TPTP Problem File: SLH0393^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Cotangent_PFD_Formula/0007_Cotangent_PFD_Formula/prob_00287_011139__14035418_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1449 ( 728 unt; 169 typ; 0 def)
% Number of atoms : 2980 (1570 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 9019 ( 135 ~; 40 |; 141 &;7784 @)
% ( 0 <=>; 919 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 5 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 1283 (1283 >; 0 *; 0 +; 0 <<)
% Number of symbols : 158 ( 155 usr; 16 con; 0-3 aty)
% Number of variables : 3044 ( 107 ^;2878 !; 59 ?;3044 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:58:05.159
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
produc4411394909380815293omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
produc6979889472282505531omplex: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
produc8892588492097263291x_real: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
produc2422161461964618553l_real: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
formal670952693614245302omplex: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (155)
thf(sy_c_BNF__Def_Oconvol_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
bNF_co6028989446319287285omplex: ( complex > complex ) > ( complex > complex ) > complex > produc4411394909380815293omplex ).
thf(sy_c_BNF__Def_Oconvol_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Real__Oreal,type,
bNF_co2757359416308857971x_real: ( complex > complex ) > ( complex > real ) > complex > produc8892588492097263291x_real ).
thf(sy_c_BNF__Def_Oconvol_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal,type,
bNF_co2841340654916505457l_real: ( complex > real ) > ( complex > real ) > complex > produc2422161461964618553l_real ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
bNF_co5645111169341782295omplex: ( nat > complex ) > ( nat > complex ) > nat > produc4411394909380815293omplex ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Complex__Ocomplex_001t__Real__Oreal,type,
bNF_co314614023501618325x_real: ( nat > complex ) > ( nat > real ) > nat > produc8892588492097263291x_real ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Real__Oreal_001t__Complex__Ocomplex,type,
bNF_co261025522997297685omplex: ( nat > real ) > ( nat > complex ) > nat > produc6979889472282505531omplex ).
thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal,type,
bNF_co8045384917381939859l_real: ( nat > real ) > ( nat > real ) > nat > produc2422161461964618553l_real ).
thf(sy_c_BNF__Def_Oconvol_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
bNF_co1531800018262148979omplex: ( real > complex ) > ( real > complex ) > real > produc4411394909380815293omplex ).
thf(sy_c_BNF__Def_Oconvol_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Real__Oreal,type,
bNF_co1476410042518385905x_real: ( real > complex ) > ( real > real ) > real > produc8892588492097263291x_real ).
thf(sy_c_BNF__Def_Oconvol_001t__Real__Oreal_001t__Real__Oreal_001t__Complex__Ocomplex,type,
bNF_co1422821542014065265omplex: ( real > real ) > ( real > complex ) > real > produc6979889472282505531omplex ).
thf(sy_c_BNF__Def_Oconvol_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
bNF_co7388595451723801839l_real: ( real > real ) > ( real > real ) > real > produc2422161461964618553l_real ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex__Transcendental_OArccos,type,
complex_Arccos: complex > complex ).
thf(sy_c_Complex__Transcendental_OArcsin,type,
complex_Arcsin: complex > complex ).
thf(sy_c_Complex__Transcendental_OArctan,type,
complex_Arctan: complex > complex ).
thf(sy_c_Cotangent__PFD__Formula_Ocot__pfd_001t__Complex__Ocomplex,type,
cotang8298477626502807258omplex: complex > complex ).
thf(sy_c_Cotangent__PFD__Formula_Ocot__pfd_001t__Real__Oreal,type,
cotang1502006655779026648d_real: real > real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
invers8013647133539491842omplex: complex > complex ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
invers8029835185024579338omplex: formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
invers68952373231134600s_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Complex__Ocomplex,type,
formal6482914284900457064omplex: complex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
comp_c5959460852125409263omplex: ( complex > complex ) > ( complex > complex ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Nat__Onat,type,
comp_c7268011922939458833ex_nat: ( complex > complex ) > ( nat > complex ) > nat > complex ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Real__Oreal,type,
comp_c2117349707075585901x_real: ( complex > complex ) > ( real > complex ) > real > complex ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Complex__Ocomplex,type,
comp_c3648228598504721169omplex: ( complex > nat ) > ( complex > complex ) > complex > nat ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Nat__Onat,type,
comp_complex_nat_nat: ( complex > nat ) > ( nat > complex ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Real__Oreal,type,
comp_c3423117485846644111t_real: ( complex > nat ) > ( real > complex ) > real > nat ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Complex__Ocomplex,type,
comp_c5081751395193823254omplex: ( complex > produc2422161461964618553l_real ) > ( complex > complex ) > complex > produc2422161461964618553l_real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal,type,
comp_c2864525590522327572l_real: ( complex > produc2422161461964618553l_real ) > ( real > complex ) > real > produc2422161461964618553l_real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Complex__Ocomplex,type,
comp_c2063761206571265261omplex: ( complex > real ) > ( complex > complex ) > complex > real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Nat__Onat,type,
comp_c7990426058975542799al_nat: ( complex > real ) > ( nat > complex ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Real__Oreal,type,
comp_c3333796836230738283l_real: ( complex > real ) > ( real > complex ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
comp_n4415746728862867729omplex: ( nat > complex ) > ( complex > nat ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Complex__Ocomplex_001t__Nat__Onat,type,
comp_nat_complex_nat: ( nat > complex ) > ( nat > nat ) > nat > complex ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Complex__Ocomplex_001t__Real__Oreal,type,
comp_n4215249288434654095x_real: ( nat > complex ) > ( real > nat ) > real > complex ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
comp_nat_nat_complex: ( nat > nat ) > ( complex > nat ) > complex > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
comp_n1008163678071708920omplex: ( nat > produc4411394909380815293omplex ) > ( complex > nat ) > complex > produc4411394909380815293omplex ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Nat__Onat,type,
comp_n1197726116097606682ex_nat: ( nat > produc4411394909380815293omplex ) > ( nat > nat ) > nat > produc4411394909380815293omplex ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Real__Oreal_J_001t__Complex__Ocomplex,type,
comp_n7862994715949103094omplex: ( nat > produc8892588492097263291x_real ) > ( complex > nat ) > complex > produc8892588492097263291x_real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Complex__Ocomplex,type,
comp_n4161660787930333455omplex: ( nat > real ) > ( complex > nat ) > complex > real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal,type,
comp_nat_real_real: ( nat > real ) > ( real > nat ) > real > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
comp_r891790309028876909omplex: ( real > complex ) > ( complex > real ) > complex > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Nat__Onat,type,
comp_r1225911664865567631ex_nat: ( real > complex ) > ( nat > real ) > nat > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Real__Oreal,type,
comp_r1968866223832618731x_real: ( real > complex ) > ( real > real ) > real > complex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Complex__Ocomplex,type,
comp_r6829500377285605775omplex: ( real > nat ) > ( complex > real ) > complex > nat ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
comp_real_nat_nat: ( real > nat ) > ( nat > real ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
comp_r501483716643258524omplex: ( real > produc4411394909380815293omplex ) > ( complex > real ) > complex > produc4411394909380815293omplex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_001t__Nat__Onat,type,
comp_r8895249581614084030ex_nat: ( real > produc4411394909380815293omplex ) > ( nat > real ) > nat > produc4411394909380815293omplex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Real__Oreal_J_001t__Nat__Onat,type,
comp_r137067546312441020al_nat: ( real > produc8892588492097263291x_real ) > ( nat > real ) > nat > produc8892588492097263291x_real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Complex__Ocomplex_J_001t__Nat__Onat,type,
comp_r2603982079200466748ex_nat: ( real > produc6979889472282505531omplex ) > ( nat > real ) > nat > produc6979889472282505531omplex ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Nat__Onat,type,
comp_r5566512450771222842al_nat: ( real > produc2422161461964618553l_real ) > ( nat > real ) > nat > produc2422161461964618553l_real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Complex__Ocomplex,type,
comp_r1915277723328298091omplex: ( real > real ) > ( complex > real ) > complex > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat,type,
comp_real_real_nat: ( real > real ) > ( nat > real ) > nat > real ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).
thf(sy_c_Fun_Ofcomp_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001t__Real__Oreal,type,
fcomp_423701733139085997x_real: ( complex > complex ) > ( complex > real ) > complex > real ).
thf(sy_c_Fun_Ofcomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Complex__Ocomplex,type,
fcomp_877936951185219153omplex: ( complex > nat ) > ( nat > complex ) > complex > complex ).
thf(sy_c_Fun_Ofcomp_001t__Complex__Ocomplex_001t__Nat__Onat_001t__Real__Oreal,type,
fcomp_1779788810238572751t_real: ( complex > nat ) > ( nat > real ) > complex > real ).
thf(sy_c_Fun_Ofcomp_001t__Complex__Ocomplex_001t__Real__Oreal_001t__Complex__Ocomplex,type,
fcomp_370113232634765357omplex: ( complex > real ) > ( real > complex ) > complex > complex ).
thf(sy_c_Fun_Ofcomp_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
fcomp_8538492730942792563omplex: ( nat > nat ) > ( nat > complex ) > nat > complex ).
thf(sy_c_Fun_Ofcomp_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
fcomp_nat_nat_real: ( nat > nat ) > ( nat > real ) > nat > real ).
thf(sy_c_Fun_Ofcomp_001t__Nat__Onat_001t__Real__Oreal_001t__Complex__Ocomplex,type,
fcomp_2518332112322262095omplex: ( nat > real ) > ( real > complex ) > nat > complex ).
thf(sy_c_Fun_Ofcomp_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal,type,
fcomp_nat_real_real: ( nat > real ) > ( real > real ) > nat > real ).
thf(sy_c_Fun_Ofcomp_001t__Real__Oreal_001t__Complex__Ocomplex_001t__Real__Oreal,type,
fcomp_8157823527453731371x_real: ( real > complex ) > ( complex > real ) > real > real ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
abs_abs_complex: complex > complex ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
uminus8566677241136511917omplex: set_complex > set_complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
zero_z1877163951443063103omplex: formal670952693614245302omplex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Infinite__Products_Oraw__has__prod_001t__Complex__Ocomplex,type,
infini5805797430451707772omplex: ( nat > complex ) > nat > complex > $o ).
thf(sy_c_Infinite__Products_Oraw__has__prod_001t__Int__Oint,type,
infini1033036620519412986od_int: ( nat > int ) > nat > int > $o ).
thf(sy_c_Infinite__Products_Oraw__has__prod_001t__Nat__Onat,type,
infini1035527091028463262od_nat: ( nat > nat ) > nat > nat > $o ).
thf(sy_c_Infinite__Products_Oraw__has__prod_001t__Real__Oreal,type,
infini2923794516677094010d_real: ( nat > real ) > nat > real > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Complex__Ocomplex,type,
ord_less_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
real_V2521375963428798218omplex: set_complex ).
thf(sy_c_Real__Vector__Spaces_OReals_001t__Real__Oreal,type,
real_V470468836141973256s_real: set_real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
real_V4546457046886955230omplex: real > complex ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
real_V1803761363581548252l_real: real > real ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V1485227260804924795R_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
image_1468599708987790691omplex: ( complex > complex ) > set_complex > set_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Nat__Onat,type,
image_complex_nat: ( complex > nat ) > set_complex > set_nat ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Real__Oreal,type,
image_complex_real: ( complex > real ) > set_complex > set_real ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Complex__Ocomplex,type,
image_nat_complex: ( nat > complex ) > set_nat > set_complex ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Complex__Ocomplex,type,
image_real_complex: ( real > complex ) > set_real > set_complex ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
topolo9015423870875150044omplex: set_complex > ( complex > complex ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
topolo3759945079839938046ex_nat: set_complex > ( complex > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
topolo8674095878704923098x_real: set_complex > ( complex > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001t__Complex__Ocomplex,type,
topolo140161755405200382omplex: set_nat > ( nat > complex ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001t__Nat__Onat,type,
topolo1182047505939668768at_nat: set_nat > ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Nat__Onat_001t__Real__Oreal,type,
topolo6943266826644216316t_real: set_nat > ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Complex__Ocomplex,type,
topolo8620507378200602458omplex: set_real > ( real > complex ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(sy_c_Transcendental_Oarsinh_001t__Complex__Ocomplex,type,
arsinh_complex: complex > complex ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Complex__Ocomplex,type,
artanh_complex: complex > complex ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
cos_complex: complex > complex ).
thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
cos_real: real > real ).
thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
exp_complex: complex > complex ).
thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
exp_real: real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
sin_complex: complex > complex ).
thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
sin_real: real > real ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_A,type,
a: set_real ).
% Relevant facts (1274)
thf(fact_0_cot__pfd__complex__of__real,axiom,
! [X: real] :
( ( cotang8298477626502807258omplex @ ( real_V4546457046886955230omplex @ X ) )
= ( real_V4546457046886955230omplex @ ( cotang1502006655779026648d_real @ X ) ) ) ).
% cot_pfd_complex_of_real
thf(fact_1_Re__complex__of__real,axiom,
! [Z: real] :
( ( re @ ( real_V4546457046886955230omplex @ Z ) )
= Z ) ).
% Re_complex_of_real
thf(fact_2_calculation,axiom,
topolo5044208981011980120l_real @ a @ ( comp_c3333796836230738283l_real @ ( comp_c2063761206571265261omplex @ re @ cotang8298477626502807258omplex ) @ real_V4546457046886955230omplex ) ).
% calculation
thf(fact_3_of__real__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( real_V4546457046886955230omplex @ X )
= ( real_V4546457046886955230omplex @ Y ) )
= ( X = Y ) ) ).
