sum_sum_c_c,Y6: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X4 @ Y6 )
% 0.19/0.46 & ( ord_le1772180283um_c_c @ Y6 @ X4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_149_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [X4: real,Y6: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X4 @ Y6 )
% 0.19/0.46 & ( ord_less_eq_real @ Y6 @ X4 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_150_antisym,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X3 @ Y3 )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ Y3 @ X3 )
% 0.19/0.46 => ( X3 = Y3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_151_antisym,axiom,
% 0.19/0.46 ! [X3: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.19/0.46 => ( ( ord_less_eq_real @ Y3 @ X3 )
% 0.19/0.46 => ( X3 = Y3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_152_linear,axiom,
% 0.19/0.46 ! [X3: real,Y3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.19/0.46 | ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_153_eq__refl,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( X3 = Y3 )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ X3 @ Y3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_154_eq__refl,axiom,
% 0.19/0.46 ! [X3: real,Y3: real] :
% 0.19/0.46 ( ( X3 = Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_155_le__cases,axiom,
% 0.19/0.46 ! [X3: real,Y3: real] :
% 0.19/0.46 ( ~ ( ord_less_eq_real @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_156_order_Otrans,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A @ B )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ B @ C )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_157_order_Otrans,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_158_le__cases3,axiom,
% 0.19/0.46 ! [X3: real,Y3: real,Z2: real] :
% 0.19/0.46 ( ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ Y3 @ Z2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_real @ Y3 @ X3 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ X3 @ Z2 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_real @ X3 @ Z2 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ Z2 @ Y3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_real @ Z2 @ Y3 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ Y3 @ X3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_real @ Y3 @ Z2 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ Z2 @ X3 ) )
% 0.19/0.46 => ~ ( ( ord_less_eq_real @ Z2 @ X3 )
% 0.19/0.46 => ~ ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_159_antisym__conv,axiom,
% 0.19/0.46 ! [Y3: set_Sum_sum_c_c,X3: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ Y3 @ X3 )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ X3 @ Y3 )
% 0.19/0.46 = ( X3 = Y3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_160_antisym__conv,axiom,
% 0.19/0.46 ! [Y3: real,X3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ Y3 @ X3 )
% 0.19/0.46 => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.19/0.46 = ( X3 = Y3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_161_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [A7: set_Sum_sum_c_c,B7: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A7 @ B7 )
% 0.19/0.46 & ( ord_le1772180283um_c_c @ B7 @ A7 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_162_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [A7: real,B7: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A7 @ B7 )
% 0.19/0.46 & ( ord_less_eq_real @ B7 @ A7 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_163_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ B @ C )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_164_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ C )
% 0.19/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_165_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_166_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_167_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ A @ B )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_168_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: real,B: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_real @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_169_order__trans,axiom,
% 0.19/0.46 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c,Z2: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ X3 @ Y3 )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ Y3 @ Z2 )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ X3 @ Z2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_170_order__trans,axiom,
% 0.19/0.46 ! [X3: real,Y3: real,Z2: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.19/0.46 => ( ( ord_less_eq_real @ Y3 @ Z2 )
% 0.19/0.46 => ( ord_less_eq_real @ X3 @ Z2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_171_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_172_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_173_linorder__wlog,axiom,
% 0.19/0.46 ! [P2: real > real > $o,A: real,B: real] :
% 0.19/0.46 ( ! [A4: real,B4: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A4 @ B4 )
% 0.19/0.46 => ( P2 @ A4 @ B4 ) )
% 0.19/0.46 => ( ! [A4: real,B4: real] :
% 0.19/0.46 ( ( P2 @ B4 @ A4 )
% 0.19/0.46 => ( P2 @ A4 @ B4 ) )
% 0.19/0.46 => ( P2 @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_174_dual__order_Otrans,axiom,
% 0.19/0.46 ! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ B @ A )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ C @ B )
% 0.19/0.46 => ( ord_le1772180283um_c_c @ C @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.trans
% 0.19/0.46 thf(fact_175_dual__order_Otrans,axiom,
% 0.19/0.46 ! [B: real,A: real,C: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.19/0.46 => ( ( ord_less_eq_real @ C @ B )
% 0.19/0.46 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.trans
% 0.19/0.46 thf(fact_176_dual__order_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [A7: set_Sum_sum_c_c,B7: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ B7 @ A7 )
% 0.19/0.46 & ( ord_le1772180283um_c_c @ A7 @ B7 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.eq_iff
% 0.19/0.46 thf(fact_177_dual__order_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [A7: real,B7: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ B7 @ A7 )
% 0.19/0.46 & ( ord_less_eq_real @ A7 @ B7 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.eq_iff
% 0.19/0.46 thf(fact_178_dual__order_Oantisym,axiom,
% 0.19/0.46 ! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1772180283um_c_c @ B @ A )
% 0.19/0.46 => ( ( ord_le1772180283um_c_c @ A @ B )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.antisym
% 0.19/0.46 thf(fact_179_dual__order_Oantisym,axiom,
% 0.19/0.46 ! [B: real,A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ B @ A )
% 0.19/0.46 => ( ( ord_less_eq_real @ A @ B )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.antisym
% 0.19/0.46 thf(fact_180_iff__sem,axiom,
% 0.19/0.46 ! [Nu: produc190496183real_c,I: denota231621370t_unit,A3: formula_a_b_c,B3: formula_a_b_c] :
% 0.19/0.46 ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ ( equiv_a_b_c @ A3 @ B3 ) ) )
% 0.19/0.46 = ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
% 0.19/0.46 = ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % iff_sem
% 0.19/0.46 thf(fact_181_IHF,axiom,
% 0.19/0.46 ! [A6: finite1398487019real_c,B6: finite1398487019real_c,Aa: finite1398487019real_c,Ba: finite1398487019real_c] :
% 0.19/0.46 ( ( denota69106024_a_b_c @ i @ j @ ( static_SIGF_a_b_c @ p ) )
% 0.19/0.46 => ( ( denota1997846518gree_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( produc394644079real_c @ Aa @ Ba ) @ ( static_FVF_a_b_c @ p ) )
% 0.19/0.46 => ( ( member1895684704real_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( denota968303861_a_b_c @ i @ p ) )
% 0.19/0.46 = ( member1895684704real_c @ ( produc394644079real_c @ Aa @ Ba ) @ ( denota968303861_a_b_c @ j @ p ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % IHF
% 0.19/0.46 thf(fact_182_minus__le__self__iff,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.19/0.46 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % minus_le_self_iff
% 0.19/0.46 thf(fact_183_le__minus__self__iff,axiom,
% 0.19/0.46 ! [A: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.19/0.46 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_minus_self_iff
% 0.19/0.46 thf(fact_184_real__eq__0__iff__le__ge__0,axiom,
% 0.19/0.46 ! [X3: real] :
% 0.19/0.46 ( ( X3 = zero_zero_real )
% 0.19/0.46 = ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.19/0.46 & ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % real_eq_0_iff_le_ge_0
% 0.19/0.46 thf(fact_185_le__numeral__extra_I3_J,axiom,
% 0.19/0.46 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.19/0.46
% 0.19/0.46 % le_numeral_extra(3)
% 0.19/0.46 thf(fact_186_coincide__fml__def,axiom,
% 0.19/0.46 ( coinci501627771_a_b_c
% 0.19/0.46 = ( ^ [Phi: formula_a_b_c] :
% 0.19/0.46 ! [Nu3: produc190496183real_c,Nu4: produc190496183real_c,I3: denota231621370t_unit,J2: denota231621370t_unit] :
% 0.19/0.46 ( ( denota69106024_a_b_c @ I3 @ J2 @ ( static_SIGF_a_b_c @ Phi ) )
% 0.19/0.46 => ( ( denota1997846518gree_c @ Nu3 @ Nu4 @ ( static_FVF_a_b_c @ Phi ) )
% 0.19/0.46 => ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I3 @ Phi ) )
% 0.19/0.46 = ( member1895684704real_c @ Nu4 @ ( denota968303861_a_b_c @ J2 @ Phi ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % coincide_fml_def
% 0.19/0.46 thf(fact_187_IAP,axiom,
% 0.19/0.46 denota69106024_a_b_c @ i @ j @ ( static_SIGF_a_b_c @ p ) ).
% 0.19/0.46
% 0.19/0.46 % IAP
% 0.19/0.46 thf(fact_188_Iagree__sub,axiom,
% 0.19/0.46 ! [A3: set_Su1783761653um_b_c,B3: set_Su1783761653um_b_c,I: denota231621370t_unit,J3: denota231621370t_unit] :
% 0.19/0.46 ( ( ord_le929608341um_b_c @ A3 @ B3 )
% 0.19/0.46 => ( ( denota69106024_a_b_c @ I @ J3 @ B3 )
% 0.19/0.46 => ( denota69106024_a_b_c @ I @ J3 @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Iagree_sub
% 0.19/0.46 thf(fact_189_Iagree__comm,axiom,
% 0.19/0.46 ! [A3: denota231621370t_unit,B3: denota231621370t_unit,V: set_Su1783761653um_b_c] :
% 0.19/0.46 ( ( denota69106024_a_b_c @ A3 @ B3 @ V )
% 0.19/0.46 => ( denota69106024_a_b_c @ B3 @ A3 @ V ) ) ).
% 0.19/0.46
% 0.19/0.46 % Iagree_comm
% 0.19/0.46 thf(fact_190_Iagree__refl,axiom,
% 0.19/0.46 ! [I: denota231621370t_unit,A3: set_Su1783761653um_b_c] : ( denota69106024_a_b_c @ I @ I @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Iagree_refl
% 0.19/0.46 thf(fact_191__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AJ_092_060close_062,axiom,
% 0.19/0.46 coinci772602909_a_b_c @ ( evolveODE_a_c_b @ ode @ p ) @ j ).
% 0.19/0.46
% 0.19/0.46 % \<open>ode_sem_equiv (EvolveODE ODE P) J\<close>
% 0.19/0.46 thf(fact_192__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AI_092_060close_062,axiom,
% 0.19/0.46 coinci772602909_a_b_c @ ( evolveODE_a_c_b @ ode @ p ) @ i ).
% 0.19/0.46
% 0.19/0.46 % \<open>ode_sem_equiv (EvolveODE ODE P) I\<close>
% 0.19/0.46 thf(fact_193_impl__sem,axiom,
% 0.19/0.46 ! [Nu: produc190496183real_c,I: denota231621370t_unit,A3: formula_a_b_c,B3: formula_a_b_c] :
% 0.19/0.46 ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ ( implies_a_b_c @ A3 @ B3 ) ) )
% 0.19/0.46 = ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
% 0.19/0.46 => ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % impl_sem
% 0.19/0.46 thf(fact_194_equiv,axiom,
% 0.19/0.46 ! [I: denota231621370t_unit] : ( coinci772602909_a_b_c @ ( evolveODE_a_c_b @ ode @ p ) @ I ) ).
% 0.19/0.46
% 0.19/0.46 % equiv
% 0.19/0.46 thf(fact_195_coincide__hp_H__def,axiom,
% 0.19/0.46 ( coinci1852866155_a_b_c
% 0.19/0.46 = ( ^ [Alpha: hp_a_b_c] :
% 0.19/0.46 ! [I3: denota231621370t_unit,J2: denota231621370t_unit] :
% 0.19/0.46 ( ( coinci354700372_a_b_c @ Alpha @ I3 @ J2 )
% 0.19/0.46 & ( coinci772602909_a_b_c @ Alpha @ I3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % coincide_hp'_def
% 0.19/0.46 thf(fact_196_hp_Oinject_I5_J,axiom,
% 0.19/0.46 ! [X51: oDE_a_c,X52: formula_a_b_c,Y51: oDE_a_c,Y52: formula_a_b_c] :
% 0.19/0.46 ( ( ( evolveODE_a_c_b @ X51 @ X52 )
% 0.19/0.46 = ( evolveODE_a_c_b @ Y51 @ Y52 ) )
% 0.19/0.46 = ( ( X51 = Y51 )
% 0.19/0.46 & ( X52 = Y52 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % hp.inject(5)
% 0.19/0.46 thf(fact_197_sol,axiom,
% 0.19/0.46 ( initia1631504802real_c @ sol
% 0.19/0.46 @ ^ [A7: real] : ( denota1275485728_a_b_c @ i @ ode )
% 0.19/0.46 @ ( set_or656347191t_real @ zero_zero_real @ t )
% 0.19/0.46 @ ( collec230941376real_c
% 0.19/0.46 @ ^ [X4: finite1398487019real_c] : ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ X4 ) @ ( denota968303861_a_b_c @ i @ p ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % sol
% 0.19/0.46 thf(fact_198_less__eq__set__def,axiom,
% 0.19/0.46 ( ord_le977353143real_c
% 0.19/0.46 = ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
% 0.19/0.46 ( ord_le1132241894al_c_o
% 0.19/0.46 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A5 )
% 0.19/0.46 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ B5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_eq_set_def
% 0.19/0.46 thf(fact_199_less__eq__set__def,axiom,
% 0.19/0.46 ( ord_le764260924um_c_d
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
% 0.19/0.46 ( ord_le2143422793_c_d_o
% 0.19/0.46 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ A5 )
% 0.19/0.46 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ B5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_eq_set_def
% 0.19/0.46 thf(fact_200_less__eq__set__def,axiom,
% 0.19/0.46 ( ord_le1772180283um_c_c
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
% 0.19/0.46 ( ord_le1315653386_c_c_o
% 0.19/0.46 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ A5 )
% 0.19/0.46 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ B5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_eq_set_def
% 0.19/0.46 thf(fact_201_Collect__subset,axiom,
% 0.19/0.46 ! [A3: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
% 0.19/0.46 ( ord_le977353143real_c
% 0.19/0.46 @ ( collec1643251106real_c
% 0.19/0.46 @ ^ [X4: produc190496183real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ X4 @ A3 )
% 0.19/0.46 & ( P2 @ X4 ) ) )
% 0.19/0.46 @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_subset
% 0.19/0.46 thf(fact_202_Collect__subset,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 0.19/0.46 ( ord_le764260924um_c_d
% 0.19/0.46 @ ( collect_Sum_sum_c_d
% 0.19/0.46 @ ^ [X4: sum_sum_c_d] :
% 0.19/0.46 ( ( member_Sum_sum_c_d @ X4 @ A3 )
% 0.19/0.46 & ( P2 @ X4 ) ) )
% 0.19/0.46 @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_subset
% 0.19/0.46 thf(fact_203_Collect__subset,axiom,
% 0.19/0.46 ! [A3: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
% 0.19/0.46 ( ord_le1327118209real_c
% 0.19/0.46 @ ( collec230941376real_c
% 0.19/0.46 @ ^ [X4: finite1398487019real_c] :
% 0.19/0.46 ( ( member1261661570real_c @ X4 @ A3 )
% 0.19/0.46 & ( P2 @ X4 ) ) )
% 0.19/0.46 @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_subset
% 0.19/0.46 thf(fact_204_Collect__subset,axiom,
% 0.19/0.46 ! [A3: set_c,P2: c > $o] :
% 0.19/0.46 ( ord_less_eq_set_c
% 0.19/0.46 @ ( collect_c
% 0.19/0.46 @ ^ [X4: c] :
% 0.19/0.46 ( ( member_c @ X4 @ A3 )
% 0.19/0.46 & ( P2 @ X4 ) ) )
% 0.19/0.46 @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_subset
% 0.19/0.46 thf(fact_205_Collect__subset,axiom,
% 0.19/0.46 ! [A3: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 0.19/0.46 ( ord_le1772180283um_c_c
% 0.19/0.46 @ ( collect_Sum_sum_c_c
% 0.19/0.46 @ ^ [X4: sum_sum_c_c] :
% 0.19/0.46 ( ( member_Sum_sum_c_c @ X4 @ A3 )
% 0.19/0.46 & ( P2 @ X4 ) ) )
% 0.19/0.46 @ A3 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_subset
% 0.19/0.46 thf(fact_206_Compl__eq,axiom,
% 0.19/0.46 ( uminus232257166real_c
% 0.19/0.46 = ( ^ [A5: set_Pr1389752855real_c] :
% 0.19/0.46 ( collec1643251106real_c
% 0.19/0.46 @ ^ [X4: produc190496183real_c] :
% 0.19/0.46 ~ ( member1895684704real_c @ X4 @ A5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_eq
% 0.19/0.46 thf(fact_207_Compl__eq,axiom,
% 0.19/0.46 ( uminus373867045um_c_d
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_d] :
% 0.19/0.46 ( collect_Sum_sum_c_d
% 0.19/0.46 @ ^ [X4: sum_sum_c_d] :
% 0.19/0.46 ~ ( member_Sum_sum_c_d @ X4 @ A5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_eq
% 0.19/0.46 thf(fact_208_Compl__eq,axiom,
% 0.19/0.46 ( uminus102402154real_c
% 0.19/0.46 = ( ^ [A5: set_Fi1407883041real_c] :
% 0.19/0.46 ( collec230941376real_c
% 0.19/0.46 @ ^ [X4: finite1398487019real_c] :
% 0.19/0.46 ~ ( member1261661570real_c @ X4 @ A5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_eq
% 0.19/0.46 thf(fact_209_Compl__eq,axiom,
% 0.19/0.46 ( uminus_uminus_set_c
% 0.19/0.46 = ( ^ [A5: set_c] :
% 0.19/0.46 ( collect_c
% 0.19/0.46 @ ^ [X4: c] :
% 0.19/0.46 ~ ( member_c @ X4 @ A5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_eq
% 0.19/0.46 thf(fact_210_Compl__eq,axiom,
% 0.19/0.46 ( uminus1381786404um_c_c
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_c] :
% 0.19/0.46 ( collect_Sum_sum_c_c
% 0.19/0.46 @ ^ [X4: sum_sum_c_c] :
% 0.19/0.46 ~ ( member_Sum_sum_c_c @ X4 @ A5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Compl_eq
% 0.19/0.46 thf(fact_211_Collect__neg__eq,axiom,
% 0.19/0.46 ! [P2: finite1398487019real_c > $o] :
% 0.19/0.46 ( ( collec230941376real_c
% 0.19/0.46 @ ^ [X4: finite1398487019real_c] :
% 0.19/0.46 ~ ( P2 @ X4 ) )
% 0.19/0.46 = ( uminus102402154real_c @ ( collec230941376real_c @ P2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_neg_eq
% 0.19/0.46 thf(fact_212_Collect__neg__eq,axiom,
% 0.19/0.46 ! [P2: c > $o] :
% 0.19/0.46 ( ( collect_c
% 0.19/0.46 @ ^ [X4: c] :
% 0.19/0.46 ~ ( P2 @ X4 ) )
% 0.19/0.46 = ( uminus_uminus_set_c @ ( collect_c @ P2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_neg_eq
% 0.19/0.46 thf(fact_213_Collect__neg__eq,axiom,
% 0.19/0.46 ! [P2: sum_sum_c_c > $o] :
% 0.19/0.46 ( ( collect_Sum_sum_c_c
% 0.19/0.46 @ ^ [X4: sum_sum_c_c] :
% 0.19/0.46 ~ ( P2 @ X4 ) )
% 0.19/0.46 = ( uminus1381786404um_c_c @ ( collect_Sum_sum_c_c @ P2 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_neg_eq
% 0.19/0.46 thf(fact_214_uminus__set__def,axiom,
% 0.19/0.46 ( uminus232257166real_c
% 0.19/0.46 = ( ^ [A5: set_Pr1389752855real_c] :
% 0.19/0.46 ( collec1643251106real_c
% 0.19/0.46 @ ( uminus32155087al_c_o
% 0.19/0.46 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_set_def
% 0.19/0.46 thf(fact_215_uminus__set__def,axiom,
% 0.19/0.46 ( uminus373867045um_c_d
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_d] :
% 0.19/0.46 ( collect_Sum_sum_c_d
% 0.19/0.46 @ ( uminus916499104_c_d_o
% 0.19/0.46 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ A5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_set_def
% 0.19/0.46 thf(fact_216_uminus__set__def,axiom,
% 0.19/0.46 ( uminus102402154real_c
% 0.19/0.46 = ( ^ [A5: set_Fi1407883041real_c] :
% 0.19/0.46 ( collec230941376real_c
% 0.19/0.46 @ ( uminus23580315al_c_o
% 0.19/0.46 @ ^ [X4: finite1398487019real_c] : ( member1261661570real_c @ X4 @ A5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_set_def
% 0.19/0.46 thf(fact_217_uminus__set__def,axiom,
% 0.19/0.46 ( uminus_uminus_set_c
% 0.19/0.46 = ( ^ [A5: set_c] :
% 0.19/0.46 ( collect_c
% 0.19/0.46 @ ( uminus_uminus_c_o
% 0.19/0.46 @ ^ [X4: c] : ( member_c @ X4 @ A5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_set_def
% 0.19/0.46 thf(fact_218_uminus__set__def,axiom,
% 0.19/0.46 ( uminus1381786404um_c_c
% 0.19/0.46 = ( ^ [A5: set_Sum_sum_c_c] :
% 0.19/0.46 ( collect_Sum_sum_c_c
% 0.19/0.46 @ ( uminus88729697_c_c_o
% 0.19/0.46 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ A5 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % uminus_set_def
% 0.19/0.46 thf(fact_219_mk__xode_Oelims,axiom,
% 0.19/0.46 ! [X3: denota231621370t_unit,Xa: oDE_a_c,Xb: finite1398487019real_c,Y3: produc190496183real_c] :
% 0.19/0.46 ( ( ( denota1896987029_a_b_c @ X3 @ Xa @ Xb )
% 0.19/0.46 = Y3 )
% 0.19/0.46 => ( Y3
% 0.19/0.46 = ( produc394644079real_c @ Xb @ ( denota1275485728_a_b_c @ X3 @ Xa @ Xb ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mk_xode.elims
% 0.19/0.46 thf(fact_220_mk__xode_Osimps,axiom,
% 0.19/0.46 ( denota1896987029_a_b_c
% 0.19/0.46 = ( ^ [I3: denota231621370t_unit,ODE2: oDE_a_c,Sol2: finite1398487019real_c] : ( produc394644079real_c @ Sol2 @ ( denota1275485728_a_b_c @ I3 @ ODE2 @ Sol2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mk_xode.simps
% 0.19/0.46 thf(fact_221_atLeastatMost__subset__iff,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,D: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le657554289um_c_c @ ( set_or935154662um_c_c @ A @ B ) @ ( set_or935154662um_c_c @ C @ D ) )
% 0.19/0.46 = ( ~ ( ord_le1772180283um_c_c @ A @ B )
% 0.19/0.46 | ( ( ord_le1772180283um_c_c @ C @ A )
% 0.19/0.46 & ( ord_le1772180283um_c_c @ B @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % atLeastatMost_subset_iff
% 0.19/0.46 thf(fact_222_atLeastatMost__subset__iff,axiom,
% 0.19/0.46 ! [A: real,B: real,C: real,D: real] :
% 0.19/0.46 ( ( ord_less_eq_set_real @ ( set_or656347191t_real @ A @ B ) @ ( set_or656347191t_real @ C @ D ) )
% 0.19/0.46 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.19/0.46 | ( ( ord_less_eq_real @ C @ A )
% 0.19/0.46 & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % atLeastatMost_subset_iff
% 0.19/0.46 thf(fact_223_atLeastAtMost__iff,axiom,
% 0.19/0.46 ! [I4: produc190496183real_c,L: produc190496183real_c,U: produc190496183real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ I4 @ ( set_or385782508real_c @ L @ U ) )
% 0.19/0.46 = ( ( ord_le850691415real_c @ L @ I4 )
% 0.19/0.46 & ( ord_le850691415real_c @ I4 @ U ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % atLeastAtMost_iff
% 0.19/0.46 thf(fact_224_atLeastAtMost__iff,axiom,
% 0.19/0.46 ! [I4: set_Sum_sum_c_c,L: set_Sum_sum_c_c,U: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( member2124950898um_c_c @ I4 @ ( set_or935154662um_c_c @ L @ U ) )
% 0.19/0.46 = ( ( ord_le1772180283um_c_c @ L @ I4 )
% 0.19/0.46 & ( ord_le1772180283um_c_c @ I4 @ U ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % atLeastAtMost_iff
% 0.19/0.46 thf(fact_225_atLeastAtMost__iff,axiom,
% 0.19/0.46 ! [I4: real,L: real,U: real] :
% 0.19/0.46 ( ( member_real @ I4 @ ( set_or656347191t_real @ L @ U ) )
% 0.19/0.46 = ( ( ord_less_eq_real @ L @ I4 )
% 0.19/0.46 & ( ord_less_eq_real @ I4 @ U ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % atLeastAtMost_iff
% 0.19/0.46 thf(fact_226_Icc__eq__Icc,axiom,
% 0.19/0.46 ! [L: set_Sum_sum_c_c,H: set_Sum_sum_c_c,L2: set_Sum_sum_c_c,H2: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ( set_or935154662um_c_c @ L @ H )
% 0.19/0.46 = ( set_or935154662um_c_c @ L2 @ H2 ) )
% 0.19/0.46 = ( ( ( L = L2 )
% 0.19/0.46 & ( H = H2 ) )
% 0.19/0.46 | ( ~ ( ord_le1772180283um_c_c @ L @ H )
% 0.19/0.46 & ~ ( ord_le1772180283um_c_c @ L2 @ H2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Icc_eq_Icc
% 0.19/0.46 thf(fact_227_Icc__eq__Icc,axiom,
% 0.19/0.46 ! [L: real,H: real,L2: real,H2: real] :
% 0.19/0.46 ( ( ( set_or656347191t_real @ L @ H )
% 0.19/0.46 = ( set_or656347191t_real @ L2 @ H2 ) )
% 0.19/0.46 = ( ( ( L = L2 )
% 0.19/0.46 & ( H = H2 ) )
% 0.19/0.46 | ( ~ ( ord_less_eq_real @ L @ H )
% 0.19/0.46 & ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Icc_eq_Icc
% 0.19/0.46 thf(fact_228_pred__subset__eq,axiom,
% 0.19/0.46 ! [R2: set_Pr1389752855real_c,S2: set_Pr1389752855real_c] :
% 0.19/0.46 ( ( ord_le1132241894al_c_o
% 0.19/0.46 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ R2 )
% 0.19/0.46 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ S2 ) )
% 0.19/0.46 = ( ord_le977353143real_c @ R2 @ S2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % pred_subset_eq
% 0.19/0.46 thf(fact_229_pred__subset__eq,axiom,
% 0.19/0.46 ! [R2: set_Sum_sum_c_d,S2: set_Sum_sum_c_d] :
% 0.19/0.46 ( ( ord_le2143422793_c_d_o
% 0.19/0.46 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ R2 )
% 0.19/0.46 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ S2 ) )
% 0.19/0.46 = ( ord_le764260924um_c_d @ R2 @ S2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % pred_subset_eq
% 0.19/0.46 thf(fact_230_pred__subset__eq,axiom,
% 0.19/0.46 ! [R2: set_Sum_sum_c_c,S2: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ord_le1315653386_c_c_o
% 0.19/0.46 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ R2 )
% 0.19/0.46 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ S2 ) )
% 0.19/0.46 = ( ord_le1772180283um_c_c @ R2 @ S2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % pred_subset_eq
% 0.19/0.46 thf(fact_231_solDeriv,axiom,
% 0.19/0.46 ! [X3: real] :
% 0.19/0.46 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.19/0.46 => ( ( ord_less_eq_real @ X3 @ t )
% 0.19/0.46 => ( has_ve1904308670real_c @ sol @ ( denota1275485728_a_b_c @ i @ ode @ ( sol @ X3 ) ) @ ( topolo1830106092n_real @ X3 @ ( set_or656347191t_real @ zero_zero_real @ t ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % solDeriv
% 0.19/0.46 thf(fact_232_mk__v__def,axiom,
% 0.19/0.46 ( denota161327353_a_b_c
% 0.19/0.46 = ( ^ [I3: denota231621370t_unit,ODE2: oDE_a_c,Nu3: produc190496183real_c,Sol2: finite1398487019real_c] :
% 0.19/0.46 ( the_Pr1290640446real_c
% 0.19/0.46 @ ^ [Omega3: produc190496183real_c] :
% 0.19/0.46 ( ( denota1997846518gree_c @ Omega3 @ Nu3 @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I3 @ ODE2 ) ) )
% 0.19/0.46 & ( denota1997846518gree_c @ Omega3 @ ( denota1896987029_a_b_c @ I3 @ ODE2 @ Sol2 ) @ ( denota1293057250_a_b_c @ I3 @ ODE2 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mk_v_def
% 0.19/0.46 thf(fact_233_subrelI,axiom,
% 0.19/0.46 ! [R3: set_Pr1389752855real_c,S: set_Pr1389752855real_c] :
% 0.19/0.46 ( ! [X: finite1398487019real_c,Y: finite1398487019real_c] :
% 0.19/0.46 ( ( member1895684704real_c @ ( produc394644079real_c @ X @ Y ) @ R3 )
% 0.19/0.46 => ( member1895684704real_c @ ( produc394644079real_c @ X @ Y ) @ S ) )
% 0.19/0.46 => ( ord_le977353143real_c @ R3 @ S ) ) ).
% 0.19/0.46
% 0.19/0.46 % subrelI
% 0.19/0.46 thf(fact_234_pred__subset__eq2,axiom,
% 0.19/0.46 ! [R2: set_Pr1389752855real_c,S2: set_Pr1389752855real_c] :
% 0.19/0.46 ( ( ord_le1382916304al_c_o
% 0.19/0.46 @ ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ R2 )
% 0.19/0.46 @ ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ S2 ) )
% 0.19/0.46 = ( ord_le977353143real_c @ R2 @ S2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % pred_subset_eq2
% 0.19/0.46 thf(fact_235_pred__equals__eq2,axiom,
% 0.19/0.46 ! [R2: set_Pr1389752855real_c,S2: set_Pr1389752855real_c] :
% 0.19/0.46 ( ( ( ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ R2 ) )
% 0.19/0.46 = ( ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ S2 ) ) )
% 0.19/0.46 = ( R2 = S2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % pred_equals_eq2
% 0.19/0.46 thf(fact_236_has__vector__derivative__Pair,axiom,
% 0.19/0.46 ! [F: real > finite1398487019real_c,F2: finite1398487019real_c,X3: real,S: set_real,G: real > finite1398487019real_c,G2: finite1398487019real_c] :
% 0.19/0.46 ( ( has_ve1904308670real_c @ F @ F2 @ ( topolo1830106092n_real @ X3 @ S ) )
% 0.19/0.46 => ( ( has_ve1904308670real_c @ G @ G2 @ ( topolo1830106092n_real @ X3 @ S ) )
% 0.19/0.46 => ( has_ve373335844real_c
% 0.19/0.46 @ ^ [X4: real] : ( produc394644079real_c @ ( F @ X4 ) @ ( G @ X4 ) )
% 0.19/0.46 @ ( produc394644079real_c @ F2 @ G2 )
% 0.19/0.46 @ ( topolo1830106092n_real @ X3 @ S ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % has_vector_derivative_Pair
% 0.19/0.46 thf(fact_237_has__vector__derivative__minus,axiom,
% 0.19/0.46 ! [F: real > real,F2: real,Net: filter_real] :
% 0.19/0.46 ( ( has_ve217091583e_real @ F @ F2 @ Net )
% 0.19/0.46 => ( has_ve217091583e_real
% 0.19/0.46 @ ^ [X4: real] : ( uminus_uminus_real @ ( F @ X4 ) )
% 0.19/0.46 @ ( uminus_uminus_real @ F2 )
% 0.19/0.46 @ Net ) ) ).
% 0.19/0.46
% 0.19/0.46 % has_vector_derivative_minus
% 0.19/0.46 thf(fact_238_has__vector__derivative__minus,axiom,
% 0.19/0.46 ! [F: real > finite1398487019real_c,F2: finite1398487019real_c,Net: filter_real] :
% 0.19/0.46 ( ( has_ve1904308670real_c @ F @ F2 @ Net )
% 0.19/0.46 => ( has_ve1904308670real_c
% 0.19/0.46 @ ^ [X4: real] : ( uminus1737280820real_c @ ( F @ X4 ) )
% 0.19/0.46 @ ( uminus1737280820real_c @ F2 )
% 0.19/0.46 @ Net ) ) ).
% 0.19/0.46
% 0.19/0.46 % has_vector_derivative_minus
% 0.19/0.46 thf(fact_239_has__vector__derivative__const,axiom,
% 0.19/0.46 ! [C: real,Net: filter_real] :
% 0.19/0.46 ( has_ve217091583e_real
% 0.19/0.46 @ ^ [X4: real] : C
% 0.19/0.46 @ zero_zero_real
% 0.19/0.46 @ Net ) ).
% 0.19/0.46
% 0.19/0.46 % has_vector_derivative_const
% 0.19/0.46 thf(fact_240_has__vector__derivative__const,axiom,
% 0.19/0.46 ! [C: finite1398487019real_c,Net: filter_real] :
% 0.19/0.46 ( has_ve1904308670real_c
% 0.19/0.46 @ ^ [X4: real] : C
% 0.19/0.46 @ zero_z109254132real_c
% 0.19/0.46 @ Net ) ).
% 0.19/0.46
% 0.19/0.46 % has_vector_derivative_const
% 0.19/0.46 thf(fact_241_has__vector__derivative__within__subset,axiom,
% 0.19/0.46 ! [F: real > finite1398487019real_c,F2: finite1398487019real_c,X3: real,S2: set_real,T2: set_real] :
% 0.19/0.46 ( ( has_ve1904308670real_c @ F @ F2 @ ( topolo1830106092n_real @ X3 @ S2 ) )
% 0.19/0.46 => ( ( ord_less_eq_set_real @ T2 @ S2 )
% 0.19/0.46 => ( has_ve1904308670real_c @ F @ F2 @ ( topolo1830106092n_real @ X3 @ T2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % has_vector_derivative_within_subset
% 0.19/0.46 thf(fact_242_has__vector__derivative__weaken,axiom,
% 0.19/0.46 ! [F: real > finite1398487019real_c,D2: finite1398487019real_c,X3: real,T2: set_real,S2: set_real,G: real > finite1398487019real_c] :
% 0.19/0.46 ( ( has_ve1904308670real_c @ F @ D2 @ ( topolo1830106092n_real @ X3 @ T2 ) )
% 0.19/0.46 => ( ( member_real @ X3 @ S2 )
% 0.19/0.46 => ( ( ord_less_eq_set_real @ S2 @ T2 )
% 0.19/0.46 => ( ! [X: real] :
% 0.19/0.46 ( ( member_real @ X @ S2 )
% 0.19/0.46 => ( ( F @ X )
% 0.19/0.46 = ( G @ X ) ) )
% 0.19/0.46 => ( has_ve1904308670real_c @ G @ D2 @ ( topolo1830106092n_real @ X3 @ S2 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % has_vector_derivative_weaken
% 0.19/0.46 thf(fact_243_solves__ode__supset__range,axiom,
% 0.19/0.46 ! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,T2: set_real,X6: set_Fi1407883041real_c,Y7: set_Fi1407883041real_c] :
% 0.19/0.46 ( ( initia1631504802real_c @ X3 @ F @ T2 @ X6 )
% 0.19/0.46 => ( ( ord_le1327118209real_c @ X6 @ Y7 )
% 0.19/0.46 => ( initia1631504802real_c @ X3 @ F @ T2 @ Y7 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % solves_ode_supset_range
% 0.19/0.46 thf(fact_244_solves__ode__subset,axiom,
% 0.19/0.46 ! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,T2: set_real,X6: set_Fi1407883041real_c,S2: set_real] :
% 0.19/0.46 ( ( initia1631504802real_c @ X3 @ F @ T2 @ X6 )
% 0.19/0.46 => ( ( ord_less_eq_set_real @ S2 @ T2 )
% 0.19/0.46 => ( initia1631504802real_c @ X3 @ F @ S2 @ X6 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % solves_ode_subset
% 0.19/0.46 thf(fact_245_solves__ode__on__subset,axiom,
% 0.19/0.46 ! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,S2: set_real,Y7: set_Fi1407883041real_c,T2: set_real,X6: set_Fi1407883041real_c] :
% 0.19/0.46 ( ( initia1631504802real_c @ X3 @ F @ S2 @ Y7 )
% 0.19/0.46 => ( ( ord_less_eq_set_real @ T2 @ S2 )
% 0.19/0.46 => ( ( ord_le1327118209real_c @ Y7 @ X6 )
% 0.19/0.46 => ( initia1631504802real_c @ X3 @ F @ T2 @ X6 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % solves_ode_on_subset
% 0.19/0.46 thf(fact_246_at__le,axiom,
% 0.19/0.46 ! [S: set_real,T3: set_real,X3: real] :
% 0.19/0.46 ( ( ord_less_eq_set_real @ S @ T3 )
% 0.19/0.46 => ( ord_le132810396r_real @ ( topolo1830106092n_real @ X3 @ S ) @ ( topolo1830106092n_real @ X3 @ T3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % at_le
% 0.19/0.46 thf(fact_247_intervalE,axiom,
% 0.19/0.46 ! [A: produc190496183real_c,X3: produc190496183real_c,B: produc190496183real_c] :
% 0.19/0.46 ( ( ( ord_le850691415real_c @ A @ X3 )
% 0.19/0.46 & ( ord_le850691415real_c @ X3 @ B ) )
% 0.19/0.46 => ( member1895684704real_c @ X3 @ ( set_or385782508real_c @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % intervalE
% 0.19/0.46 thf(fact_248_intervalE,axiom,
% 0.19/0.46 ! [A: set_Sum_sum_c_c,X3: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
% 0.19/0.46 ( ( ( ord_le1772180283um_c_c @ A @ X3 )
% 0.19/0.46 & ( ord_le1772180283um_c_c @ X3 @ B ) )
% 0.19/0.46 => ( member2124950898um_c_c @ X3 @ ( set_or935154662um_c_c @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % intervalE
% 0.19/0.46 thf(fact_249_intervalE,axiom,
% 0.19/0.46 ! [A: real,X3: real,B: real] :
% 0.19/0.46 ( ( ( ord_less_eq_real @ A @ X3 )
% 0.19/0.46 & ( ord_less_eq_real @ X3 @ B ) )
% 0.19/0.46 => ( member_real @ X3 @ ( set_or656347191t_real @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % intervalE
% 0.19/0.46 thf(fact_250_conj__subset__def,axiom,
% 0.19/0.46 ! [A3: set_Fi1407883041real_c,P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
% 0.19/0.46 ( ( ord_le1327118209real_c @ A3
% 0.19/0.46 @ ( collec230941376real_c
% 0.19/0.46 @ ^ [X4: finite1398487019real_c] :
% 0.19/0.46 ( ( P2 @ X4 )
% 0.19/0.46 & ( Q @ X4 ) ) ) )
% 0.19/0.46 = ( ( ord_le1327118209real_c @ A3 @ ( collec230941376real_c @ P2 ) )
% 0.19/0.46 & ( ord_le1327118209real_c @ A3 @ ( collec230941376real_c @ Q ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % conj_subset_def
% 0.19/0.46 thf(fact_251_conj__subset__def,axiom,
% 0.19/0.46 ! [A3: set_c,P2: c > $o,Q: c > $o] :
% 0.19/0.46 ( ( ord_less_eq_set_c @ A3
% 0.19/0.46 @ ( collect_c
% 0.19/0.46 @ ^ [X4: c] :
% 0.19/0.46 ( ( P2 @ X4 )
% 0.19/0.46 & ( Q @ X4 ) ) ) )
% 0.19/0.46 = ( ( ord_less_eq_set_c @ A3 @ ( collect_c @ P2 ) )
% 0.19/0.46 & ( ord_less_eq_set_c @ A3 @ ( collect_c @ Q ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % conj_subset_def
% 0.19/0.47 thf(fact_252_conj__subset__def,axiom,
% 0.19/0.47 ! [A3: set_Sum_sum_c_c,P2: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ A3
% 0.19/0.47 @ ( collect_Sum_sum_c_c
% 0.19/0.47 @ ^ [X4: sum_sum_c_c] :
% 0.19/0.47 ( ( P2 @ X4 )
% 0.19/0.47 & ( Q @ X4 ) ) ) )
% 0.19/0.47 = ( ( ord_le1772180283um_c_c @ A3 @ ( collect_Sum_sum_c_c @ P2 ) )
% 0.19/0.47 & ( ord_le1772180283um_c_c @ A3 @ ( collect_Sum_sum_c_c @ Q ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % conj_subset_def
% 0.19/0.47 thf(fact_253_subset__Collect__iff,axiom,
% 0.19/0.47 ! [B3: set_Pr1389752855real_c,A3: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
% 0.19/0.47 ( ( ord_le977353143real_c @ B3 @ A3 )
% 0.19/0.47 => ( ( ord_le977353143real_c @ B3
% 0.19/0.47 @ ( collec1643251106real_c
% 0.19/0.47 @ ^ [X4: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 = ( ! [X4: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ X4 @ B3 )
% 0.19/0.47 => ( P2 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_Collect_iff
% 0.19/0.47 thf(fact_254_subset__Collect__iff,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_d,A3: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 0.19/0.47 ( ( ord_le764260924um_c_d @ B3 @ A3 )
% 0.19/0.47 => ( ( ord_le764260924um_c_d @ B3
% 0.19/0.47 @ ( collect_Sum_sum_c_d
% 0.19/0.47 @ ^ [X4: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 = ( ! [X4: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X4 @ B3 )
% 0.19/0.47 => ( P2 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_Collect_iff
% 0.19/0.47 thf(fact_255_subset__Collect__iff,axiom,
% 0.19/0.47 ! [B3: set_Fi1407883041real_c,A3: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
% 0.19/0.47 ( ( ord_le1327118209real_c @ B3 @ A3 )
% 0.19/0.47 => ( ( ord_le1327118209real_c @ B3
% 0.19/0.47 @ ( collec230941376real_c
% 0.19/0.47 @ ^ [X4: finite1398487019real_c] :
% 0.19/0.47 ( ( member1261661570real_c @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 = ( ! [X4: finite1398487019real_c] :
% 0.19/0.47 ( ( member1261661570real_c @ X4 @ B3 )
% 0.19/0.47 => ( P2 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_Collect_iff
% 0.19/0.47 thf(fact_256_subset__Collect__iff,axiom,
% 0.19/0.47 ! [B3: set_c,A3: set_c,P2: c > $o] :
% 0.19/0.47 ( ( ord_less_eq_set_c @ B3 @ A3 )
% 0.19/0.47 => ( ( ord_less_eq_set_c @ B3
% 0.19/0.47 @ ( collect_c
% 0.19/0.47 @ ^ [X4: c] :
% 0.19/0.47 ( ( member_c @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 = ( ! [X4: c] :
% 0.19/0.47 ( ( member_c @ X4 @ B3 )
% 0.19/0.47 => ( P2 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_Collect_iff
% 0.19/0.47 thf(fact_257_subset__Collect__iff,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_c,A3: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ B3 @ A3 )
% 0.19/0.47 => ( ( ord_le1772180283um_c_c @ B3
% 0.19/0.47 @ ( collect_Sum_sum_c_c
% 0.19/0.47 @ ^ [X4: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 = ( ! [X4: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X4 @ B3 )
% 0.19/0.47 => ( P2 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_Collect_iff
% 0.19/0.47 thf(fact_258_subset__CollectI,axiom,
% 0.19/0.47 ! [B3: set_Pr1389752855real_c,A3: set_Pr1389752855real_c,Q: produc190496183real_c > $o,P2: produc190496183real_c > $o] :
% 0.19/0.47 ( ( ord_le977353143real_c @ B3 @ A3 )
% 0.19/0.47 => ( ! [X: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ X @ B3 )
% 0.19/0.47 => ( ( Q @ X )
% 0.19/0.47 => ( P2 @ X ) ) )
% 0.19/0.47 => ( ord_le977353143real_c
% 0.19/0.47 @ ( collec1643251106real_c
% 0.19/0.47 @ ^ [X4: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ X4 @ B3 )
% 0.19/0.47 & ( Q @ X4 ) ) )
% 0.19/0.47 @ ( collec1643251106real_c
% 0.19/0.47 @ ^ [X4: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_CollectI
% 0.19/0.47 thf(fact_259_subset__CollectI,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_d,A3: set_Sum_sum_c_d,Q: sum_sum_c_d > $o,P2: sum_sum_c_d > $o] :
% 0.19/0.47 ( ( ord_le764260924um_c_d @ B3 @ A3 )
% 0.19/0.47 => ( ! [X: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X @ B3 )
% 0.19/0.47 => ( ( Q @ X )
% 0.19/0.47 => ( P2 @ X ) ) )
% 0.19/0.47 => ( ord_le764260924um_c_d
% 0.19/0.47 @ ( collect_Sum_sum_c_d
% 0.19/0.47 @ ^ [X4: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X4 @ B3 )
% 0.19/0.47 & ( Q @ X4 ) ) )
% 0.19/0.47 @ ( collect_Sum_sum_c_d
% 0.19/0.47 @ ^ [X4: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_CollectI
% 0.19/0.47 thf(fact_260_subset__CollectI,axiom,
% 0.19/0.47 ! [B3: set_Fi1407883041real_c,A3: set_Fi1407883041real_c,Q: finite1398487019real_c > $o,P2: finite1398487019real_c > $o] :
% 0.19/0.47 ( ( ord_le1327118209real_c @ B3 @ A3 )
% 0.19/0.47 => ( ! [X: finite1398487019real_c] :
% 0.19/0.47 ( ( member1261661570real_c @ X @ B3 )
% 0.19/0.47 => ( ( Q @ X )
% 0.19/0.47 => ( P2 @ X ) ) )
% 0.19/0.47 => ( ord_le1327118209real_c
% 0.19/0.47 @ ( collec230941376real_c
% 0.19/0.47 @ ^ [X4: finite1398487019real_c] :
% 0.19/0.47 ( ( member1261661570real_c @ X4 @ B3 )
% 0.19/0.47 & ( Q @ X4 ) ) )
% 0.19/0.47 @ ( collec230941376real_c
% 0.19/0.47 @ ^ [X4: finite1398487019real_c] :
% 0.19/0.47 ( ( member1261661570real_c @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_CollectI
% 0.19/0.47 thf(fact_261_subset__CollectI,axiom,
% 0.19/0.47 ! [B3: set_c,A3: set_c,Q: c > $o,P2: c > $o] :
% 0.19/0.47 ( ( ord_less_eq_set_c @ B3 @ A3 )
% 0.19/0.47 => ( ! [X: c] :
% 0.19/0.47 ( ( member_c @ X @ B3 )
% 0.19/0.47 => ( ( Q @ X )
% 0.19/0.47 => ( P2 @ X ) ) )
% 0.19/0.47 => ( ord_less_eq_set_c
% 0.19/0.47 @ ( collect_c
% 0.19/0.47 @ ^ [X4: c] :
% 0.19/0.47 ( ( member_c @ X4 @ B3 )
% 0.19/0.47 & ( Q @ X4 ) ) )
% 0.19/0.47 @ ( collect_c
% 0.19/0.47 @ ^ [X4: c] :
% 0.19/0.47 ( ( member_c @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_CollectI
% 0.19/0.47 thf(fact_262_subset__CollectI,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_c,A3: set_Sum_sum_c_c,Q: sum_sum_c_c > $o,P2: sum_sum_c_c > $o] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ B3 @ A3 )
% 0.19/0.47 => ( ! [X: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X @ B3 )
% 0.19/0.47 => ( ( Q @ X )
% 0.19/0.47 => ( P2 @ X ) ) )
% 0.19/0.47 => ( ord_le1772180283um_c_c
% 0.19/0.47 @ ( collect_Sum_sum_c_c
% 0.19/0.47 @ ^ [X4: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X4 @ B3 )
% 0.19/0.47 & ( Q @ X4 ) ) )
% 0.19/0.47 @ ( collect_Sum_sum_c_c
% 0.19/0.47 @ ^ [X4: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X4 @ A3 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_CollectI
% 0.19/0.47 thf(fact_263_prop__restrict,axiom,
% 0.19/0.47 ! [X3: produc190496183real_c,Z3: set_Pr1389752855real_c,X6: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
% 0.19/0.47 ( ( member1895684704real_c @ X3 @ Z3 )
% 0.19/0.47 => ( ( ord_le977353143real_c @ Z3
% 0.19/0.47 @ ( collec1643251106real_c
% 0.19/0.47 @ ^ [X4: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 => ( P2 @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prop_restrict
% 0.19/0.47 thf(fact_264_prop__restrict,axiom,
% 0.19/0.47 ! [X3: sum_sum_c_d,Z3: set_Sum_sum_c_d,X6: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X3 @ Z3 )
% 0.19/0.47 => ( ( ord_le764260924um_c_d @ Z3
% 0.19/0.47 @ ( collect_Sum_sum_c_d
% 0.19/0.47 @ ^ [X4: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 => ( P2 @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prop_restrict
% 0.19/0.47 thf(fact_265_prop__restrict,axiom,
% 0.19/0.47 ! [X3: finite1398487019real_c,Z3: set_Fi1407883041real_c,X6: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
% 0.19/0.47 ( ( member1261661570real_c @ X3 @ Z3 )
% 0.19/0.47 => ( ( ord_le1327118209real_c @ Z3
% 0.19/0.47 @ ( collec230941376real_c
% 0.19/0.47 @ ^ [X4: finite1398487019real_c] :
% 0.19/0.47 ( ( member1261661570real_c @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 => ( P2 @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prop_restrict
% 0.19/0.47 thf(fact_266_prop__restrict,axiom,
% 0.19/0.47 ! [X3: c,Z3: set_c,X6: set_c,P2: c > $o] :
% 0.19/0.47 ( ( member_c @ X3 @ Z3 )
% 0.19/0.47 => ( ( ord_less_eq_set_c @ Z3
% 0.19/0.47 @ ( collect_c
% 0.19/0.47 @ ^ [X4: c] :
% 0.19/0.47 ( ( member_c @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 => ( P2 @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prop_restrict
% 0.19/0.47 thf(fact_267_prop__restrict,axiom,
% 0.19/0.47 ! [X3: sum_sum_c_c,Z3: set_Sum_sum_c_c,X6: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X3 @ Z3 )
% 0.19/0.47 => ( ( ord_le1772180283um_c_c @ Z3
% 0.19/0.47 @ ( collect_Sum_sum_c_c
% 0.19/0.47 @ ^ [X4: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) ) )
% 0.19/0.47 => ( P2 @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prop_restrict
% 0.19/0.47 thf(fact_268_Collect__restrict,axiom,
% 0.19/0.47 ! [X6: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
% 0.19/0.47 ( ord_le977353143real_c
% 0.19/0.47 @ ( collec1643251106real_c
% 0.19/0.47 @ ^ [X4: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) )
% 0.19/0.47 @ X6 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_restrict
% 0.19/0.47 thf(fact_269_Collect__restrict,axiom,
% 0.19/0.47 ! [X6: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 0.19/0.47 ( ord_le764260924um_c_d
% 0.19/0.47 @ ( collect_Sum_sum_c_d
% 0.19/0.47 @ ^ [X4: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) )
% 0.19/0.47 @ X6 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_restrict
% 0.19/0.47 thf(fact_270_Collect__restrict,axiom,
% 0.19/0.47 ! [X6: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
% 0.19/0.47 ( ord_le1327118209real_c
% 0.19/0.47 @ ( collec230941376real_c
% 0.19/0.47 @ ^ [X4: finite1398487019real_c] :
% 0.19/0.47 ( ( member1261661570real_c @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) )
% 0.19/0.47 @ X6 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_restrict
% 0.19/0.47 thf(fact_271_Collect__restrict,axiom,
% 0.19/0.47 ! [X6: set_c,P2: c > $o] :
% 0.19/0.47 ( ord_less_eq_set_c
% 0.19/0.47 @ ( collect_c
% 0.19/0.47 @ ^ [X4: c] :
% 0.19/0.47 ( ( member_c @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) )
% 0.19/0.47 @ X6 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_restrict
% 0.19/0.47 thf(fact_272_Collect__restrict,axiom,
% 0.19/0.47 ! [X6: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 0.19/0.47 ( ord_le1772180283um_c_c
% 0.19/0.47 @ ( collect_Sum_sum_c_c
% 0.19/0.47 @ ^ [X4: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X4 @ X6 )
% 0.19/0.47 & ( P2 @ X4 ) ) )
% 0.19/0.47 @ X6 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_restrict
% 0.19/0.47 thf(fact_273_ssubst__Pair__rhs,axiom,
% 0.19/0.47 ! [R3: finite1398487019real_c,S: finite1398487019real_c,R2: set_Pr1389752855real_c,S3: finite1398487019real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ ( produc394644079real_c @ R3 @ S ) @ R2 )
% 0.19/0.47 => ( ( S3 = S )
% 0.19/0.47 => ( member1895684704real_c @ ( produc394644079real_c @ R3 @ S3 ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ssubst_Pair_rhs
% 0.19/0.47 thf(fact_274_hpsafe__ODE__simps,axiom,
% 0.19/0.47 ! [ODE: oDE_a_c,P: formula_a_b_c] :
% 0.19/0.47 ( ( hpsafe_a_b_c @ ( evolveODE_a_c_b @ ODE @ P ) )
% 0.19/0.47 = ( ( osafe_a_c @ ODE )
% 0.19/0.47 & ( fsafe_a_b_c @ P ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % hpsafe_ODE_simps
% 0.19/0.47 thf(fact_275_hpsafe__fsafe_Ohpsafe__Evolve,axiom,
% 0.19/0.47 ! [ODE: oDE_a_c,P2: formula_a_b_c] :
% 0.19/0.47 ( ( osafe_a_c @ ODE )
% 0.19/0.47 => ( ( fsafe_a_b_c @ P2 )
% 0.19/0.47 => ( hpsafe_a_b_c @ ( evolveODE_a_c_b @ ODE @ P2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % hpsafe_fsafe.hpsafe_Evolve
% 0.19/0.47 thf(fact_276_Icc__subset__Ici__iff,axiom,
% 0.19/0.47 ! [L: set_Sum_sum_c_c,H: set_Sum_sum_c_c,L2: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le657554289um_c_c @ ( set_or935154662um_c_c @ L @ H ) @ ( set_or2028906832um_c_c @ L2 ) )
% 0.19/0.47 = ( ~ ( ord_le1772180283um_c_c @ L @ H )
% 0.19/0.47 | ( ord_le1772180283um_c_c @ L2 @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Icc_subset_Ici_iff
% 0.19/0.47 thf(fact_277_Icc__subset__Ici__iff,axiom,
% 0.19/0.47 ! [L: real,H: real,L2: real] :
% 0.19/0.47 ( ( ord_less_eq_set_real @ ( set_or656347191t_real @ L @ H ) @ ( set_ord_atLeast_real @ L2 ) )
% 0.19/0.47 = ( ~ ( ord_less_eq_real @ L @ H )
% 0.19/0.47 | ( ord_less_eq_real @ L2 @ L ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Icc_subset_Ici_iff
% 0.19/0.47 thf(fact_278_Icc__subset__Iic__iff,axiom,
% 0.19/0.47 ! [L: set_Sum_sum_c_c,H: set_Sum_sum_c_c,H2: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le657554289um_c_c @ ( set_or935154662um_c_c @ L @ H ) @ ( set_or326480468um_c_c @ H2 ) )
% 0.19/0.47 = ( ~ ( ord_le1772180283um_c_c @ L @ H )
% 0.19/0.47 | ( ord_le1772180283um_c_c @ H @ H2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Icc_subset_Iic_iff
% 0.19/0.47 thf(fact_279_Icc__subset__Iic__iff,axiom,
% 0.19/0.47 ! [L: real,H: real,H2: real] :
% 0.19/0.47 ( ( ord_less_eq_set_real @ ( set_or656347191t_real @ L @ H ) @ ( set_ord_atMost_real @ H2 ) )
% 0.19/0.47 = ( ~ ( ord_less_eq_real @ L @ H )
% 0.19/0.47 | ( ord_less_eq_real @ H @ H2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Icc_subset_Iic_iff
% 0.19/0.47 thf(fact_280_atMost__iff,axiom,
% 0.19/0.47 ! [I4: produc190496183real_c,K: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ I4 @ ( set_or248124158real_c @ K ) )
% 0.19/0.47 = ( ord_le850691415real_c @ I4 @ K ) ) ).