% of_real_eq_iff
thf(fact_4_comp__apply,axiom,
( comp_real_real_nat
= ( ^ [F: real > real,G: nat > real,X2: nat] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_5_comp__apply,axiom,
( comp_n4415746728862867729omplex
= ( ^ [F: nat > complex,G: complex > nat,X2: complex] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_6_comp__apply,axiom,
( comp_nat_complex_nat
= ( ^ [F: nat > complex,G: nat > nat,X2: nat] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_7_comp__apply,axiom,
( comp_n4161660787930333455omplex
= ( ^ [F: nat > real,G: complex > nat,X2: complex] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_8_comp__apply,axiom,
( comp_nat_real_nat
= ( ^ [F: nat > real,G: nat > nat,X2: nat] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_9_comp__apply,axiom,
( comp_c3333796836230738283l_real
= ( ^ [F: complex > real,G: real > complex,X2: real] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_10_comp__apply,axiom,
( comp_c2063761206571265261omplex
= ( ^ [F: complex > real,G: complex > complex,X2: complex] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_11_comp__apply,axiom,
( comp_r1225911664865567631ex_nat
= ( ^ [F: real > complex,G: nat > real,X2: nat] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_12_comp__apply,axiom,
( comp_r891790309028876909omplex
= ( ^ [F: real > complex,G: complex > real,X2: complex] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_13_fun_Omap__comp,axiom,
! [G2: real > complex,F2: complex > real,V: real > complex] :
( ( comp_r1968866223832618731x_real @ G2 @ ( comp_c3333796836230738283l_real @ F2 @ V ) )
= ( comp_c2117349707075585901x_real @ ( comp_r891790309028876909omplex @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_14_fun_Omap__comp,axiom,
! [G2: real > real,F2: complex > real,V: real > complex] :
( ( comp_real_real_real @ G2 @ ( comp_c3333796836230738283l_real @ F2 @ V ) )
= ( comp_c3333796836230738283l_real @ ( comp_r1915277723328298091omplex @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_15_fun_Omap__comp,axiom,
! [G2: real > real,F2: complex > real,V: complex > complex] :
( ( comp_r1915277723328298091omplex @ G2 @ ( comp_c2063761206571265261omplex @ F2 @ V ) )
= ( comp_c2063761206571265261omplex @ ( comp_r1915277723328298091omplex @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_16_fun_Omap__comp,axiom,
! [G2: complex > real,F2: real > complex,V: nat > real] :
( ( comp_c7990426058975542799al_nat @ G2 @ ( comp_r1225911664865567631ex_nat @ F2 @ V ) )
= ( comp_real_real_nat @ ( comp_c3333796836230738283l_real @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_17_fun_Omap__comp,axiom,
! [G2: complex > complex,F2: real > complex,V: nat > real] :
( ( comp_c7268011922939458833ex_nat @ G2 @ ( comp_r1225911664865567631ex_nat @ F2 @ V ) )
= ( comp_r1225911664865567631ex_nat @ ( comp_c2117349707075585901x_real @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_18_fun_Omap__comp,axiom,
! [G2: complex > complex,F2: real > complex,V: complex > real] :
( ( comp_c5959460852125409263omplex @ G2 @ ( comp_r891790309028876909omplex @ F2 @ V ) )
= ( comp_r891790309028876909omplex @ ( comp_c2117349707075585901x_real @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_19_fun_Omap__comp,axiom,
! [G2: complex > real,F2: real > complex,V: real > real] :
( ( comp_c3333796836230738283l_real @ G2 @ ( comp_r1968866223832618731x_real @ F2 @ V ) )
= ( comp_real_real_real @ ( comp_c3333796836230738283l_real @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_20_fun_Omap__comp,axiom,
! [G2: complex > real,F2: complex > complex,V: real > complex] :
( ( comp_c3333796836230738283l_real @ G2 @ ( comp_c2117349707075585901x_real @ F2 @ V ) )
= ( comp_c3333796836230738283l_real @ ( comp_c2063761206571265261omplex @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_21_fun_Omap__comp,axiom,
! [G2: complex > real,F2: complex > complex,V: complex > complex] :
( ( comp_c2063761206571265261omplex @ G2 @ ( comp_c5959460852125409263omplex @ F2 @ V ) )
= ( comp_c2063761206571265261omplex @ ( comp_c2063761206571265261omplex @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_22_fun_Omap__comp,axiom,
! [G2: complex > real,F2: real > complex,V: complex > real] :
( ( comp_c2063761206571265261omplex @ G2 @ ( comp_r891790309028876909omplex @ F2 @ V ) )
= ( comp_r1915277723328298091omplex @ ( comp_c3333796836230738283l_real @ G2 @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_23_comp__def,axiom,
( comp_real_real_nat
= ( ^ [F: real > real,G: nat > real,X2: nat] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_24_comp__def,axiom,
( comp_n4415746728862867729omplex
= ( ^ [F: nat > complex,G: complex > nat,X2: complex] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_25_comp__def,axiom,
( comp_nat_complex_nat
= ( ^ [F: nat > complex,G: nat > nat,X2: nat] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_26_comp__def,axiom,
( comp_n4161660787930333455omplex
= ( ^ [F: nat > real,G: complex > nat,X2: complex] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_27_comp__def,axiom,
( comp_nat_real_nat
= ( ^ [F: nat > real,G: nat > nat,X2: nat] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_28_comp__def,axiom,
( comp_c3333796836230738283l_real
= ( ^ [F: complex > real,G: real > complex,X2: real] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_29_comp__def,axiom,
( comp_c2063761206571265261omplex
= ( ^ [F: complex > real,G: complex > complex,X2: complex] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_30_comp__def,axiom,
( comp_r1225911664865567631ex_nat
= ( ^ [F: real > complex,G: nat > real,X2: nat] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_31_comp__def,axiom,
( comp_r891790309028876909omplex
= ( ^ [F: real > complex,G: complex > real,X2: complex] : ( F @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_32_comp__assoc,axiom,
! [F2: complex > real,G2: real > complex,H: nat > real] :
( ( comp_real_real_nat @ ( comp_c3333796836230738283l_real @ F2 @ G2 ) @ H )
= ( comp_c7990426058975542799al_nat @ F2 @ ( comp_r1225911664865567631ex_nat @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_33_comp__assoc,axiom,
! [F2: complex > real,G2: real > complex,H: real > real] :
( ( comp_real_real_real @ ( comp_c3333796836230738283l_real @ F2 @ G2 ) @ H )
= ( comp_c3333796836230738283l_real @ F2 @ ( comp_r1968866223832618731x_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_34_comp__assoc,axiom,
! [F2: complex > real,G2: real > complex,H: complex > real] :
( ( comp_r1915277723328298091omplex @ ( comp_c3333796836230738283l_real @ F2 @ G2 ) @ H )
= ( comp_c2063761206571265261omplex @ F2 @ ( comp_r891790309028876909omplex @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_35_comp__assoc,axiom,
! [F2: real > complex,G2: nat > real,H: nat > nat] :
( ( comp_nat_complex_nat @ ( comp_r1225911664865567631ex_nat @ F2 @ G2 ) @ H )
= ( comp_r1225911664865567631ex_nat @ F2 @ ( comp_nat_real_nat @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_36_comp__assoc,axiom,
! [F2: real > complex,G2: nat > real,H: complex > nat] :
( ( comp_n4415746728862867729omplex @ ( comp_r1225911664865567631ex_nat @ F2 @ G2 ) @ H )
= ( comp_r891790309028876909omplex @ F2 @ ( comp_n4161660787930333455omplex @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_37_comp__assoc,axiom,
! [F2: real > complex,G2: complex > real,H: real > complex] :
( ( comp_c2117349707075585901x_real @ ( comp_r891790309028876909omplex @ F2 @ G2 ) @ H )
= ( comp_r1968866223832618731x_real @ F2 @ ( comp_c3333796836230738283l_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_38_comp__assoc,axiom,
! [F2: real > complex,G2: complex > real,H: nat > complex] :
( ( comp_c7268011922939458833ex_nat @ ( comp_r891790309028876909omplex @ F2 @ G2 ) @ H )
= ( comp_r1225911664865567631ex_nat @ F2 @ ( comp_c7990426058975542799al_nat @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_39_comp__assoc,axiom,
! [F2: real > complex,G2: complex > real,H: complex > complex] :
( ( comp_c5959460852125409263omplex @ ( comp_r891790309028876909omplex @ F2 @ G2 ) @ H )
= ( comp_r891790309028876909omplex @ F2 @ ( comp_c2063761206571265261omplex @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_40_comp__assoc,axiom,
! [F2: real > real,G2: complex > real,H: real > complex] :
( ( comp_c3333796836230738283l_real @ ( comp_r1915277723328298091omplex @ F2 @ G2 ) @ H )
= ( comp_real_real_real @ F2 @ ( comp_c3333796836230738283l_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_41_comp__assoc,axiom,
! [F2: complex > real,G2: complex > complex,H: real > complex] :
( ( comp_c3333796836230738283l_real @ ( comp_c2063761206571265261omplex @ F2 @ G2 ) @ H )
= ( comp_c3333796836230738283l_real @ F2 @ ( comp_c2117349707075585901x_real @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_42_comp__eq__dest,axiom,
! [A: complex > real,B: real > complex,C: complex > real,D: real > complex,V: real] :
( ( ( comp_c3333796836230738283l_real @ A @ B )
= ( comp_c3333796836230738283l_real @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_43_comp__eq__dest,axiom,
! [A: complex > real,B: complex > complex,C: complex > real,D: complex > complex,V: complex] :
( ( ( comp_c2063761206571265261omplex @ A @ B )
= ( comp_c2063761206571265261omplex @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_44_comp__eq__dest,axiom,
! [A: real > complex,B: nat > real,C: real > complex,D: nat > real,V: nat] :
( ( ( comp_r1225911664865567631ex_nat @ A @ B )
= ( comp_r1225911664865567631ex_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_45_comp__eq__dest,axiom,
! [A: real > complex,B: complex > real,C: real > complex,D: complex > real,V: complex] :
( ( ( comp_r891790309028876909omplex @ A @ B )
= ( comp_r891790309028876909omplex @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_46_comp__eq__dest,axiom,
! [A: complex > real,B: complex > complex,C: nat > real,D: complex > nat,V: complex] :
( ( ( comp_c2063761206571265261omplex @ A @ B )
= ( comp_n4161660787930333455omplex @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_47_comp__eq__dest,axiom,
! [A: real > complex,B: nat > real,C: nat > complex,D: nat > nat,V: nat] :
( ( ( comp_r1225911664865567631ex_nat @ A @ B )
= ( comp_nat_complex_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_48_comp__eq__dest,axiom,
! [A: real > complex,B: complex > real,C: nat > complex,D: complex > nat,V: complex] :
( ( ( comp_r891790309028876909omplex @ A @ B )
= ( comp_n4415746728862867729omplex @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_49_comp__eq__dest,axiom,
! [A: real > real,B: nat > real,C: real > real,D: nat > real,V: nat] :
( ( ( comp_real_real_nat @ A @ B )
= ( comp_real_real_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_50_comp__eq__dest,axiom,
! [A: real > real,B: nat > real,C: nat > real,D: nat > nat,V: nat] :
( ( ( comp_real_real_nat @ A @ B )
= ( comp_nat_real_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_51_comp__eq__dest,axiom,
! [A: nat > complex,B: complex > nat,C: real > complex,D: complex > real,V: complex] :
( ( ( comp_n4415746728862867729omplex @ A @ B )
= ( comp_r891790309028876909omplex @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_52_comp__eq__elim,axiom,
! [A: complex > real,B: real > complex,C: complex > real,D: real > complex] :
( ( ( comp_c3333796836230738283l_real @ A @ B )
= ( comp_c3333796836230738283l_real @ C @ D ) )
=> ! [V2: real] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_53_comp__eq__elim,axiom,
! [A: complex > real,B: complex > complex,C: complex > real,D: complex > complex] :
( ( ( comp_c2063761206571265261omplex @ A @ B )
= ( comp_c2063761206571265261omplex @ C @ D ) )
=> ! [V2: complex] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_54_comp__eq__elim,axiom,
! [A: real > complex,B: nat > real,C: real > complex,D: nat > real] :
( ( ( comp_r1225911664865567631ex_nat @ A @ B )
= ( comp_r1225911664865567631ex_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_55_comp__eq__elim,axiom,
! [A: real > complex,B: complex > real,C: real > complex,D: complex > real] :
( ( ( comp_r891790309028876909omplex @ A @ B )
= ( comp_r891790309028876909omplex @ C @ D ) )
=> ! [V2: complex] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_56_comp__eq__elim,axiom,
! [A: complex > real,B: complex > complex,C: nat > real,D: complex > nat] :
( ( ( comp_c2063761206571265261omplex @ A @ B )
= ( comp_n4161660787930333455omplex @ C @ D ) )
=> ! [V2: complex] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_57_comp__eq__elim,axiom,
! [A: real > complex,B: nat > real,C: nat > complex,D: nat > nat] :
( ( ( comp_r1225911664865567631ex_nat @ A @ B )
= ( comp_nat_complex_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_58_comp__eq__elim,axiom,
! [A: real > complex,B: complex > real,C: nat > complex,D: complex > nat] :
( ( ( comp_r891790309028876909omplex @ A @ B )
= ( comp_n4415746728862867729omplex @ C @ D ) )
=> ! [V2: complex] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_59_comp__eq__elim,axiom,
! [A: real > real,B: nat > real,C: real > real,D: nat > real] :
( ( ( comp_real_real_nat @ A @ B )
= ( comp_real_real_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_60_comp__eq__elim,axiom,
! [A: real > real,B: nat > real,C: nat > real,D: nat > nat] :
( ( ( comp_real_real_nat @ A @ B )
= ( comp_nat_real_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_61_comp__eq__elim,axiom,
! [A: nat > complex,B: complex > nat,C: real > complex,D: complex > real] :
( ( ( comp_n4415746728862867729omplex @ A @ B )
= ( comp_r891790309028876909omplex @ C @ D ) )
=> ! [V2: complex] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_62_comp__cong,axiom,
! [F2: complex > real,G2: real > complex,X: real,F3: complex > real,G3: real > complex,X3: real] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c3333796836230738283l_real @ F2 @ G2 @ X )
= ( comp_c3333796836230738283l_real @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_63_comp__cong,axiom,
! [F2: complex > real,G2: real > complex,X: real,F3: complex > real,G3: complex > complex,X3: complex] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c3333796836230738283l_real @ F2 @ G2 @ X )
= ( comp_c2063761206571265261omplex @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_64_comp__cong,axiom,
! [F2: complex > real,G2: real > complex,X: real,F3: real > real,G3: nat > real,X3: nat] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c3333796836230738283l_real @ F2 @ G2 @ X )
= ( comp_real_real_nat @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_65_comp__cong,axiom,
! [F2: complex > real,G2: real > complex,X: real,F3: nat > real,G3: complex > nat,X3: complex] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c3333796836230738283l_real @ F2 @ G2 @ X )
= ( comp_n4161660787930333455omplex @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_66_comp__cong,axiom,
! [F2: complex > real,G2: real > complex,X: real,F3: nat > real,G3: nat > nat,X3: nat] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c3333796836230738283l_real @ F2 @ G2 @ X )
= ( comp_nat_real_nat @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_67_comp__cong,axiom,
! [F2: complex > real,G2: complex > complex,X: complex,F3: complex > real,G3: real > complex,X3: real] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c2063761206571265261omplex @ F2 @ G2 @ X )
= ( comp_c3333796836230738283l_real @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_68_comp__cong,axiom,
! [F2: complex > real,G2: complex > complex,X: complex,F3: complex > real,G3: complex > complex,X3: complex] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c2063761206571265261omplex @ F2 @ G2 @ X )
= ( comp_c2063761206571265261omplex @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_69_comp__cong,axiom,
! [F2: complex > real,G2: complex > complex,X: complex,F3: real > real,G3: nat > real,X3: nat] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c2063761206571265261omplex @ F2 @ G2 @ X )
= ( comp_real_real_nat @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_70_comp__cong,axiom,
! [F2: complex > real,G2: complex > complex,X: complex,F3: nat > real,G3: complex > nat,X3: complex] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c2063761206571265261omplex @ F2 @ G2 @ X )
= ( comp_n4161660787930333455omplex @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_71_comp__cong,axiom,
! [F2: complex > real,G2: complex > complex,X: complex,F3: nat > real,G3: nat > nat,X3: nat] :
( ( ( F2 @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp_c2063761206571265261omplex @ F2 @ G2 @ X )
= ( comp_nat_real_nat @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_72_comp__eq__dest__lhs,axiom,
! [A: complex > real,B: real > complex,C: real > real,V: real] :
( ( ( comp_c3333796836230738283l_real @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_73_comp__eq__dest__lhs,axiom,
! [A: complex > real,B: complex > complex,C: complex > real,V: complex] :
( ( ( comp_c2063761206571265261omplex @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_74_comp__eq__dest__lhs,axiom,
! [A: real > complex,B: nat > real,C: nat > complex,V: nat] :
( ( ( comp_r1225911664865567631ex_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_75_comp__eq__dest__lhs,axiom,
! [A: real > complex,B: complex > real,C: complex > complex,V: complex] :
( ( ( comp_r891790309028876909omplex @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_76_comp__eq__dest__lhs,axiom,
! [A: real > real,B: nat > real,C: nat > real,V: nat] :
( ( ( comp_real_real_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_77_comp__eq__dest__lhs,axiom,
! [A: nat > complex,B: complex > nat,C: complex > complex,V: complex] :
( ( ( comp_n4415746728862867729omplex @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_78_comp__eq__dest__lhs,axiom,
! [A: nat > complex,B: nat > nat,C: nat > complex,V: nat] :
( ( ( comp_nat_complex_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_79_comp__eq__dest__lhs,axiom,
! [A: nat > real,B: complex > nat,C: complex > real,V: complex] :
( ( ( comp_n4161660787930333455omplex @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_80_comp__eq__dest__lhs,axiom,
! [A: nat > real,B: nat > nat,C: nat > real,V: nat] :
( ( ( comp_nat_real_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_81_comp__apply__eq,axiom,
! [F2: complex > real,G2: real > complex,X: real,H: complex > real,K: real > complex] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_c3333796836230738283l_real @ F2 @ G2 @ X )
= ( comp_c3333796836230738283l_real @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_82_comp__apply__eq,axiom,
! [F2: complex > real,G2: complex > complex,X: complex,H: complex > real,K: complex > complex] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_c2063761206571265261omplex @ F2 @ G2 @ X )
= ( comp_c2063761206571265261omplex @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_83_comp__apply__eq,axiom,
! [F2: complex > real,G2: complex > complex,X: complex,H: nat > real,K: complex > nat] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_c2063761206571265261omplex @ F2 @ G2 @ X )
= ( comp_n4161660787930333455omplex @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_84_comp__apply__eq,axiom,
! [F2: real > complex,G2: nat > real,X: nat,H: real > complex,K: nat > real] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_r1225911664865567631ex_nat @ F2 @ G2 @ X )
= ( comp_r1225911664865567631ex_nat @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_85_comp__apply__eq,axiom,
! [F2: real > complex,G2: nat > real,X: nat,H: nat > complex,K: nat > nat] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_r1225911664865567631ex_nat @ F2 @ G2 @ X )
= ( comp_nat_complex_nat @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_86_comp__apply__eq,axiom,
! [F2: real > complex,G2: complex > real,X: complex,H: real > complex,K: complex > real] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_r891790309028876909omplex @ F2 @ G2 @ X )
= ( comp_r891790309028876909omplex @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_87_comp__apply__eq,axiom,
! [F2: real > complex,G2: complex > real,X: complex,H: nat > complex,K: complex > nat] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_r891790309028876909omplex @ F2 @ G2 @ X )
= ( comp_n4415746728862867729omplex @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_88_comp__apply__eq,axiom,
! [F2: real > real,G2: nat > real,X: nat,H: real > real,K: nat > real] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_real_real_nat @ F2 @ G2 @ X )
= ( comp_real_real_nat @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_89_comp__apply__eq,axiom,
! [F2: real > real,G2: nat > real,X: nat,H: nat > real,K: nat > nat] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_real_real_nat @ F2 @ G2 @ X )
= ( comp_nat_real_nat @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_90_comp__apply__eq,axiom,
! [F2: nat > complex,G2: complex > nat,X: complex,H: real > complex,K: complex > real] :
( ( ( F2 @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp_n4415746728862867729omplex @ F2 @ G2 @ X )
= ( comp_r891790309028876909omplex @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_91_continuous__on__of__real__id,axiom,
! [S: set_real] : ( topolo8620507378200602458omplex @ S @ real_V4546457046886955230omplex ) ).
% continuous_on_of_real_id
thf(fact_92_continuous__on__of__real__id,axiom,
! [S: set_real] : ( topolo5044208981011980120l_real @ S @ real_V1803761363581548252l_real ) ).