% 0.19/0.47
% 0.19/0.47 % atMost_iff
% 0.19/0.47 thf(fact_281_atMost__iff,axiom,
% 0.19/0.47 ! [I4: set_Sum_sum_c_c,K: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( member2124950898um_c_c @ I4 @ ( set_or326480468um_c_c @ K ) )
% 0.19/0.47 = ( ord_le1772180283um_c_c @ I4 @ K ) ) ).
% 0.19/0.47
% 0.19/0.47 % atMost_iff
% 0.19/0.47 thf(fact_282_atMost__iff,axiom,
% 0.19/0.47 ! [I4: real,K: real] :
% 0.19/0.47 ( ( member_real @ I4 @ ( set_ord_atMost_real @ K ) )
% 0.19/0.47 = ( ord_less_eq_real @ I4 @ K ) ) ).
% 0.19/0.47
% 0.19/0.47 % atMost_iff
% 0.19/0.47 thf(fact_283_atLeast__iff,axiom,
% 0.19/0.47 ! [I4: produc190496183real_c,K: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ I4 @ ( set_or10334466real_c @ K ) )
% 0.19/0.47 = ( ord_le850691415real_c @ K @ I4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeast_iff
% 0.19/0.47 thf(fact_284_atLeast__iff,axiom,
% 0.19/0.47 ! [I4: set_Sum_sum_c_c,K: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( member2124950898um_c_c @ I4 @ ( set_or2028906832um_c_c @ K ) )
% 0.19/0.47 = ( ord_le1772180283um_c_c @ K @ I4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeast_iff
% 0.19/0.47 thf(fact_285_atLeast__iff,axiom,
% 0.19/0.47 ! [I4: real,K: real] :
% 0.19/0.47 ( ( member_real @ I4 @ ( set_ord_atLeast_real @ K ) )
% 0.19/0.47 = ( ord_less_eq_real @ K @ I4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeast_iff
% 0.19/0.47 thf(fact_286_atMost__subset__iff,axiom,
% 0.19/0.47 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le657554289um_c_c @ ( set_or326480468um_c_c @ X3 ) @ ( set_or326480468um_c_c @ Y3 ) )
% 0.19/0.47 = ( ord_le1772180283um_c_c @ X3 @ Y3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % atMost_subset_iff
% 0.19/0.47 thf(fact_287_atMost__subset__iff,axiom,
% 0.19/0.47 ! [X3: real,Y3: real] :
% 0.19/0.47 ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X3 ) @ ( set_ord_atMost_real @ Y3 ) )
% 0.19/0.47 = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % atMost_subset_iff
% 0.19/0.47 thf(fact_288_atLeast__subset__iff,axiom,
% 0.19/0.47 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le657554289um_c_c @ ( set_or2028906832um_c_c @ X3 ) @ ( set_or2028906832um_c_c @ Y3 ) )
% 0.19/0.47 = ( ord_le1772180283um_c_c @ Y3 @ X3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeast_subset_iff
% 0.19/0.47 thf(fact_289_atLeast__subset__iff,axiom,
% 0.19/0.47 ! [X3: real,Y3: real] :
% 0.19/0.47 ( ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ X3 ) @ ( set_ord_atLeast_real @ Y3 ) )
% 0.19/0.47 = ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeast_subset_iff
% 0.19/0.47 thf(fact_290_not__Ici__le__Icc,axiom,
% 0.19/0.47 ! [L: real,L2: real,H2: real] :
% 0.19/0.47 ~ ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ L ) @ ( set_or656347191t_real @ L2 @ H2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_Ici_le_Icc
% 0.19/0.47 thf(fact_291_not__Iic__le__Icc,axiom,
% 0.19/0.47 ! [H: real,L2: real,H2: real] :
% 0.19/0.47 ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H ) @ ( set_or656347191t_real @ L2 @ H2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % not_Iic_le_Icc
% 0.19/0.47 thf(fact_292_atLeast__def,axiom,
% 0.19/0.47 ( set_or2003963744real_c
% 0.19/0.47 = ( ^ [L3: finite1398487019real_c] : ( collec230941376real_c @ ( ord_le775706699real_c @ L3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeast_def
% 0.19/0.47 thf(fact_293_atLeast__def,axiom,
% 0.19/0.47 ( set_or2028906832um_c_c
% 0.19/0.47 = ( ^ [L3: set_Sum_sum_c_c] : ( collec95329456um_c_c @ ( ord_le1772180283um_c_c @ L3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeast_def
% 0.19/0.47 thf(fact_294_atLeast__def,axiom,
% 0.19/0.47 ( set_ord_atLeast_real
% 0.19/0.47 = ( ^ [L3: real] : ( collect_real @ ( ord_less_eq_real @ L3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % atLeast_def
% 0.19/0.47 thf(fact_295_atMost__def,axiom,
% 0.19/0.47 ( set_or1478397028real_c
% 0.19/0.47 = ( ^ [U2: finite1398487019real_c] :
% 0.19/0.47 ( collec230941376real_c
% 0.19/0.47 @ ^ [X4: finite1398487019real_c] : ( ord_le775706699real_c @ X4 @ U2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % atMost_def
% 0.19/0.47 thf(fact_296_atMost__def,axiom,
% 0.19/0.47 ( set_or326480468um_c_c
% 0.19/0.47 = ( ^ [U2: set_Sum_sum_c_c] :
% 0.19/0.47 ( collec95329456um_c_c
% 0.19/0.47 @ ^ [X4: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X4 @ U2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % atMost_def
% 0.19/0.47 thf(fact_297_atMost__def,axiom,
% 0.19/0.47 ( set_ord_atMost_real
% 0.19/0.47 = ( ^ [U2: real] :
% 0.19/0.47 ( collect_real
% 0.19/0.47 @ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % atMost_def
% 0.19/0.47 thf(fact_298_image__uminus__atMost,axiom,
% 0.19/0.47 ! [X3: real] :
% 0.19/0.47 ( ( image_real_real @ uminus_uminus_real @ ( set_ord_atMost_real @ X3 ) )
% 0.19/0.47 = ( set_ord_atLeast_real @ ( uminus_uminus_real @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_uminus_atMost
% 0.19/0.47 thf(fact_299_image__uminus__atLeast,axiom,
% 0.19/0.47 ! [X3: real] :
% 0.19/0.47 ( ( image_real_real @ uminus_uminus_real @ ( set_ord_atLeast_real @ X3 ) )
% 0.19/0.47 = ( set_ord_atMost_real @ ( uminus_uminus_real @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_uminus_atLeast
% 0.19/0.47 thf(fact_300_image__eqI,axiom,
% 0.19/0.47 ! [B: sum_sum_c_c,F: c > sum_sum_c_c,X3: c,A3: set_c] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member_c @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ B @ ( image_c_Sum_sum_c_c @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_301_image__eqI,axiom,
% 0.19/0.47 ! [B: sum_sum_c_d,F: c > sum_sum_c_d,X3: c,A3: set_c] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member_c @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ B @ ( image_c_Sum_sum_c_d @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_302_image__eqI,axiom,
% 0.19/0.47 ! [B: sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,X3: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ B @ ( image_666880337um_c_c @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_303_image__eqI,axiom,
% 0.19/0.47 ! [B: sum_sum_c_d,F: sum_sum_c_c > sum_sum_c_d,X3: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ B @ ( image_675141842um_c_d @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_304_image__eqI,axiom,
% 0.19/0.47 ! [B: sum_sum_c_c,F: sum_sum_c_d > sum_sum_c_c,X3: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ B @ ( image_1558941394um_c_c @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_305_image__eqI,axiom,
% 0.19/0.47 ! [B: sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d,X3: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ B @ ( image_1567202899um_c_d @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_306_image__eqI,axiom,
% 0.19/0.47 ! [B: sum_sum_c_c,F: produc190496183real_c > sum_sum_c_c,X3: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member1895684704real_c @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ B @ ( image_1467045143um_c_c @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_307_image__eqI,axiom,
% 0.19/0.47 ! [B: sum_sum_c_d,F: produc190496183real_c > sum_sum_c_d,X3: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member1895684704real_c @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ B @ ( image_1475306648um_c_d @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_308_image__eqI,axiom,
% 0.19/0.47 ! [B: produc190496183real_c,F: sum_sum_c_c > produc190496183real_c,X3: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 0.19/0.47 => ( member1895684704real_c @ B @ ( image_1146069259real_c @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_309_image__eqI,axiom,
% 0.19/0.47 ! [B: produc190496183real_c,F: sum_sum_c_d > produc190496183real_c,X3: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 0.19/0.47 ( ( B
% 0.19/0.47 = ( F @ X3 ) )
% 0.19/0.47 => ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 0.19/0.47 => ( member1895684704real_c @ B @ ( image_1980705354real_c @ F @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_eqI
% 0.19/0.47 thf(fact_310_image__uminus__atLeastAtMost,axiom,
% 0.19/0.47 ! [X3: real,Y3: real] :
% 0.19/0.47 ( ( image_real_real @ uminus_uminus_real @ ( set_or656347191t_real @ X3 @ Y3 ) )
% 0.19/0.47 = ( set_or656347191t_real @ ( uminus_uminus_real @ Y3 ) @ ( uminus_uminus_real @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_uminus_atLeastAtMost
% 0.19/0.47 thf(fact_311_subset__image__iff,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_d,F: c > sum_sum_c_d,A3: set_c] :
% 0.19/0.47 ( ( ord_le764260924um_c_d @ B3 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
% 0.19/0.47 = ( ? [AA: set_c] :
% 0.19/0.47 ( ( ord_less_eq_set_c @ AA @ A3 )
% 0.19/0.47 & ( B3
% 0.19/0.47 = ( image_c_Sum_sum_c_d @ F @ AA ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_image_iff
% 0.19/0.47 thf(fact_312_subset__image__iff,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_c,F: c > sum_sum_c_c,A3: set_c] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ B3 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
% 0.19/0.47 = ( ? [AA: set_c] :
% 0.19/0.47 ( ( ord_less_eq_set_c @ AA @ A3 )
% 0.19/0.47 & ( B3
% 0.19/0.47 = ( image_c_Sum_sum_c_c @ F @ AA ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_image_iff
% 0.19/0.47 thf(fact_313_subset__image__iff,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ B3 @ ( image_666880337um_c_c @ F @ A3 ) )
% 0.19/0.47 = ( ? [AA: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ AA @ A3 )
% 0.19/0.47 & ( B3
% 0.19/0.47 = ( image_666880337um_c_c @ F @ AA ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_image_iff
% 0.19/0.47 thf(fact_314_image__subset__iff,axiom,
% 0.19/0.47 ! [F: c > sum_sum_c_d,A3: set_c,B3: set_Sum_sum_c_d] :
% 0.19/0.47 ( ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A3 ) @ B3 )
% 0.19/0.47 = ( ! [X4: c] :
% 0.19/0.47 ( ( member_c @ X4 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ ( F @ X4 ) @ B3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subset_iff
% 0.19/0.47 thf(fact_315_image__subset__iff,axiom,
% 0.19/0.47 ! [F: c > sum_sum_c_c,A3: set_c,B3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A3 ) @ B3 )
% 0.19/0.47 = ( ! [X4: c] :
% 0.19/0.47 ( ( member_c @ X4 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ ( F @ X4 ) @ B3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subset_iff
% 0.19/0.47 thf(fact_316_subset__imageE,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_d,F: c > sum_sum_c_d,A3: set_c] :
% 0.19/0.47 ( ( ord_le764260924um_c_d @ B3 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
% 0.19/0.47 => ~ ! [C3: set_c] :
% 0.19/0.47 ( ( ord_less_eq_set_c @ C3 @ A3 )
% 0.19/0.47 => ( B3
% 0.19/0.47 != ( image_c_Sum_sum_c_d @ F @ C3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_imageE
% 0.19/0.47 thf(fact_317_subset__imageE,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_c,F: c > sum_sum_c_c,A3: set_c] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ B3 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
% 0.19/0.47 => ~ ! [C3: set_c] :
% 0.19/0.47 ( ( ord_less_eq_set_c @ C3 @ A3 )
% 0.19/0.47 => ( B3
% 0.19/0.47 != ( image_c_Sum_sum_c_c @ F @ C3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_imageE
% 0.19/0.47 thf(fact_318_subset__imageE,axiom,
% 0.19/0.47 ! [B3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ B3 @ ( image_666880337um_c_c @ F @ A3 ) )
% 0.19/0.47 => ~ ! [C3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ C3 @ A3 )
% 0.19/0.47 => ( B3
% 0.19/0.47 != ( image_666880337um_c_c @ F @ C3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_imageE
% 0.19/0.47 thf(fact_319_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_c,F: c > sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 0.19/0.47 ( ! [X: c] :
% 0.19/0.47 ( ( member_c @ X @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_320_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_c,F: c > sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ! [X: c] :
% 0.19/0.47 ( ( member_c @ X @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_321_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 0.19/0.47 ( ! [X: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le764260924um_c_d @ ( image_675141842um_c_d @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_322_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 0.19/0.47 ( ! [X: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le764260924um_c_d @ ( image_1567202899um_c_d @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_323_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ! [X: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le1772180283um_c_c @ ( image_666880337um_c_c @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_324_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ! [X: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le1772180283um_c_c @ ( image_1558941394um_c_c @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_325_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 0.19/0.47 ( ! [X: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ X @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le764260924um_c_d @ ( image_1475306648um_c_d @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_326_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_Sum_sum_c_c,F: sum_sum_c_c > produc190496183real_c,B3: set_Pr1389752855real_c] :
% 0.19/0.47 ( ! [X: sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X @ A3 )
% 0.19/0.47 => ( member1895684704real_c @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le977353143real_c @ ( image_1146069259real_c @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_327_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_Sum_sum_c_d,F: sum_sum_c_d > produc190496183real_c,B3: set_Pr1389752855real_c] :
% 0.19/0.47 ( ! [X: sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_d @ X @ A3 )
% 0.19/0.47 => ( member1895684704real_c @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le977353143real_c @ ( image_1980705354real_c @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_328_image__subsetI,axiom,
% 0.19/0.47 ! [A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 0.19/0.47 ( ! [X: produc190496183real_c] :
% 0.19/0.47 ( ( member1895684704real_c @ X @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
% 0.19/0.47 => ( ord_le1772180283um_c_c @ ( image_1467045143um_c_c @ F @ A3 ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_subsetI
% 0.19/0.47 thf(fact_329_image__mono,axiom,
% 0.19/0.47 ! [A3: set_c,B3: set_c,F: c > sum_sum_c_d] :
% 0.19/0.47 ( ( ord_less_eq_set_c @ A3 @ B3 )
% 0.19/0.47 => ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A3 ) @ ( image_c_Sum_sum_c_d @ F @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_mono
% 0.19/0.47 thf(fact_330_image__mono,axiom,
% 0.19/0.47 ! [A3: set_c,B3: set_c,F: c > sum_sum_c_c] :
% 0.19/0.47 ( ( ord_less_eq_set_c @ A3 @ B3 )
% 0.19/0.47 => ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A3 ) @ ( image_c_Sum_sum_c_c @ F @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_mono
% 0.19/0.47 thf(fact_331_image__mono,axiom,
% 0.19/0.47 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c] :
% 0.19/0.47 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 0.19/0.47 => ( ord_le1772180283um_c_c @ ( image_666880337um_c_c @ F @ A3 ) @ ( image_666880337um_c_c @ F @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % image_mono
% 0.19/0.47 thf(fact_332_solves__ode__subset__range,axiom,
% 0.19/0.47 ! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,T2: set_real,X6: set_Fi1407883041real_c,Y7: set_Fi1407883041real_c] :
% 0.19/0.47 ( ( initia1631504802real_c @ X3 @ F @ T2 @ X6 )
% 0.19/0.47 => ( ( ord_le1327118209real_c @ ( image_327677598real_c @ X3 @ T2 ) @ Y7 )
% 0.19/0.47 => ( initia1631504802real_c @ X3 @ F @ T2 @ Y7 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % solves_ode_subset_range
% 0.19/0.47 thf(fact_333_imageI,axiom,
% 0.19/0.47 ! [X3: c,A3: set_c,F: c > sum_sum_c_c] :
% 0.19/0.47 ( ( member_c @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_c_Sum_sum_c_c @ F @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % imageI
% 0.19/0.47 thf(fact_334_imageI,axiom,
% 0.19/0.47 ! [X3: c,A3: set_c,F: c > sum_sum_c_d] :
% 0.19/0.47 ( ( member_c @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_c_Sum_sum_c_d @ F @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % imageI
% 0.19/0.47 thf(fact_335_imageI,axiom,
% 0.19/0.47 ! [X3: sum_sum_c_c,A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 0.19/0.47 => ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_666880337um_c_c @ F @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % imageI
% 0.19/0.47 thf(fact_336_imageI,axiom,
% 0.19/0.47 ! [X3: sum_sum_c_c,A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d] :
% 0.19/0.47 ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 0.30/0.47 => ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_675141842um_c_d @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_337_imageI,axiom,
% 0.30/0.47 ! [X3: sum_sum_c_d,A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c] :
% 0.30/0.47 ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 0.30/0.47 => ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_1558941394um_c_c @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_338_imageI,axiom,
% 0.30/0.47 ! [X3: sum_sum_c_d,A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d] :
% 0.30/0.47 ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 0.30/0.47 => ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_1567202899um_c_d @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_339_imageI,axiom,
% 0.30/0.47 ! [X3: produc190496183real_c,A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_c] :
% 0.30/0.47 ( ( member1895684704real_c @ X3 @ A3 )
% 0.30/0.47 => ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_1467045143um_c_c @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_340_imageI,axiom,
% 0.30/0.47 ! [X3: produc190496183real_c,A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_d] :
% 0.30/0.47 ( ( member1895684704real_c @ X3 @ A3 )
% 0.30/0.47 => ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_1475306648um_c_d @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_341_imageI,axiom,
% 0.30/0.47 ! [X3: sum_sum_c_c,A3: set_Sum_sum_c_c,F: sum_sum_c_c > produc190496183real_c] :
% 0.30/0.47 ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 0.30/0.47 => ( member1895684704real_c @ ( F @ X3 ) @ ( image_1146069259real_c @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_342_imageI,axiom,
% 0.30/0.47 ! [X3: sum_sum_c_d,A3: set_Sum_sum_c_d,F: sum_sum_c_d > produc190496183real_c] :
% 0.30/0.47 ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 0.30/0.47 => ( member1895684704real_c @ ( F @ X3 ) @ ( image_1980705354real_c @ F @ A3 ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % imageI
% 0.30/0.47 thf(fact_343_image__iff,axiom,
% 0.30/0.47 ! [Z2: sum_sum_c_c,F: c > sum_sum_c_c,A3: set_c] :
% 0.30/0.47 ( ( member_Sum_sum_c_c @ Z2 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
% 0.30/0.47 = ( ? [X4: c] :
% 0.30/0.47 ( ( member_c @ X4 @ A3 )
% 0.30/0.47 & ( Z2
% 0.30/0.47 = ( F @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_iff
% 0.30/0.47 thf(fact_344_image__iff,axiom,
% 0.30/0.47 ! [Z2: sum_sum_c_d,F: c > sum_sum_c_d,A3: set_c] :
% 0.30/0.47 ( ( member_Sum_sum_c_d @ Z2 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
% 0.30/0.47 = ( ? [X4: c] :
% 0.30/0.47 ( ( member_c @ X4 @ A3 )
% 0.30/0.47 & ( Z2
% 0.30/0.47 = ( F @ X4 ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_iff
% 0.30/0.47 thf(fact_345_bex__imageD,axiom,
% 0.30/0.47 ! [F: c > sum_sum_c_c,A3: set_c,P2: sum_sum_c_c > $o] :
% 0.30/0.47 ( ? [X7: sum_sum_c_c] :
% 0.30/0.47 ( ( member_Sum_sum_c_c @ X7 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
% 0.30/0.47 & ( P2 @ X7 ) )
% 0.30/0.47 => ? [X: c] :
% 0.30/0.47 ( ( member_c @ X @ A3 )
% 0.30/0.47 & ( P2 @ ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_imageD
% 0.30/0.47 thf(fact_346_bex__imageD,axiom,
% 0.30/0.47 ! [F: c > sum_sum_c_d,A3: set_c,P2: sum_sum_c_d > $o] :
% 0.30/0.47 ( ? [X7: sum_sum_c_d] :
% 0.30/0.47 ( ( member_Sum_sum_c_d @ X7 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
% 0.30/0.47 & ( P2 @ X7 ) )
% 0.30/0.47 => ? [X: c] :
% 0.30/0.47 ( ( member_c @ X @ A3 )
% 0.30/0.47 & ( P2 @ ( F @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % bex_imageD
% 0.30/0.47 thf(fact_347_image__cong,axiom,
% 0.30/0.47 ! [M: set_c,N: set_c,F: c > sum_sum_c_c,G: c > sum_sum_c_c] :
% 0.30/0.47 ( ( M = N )
% 0.30/0.47 => ( ! [X: c] :
% 0.30/0.47 ( ( member_c @ X @ N )
% 0.30/0.47 => ( ( F @ X )
% 0.30/0.47 = ( G @ X ) ) )
% 0.30/0.47 => ( ( image_c_Sum_sum_c_c @ F @ M )
% 0.30/0.47 = ( image_c_Sum_sum_c_c @ G @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_cong
% 0.30/0.47 thf(fact_348_image__cong,axiom,
% 0.30/0.47 ! [M: set_c,N: set_c,F: c > sum_sum_c_d,G: c > sum_sum_c_d] :
% 0.30/0.47 ( ( M = N )
% 0.30/0.47 => ( ! [X: c] :
% 0.30/0.47 ( ( member_c @ X @ N )
% 0.30/0.47 => ( ( F @ X )
% 0.30/0.47 = ( G @ X ) ) )
% 0.30/0.47 => ( ( image_c_Sum_sum_c_d @ F @ M )
% 0.30/0.47 = ( image_c_Sum_sum_c_d @ G @ N ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % image_cong
% 0.30/0.47 thf(fact_349_Osub,axiom,
% 0.30/0.47 ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( static_FVO_a_c @ ode ) ) @ v ).
% 0.30/0.47
% 0.30/0.47 % Osub
% 0.30/0.47 thf(fact_350_VSA,axiom,
% 0.30/0.47 ( denota256060419gree_c @ ( sol @ zero_zero_real ) @ a
% 0.30/0.47 @ ( collect_c
% 0.30/0.47 @ ^ [Uu: c] :
% 0.30/0.47 ( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_BVO_a_c @ ode ) )
% 0.30/0.47 | ( member_Sum_sum_c_d @ ( sum_Inl_c_d @ Uu ) @ ( image_c_Sum_sum_c_d @ sum_Inl_c_d @ ( static_FVO_a_c @ ode ) ) )
% 0.30/0.47 | ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_FVF_a_b_c @ p ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 % VSA
% 0.30/0.47 thf(fact_351_OVsub_H,axiom,
% 0.30/0.47 ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( oDE_dom_a_c @ ode ) ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( oDE_dom_a_c @ ode ) ) ) @ v ).
% 0.30/0.47
% 0.30/0.47 % OVsub'
% 0.30/0.47
% 0.30/0.47 % Conjectures (1)
% 0.30/0.47 thf(conj_0,conjecture,
% 0.30/0.47 denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.43/1.66 ---- Embedded in THF ---
% 2.43/1.66 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.43/1.66 %%% Generated on Tue Feb 24 08:24:13 EST 2026
% 2.43/1.66 %%% using '$$dhol' embedding, version 1.3.0.
% 2.43/1.66 %%% Logic specification used:
% 2.43/1.66 %%% thf(spec, logic, $$dhol).
% 2.43/1.66
% 2.43/1.66 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.VZsX1DUqbC/DTF2DTF10391.p
% See solution above
% 2.43/1.66 ------------------------
% 2.60/1.71 ---- Cleaned THF ---
% 2.60/1.71 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_J,type,
% 2.60/1.71 set_Pr1389752855real_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 2.60/1.71 produc190496183real_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_Mt__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_J,type,
% 2.60/1.71 produc214722967um_c_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__a_Mtf__b_Mtf__c_Mt__Product____Type__Ounit_J,type,
% 2.60/1.71 denota231621370t_unit: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_Mt__Real__Oreal_J,type,
% 2.60/1.71 produc1398711912c_real: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_J,type,
% 2.60/1.71 produc754691304um_c_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Sum____Type__Osum_Itf__b_Mtf__c_J_J_J,type,
% 2.60/1.71 set_Su1783761653um_b_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 2.60/1.71 set_Fi1407883041real_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_J,type,
% 2.60/1.71 set_set_Sum_sum_c_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
% 2.60/1.71 finite1398487019real_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 2.60/1.71 produc957004601l_real: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__d_J_J,type,
% 2.60/1.71 set_Sum_sum_c_d: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
% 2.60/1.71 set_Sum_sum_c_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Syntax__Oformula_Itf__a_Mtf__b_Mtf__c_J,type,
% 2.60/1.71 formula_a_b_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Syntax__Ohp_Itf__a_Mtf__b_Mtf__c_J,type,
% 2.60/1.71 hp_a_b_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
% 2.60/1.71 sum_sum_c_d: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
% 2.60/1.71 sum_sum_c_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 2.60/1.71 filter_real: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Syntax__OODE_Itf__a_Mtf__c_J,type,
% 2.60/1.71 oDE_a_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 2.60/1.71 set_real: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Set__Oset_Itf__c_J,type,
% 2.60/1.71 set_c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_t__Real__Oreal,type,
% 2.60/1.71 real: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(ty_n_tf__c,type,
% 2.60/1.71 c: $tType ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_001t__Real__Oreal,type,
% 2.60/1.71 bNF_Ca449034301c_real: set_Pr1389752855real_c > ( finite1398487019real_c > real ) > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
% 2.60/1.71 bNF_Ca1693016812um_c_c: set_Pr1389752855real_c > ( finite1398487019real_c > set_Sum_sum_c_c ) > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Coincidence__Mirabelle__veeocdlawq_Oids_Ocoincide__fml_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 coinci501627771_a_b_c: formula_a_b_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Coincidence__Mirabelle__veeocdlawq_Oids_Ocoincide__hp_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 coinci354700372_a_b_c: hp_a_b_c > denota231621370t_unit > denota231621370t_unit > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Coincidence__Mirabelle__veeocdlawq_Oids_Ocoincide__hp_H_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 coinci1852866155_a_b_c: hp_a_b_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Coincidence__Mirabelle__veeocdlawq_Oids_Oode__sem__equiv_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 coinci772602909_a_b_c: hp_a_b_c > denota231621370t_unit > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Denotational__Semantics_OIagree_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 denota69106024_a_b_c: denota231621370t_unit > denota231621370t_unit > set_Su1783761653um_b_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Denotational__Semantics_OODE__sem_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 denota1275485728_a_b_c: denota231621370t_unit > oDE_a_c > finite1398487019real_c > finite1398487019real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Denotational__Semantics_OVSagree_001tf__c,type,
% 2.60/1.71 denota256060419gree_c: finite1398487019real_c > finite1398487019real_c > set_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Denotational__Semantics_OVagree_001tf__c,type,
% 2.60/1.71 denota1997846518gree_c: produc190496183real_c > produc190496183real_c > set_Sum_sum_c_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Denotational__Semantics_Ofml__sem_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 denota968303861_a_b_c: denota231621370t_unit > formula_a_b_c > set_Pr1389752855real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Denotational__Semantics_Omk__v_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 denota161327353_a_b_c: denota231621370t_unit > oDE_a_c > produc190496183real_c > finite1398487019real_c > produc190496183real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Denotational__Semantics_Omk__xode_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 denota1896987029_a_b_c: denota231621370t_unit > oDE_a_c > finite1398487019real_c > produc190496183real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Denotational__Semantics_OsemBV_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 denota1293057250_a_b_c: denota231621370t_unit > oDE_a_c > set_Sum_sum_c_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Deriv_Ohas__vector__derivative_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
% 2.60/1.71 has_ve1904308670real_c: ( real > finite1398487019real_c ) > finite1398487019real_c > filter_real > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Deriv_Ohas__vector__derivative_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 2.60/1.71 has_ve373335844real_c: ( real > produc190496183real_c ) > produc190496183real_c > filter_real > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Deriv_Ohas__vector__derivative_001t__Real__Oreal,type,
% 2.60/1.71 has_ve217091583e_real: ( real > real ) > real > filter_real > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_M_Eo_J,type,
% 2.60/1.71 uminus23580315al_c_o: ( finite1398487019real_c > $o ) > finite1398487019real_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_M_Eo_J,type,
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% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Sum____Type__Osum_Itf__c_Mtf__c_J_M_Eo_J,type,
% 2.60/1.71 uminus88729697_c_c_o: ( sum_sum_c_c > $o ) > sum_sum_c_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Sum____Type__Osum_Itf__c_Mtf__d_J_M_Eo_J,type,
% 2.60/1.71 uminus916499104_c_d_o: ( sum_sum_c_d > $o ) > sum_sum_c_d > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_Itf__c_M_Eo_J,type,
% 2.60/1.71 uminus_uminus_c_o: ( c > $o ) > c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
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% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
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% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J_J,type,
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% 2.60/1.71
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% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
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% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
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% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__c_J,type,
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% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ozero__class_Ozero_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
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% 2.60/1.71
% 2.60/1.71 thf(sy_c_Groups_Ozero__class_Ozero_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
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% 2.60/1.71 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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% 2.60/1.71 thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
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% 2.60/1.71
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% 2.60/1.71 thf(sy_c_Syntax_Ohp_OEvolveODE_001tf__a_001tf__c_001tf__b,type,
% 2.60/1.71 evolveODE_a_c_b: oDE_a_c > formula_a_b_c > hp_a_b_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Syntax_Ohpsafe_001tf__a_001tf__b_001tf__c,type,
% 2.60/1.71 hpsafe_a_b_c: hp_a_b_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Syntax_Oosafe_001tf__a_001tf__c,type,
% 2.60/1.71 osafe_a_c: oDE_a_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 2.60/1.71 topolo1830106092n_real: real > set_real > filter_real ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
% 2.60/1.71 member1261661570real_c: finite1398487019real_c > set_Fi1407883041real_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_member_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
% 2.60/1.71 member1895684704real_c: produc190496183real_c > set_Pr1389752855real_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_member_001t__Real__Oreal,type,
% 2.60/1.71 member_real: real > set_real > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_Itf__c_Mtf__c_J_J,type,
% 2.60/1.71 member2124950898um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_member_001t__Sum____Type__Osum_Itf__c_Mtf__c_J,type,
% 2.60/1.71 member_Sum_sum_c_c: sum_sum_c_c > set_Sum_sum_c_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_member_001t__Sum____Type__Osum_Itf__c_Mtf__d_J,type,
% 2.60/1.71 member_Sum_sum_c_d: sum_sum_c_d > set_Sum_sum_c_d > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_c_member_001tf__c,type,
% 2.60/1.71 member_c: c > set_c > $o ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_I____,type,
% 2.60/1.71 i: denota231621370t_unit ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_J____,type,
% 2.60/1.71 j: denota231621370t_unit ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_ODE____,type,
% 2.60/1.71 ode: oDE_a_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_P____,type,
% 2.60/1.71 p: formula_a_b_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_V____,type,
% 2.60/1.71 v: set_Sum_sum_c_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_a____,type,
% 2.60/1.71 a: finite1398487019real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_aa____,type,
% 2.60/1.71 aa: finite1398487019real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_ab____,type,
% 2.60/1.71 ab: finite1398487019real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_b____,type,
% 2.60/1.71 b: finite1398487019real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_ba____,type,
% 2.60/1.71 ba: finite1398487019real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_bb____,type,
% 2.60/1.71 bb: finite1398487019real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_s____,type,
% 2.60/1.71 s: real ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_sol____,type,
% 2.60/1.71 sol: real > finite1398487019real_c ).