% continuous_on_of_real_id
thf(fact_93_continuous__on__eq,axiom,
! [S2: set_real,F2: real > real,G2: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F2 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ( F2 @ X4 )
= ( G2 @ X4 ) ) )
=> ( topolo5044208981011980120l_real @ S2 @ G2 ) ) ) ).
% continuous_on_eq
thf(fact_94_continuous__on__eq,axiom,
! [S2: set_complex,F2: complex > complex,G2: complex > complex] :
( ( topolo9015423870875150044omplex @ S2 @ F2 )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ S2 )
=> ( ( F2 @ X4 )
= ( G2 @ X4 ) ) )
=> ( topolo9015423870875150044omplex @ S2 @ G2 ) ) ) ).
% continuous_on_eq
thf(fact_95_continuous__on__cong,axiom,
! [S2: set_real,T: set_real,F2: real > real,G2: real > real] :
( ( S2 = T )
=> ( ! [X4: real] :
( ( member_real @ X4 @ T )
=> ( ( F2 @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( topolo5044208981011980120l_real @ S2 @ F2 )
= ( topolo5044208981011980120l_real @ T @ G2 ) ) ) ) ).
% continuous_on_cong
thf(fact_96_continuous__on__cong,axiom,
! [S2: set_complex,T: set_complex,F2: complex > complex,G2: complex > complex] :
( ( S2 = T )
=> ( ! [X4: complex] :
( ( member_complex @ X4 @ T )
=> ( ( F2 @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( topolo9015423870875150044omplex @ S2 @ F2 )
= ( topolo9015423870875150044omplex @ T @ G2 ) ) ) ) ).
% continuous_on_cong
thf(fact_97_rewriteR__comp__comp2,axiom,
! [G2: complex > nat,H: nat > complex,R1: real > nat,R2: nat > real,F2: nat > complex,L: real > complex] :
( ( ( comp_complex_nat_nat @ G2 @ H )
= ( comp_real_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_n4215249288434654095x_real @ F2 @ R1 )
= L )
=> ( ( comp_c7268011922939458833ex_nat @ ( comp_n4415746728862867729omplex @ F2 @ G2 ) @ H )
= ( comp_r1225911664865567631ex_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_98_rewriteR__comp__comp2,axiom,
! [G2: complex > nat,H: complex > complex,R1: real > nat,R2: complex > real,F2: nat > complex,L: real > complex] :
( ( ( comp_c3648228598504721169omplex @ G2 @ H )
= ( comp_r6829500377285605775omplex @ R1 @ R2 ) )
=> ( ( ( comp_n4215249288434654095x_real @ F2 @ R1 )
= L )
=> ( ( comp_c5959460852125409263omplex @ ( comp_n4415746728862867729omplex @ F2 @ G2 ) @ H )
= ( comp_r891790309028876909omplex @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_99_rewriteR__comp__comp2,axiom,
! [G2: complex > nat,H: nat > complex,R1: real > nat,R2: nat > real,F2: nat > real,L: real > real] :
( ( ( comp_complex_nat_nat @ G2 @ H )
= ( comp_real_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_nat_real_real @ F2 @ R1 )
= L )
=> ( ( comp_c7990426058975542799al_nat @ ( comp_n4161660787930333455omplex @ F2 @ G2 ) @ H )
= ( comp_real_real_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_100_rewriteR__comp__comp2,axiom,
! [G2: nat > nat,H: complex > nat,R1: real > nat,R2: complex > real,F2: nat > complex,L: real > complex] :
( ( ( comp_nat_nat_complex @ G2 @ H )
= ( comp_r6829500377285605775omplex @ R1 @ R2 ) )
=> ( ( ( comp_n4215249288434654095x_real @ F2 @ R1 )
= L )
=> ( ( comp_n4415746728862867729omplex @ ( comp_nat_complex_nat @ F2 @ G2 ) @ H )
= ( comp_r891790309028876909omplex @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_101_rewriteR__comp__comp2,axiom,
! [G2: nat > nat,H: nat > nat,R1: real > nat,R2: nat > real,F2: nat > complex,L: real > complex] :
( ( ( comp_nat_nat_nat @ G2 @ H )
= ( comp_real_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_n4215249288434654095x_real @ F2 @ R1 )
= L )
=> ( ( comp_nat_complex_nat @ ( comp_nat_complex_nat @ F2 @ G2 ) @ H )
= ( comp_r1225911664865567631ex_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_102_rewriteR__comp__comp2,axiom,
! [G2: nat > nat,H: nat > nat,R1: real > nat,R2: nat > real,F2: nat > real,L: real > real] :
( ( ( comp_nat_nat_nat @ G2 @ H )
= ( comp_real_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_nat_real_real @ F2 @ R1 )
= L )
=> ( ( comp_nat_real_nat @ ( comp_nat_real_nat @ F2 @ G2 ) @ H )
= ( comp_real_real_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_103_rewriteR__comp__comp2,axiom,
! [G2: complex > complex,H: real > complex,R1: real > complex,R2: real > real,F2: complex > real,L: real > real] :
( ( ( comp_c2117349707075585901x_real @ G2 @ H )
= ( comp_r1968866223832618731x_real @ R1 @ R2 ) )
=> ( ( ( comp_c3333796836230738283l_real @ F2 @ R1 )
= L )
=> ( ( comp_c3333796836230738283l_real @ ( comp_c2063761206571265261omplex @ F2 @ G2 ) @ H )
= ( comp_real_real_real @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_104_rewriteR__comp__comp2,axiom,
! [G2: real > complex,H: real > real,R1: complex > complex,R2: real > complex,F2: complex > real,L: complex > real] :
( ( ( comp_r1968866223832618731x_real @ G2 @ H )
= ( comp_c2117349707075585901x_real @ R1 @ R2 ) )
=> ( ( ( comp_c2063761206571265261omplex @ F2 @ R1 )
= L )
=> ( ( comp_real_real_real @ ( comp_c3333796836230738283l_real @ F2 @ G2 ) @ H )
= ( comp_c3333796836230738283l_real @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_105_rewriteR__comp__comp2,axiom,
! [G2: complex > complex,H: real > complex,R1: complex > complex,R2: real > complex,F2: complex > real,L: complex > real] :
( ( ( comp_c2117349707075585901x_real @ G2 @ H )
= ( comp_c2117349707075585901x_real @ R1 @ R2 ) )
=> ( ( ( comp_c2063761206571265261omplex @ F2 @ R1 )
= L )
=> ( ( comp_c3333796836230738283l_real @ ( comp_c2063761206571265261omplex @ F2 @ G2 ) @ H )
= ( comp_c3333796836230738283l_real @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_106_rewriteR__comp__comp2,axiom,
! [G2: complex > complex,H: complex > complex,R1: complex > complex,R2: complex > complex,F2: complex > real,L: complex > real] :
( ( ( comp_c5959460852125409263omplex @ G2 @ H )
= ( comp_c5959460852125409263omplex @ R1 @ R2 ) )
=> ( ( ( comp_c2063761206571265261omplex @ F2 @ R1 )
= L )
=> ( ( comp_c2063761206571265261omplex @ ( comp_c2063761206571265261omplex @ F2 @ G2 ) @ H )
= ( comp_c2063761206571265261omplex @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_107_rewriteL__comp__comp2,axiom,
! [F2: complex > complex,G2: real > complex,L1: nat > complex,L2: real > nat,H: nat > real,R: nat > nat] :
( ( ( comp_c2117349707075585901x_real @ F2 @ G2 )
= ( comp_n4215249288434654095x_real @ L1 @ L2 ) )
=> ( ( ( comp_real_nat_nat @ L2 @ H )
= R )
=> ( ( comp_c7268011922939458833ex_nat @ F2 @ ( comp_r1225911664865567631ex_nat @ G2 @ H ) )
= ( comp_nat_complex_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_108_rewriteL__comp__comp2,axiom,
! [F2: complex > complex,G2: real > complex,L1: real > complex,L2: real > real,H: complex > real,R: complex > real] :
( ( ( comp_c2117349707075585901x_real @ F2 @ G2 )
= ( comp_r1968866223832618731x_real @ L1 @ L2 ) )
=> ( ( ( comp_r1915277723328298091omplex @ L2 @ H )
= R )
=> ( ( comp_c5959460852125409263omplex @ F2 @ ( comp_r891790309028876909omplex @ G2 @ H ) )
= ( comp_r891790309028876909omplex @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_109_rewriteL__comp__comp2,axiom,
! [F2: complex > complex,G2: real > complex,L1: nat > complex,L2: real > nat,H: complex > real,R: complex > nat] :
( ( ( comp_c2117349707075585901x_real @ F2 @ G2 )
= ( comp_n4215249288434654095x_real @ L1 @ L2 ) )
=> ( ( ( comp_r6829500377285605775omplex @ L2 @ H )
= R )
=> ( ( comp_c5959460852125409263omplex @ F2 @ ( comp_r891790309028876909omplex @ G2 @ H ) )
= ( comp_n4415746728862867729omplex @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_110_rewriteL__comp__comp2,axiom,
! [F2: real > complex,G2: real > real,L1: nat > complex,L2: real > nat,H: nat > real,R: nat > nat] :
( ( ( comp_r1968866223832618731x_real @ F2 @ G2 )
= ( comp_n4215249288434654095x_real @ L1 @ L2 ) )
=> ( ( ( comp_real_nat_nat @ L2 @ H )
= R )
=> ( ( comp_r1225911664865567631ex_nat @ F2 @ ( comp_real_real_nat @ G2 @ H ) )
= ( comp_nat_complex_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_111_rewriteL__comp__comp2,axiom,
! [F2: real > real,G2: real > real,L1: nat > real,L2: real > nat,H: nat > real,R: nat > nat] :
( ( ( comp_real_real_real @ F2 @ G2 )
= ( comp_nat_real_real @ L1 @ L2 ) )
=> ( ( ( comp_real_nat_nat @ L2 @ H )
= R )
=> ( ( comp_real_real_nat @ F2 @ ( comp_real_real_nat @ G2 @ H ) )
= ( comp_nat_real_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_112_rewriteL__comp__comp2,axiom,
! [F2: real > complex,G2: real > real,L1: complex > complex,L2: real > complex,H: nat > real,R: nat > complex] :
( ( ( comp_r1968866223832618731x_real @ F2 @ G2 )
= ( comp_c2117349707075585901x_real @ L1 @ L2 ) )
=> ( ( ( comp_r1225911664865567631ex_nat @ L2 @ H )
= R )
=> ( ( comp_r1225911664865567631ex_nat @ F2 @ ( comp_real_real_nat @ G2 @ H ) )
= ( comp_c7268011922939458833ex_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_113_rewriteL__comp__comp2,axiom,
! [F2: nat > complex,G2: real > nat,L1: complex > complex,L2: real > complex,H: nat > real,R: nat > complex] :
( ( ( comp_n4215249288434654095x_real @ F2 @ G2 )
= ( comp_c2117349707075585901x_real @ L1 @ L2 ) )
=> ( ( ( comp_r1225911664865567631ex_nat @ L2 @ H )
= R )
=> ( ( comp_nat_complex_nat @ F2 @ ( comp_real_nat_nat @ G2 @ H ) )
= ( comp_c7268011922939458833ex_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_114_rewriteL__comp__comp2,axiom,
! [F2: real > complex,G2: real > real,L1: complex > complex,L2: real > complex,H: complex > real,R: complex > complex] :
( ( ( comp_r1968866223832618731x_real @ F2 @ G2 )
= ( comp_c2117349707075585901x_real @ L1 @ L2 ) )
=> ( ( ( comp_r891790309028876909omplex @ L2 @ H )
= R )
=> ( ( comp_r891790309028876909omplex @ F2 @ ( comp_r1915277723328298091omplex @ G2 @ H ) )
= ( comp_c5959460852125409263omplex @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_115_rewriteL__comp__comp2,axiom,
! [F2: nat > complex,G2: real > nat,L1: complex > complex,L2: real > complex,H: complex > real,R: complex > complex] :
( ( ( comp_n4215249288434654095x_real @ F2 @ G2 )
= ( comp_c2117349707075585901x_real @ L1 @ L2 ) )
=> ( ( ( comp_r891790309028876909omplex @ L2 @ H )
= R )
=> ( ( comp_n4415746728862867729omplex @ F2 @ ( comp_r6829500377285605775omplex @ G2 @ H ) )
= ( comp_c5959460852125409263omplex @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_116_rewriteL__comp__comp2,axiom,
! [F2: complex > complex,G2: real > complex,L1: real > complex,L2: real > real,H: nat > real,R: nat > real] :
( ( ( comp_c2117349707075585901x_real @ F2 @ G2 )
= ( comp_r1968866223832618731x_real @ L1 @ L2 ) )
=> ( ( ( comp_real_real_nat @ L2 @ H )
= R )
=> ( ( comp_c7268011922939458833ex_nat @ F2 @ ( comp_r1225911664865567631ex_nat @ G2 @ H ) )
= ( comp_r1225911664865567631ex_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_117_rewriteR__comp__comp,axiom,
! [G2: real > complex,H: real > real,R: real > complex,F2: complex > real] :
( ( ( comp_r1968866223832618731x_real @ G2 @ H )
= R )
=> ( ( comp_real_real_real @ ( comp_c3333796836230738283l_real @ F2 @ G2 ) @ H )
= ( comp_c3333796836230738283l_real @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_118_rewriteR__comp__comp,axiom,
! [G2: complex > real,H: nat > complex,R: nat > real,F2: real > complex] :
( ( ( comp_c7990426058975542799al_nat @ G2 @ H )
= R )
=> ( ( comp_c7268011922939458833ex_nat @ ( comp_r891790309028876909omplex @ F2 @ G2 ) @ H )
= ( comp_r1225911664865567631ex_nat @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_119_rewriteR__comp__comp,axiom,
! [G2: complex > nat,H: complex > complex,R: complex > nat,F2: nat > complex] :
( ( ( comp_c3648228598504721169omplex @ G2 @ H )
= R )
=> ( ( comp_c5959460852125409263omplex @ ( comp_n4415746728862867729omplex @ F2 @ G2 ) @ H )
= ( comp_n4415746728862867729omplex @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_120_rewriteR__comp__comp,axiom,
! [G2: complex > nat,H: nat > complex,R: nat > nat,F2: nat > complex] :
( ( ( comp_complex_nat_nat @ G2 @ H )
= R )
=> ( ( comp_c7268011922939458833ex_nat @ ( comp_n4415746728862867729omplex @ F2 @ G2 ) @ H )
= ( comp_nat_complex_nat @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_121_rewriteR__comp__comp,axiom,
! [G2: complex > nat,H: nat > complex,R: nat > nat,F2: nat > real] :
( ( ( comp_complex_nat_nat @ G2 @ H )
= R )
=> ( ( comp_c7990426058975542799al_nat @ ( comp_n4161660787930333455omplex @ F2 @ G2 ) @ H )
= ( comp_nat_real_nat @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_122_rewriteR__comp__comp,axiom,
! [G2: complex > complex,H: real > complex,R: real > complex,F2: complex > real] :
( ( ( comp_c2117349707075585901x_real @ G2 @ H )
= R )
=> ( ( comp_c3333796836230738283l_real @ ( comp_c2063761206571265261omplex @ F2 @ G2 ) @ H )
= ( comp_c3333796836230738283l_real @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_123_rewriteR__comp__comp,axiom,
! [G2: complex > nat,H: real > complex,R: real > nat,F2: nat > real] :
( ( ( comp_c3423117485846644111t_real @ G2 @ H )
= R )
=> ( ( comp_c3333796836230738283l_real @ ( comp_n4161660787930333455omplex @ F2 @ G2 ) @ H )
= ( comp_nat_real_real @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_124_rewriteR__comp__comp,axiom,
! [G2: complex > complex,H: complex > complex,R: complex > complex,F2: complex > real] :
( ( ( comp_c5959460852125409263omplex @ G2 @ H )
= R )
=> ( ( comp_c2063761206571265261omplex @ ( comp_c2063761206571265261omplex @ F2 @ G2 ) @ H )
= ( comp_c2063761206571265261omplex @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_125_rewriteR__comp__comp,axiom,
! [G2: complex > nat,H: complex > complex,R: complex > nat,F2: nat > real] :
( ( ( comp_c3648228598504721169omplex @ G2 @ H )
= R )
=> ( ( comp_c2063761206571265261omplex @ ( comp_n4161660787930333455omplex @ F2 @ G2 ) @ H )
= ( comp_n4161660787930333455omplex @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_126_rewriteR__comp__comp,axiom,
! [G2: real > nat,H: nat > real,R: nat > nat,F2: nat > complex] :
( ( ( comp_real_nat_nat @ G2 @ H )
= R )
=> ( ( comp_r1225911664865567631ex_nat @ ( comp_n4215249288434654095x_real @ F2 @ G2 ) @ H )
= ( comp_nat_complex_nat @ F2 @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_127_rewriteL__comp__comp,axiom,
! [F2: real > real,G2: complex > real,L: complex > real,H: real > complex] :
( ( ( comp_r1915277723328298091omplex @ F2 @ G2 )
= L )
=> ( ( comp_real_real_real @ F2 @ ( comp_c3333796836230738283l_real @ G2 @ H ) )
= ( comp_c3333796836230738283l_real @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_128_rewriteL__comp__comp,axiom,
! [F2: real > real,G2: complex > real,L: complex > real,H: complex > complex] :
( ( ( comp_r1915277723328298091omplex @ F2 @ G2 )
= L )
=> ( ( comp_r1915277723328298091omplex @ F2 @ ( comp_c2063761206571265261omplex @ G2 @ H ) )
= ( comp_c2063761206571265261omplex @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_129_rewriteL__comp__comp,axiom,
! [F2: complex > complex,G2: real > complex,L: real > complex,H: nat > real] :
( ( ( comp_c2117349707075585901x_real @ F2 @ G2 )
= L )
=> ( ( comp_c7268011922939458833ex_nat @ F2 @ ( comp_r1225911664865567631ex_nat @ G2 @ H ) )
= ( comp_r1225911664865567631ex_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_130_rewriteL__comp__comp,axiom,
! [F2: complex > complex,G2: real > complex,L: real > complex,H: complex > real] :
( ( ( comp_c2117349707075585901x_real @ F2 @ G2 )
= L )
=> ( ( comp_c5959460852125409263omplex @ F2 @ ( comp_r891790309028876909omplex @ G2 @ H ) )
= ( comp_r891790309028876909omplex @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_131_rewriteL__comp__comp,axiom,
! [F2: complex > complex,G2: nat > complex,L: nat > complex,H: complex > nat] :
( ( ( comp_c7268011922939458833ex_nat @ F2 @ G2 )
= L )
=> ( ( comp_c5959460852125409263omplex @ F2 @ ( comp_n4415746728862867729omplex @ G2 @ H ) )
= ( comp_n4415746728862867729omplex @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_132_rewriteL__comp__comp,axiom,
! [F2: complex > complex,G2: nat > complex,L: nat > complex,H: nat > nat] :
( ( ( comp_c7268011922939458833ex_nat @ F2 @ G2 )
= L )
=> ( ( comp_c7268011922939458833ex_nat @ F2 @ ( comp_nat_complex_nat @ G2 @ H ) )
= ( comp_nat_complex_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_133_rewriteL__comp__comp,axiom,
! [F2: complex > real,G2: nat > complex,L: nat > real,H: nat > nat] :
( ( ( comp_c7990426058975542799al_nat @ F2 @ G2 )
= L )
=> ( ( comp_c7990426058975542799al_nat @ F2 @ ( comp_nat_complex_nat @ G2 @ H ) )
= ( comp_nat_real_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_134_rewriteL__comp__comp,axiom,
! [F2: complex > real,G2: nat > complex,L: nat > real,H: complex > nat] :
( ( ( comp_c7990426058975542799al_nat @ F2 @ G2 )
= L )
=> ( ( comp_c2063761206571265261omplex @ F2 @ ( comp_n4415746728862867729omplex @ G2 @ H ) )
= ( comp_n4161660787930333455omplex @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_135_rewriteL__comp__comp,axiom,
! [F2: real > complex,G2: real > real,L: real > complex,H: nat > real] :
( ( ( comp_r1968866223832618731x_real @ F2 @ G2 )
= L )
=> ( ( comp_r1225911664865567631ex_nat @ F2 @ ( comp_real_real_nat @ G2 @ H ) )
= ( comp_r1225911664865567631ex_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_136_rewriteL__comp__comp,axiom,
! [F2: real > complex,G2: real > real,L: real > complex,H: complex > real] :
( ( ( comp_r1968866223832618731x_real @ F2 @ G2 )
= L )
=> ( ( comp_r891790309028876909omplex @ F2 @ ( comp_r1915277723328298091omplex @ G2 @ H ) )
= ( comp_r891790309028876909omplex @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_137_type__copy__map__cong0,axiom,
! [M: real > complex,G2: real > real,X: real,N: complex > complex,H: real > complex,F2: complex > real] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_real_real_real @ ( comp_c3333796836230738283l_real @ F2 @ M ) @ G2 @ X )
= ( comp_c3333796836230738283l_real @ ( comp_c2063761206571265261omplex @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_138_type__copy__map__cong0,axiom,
! [M: real > complex,G2: complex > real,X: complex,N: complex > complex,H: complex > complex,F2: complex > real] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_r1915277723328298091omplex @ ( comp_c3333796836230738283l_real @ F2 @ M ) @ G2 @ X )
= ( comp_c2063761206571265261omplex @ ( comp_c2063761206571265261omplex @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_139_type__copy__map__cong0,axiom,
! [M: real > complex,G2: complex > real,X: complex,N: nat > complex,H: complex > nat,F2: complex > real] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_r1915277723328298091omplex @ ( comp_c3333796836230738283l_real @ F2 @ M ) @ G2 @ X )
= ( comp_n4161660787930333455omplex @ ( comp_c7990426058975542799al_nat @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_140_type__copy__map__cong0,axiom,
! [M: complex > complex,G2: nat > complex,X: nat,N: real > complex,H: nat > real,F2: complex > real] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_c7990426058975542799al_nat @ ( comp_c2063761206571265261omplex @ F2 @ M ) @ G2 @ X )
= ( comp_real_real_nat @ ( comp_c3333796836230738283l_real @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_141_type__copy__map__cong0,axiom,
! [M: complex > complex,G2: nat > complex,X: nat,N: nat > complex,H: nat > nat,F2: complex > real] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_c7990426058975542799al_nat @ ( comp_c2063761206571265261omplex @ F2 @ M ) @ G2 @ X )
= ( comp_nat_real_nat @ ( comp_c7990426058975542799al_nat @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_142_type__copy__map__cong0,axiom,
! [M: complex > real,G2: nat > complex,X: nat,N: real > real,H: nat > real,F2: real > complex] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_c7268011922939458833ex_nat @ ( comp_r891790309028876909omplex @ F2 @ M ) @ G2 @ X )
= ( comp_r1225911664865567631ex_nat @ ( comp_r1968866223832618731x_real @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_143_type__copy__map__cong0,axiom,
! [M: complex > real,G2: complex > complex,X: complex,N: real > real,H: complex > real,F2: real > complex] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_c5959460852125409263omplex @ ( comp_r891790309028876909omplex @ F2 @ M ) @ G2 @ X )
= ( comp_r891790309028876909omplex @ ( comp_r1968866223832618731x_real @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_144_type__copy__map__cong0,axiom,
! [M: complex > real,G2: complex > complex,X: complex,N: nat > real,H: complex > nat,F2: real > complex] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_c5959460852125409263omplex @ ( comp_r891790309028876909omplex @ F2 @ M ) @ G2 @ X )
= ( comp_n4415746728862867729omplex @ ( comp_r1225911664865567631ex_nat @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_145_type__copy__map__cong0,axiom,
! [M: complex > real,G2: nat > complex,X: nat,N: nat > real,H: nat > nat,F2: real > complex] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_c7268011922939458833ex_nat @ ( comp_r891790309028876909omplex @ F2 @ M ) @ G2 @ X )
= ( comp_nat_complex_nat @ ( comp_r1225911664865567631ex_nat @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_146_type__copy__map__cong0,axiom,
! [M: nat > real,G2: real > nat,X: real,N: complex > real,H: real > complex,F2: real > real] :
( ( ( M @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp_nat_real_real @ ( comp_real_real_nat @ F2 @ M ) @ G2 @ X )
= ( comp_c3333796836230738283l_real @ ( comp_r1915277723328298091omplex @ F2 @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_147_function__factors__right,axiom,
! [G2: complex > real,F2: real > real] :
( ( ! [X2: real] :
? [Y2: complex] :
( ( G2 @ Y2 )
= ( F2 @ X2 ) ) )
= ( ? [H2: real > complex] :
( F2
= ( comp_c3333796836230738283l_real @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_148_function__factors__right,axiom,
! [G2: complex > real,F2: complex > real] :
( ( ! [X2: complex] :
? [Y2: complex] :
( ( G2 @ Y2 )
= ( F2 @ X2 ) ) )
= ( ? [H2: complex > complex] :
( F2
= ( comp_c2063761206571265261omplex @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_149_function__factors__right,axiom,
! [G2: real > complex,F2: nat > complex] :
( ( ! [X2: nat] :
? [Y2: real] :
( ( G2 @ Y2 )
= ( F2 @ X2 ) ) )
= ( ? [H2: nat > real] :
( F2
= ( comp_r1225911664865567631ex_nat @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_150_function__factors__right,axiom,
! [G2: real > complex,F2: complex > complex] :
( ( ! [X2: complex] :
? [Y2: real] :
( ( G2 @ Y2 )
= ( F2 @ X2 ) ) )
= ( ? [H2: complex > real] :
( F2
= ( comp_r891790309028876909omplex @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_151_function__factors__right,axiom,
! [G2: real > real,F2: nat > real] :
( ( ! [X2: nat] :
? [Y2: real] :
( ( G2 @ Y2 )
= ( F2 @ X2 ) ) )
= ( ? [H2: nat > real] :
( F2
= ( comp_real_real_nat @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_152_function__factors__right,axiom,
! [G2: nat > complex,F2: complex > complex] :
( ( ! [X2: complex] :
? [Y2: nat] :
( ( G2 @ Y2 )
= ( F2 @ X2 ) ) )
= ( ? [H2: complex > nat] :
( F2
= ( comp_n4415746728862867729omplex @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_153_function__factors__right,axiom,
! [G2: nat > complex,F2: nat > complex] :
( ( ! [X2: nat] :
? [Y2: nat] :
( ( G2 @ Y2 )
= ( F2 @ X2 ) ) )
= ( ? [H2: nat > nat] :
( F2
= ( comp_nat_complex_nat @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_154_function__factors__right,axiom,
! [G2: nat > real,F2: complex > real] :
( ( ! [X2: complex] :
? [Y2: nat] :
( ( G2 @ Y2 )
= ( F2 @ X2 ) ) )
= ( ? [H2: complex > nat] :
( F2
= ( comp_n4161660787930333455omplex @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_155_function__factors__right,axiom,
! [G2: nat > real,F2: nat > real] :
( ( ! [X2: nat] :
? [Y2: nat] :
( ( G2 @ Y2 )
= ( F2 @ X2 ) ) )
= ( ? [H2: nat > nat] :
( F2
= ( comp_nat_real_nat @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_156_function__factors__left,axiom,
! [G2: real > complex,F2: real > real] :
( ( ! [X2: real,Y2: real] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) )
= ( ? [H2: complex > real] :
( F2
= ( comp_c3333796836230738283l_real @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_157_function__factors__left,axiom,
! [G2: complex > complex,F2: complex > real] :
( ( ! [X2: complex,Y2: complex] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) )
= ( ? [H2: complex > real] :
( F2
= ( comp_c2063761206571265261omplex @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_158_function__factors__left,axiom,
! [G2: nat > real,F2: nat > complex] :
( ( ! [X2: nat,Y2: nat] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) )
= ( ? [H2: real > complex] :
( F2
= ( comp_r1225911664865567631ex_nat @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_159_function__factors__left,axiom,
! [G2: complex > real,F2: complex > complex] :
( ( ! [X2: complex,Y2: complex] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) )
= ( ? [H2: real > complex] :
( F2
= ( comp_r891790309028876909omplex @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_160_function__factors__left,axiom,
! [G2: nat > real,F2: nat > real] :
( ( ! [X2: nat,Y2: nat] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) )
= ( ? [H2: real > real] :
( F2
= ( comp_real_real_nat @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_161_function__factors__left,axiom,
! [G2: complex > nat,F2: complex > complex] :
( ( ! [X2: complex,Y2: complex] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) )
= ( ? [H2: nat > complex] :
( F2
= ( comp_n4415746728862867729omplex @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_162_function__factors__left,axiom,
! [G2: nat > nat,F2: nat > complex] :
( ( ! [X2: nat,Y2: nat] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) )
= ( ? [H2: nat > complex] :
( F2
= ( comp_nat_complex_nat @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_163_function__factors__left,axiom,
! [G2: complex > nat,F2: complex > real] :
( ( ! [X2: complex,Y2: complex] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) )
= ( ? [H2: nat > real] :
( F2
= ( comp_n4161660787930333455omplex @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_164_function__factors__left,axiom,
! [G2: nat > nat,F2: nat > real] :
( ( ! [X2: nat,Y2: nat] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F2 @ X2 )
= ( F2 @ Y2 ) ) ) )
= ( ? [H2: nat > real] :
( F2
= ( comp_nat_real_nat @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_165_of__real__Re,axiom,
! [Z: complex] :
( ( member_complex @ Z @ real_V2521375963428798218omplex )
=> ( ( real_V4546457046886955230omplex @ ( re @ Z ) )
= Z ) ) ).
% of_real_Re
thf(fact_166_Reals__cases,axiom,
! [Q: complex] :
( ( member_complex @ Q @ real_V2521375963428798218omplex )
=> ~ ! [R3: real] :
( Q
!= ( real_V4546457046886955230omplex @ R3 ) ) ) ).
% Reals_cases
thf(fact_167_Reals__induct,axiom,
! [Q: complex,P: complex > $o] :
( ( member_complex @ Q @ real_V2521375963428798218omplex )
=> ( ! [R3: real] : ( P @ ( real_V4546457046886955230omplex @ R3 ) )
=> ( P @ Q ) ) ) ).
% Reals_induct
thf(fact_168_Reals__of__real,axiom,
! [R: real] : ( member_complex @ ( real_V4546457046886955230omplex @ R ) @ real_V2521375963428798218omplex ) ).
% Reals_of_real
thf(fact_169_continuous__on__compose,axiom,
! [S2: set_nat,F2: nat > real,G2: real > complex] :
( ( topolo6943266826644216316t_real @ S2 @ F2 )
=> ( ( topolo8620507378200602458omplex @ ( image_nat_real @ F2 @ S2 ) @ G2 )
=> ( topolo140161755405200382omplex @ S2 @ ( comp_r1225911664865567631ex_nat @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_170_continuous__on__compose,axiom,
! [S2: set_nat,F2: nat > nat,G2: nat > complex] :
( ( topolo1182047505939668768at_nat @ S2 @ F2 )
=> ( ( topolo140161755405200382omplex @ ( image_nat_nat @ F2 @ S2 ) @ G2 )
=> ( topolo140161755405200382omplex @ S2 @ ( comp_nat_complex_nat @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_171_continuous__on__compose,axiom,
! [S2: set_complex,F2: complex > nat,G2: nat > real] :
( ( topolo3759945079839938046ex_nat @ S2 @ F2 )
=> ( ( topolo6943266826644216316t_real @ ( image_complex_nat @ F2 @ S2 ) @ G2 )
=> ( topolo8674095878704923098x_real @ S2 @ ( comp_n4161660787930333455omplex @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_172_continuous__on__compose,axiom,
! [S2: set_nat,F2: nat > nat,G2: nat > real] :
( ( topolo1182047505939668768at_nat @ S2 @ F2 )
=> ( ( topolo6943266826644216316t_real @ ( image_nat_nat @ F2 @ S2 ) @ G2 )
=> ( topolo6943266826644216316t_real @ S2 @ ( comp_nat_real_nat @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_173_continuous__on__compose,axiom,
! [S2: set_real,F2: real > complex,G2: complex > real] :
( ( topolo8620507378200602458omplex @ S2 @ F2 )
=> ( ( topolo8674095878704923098x_real @ ( image_real_complex @ F2 @ S2 ) @ G2 )
=> ( topolo5044208981011980120l_real @ S2 @ ( comp_c3333796836230738283l_real @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_174_continuous__on__compose,axiom,
! [S2: set_complex,F2: complex > real,G2: real > complex] :
( ( topolo8674095878704923098x_real @ S2 @ F2 )
=> ( ( topolo8620507378200602458omplex @ ( image_complex_real @ F2 @ S2 ) @ G2 )
=> ( topolo9015423870875150044omplex @ S2 @ ( comp_r891790309028876909omplex @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_175_continuous__on__compose,axiom,
! [S2: set_complex,F2: complex > nat,G2: nat > complex] :
( ( topolo3759945079839938046ex_nat @ S2 @ F2 )
=> ( ( topolo140161755405200382omplex @ ( image_complex_nat @ F2 @ S2 ) @ G2 )
=> ( topolo9015423870875150044omplex @ S2 @ ( comp_n4415746728862867729omplex @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_176_continuous__on__compose,axiom,
! [S2: set_nat,F2: nat > real,G2: real > real] :
( ( topolo6943266826644216316t_real @ S2 @ F2 )
=> ( ( topolo5044208981011980120l_real @ ( image_nat_real @ F2 @ S2 ) @ G2 )
=> ( topolo6943266826644216316t_real @ S2 @ ( comp_real_real_nat @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_177_continuous__on__compose,axiom,
! [S2: set_real,F2: real > real,G2: real > real] :
( ( topolo5044208981011980120l_real @ S2 @ F2 )
=> ( ( topolo5044208981011980120l_real @ ( image_real_real @ F2 @ S2 ) @ G2 )
=> ( topolo5044208981011980120l_real @ S2 @ ( comp_real_real_real @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_178_continuous__on__compose,axiom,
! [S2: set_complex,F2: complex > complex,G2: complex > real] :
( ( topolo9015423870875150044omplex @ S2 @ F2 )
=> ( ( topolo8674095878704923098x_real @ ( image_1468599708987790691omplex @ F2 @ S2 ) @ G2 )
=> ( topolo8674095878704923098x_real @ S2 @ ( comp_c2063761206571265261omplex @ G2 @ F2 ) ) ) ) ).