% 2.60/1.71
% 2.60/1.71 thf(sy_v_t____,type,
% 2.60/1.71 t: real ).
% 2.60/1.71
% 2.60/1.71 thf(fact_0_osafe,axiom,
% 2.60/1.71 osafe_a_c @ ode ).
% 2.60/1.71
% 2.60/1.71 thf(fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062,axiom,
% 2.60/1.71 ( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( produc394644079real_c @ aa @ ba ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ j @ ode ) ) )
% 2.60/1.71 & ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_2__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_Asa_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_092_060close_062,axiom,
% 2.60/1.71 ! [S: real] : ( denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ S ) ) @ ( produc394644079real_c @ aa @ ba ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ j @ ode ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_3_mk__v__agree,axiom,
% 2.60/1.71 ! [I: denota231621370t_unit,ODE: oDE_a_c,Nu: produc190496183real_c,Sol: finite1398487019real_c] :
% 2.60/1.71 ( ( denota1997846518gree_c @ ( denota161327353_a_b_c @ I @ ODE @ Nu @ Sol ) @ Nu @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I @ ODE ) ) )
% 2.60/1.71 & ( denota1997846518gree_c @ ( denota161327353_a_b_c @ I @ ODE @ Nu @ Sol ) @ ( denota1896987029_a_b_c @ I @ ODE @ Sol ) @ ( denota1293057250_a_b_c @ I @ ODE ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_4_prod_Oinject,axiom,
% 2.60/1.71 ! [X1: finite1398487019real_c,X2: finite1398487019real_c,Y1: finite1398487019real_c,Y2: finite1398487019real_c] :
% 2.60/1.71 ( ( ( produc394644079real_c @ X1 @ X2 )
% 2.60/1.71 = ( produc394644079real_c @ Y1 @ Y2 ) )
% 2.60/1.71 = ( ( X1 = Y1 )
% 2.60/1.71 & ( X2 = Y2 ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_5_old_Oprod_Oinject,axiom,
% 2.60/1.71 ! [A: finite1398487019real_c,B: finite1398487019real_c,A2: finite1398487019real_c,B2: finite1398487019real_c] :
% 2.60/1.71 ( ( ( produc394644079real_c @ A @ B )
% 2.60/1.71 = ( produc394644079real_c @ A2 @ B2 ) )
% 2.60/1.71 = ( ( A = A2 )
% 2.60/1.71 & ( B = B2 ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_6_mk__v__exists,axiom,
% 2.60/1.71 ! [Nu: produc190496183real_c,I: denota231621370t_unit,ODE: oDE_a_c,Sol: finite1398487019real_c] :
% 2.60/1.71 ? [Omega: produc190496183real_c] :
% 2.60/1.71 ( ( denota1997846518gree_c @ Omega @ Nu @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I @ ODE ) ) )
% 2.60/1.71 & ( denota1997846518gree_c @ Omega @ ( denota1896987029_a_b_c @ I @ ODE @ Sol ) @ ( denota1293057250_a_b_c @ I @ ODE ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_7_VA,axiom,
% 2.60/1.71 denota1997846518gree_c @ ( produc394644079real_c @ a @ b ) @ ( produc394644079real_c @ aa @ ba ) @ v ).
% 2.60/1.71
% 2.60/1.71 thf(fact_8__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Imk__xode_AI_AODE_A_Isol_Asa_J_J_A_IsemBV_AI_AODE_J_092_060close_062,axiom,
% 2.60/1.71 ! [S: real] : ( denota1997846518gree_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ S ) ) @ ( denota1896987029_a_b_c @ i @ ode @ ( sol @ S ) ) @ ( denota1293057250_a_b_c @ i @ ode ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_9_VAOV,axiom,
% 2.60/1.71 denota1997846518gree_c @ ( produc394644079real_c @ a @ b ) @ ( produc394644079real_c @ aa @ ba ) @ ( static_BVO_a_c @ ode ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_10_agree__comm,axiom,
% 2.60/1.71 ! [A3: produc190496183real_c,B3: produc190496183real_c,V: set_Sum_sum_c_c] :
% 2.60/1.71 ( ( denota1997846518gree_c @ A3 @ B3 @ V )
% 2.60/1.71 => ( denota1997846518gree_c @ B3 @ A3 @ V ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_11_agree__refl,axiom,
% 2.60/1.71 ! [Nu: produc190496183real_c,A3: set_Sum_sum_c_c] : ( denota1997846518gree_c @ Nu @ Nu @ A3 ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_12__092_060open_062_092_060And_062sa_O_AVagree_A_Imk__v_AI_AODE_A_Ia_M_Ab_J_A_Isol_Asa_J_J_A_Ia_M_Ab_J_A_I_N_AsemBV_AI_AODE_J_092_060close_062,axiom,
% 2.60/1.71 ! [S: real] : ( denota1997846518gree_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ S ) ) @ ( produc394644079real_c @ a @ b ) @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ i @ ode ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_13_surj__pair,axiom,
% 2.60/1.71 ! [P: produc190496183real_c] :
% 2.60/1.71 ? [X: finite1398487019real_c,Y: finite1398487019real_c] :
% 2.60/1.71 ( P
% 2.60/1.71 = ( produc394644079real_c @ X @ Y ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_14_prod__cases,axiom,
% 2.60/1.71 ! [P2: produc190496183real_c > $o,P: produc190496183real_c] :
% 2.60/1.71 ( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P2 @ ( produc394644079real_c @ A4 @ B4 ) )
% 2.60/1.71 => ( P2 @ P ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_15_OVsub,axiom,
% 2.60/1.71 ord_le1772180283um_c_c @ ( static_BVO_a_c @ ode ) @ v ).
% 2.60/1.71
% 2.60/1.71 thf(fact_16_veq,axiom,
% 2.60/1.71 ( ( produc394644079real_c @ ab @ bb )
% 2.60/1.71 = ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ t ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_17_old_Oprod_Oinducts,axiom,
% 2.60/1.71 ! [P2: produc190496183real_c > $o,Prod: produc190496183real_c] :
% 2.60/1.71 ( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P2 @ ( produc394644079real_c @ A4 @ B4 ) )
% 2.60/1.71 => ( P2 @ Prod ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_18_old_Oprod_Oexhaust,axiom,
% 2.60/1.71 ! [Y3: produc190496183real_c] :
% 2.60/1.71 ~ ! [A4: finite1398487019real_c,B4: finite1398487019real_c] :
% 2.60/1.71 ( Y3
% 2.60/1.71 != ( produc394644079real_c @ A4 @ B4 ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_19_Pair__inject,axiom,
% 2.60/1.71 ! [A: finite1398487019real_c,B: finite1398487019real_c,A2: finite1398487019real_c,B2: finite1398487019real_c] :
% 2.60/1.71 ( ( ( produc394644079real_c @ A @ B )
% 2.60/1.71 = ( produc394644079real_c @ A2 @ B2 ) )
% 2.60/1.71 => ~ ( ( A = A2 )
% 2.60/1.71 => ( B != B2 ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_20_OVsub_H_H,axiom,
% 2.60/1.71 ord_le1772180283um_c_c @ ( denota1293057250_a_b_c @ i @ ode ) @ v ).
% 2.60/1.71
% 2.60/1.71 thf(fact_21_semBVsub,axiom,
% 2.60/1.71 ord_le1772180283um_c_c @ ( denota1293057250_a_b_c @ i @ ode ) @ ( static_BVO_a_c @ ode ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_22_uminus__Pair,axiom,
% 2.60/1.71 ! [A: finite1398487019real_c,B: finite1398487019real_c] :
% 2.60/1.71 ( ( uminus17964590real_c @ ( produc394644079real_c @ A @ B ) )
% 2.60/1.71 = ( produc394644079real_c @ ( uminus1737280820real_c @ A ) @ ( uminus1737280820real_c @ B ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_23_uminus__Pair,axiom,
% 2.60/1.71 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
% 2.60/1.71 ( ( uminus1225118734um_c_c @ ( produc1439061071um_c_c @ A @ B ) )
% 2.60/1.71 = ( produc1439061071um_c_c @ ( uminus1381786404um_c_c @ A ) @ ( uminus1381786404um_c_c @ B ) ) ) ).
% 2.60/1.71
% 2.60/1.71 thf(fact_24_uminus__Pair,axiom,
% 2.60/1.71 ! [A: set_Sum_sum_c_c,B: real] :
% 2.60/1.71 ( ( uminus125610847c_real @ ( produc1528874528c_real @ A @ B ) )
% 2.60/1.72 = ( produc1528874528c_real @ ( uminus1381786404um_c_c @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_25_uminus__Pair,axiom,
% 2.60/1.72 ! [A: real,B: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( uminus1629073887um_c_c @ ( produc1186709024um_c_c @ A @ B ) )
% 2.60/1.72 = ( produc1186709024um_c_c @ ( uminus_uminus_real @ A ) @ ( uminus1381786404um_c_c @ B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_26_uminus__Pair,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( uminus856758960l_real @ ( produc705216881l_real @ A @ B ) )
% 2.60/1.72 = ( produc705216881l_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_27_ComplI,axiom,
% 2.60/1.72 ! [C: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 2.60/1.72 ( ~ ( member1895684704real_c @ C @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ C @ ( uminus232257166real_c @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_28_ComplI,axiom,
% 2.60/1.72 ! [C: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ~ ( member_Sum_sum_c_d @ C @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_29_ComplI,axiom,
% 2.60/1.72 ! [C: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ~ ( member_Sum_sum_c_c @ C @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_30_Compl__iff,axiom,
% 2.60/1.72 ! [C: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ C @ ( uminus232257166real_c @ A3 ) )
% 2.60/1.72 = ( ~ ( member1895684704real_c @ C @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_31_Compl__iff,axiom,
% 2.60/1.72 ! [C: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A3 ) )
% 2.60/1.72 = ( ~ ( member_Sum_sum_c_d @ C @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_32_Compl__iff,axiom,
% 2.60/1.72 ! [C: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A3 ) )
% 2.60/1.72 = ( ~ ( member_Sum_sum_c_c @ C @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_33_Compl__eq__Compl__iff,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ( uminus1381786404um_c_c @ A3 )
% 2.60/1.72 = ( uminus1381786404um_c_c @ B3 ) )
% 2.60/1.72 = ( A3 = B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_34_verit__minus__simplify_I4_J,axiom,
% 2.60/1.72 ! [B: real] :
% 2.60/1.72 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 2.60/1.72 = B ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_35_add_Oinverse__inverse,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 2.60/1.72 = A ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_36_subset__antisym,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B3 @ A3 )
% 2.60/1.72 => ( A3 = B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_37_subsetI,axiom,
% 2.60/1.72 ! [A3: set_Pr1389752855real_c,B3: set_Pr1389752855real_c] :
% 2.60/1.72 ( ! [X: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ X @ B3 ) )
% 2.60/1.72 => ( ord_le977353143real_c @ A3 @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_38_subsetI,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ! [X: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ X @ B3 ) )
% 2.60/1.72 => ( ord_le764260924um_c_d @ A3 @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_39_subsetI,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ! [X: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ X @ B3 ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A3 @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_40_compl__eq__compl__iff,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ( uminus1381786404um_c_c @ X3 )
% 2.60/1.72 = ( uminus1381786404um_c_c @ Y3 ) )
% 2.60/1.72 = ( X3 = Y3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_41_double__compl,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( uminus1381786404um_c_c @ ( uminus1381786404um_c_c @ X3 ) )
% 2.60/1.72 = X3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_42_neg__equal__iff__equal,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( ( uminus_uminus_real @ A )
% 2.60/1.72 = ( uminus_uminus_real @ B ) )
% 2.60/1.72 = ( A = B ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_43_mem__Collect__eq,axiom,
% 2.60/1.72 ! [A: produc190496183real_c,P2: produc190496183real_c > $o] :
% 2.60/1.72 ( ( member1895684704real_c @ A @ ( collec1643251106real_c @ P2 ) )
% 2.60/1.72 = ( P2 @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_44_mem__Collect__eq,axiom,
% 2.60/1.72 ! [A: sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ A @ ( collect_Sum_sum_c_c @ P2 ) )
% 2.60/1.72 = ( P2 @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_45_mem__Collect__eq,axiom,
% 2.60/1.72 ! [A: sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ A @ ( collect_Sum_sum_c_d @ P2 ) )
% 2.60/1.72 = ( P2 @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_46_mem__Collect__eq,axiom,
% 2.60/1.72 ! [A: finite1398487019real_c,P2: finite1398487019real_c > $o] :
% 2.60/1.72 ( ( member1261661570real_c @ A @ ( collec230941376real_c @ P2 ) )
% 2.60/1.72 = ( P2 @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_47_mem__Collect__eq,axiom,
% 2.60/1.72 ! [A: c,P2: c > $o] :
% 2.60/1.72 ( ( member_c @ A @ ( collect_c @ P2 ) )
% 2.60/1.72 = ( P2 @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_48_Collect__mem__eq,axiom,
% 2.60/1.72 ! [A3: set_Pr1389752855real_c] :
% 2.60/1.72 ( ( collec1643251106real_c
% 2.60/1.72 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A3 ) )
% 2.60/1.72 = A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_49_Collect__mem__eq,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ A3 ) )
% 2.60/1.72 = A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_50_Collect__mem__eq,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ( collect_Sum_sum_c_d
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ A3 ) )
% 2.60/1.72 = A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_51_Collect__mem__eq,axiom,
% 2.60/1.72 ! [A3: set_Fi1407883041real_c] :
% 2.60/1.72 ( ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] : ( member1261661570real_c @ X4 @ A3 ) )
% 2.60/1.72 = A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_52_Collect__mem__eq,axiom,
% 2.60/1.72 ! [A3: set_c] :
% 2.60/1.72 ( ( collect_c
% 2.60/1.72 @ ^ [X4: c] : ( member_c @ X4 @ A3 ) )
% 2.60/1.72 = A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_53_Collect__cong,axiom,
% 2.60/1.72 ! [P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
% 2.60/1.72 ( ! [X: finite1398487019real_c] :
% 2.60/1.72 ( ( P2 @ X )
% 2.60/1.72 = ( Q @ X ) )
% 2.60/1.72 => ( ( collec230941376real_c @ P2 )
% 2.60/1.72 = ( collec230941376real_c @ Q ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_54_Collect__cong,axiom,
% 2.60/1.72 ! [P2: c > $o,Q: c > $o] :
% 2.60/1.72 ( ! [X: c] :
% 2.60/1.72 ( ( P2 @ X )
% 2.60/1.72 = ( Q @ X ) )
% 2.60/1.72 => ( ( collect_c @ P2 )
% 2.60/1.72 = ( collect_c @ Q ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_55_compl__le__compl__iff,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ X3 ) @ ( uminus1381786404um_c_c @ Y3 ) )
% 2.60/1.72 = ( ord_le1772180283um_c_c @ Y3 @ X3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_56_neg__le__iff__le,axiom,
% 2.60/1.72 ! [B: real,A: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 2.60/1.72 = ( ord_less_eq_real @ A @ B ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_57_Compl__subset__Compl__iff,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ A3 ) @ ( uminus1381786404um_c_c @ B3 ) )
% 2.60/1.72 = ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_58_Compl__anti__mono,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ B3 ) @ ( uminus1381786404um_c_c @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_59_verit__la__disequality,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( A = B )
% 2.60/1.72 | ~ ( ord_less_eq_real @ A @ B )
% 2.60/1.72 | ~ ( ord_less_eq_real @ B @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_60_Collect__mono__iff,axiom,
% 2.60/1.72 ! [P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
% 2.60/1.72 ( ( ord_le1327118209real_c @ ( collec230941376real_c @ P2 ) @ ( collec230941376real_c @ Q ) )
% 2.60/1.72 = ( ! [X4: finite1398487019real_c] :
% 2.60/1.72 ( ( P2 @ X4 )
% 2.60/1.72 => ( Q @ X4 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_61_Collect__mono__iff,axiom,
% 2.60/1.72 ! [P2: c > $o,Q: c > $o] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ ( collect_c @ P2 ) @ ( collect_c @ Q ) )
% 2.60/1.72 = ( ! [X4: c] :
% 2.60/1.72 ( ( P2 @ X4 )
% 2.60/1.72 => ( Q @ X4 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_62_Collect__mono__iff,axiom,
% 2.60/1.72 ! [P2: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ ( collect_Sum_sum_c_c @ P2 ) @ ( collect_Sum_sum_c_c @ Q ) )
% 2.60/1.72 = ( ! [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( P2 @ X4 )
% 2.60/1.72 => ( Q @ X4 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_63_set__eq__subset,axiom,
% 2.60/1.72 ( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : ( Y4 = Z ) )
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A5 @ B5 )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ B5 @ A5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_64_subset__trans,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,C2: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B3 @ C2 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A3 @ C2 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_65_Collect__mono,axiom,
% 2.60/1.72 ! [P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
% 2.60/1.72 ( ! [X: finite1398487019real_c] :
% 2.60/1.72 ( ( P2 @ X )
% 2.60/1.72 => ( Q @ X ) )
% 2.60/1.72 => ( ord_le1327118209real_c @ ( collec230941376real_c @ P2 ) @ ( collec230941376real_c @ Q ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_66_Collect__mono,axiom,
% 2.60/1.72 ! [P2: c > $o,Q: c > $o] :
% 2.60/1.72 ( ! [X: c] :
% 2.60/1.72 ( ( P2 @ X )
% 2.60/1.72 => ( Q @ X ) )
% 2.60/1.72 => ( ord_less_eq_set_c @ ( collect_c @ P2 ) @ ( collect_c @ Q ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_67_Collect__mono,axiom,
% 2.60/1.72 ! [P2: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
% 2.60/1.72 ( ! [X: sum_sum_c_c] :
% 2.60/1.72 ( ( P2 @ X )
% 2.60/1.72 => ( Q @ X ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( collect_Sum_sum_c_c @ P2 ) @ ( collect_Sum_sum_c_c @ Q ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_68_subset__refl,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A3 @ A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_69_subset__iff,axiom,
% 2.60/1.72 ( ord_le977353143real_c
% 2.60/1.72 = ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
% 2.60/1.72 ! [T: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ T @ A5 )
% 2.60/1.72 => ( member1895684704real_c @ T @ B5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_70_subset__iff,axiom,
% 2.60/1.72 ( ord_le764260924um_c_d
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
% 2.60/1.72 ! [T: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ T @ A5 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ T @ B5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_71_subset__iff,axiom,
% 2.60/1.72 ( ord_le1772180283um_c_c
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
% 2.60/1.72 ! [T: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ T @ A5 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ T @ B5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_72_equalityD2,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( A3 = B3 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_73_equalityD1,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( A3 = B3 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A3 @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_74_subset__eq,axiom,
% 2.60/1.72 ( ord_le977353143real_c
% 2.60/1.72 = ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
% 2.60/1.72 ! [X4: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X4 @ A5 )
% 2.60/1.72 => ( member1895684704real_c @ X4 @ B5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_75_subset__eq,axiom,
% 2.60/1.72 ( ord_le764260924um_c_d
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
% 2.60/1.72 ! [X4: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X4 @ A5 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ X4 @ B5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_76_subset__eq,axiom,
% 2.60/1.72 ( ord_le1772180283um_c_c
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
% 2.60/1.72 ! [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X4 @ A5 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ X4 @ B5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_77_equalityE,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( A3 = B3 )
% 2.60/1.72 => ~ ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 2.60/1.72 => ~ ( ord_le1772180283um_c_c @ B3 @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_78_subsetD,axiom,
% 2.60/1.72 ! [A3: set_Pr1389752855real_c,B3: set_Pr1389752855real_c,C: produc190496183real_c] :
% 2.60/1.72 ( ( ord_le977353143real_c @ A3 @ B3 )
% 2.60/1.72 => ( ( member1895684704real_c @ C @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ C @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_79_subsetD,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_d,B3: set_Sum_sum_c_d,C: sum_sum_c_d] :
% 2.60/1.72 ( ( ord_le764260924um_c_d @ A3 @ B3 )
% 2.60/1.72 => ( ( member_Sum_sum_c_d @ C @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ C @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_80_subsetD,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,C: sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 2.60/1.72 => ( ( member_Sum_sum_c_c @ C @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ C @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_81_in__mono,axiom,
% 2.60/1.72 ! [A3: set_Pr1389752855real_c,B3: set_Pr1389752855real_c,X3: produc190496183real_c] :
% 2.60/1.72 ( ( ord_le977353143real_c @ A3 @ B3 )
% 2.60/1.72 => ( ( member1895684704real_c @ X3 @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ X3 @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_82_in__mono,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_d,B3: set_Sum_sum_c_d,X3: sum_sum_c_d] :
% 2.60/1.72 ( ( ord_le764260924um_c_d @ A3 @ B3 )
% 2.60/1.72 => ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ X3 @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_83_in__mono,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,X3: sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 2.60/1.72 => ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ X3 @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_84_le__imp__neg__le,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ B )
% 2.60/1.72 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_85_minus__le__iff,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 2.60/1.72 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_86_le__minus__iff,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 2.60/1.72 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_87_compl__le__swap2,axiom,
% 2.60/1.72 ! [Y3: set_Sum_sum_c_c,X3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ Y3 ) @ X3 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ X3 ) @ Y3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_88_compl__le__swap1,axiom,
% 2.60/1.72 ! [Y3: set_Sum_sum_c_c,X3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ Y3 @ ( uminus1381786404um_c_c @ X3 ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ X3 @ ( uminus1381786404um_c_c @ Y3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_89_compl__mono,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X3 @ Y3 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( uminus1381786404um_c_c @ Y3 ) @ ( uminus1381786404um_c_c @ X3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_90_agree__sub,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,Nu: produc190496183real_c,Omega2: produc190496183real_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 2.60/1.72 => ( ( denota1997846518gree_c @ Nu @ Omega2 @ B3 )
% 2.60/1.72 => ( denota1997846518gree_c @ Nu @ Omega2 @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_91_agree__supset,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_c,A3: set_Sum_sum_c_c,Nu: produc190496183real_c,Nu2: produc190496183real_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B3 @ A3 )
% 2.60/1.72 => ( ( denota1997846518gree_c @ Nu @ Nu2 @ A3 )
% 2.60/1.72 => ( denota1997846518gree_c @ Nu @ Nu2 @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_92_minus__equation__iff,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( ( uminus_uminus_real @ A )
% 2.60/1.72 = B )
% 2.60/1.72 = ( ( uminus_uminus_real @ B )
% 2.60/1.72 = A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_93_equation__minus__iff,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( A
% 2.60/1.72 = ( uminus_uminus_real @ B ) )
% 2.60/1.72 = ( B
% 2.60/1.72 = ( uminus_uminus_real @ A ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_94_verit__negate__coefficient_I3_J,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( A = B )
% 2.60/1.72 => ( ( uminus_uminus_real @ A )
% 2.60/1.72 = ( uminus_uminus_real @ B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_95_double__complement,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( uminus1381786404um_c_c @ ( uminus1381786404um_c_c @ A3 ) )
% 2.60/1.72 = A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_96_ComplD,axiom,
% 2.60/1.72 ! [C: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ C @ ( uminus232257166real_c @ A3 ) )
% 2.60/1.72 => ~ ( member1895684704real_c @ C @ A3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_97_ComplD,axiom,
% 2.60/1.72 ! [C: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ C @ ( uminus373867045um_c_d @ A3 ) )
% 2.60/1.72 => ~ ( member_Sum_sum_c_d @ C @ A3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_98_ComplD,axiom,
% 2.60/1.72 ! [C: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ C @ ( uminus1381786404um_c_c @ A3 ) )
% 2.60/1.72 => ~ ( member_Sum_sum_c_c @ C @ A3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_99_Pair__le,axiom,
% 2.60/1.72 ! [A: finite1398487019real_c,B: finite1398487019real_c,C: finite1398487019real_c,D: finite1398487019real_c] :
% 2.60/1.72 ( ( ord_le850691415real_c @ ( produc394644079real_c @ A @ B ) @ ( produc394644079real_c @ C @ D ) )
% 2.60/1.72 = ( ( ord_le775706699real_c @ A @ C )
% 2.60/1.72 & ( ord_le775706699real_c @ B @ D ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_100_Pair__le,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,D: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le684454199um_c_c @ ( produc1439061071um_c_c @ A @ B ) @ ( produc1439061071um_c_c @ C @ D ) )
% 2.60/1.72 = ( ( ord_le1772180283um_c_c @ A @ C )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ B @ D ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_101_Pair__le,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: real,C: set_Sum_sum_c_c,D: real] :
% 2.60/1.72 ( ( ord_le835231240c_real @ ( produc1528874528c_real @ A @ B ) @ ( produc1528874528c_real @ C @ D ) )
% 2.60/1.72 = ( ( ord_le1772180283um_c_c @ A @ C )
% 2.60/1.72 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_102_Pair__le,axiom,
% 2.60/1.72 ! [A: real,B: set_Sum_sum_c_c,C: real,D: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le191210632um_c_c @ ( produc1186709024um_c_c @ A @ B ) @ ( produc1186709024um_c_c @ C @ D ) )
% 2.60/1.72 = ( ( ord_less_eq_real @ A @ C )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ B @ D ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_103_Pair__le,axiom,
% 2.60/1.72 ! [A: real,B: real,C: real,D: real] :
% 2.60/1.72 ( ( ord_le1342644953l_real @ ( produc705216881l_real @ A @ B ) @ ( produc705216881l_real @ C @ D ) )
% 2.60/1.72 = ( ( ord_less_eq_real @ A @ C )
% 2.60/1.72 & ( ord_less_eq_real @ B @ D ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_104_order__refl,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X3 @ X3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_105_order__refl,axiom,
% 2.60/1.72 ! [X3: real] : ( ord_less_eq_real @ X3 @ X3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_106_Pair__mono,axiom,
% 2.60/1.72 ! [X3: finite1398487019real_c,X5: finite1398487019real_c,Y3: finite1398487019real_c,Y5: finite1398487019real_c] :
% 2.60/1.72 ( ( ord_le775706699real_c @ X3 @ X5 )
% 2.60/1.72 => ( ( ord_le775706699real_c @ Y3 @ Y5 )
% 2.60/1.72 => ( ord_le850691415real_c @ ( produc394644079real_c @ X3 @ Y3 ) @ ( produc394644079real_c @ X5 @ Y5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_107_Pair__mono,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,X5: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X3 @ X5 )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ Y3 @ Y5 )
% 2.60/1.72 => ( ord_le684454199um_c_c @ ( produc1439061071um_c_c @ X3 @ Y3 ) @ ( produc1439061071um_c_c @ X5 @ Y5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_108_Pair__mono,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,X5: set_Sum_sum_c_c,Y3: real,Y5: real] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X3 @ X5 )
% 2.60/1.72 => ( ( ord_less_eq_real @ Y3 @ Y5 )
% 2.60/1.72 => ( ord_le835231240c_real @ ( produc1528874528c_real @ X3 @ Y3 ) @ ( produc1528874528c_real @ X5 @ Y5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_109_Pair__mono,axiom,
% 2.60/1.72 ! [X3: real,X5: real,Y3: set_Sum_sum_c_c,Y5: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_less_eq_real @ X3 @ X5 )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ Y3 @ Y5 )
% 2.60/1.72 => ( ord_le191210632um_c_c @ ( produc1186709024um_c_c @ X3 @ Y3 ) @ ( produc1186709024um_c_c @ X5 @ Y5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_110_Pair__mono,axiom,
% 2.60/1.72 ! [X3: real,X5: real,Y3: real,Y5: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X3 @ X5 )
% 2.60/1.72 => ( ( ord_less_eq_real @ Y3 @ Y5 )
% 2.60/1.72 => ( ord_le1342644953l_real @ ( produc705216881l_real @ X3 @ Y3 ) @ ( produc705216881l_real @ X5 @ Y5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_111_Fsub,axiom,
% 2.60/1.72 ord_le1772180283um_c_c @ ( static_FVF_a_b_c @ p ) @ v ).