% continuous_on_compose
thf(fact_179_fcomp__comp,axiom,
( fcomp_8157823527453731371x_real
= ( ^ [F: real > complex,G: complex > real] : ( comp_c3333796836230738283l_real @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_180_fcomp__comp,axiom,
( fcomp_423701733139085997x_real
= ( ^ [F: complex > complex,G: complex > real] : ( comp_c2063761206571265261omplex @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_181_fcomp__comp,axiom,
( fcomp_2518332112322262095omplex
= ( ^ [F: nat > real,G: real > complex] : ( comp_r1225911664865567631ex_nat @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_182_fcomp__comp,axiom,
( fcomp_370113232634765357omplex
= ( ^ [F: complex > real,G: real > complex] : ( comp_r891790309028876909omplex @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_183_fcomp__comp,axiom,
( fcomp_nat_real_real
= ( ^ [F: nat > real,G: real > real] : ( comp_real_real_nat @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_184_fcomp__comp,axiom,
( fcomp_877936951185219153omplex
= ( ^ [F: complex > nat,G: nat > complex] : ( comp_n4415746728862867729omplex @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_185_fcomp__comp,axiom,
( fcomp_8538492730942792563omplex
= ( ^ [F: nat > nat,G: nat > complex] : ( comp_nat_complex_nat @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_186_fcomp__comp,axiom,
( fcomp_1779788810238572751t_real
= ( ^ [F: complex > nat,G: nat > real] : ( comp_n4161660787930333455omplex @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_187_fcomp__comp,axiom,
( fcomp_nat_nat_real
= ( ^ [F: nat > nat,G: nat > real] : ( comp_nat_real_nat @ G @ F ) ) ) ).
% fcomp_comp
thf(fact_188_raw__has__prod__in__Reals,axiom,
! [Z: nat > real,M: nat,P2: complex] :
( ( infini5805797430451707772omplex @ ( comp_r1225911664865567631ex_nat @ real_V4546457046886955230omplex @ Z ) @ M @ P2 )
=> ( member_complex @ P2 @ real_V2521375963428798218omplex ) ) ).
% raw_has_prod_in_Reals
thf(fact_189_convol__o,axiom,
! [F2: complex > real,G2: complex > real,H: real > complex] :
( ( comp_c2864525590522327572l_real @ ( bNF_co2841340654916505457l_real @ F2 @ G2 ) @ H )
= ( bNF_co7388595451723801839l_real @ ( comp_c3333796836230738283l_real @ F2 @ H ) @ ( comp_c3333796836230738283l_real @ G2 @ H ) ) ) ).
% convol_o
thf(fact_190_convol__o,axiom,
! [F2: complex > real,G2: complex > real,H: complex > complex] :
( ( comp_c5081751395193823254omplex @ ( bNF_co2841340654916505457l_real @ F2 @ G2 ) @ H )
= ( bNF_co2841340654916505457l_real @ ( comp_c2063761206571265261omplex @ F2 @ H ) @ ( comp_c2063761206571265261omplex @ G2 @ H ) ) ) ).
% convol_o
thf(fact_191_convol__o,axiom,
! [F2: real > complex,G2: real > complex,H: nat > real] :
( ( comp_r8895249581614084030ex_nat @ ( bNF_co1531800018262148979omplex @ F2 @ G2 ) @ H )
= ( bNF_co5645111169341782295omplex @ ( comp_r1225911664865567631ex_nat @ F2 @ H ) @ ( comp_r1225911664865567631ex_nat @ G2 @ H ) ) ) ).
% convol_o
thf(fact_192_convol__o,axiom,
! [F2: real > complex,G2: real > real,H: nat > real] :
( ( comp_r137067546312441020al_nat @ ( bNF_co1476410042518385905x_real @ F2 @ G2 ) @ H )
= ( bNF_co314614023501618325x_real @ ( comp_r1225911664865567631ex_nat @ F2 @ H ) @ ( comp_real_real_nat @ G2 @ H ) ) ) ).
% convol_o
thf(fact_193_convol__o,axiom,
! [F2: real > complex,G2: real > complex,H: complex > real] :
( ( comp_r501483716643258524omplex @ ( bNF_co1531800018262148979omplex @ F2 @ G2 ) @ H )
= ( bNF_co6028989446319287285omplex @ ( comp_r891790309028876909omplex @ F2 @ H ) @ ( comp_r891790309028876909omplex @ G2 @ H ) ) ) ).
% convol_o
thf(fact_194_convol__o,axiom,
! [F2: real > real,G2: real > complex,H: nat > real] :
( ( comp_r2603982079200466748ex_nat @ ( bNF_co1422821542014065265omplex @ F2 @ G2 ) @ H )
= ( bNF_co261025522997297685omplex @ ( comp_real_real_nat @ F2 @ H ) @ ( comp_r1225911664865567631ex_nat @ G2 @ H ) ) ) ).
% convol_o
thf(fact_195_convol__o,axiom,
! [F2: real > real,G2: real > real,H: nat > real] :
( ( comp_r5566512450771222842al_nat @ ( bNF_co7388595451723801839l_real @ F2 @ G2 ) @ H )
= ( bNF_co8045384917381939859l_real @ ( comp_real_real_nat @ F2 @ H ) @ ( comp_real_real_nat @ G2 @ H ) ) ) ).
% convol_o
thf(fact_196_convol__o,axiom,
! [F2: nat > complex,G2: nat > complex,H: complex > nat] :
( ( comp_n1008163678071708920omplex @ ( bNF_co5645111169341782295omplex @ F2 @ G2 ) @ H )
= ( bNF_co6028989446319287285omplex @ ( comp_n4415746728862867729omplex @ F2 @ H ) @ ( comp_n4415746728862867729omplex @ G2 @ H ) ) ) ).
% convol_o
thf(fact_197_convol__o,axiom,
! [F2: nat > complex,G2: nat > real,H: complex > nat] :
( ( comp_n7862994715949103094omplex @ ( bNF_co314614023501618325x_real @ F2 @ G2 ) @ H )
= ( bNF_co2757359416308857971x_real @ ( comp_n4415746728862867729omplex @ F2 @ H ) @ ( comp_n4161660787930333455omplex @ G2 @ H ) ) ) ).
% convol_o
thf(fact_198_convol__o,axiom,
! [F2: nat > complex,G2: nat > complex,H: nat > nat] :
( ( comp_n1197726116097606682ex_nat @ ( bNF_co5645111169341782295omplex @ F2 @ G2 ) @ H )
= ( bNF_co5645111169341782295omplex @ ( comp_nat_complex_nat @ F2 @ H ) @ ( comp_nat_complex_nat @ G2 @ H ) ) ) ).
% convol_o
thf(fact_199_Reals__minus__iff,axiom,
! [A: complex] :
( ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ real_V2521375963428798218omplex )
= ( member_complex @ A @ real_V2521375963428798218omplex ) ) ).
% Reals_minus_iff
thf(fact_200_Reals__minus__iff,axiom,
! [A: real] :
( ( member_real @ ( uminus_uminus_real @ A ) @ real_V470468836141973256s_real )
= ( member_real @ A @ real_V470468836141973256s_real ) ) ).
% Reals_minus_iff
thf(fact_201_Reals__inverse__iff,axiom,
! [X: complex] :
( ( member_complex @ ( invers8013647133539491842omplex @ X ) @ real_V2521375963428798218omplex )
= ( member_complex @ X @ real_V2521375963428798218omplex ) ) ).
% Reals_inverse_iff
thf(fact_202_Reals__inverse__iff,axiom,
! [X: real] :
( ( member_real @ ( inverse_inverse_real @ X ) @ real_V470468836141973256s_real )
= ( member_real @ X @ real_V470468836141973256s_real ) ) ).
% Reals_inverse_iff
thf(fact_203_exp__Reals__eq,axiom,
! [Z: complex] :
( ( member_complex @ Z @ real_V2521375963428798218omplex )
=> ( ( exp_complex @ Z )
= ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z ) ) ) ) ) ).
% exp_Reals_eq
thf(fact_204_complex_Oinject,axiom,
! [X1: real,X22: real,Y1: real,Y22: real] :
( ( ( complex2 @ X1 @ X22 )
= ( complex2 @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% complex.inject
thf(fact_205_of__real__minus,axiom,
! [X: real] :
( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% of_real_minus
thf(fact_206_of__real__minus,axiom,
! [X: real] :
( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ X ) )
= ( uminus1482373934393186551omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% of_real_minus
thf(fact_207_minus__of__real__eq__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ X ) )
= ( real_V1803761363581548252l_real @ Y ) )
= ( ( uminus_uminus_real @ X )
= Y ) ) ).
% minus_of_real_eq_of_real_iff
thf(fact_208_minus__of__real__eq__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( uminus1482373934393186551omplex @ ( real_V4546457046886955230omplex @ X ) )
= ( real_V4546457046886955230omplex @ Y ) )
= ( ( uminus_uminus_real @ X )
= Y ) ) ).
% minus_of_real_eq_of_real_iff
thf(fact_209_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_210_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_211_of__real__eq__minus__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( real_V1803761363581548252l_real @ X )
= ( uminus_uminus_real @ ( real_V1803761363581548252l_real @ Y ) ) )
= ( X
= ( uminus_uminus_real @ Y ) ) ) ).
% of_real_eq_minus_of_real_iff
thf(fact_212_of__real__eq__minus__of__real__iff,axiom,
! [X: real,Y: real] :
( ( ( real_V4546457046886955230omplex @ X )
= ( uminus1482373934393186551omplex @ ( real_V4546457046886955230omplex @ Y ) ) )
= ( X
= ( uminus_uminus_real @ Y ) ) ) ).
% of_real_eq_minus_of_real_iff
thf(fact_213_of__real__inverse,axiom,
! [X: real] :
( ( real_V4546457046886955230omplex @ ( inverse_inverse_real @ X ) )
= ( invers8013647133539491842omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% of_real_inverse
thf(fact_214_of__real__inverse,axiom,
! [X: real] :
( ( real_V1803761363581548252l_real @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% of_real_inverse
thf(fact_215_Re__inverse,axiom,
! [R: complex] :
( ( member_complex @ R @ real_V2521375963428798218omplex )
=> ( ( re @ ( invers8013647133539491842omplex @ R ) )
= ( inverse_inverse_real @ ( re @ R ) ) ) ) ).
% Re_inverse
thf(fact_216_complex_Ocollapse,axiom,
! [Complex: complex] :
( ( complex2 @ ( re @ Complex ) @ ( im @ Complex ) )
= Complex ) ).
% complex.collapse
thf(fact_217_complex__surj,axiom,
! [Z: complex] :
( ( complex2 @ ( re @ Z ) @ ( im @ Z ) )
= Z ) ).
% complex_surj
thf(fact_218_complex__minus,axiom,
! [A: real,B: real] :
( ( uminus1482373934393186551omplex @ ( complex2 @ A @ B ) )
= ( complex2 @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% complex_minus
thf(fact_219_uminus__complex_Ocode,axiom,
( uminus1482373934393186551omplex
= ( ^ [X2: complex] : ( complex2 @ ( uminus_uminus_real @ ( re @ X2 ) ) @ ( uminus_uminus_real @ ( im @ X2 ) ) ) ) ) ).
% uminus_complex.code
thf(fact_220_complex_Oexhaust,axiom,
! [Y: complex] :
~ ! [X12: real,X23: real] :
( Y
!= ( complex2 @ X12 @ X23 ) ) ).
% complex.exhaust
thf(fact_221_uminus__complex_Osimps_I2_J,axiom,
! [X: complex] :
( ( im @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus_uminus_real @ ( im @ X ) ) ) ).
% uminus_complex.simps(2)
thf(fact_222_complex_Osel_I2_J,axiom,
! [X1: real,X22: real] :
( ( im @ ( complex2 @ X1 @ X22 ) )
= X22 ) ).
% complex.sel(2)
thf(fact_223_complex_Oexhaust__sel,axiom,
! [Complex: complex] :
( Complex
= ( complex2 @ ( re @ Complex ) @ ( im @ Complex ) ) ) ).
% complex.exhaust_sel
thf(fact_224_uminus__complex_Osimps_I1_J,axiom,
! [X: complex] :
( ( re @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus_uminus_real @ ( re @ X ) ) ) ).
% uminus_complex.simps(1)
thf(fact_225_complex_Osel_I1_J,axiom,
! [X1: real,X22: real] :
( ( re @ ( complex2 @ X1 @ X22 ) )
= X1 ) ).
% complex.sel(1)
thf(fact_226_image__eq__imp__comp,axiom,
! [F2: real > complex,A2: set_real,G2: real > complex,B2: set_real,H: complex > real] :
( ( ( image_real_complex @ F2 @ A2 )
= ( image_real_complex @ G2 @ B2 ) )
=> ( ( image_real_real @ ( comp_c3333796836230738283l_real @ H @ F2 ) @ A2 )
= ( image_real_real @ ( comp_c3333796836230738283l_real @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_227_image__eq__imp__comp,axiom,
! [F2: real > complex,A2: set_real,G2: complex > complex,B2: set_complex,H: complex > real] :
( ( ( image_real_complex @ F2 @ A2 )
= ( image_1468599708987790691omplex @ G2 @ B2 ) )
=> ( ( image_real_real @ ( comp_c3333796836230738283l_real @ H @ F2 ) @ A2 )
= ( image_complex_real @ ( comp_c2063761206571265261omplex @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_228_image__eq__imp__comp,axiom,
! [F2: complex > complex,A2: set_complex,G2: real > complex,B2: set_real,H: complex > real] :
( ( ( image_1468599708987790691omplex @ F2 @ A2 )
= ( image_real_complex @ G2 @ B2 ) )
=> ( ( image_complex_real @ ( comp_c2063761206571265261omplex @ H @ F2 ) @ A2 )
= ( image_real_real @ ( comp_c3333796836230738283l_real @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_229_image__eq__imp__comp,axiom,
! [F2: complex > complex,A2: set_complex,G2: complex > complex,B2: set_complex,H: complex > real] :
( ( ( image_1468599708987790691omplex @ F2 @ A2 )
= ( image_1468599708987790691omplex @ G2 @ B2 ) )
=> ( ( image_complex_real @ ( comp_c2063761206571265261omplex @ H @ F2 ) @ A2 )
= ( image_complex_real @ ( comp_c2063761206571265261omplex @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_230_image__eq__imp__comp,axiom,
! [F2: nat > real,A2: set_nat,G2: nat > real,B2: set_nat,H: real > complex] :
( ( ( image_nat_real @ F2 @ A2 )
= ( image_nat_real @ G2 @ B2 ) )
=> ( ( image_nat_complex @ ( comp_r1225911664865567631ex_nat @ H @ F2 ) @ A2 )
= ( image_nat_complex @ ( comp_r1225911664865567631ex_nat @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_231_image__eq__imp__comp,axiom,
! [F2: nat > real,A2: set_nat,G2: complex > real,B2: set_complex,H: real > complex] :
( ( ( image_nat_real @ F2 @ A2 )
= ( image_complex_real @ G2 @ B2 ) )
=> ( ( image_nat_complex @ ( comp_r1225911664865567631ex_nat @ H @ F2 ) @ A2 )
= ( image_1468599708987790691omplex @ ( comp_r891790309028876909omplex @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_232_image__eq__imp__comp,axiom,
! [F2: complex > real,A2: set_complex,G2: nat > real,B2: set_nat,H: real > complex] :
( ( ( image_complex_real @ F2 @ A2 )
= ( image_nat_real @ G2 @ B2 ) )
=> ( ( image_1468599708987790691omplex @ ( comp_r891790309028876909omplex @ H @ F2 ) @ A2 )
= ( image_nat_complex @ ( comp_r1225911664865567631ex_nat @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_233_image__eq__imp__comp,axiom,
! [F2: complex > real,A2: set_complex,G2: complex > real,B2: set_complex,H: real > complex] :
( ( ( image_complex_real @ F2 @ A2 )
= ( image_complex_real @ G2 @ B2 ) )
=> ( ( image_1468599708987790691omplex @ ( comp_r891790309028876909omplex @ H @ F2 ) @ A2 )
= ( image_1468599708987790691omplex @ ( comp_r891790309028876909omplex @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_234_image__eq__imp__comp,axiom,
! [F2: nat > real,A2: set_nat,G2: nat > real,B2: set_nat,H: real > real] :
( ( ( image_nat_real @ F2 @ A2 )
= ( image_nat_real @ G2 @ B2 ) )
=> ( ( image_nat_real @ ( comp_real_real_nat @ H @ F2 ) @ A2 )
= ( image_nat_real @ ( comp_real_real_nat @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_235_image__eq__imp__comp,axiom,
! [F2: complex > nat,A2: set_complex,G2: complex > nat,B2: set_complex,H: nat > complex] :
( ( ( image_complex_nat @ F2 @ A2 )
= ( image_complex_nat @ G2 @ B2 ) )
=> ( ( image_1468599708987790691omplex @ ( comp_n4415746728862867729omplex @ H @ F2 ) @ A2 )
= ( image_1468599708987790691omplex @ ( comp_n4415746728862867729omplex @ H @ G2 ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_236_image__comp,axiom,
! [F2: complex > real,G2: real > complex,R: set_real] :
( ( image_complex_real @ F2 @ ( image_real_complex @ G2 @ R ) )
= ( image_real_real @ ( comp_c3333796836230738283l_real @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_237_image__comp,axiom,
! [F2: complex > real,G2: complex > complex,R: set_complex] :
( ( image_complex_real @ F2 @ ( image_1468599708987790691omplex @ G2 @ R ) )
= ( image_complex_real @ ( comp_c2063761206571265261omplex @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_238_image__comp,axiom,
! [F2: real > complex,G2: nat > real,R: set_nat] :
( ( image_real_complex @ F2 @ ( image_nat_real @ G2 @ R ) )
= ( image_nat_complex @ ( comp_r1225911664865567631ex_nat @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_239_image__comp,axiom,
! [F2: real > complex,G2: complex > real,R: set_complex] :
( ( image_real_complex @ F2 @ ( image_complex_real @ G2 @ R ) )
= ( image_1468599708987790691omplex @ ( comp_r891790309028876909omplex @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_240_image__comp,axiom,
! [F2: real > real,G2: nat > real,R: set_nat] :
( ( image_real_real @ F2 @ ( image_nat_real @ G2 @ R ) )
= ( image_nat_real @ ( comp_real_real_nat @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_241_image__comp,axiom,
! [F2: nat > complex,G2: complex > nat,R: set_complex] :
( ( image_nat_complex @ F2 @ ( image_complex_nat @ G2 @ R ) )
= ( image_1468599708987790691omplex @ ( comp_n4415746728862867729omplex @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_242_image__comp,axiom,
! [F2: nat > complex,G2: nat > nat,R: set_nat] :
( ( image_nat_complex @ F2 @ ( image_nat_nat @ G2 @ R ) )
= ( image_nat_complex @ ( comp_nat_complex_nat @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_243_image__comp,axiom,
! [F2: nat > real,G2: complex > nat,R: set_complex] :
( ( image_nat_real @ F2 @ ( image_complex_nat @ G2 @ R ) )
= ( image_complex_real @ ( comp_n4161660787930333455omplex @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_244_image__comp,axiom,
! [F2: nat > real,G2: nat > nat,R: set_nat] :
( ( image_nat_real @ F2 @ ( image_nat_nat @ G2 @ R ) )
= ( image_nat_real @ ( comp_nat_real_nat @ F2 @ G2 ) @ R ) ) ).
% image_comp
thf(fact_245_complex__eq__iff,axiom,
( ( ^ [Y3: complex,Z2: complex] : ( Y3 = Z2 ) )
= ( ^ [X2: complex,Y2: complex] :
( ( ( re @ X2 )
= ( re @ Y2 ) )
& ( ( im @ X2 )
= ( im @ Y2 ) ) ) ) ) ).
% complex_eq_iff
thf(fact_246_complex__eqI,axiom,
! [X: complex,Y: complex] :
( ( ( re @ X )
= ( re @ Y ) )
=> ( ( ( im @ X )
= ( im @ Y ) )
=> ( X = Y ) ) ) ).
% complex_eqI
thf(fact_247_complex_Ocoinduct__strong,axiom,
! [R4: complex > complex > $o,Complex: complex,Complex2: complex] :
( ( R4 @ Complex @ Complex2 )
=> ( ! [Complex3: complex,Complex4: complex] :
( ( R4 @ Complex3 @ Complex4 )
=> ( ( ( re @ Complex3 )
= ( re @ Complex4 ) )
& ( ( im @ Complex3 )
= ( im @ Complex4 ) ) ) )
=> ( Complex = Complex2 ) ) ) ).
% complex.coinduct_strong
thf(fact_248_complex_Oexpand,axiom,
! [Complex: complex,Complex2: complex] :
( ( ( ( re @ Complex )
= ( re @ Complex2 ) )
& ( ( im @ Complex )
= ( im @ Complex2 ) ) )
=> ( Complex = Complex2 ) ) ).
% complex.expand
thf(fact_249_Reals__inverse,axiom,
! [A: complex] :
( ( member_complex @ A @ real_V2521375963428798218omplex )
=> ( member_complex @ ( invers8013647133539491842omplex @ A ) @ real_V2521375963428798218omplex ) ) ).
% Reals_inverse
thf(fact_250_Reals__inverse,axiom,
! [A: real] :
( ( member_real @ A @ real_V470468836141973256s_real )
=> ( member_real @ ( inverse_inverse_real @ A ) @ real_V470468836141973256s_real ) ) ).
% Reals_inverse
thf(fact_251_Reals__minus,axiom,
! [A: complex] :
( ( member_complex @ A @ real_V2521375963428798218omplex )
=> ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ real_V2521375963428798218omplex ) ) ).
% Reals_minus
thf(fact_252_Reals__minus,axiom,
! [A: real] :
( ( member_real @ A @ real_V470468836141973256s_real )
=> ( member_real @ ( uminus_uminus_real @ A ) @ real_V470468836141973256s_real ) ) ).
% Reals_minus
thf(fact_253_inverse__minus__eq,axiom,
! [A: complex] :
( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
= ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).
% inverse_minus_eq
thf(fact_254_inverse__minus__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).
% inverse_minus_eq
thf(fact_255_of__real__exp,axiom,
! [X: real] :
( ( real_V1803761363581548252l_real @ ( exp_real @ X ) )
= ( exp_real @ ( real_V1803761363581548252l_real @ X ) ) ) ).
% of_real_exp
thf(fact_256_of__real__exp,axiom,
! [X: real] :
( ( real_V4546457046886955230omplex @ ( exp_real @ X ) )
= ( exp_complex @ ( real_V4546457046886955230omplex @ X ) ) ) ).
% of_real_exp
thf(fact_257_exp__minus,axiom,
! [X: complex] :
( ( exp_complex @ ( uminus1482373934393186551omplex @ X ) )
= ( invers8013647133539491842omplex @ ( exp_complex @ X ) ) ) ).
% exp_minus
thf(fact_258_exp__minus,axiom,
! [X: real] :
( ( exp_real @ ( uminus_uminus_real @ X ) )
= ( inverse_inverse_real @ ( exp_real @ X ) ) ) ).
% exp_minus
thf(fact_259_exp__inj__iff,axiom,
! [X: real,Y: real] :
( ( ( exp_real @ X )
= ( exp_real @ Y ) )
= ( X = Y ) ) ).
% exp_inj_iff
thf(fact_260_inverse__eq__iff__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_261_inverse__eq__iff__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_262_inverse__inverse__eq,axiom,
! [A: complex] :
( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_263_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_264_verit__minus__simplify_I4_J,axiom,
! [B: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_265_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_266_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_267_add_Oinverse__inverse,axiom,
! [A: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_268_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_269_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_270_neg__equal__iff__equal,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= ( uminus1482373934393186551omplex @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_271_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_272_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_273_minus__equation__iff,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( uminus1482373934393186551omplex @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_274_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_275_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_276_equation__minus__iff,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% equation_minus_iff
thf(fact_277_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_278_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_279_verit__negate__coefficient_I3_J,axiom,
! [A: complex,B: complex] :
( ( A = B )
=> ( ( uminus1482373934393186551omplex @ A )
= ( uminus1482373934393186551omplex @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_280_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_281_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_282_inverse__eq__imp__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_283_inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_284_exp__in__Reals,axiom,
! [Z: real] :
( ( member_real @ Z @ real_V470468836141973256s_real )
=> ( member_real @ ( exp_real @ Z ) @ real_V470468836141973256s_real ) ) ).
% exp_in_Reals
thf(fact_285_exp__in__Reals,axiom,
! [Z: complex] :
( ( member_complex @ Z @ real_V2521375963428798218omplex )
=> ( member_complex @ ( exp_complex @ Z ) @ real_V2521375963428798218omplex ) ) ).
% exp_in_Reals
thf(fact_286_image__eqI,axiom,
! [B: complex,F2: complex > complex,X: complex,A2: set_complex] :
( ( B
= ( F2 @ X ) )
=> ( ( member_complex @ X @ A2 )
=> ( member_complex @ B @ ( image_1468599708987790691omplex @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_287_Inf_OINF__image,axiom,
! [Inf: set_real > real,G2: complex > real,F2: real > complex,A2: set_real] :
( ( Inf @ ( image_complex_real @ G2 @ ( image_real_complex @ F2 @ A2 ) ) )
= ( Inf @ ( image_real_real @ ( comp_c3333796836230738283l_real @ G2 @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_288_Inf_OINF__image,axiom,
! [Inf: set_real > real,G2: complex > real,F2: complex > complex,A2: set_complex] :
( ( Inf @ ( image_complex_real @ G2 @ ( image_1468599708987790691omplex @ F2 @ A2 ) ) )
= ( Inf @ ( image_complex_real @ ( comp_c2063761206571265261omplex @ G2 @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_289_Inf_OINF__image,axiom,
! [Inf: set_complex > complex,G2: real > complex,F2: nat > real,A2: set_nat] :
( ( Inf @ ( image_real_complex @ G2 @ ( image_nat_real @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_complex @ ( comp_r1225911664865567631ex_nat @ G2 @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_290_Inf_OINF__image,axiom,
! [Inf: set_complex > complex,G2: real > complex,F2: complex > real,A2: set_complex] :
( ( Inf @ ( image_real_complex @ G2 @ ( image_complex_real @ F2 @ A2 ) ) )
= ( Inf @ ( image_1468599708987790691omplex @ ( comp_r891790309028876909omplex @ G2 @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_291_Inf_OINF__image,axiom,
! [Inf: set_real > real,G2: real > real,F2: nat > real,A2: set_nat] :
( ( Inf @ ( image_real_real @ G2 @ ( image_nat_real @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_real @ ( comp_real_real_nat @ G2 @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_292_Inf_OINF__image,axiom,
! [Inf: set_complex > complex,G2: nat > complex,F2: complex > nat,A2: set_complex] :
( ( Inf @ ( image_nat_complex @ G2 @ ( image_complex_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_1468599708987790691omplex @ ( comp_n4415746728862867729omplex @ G2 @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_293_Inf_OINF__image,axiom,
! [Inf: set_complex > complex,G2: nat > complex,F2: nat > nat,A2: set_nat] :
( ( Inf @ ( image_nat_complex @ G2 @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_complex @ ( comp_nat_complex_nat @ G2 @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_294_Inf_OINF__image,axiom,
! [Inf: set_real > real,G2: nat > real,F2: complex > nat,A2: set_complex] :
( ( Inf @ ( image_nat_real @ G2 @ ( image_complex_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_complex_real @ ( comp_n4161660787930333455omplex @ G2 @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_295_Inf_OINF__image,axiom,
! [Inf: set_real > real,G2: nat > real,F2: nat > nat,A2: set_nat] :
( ( Inf @ ( image_nat_real @ G2 @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_real @ ( comp_nat_real_nat @ G2 @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_296_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G2: complex > real,F2: real > complex,A2: set_real] :
( ( Sup @ ( image_complex_real @ G2 @ ( image_real_complex @ F2 @ A2 ) ) )
= ( Sup @ ( image_real_real @ ( comp_c3333796836230738283l_real @ G2 @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_297_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G2: complex > real,F2: complex > complex,A2: set_complex] :
( ( Sup @ ( image_complex_real @ G2 @ ( image_1468599708987790691omplex @ F2 @ A2 ) ) )
= ( Sup @ ( image_complex_real @ ( comp_c2063761206571265261omplex @ G2 @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_298_Sup_OSUP__image,axiom,
! [Sup: set_complex > complex,G2: real > complex,F2: nat > real,A2: set_nat] :
( ( Sup @ ( image_real_complex @ G2 @ ( image_nat_real @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_complex @ ( comp_r1225911664865567631ex_nat @ G2 @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_299_Sup_OSUP__image,axiom,
! [Sup: set_complex > complex,G2: real > complex,F2: complex > real,A2: set_complex] :
( ( Sup @ ( image_real_complex @ G2 @ ( image_complex_real @ F2 @ A2 ) ) )
= ( Sup @ ( image_1468599708987790691omplex @ ( comp_r891790309028876909omplex @ G2 @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_300_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G2: real > real,F2: nat > real,A2: set_nat] :
( ( Sup @ ( image_real_real @ G2 @ ( image_nat_real @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_real @ ( comp_real_real_nat @ G2 @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_301_Sup_OSUP__image,axiom,
! [Sup: set_complex > complex,G2: nat > complex,F2: complex > nat,A2: set_complex] :
( ( Sup @ ( image_nat_complex @ G2 @ ( image_complex_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_1468599708987790691omplex @ ( comp_n4415746728862867729omplex @ G2 @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_302_Sup_OSUP__image,axiom,
! [Sup: set_complex > complex,G2: nat > complex,F2: nat > nat,A2: set_nat] :
( ( Sup @ ( image_nat_complex @ G2 @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_complex @ ( comp_nat_complex_nat @ G2 @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_303_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G2: nat > real,F2: complex > nat,A2: set_complex] :
( ( Sup @ ( image_nat_real @ G2 @ ( image_complex_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_complex_real @ ( comp_n4161660787930333455omplex @ G2 @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_304_Sup_OSUP__image,axiom,
! [Sup: set_real > real,G2: nat > real,F2: nat > nat,A2: set_nat] :
( ( Sup @ ( image_nat_real @ G2 @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_real @ ( comp_nat_real_nat @ G2 @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_305_raw__has__prod__of__real__iff,axiom,
! [Z: nat > real,M: nat,P2: real] :
( ( infini5805797430451707772omplex @ ( comp_r1225911664865567631ex_nat @ real_V4546457046886955230omplex @ Z ) @ M @ ( real_V4546457046886955230omplex @ P2 ) )
= ( infini2923794516677094010d_real @ Z @ M @ P2 ) ) ).
% raw_has_prod_of_real_iff
thf(fact_306_Im__inverse,axiom,
! [R: complex] :
( ( member_complex @ R @ real_V2521375963428798218omplex )
=> ( ( im @ ( invers8013647133539491842omplex @ R ) )
= zero_zero_real ) ) ).
% Im_inverse
thf(fact_307_norm__exp__eq__Re,axiom,
! [Z: complex] :
( ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) )
= ( exp_real @ ( re @ Z ) ) ) ).
% norm_exp_eq_Re
thf(fact_308_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_309_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_310_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_311_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_312_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_313_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_314_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_315_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_316_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_317_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_318_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_319_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_320_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_321_inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% inverse_zero
thf(fact_322_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_323_inverse__nonzero__iff__nonzero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_324_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_325_norm__minus__cancel,axiom,
! [X: complex] :
( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ X ) )
= ( real_V1022390504157884413omplex @ X ) ) ).
% norm_minus_cancel
thf(fact_326_norm__minus__cancel,axiom,
! [X: real] :
( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ X ) )
= ( real_V7735802525324610683m_real @ X ) ) ).
% norm_minus_cancel
thf(fact_327_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_328_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_329_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_330_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_331_of__real__0,axiom,
( ( real_V1803761363581548252l_real @ zero_zero_real )
= zero_zero_real ) ).
% of_real_0
thf(fact_332_of__real__0,axiom,
( ( real_V4546457046886955230omplex @ zero_zero_real )
= zero_zero_complex ) ).
% of_real_0
thf(fact_333_of__real__eq__0__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_334_of__real__eq__0__iff,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= zero_zero_complex )
= ( X = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_335_Im__complex__of__real,axiom,
! [Z: real] :
( ( im @ ( real_V4546457046886955230omplex @ Z ) )
= zero_zero_real ) ).
% Im_complex_of_real
thf(fact_336_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_337_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_338_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_339_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_340_nonzero__norm__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
= ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_341_nonzero__norm__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_342_field__class_Ofield__inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% field_class.field_inverse_zero
thf(fact_343_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_344_inverse__zero__imp__zero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ).
% inverse_zero_imp_zero
thf(fact_345_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_346_nonzero__inverse__eq__imp__eq,axiom,
! [A: complex,B: complex] :
( ( ( invers8013647133539491842omplex @ A )
= ( invers8013647133539491842omplex @ B ) )
=> ( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_347_nonzero__inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_348_nonzero__inverse__inverse__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_349_nonzero__inverse__inverse__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_350_nonzero__imp__inverse__nonzero,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
!= zero_zero_complex ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_351_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_352_exp__not__eq__zero,axiom,
! [X: real] :
( ( exp_real @ X )
!= zero_zero_real ) ).
% exp_not_eq_zero
thf(fact_353_exp__not__eq__zero,axiom,
! [X: complex] :
( ( exp_complex @ X )
!= zero_zero_complex ) ).
% exp_not_eq_zero
thf(fact_354_Reals__0,axiom,
member_real @ zero_zero_real @ real_V470468836141973256s_real ).
% Reals_0
thf(fact_355_Reals__0,axiom,
member_complex @ zero_zero_complex @ real_V2521375963428798218omplex ).
% Reals_0
thf(fact_356_raw__has__prod__nonzero,axiom,
! [F2: nat > nat,M: nat] :
~ ( infini1035527091028463262od_nat @ F2 @ M @ zero_zero_nat ) ).