% 2.60/1.72
% 2.60/1.72 thf(fact_112_solSem,axiom,
% 2.60/1.72 ! [X3: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 2.60/1.72 => ( ( ord_less_eq_real @ X3 @ t )
% 2.60/1.72 => ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ ( sol @ X3 ) ) @ ( denota968303861_a_b_c @ i @ p ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_113_relChain__def,axiom,
% 2.60/1.72 ( bNF_Ca1693016812um_c_c
% 2.60/1.72 = ( ^ [R: set_Pr1389752855real_c,As: finite1398487019real_c > set_Sum_sum_c_c] :
% 2.60/1.72 ! [I2: finite1398487019real_c,J: finite1398487019real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ ( produc394644079real_c @ I2 @ J ) @ R )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_114_relChain__def,axiom,
% 2.60/1.72 ( bNF_Ca449034301c_real
% 2.60/1.72 = ( ^ [R: set_Pr1389752855real_c,As: finite1398487019real_c > real] :
% 2.60/1.72 ! [I2: finite1398487019real_c,J: finite1398487019real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ ( produc394644079real_c @ I2 @ J ) @ R )
% 2.60/1.72 => ( ord_less_eq_real @ ( As @ I2 ) @ ( As @ J ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_115_hpsafe__Evolve_OIH,axiom,
% 2.60/1.72 coinci501627771_a_b_c @ p ).
% 2.60/1.72
% 2.60/1.72 thf(fact_116_t,axiom,
% 2.60/1.72 ord_less_eq_real @ zero_zero_real @ t ).
% 2.60/1.72
% 2.60/1.72 thf(fact_117_add_Oinverse__neutral,axiom,
% 2.60/1.72 ( ( uminus_uminus_real @ zero_zero_real )
% 2.60/1.72 = zero_zero_real ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_118_neg__0__equal__iff__equal,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( zero_zero_real
% 2.60/1.72 = ( uminus_uminus_real @ A ) )
% 2.60/1.72 = ( zero_zero_real = A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_119_neg__equal__0__iff__equal,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( ( uminus_uminus_real @ A )
% 2.60/1.72 = zero_zero_real )
% 2.60/1.72 = ( A = zero_zero_real ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_120_equal__neg__zero,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( A
% 2.60/1.72 = ( uminus_uminus_real @ A ) )
% 2.60/1.72 = ( A = zero_zero_real ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_121_neg__equal__zero,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( ( uminus_uminus_real @ A )
% 2.60/1.72 = A )
% 2.60/1.72 = ( A = zero_zero_real ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_122_IH_H,axiom,
% 2.60/1.72 ! [A6: finite1398487019real_c,B6: finite1398487019real_c,Aa: finite1398487019real_c,Ba: finite1398487019real_c] :
% 2.60/1.72 ( ( denota1997846518gree_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( produc394644079real_c @ Aa @ Ba ) @ ( static_FVF_a_b_c @ p ) )
% 2.60/1.72 => ( ( member1895684704real_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( denota968303861_a_b_c @ i @ p ) )
% 2.60/1.72 = ( member1895684704real_c @ ( produc394644079real_c @ Aa @ Ba ) @ ( denota968303861_a_b_c @ j @ p ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_123_neg__less__eq__nonneg,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 2.60/1.72 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_124_less__eq__neg__nonpos,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 2.60/1.72 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_125_neg__le__0__iff__le,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 2.60/1.72 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_126_neg__0__le__iff__le,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 2.60/1.72 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_127_fsafe,axiom,
% 2.60/1.72 fsafe_a_b_c @ p ).
% 2.60/1.72
% 2.60/1.72 thf(fact_128_zero__prod__def,axiom,
% 2.60/1.72 ( zero_z1506780526real_c
% 2.60/1.72 = ( produc394644079real_c @ zero_z109254132real_c @ zero_z109254132real_c ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_129_zero__prod__def,axiom,
% 2.60/1.72 ( zero_z659284464l_real
% 2.60/1.72 = ( produc705216881l_real @ zero_zero_real @ zero_zero_real ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_130_zero__reorient,axiom,
% 2.60/1.72 ! [X3: real] :
% 2.60/1.72 ( ( zero_zero_real = X3 )
% 2.60/1.72 = ( X3 = zero_zero_real ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_131_iff__to__impl,axiom,
% 2.60/1.72 ! [Nu: produc190496183real_c,I: denota231621370t_unit,A3: formula_a_b_c,B3: formula_a_b_c] :
% 2.60/1.72 ( ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
% 2.60/1.72 = ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) )
% 2.60/1.72 = ( ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
% 2.60/1.72 => ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) )
% 2.60/1.72 & ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) )
% 2.60/1.72 => ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_132_order__subst1,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A @ ( F @ B ) )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B @ C )
% 2.60/1.72 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X @ Y )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_133_order__subst1,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,F: real > set_Sum_sum_c_c,B: real,C: real] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A @ ( F @ B ) )
% 2.60/1.72 => ( ( ord_less_eq_real @ B @ C )
% 2.60/1.72 => ( ! [X: real,Y: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X @ Y )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_134_order__subst1,axiom,
% 2.60/1.72 ! [A: real,F: set_Sum_sum_c_c > real,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B @ C )
% 2.60/1.72 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X @ Y )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_135_order__subst1,axiom,
% 2.60/1.72 ! [A: real,F: real > real,B: real,C: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 2.60/1.72 => ( ( ord_less_eq_real @ B @ C )
% 2.60/1.72 => ( ! [X: real,Y: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X @ Y )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_136_order__subst2,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A @ B )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ ( F @ B ) @ C )
% 2.60/1.72 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X @ Y )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_137_order__subst2,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > real,C: real] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A @ B )
% 2.60/1.72 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.60/1.72 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X @ Y )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_138_order__subst2,axiom,
% 2.60/1.72 ! [A: real,B: real,F: real > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ B )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ ( F @ B ) @ C )
% 2.60/1.72 => ( ! [X: real,Y: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X @ Y )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_139_order__subst2,axiom,
% 2.60/1.72 ! [A: real,B: real,F: real > real,C: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ B )
% 2.60/1.72 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 2.60/1.72 => ( ! [X: real,Y: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X @ Y )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_140_ord__eq__le__subst,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( A
% 2.60/1.72 = ( F @ B ) )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B @ C )
% 2.60/1.72 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X @ Y )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_141_ord__eq__le__subst,axiom,
% 2.60/1.72 ! [A: real,F: set_Sum_sum_c_c > real,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( A
% 2.60/1.72 = ( F @ B ) )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B @ C )
% 2.60/1.72 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X @ Y )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_142_ord__eq__le__subst,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,F: real > set_Sum_sum_c_c,B: real,C: real] :
% 2.60/1.72 ( ( A
% 2.60/1.72 = ( F @ B ) )
% 2.60/1.72 => ( ( ord_less_eq_real @ B @ C )
% 2.60/1.72 => ( ! [X: real,Y: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X @ Y )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A @ ( F @ C ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_143_ord__eq__le__subst,axiom,
% 2.60/1.72 ! [A: real,F: real > real,B: real,C: real] :
% 2.60/1.72 ( ( A
% 2.60/1.72 = ( F @ B ) )
% 2.60/1.72 => ( ( ord_less_eq_real @ B @ C )
% 2.60/1.72 => ( ! [X: real,Y: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X @ Y )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_144_ord__le__eq__subst,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A @ B )
% 2.60/1.72 => ( ( ( F @ B )
% 2.60/1.72 = C )
% 2.60/1.72 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X @ Y )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_145_ord__le__eq__subst,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,F: set_Sum_sum_c_c > real,C: real] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A @ B )
% 2.60/1.72 => ( ( ( F @ B )
% 2.60/1.72 = C )
% 2.60/1.72 => ( ! [X: set_Sum_sum_c_c,Y: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X @ Y )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_146_ord__le__eq__subst,axiom,
% 2.60/1.72 ! [A: real,B: real,F: real > set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ B )
% 2.60/1.72 => ( ( ( F @ B )
% 2.60/1.72 = C )
% 2.60/1.72 => ( ! [X: real,Y: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X @ Y )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( F @ A ) @ C ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_147_ord__le__eq__subst,axiom,
% 2.60/1.72 ! [A: real,B: real,F: real > real,C: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ B )
% 2.60/1.72 => ( ( ( F @ B )
% 2.60/1.72 = C )
% 2.60/1.72 => ( ! [X: real,Y: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X @ Y )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y ) ) )
% 2.60/1.72 => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_148_eq__iff,axiom,
% 2.60/1.72 ( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : ( Y4 = Z ) )
% 2.60/1.72 = ( ^ [X4: set_Sum_sum_c_c,Y6: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X4 @ Y6 )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ Y6 @ X4 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_149_eq__iff,axiom,
% 2.60/1.72 ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
% 2.60/1.72 = ( ^ [X4: real,Y6: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X4 @ Y6 )
% 2.60/1.72 & ( ord_less_eq_real @ Y6 @ X4 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_150_antisym,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X3 @ Y3 )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ Y3 @ X3 )
% 2.60/1.72 => ( X3 = Y3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_151_antisym,axiom,
% 2.60/1.72 ! [X3: real,Y3: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X3 @ Y3 )
% 2.60/1.72 => ( ( ord_less_eq_real @ Y3 @ X3 )
% 2.60/1.72 => ( X3 = Y3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_152_linear,axiom,
% 2.60/1.72 ! [X3: real,Y3: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X3 @ Y3 )
% 2.60/1.72 | ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_153_eq__refl,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( X3 = Y3 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ X3 @ Y3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_154_eq__refl,axiom,
% 2.60/1.72 ! [X3: real,Y3: real] :
% 2.60/1.72 ( ( X3 = Y3 )
% 2.60/1.72 => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_155_le__cases,axiom,
% 2.60/1.72 ! [X3: real,Y3: real] :
% 2.60/1.72 ( ~ ( ord_less_eq_real @ X3 @ Y3 )
% 2.60/1.72 => ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_156_order_Otrans,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A @ B )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B @ C )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_157_order_Otrans,axiom,
% 2.60/1.72 ! [A: real,B: real,C: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ B )
% 2.60/1.72 => ( ( ord_less_eq_real @ B @ C )
% 2.60/1.72 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_158_le__cases3,axiom,
% 2.60/1.72 ! [X3: real,Y3: real,Z2: real] :
% 2.60/1.72 ( ( ( ord_less_eq_real @ X3 @ Y3 )
% 2.60/1.72 => ~ ( ord_less_eq_real @ Y3 @ Z2 ) )
% 2.60/1.72 => ( ( ( ord_less_eq_real @ Y3 @ X3 )
% 2.60/1.72 => ~ ( ord_less_eq_real @ X3 @ Z2 ) )
% 2.60/1.72 => ( ( ( ord_less_eq_real @ X3 @ Z2 )
% 2.60/1.72 => ~ ( ord_less_eq_real @ Z2 @ Y3 ) )
% 2.60/1.72 => ( ( ( ord_less_eq_real @ Z2 @ Y3 )
% 2.60/1.72 => ~ ( ord_less_eq_real @ Y3 @ X3 ) )
% 2.60/1.72 => ( ( ( ord_less_eq_real @ Y3 @ Z2 )
% 2.60/1.72 => ~ ( ord_less_eq_real @ Z2 @ X3 ) )
% 2.60/1.72 => ~ ( ( ord_less_eq_real @ Z2 @ X3 )
% 2.60/1.72 => ~ ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_159_antisym__conv,axiom,
% 2.60/1.72 ! [Y3: set_Sum_sum_c_c,X3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ Y3 @ X3 )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ X3 @ Y3 )
% 2.60/1.72 = ( X3 = Y3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_160_antisym__conv,axiom,
% 2.60/1.72 ! [Y3: real,X3: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ Y3 @ X3 )
% 2.60/1.72 => ( ( ord_less_eq_real @ X3 @ Y3 )
% 2.60/1.72 = ( X3 = Y3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_161_order__class_Oorder_Oeq__iff,axiom,
% 2.60/1.72 ( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : ( Y4 = Z ) )
% 2.60/1.72 = ( ^ [A7: set_Sum_sum_c_c,B7: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A7 @ B7 )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ B7 @ A7 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_162_order__class_Oorder_Oeq__iff,axiom,
% 2.60/1.72 ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
% 2.60/1.72 = ( ^ [A7: real,B7: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A7 @ B7 )
% 2.60/1.72 & ( ord_less_eq_real @ B7 @ A7 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_163_ord__eq__le__trans,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( A = B )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B @ C )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_164_ord__eq__le__trans,axiom,
% 2.60/1.72 ! [A: real,B: real,C: real] :
% 2.60/1.72 ( ( A = B )
% 2.60/1.72 => ( ( ord_less_eq_real @ B @ C )
% 2.60/1.72 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_165_ord__le__eq__trans,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A @ B )
% 2.60/1.72 => ( ( B = C )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ A @ C ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_166_ord__le__eq__trans,axiom,
% 2.60/1.72 ! [A: real,B: real,C: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ B )
% 2.60/1.72 => ( ( B = C )
% 2.60/1.72 => ( ord_less_eq_real @ A @ C ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_167_order__class_Oorder_Oantisym,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A @ B )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B @ A )
% 2.60/1.72 => ( A = B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_168_order__class_Oorder_Oantisym,axiom,
% 2.60/1.72 ! [A: real,B: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ B )
% 2.60/1.72 => ( ( ord_less_eq_real @ B @ A )
% 2.60/1.72 => ( A = B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_169_order__trans,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c,Z2: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ X3 @ Y3 )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ Y3 @ Z2 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ X3 @ Z2 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_170_order__trans,axiom,
% 2.60/1.72 ! [X3: real,Y3: real,Z2: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ X3 @ Y3 )
% 2.60/1.72 => ( ( ord_less_eq_real @ Y3 @ Z2 )
% 2.60/1.72 => ( ord_less_eq_real @ X3 @ Z2 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_171_dual__order_Orefl,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ A @ A ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_172_dual__order_Orefl,axiom,
% 2.60/1.72 ! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_173_linorder__wlog,axiom,
% 2.60/1.72 ! [P2: real > real > $o,A: real,B: real] :
% 2.60/1.72 ( ! [A4: real,B4: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A4 @ B4 )
% 2.60/1.72 => ( P2 @ A4 @ B4 ) )
% 2.60/1.72 => ( ! [A4: real,B4: real] :
% 2.60/1.72 ( ( P2 @ B4 @ A4 )
% 2.60/1.72 => ( P2 @ A4 @ B4 ) )
% 2.60/1.72 => ( P2 @ A @ B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_174_dual__order_Otrans,axiom,
% 2.60/1.72 ! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c,C: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B @ A )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ C @ B )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ C @ A ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_175_dual__order_Otrans,axiom,
% 2.60/1.72 ! [B: real,A: real,C: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ B @ A )
% 2.60/1.72 => ( ( ord_less_eq_real @ C @ B )
% 2.60/1.72 => ( ord_less_eq_real @ C @ A ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_176_dual__order_Oeq__iff,axiom,
% 2.60/1.72 ( ( ^ [Y4: set_Sum_sum_c_c,Z: set_Sum_sum_c_c] : ( Y4 = Z ) )
% 2.60/1.72 = ( ^ [A7: set_Sum_sum_c_c,B7: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B7 @ A7 )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ A7 @ B7 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_177_dual__order_Oeq__iff,axiom,
% 2.60/1.72 ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
% 2.60/1.72 = ( ^ [A7: real,B7: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ B7 @ A7 )
% 2.60/1.72 & ( ord_less_eq_real @ A7 @ B7 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_178_dual__order_Oantisym,axiom,
% 2.60/1.72 ! [B: set_Sum_sum_c_c,A: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B @ A )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ A @ B )
% 2.60/1.72 => ( A = B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_179_dual__order_Oantisym,axiom,
% 2.60/1.72 ! [B: real,A: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ B @ A )
% 2.60/1.72 => ( ( ord_less_eq_real @ A @ B )
% 2.60/1.72 => ( A = B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_180_iff__sem,axiom,
% 2.60/1.72 ! [Nu: produc190496183real_c,I: denota231621370t_unit,A3: formula_a_b_c,B3: formula_a_b_c] :
% 2.60/1.72 ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ ( equiv_a_b_c @ A3 @ B3 ) ) )
% 2.60/1.72 = ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
% 2.60/1.72 = ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_181_IHF,axiom,
% 2.60/1.72 ! [A6: finite1398487019real_c,B6: finite1398487019real_c,Aa: finite1398487019real_c,Ba: finite1398487019real_c] :
% 2.60/1.72 ( ( denota69106024_a_b_c @ i @ j @ ( static_SIGF_a_b_c @ p ) )
% 2.60/1.72 => ( ( denota1997846518gree_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( produc394644079real_c @ Aa @ Ba ) @ ( static_FVF_a_b_c @ p ) )
% 2.60/1.72 => ( ( member1895684704real_c @ ( produc394644079real_c @ A6 @ B6 ) @ ( denota968303861_a_b_c @ i @ p ) )
% 2.60/1.72 = ( member1895684704real_c @ ( produc394644079real_c @ Aa @ Ba ) @ ( denota968303861_a_b_c @ j @ p ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_182_minus__le__self__iff,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 2.60/1.72 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_183_le__minus__self__iff,axiom,
% 2.60/1.72 ! [A: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 2.60/1.72 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_184_real__eq__0__iff__le__ge__0,axiom,
% 2.60/1.72 ! [X3: real] :
% 2.60/1.72 ( ( X3 = zero_zero_real )
% 2.60/1.72 = ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 2.60/1.72 & ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_185_le__numeral__extra_I3_J,axiom,
% 2.60/1.72 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 2.60/1.72
% 2.60/1.72 thf(fact_186_coincide__fml__def,axiom,
% 2.60/1.72 ( coinci501627771_a_b_c
% 2.60/1.72 = ( ^ [Phi: formula_a_b_c] :
% 2.60/1.72 ! [Nu3: produc190496183real_c,Nu4: produc190496183real_c,I3: denota231621370t_unit,J2: denota231621370t_unit] :
% 2.60/1.72 ( ( denota69106024_a_b_c @ I3 @ J2 @ ( static_SIGF_a_b_c @ Phi ) )
% 2.60/1.72 => ( ( denota1997846518gree_c @ Nu3 @ Nu4 @ ( static_FVF_a_b_c @ Phi ) )
% 2.60/1.72 => ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I3 @ Phi ) )
% 2.60/1.72 = ( member1895684704real_c @ Nu4 @ ( denota968303861_a_b_c @ J2 @ Phi ) ) ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_187_IAP,axiom,
% 2.60/1.72 denota69106024_a_b_c @ i @ j @ ( static_SIGF_a_b_c @ p ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_188_Iagree__sub,axiom,
% 2.60/1.72 ! [A3: set_Su1783761653um_b_c,B3: set_Su1783761653um_b_c,I: denota231621370t_unit,J3: denota231621370t_unit] :
% 2.60/1.72 ( ( ord_le929608341um_b_c @ A3 @ B3 )
% 2.60/1.72 => ( ( denota69106024_a_b_c @ I @ J3 @ B3 )
% 2.60/1.72 => ( denota69106024_a_b_c @ I @ J3 @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_189_Iagree__comm,axiom,
% 2.60/1.72 ! [A3: denota231621370t_unit,B3: denota231621370t_unit,V: set_Su1783761653um_b_c] :
% 2.60/1.72 ( ( denota69106024_a_b_c @ A3 @ B3 @ V )
% 2.60/1.72 => ( denota69106024_a_b_c @ B3 @ A3 @ V ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_190_Iagree__refl,axiom,
% 2.60/1.72 ! [I: denota231621370t_unit,A3: set_Su1783761653um_b_c] : ( denota69106024_a_b_c @ I @ I @ A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_191__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AJ_092_060close_062,axiom,
% 2.60/1.72 coinci772602909_a_b_c @ ( evolveODE_a_c_b @ ode @ p ) @ j ).
% 2.60/1.72
% 2.60/1.72 thf(fact_192__092_060open_062ode__sem__equiv_A_IEvolveODE_AODE_AP_J_AI_092_060close_062,axiom,
% 2.60/1.72 coinci772602909_a_b_c @ ( evolveODE_a_c_b @ ode @ p ) @ i ).