% raw_has_prod_nonzero
thf(fact_357_raw__has__prod__nonzero,axiom,
! [F2: nat > int,M: nat] :
~ ( infini1033036620519412986od_int @ F2 @ M @ zero_zero_int ) ).
% raw_has_prod_nonzero
thf(fact_358_raw__has__prod__nonzero,axiom,
! [F2: nat > complex,M: nat] :
~ ( infini5805797430451707772omplex @ F2 @ M @ zero_zero_complex ) ).
% raw_has_prod_nonzero
thf(fact_359_raw__has__prod__nonzero,axiom,
! [F2: nat > real,M: nat] :
~ ( infini2923794516677094010d_real @ F2 @ M @ zero_zero_real ) ).
% raw_has_prod_nonzero
thf(fact_360_norm__inverse,axiom,
! [A: complex] :
( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
= ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% norm_inverse
thf(fact_361_norm__inverse,axiom,
! [A: real] :
( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% norm_inverse
thf(fact_362_nonzero__inverse__minus__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
= ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_363_nonzero__inverse__minus__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_364_nonzero__Reals__inverse,axiom,
! [A: complex] :
( ( member_complex @ A @ real_V2521375963428798218omplex )
=> ( ( A != zero_zero_complex )
=> ( member_complex @ ( invers8013647133539491842omplex @ A ) @ real_V2521375963428798218omplex ) ) ) ).
% nonzero_Reals_inverse
thf(fact_365_nonzero__Reals__inverse,axiom,
! [A: real] :
( ( member_real @ A @ real_V470468836141973256s_real )
=> ( ( A != zero_zero_real )
=> ( member_real @ ( inverse_inverse_real @ A ) @ real_V470468836141973256s_real ) ) ) ).
% nonzero_Reals_inverse
thf(fact_366_complex__is__Real__iff,axiom,
! [Z: complex] :
( ( member_complex @ Z @ real_V2521375963428798218omplex )
= ( ( im @ Z )
= zero_zero_real ) ) ).
% complex_is_Real_iff
thf(fact_367_complex__of__real__def,axiom,
( real_V4546457046886955230omplex
= ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).
% complex_of_real_def
thf(fact_368_complex__of__real__code,axiom,
( real_V4546457046886955230omplex
= ( ^ [X2: real] : ( complex2 @ X2 @ zero_zero_real ) ) ) ).
% complex_of_real_code
thf(fact_369_complex__eq__cancel__iff2,axiom,
! [X: real,Y: real,Xa: real] :
( ( ( complex2 @ X @ Y )
= ( real_V4546457046886955230omplex @ Xa ) )
= ( ( X = Xa )
& ( Y = zero_zero_real ) ) ) ).
% complex_eq_cancel_iff2
thf(fact_370_Complex__in__Reals,axiom,
! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).
% Complex_in_Reals
thf(fact_371_imageI,axiom,
! [X: complex,A2: set_complex,F2: complex > complex] :
( ( member_complex @ X @ A2 )
=> ( member_complex @ ( F2 @ X ) @ ( image_1468599708987790691omplex @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_372_rev__image__eqI,axiom,
! [X: complex,A2: set_complex,B: complex,F2: complex > complex] :
( ( member_complex @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_complex @ B @ ( image_1468599708987790691omplex @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_373_nonzero__of__real__inverse,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( real_V4546457046886955230omplex @ ( inverse_inverse_real @ X ) )
= ( invers8013647133539491842omplex @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% nonzero_of_real_inverse
thf(fact_374_nonzero__of__real__inverse,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( real_V1803761363581548252l_real @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( real_V1803761363581548252l_real @ X ) ) ) ) ).
% nonzero_of_real_inverse
thf(fact_375_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_376_artanh__0,axiom,
( ( artanh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% artanh_0
thf(fact_377_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_378_arsinh__0,axiom,
( ( arsinh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% arsinh_0
thf(fact_379_continuous__on__norm__id,axiom,
! [S: set_complex] : ( topolo8674095878704923098x_real @ S @ real_V1022390504157884413omplex ) ).
% continuous_on_norm_id
thf(fact_380_continuous__on__norm__id,axiom,
! [S: set_real] : ( topolo5044208981011980120l_real @ S @ real_V7735802525324610683m_real ) ).
% continuous_on_norm_id
thf(fact_381_fps__inverse__zero_H,axiom,
( ( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex )
=> ( ( invers8029835185024579338omplex @ zero_z1877163951443063103omplex )
= zero_z1877163951443063103omplex ) ) ).
% fps_inverse_zero'
thf(fact_382_fps__inverse__zero_H,axiom,
( ( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ zero_z7760665558314615101s_real )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_zero'
thf(fact_383_cmod__eq__Re,axiom,
! [Z: complex] :
( ( ( im @ Z )
= zero_zero_real )
=> ( ( real_V1022390504157884413omplex @ Z )
= ( abs_abs_real @ ( re @ Z ) ) ) ) ).
% cmod_eq_Re
thf(fact_384_cmod__eq__Im,axiom,
! [Z: complex] :
( ( ( re @ Z )
= zero_zero_real )
=> ( ( real_V1022390504157884413omplex @ Z )
= ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% cmod_eq_Im
thf(fact_385_Im__eq__0,axiom,
! [Z: complex] :
( ( ( abs_abs_real @ ( re @ Z ) )
= ( real_V1022390504157884413omplex @ Z ) )
=> ( ( im @ Z )
= zero_zero_real ) ) ).
% Im_eq_0
thf(fact_386_nonzero__inverse__scaleR__distrib,axiom,
! [A: real,X: complex] :
( ( A != zero_zero_real )
=> ( ( X != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
= ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_387_nonzero__inverse__scaleR__distrib,axiom,
! [A: real,X: real] :
( ( A != zero_zero_real )
=> ( ( X != zero_zero_real )
=> ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
= ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_388_cnj_Ocode,axiom,
( cnj
= ( ^ [Z3: complex] : ( complex2 @ ( re @ Z3 ) @ ( uminus_uminus_real @ ( im @ Z3 ) ) ) ) ) ).
% cnj.code
thf(fact_389_norm__le__zero__iff,axiom,
! [X: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_390_norm__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_391_abs__idempotent,axiom,
! [A: real] :
( ( abs_abs_real @ ( abs_abs_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_idempotent
thf(fact_392_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_393_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_394_complex__cnj__zero__iff,axiom,
! [Z: complex] :
( ( ( cnj @ Z )
= zero_zero_complex )
= ( Z = zero_zero_complex ) ) ).
% complex_cnj_zero_iff
thf(fact_395_complex__cnj__zero,axiom,
( ( cnj @ zero_zero_complex )
= zero_zero_complex ) ).
% complex_cnj_zero
thf(fact_396_complex__cnj__cnj,axiom,
! [Z: complex] :
( ( cnj @ ( cnj @ Z ) )
= Z ) ).
% complex_cnj_cnj
thf(fact_397_complex__cnj__scaleR,axiom,
! [R: real,X: complex] :
( ( cnj @ ( real_V2046097035970521341omplex @ R @ X ) )
= ( real_V2046097035970521341omplex @ R @ ( cnj @ X ) ) ) ).
% complex_cnj_scaleR
thf(fact_398_complex__cnj__cancel__iff,axiom,
! [X: complex,Y: complex] :
( ( ( cnj @ X )
= ( cnj @ Y ) )
= ( X = Y ) ) ).
% complex_cnj_cancel_iff
thf(fact_399_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_400_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_401_neg__le__iff__le,axiom,
! [B: complex,A: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) )
= ( ord_less_eq_complex @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_402_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_403_scaleR__zero__right,axiom,
! [A: real] :
( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% scaleR_zero_right
thf(fact_404_scaleR__zero__right,axiom,
! [A: real] :
( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% scaleR_zero_right
thf(fact_405_scaleR__cancel__right,axiom,
! [A: real,X: complex,B: real] :
( ( ( real_V2046097035970521341omplex @ A @ X )
= ( real_V2046097035970521341omplex @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_complex ) ) ) ).
% scaleR_cancel_right
thf(fact_406_scaleR__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% scaleR_cancel_right
thf(fact_407_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real
= ( abs_abs_real @ A ) )
= ( A = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_408_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_409_abs__eq__0,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_410_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_411_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_412_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_413_scaleR__cancel__left,axiom,
! [A: real,X: complex,Y: complex] :
( ( ( real_V2046097035970521341omplex @ A @ X )
= ( real_V2046097035970521341omplex @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% scaleR_cancel_left
thf(fact_414_scaleR__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% scaleR_cancel_left
thf(fact_415_scaleR__minus__right,axiom,
! [A: real,X: complex] :
( ( real_V2046097035970521341omplex @ A @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ A @ X ) ) ) ).
% scaleR_minus_right
thf(fact_416_scaleR__minus__right,axiom,
! [A: real,X: real] :
( ( real_V1485227260804924795R_real @ A @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ).
% scaleR_minus_right
thf(fact_417_abs__minus__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus_cancel
thf(fact_418_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_419_abs__inverse,axiom,
! [A: complex] :
( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
= ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).
% abs_inverse
thf(fact_420_abs__inverse,axiom,
! [A: real] :
( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).
% abs_inverse
thf(fact_421_abs__norm__cancel,axiom,
! [A: complex] :
( ( abs_abs_real @ ( real_V1022390504157884413omplex @ A ) )
= ( real_V1022390504157884413omplex @ A ) ) ).
% abs_norm_cancel
thf(fact_422_abs__norm__cancel,axiom,
! [A: real] :
( ( abs_abs_real @ ( real_V7735802525324610683m_real @ A ) )
= ( real_V7735802525324610683m_real @ A ) ) ).
% abs_norm_cancel
thf(fact_423_exp__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% exp_le_cancel_iff
thf(fact_424_abs__exp__cancel,axiom,
! [X: real] :
( ( abs_abs_real @ ( exp_real @ X ) )
= ( exp_real @ X ) ) ).
% abs_exp_cancel
thf(fact_425_complex__cnj__complex__of__real,axiom,
! [X: real] :
( ( cnj @ ( real_V4546457046886955230omplex @ X ) )
= ( real_V4546457046886955230omplex @ X ) ) ).
% complex_cnj_complex_of_real
thf(fact_426_complex__mod__cnj,axiom,
! [Z: complex] :
( ( real_V1022390504157884413omplex @ ( cnj @ Z ) )
= ( real_V1022390504157884413omplex @ Z ) ) ).
% complex_mod_cnj
thf(fact_427_complex__cnj__minus,axiom,
! [X: complex] :
( ( cnj @ ( uminus1482373934393186551omplex @ X ) )
= ( uminus1482373934393186551omplex @ ( cnj @ X ) ) ) ).
% complex_cnj_minus
thf(fact_428_complex__cnj__inverse,axiom,
! [X: complex] :
( ( cnj @ ( invers8013647133539491842omplex @ X ) )
= ( invers8013647133539491842omplex @ ( cnj @ X ) ) ) ).
% complex_cnj_inverse
thf(fact_429_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_430_neg__0__le__iff__le,axiom,
! [A: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ ( uminus1482373934393186551omplex @ A ) )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% neg_0_le_iff_le
thf(fact_431_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_432_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_433_neg__le__0__iff__le,axiom,
! [A: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ A ) @ zero_zero_complex )
= ( ord_less_eq_complex @ zero_zero_complex @ A ) ) ).
% neg_le_0_iff_le
thf(fact_434_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_435_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_436_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_437_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_438_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_439_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_440_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_441_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_442_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_443_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_444_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_445_inverse__nonnegative__iff__nonnegative,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_446_inverse__nonpositive__iff__nonpositive,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_447_scaleR__eq__0__iff,axiom,
! [A: real,X: complex] :
( ( ( real_V2046097035970521341omplex @ A @ X )
= zero_zero_complex )
= ( ( A = zero_zero_real )
| ( X = zero_zero_complex ) ) ) ).
% scaleR_eq_0_iff
thf(fact_448_scaleR__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% scaleR_eq_0_iff
thf(fact_449_scaleR__zero__left,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
= zero_zero_complex ) ).
% scaleR_zero_left
thf(fact_450_scaleR__zero__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% scaleR_zero_left
thf(fact_451_scaleR__left_Ominus,axiom,
! [X: real,Xa: complex] :
( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ X ) @ Xa )
= ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ X @ Xa ) ) ) ).
% scaleR_left.minus
thf(fact_452_scaleR__left_Ominus,axiom,
! [X: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ X ) @ Xa )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) ) ) ).
% scaleR_left.minus
thf(fact_453_scaleR__minus__left,axiom,
! [A: real,X: complex] :
( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ A ) @ X )
= ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ A @ X ) ) ) ).
% scaleR_minus_left
thf(fact_454_scaleR__minus__left,axiom,
! [A: real,X: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ A ) @ X )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ).
% scaleR_minus_left
thf(fact_455_norm__of__real,axiom,
! [R: real] :
( ( real_V1022390504157884413omplex @ ( real_V4546457046886955230omplex @ R ) )
= ( abs_abs_real @ R ) ) ).
% norm_of_real
thf(fact_456_norm__of__real,axiom,
! [R: real] :
( ( real_V7735802525324610683m_real @ ( real_V1803761363581548252l_real @ R ) )
= ( abs_abs_real @ R ) ) ).
% norm_of_real
thf(fact_457_abs__of__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_nonpos
thf(fact_458_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_459_abs__leI,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
=> ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_460_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_461_abs__le__D2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_462_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_463_abs__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_464_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_465_abs__ge__minus__self,axiom,
! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% abs_ge_minus_self
thf(fact_466_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_467_abs__ge__zero,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% abs_ge_zero
thf(fact_468_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_469_scaleR__right__mono__neg,axiom,
! [B: real,A: real,C: complex] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A @ C ) @ ( real_V2046097035970521341omplex @ B @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_470_scaleR__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_471_scaleR__left__mono__neg,axiom,
! [B: complex,A: complex,C: real] :
( ( ord_less_eq_complex @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ C @ A ) @ ( real_V2046097035970521341omplex @ C @ B ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_472_scaleR__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_473_real__norm__def,axiom,
real_V7735802525324610683m_real = abs_abs_real ).
% real_norm_def
thf(fact_474_scaleR__right__mono,axiom,
! [A: real,B: real,X: complex] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ) ).
% scaleR_right_mono
thf(fact_475_scaleR__right__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).
% scaleR_right_mono
thf(fact_476_scaleR__left__mono,axiom,
! [X: complex,Y: complex,A: real] :
( ( ord_less_eq_complex @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y ) ) ) ) ).
% scaleR_left_mono
thf(fact_477_scaleR__left__mono,axiom,
! [X: real,Y: real,A: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ) ).
% scaleR_left_mono
thf(fact_478_scaleR__left__commute,axiom,
! [A: real,B: real,X: complex] :
( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B @ X ) )
= ( real_V2046097035970521341omplex @ B @ ( real_V2046097035970521341omplex @ A @ X ) ) ) ).
% scaleR_left_commute
thf(fact_479_scaleR__left__commute,axiom,
! [A: real,B: real,X: real] :
( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B @ X ) )
= ( real_V1485227260804924795R_real @ B @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ).
% scaleR_left_commute
thf(fact_480_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_481_verit__comp__simplify1_I2_J,axiom,
! [A: complex] : ( ord_less_eq_complex @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_482_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_483_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_484_abs__le__D1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% abs_le_D1
thf(fact_485_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_486_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_487_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_488_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_489_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_490_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_491_scaleR__mono,axiom,
! [A: real,B: real,X: complex,Y: complex] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_complex @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ Y ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_492_scaleR__mono,axiom,
! [A: real,B: real,X: real,Y: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_493_scaleR__mono_H,axiom,
! [A: real,B: real,C: complex,D: complex] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A @ C ) @ ( real_V2046097035970521341omplex @ B @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_494_scaleR__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_495_split__scaleR__neg__le,axiom,
! [A: real,X: complex] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_complex @ X @ zero_zero_complex ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_complex @ zero_zero_complex @ X ) ) )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ zero_zero_complex ) ) ).
% split_scaleR_neg_le
thf(fact_496_split__scaleR__neg__le,axiom,
! [A: real,X: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ X @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ X ) ) )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% split_scaleR_neg_le
thf(fact_497_split__scaleR__pos__le,axiom,
! [A: real,B: complex] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_complex @ zero_zero_complex @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_complex @ B @ zero_zero_complex ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( real_V2046097035970521341omplex @ A @ B ) ) ) ).
% split_scaleR_pos_le
thf(fact_498_split__scaleR__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).
% split_scaleR_pos_le
thf(fact_499_scaleR__nonneg__nonneg,axiom,
! [A: real,X: complex] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( real_V2046097035970521341omplex @ A @ X ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_500_scaleR__nonneg__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_501_scaleR__nonneg__nonpos,axiom,
! [A: real,X: complex] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_complex @ X @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ zero_zero_complex ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_502_scaleR__nonneg__nonpos,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_503_scaleR__nonpos__nonneg,axiom,
! [A: real,X: complex] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( ord_less_eq_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ zero_zero_complex ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_504_scaleR__nonpos__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_505_scaleR__nonpos__nonpos,axiom,
! [A: real,B: complex] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_complex @ B @ zero_zero_complex )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( real_V2046097035970521341omplex @ A @ B ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_506_scaleR__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_507_abs__minus__le__zero,axiom,
! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% abs_minus_le_zero
thf(fact_508_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_509_abs__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% abs_Re_le_cmod
thf(fact_510_abs__Im__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% abs_Im_le_cmod
thf(fact_511_cnj_Osimps_I1_J,axiom,
! [Z: complex] :
( ( re @ ( cnj @ Z ) )
= ( re @ Z ) ) ).