% 2.60/1.72
% 2.60/1.72 thf(fact_193_impl__sem,axiom,
% 2.60/1.72 ! [Nu: produc190496183real_c,I: denota231621370t_unit,A3: formula_a_b_c,B3: formula_a_b_c] :
% 2.60/1.72 ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ ( implies_a_b_c @ A3 @ B3 ) ) )
% 2.60/1.72 = ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ A3 ) )
% 2.60/1.72 => ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ B3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_194_equiv,axiom,
% 2.60/1.72 ! [I: denota231621370t_unit] : ( coinci772602909_a_b_c @ ( evolveODE_a_c_b @ ode @ p ) @ I ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_195_coincide__hp_H__def,axiom,
% 2.60/1.72 ( coinci1852866155_a_b_c
% 2.60/1.72 = ( ^ [Alpha: hp_a_b_c] :
% 2.60/1.72 ! [I3: denota231621370t_unit,J2: denota231621370t_unit] :
% 2.60/1.72 ( ( coinci354700372_a_b_c @ Alpha @ I3 @ J2 )
% 2.60/1.72 & ( coinci772602909_a_b_c @ Alpha @ I3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_196_hp_Oinject_I5_J,axiom,
% 2.60/1.72 ! [X51: oDE_a_c,X52: formula_a_b_c,Y51: oDE_a_c,Y52: formula_a_b_c] :
% 2.60/1.72 ( ( ( evolveODE_a_c_b @ X51 @ X52 )
% 2.60/1.72 = ( evolveODE_a_c_b @ Y51 @ Y52 ) )
% 2.60/1.72 = ( ( X51 = Y51 )
% 2.60/1.72 & ( X52 = Y52 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_197_sol,axiom,
% 2.60/1.72 ( initia1631504802real_c @ sol
% 2.60/1.72 @ ^ [A7: real] : ( denota1275485728_a_b_c @ i @ ode )
% 2.60/1.72 @ ( set_or656347191t_real @ zero_zero_real @ t )
% 2.60/1.72 @ ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] : ( member1895684704real_c @ ( denota161327353_a_b_c @ i @ ode @ ( produc394644079real_c @ a @ b ) @ X4 ) @ ( denota968303861_a_b_c @ i @ p ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_198_less__eq__set__def,axiom,
% 2.60/1.72 ( ord_le977353143real_c
% 2.60/1.72 = ( ^ [A5: set_Pr1389752855real_c,B5: set_Pr1389752855real_c] :
% 2.60/1.72 ( ord_le1132241894al_c_o
% 2.60/1.72 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A5 )
% 2.60/1.72 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ B5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_199_less__eq__set__def,axiom,
% 2.60/1.72 ( ord_le764260924um_c_d
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_d,B5: set_Sum_sum_c_d] :
% 2.60/1.72 ( ord_le2143422793_c_d_o
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ A5 )
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ B5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_200_less__eq__set__def,axiom,
% 2.60/1.72 ( ord_le1772180283um_c_c
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_c,B5: set_Sum_sum_c_c] :
% 2.60/1.72 ( ord_le1315653386_c_c_o
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ A5 )
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ B5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_201_Collect__subset,axiom,
% 2.60/1.72 ! [A3: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
% 2.60/1.72 ( ord_le977353143real_c
% 2.60/1.72 @ ( collec1643251106real_c
% 2.60/1.72 @ ^ [X4: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_202_Collect__subset,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 2.60/1.72 ( ord_le764260924um_c_d
% 2.60/1.72 @ ( collect_Sum_sum_c_d
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_203_Collect__subset,axiom,
% 2.60/1.72 ! [A3: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
% 2.60/1.72 ( ord_le1327118209real_c
% 2.60/1.72 @ ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] :
% 2.60/1.72 ( ( member1261661570real_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_204_Collect__subset,axiom,
% 2.60/1.72 ! [A3: set_c,P2: c > $o] :
% 2.60/1.72 ( ord_less_eq_set_c
% 2.60/1.72 @ ( collect_c
% 2.60/1.72 @ ^ [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_205_Collect__subset,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 2.60/1.72 ( ord_le1772180283um_c_c
% 2.60/1.72 @ ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ A3 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_206_Compl__eq,axiom,
% 2.60/1.72 ( uminus232257166real_c
% 2.60/1.72 = ( ^ [A5: set_Pr1389752855real_c] :
% 2.60/1.72 ( collec1643251106real_c
% 2.60/1.72 @ ^ [X4: produc190496183real_c] :
% 2.60/1.72 ~ ( member1895684704real_c @ X4 @ A5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_207_Compl__eq,axiom,
% 2.60/1.72 ( uminus373867045um_c_d
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_d] :
% 2.60/1.72 ( collect_Sum_sum_c_d
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] :
% 2.60/1.72 ~ ( member_Sum_sum_c_d @ X4 @ A5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_208_Compl__eq,axiom,
% 2.60/1.72 ( uminus102402154real_c
% 2.60/1.72 = ( ^ [A5: set_Fi1407883041real_c] :
% 2.60/1.72 ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] :
% 2.60/1.72 ~ ( member1261661570real_c @ X4 @ A5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_209_Compl__eq,axiom,
% 2.60/1.72 ( uminus_uminus_set_c
% 2.60/1.72 = ( ^ [A5: set_c] :
% 2.60/1.72 ( collect_c
% 2.60/1.72 @ ^ [X4: c] :
% 2.60/1.72 ~ ( member_c @ X4 @ A5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_210_Compl__eq,axiom,
% 2.60/1.72 ( uminus1381786404um_c_c
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_c] :
% 2.60/1.72 ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] :
% 2.60/1.72 ~ ( member_Sum_sum_c_c @ X4 @ A5 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_211_Collect__neg__eq,axiom,
% 2.60/1.72 ! [P2: finite1398487019real_c > $o] :
% 2.60/1.72 ( ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] :
% 2.60/1.72 ~ ( P2 @ X4 ) )
% 2.60/1.72 = ( uminus102402154real_c @ ( collec230941376real_c @ P2 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_212_Collect__neg__eq,axiom,
% 2.60/1.72 ! [P2: c > $o] :
% 2.60/1.72 ( ( collect_c
% 2.60/1.72 @ ^ [X4: c] :
% 2.60/1.72 ~ ( P2 @ X4 ) )
% 2.60/1.72 = ( uminus_uminus_set_c @ ( collect_c @ P2 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_213_Collect__neg__eq,axiom,
% 2.60/1.72 ! [P2: sum_sum_c_c > $o] :
% 2.60/1.72 ( ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] :
% 2.60/1.72 ~ ( P2 @ X4 ) )
% 2.60/1.72 = ( uminus1381786404um_c_c @ ( collect_Sum_sum_c_c @ P2 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_214_uminus__set__def,axiom,
% 2.60/1.72 ( uminus232257166real_c
% 2.60/1.72 = ( ^ [A5: set_Pr1389752855real_c] :
% 2.60/1.72 ( collec1643251106real_c
% 2.60/1.72 @ ( uminus32155087al_c_o
% 2.60/1.72 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A5 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_215_uminus__set__def,axiom,
% 2.60/1.72 ( uminus373867045um_c_d
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_d] :
% 2.60/1.72 ( collect_Sum_sum_c_d
% 2.60/1.72 @ ( uminus916499104_c_d_o
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ A5 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_216_uminus__set__def,axiom,
% 2.60/1.72 ( uminus102402154real_c
% 2.60/1.72 = ( ^ [A5: set_Fi1407883041real_c] :
% 2.60/1.72 ( collec230941376real_c
% 2.60/1.72 @ ( uminus23580315al_c_o
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] : ( member1261661570real_c @ X4 @ A5 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_217_uminus__set__def,axiom,
% 2.60/1.72 ( uminus_uminus_set_c
% 2.60/1.72 = ( ^ [A5: set_c] :
% 2.60/1.72 ( collect_c
% 2.60/1.72 @ ( uminus_uminus_c_o
% 2.60/1.72 @ ^ [X4: c] : ( member_c @ X4 @ A5 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_218_uminus__set__def,axiom,
% 2.60/1.72 ( uminus1381786404um_c_c
% 2.60/1.72 = ( ^ [A5: set_Sum_sum_c_c] :
% 2.60/1.72 ( collect_Sum_sum_c_c
% 2.60/1.72 @ ( uminus88729697_c_c_o
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ A5 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_219_mk__xode_Oelims,axiom,
% 2.60/1.72 ! [X3: denota231621370t_unit,Xa: oDE_a_c,Xb: finite1398487019real_c,Y3: produc190496183real_c] :
% 2.60/1.72 ( ( ( denota1896987029_a_b_c @ X3 @ Xa @ Xb )
% 2.60/1.72 = Y3 )
% 2.60/1.72 => ( Y3
% 2.60/1.72 = ( produc394644079real_c @ Xb @ ( denota1275485728_a_b_c @ X3 @ Xa @ Xb ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_220_mk__xode_Osimps,axiom,
% 2.60/1.72 ( denota1896987029_a_b_c
% 2.60/1.72 = ( ^ [I3: denota231621370t_unit,ODE2: oDE_a_c,Sol2: finite1398487019real_c] : ( produc394644079real_c @ Sol2 @ ( denota1275485728_a_b_c @ I3 @ ODE2 @ Sol2 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_221_atLeastatMost__subset__iff,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,B: set_Sum_sum_c_c,C: set_Sum_sum_c_c,D: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le657554289um_c_c @ ( set_or935154662um_c_c @ A @ B ) @ ( set_or935154662um_c_c @ C @ D ) )
% 2.60/1.72 = ( ~ ( ord_le1772180283um_c_c @ A @ B )
% 2.60/1.72 | ( ( ord_le1772180283um_c_c @ C @ A )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ B @ D ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_222_atLeastatMost__subset__iff,axiom,
% 2.60/1.72 ! [A: real,B: real,C: real,D: real] :
% 2.60/1.72 ( ( ord_less_eq_set_real @ ( set_or656347191t_real @ A @ B ) @ ( set_or656347191t_real @ C @ D ) )
% 2.60/1.72 = ( ~ ( ord_less_eq_real @ A @ B )
% 2.60/1.72 | ( ( ord_less_eq_real @ C @ A )
% 2.60/1.72 & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_223_atLeastAtMost__iff,axiom,
% 2.60/1.72 ! [I4: produc190496183real_c,L: produc190496183real_c,U: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ I4 @ ( set_or385782508real_c @ L @ U ) )
% 2.60/1.72 = ( ( ord_le850691415real_c @ L @ I4 )
% 2.60/1.72 & ( ord_le850691415real_c @ I4 @ U ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_224_atLeastAtMost__iff,axiom,
% 2.60/1.72 ! [I4: set_Sum_sum_c_c,L: set_Sum_sum_c_c,U: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( member2124950898um_c_c @ I4 @ ( set_or935154662um_c_c @ L @ U ) )
% 2.60/1.72 = ( ( ord_le1772180283um_c_c @ L @ I4 )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ I4 @ U ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_225_atLeastAtMost__iff,axiom,
% 2.60/1.72 ! [I4: real,L: real,U: real] :
% 2.60/1.72 ( ( member_real @ I4 @ ( set_or656347191t_real @ L @ U ) )
% 2.60/1.72 = ( ( ord_less_eq_real @ L @ I4 )
% 2.60/1.72 & ( ord_less_eq_real @ I4 @ U ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_226_Icc__eq__Icc,axiom,
% 2.60/1.72 ! [L: set_Sum_sum_c_c,H: set_Sum_sum_c_c,L2: set_Sum_sum_c_c,H2: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ( set_or935154662um_c_c @ L @ H )
% 2.60/1.72 = ( set_or935154662um_c_c @ L2 @ H2 ) )
% 2.60/1.72 = ( ( ( L = L2 )
% 2.60/1.72 & ( H = H2 ) )
% 2.60/1.72 | ( ~ ( ord_le1772180283um_c_c @ L @ H )
% 2.60/1.72 & ~ ( ord_le1772180283um_c_c @ L2 @ H2 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_227_Icc__eq__Icc,axiom,
% 2.60/1.72 ! [L: real,H: real,L2: real,H2: real] :
% 2.60/1.72 ( ( ( set_or656347191t_real @ L @ H )
% 2.60/1.72 = ( set_or656347191t_real @ L2 @ H2 ) )
% 2.60/1.72 = ( ( ( L = L2 )
% 2.60/1.72 & ( H = H2 ) )
% 2.60/1.72 | ( ~ ( ord_less_eq_real @ L @ H )
% 2.60/1.72 & ~ ( ord_less_eq_real @ L2 @ H2 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_228_pred__subset__eq,axiom,
% 2.60/1.72 ! [R2: set_Pr1389752855real_c,S2: set_Pr1389752855real_c] :
% 2.60/1.72 ( ( ord_le1132241894al_c_o
% 2.60/1.72 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ R2 )
% 2.60/1.72 @ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ S2 ) )
% 2.60/1.72 = ( ord_le977353143real_c @ R2 @ S2 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_229_pred__subset__eq,axiom,
% 2.60/1.72 ! [R2: set_Sum_sum_c_d,S2: set_Sum_sum_c_d] :
% 2.60/1.72 ( ( ord_le2143422793_c_d_o
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ R2 )
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] : ( member_Sum_sum_c_d @ X4 @ S2 ) )
% 2.60/1.72 = ( ord_le764260924um_c_d @ R2 @ S2 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_230_pred__subset__eq,axiom,
% 2.60/1.72 ! [R2: set_Sum_sum_c_c,S2: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1315653386_c_c_o
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ R2 )
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] : ( member_Sum_sum_c_c @ X4 @ S2 ) )
% 2.60/1.72 = ( ord_le1772180283um_c_c @ R2 @ S2 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_231_solDeriv,axiom,
% 2.60/1.72 ! [X3: real] :
% 2.60/1.72 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 2.60/1.72 => ( ( ord_less_eq_real @ X3 @ t )
% 2.60/1.72 => ( has_ve1904308670real_c @ sol @ ( denota1275485728_a_b_c @ i @ ode @ ( sol @ X3 ) ) @ ( topolo1830106092n_real @ X3 @ ( set_or656347191t_real @ zero_zero_real @ t ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_232_mk__v__def,axiom,
% 2.60/1.72 ( denota161327353_a_b_c
% 2.60/1.72 = ( ^ [I3: denota231621370t_unit,ODE2: oDE_a_c,Nu3: produc190496183real_c,Sol2: finite1398487019real_c] :
% 2.60/1.72 ( the_Pr1290640446real_c
% 2.60/1.72 @ ^ [Omega3: produc190496183real_c] :
% 2.60/1.72 ( ( denota1997846518gree_c @ Omega3 @ Nu3 @ ( uminus1381786404um_c_c @ ( denota1293057250_a_b_c @ I3 @ ODE2 ) ) )
% 2.60/1.72 & ( denota1997846518gree_c @ Omega3 @ ( denota1896987029_a_b_c @ I3 @ ODE2 @ Sol2 ) @ ( denota1293057250_a_b_c @ I3 @ ODE2 ) ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_233_subrelI,axiom,
% 2.60/1.72 ! [R3: set_Pr1389752855real_c,S: set_Pr1389752855real_c] :
% 2.60/1.72 ( ! [X: finite1398487019real_c,Y: finite1398487019real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ ( produc394644079real_c @ X @ Y ) @ R3 )
% 2.60/1.72 => ( member1895684704real_c @ ( produc394644079real_c @ X @ Y ) @ S ) )
% 2.60/1.72 => ( ord_le977353143real_c @ R3 @ S ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_234_pred__subset__eq2,axiom,
% 2.60/1.72 ! [R2: set_Pr1389752855real_c,S2: set_Pr1389752855real_c] :
% 2.60/1.72 ( ( ord_le1382916304al_c_o
% 2.60/1.72 @ ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ R2 )
% 2.60/1.72 @ ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ S2 ) )
% 2.60/1.72 = ( ord_le977353143real_c @ R2 @ S2 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_235_pred__equals__eq2,axiom,
% 2.60/1.72 ! [R2: set_Pr1389752855real_c,S2: set_Pr1389752855real_c] :
% 2.60/1.72 ( ( ( ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ R2 ) )
% 2.60/1.72 = ( ^ [X4: finite1398487019real_c,Y6: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y6 ) @ S2 ) ) )
% 2.60/1.72 = ( R2 = S2 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_236_has__vector__derivative__Pair,axiom,
% 2.60/1.72 ! [F: real > finite1398487019real_c,F2: finite1398487019real_c,X3: real,S: set_real,G: real > finite1398487019real_c,G2: finite1398487019real_c] :
% 2.60/1.72 ( ( has_ve1904308670real_c @ F @ F2 @ ( topolo1830106092n_real @ X3 @ S ) )
% 2.60/1.72 => ( ( has_ve1904308670real_c @ G @ G2 @ ( topolo1830106092n_real @ X3 @ S ) )
% 2.60/1.72 => ( has_ve373335844real_c
% 2.60/1.72 @ ^ [X4: real] : ( produc394644079real_c @ ( F @ X4 ) @ ( G @ X4 ) )
% 2.60/1.72 @ ( produc394644079real_c @ F2 @ G2 )
% 2.60/1.72 @ ( topolo1830106092n_real @ X3 @ S ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_237_has__vector__derivative__minus,axiom,
% 2.60/1.72 ! [F: real > real,F2: real,Net: filter_real] :
% 2.60/1.72 ( ( has_ve217091583e_real @ F @ F2 @ Net )
% 2.60/1.72 => ( has_ve217091583e_real
% 2.60/1.72 @ ^ [X4: real] : ( uminus_uminus_real @ ( F @ X4 ) )
% 2.60/1.72 @ ( uminus_uminus_real @ F2 )
% 2.60/1.72 @ Net ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_238_has__vector__derivative__minus,axiom,
% 2.60/1.72 ! [F: real > finite1398487019real_c,F2: finite1398487019real_c,Net: filter_real] :
% 2.60/1.72 ( ( has_ve1904308670real_c @ F @ F2 @ Net )
% 2.60/1.72 => ( has_ve1904308670real_c
% 2.60/1.72 @ ^ [X4: real] : ( uminus1737280820real_c @ ( F @ X4 ) )
% 2.60/1.72 @ ( uminus1737280820real_c @ F2 )
% 2.60/1.72 @ Net ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_239_has__vector__derivative__const,axiom,
% 2.60/1.72 ! [C: real,Net: filter_real] :
% 2.60/1.72 ( has_ve217091583e_real
% 2.60/1.72 @ ^ [X4: real] : C
% 2.60/1.72 @ zero_zero_real
% 2.60/1.72 @ Net ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_240_has__vector__derivative__const,axiom,
% 2.60/1.72 ! [C: finite1398487019real_c,Net: filter_real] :
% 2.60/1.72 ( has_ve1904308670real_c
% 2.60/1.72 @ ^ [X4: real] : C
% 2.60/1.72 @ zero_z109254132real_c
% 2.60/1.72 @ Net ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_241_has__vector__derivative__within__subset,axiom,
% 2.60/1.72 ! [F: real > finite1398487019real_c,F2: finite1398487019real_c,X3: real,S2: set_real,T2: set_real] :
% 2.60/1.72 ( ( has_ve1904308670real_c @ F @ F2 @ ( topolo1830106092n_real @ X3 @ S2 ) )
% 2.60/1.72 => ( ( ord_less_eq_set_real @ T2 @ S2 )
% 2.60/1.72 => ( has_ve1904308670real_c @ F @ F2 @ ( topolo1830106092n_real @ X3 @ T2 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_242_has__vector__derivative__weaken,axiom,
% 2.60/1.72 ! [F: real > finite1398487019real_c,D2: finite1398487019real_c,X3: real,T2: set_real,S2: set_real,G: real > finite1398487019real_c] :
% 2.60/1.72 ( ( has_ve1904308670real_c @ F @ D2 @ ( topolo1830106092n_real @ X3 @ T2 ) )
% 2.60/1.72 => ( ( member_real @ X3 @ S2 )
% 2.60/1.72 => ( ( ord_less_eq_set_real @ S2 @ T2 )
% 2.60/1.72 => ( ! [X: real] :
% 2.60/1.72 ( ( member_real @ X @ S2 )
% 2.60/1.72 => ( ( F @ X )
% 2.60/1.72 = ( G @ X ) ) )
% 2.60/1.72 => ( has_ve1904308670real_c @ G @ D2 @ ( topolo1830106092n_real @ X3 @ S2 ) ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_243_solves__ode__supset__range,axiom,
% 2.60/1.72 ! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,T2: set_real,X6: set_Fi1407883041real_c,Y7: set_Fi1407883041real_c] :
% 2.60/1.72 ( ( initia1631504802real_c @ X3 @ F @ T2 @ X6 )
% 2.60/1.72 => ( ( ord_le1327118209real_c @ X6 @ Y7 )
% 2.60/1.72 => ( initia1631504802real_c @ X3 @ F @ T2 @ Y7 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_244_solves__ode__subset,axiom,
% 2.60/1.72 ! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,T2: set_real,X6: set_Fi1407883041real_c,S2: set_real] :
% 2.60/1.72 ( ( initia1631504802real_c @ X3 @ F @ T2 @ X6 )
% 2.60/1.72 => ( ( ord_less_eq_set_real @ S2 @ T2 )
% 2.60/1.72 => ( initia1631504802real_c @ X3 @ F @ S2 @ X6 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_245_solves__ode__on__subset,axiom,
% 2.60/1.72 ! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,S2: set_real,Y7: set_Fi1407883041real_c,T2: set_real,X6: set_Fi1407883041real_c] :
% 2.60/1.72 ( ( initia1631504802real_c @ X3 @ F @ S2 @ Y7 )
% 2.60/1.72 => ( ( ord_less_eq_set_real @ T2 @ S2 )
% 2.60/1.72 => ( ( ord_le1327118209real_c @ Y7 @ X6 )
% 2.60/1.72 => ( initia1631504802real_c @ X3 @ F @ T2 @ X6 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_246_at__le,axiom,
% 2.60/1.72 ! [S: set_real,T3: set_real,X3: real] :
% 2.60/1.72 ( ( ord_less_eq_set_real @ S @ T3 )
% 2.60/1.72 => ( ord_le132810396r_real @ ( topolo1830106092n_real @ X3 @ S ) @ ( topolo1830106092n_real @ X3 @ T3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_247_intervalE,axiom,
% 2.60/1.72 ! [A: produc190496183real_c,X3: produc190496183real_c,B: produc190496183real_c] :
% 2.60/1.72 ( ( ( ord_le850691415real_c @ A @ X3 )
% 2.60/1.72 & ( ord_le850691415real_c @ X3 @ B ) )
% 2.60/1.72 => ( member1895684704real_c @ X3 @ ( set_or385782508real_c @ A @ B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_248_intervalE,axiom,
% 2.60/1.72 ! [A: set_Sum_sum_c_c,X3: set_Sum_sum_c_c,B: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ( ord_le1772180283um_c_c @ A @ X3 )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ X3 @ B ) )
% 2.60/1.72 => ( member2124950898um_c_c @ X3 @ ( set_or935154662um_c_c @ A @ B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_249_intervalE,axiom,
% 2.60/1.72 ! [A: real,X3: real,B: real] :
% 2.60/1.72 ( ( ( ord_less_eq_real @ A @ X3 )
% 2.60/1.72 & ( ord_less_eq_real @ X3 @ B ) )
% 2.60/1.72 => ( member_real @ X3 @ ( set_or656347191t_real @ A @ B ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_250_conj__subset__def,axiom,
% 2.60/1.72 ! [A3: set_Fi1407883041real_c,P2: finite1398487019real_c > $o,Q: finite1398487019real_c > $o] :
% 2.60/1.72 ( ( ord_le1327118209real_c @ A3
% 2.60/1.72 @ ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] :
% 2.60/1.72 ( ( P2 @ X4 )
% 2.60/1.72 & ( Q @ X4 ) ) ) )
% 2.60/1.72 = ( ( ord_le1327118209real_c @ A3 @ ( collec230941376real_c @ P2 ) )
% 2.60/1.72 & ( ord_le1327118209real_c @ A3 @ ( collec230941376real_c @ Q ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_251_conj__subset__def,axiom,
% 2.60/1.72 ! [A3: set_c,P2: c > $o,Q: c > $o] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ A3
% 2.60/1.72 @ ( collect_c
% 2.60/1.72 @ ^ [X4: c] :
% 2.60/1.72 ( ( P2 @ X4 )
% 2.60/1.72 & ( Q @ X4 ) ) ) )
% 2.60/1.72 = ( ( ord_less_eq_set_c @ A3 @ ( collect_c @ P2 ) )
% 2.60/1.72 & ( ord_less_eq_set_c @ A3 @ ( collect_c @ Q ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_252_conj__subset__def,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,P2: sum_sum_c_c > $o,Q: sum_sum_c_c > $o] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A3
% 2.60/1.72 @ ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( P2 @ X4 )
% 2.60/1.72 & ( Q @ X4 ) ) ) )
% 2.60/1.72 = ( ( ord_le1772180283um_c_c @ A3 @ ( collect_Sum_sum_c_c @ P2 ) )
% 2.60/1.72 & ( ord_le1772180283um_c_c @ A3 @ ( collect_Sum_sum_c_c @ Q ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_253_subset__Collect__iff,axiom,
% 2.60/1.72 ! [B3: set_Pr1389752855real_c,A3: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
% 2.60/1.72 ( ( ord_le977353143real_c @ B3 @ A3 )
% 2.60/1.72 => ( ( ord_le977353143real_c @ B3
% 2.60/1.72 @ ( collec1643251106real_c
% 2.60/1.72 @ ^ [X4: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 = ( ! [X4: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X4 @ B3 )
% 2.60/1.72 => ( P2 @ X4 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_254_subset__Collect__iff,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_d,A3: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 2.60/1.72 ( ( ord_le764260924um_c_d @ B3 @ A3 )
% 2.60/1.72 => ( ( ord_le764260924um_c_d @ B3
% 2.60/1.72 @ ( collect_Sum_sum_c_d
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 = ( ! [X4: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X4 @ B3 )
% 2.60/1.72 => ( P2 @ X4 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_255_subset__Collect__iff,axiom,
% 2.60/1.72 ! [B3: set_Fi1407883041real_c,A3: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
% 2.60/1.72 ( ( ord_le1327118209real_c @ B3 @ A3 )
% 2.60/1.72 => ( ( ord_le1327118209real_c @ B3
% 2.60/1.72 @ ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] :
% 2.60/1.72 ( ( member1261661570real_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 = ( ! [X4: finite1398487019real_c] :
% 2.60/1.72 ( ( member1261661570real_c @ X4 @ B3 )
% 2.60/1.72 => ( P2 @ X4 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_256_subset__Collect__iff,axiom,
% 2.60/1.72 ! [B3: set_c,A3: set_c,P2: c > $o] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ B3 @ A3 )
% 2.60/1.72 => ( ( ord_less_eq_set_c @ B3
% 2.60/1.72 @ ( collect_c
% 2.60/1.72 @ ^ [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 = ( ! [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ B3 )
% 2.60/1.72 => ( P2 @ X4 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_257_subset__Collect__iff,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_c,A3: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B3 @ A3 )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ B3
% 2.60/1.72 @ ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 = ( ! [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X4 @ B3 )
% 2.60/1.72 => ( P2 @ X4 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_258_subset__CollectI,axiom,
% 2.60/1.72 ! [B3: set_Pr1389752855real_c,A3: set_Pr1389752855real_c,Q: produc190496183real_c > $o,P2: produc190496183real_c > $o] :
% 2.60/1.72 ( ( ord_le977353143real_c @ B3 @ A3 )
% 2.60/1.72 => ( ! [X: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X @ B3 )
% 2.60/1.72 => ( ( Q @ X )
% 2.60/1.72 => ( P2 @ X ) ) )
% 2.60/1.72 => ( ord_le977353143real_c
% 2.60/1.72 @ ( collec1643251106real_c
% 2.60/1.72 @ ^ [X4: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X4 @ B3 )
% 2.60/1.72 & ( Q @ X4 ) ) )
% 2.60/1.72 @ ( collec1643251106real_c
% 2.60/1.72 @ ^ [X4: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_259_subset__CollectI,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_d,A3: set_Sum_sum_c_d,Q: sum_sum_c_d > $o,P2: sum_sum_c_d > $o] :
% 2.60/1.72 ( ( ord_le764260924um_c_d @ B3 @ A3 )
% 2.60/1.72 => ( ! [X: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X @ B3 )
% 2.60/1.72 => ( ( Q @ X )
% 2.60/1.72 => ( P2 @ X ) ) )
% 2.60/1.72 => ( ord_le764260924um_c_d
% 2.60/1.72 @ ( collect_Sum_sum_c_d
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X4 @ B3 )
% 2.60/1.72 & ( Q @ X4 ) ) )
% 2.60/1.72 @ ( collect_Sum_sum_c_d
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_260_subset__CollectI,axiom,
% 2.60/1.72 ! [B3: set_Fi1407883041real_c,A3: set_Fi1407883041real_c,Q: finite1398487019real_c > $o,P2: finite1398487019real_c > $o] :
% 2.60/1.72 ( ( ord_le1327118209real_c @ B3 @ A3 )
% 2.60/1.72 => ( ! [X: finite1398487019real_c] :
% 2.60/1.72 ( ( member1261661570real_c @ X @ B3 )
% 2.60/1.72 => ( ( Q @ X )
% 2.60/1.72 => ( P2 @ X ) ) )
% 2.60/1.72 => ( ord_le1327118209real_c
% 2.60/1.72 @ ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] :
% 2.60/1.72 ( ( member1261661570real_c @ X4 @ B3 )
% 2.60/1.72 & ( Q @ X4 ) ) )
% 2.60/1.72 @ ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] :
% 2.60/1.72 ( ( member1261661570real_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_261_subset__CollectI,axiom,
% 2.60/1.72 ! [B3: set_c,A3: set_c,Q: c > $o,P2: c > $o] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ B3 @ A3 )
% 2.60/1.72 => ( ! [X: c] :
% 2.60/1.72 ( ( member_c @ X @ B3 )
% 2.60/1.72 => ( ( Q @ X )
% 2.60/1.72 => ( P2 @ X ) ) )
% 2.60/1.72 => ( ord_less_eq_set_c
% 2.60/1.72 @ ( collect_c
% 2.60/1.72 @ ^ [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ B3 )
% 2.60/1.72 & ( Q @ X4 ) ) )
% 2.60/1.72 @ ( collect_c
% 2.60/1.72 @ ^ [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_262_subset__CollectI,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_c,A3: set_Sum_sum_c_c,Q: sum_sum_c_c > $o,P2: sum_sum_c_c > $o] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B3 @ A3 )
% 2.60/1.72 => ( ! [X: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X @ B3 )
% 2.60/1.72 => ( ( Q @ X )
% 2.60/1.72 => ( P2 @ X ) ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c
% 2.60/1.72 @ ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X4 @ B3 )
% 2.60/1.72 & ( Q @ X4 ) ) )
% 2.60/1.72 @ ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X4 @ A3 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_263_prop__restrict,axiom,
% 2.60/1.72 ! [X3: produc190496183real_c,Z3: set_Pr1389752855real_c,X6: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
% 2.60/1.72 ( ( member1895684704real_c @ X3 @ Z3 )
% 2.60/1.72 => ( ( ord_le977353143real_c @ Z3
% 2.60/1.72 @ ( collec1643251106real_c
% 2.60/1.72 @ ^ [X4: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 => ( P2 @ X3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_264_prop__restrict,axiom,
% 2.60/1.72 ! [X3: sum_sum_c_d,Z3: set_Sum_sum_c_d,X6: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X3 @ Z3 )
% 2.60/1.72 => ( ( ord_le764260924um_c_d @ Z3
% 2.60/1.72 @ ( collect_Sum_sum_c_d
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 => ( P2 @ X3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_265_prop__restrict,axiom,
% 2.60/1.72 ! [X3: finite1398487019real_c,Z3: set_Fi1407883041real_c,X6: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
% 2.60/1.72 ( ( member1261661570real_c @ X3 @ Z3 )
% 2.60/1.72 => ( ( ord_le1327118209real_c @ Z3
% 2.60/1.72 @ ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] :
% 2.60/1.72 ( ( member1261661570real_c @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 => ( P2 @ X3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_266_prop__restrict,axiom,
% 2.60/1.72 ! [X3: c,Z3: set_c,X6: set_c,P2: c > $o] :
% 2.60/1.72 ( ( member_c @ X3 @ Z3 )
% 2.60/1.72 => ( ( ord_less_eq_set_c @ Z3
% 2.60/1.72 @ ( collect_c
% 2.60/1.72 @ ^ [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 => ( P2 @ X3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_267_prop__restrict,axiom,
% 2.60/1.72 ! [X3: sum_sum_c_c,Z3: set_Sum_sum_c_c,X6: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X3 @ Z3 )
% 2.60/1.72 => ( ( ord_le1772180283um_c_c @ Z3
% 2.60/1.72 @ ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) ) )
% 2.60/1.72 => ( P2 @ X3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_268_Collect__restrict,axiom,
% 2.60/1.72 ! [X6: set_Pr1389752855real_c,P2: produc190496183real_c > $o] :
% 2.60/1.72 ( ord_le977353143real_c
% 2.60/1.72 @ ( collec1643251106real_c
% 2.60/1.72 @ ^ [X4: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ X6 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_269_Collect__restrict,axiom,
% 2.60/1.72 ! [X6: set_Sum_sum_c_d,P2: sum_sum_c_d > $o] :
% 2.60/1.72 ( ord_le764260924um_c_d
% 2.60/1.72 @ ( collect_Sum_sum_c_d
% 2.60/1.72 @ ^ [X4: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ X6 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_270_Collect__restrict,axiom,
% 2.60/1.72 ! [X6: set_Fi1407883041real_c,P2: finite1398487019real_c > $o] :
% 2.60/1.72 ( ord_le1327118209real_c
% 2.60/1.72 @ ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] :
% 2.60/1.72 ( ( member1261661570real_c @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ X6 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_271_Collect__restrict,axiom,
% 2.60/1.72 ! [X6: set_c,P2: c > $o] :
% 2.60/1.72 ( ord_less_eq_set_c
% 2.60/1.72 @ ( collect_c
% 2.60/1.72 @ ^ [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ X6 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_272_Collect__restrict,axiom,
% 2.60/1.72 ! [X6: set_Sum_sum_c_c,P2: sum_sum_c_c > $o] :
% 2.60/1.72 ( ord_le1772180283um_c_c
% 2.60/1.72 @ ( collect_Sum_sum_c_c
% 2.60/1.72 @ ^ [X4: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X4 @ X6 )
% 2.60/1.72 & ( P2 @ X4 ) ) )
% 2.60/1.72 @ X6 ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_273_ssubst__Pair__rhs,axiom,
% 2.60/1.72 ! [R3: finite1398487019real_c,S: finite1398487019real_c,R2: set_Pr1389752855real_c,S3: finite1398487019real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ ( produc394644079real_c @ R3 @ S ) @ R2 )
% 2.60/1.72 => ( ( S3 = S )
% 2.60/1.72 => ( member1895684704real_c @ ( produc394644079real_c @ R3 @ S3 ) @ R2 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_274_hpsafe__ODE__simps,axiom,
% 2.60/1.72 ! [ODE: oDE_a_c,P: formula_a_b_c] :
% 2.60/1.72 ( ( hpsafe_a_b_c @ ( evolveODE_a_c_b @ ODE @ P ) )
% 2.60/1.72 = ( ( osafe_a_c @ ODE )
% 2.60/1.72 & ( fsafe_a_b_c @ P ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_275_hpsafe__fsafe_Ohpsafe__Evolve,axiom,
% 2.60/1.72 ! [ODE: oDE_a_c,P2: formula_a_b_c] :
% 2.60/1.72 ( ( osafe_a_c @ ODE )
% 2.60/1.72 => ( ( fsafe_a_b_c @ P2 )
% 2.60/1.72 => ( hpsafe_a_b_c @ ( evolveODE_a_c_b @ ODE @ P2 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_276_Icc__subset__Ici__iff,axiom,
% 2.60/1.72 ! [L: set_Sum_sum_c_c,H: set_Sum_sum_c_c,L2: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le657554289um_c_c @ ( set_or935154662um_c_c @ L @ H ) @ ( set_or2028906832um_c_c @ L2 ) )
% 2.60/1.72 = ( ~ ( ord_le1772180283um_c_c @ L @ H )
% 2.60/1.72 | ( ord_le1772180283um_c_c @ L2 @ L ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_277_Icc__subset__Ici__iff,axiom,
% 2.60/1.72 ! [L: real,H: real,L2: real] :
% 2.60/1.72 ( ( ord_less_eq_set_real @ ( set_or656347191t_real @ L @ H ) @ ( set_ord_atLeast_real @ L2 ) )
% 2.60/1.72 = ( ~ ( ord_less_eq_real @ L @ H )
% 2.60/1.72 | ( ord_less_eq_real @ L2 @ L ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_278_Icc__subset__Iic__iff,axiom,
% 2.60/1.72 ! [L: set_Sum_sum_c_c,H: set_Sum_sum_c_c,H2: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le657554289um_c_c @ ( set_or935154662um_c_c @ L @ H ) @ ( set_or326480468um_c_c @ H2 ) )
% 2.60/1.72 = ( ~ ( ord_le1772180283um_c_c @ L @ H )
% 2.60/1.72 | ( ord_le1772180283um_c_c @ H @ H2 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_279_Icc__subset__Iic__iff,axiom,
% 2.60/1.72 ! [L: real,H: real,H2: real] :
% 2.60/1.72 ( ( ord_less_eq_set_real @ ( set_or656347191t_real @ L @ H ) @ ( set_ord_atMost_real @ H2 ) )
% 2.60/1.72 = ( ~ ( ord_less_eq_real @ L @ H )
% 2.60/1.72 | ( ord_less_eq_real @ H @ H2 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_280_atMost__iff,axiom,
% 2.60/1.72 ! [I4: produc190496183real_c,K: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ I4 @ ( set_or248124158real_c @ K ) )
% 2.60/1.72 = ( ord_le850691415real_c @ I4 @ K ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_281_atMost__iff,axiom,
% 2.60/1.72 ! [I4: set_Sum_sum_c_c,K: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( member2124950898um_c_c @ I4 @ ( set_or326480468um_c_c @ K ) )
% 2.60/1.72 = ( ord_le1772180283um_c_c @ I4 @ K ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_282_atMost__iff,axiom,
% 2.60/1.72 ! [I4: real,K: real] :
% 2.60/1.72 ( ( member_real @ I4 @ ( set_ord_atMost_real @ K ) )
% 2.60/1.72 = ( ord_less_eq_real @ I4 @ K ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_283_atLeast__iff,axiom,
% 2.60/1.72 ! [I4: produc190496183real_c,K: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ I4 @ ( set_or10334466real_c @ K ) )
% 2.60/1.72 = ( ord_le850691415real_c @ K @ I4 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_284_atLeast__iff,axiom,
% 2.60/1.72 ! [I4: set_Sum_sum_c_c,K: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( member2124950898um_c_c @ I4 @ ( set_or2028906832um_c_c @ K ) )
% 2.60/1.72 = ( ord_le1772180283um_c_c @ K @ I4 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_285_atLeast__iff,axiom,
% 2.60/1.72 ! [I4: real,K: real] :
% 2.60/1.72 ( ( member_real @ I4 @ ( set_ord_atLeast_real @ K ) )
% 2.60/1.72 = ( ord_less_eq_real @ K @ I4 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_286_atMost__subset__iff,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le657554289um_c_c @ ( set_or326480468um_c_c @ X3 ) @ ( set_or326480468um_c_c @ Y3 ) )
% 2.60/1.72 = ( ord_le1772180283um_c_c @ X3 @ Y3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_287_atMost__subset__iff,axiom,
% 2.60/1.72 ! [X3: real,Y3: real] :
% 2.60/1.72 ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X3 ) @ ( set_ord_atMost_real @ Y3 ) )
% 2.60/1.72 = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_288_atLeast__subset__iff,axiom,
% 2.60/1.72 ! [X3: set_Sum_sum_c_c,Y3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le657554289um_c_c @ ( set_or2028906832um_c_c @ X3 ) @ ( set_or2028906832um_c_c @ Y3 ) )
% 2.60/1.72 = ( ord_le1772180283um_c_c @ Y3 @ X3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_289_atLeast__subset__iff,axiom,
% 2.60/1.72 ! [X3: real,Y3: real] :
% 2.60/1.72 ( ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ X3 ) @ ( set_ord_atLeast_real @ Y3 ) )
% 2.60/1.72 = ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_290_not__Ici__le__Icc,axiom,
% 2.60/1.72 ! [L: real,L2: real,H2: real] :
% 2.60/1.72 ~ ( ord_less_eq_set_real @ ( set_ord_atLeast_real @ L ) @ ( set_or656347191t_real @ L2 @ H2 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_291_not__Iic__le__Icc,axiom,
% 2.60/1.72 ! [H: real,L2: real,H2: real] :
% 2.60/1.72 ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H ) @ ( set_or656347191t_real @ L2 @ H2 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_292_atLeast__def,axiom,
% 2.60/1.72 ( set_or2003963744real_c
% 2.60/1.72 = ( ^ [L3: finite1398487019real_c] : ( collec230941376real_c @ ( ord_le775706699real_c @ L3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_293_atLeast__def,axiom,
% 2.60/1.72 ( set_or2028906832um_c_c
% 2.60/1.72 = ( ^ [L3: set_Sum_sum_c_c] : ( collec95329456um_c_c @ ( ord_le1772180283um_c_c @ L3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_294_atLeast__def,axiom,
% 2.60/1.72 ( set_ord_atLeast_real
% 2.60/1.72 = ( ^ [L3: real] : ( collect_real @ ( ord_less_eq_real @ L3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_295_atMost__def,axiom,
% 2.60/1.72 ( set_or1478397028real_c
% 2.60/1.72 = ( ^ [U2: finite1398487019real_c] :
% 2.60/1.72 ( collec230941376real_c
% 2.60/1.72 @ ^ [X4: finite1398487019real_c] : ( ord_le775706699real_c @ X4 @ U2 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_296_atMost__def,axiom,
% 2.60/1.72 ( set_or326480468um_c_c
% 2.60/1.72 = ( ^ [U2: set_Sum_sum_c_c] :
% 2.60/1.72 ( collec95329456um_c_c
% 2.60/1.72 @ ^ [X4: set_Sum_sum_c_c] : ( ord_le1772180283um_c_c @ X4 @ U2 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_297_atMost__def,axiom,
% 2.60/1.72 ( set_ord_atMost_real
% 2.60/1.72 = ( ^ [U2: real] :
% 2.60/1.72 ( collect_real
% 2.60/1.72 @ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U2 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_298_image__uminus__atMost,axiom,
% 2.60/1.72 ! [X3: real] :
% 2.60/1.72 ( ( image_real_real @ uminus_uminus_real @ ( set_ord_atMost_real @ X3 ) )
% 2.60/1.72 = ( set_ord_atLeast_real @ ( uminus_uminus_real @ X3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_299_image__uminus__atLeast,axiom,
% 2.60/1.72 ! [X3: real] :
% 2.60/1.72 ( ( image_real_real @ uminus_uminus_real @ ( set_ord_atLeast_real @ X3 ) )
% 2.60/1.72 = ( set_ord_atMost_real @ ( uminus_uminus_real @ X3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_300_image__eqI,axiom,
% 2.60/1.72 ! [B: sum_sum_c_c,F: c > sum_sum_c_c,X3: c,A3: set_c] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ B @ ( image_c_Sum_sum_c_c @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_301_image__eqI,axiom,
% 2.60/1.72 ! [B: sum_sum_c_d,F: c > sum_sum_c_d,X3: c,A3: set_c] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ B @ ( image_c_Sum_sum_c_d @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_302_image__eqI,axiom,
% 2.60/1.72 ! [B: sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,X3: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ B @ ( image_666880337um_c_c @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_303_image__eqI,axiom,
% 2.60/1.72 ! [B: sum_sum_c_d,F: sum_sum_c_c > sum_sum_c_d,X3: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ B @ ( image_675141842um_c_d @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_304_image__eqI,axiom,
% 2.60/1.72 ! [B: sum_sum_c_c,F: sum_sum_c_d > sum_sum_c_c,X3: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ B @ ( image_1558941394um_c_c @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_305_image__eqI,axiom,
% 2.60/1.72 ! [B: sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d,X3: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ B @ ( image_1567202899um_c_d @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_306_image__eqI,axiom,
% 2.60/1.72 ! [B: sum_sum_c_c,F: produc190496183real_c > sum_sum_c_c,X3: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member1895684704real_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ B @ ( image_1467045143um_c_c @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_307_image__eqI,axiom,
% 2.60/1.72 ! [B: sum_sum_c_d,F: produc190496183real_c > sum_sum_c_d,X3: produc190496183real_c,A3: set_Pr1389752855real_c] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member1895684704real_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ B @ ( image_1475306648um_c_d @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_308_image__eqI,axiom,
% 2.60/1.72 ! [B: produc190496183real_c,F: sum_sum_c_c > produc190496183real_c,X3: sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ B @ ( image_1146069259real_c @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_309_image__eqI,axiom,
% 2.60/1.72 ! [B: produc190496183real_c,F: sum_sum_c_d > produc190496183real_c,X3: sum_sum_c_d,A3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ( B
% 2.60/1.72 = ( F @ X3 ) )
% 2.60/1.72 => ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ B @ ( image_1980705354real_c @ F @ A3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_310_image__uminus__atLeastAtMost,axiom,
% 2.60/1.72 ! [X3: real,Y3: real] :
% 2.60/1.72 ( ( image_real_real @ uminus_uminus_real @ ( set_or656347191t_real @ X3 @ Y3 ) )
% 2.60/1.72 = ( set_or656347191t_real @ ( uminus_uminus_real @ Y3 ) @ ( uminus_uminus_real @ X3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_311_subset__image__iff,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_d,F: c > sum_sum_c_d,A3: set_c] :
% 2.60/1.72 ( ( ord_le764260924um_c_d @ B3 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
% 2.60/1.72 = ( ? [AA: set_c] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ AA @ A3 )
% 2.60/1.72 & ( B3
% 2.60/1.72 = ( image_c_Sum_sum_c_d @ F @ AA ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_312_subset__image__iff,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_c,F: c > sum_sum_c_c,A3: set_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B3 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
% 2.60/1.72 = ( ? [AA: set_c] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ AA @ A3 )
% 2.60/1.72 & ( B3
% 2.60/1.72 = ( image_c_Sum_sum_c_c @ F @ AA ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_313_subset__image__iff,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B3 @ ( image_666880337um_c_c @ F @ A3 ) )
% 2.60/1.72 = ( ? [AA: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ AA @ A3 )
% 2.60/1.72 & ( B3
% 2.60/1.72 = ( image_666880337um_c_c @ F @ AA ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_314_image__subset__iff,axiom,
% 2.60/1.72 ! [F: c > sum_sum_c_d,A3: set_c,B3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A3 ) @ B3 )
% 2.60/1.72 = ( ! [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ ( F @ X4 ) @ B3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_315_image__subset__iff,axiom,
% 2.60/1.72 ! [F: c > sum_sum_c_c,A3: set_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A3 ) @ B3 )
% 2.60/1.72 = ( ! [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ ( F @ X4 ) @ B3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_316_subset__imageE,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_d,F: c > sum_sum_c_d,A3: set_c] :
% 2.60/1.72 ( ( ord_le764260924um_c_d @ B3 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
% 2.60/1.72 => ~ ! [C3: set_c] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ C3 @ A3 )
% 2.60/1.72 => ( B3
% 2.60/1.72 != ( image_c_Sum_sum_c_d @ F @ C3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_317_subset__imageE,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_c,F: c > sum_sum_c_c,A3: set_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B3 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
% 2.60/1.72 => ~ ! [C3: set_c] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ C3 @ A3 )
% 2.60/1.72 => ( B3
% 2.60/1.72 != ( image_c_Sum_sum_c_c @ F @ C3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_318_subset__imageE,axiom,
% 2.60/1.72 ! [B3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,A3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ B3 @ ( image_666880337um_c_c @ F @ A3 ) )
% 2.60/1.72 => ~ ! [C3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ C3 @ A3 )
% 2.60/1.72 => ( B3
% 2.60/1.72 != ( image_666880337um_c_c @ F @ C3 ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_319_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_c,F: c > sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ! [X: c] :
% 2.60/1.72 ( ( member_c @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_320_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_c,F: c > sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ! [X: c] :
% 2.60/1.72 ( ( member_c @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_321_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ! [X: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le764260924um_c_d @ ( image_675141842um_c_d @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_322_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ! [X: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le764260924um_c_d @ ( image_1567202899um_c_d @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_323_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ! [X: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( image_666880337um_c_c @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_324_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ! [X: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( image_1558941394um_c_c @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_325_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_d,B3: set_Sum_sum_c_d] :
% 2.60/1.72 ( ! [X: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le764260924um_c_d @ ( image_1475306648um_c_d @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_326_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,F: sum_sum_c_c > produc190496183real_c,B3: set_Pr1389752855real_c] :
% 2.60/1.72 ( ! [X: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le977353143real_c @ ( image_1146069259real_c @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_327_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_d,F: sum_sum_c_d > produc190496183real_c,B3: set_Pr1389752855real_c] :
% 2.60/1.72 ( ! [X: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le977353143real_c @ ( image_1980705354real_c @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_328_image__subsetI,axiom,
% 2.60/1.72 ! [A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_c,B3: set_Sum_sum_c_c] :
% 2.60/1.72 ( ! [X: produc190496183real_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ ( F @ X ) @ B3 ) )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( image_1467045143um_c_c @ F @ A3 ) @ B3 ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_329_image__mono,axiom,
% 2.60/1.72 ! [A3: set_c,B3: set_c,F: c > sum_sum_c_d] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ A3 @ B3 )
% 2.60/1.72 => ( ord_le764260924um_c_d @ ( image_c_Sum_sum_c_d @ F @ A3 ) @ ( image_c_Sum_sum_c_d @ F @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_330_image__mono,axiom,
% 2.60/1.72 ! [A3: set_c,B3: set_c,F: c > sum_sum_c_c] :
% 2.60/1.72 ( ( ord_less_eq_set_c @ A3 @ B3 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ F @ A3 ) @ ( image_c_Sum_sum_c_c @ F @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_331_image__mono,axiom,
% 2.60/1.72 ! [A3: set_Sum_sum_c_c,B3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c] :
% 2.60/1.72 ( ( ord_le1772180283um_c_c @ A3 @ B3 )
% 2.60/1.72 => ( ord_le1772180283um_c_c @ ( image_666880337um_c_c @ F @ A3 ) @ ( image_666880337um_c_c @ F @ B3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_332_solves__ode__subset__range,axiom,
% 2.60/1.72 ! [X3: real > finite1398487019real_c,F: real > finite1398487019real_c > finite1398487019real_c,T2: set_real,X6: set_Fi1407883041real_c,Y7: set_Fi1407883041real_c] :
% 2.60/1.72 ( ( initia1631504802real_c @ X3 @ F @ T2 @ X6 )
% 2.60/1.72 => ( ( ord_le1327118209real_c @ ( image_327677598real_c @ X3 @ T2 ) @ Y7 )
% 2.60/1.72 => ( initia1631504802real_c @ X3 @ F @ T2 @ Y7 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_333_imageI,axiom,
% 2.60/1.72 ! [X3: c,A3: set_c,F: c > sum_sum_c_c] :
% 2.60/1.72 ( ( member_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_c_Sum_sum_c_c @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_334_imageI,axiom,
% 2.60/1.72 ! [X3: c,A3: set_c,F: c > sum_sum_c_d] :
% 2.60/1.72 ( ( member_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_c_Sum_sum_c_d @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_335_imageI,axiom,
% 2.60/1.72 ! [X3: sum_sum_c_c,A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_666880337um_c_c @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_336_imageI,axiom,
% 2.60/1.72 ! [X3: sum_sum_c_c,A3: set_Sum_sum_c_c,F: sum_sum_c_c > sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_675141842um_c_d @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_337_imageI,axiom,
% 2.60/1.72 ! [X3: sum_sum_c_d,A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_1558941394um_c_c @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_338_imageI,axiom,
% 2.60/1.72 ! [X3: sum_sum_c_d,A3: set_Sum_sum_c_d,F: sum_sum_c_d > sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_1567202899um_c_d @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_339_imageI,axiom,
% 2.60/1.72 ! [X3: produc190496183real_c,A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_c] :
% 2.60/1.72 ( ( member1895684704real_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_c @ ( F @ X3 ) @ ( image_1467045143um_c_c @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_340_imageI,axiom,
% 2.60/1.72 ! [X3: produc190496183real_c,A3: set_Pr1389752855real_c,F: produc190496183real_c > sum_sum_c_d] :
% 2.60/1.72 ( ( member1895684704real_c @ X3 @ A3 )
% 2.60/1.72 => ( member_Sum_sum_c_d @ ( F @ X3 ) @ ( image_1475306648um_c_d @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_341_imageI,axiom,
% 2.60/1.72 ! [X3: sum_sum_c_c,A3: set_Sum_sum_c_c,F: sum_sum_c_c > produc190496183real_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X3 @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ ( F @ X3 ) @ ( image_1146069259real_c @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_342_imageI,axiom,
% 2.60/1.72 ! [X3: sum_sum_c_d,A3: set_Sum_sum_c_d,F: sum_sum_c_d > produc190496183real_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X3 @ A3 )
% 2.60/1.72 => ( member1895684704real_c @ ( F @ X3 ) @ ( image_1980705354real_c @ F @ A3 ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_343_image__iff,axiom,
% 2.60/1.72 ! [Z2: sum_sum_c_c,F: c > sum_sum_c_c,A3: set_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ Z2 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
% 2.60/1.72 = ( ? [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ A3 )
% 2.60/1.72 & ( Z2
% 2.60/1.72 = ( F @ X4 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_344_image__iff,axiom,
% 2.60/1.72 ! [Z2: sum_sum_c_d,F: c > sum_sum_c_d,A3: set_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ Z2 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
% 2.60/1.72 = ( ? [X4: c] :
% 2.60/1.72 ( ( member_c @ X4 @ A3 )
% 2.60/1.72 & ( Z2
% 2.60/1.72 = ( F @ X4 ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_345_bex__imageD,axiom,
% 2.60/1.72 ! [F: c > sum_sum_c_c,A3: set_c,P2: sum_sum_c_c > $o] :
% 2.60/1.72 ( ? [X7: sum_sum_c_c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ X7 @ ( image_c_Sum_sum_c_c @ F @ A3 ) )
% 2.60/1.72 & ( P2 @ X7 ) )
% 2.60/1.72 => ? [X: c] :
% 2.60/1.72 ( ( member_c @ X @ A3 )
% 2.60/1.72 & ( P2 @ ( F @ X ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_346_bex__imageD,axiom,
% 2.60/1.72 ! [F: c > sum_sum_c_d,A3: set_c,P2: sum_sum_c_d > $o] :
% 2.60/1.72 ( ? [X7: sum_sum_c_d] :
% 2.60/1.72 ( ( member_Sum_sum_c_d @ X7 @ ( image_c_Sum_sum_c_d @ F @ A3 ) )
% 2.60/1.72 & ( P2 @ X7 ) )
% 2.60/1.72 => ? [X: c] :
% 2.60/1.72 ( ( member_c @ X @ A3 )
% 2.60/1.72 & ( P2 @ ( F @ X ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_347_image__cong,axiom,
% 2.60/1.72 ! [M: set_c,N: set_c,F: c > sum_sum_c_c,G: c > sum_sum_c_c] :
% 2.60/1.72 ( ( M = N )
% 2.60/1.72 => ( ! [X: c] :
% 2.60/1.72 ( ( member_c @ X @ N )
% 2.60/1.72 => ( ( F @ X )
% 2.60/1.72 = ( G @ X ) ) )
% 2.60/1.72 => ( ( image_c_Sum_sum_c_c @ F @ M )
% 2.60/1.72 = ( image_c_Sum_sum_c_c @ G @ N ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_348_image__cong,axiom,
% 2.60/1.72 ! [M: set_c,N: set_c,F: c > sum_sum_c_d,G: c > sum_sum_c_d] :
% 2.60/1.72 ( ( M = N )
% 2.60/1.72 => ( ! [X: c] :
% 2.60/1.72 ( ( member_c @ X @ N )
% 2.60/1.72 => ( ( F @ X )
% 2.60/1.72 = ( G @ X ) ) )
% 2.60/1.72 => ( ( image_c_Sum_sum_c_d @ F @ M )
% 2.60/1.72 = ( image_c_Sum_sum_c_d @ G @ N ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_349_Osub,axiom,
% 2.60/1.72 ord_le1772180283um_c_c @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( static_FVO_a_c @ ode ) ) @ v ).
% 2.60/1.72
% 2.60/1.72 thf(fact_350_VSA,axiom,
% 2.60/1.72 ( denota256060419gree_c @ ( sol @ zero_zero_real ) @ a
% 2.60/1.72 @ ( collect_c
% 2.60/1.72 @ ^ [Uu: c] :
% 2.60/1.72 ( ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_BVO_a_c @ ode ) )
% 2.60/1.72 | ( member_Sum_sum_c_d @ ( sum_Inl_c_d @ Uu ) @ ( image_c_Sum_sum_c_d @ sum_Inl_c_d @ ( static_FVO_a_c @ ode ) ) )
% 2.60/1.72 | ( member_Sum_sum_c_c @ ( sum_Inl_c_c @ Uu ) @ ( static_FVF_a_b_c @ p ) ) ) ) ) ).
% 2.60/1.72
% 2.60/1.72 thf(fact_351_OVsub_H,axiom,
% 2.60/1.72 ord_le1772180283um_c_c @ ( sup_su1349021703um_c_c @ ( image_c_Sum_sum_c_c @ sum_Inl_c_c @ ( oDE_dom_a_c @ ode ) ) @ ( image_c_Sum_sum_c_c @ sum_Inr_c_c @ ( oDE_dom_a_c @ ode ) ) ) @ v ).
% 2.60/1.72
% 2.60/1.72 thf(conj_0,conjecture,
% 2.60/1.72 denota1997846518gree_c @ ( denota161327353_a_b_c @ j @ ode @ ( produc394644079real_c @ aa @ ba ) @ ( sol @ s ) ) @ ( denota1896987029_a_b_c @ j @ ode @ ( sol @ s ) ) @ ( denota1293057250_a_b_c @ j @ ode ) ).
% 2.60/1.72 ------------------------
% 2.60/1.74 % (10502)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_10391 for (2999ds/275Mi)
% 2.60/1.74 % (10497)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_10391 for (2999ds/183Mi)
% 2.60/1.74 % (10503)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_10391 for (2999ds/18Mi)
% 2.60/1.74 % (10498)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_10391 for (2999ds/4Mi)
% 2.60/1.74 % (10498)Instruction limit reached!
% 2.60/1.74 % (10498)------------------------------
% 2.60/1.74 % (10498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.74 % (10498)Termination reason: Unknown
% 2.60/1.74 % (10498)Termination phase: shuffling
% 2.60/1.74
% 2.60/1.74 % (10498)Memory used [KB]: 1535
% 2.60/1.74 % (10498)Time elapsed: 0.003 s
% 2.60/1.74 % (10498)Instructions burned: 4 (million)
% 2.60/1.74 % (10498)------------------------------
% 2.60/1.74 % (10498)------------------------------
% 2.60/1.74 % (10501)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_10391 for (2999ds/2Mi)
% 2.60/1.74 % (10501)Instruction limit reached!
% 2.60/1.74 % (10501)------------------------------
% 2.60/1.74 % (10501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.74 % (10501)Termination reason: Unknown
% 2.60/1.74 % (10501)Termination phase: shuffling
% 2.60/1.74
% 2.60/1.74 % (10501)Memory used [KB]: 1535
% 2.60/1.74 % (10501)Time elapsed: 0.002 s
% 2.60/1.74 % (10501)Instructions burned: 2 (million)
% 2.60/1.74 % (10501)------------------------------
% 2.60/1.74 % (10501)------------------------------
% 2.60/1.75 % (10503)Instruction limit reached!
% 2.60/1.75 % (10503)------------------------------
% 2.60/1.75 % (10503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.75 % (10503)Termination reason: Unknown
% 2.60/1.75 % (10503)Termination phase: shuffling
% 2.60/1.75
% 2.60/1.75 % (10503)Memory used [KB]: 1918
% 2.60/1.75 % (10503)Time elapsed: 0.007 s
% 2.60/1.75 % (10503)Instructions burned: 18 (million)
% 2.60/1.75 % (10503)------------------------------
% 2.60/1.75 % (10503)------------------------------
% 2.60/1.75 % (10499)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_10391 for (2999ds/27Mi)
% 2.60/1.75 % (10500)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_10391 for (2999ds/2Mi)
% 2.60/1.75 % (10500)Instruction limit reached!
% 2.60/1.75 % (10500)------------------------------
% 2.60/1.75 % (10500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.75 % (10500)Termination reason: Unknown
% 2.60/1.75 % (10500)Termination phase: shuffling
% 2.60/1.75
% 2.60/1.75 % (10500)Memory used [KB]: 1535
% 2.60/1.75 % (10500)Time elapsed: 0.004 s
% 2.60/1.75 % (10500)Instructions burned: 3 (million)
% 2.60/1.75 % (10500)------------------------------
% 2.60/1.75 % (10500)------------------------------
% 2.60/1.75 % (10507)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10391 for (2999ds/3Mi)
% 2.60/1.75 % (10507)Instruction limit reached!
% 2.60/1.75 % (10507)------------------------------
% 2.60/1.75 % (10507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.75 % (10507)Termination reason: Unknown
% 2.60/1.75 % (10507)Termination phase: shuffling
% 2.60/1.75
% 2.60/1.75 % (10507)Memory used [KB]: 1535
% 2.60/1.75 % (10507)Time elapsed: 0.003 s
% 2.60/1.75 % (10507)Instructions burned: 5 (million)
% 2.60/1.75 % (10507)------------------------------
% 2.60/1.75 % (10507)------------------------------
% 2.60/1.75 % (10508)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_10391 for (2999ds/37Mi)
% 2.60/1.76 % (10510)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_10391 for (2999ds/15Mi)
% 2.60/1.76 % (10502)First to succeed.
% 2.60/1.76 % (10510)Instruction limit reached!
% 2.60/1.76 % (10510)------------------------------
% 2.60/1.76 % (10510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.76 % (10510)Termination reason: Unknown
% 2.60/1.76 % (10510)Termination phase: shuffling
% 2.60/1.76
% 2.60/1.76 % (10510)Memory used [KB]: 1791
% 2.60/1.76 % (10510)Time elapsed: 0.006 s
% 2.60/1.76 % (10510)Instructions burned: 15 (million)
% 2.60/1.76 % (10510)------------------------------
% 2.60/1.76 % (10510)------------------------------
% 2.60/1.76 % (10502)Refutation found. Thanks to Tanya!
% 2.60/1.76 % SZS status Theorem for DTF2THF_10391
% 2.60/1.76 % SZS output start Proof for DTF2THF_10391
% 2.60/1.76 thf(type_def_5, type, set_Pr1389752855real_c: $tType).
% 2.60/1.76 thf(type_def_6, type, finite1398487019real_c: $tType).
% 2.60/1.76 thf(type_def_7, type, real: $tType).
% 2.60/1.76 thf(type_def_9, type, set_Sum_sum_c_c: $tType).
% 2.60/1.76 thf(type_def_10, type, formula_a_b_c: $tType).
% 2.60/1.76 thf(type_def_11, type, hp_a_b_c: $tType).
% 2.60/1.76 thf(type_def_12, type, denota231621370t_unit: $tType).
% 2.60/1.76 thf(type_def_13, type, set_Su1783761653um_b_c: $tType).
% 2.60/1.76 thf(type_def_14, type, oDE_a_c: $tType).
% 2.60/1.76 thf(type_def_15, type, set_c: $tType).
% 2.60/1.76 thf(type_def_16, type, produc190496183real_c: $tType).
% 2.60/1.76 thf(type_def_17, type, filter_real: $tType).
% 2.60/1.76 thf(type_def_18, type, sum_sum_c_c: $tType).
% 2.60/1.76 thf(type_def_19, type, sum_sum_c_d: $tType).
% 2.60/1.76 thf(type_def_20, type, c: $tType).
% 2.60/1.76 thf(type_def_21, type, produc957004601l_real: $tType).
% 2.60/1.76 thf(type_def_22, type, produc754691304um_c_c: $tType).
% 2.60/1.76 thf(type_def_23, type, produc1398711912c_real: $tType).
% 2.60/1.76 thf(type_def_24, type, produc214722967um_c_c: $tType).
% 2.60/1.76 thf(type_def_25, type, set_Fi1407883041real_c: $tType).
% 2.60/1.76 thf(type_def_26, type, set_Sum_sum_c_d: $tType).
% 2.60/1.76 thf(type_def_27, type, set_real: $tType).
% 2.60/1.76 thf(type_def_28, type, set_set_Sum_sum_c_c: $tType).
% 2.60/1.76 thf(func_def_0, type, set_Pr1389752855real_c: $tType).
% 2.60/1.76 thf(func_def_1, type, produc190496183real_c: $tType).
% 2.60/1.76 thf(func_def_2, type, produc214722967um_c_c: $tType).
% 2.60/1.76 thf(func_def_3, type, denota231621370t_unit: $tType).
% 2.60/1.76 thf(func_def_4, type, produc1398711912c_real: $tType).
% 2.60/1.76 thf(func_def_5, type, produc754691304um_c_c: $tType).
% 2.60/1.76 thf(func_def_6, type, set_Su1783761653um_b_c: $tType).
% 2.60/1.76 thf(func_def_7, type, set_Fi1407883041real_c: $tType).
% 2.60/1.76 thf(func_def_8, type, set_set_Sum_sum_c_c: $tType).
% 2.60/1.76 thf(func_def_9, type, finite1398487019real_c: $tType).
% 2.60/1.76 thf(func_def_10, type, produc957004601l_real: $tType).
% 2.60/1.76 thf(func_def_11, type, set_Sum_sum_c_d: $tType).
% 2.60/1.76 thf(func_def_12, type, set_Sum_sum_c_c: $tType).
% 2.60/1.76 thf(func_def_13, type, formula_a_b_c: $tType).
% 2.60/1.76 thf(func_def_14, type, hp_a_b_c: $tType).
% 2.60/1.76 thf(func_def_15, type, sum_sum_c_d: $tType).
% 2.60/1.76 thf(func_def_16, type, sum_sum_c_c: $tType).
% 2.60/1.76 thf(func_def_17, type, filter_real: $tType).
% 2.60/1.76 thf(func_def_18, type, oDE_a_c: $tType).
% 2.60/1.76 thf(func_def_19, type, set_real: $tType).
% 2.60/1.76 thf(func_def_20, type, set_c: $tType).
% 2.60/1.76 thf(func_def_21, type, real: $tType).
% 2.60/1.76 thf(func_def_22, type, c: $tType).
% 2.60/1.76 thf(func_def_23, type, bNF_Ca449034301c_real: set_Pr1389752855real_c > (finite1398487019real_c > real) > $o).
% 2.60/1.76 thf(func_def_24, type, bNF_Ca1693016812um_c_c: set_Pr1389752855real_c > (finite1398487019real_c > set_Sum_sum_c_c) > $o).
% 2.60/1.76 thf(func_def_25, type, coinci501627771_a_b_c: formula_a_b_c > $o).
% 2.60/1.76 thf(func_def_26, type, coinci354700372_a_b_c: hp_a_b_c > denota231621370t_unit > denota231621370t_unit > $o).
% 2.60/1.76 thf(func_def_27, type, coinci1852866155_a_b_c: hp_a_b_c > $o).
% 2.60/1.76 thf(func_def_28, type, coinci772602909_a_b_c: hp_a_b_c > denota231621370t_unit > $o).
% 2.60/1.76 thf(func_def_29, type, denota69106024_a_b_c: denota231621370t_unit > denota231621370t_unit > set_Su1783761653um_b_c > $o).
% 2.60/1.76 thf(func_def_30, type, denota1275485728_a_b_c: denota231621370t_unit > oDE_a_c > finite1398487019real_c > finite1398487019real_c).
% 2.60/1.76 thf(func_def_31, type, denota256060419gree_c: finite1398487019real_c > finite1398487019real_c > set_c > $o).
% 2.60/1.76 thf(func_def_32, type, denota1997846518gree_c: produc190496183real_c > produc190496183real_c > set_Sum_sum_c_c > $o).
% 2.60/1.76 thf(func_def_33, type, denota968303861_a_b_c: denota231621370t_unit > formula_a_b_c > set_Pr1389752855real_c).
% 2.60/1.76 thf(func_def_34, type, denota161327353_a_b_c: denota231621370t_unit > oDE_a_c > produc190496183real_c > finite1398487019real_c > produc190496183real_c).
% 2.60/1.76 thf(func_def_35, type, denota1896987029_a_b_c: denota231621370t_unit > oDE_a_c > finite1398487019real_c > produc190496183real_c).
% 2.60/1.76 thf(func_def_36, type, denota1293057250_a_b_c: denota231621370t_unit > oDE_a_c > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_37, type, has_ve1904308670real_c: (real > finite1398487019real_c) > finite1398487019real_c > filter_real > $o).
% 2.60/1.76 thf(func_def_38, type, has_ve373335844real_c: (real > produc190496183real_c) > produc190496183real_c > filter_real > $o).
% 2.60/1.76 thf(func_def_39, type, has_ve217091583e_real: (real > real) > real > filter_real > $o).
% 2.60/1.76 thf(func_def_40, type, uminus23580315al_c_o: (finite1398487019real_c > $o) > finite1398487019real_c > $o).
% 2.60/1.76 thf(func_def_41, type, uminus32155087al_c_o: (produc190496183real_c > $o) > produc190496183real_c > $o).
% 2.60/1.76 thf(func_def_42, type, uminus88729697_c_c_o: (sum_sum_c_c > $o) > sum_sum_c_c > $o).
% 2.60/1.76 thf(func_def_43, type, uminus916499104_c_d_o: (sum_sum_c_d > $o) > sum_sum_c_d > $o).
% 2.60/1.76 thf(func_def_44, type, uminus_uminus_c_o: (c > $o) > c > $o).
% 2.60/1.76 thf(func_def_45, type, uminus1737280820real_c: finite1398487019real_c > finite1398487019real_c).
% 2.60/1.76 thf(func_def_46, type, uminus17964590real_c: produc190496183real_c > produc190496183real_c).
% 2.60/1.76 thf(func_def_47, type, uminus856758960l_real: produc957004601l_real > produc957004601l_real).
% 2.60/1.76 thf(func_def_48, type, uminus1629073887um_c_c: produc754691304um_c_c > produc754691304um_c_c).
% 2.60/1.76 thf(func_def_49, type, uminus125610847c_real: produc1398711912c_real > produc1398711912c_real).
% 2.60/1.76 thf(func_def_50, type, uminus1225118734um_c_c: produc214722967um_c_c > produc214722967um_c_c).
% 2.60/1.76 thf(func_def_51, type, uminus_uminus_real: real > real).
% 2.60/1.76 thf(func_def_52, type, uminus102402154real_c: set_Fi1407883041real_c > set_Fi1407883041real_c).
% 2.60/1.76 thf(func_def_53, type, uminus232257166real_c: set_Pr1389752855real_c > set_Pr1389752855real_c).
% 2.60/1.76 thf(func_def_54, type, uminus1381786404um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_55, type, uminus373867045um_c_d: set_Sum_sum_c_d > set_Sum_sum_c_d).
% 2.60/1.76 thf(func_def_56, type, uminus_uminus_set_c: set_c > set_c).
% 2.60/1.76 thf(func_def_57, type, zero_z109254132real_c: finite1398487019real_c).
% 2.60/1.76 thf(func_def_58, type, zero_z1506780526real_c: produc190496183real_c).
% 2.60/1.76 thf(func_def_59, type, zero_z659284464l_real: produc957004601l_real).
% 2.60/1.76 thf(func_def_60, type, zero_zero_real: real).
% 2.60/1.76 thf(func_def_61, type, the_Pr1290640446real_c: (produc190496183real_c > $o) > produc190496183real_c).
% 2.60/1.76 thf(func_def_62, type, initia1631504802real_c: (real > finite1398487019real_c) > (real > finite1398487019real_c > finite1398487019real_c) > set_real > set_Fi1407883041real_c > $o).
% 2.60/1.76 thf(func_def_63, type, sup_su1349021703um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_64, type, ord_le1382916304al_c_o: (finite1398487019real_c > finite1398487019real_c > $o) > (finite1398487019real_c > finite1398487019real_c > $o) > $o).
% 2.60/1.76 thf(func_def_65, type, ord_le1132241894al_c_o: (produc190496183real_c > $o) > (produc190496183real_c > $o) > $o).
% 2.60/1.76 thf(func_def_66, type, ord_le1315653386_c_c_o: (sum_sum_c_c > $o) > (sum_sum_c_c > $o) > $o).
% 2.60/1.76 thf(func_def_67, type, ord_le2143422793_c_d_o: (sum_sum_c_d > $o) > (sum_sum_c_d > $o) > $o).
% 2.60/1.76 thf(func_def_68, type, ord_le132810396r_real: filter_real > filter_real > $o).
% 2.60/1.76 thf(func_def_69, type, ord_le775706699real_c: finite1398487019real_c > finite1398487019real_c > $o).
% 2.60/1.76 thf(func_def_70, type, ord_le850691415real_c: produc190496183real_c > produc190496183real_c > $o).
% 2.60/1.76 thf(func_def_71, type, ord_le1342644953l_real: produc957004601l_real > produc957004601l_real > $o).
% 2.60/1.76 thf(func_def_72, type, ord_le191210632um_c_c: produc754691304um_c_c > produc754691304um_c_c > $o).
% 2.60/1.76 thf(func_def_73, type, ord_le835231240c_real: produc1398711912c_real > produc1398711912c_real > $o).
% 2.60/1.76 thf(func_def_74, type, ord_le684454199um_c_c: produc214722967um_c_c > produc214722967um_c_c > $o).
% 2.60/1.76 thf(func_def_75, type, ord_less_eq_real: real > real > $o).
% 2.60/1.76 thf(func_def_76, type, ord_le1327118209real_c: set_Fi1407883041real_c > set_Fi1407883041real_c > $o).
% 2.60/1.76 thf(func_def_77, type, ord_le977353143real_c: set_Pr1389752855real_c > set_Pr1389752855real_c > $o).
% 2.60/1.76 thf(func_def_78, type, ord_less_eq_set_real: set_real > set_real > $o).
% 2.60/1.76 thf(func_def_79, type, ord_le657554289um_c_c: set_set_Sum_sum_c_c > set_set_Sum_sum_c_c > $o).
% 2.60/1.76 thf(func_def_80, type, ord_le929608341um_b_c: set_Su1783761653um_b_c > set_Su1783761653um_b_c > $o).
% 2.60/1.76 thf(func_def_81, type, ord_le1772180283um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > $o).
% 2.60/1.76 thf(func_def_82, type, ord_le764260924um_c_d: set_Sum_sum_c_d > set_Sum_sum_c_d > $o).
% 2.60/1.76 thf(func_def_83, type, ord_less_eq_set_c: set_c > set_c > $o).
% 2.60/1.76 thf(func_def_84, type, produc394644079real_c: finite1398487019real_c > finite1398487019real_c > produc190496183real_c).
% 2.60/1.76 thf(func_def_85, type, produc705216881l_real: real > real > produc957004601l_real).
% 2.60/1.76 thf(func_def_86, type, produc1186709024um_c_c: real > set_Sum_sum_c_c > produc754691304um_c_c).
% 2.60/1.76 thf(func_def_87, type, produc1528874528c_real: set_Sum_sum_c_c > real > produc1398711912c_real).
% 2.60/1.76 thf(func_def_88, type, produc1439061071um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > produc214722967um_c_c).
% 2.60/1.76 thf(func_def_89, type, collec230941376real_c: (finite1398487019real_c > $o) > set_Fi1407883041real_c).
% 2.60/1.76 thf(func_def_90, type, collec1643251106real_c: (produc190496183real_c > $o) > set_Pr1389752855real_c).
% 2.60/1.76 thf(func_def_91, type, collect_real: (real > $o) > set_real).
% 2.60/1.76 thf(func_def_92, type, collec95329456um_c_c: (set_Sum_sum_c_c > $o) > set_set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_93, type, collect_Sum_sum_c_c: (sum_sum_c_c > $o) > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_94, type, collect_Sum_sum_c_d: (sum_sum_c_d > $o) > set_Sum_sum_c_d).
% 2.60/1.76 thf(func_def_95, type, collect_c: (c > $o) > set_c).
% 2.60/1.76 thf(func_def_96, type, image_1467045143um_c_c: (produc190496183real_c > sum_sum_c_c) > set_Pr1389752855real_c > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_97, type, image_1475306648um_c_d: (produc190496183real_c > sum_sum_c_d) > set_Pr1389752855real_c > set_Sum_sum_c_d).
% 2.60/1.76 thf(func_def_98, type, image_327677598real_c: (real > finite1398487019real_c) > set_real > set_Fi1407883041real_c).
% 2.60/1.76 thf(func_def_99, type, image_real_real: (real > real) > set_real > set_real).
% 2.60/1.76 thf(func_def_100, type, image_1146069259real_c: (sum_sum_c_c > produc190496183real_c) > set_Sum_sum_c_c > set_Pr1389752855real_c).
% 2.60/1.76 thf(func_def_101, type, image_666880337um_c_c: (sum_sum_c_c > sum_sum_c_c) > set_Sum_sum_c_c > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_102, type, image_675141842um_c_d: (sum_sum_c_c > sum_sum_c_d) > set_Sum_sum_c_c > set_Sum_sum_c_d).
% 2.60/1.76 thf(func_def_103, type, image_1980705354real_c: (sum_sum_c_d > produc190496183real_c) > set_Sum_sum_c_d > set_Pr1389752855real_c).
% 2.60/1.76 thf(func_def_104, type, image_1558941394um_c_c: (sum_sum_c_d > sum_sum_c_c) > set_Sum_sum_c_d > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_105, type, image_1567202899um_c_d: (sum_sum_c_d > sum_sum_c_d) > set_Sum_sum_c_d > set_Sum_sum_c_d).
% 2.60/1.76 thf(func_def_106, type, image_c_Sum_sum_c_c: (c > sum_sum_c_c) > set_c > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_107, type, image_c_Sum_sum_c_d: (c > sum_sum_c_d) > set_c > set_Sum_sum_c_d).
% 2.60/1.76 thf(func_def_108, type, set_or385782508real_c: produc190496183real_c > produc190496183real_c > set_Pr1389752855real_c).
% 2.60/1.76 thf(func_def_109, type, set_or656347191t_real: real > real > set_real).
% 2.60/1.76 thf(func_def_110, type, set_or935154662um_c_c: set_Sum_sum_c_c > set_Sum_sum_c_c > set_set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_111, type, set_or2003963744real_c: finite1398487019real_c > set_Fi1407883041real_c).
% 2.60/1.76 thf(func_def_112, type, set_or10334466real_c: produc190496183real_c > set_Pr1389752855real_c).
% 2.60/1.76 thf(func_def_113, type, set_ord_atLeast_real: real > set_real).
% 2.60/1.76 thf(func_def_114, type, set_or2028906832um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_115, type, set_or1478397028real_c: finite1398487019real_c > set_Fi1407883041real_c).
% 2.60/1.76 thf(func_def_116, type, set_or248124158real_c: produc190496183real_c > set_Pr1389752855real_c).
% 2.60/1.76 thf(func_def_117, type, set_ord_atMost_real: real > set_real).
% 2.60/1.76 thf(func_def_118, type, set_or326480468um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_119, type, static_BVO_a_c: oDE_a_c > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_120, type, static_FVF_a_b_c: formula_a_b_c > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_121, type, static_FVO_a_c: oDE_a_c > set_c).
% 2.60/1.76 thf(func_def_122, type, static_SIGF_a_b_c: formula_a_b_c > set_Su1783761653um_b_c).
% 2.60/1.76 thf(func_def_123, type, sum_Inl_c_c: c > sum_sum_c_c).
% 2.60/1.76 thf(func_def_124, type, sum_Inl_c_d: c > sum_sum_c_d).
% 2.60/1.76 thf(func_def_125, type, sum_Inr_c_c: c > sum_sum_c_c).
% 2.60/1.76 thf(func_def_126, type, equiv_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c).
% 2.60/1.76 thf(func_def_127, type, implies_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c).
% 2.60/1.76 thf(func_def_128, type, oDE_dom_a_c: oDE_a_c > set_c).
% 2.60/1.76 thf(func_def_129, type, fsafe_a_b_c: formula_a_b_c > $o).
% 2.60/1.76 thf(func_def_130, type, evolveODE_a_c_b: oDE_a_c > formula_a_b_c > hp_a_b_c).
% 2.60/1.76 thf(func_def_131, type, hpsafe_a_b_c: hp_a_b_c > $o).
% 2.60/1.76 thf(func_def_132, type, osafe_a_c: oDE_a_c > $o).
% 2.60/1.76 thf(func_def_133, type, topolo1830106092n_real: real > set_real > filter_real).
% 2.60/1.76 thf(func_def_134, type, member1261661570real_c: finite1398487019real_c > set_Fi1407883041real_c > $o).
% 2.60/1.76 thf(func_def_135, type, member1895684704real_c: produc190496183real_c > set_Pr1389752855real_c > $o).
% 2.60/1.76 thf(func_def_136, type, member_real: real > set_real > $o).
% 2.60/1.76 thf(func_def_137, type, member2124950898um_c_c: set_Sum_sum_c_c > set_set_Sum_sum_c_c > $o).
% 2.60/1.76 thf(func_def_138, type, member_Sum_sum_c_c: sum_sum_c_c > set_Sum_sum_c_c > $o).
% 2.60/1.76 thf(func_def_139, type, member_Sum_sum_c_d: sum_sum_c_d > set_Sum_sum_c_d > $o).
% 2.60/1.76 thf(func_def_140, type, member_c: c > set_c > $o).
% 2.60/1.76 thf(func_def_141, type, i: denota231621370t_unit).
% 2.60/1.76 thf(func_def_142, type, j: denota231621370t_unit).
% 2.60/1.76 thf(func_def_143, type, ode: oDE_a_c).
% 2.60/1.76 thf(func_def_144, type, p: formula_a_b_c).
% 2.60/1.76 thf(func_def_145, type, v: set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_146, type, a: finite1398487019real_c).
% 2.60/1.76 thf(func_def_147, type, aa: finite1398487019real_c).
% 2.60/1.76 thf(func_def_148, type, ab: finite1398487019real_c).
% 2.60/1.76 thf(func_def_149, type, b: finite1398487019real_c).
% 2.60/1.76 thf(func_def_150, type, ba: finite1398487019real_c).
% 2.60/1.76 thf(func_def_151, type, bb: finite1398487019real_c).
% 2.60/1.76 thf(func_def_152, type, s: real).
% 2.60/1.76 thf(func_def_153, type, sol: real > finite1398487019real_c).
% 2.60/1.76 thf(func_def_154, type, t: real).
% 2.60/1.76 thf(func_def_170, type, sK0: (set_Sum_sum_c_c > set_Sum_sum_c_c) > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_171, type, sK1: (set_Sum_sum_c_c > set_Sum_sum_c_c) > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_172, type, sK2: produc190496183real_c > finite1398487019real_c).
% 2.60/1.76 thf(func_def_173, type, sK3: produc190496183real_c > finite1398487019real_c).
% 2.60/1.76 thf(func_def_174, type, sK4: (set_Sum_sum_c_c > set_Sum_sum_c_c) > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_175, type, sK5: (set_Sum_sum_c_c > set_Sum_sum_c_c) > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_176, type, sK6: (produc190496183real_c > $o) > finite1398487019real_c).
% 2.60/1.76 thf(func_def_177, type, sK7: (produc190496183real_c > $o) > finite1398487019real_c).
% 2.60/1.76 thf(func_def_178, type, sK8: (produc190496183real_c > $o) > finite1398487019real_c).
% 2.60/1.76 thf(func_def_179, type, sK9: (produc190496183real_c > $o) > finite1398487019real_c).
% 2.60/1.76 thf(func_def_180, type, sK10: (set_Sum_sum_c_c > set_Sum_sum_c_c) > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_181, type, sK11: (set_Sum_sum_c_c > set_Sum_sum_c_c) > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_182, type, sK12: produc190496183real_c > finite1398487019real_c).
% 2.60/1.76 thf(func_def_183, type, sK13: produc190496183real_c > finite1398487019real_c).
% 2.60/1.76 thf(func_def_184, type, sK14: (set_Sum_sum_c_c > set_Sum_sum_c_c) > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_185, type, sK15: (set_Sum_sum_c_c > set_Sum_sum_c_c) > set_Sum_sum_c_c).
% 2.60/1.76 thf(func_def_186, type, sK16: produc190496183real_c > finite1398487019real_c > denota231621370t_unit > oDE_a_c > produc190496183real_c).
% 2.60/1.76 thf(func_def_188, type, ph18: !>[X0: $tType]:(X0)).
% 2.60/1.76 thf(f1148,plain,(
% 2.60/1.76 $false),
% 2.60/1.76 inference(subsumption_resolution,[status(thm)],[f1069,f1066])).
% 2.60/1.76 thf(f1066,plain,(
% 2.60/1.76 ((denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode)) = $true)),
% 2.60/1.76 inference(cnf_transformation,[status(thm)],[f901])).
% 2.60/1.76 thf(f901,plain,(
% 2.60/1.76 ((denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode)) = $true) & ((denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (produc394644079real_c @ aa @ ba) @ (uminus1381786404um_c_c @ (denota1293057250_a_b_c @ j @ ode))) = $true)),
% 2.60/1.76 inference(fool_elimination,[status(thm)],[f900])).
% 2.60/1.76 thf(f900,plain,(
% 2.60/1.76 (denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode)) & (denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (produc394644079real_c @ aa @ ba) @ (uminus1381786404um_c_c @ (denota1293057250_a_b_c @ j @ ode)))),
% 2.60/1.76 inference(rectify,[status(thm)],[f2])).
% 2.60/1.76 thf(f2,axiom,(
% 2.60/1.76 (denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode)) & (denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (produc394644079real_c @ aa @ ba) @ (uminus1381786404um_c_c @ (denota1293057250_a_b_c @ j @ ode)))),
% 2.60/1.76 file('/export/starexec/sandbox/tmp/tmp.VZsX1DUqbC/DTF2THF_10391.p',fact_1__092_060open_062Vagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Iaa_M_Aba_J_A_I_N_AsemBV_AJ_AODE_J_A_092_060and_062_AVagree_A_Imk__v_AJ_AODE_A_Iaa_M_Aba_J_A_Isol_As_J_J_A_Imk__xode_AJ_AODE_A_Isol_As_J_J_A_IsemBV_AJ_AODE_J_092_060close_062)).
% 2.60/1.76 thf(f1069,plain,(
% 2.60/1.76 ((denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode)) != $true)),
% 2.60/1.76 inference(cnf_transformation,[status(thm)],[f964])).
% 2.60/1.76 thf(f964,plain,(
% 2.60/1.76 ((denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode)) != $true)),
% 2.60/1.76 inference(flattening,[status(thm)],[f455])).
% 2.60/1.76 thf(f455,plain,(
% 2.60/1.76 ~((denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode)) = $true)),
% 2.60/1.76 inference(fool_elimination,[status(thm)],[f454])).
% 2.60/1.76 thf(f454,plain,(
% 2.60/1.76 ~(denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode))),
% 2.60/1.76 inference(rectify,[status(thm)],[f354])).
% 2.60/1.76 thf(f354,negated_conjecture,(
% 2.60/1.76 ~(denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode))),
% 2.60/1.76 inference(negated_conjecture,[status(cth)],[f353])).
% 2.60/1.76 thf(f353,conjecture,(
% 2.60/1.76 (denota1997846518gree_c @ (denota161327353_a_b_c @ j @ ode @ (produc394644079real_c @ aa @ ba) @ (sol @ s)) @ (denota1896987029_a_b_c @ j @ ode @ (sol @ s)) @ (denota1293057250_a_b_c @ j @ ode))),
% 2.60/1.76 file('/export/starexec/sandbox/tmp/tmp.VZsX1DUqbC/DTF2THF_10391.p',conj_0)).
% 2.60/1.76 % SZS output end Proof for DTF2THF_10391
% 2.60/1.76 % (10502)------------------------------
% 2.60/1.76 % (10502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.60/1.76 % (10502)Termination reason: Refutation
% 2.60/1.76
% 2.60/1.76 % (10502)Memory used [KB]: 6780
% 2.60/1.76 % (10502)Time elapsed: 0.022 s
% 2.60/1.76 % (10502)Instructions burned: 64 (million)
% 2.60/1.76 % (10502)------------------------------
% 2.60/1.76 % (10502)------------------------------
% 2.60/1.76 % (10496)Success in time 0.033 s
%------------------------------------------------------------------------------