% cnj.simps(1)
thf(fact_512_scaleR__right__imp__eq,axiom,
! [X: complex,A: real,B: real] :
( ( X != zero_zero_complex )
=> ( ( ( real_V2046097035970521341omplex @ A @ X )
= ( real_V2046097035970521341omplex @ B @ X ) )
=> ( A = B ) ) ) ).
% scaleR_right_imp_eq
thf(fact_513_scaleR__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ B @ X ) )
=> ( A = B ) ) ) ).
% scaleR_right_imp_eq
thf(fact_514_scaleR__left__imp__eq,axiom,
! [A: real,X: complex,Y: complex] :
( ( A != zero_zero_real )
=> ( ( ( real_V2046097035970521341omplex @ A @ X )
= ( real_V2046097035970521341omplex @ A @ Y ) )
=> ( X = Y ) ) ) ).
% scaleR_left_imp_eq
thf(fact_515_scaleR__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% scaleR_left_imp_eq
thf(fact_516_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_517_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_518_le__imp__neg__le,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% le_imp_neg_le
thf(fact_519_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_520_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_521_minus__le__iff,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( ord_less_eq_complex @ ( uminus1482373934393186551omplex @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_522_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_523_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_524_le__minus__iff,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( ord_less_eq_complex @ B @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% le_minus_iff
thf(fact_525_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_526_Reals__cnj__iff,axiom,
! [Z: complex] :
( ( member_complex @ Z @ real_V2521375963428798218omplex )
= ( ( cnj @ Z )
= Z ) ) ).
% Reals_cnj_iff
thf(fact_527_cmod__Im__le__iff,axiom,
! [X: complex,Y: complex] :
( ( ( re @ X )
= ( re @ Y ) )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
= ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).
% cmod_Im_le_iff
thf(fact_528_cmod__Re__le__iff,axiom,
! [X: complex,Y: complex] :
( ( ( im @ X )
= ( im @ Y ) )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
= ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).
% cmod_Re_le_iff
thf(fact_529_nonzero__abs__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_530_cnj_Osimps_I2_J,axiom,
! [Z: complex] :
( ( im @ ( cnj @ Z ) )
= ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% cnj.simps(2)
thf(fact_531_continuous__on__arsinh,axiom,
! [A2: set_real] : ( topolo5044208981011980120l_real @ A2 @ arsinh_real ) ).
% continuous_on_arsinh
thf(fact_532_complex__cnj,axiom,
! [A: real,B: real] :
( ( cnj @ ( complex2 @ A @ B ) )
= ( complex2 @ A @ ( uminus_uminus_real @ B ) ) ) ).
% complex_cnj
thf(fact_533_inverse__scaleR__distrib,axiom,
! [A: real,X: complex] :
( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
= ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ).
% inverse_scaleR_distrib
thf(fact_534_inverse__scaleR__distrib,axiom,
! [A: real,X: real] :
( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
= ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ).
% inverse_scaleR_distrib
thf(fact_535_norm__ge__zero,axiom,
! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% norm_ge_zero
thf(fact_536_norm__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) ) ).
% norm_ge_zero
thf(fact_537_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% exp_ge_zero
thf(fact_538_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% not_exp_le_zero
thf(fact_539_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_540_complex__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% complex_Re_le_cmod
thf(fact_541_zero__complex_Osimps_I1_J,axiom,
( ( re @ zero_zero_complex )
= zero_zero_real ) ).
% zero_complex.simps(1)
thf(fact_542_zero__complex_Osimps_I2_J,axiom,
( ( im @ zero_zero_complex )
= zero_zero_real ) ).
% zero_complex.simps(2)
thf(fact_543_zero__complex_Ocode,axiom,
( zero_zero_complex
= ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).
% zero_complex.code
thf(fact_544_Complex__eq__0,axiom,
! [A: real,B: real] :
( ( ( complex2 @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_real )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_0
thf(fact_545_in__Reals__norm,axiom,
! [Z: complex] :
( ( member_complex @ Z @ real_V2521375963428798218omplex )
=> ( ( real_V1022390504157884413omplex @ Z )
= ( abs_abs_real @ ( re @ Z ) ) ) ) ).
% in_Reals_norm
thf(fact_546_norm__exp,axiom,
! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% norm_exp
thf(fact_547_norm__exp,axiom,
! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% norm_exp
thf(fact_548_exp__uminus__Im,axiom,
! [Z: complex] : ( ord_less_eq_real @ ( exp_real @ ( uminus_uminus_real @ ( im @ Z ) ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% exp_uminus_Im
thf(fact_549_abs__minus,axiom,
! [A: complex] :
( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
= ( abs_abs_complex @ A ) ) ).
% abs_minus
thf(fact_550_abs__minus,axiom,
! [A: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
= ( abs_abs_real @ A ) ) ).
% abs_minus
thf(fact_551_abs__minus,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus
thf(fact_552_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_553_abs__0,axiom,
( ( abs_abs_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% abs_0
thf(fact_554_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_555_eq__abs__iff_H,axiom,
! [A: real,B: real] :
( ( A
= ( abs_abs_real @ B ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_real @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_556_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_557_abs__eq__iff_H,axiom,
! [A: real,B: real] :
( ( ( abs_abs_real @ A )
= B )
= ( ( ord_less_eq_real @ zero_zero_real @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_558_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_559_divideR__right,axiom,
! [R: real,Y: complex,X: complex] :
( ( R != zero_zero_real )
=> ( ( Y
= ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ R ) @ X ) )
= ( ( real_V2046097035970521341omplex @ R @ Y )
= X ) ) ) ).
% divideR_right
thf(fact_560_divideR__right,axiom,
! [R: real,Y: real,X: real] :
( ( R != zero_zero_real )
=> ( ( Y
= ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ R ) @ X ) )
= ( ( real_V1485227260804924795R_real @ R @ Y )
= X ) ) ) ).
% divideR_right
thf(fact_561_real__eq__0__iff__le__ge__0,axiom,
! [X: real] :
( ( X = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_562_norm__imp__pos__and__ge,axiom,
! [X: complex,N2: real] :
( ( ( real_V1022390504157884413omplex @ X )
= N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
& ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ) ).
% norm_imp_pos_and_ge
thf(fact_563_norm__imp__pos__and__ge,axiom,
! [X: real,N2: real] :
( ( ( real_V7735802525324610683m_real @ X )
= N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
& ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ) ).
% norm_imp_pos_and_ge
thf(fact_564_vector__choose__size,axiom,
! [C: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ~ ! [X4: complex] :
( ( real_V1022390504157884413omplex @ X4 )
!= C ) ) ).
% vector_choose_size
thf(fact_565_vector__choose__size,axiom,
! [C: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ~ ! [X4: real] :
( ( real_V7735802525324610683m_real @ X4 )
!= C ) ) ).
% vector_choose_size
thf(fact_566_image__subsetI,axiom,
! [A2: set_complex,F2: complex > complex,B2: set_complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A2 )
=> ( member_complex @ ( F2 @ X4 ) @ B2 ) )
=> ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_567_continuous__on__subset,axiom,
! [S2: set_real,F2: real > real,T: set_real] :
( ( topolo5044208981011980120l_real @ S2 @ F2 )
=> ( ( ord_less_eq_set_real @ T @ S2 )
=> ( topolo5044208981011980120l_real @ T @ F2 ) ) ) ).
% continuous_on_subset
thf(fact_568_continuous__on__subset,axiom,
! [S2: set_complex,F2: complex > complex,T: set_complex] :
( ( topolo9015423870875150044omplex @ S2 @ F2 )
=> ( ( ord_le211207098394363844omplex @ T @ S2 )
=> ( topolo9015423870875150044omplex @ T @ F2 ) ) ) ).
% continuous_on_subset
thf(fact_569_raw__has__prod__ignore__initial__segment,axiom,
! [F2: nat > complex,M: nat,P2: complex,N: nat] :
( ( infini5805797430451707772omplex @ F2 @ M @ P2 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [Q2: complex] :
~ ( infini5805797430451707772omplex @ F2 @ N @ Q2 ) ) ) ).
% raw_has_prod_ignore_initial_segment
thf(fact_570_raw__has__prod__ignore__initial__segment,axiom,
! [F2: nat > real,M: nat,P2: real,N: nat] :
( ( infini2923794516677094010d_real @ F2 @ M @ P2 )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [Q2: real] :
~ ( infini2923794516677094010d_real @ F2 @ N @ Q2 ) ) ) ).
% raw_has_prod_ignore_initial_segment
thf(fact_571_abs__complex__def,axiom,
( abs_abs_complex
= ( comp_r891790309028876909omplex @ real_V4546457046886955230omplex @ real_V1022390504157884413omplex ) ) ).
% abs_complex_def
thf(fact_572_raw__has__prod__eq__0,axiom,
! [F2: nat > nat,M2: nat,P2: nat,I: nat] :
( ( infini1035527091028463262od_nat @ F2 @ M2 @ P2 )
=> ( ( ( F2 @ I )
= zero_zero_nat )
=> ( ( ord_less_eq_nat @ M2 @ I )
=> ( P2 = zero_zero_nat ) ) ) ) ).
% raw_has_prod_eq_0
thf(fact_573_raw__has__prod__eq__0,axiom,
! [F2: nat > int,M2: nat,P2: int,I: nat] :
( ( infini1033036620519412986od_int @ F2 @ M2 @ P2 )
=> ( ( ( F2 @ I )
= zero_zero_int )
=> ( ( ord_less_eq_nat @ M2 @ I )
=> ( P2 = zero_zero_int ) ) ) ) ).
% raw_has_prod_eq_0
thf(fact_574_raw__has__prod__eq__0,axiom,
! [F2: nat > complex,M2: nat,P2: complex,I: nat] :
( ( infini5805797430451707772omplex @ F2 @ M2 @ P2 )
=> ( ( ( F2 @ I )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ M2 @ I )
=> ( P2 = zero_zero_complex ) ) ) ) ).
% raw_has_prod_eq_0
thf(fact_575_raw__has__prod__eq__0,axiom,
! [F2: nat > real,M2: nat,P2: real,I: nat] :
( ( infini2923794516677094010d_real @ F2 @ M2 @ P2 )
=> ( ( ( F2 @ I )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ M2 @ I )
=> ( P2 = zero_zero_real ) ) ) ) ).
% raw_has_prod_eq_0
thf(fact_576_pth__4_I2_J,axiom,
! [C: real] :
( ( real_V2046097035970521341omplex @ C @ zero_zero_complex )
= zero_zero_complex ) ).
% pth_4(2)
thf(fact_577_pth__4_I2_J,axiom,
! [C: real] :
( ( real_V1485227260804924795R_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% pth_4(2)
thf(fact_578_abs__eq__0__iff,axiom,
! [A: real] :
( ( ( abs_abs_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_579_abs__eq__0__iff,axiom,
! [A: complex] :
( ( ( abs_abs_complex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% abs_eq_0_iff
thf(fact_580_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_581_abs__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( abs_abs_real @ X )
= ( abs_abs_real @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_real @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_582_abs__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( abs_abs_int @ X )
= ( abs_abs_int @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus_int @ Y ) ) ) ) ).
% abs_eq_iff
thf(fact_583_exp__cnj,axiom,
! [Z: complex] :
( ( cnj @ ( exp_complex @ Z ) )
= ( exp_complex @ ( cnj @ Z ) ) ) ).
% exp_cnj
thf(fact_584_pth__4_I1_J,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
= zero_zero_complex ) ).
% pth_4(1)
thf(fact_585_pth__4_I1_J,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% pth_4(1)
thf(fact_586_less__eq__complex__def,axiom,
( ord_less_eq_complex
= ( ^ [X2: complex,Y2: complex] :
( ( ord_less_eq_real @ ( re @ X2 ) @ ( re @ Y2 ) )
& ( ( im @ X2 )
= ( im @ Y2 ) ) ) ) ) ).
% less_eq_complex_def
thf(fact_587_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_588_fps__tan__0,axiom,
( ( formal6482914284900457064omplex @ zero_zero_complex )
= zero_z1877163951443063103omplex ) ).
% fps_tan_0
thf(fact_589_complex__is__real__iff__compare0,axiom,
! [X: complex] :
( ( member_complex @ X @ real_V2521375963428798218omplex )
= ( ( ord_less_eq_complex @ X @ zero_zero_complex )
| ( ord_less_eq_complex @ zero_zero_complex @ X ) ) ) ).
% complex_is_real_iff_compare0
thf(fact_590_nonnegative__complex__is__real,axiom,
! [X: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( member_complex @ X @ real_V2521375963428798218omplex ) ) ).
% nonnegative_complex_is_real
thf(fact_591_invertible__fixpoint__property,axiom,
! [T2: set_complex,I: complex > real,S: set_real,R: real > complex,G2: complex > complex] :
( ( topolo8674095878704923098x_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_complex_real @ I @ T2 ) @ S )
=> ( ( topolo8620507378200602458omplex @ S @ R )
=> ( ( ord_le211207098394363844omplex @ ( image_real_complex @ R @ S ) @ T2 )
=> ( ! [Y4: complex] :
( ( member_complex @ Y4 @ T2 )
=> ( ( R @ ( I @ Y4 ) )
= Y4 ) )
=> ( ! [F4: real > real] :
( ( topolo5044208981011980120l_real @ S @ F4 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F4 @ S ) @ S )
=> ? [X5: real] :
( ( member_real @ X5 @ S )
& ( ( F4 @ X5 )
= X5 ) ) ) )
=> ( ( topolo9015423870875150044omplex @ T2 @ G2 )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ G2 @ T2 ) @ T2 )
=> ~ ! [Y4: complex] :
( ( member_complex @ Y4 @ T2 )
=> ( ( G2 @ Y4 )
!= Y4 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_592_invertible__fixpoint__property,axiom,
! [T2: set_real,I: real > complex,S: set_complex,R: complex > real,G2: real > real] :
( ( topolo8620507378200602458omplex @ T2 @ I )
=> ( ( ord_le211207098394363844omplex @ ( image_real_complex @ I @ T2 ) @ S )
=> ( ( topolo8674095878704923098x_real @ S @ R )
=> ( ( ord_less_eq_set_real @ ( image_complex_real @ R @ S ) @ T2 )
=> ( ! [Y4: real] :
( ( member_real @ Y4 @ T2 )
=> ( ( R @ ( I @ Y4 ) )
= Y4 ) )
=> ( ! [F4: complex > complex] :
( ( topolo9015423870875150044omplex @ S @ F4 )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F4 @ S ) @ S )
=> ? [X5: complex] :
( ( member_complex @ X5 @ S )
& ( ( F4 @ X5 )
= X5 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T2 @ G2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G2 @ T2 ) @ T2 )
=> ~ ! [Y4: real] :
( ( member_real @ Y4 @ T2 )
=> ( ( G2 @ Y4 )
!= Y4 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_593_invertible__fixpoint__property,axiom,
! [T2: set_real,I: real > real,S: set_real,R: real > real,G2: real > real] :
( ( topolo5044208981011980120l_real @ T2 @ I )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ I @ T2 ) @ S )
=> ( ( topolo5044208981011980120l_real @ S @ R )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ R @ S ) @ T2 )
=> ( ! [Y4: real] :
( ( member_real @ Y4 @ T2 )
=> ( ( R @ ( I @ Y4 ) )
= Y4 ) )
=> ( ! [F4: real > real] :
( ( topolo5044208981011980120l_real @ S @ F4 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ F4 @ S ) @ S )
=> ? [X5: real] :
( ( member_real @ X5 @ S )
& ( ( F4 @ X5 )
= X5 ) ) ) )
=> ( ( topolo5044208981011980120l_real @ T2 @ G2 )
=> ( ( ord_less_eq_set_real @ ( image_real_real @ G2 @ T2 ) @ T2 )
=> ~ ! [Y4: real] :
( ( member_real @ Y4 @ T2 )
=> ( ( G2 @ Y4 )
!= Y4 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_594_invertible__fixpoint__property,axiom,
! [T2: set_complex,I: complex > complex,S: set_complex,R: complex > complex,G2: complex > complex] :
( ( topolo9015423870875150044omplex @ T2 @ I )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ I @ T2 ) @ S )
=> ( ( topolo9015423870875150044omplex @ S @ R )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ R @ S ) @ T2 )
=> ( ! [Y4: complex] :
( ( member_complex @ Y4 @ T2 )
=> ( ( R @ ( I @ Y4 ) )
= Y4 ) )
=> ( ! [F4: complex > complex] :
( ( topolo9015423870875150044omplex @ S @ F4 )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F4 @ S ) @ S )
=> ? [X5: complex] :
( ( member_complex @ X5 @ S )
& ( ( F4 @ X5 )
= X5 ) ) ) )
=> ( ( topolo9015423870875150044omplex @ T2 @ G2 )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ G2 @ T2 ) @ T2 )
=> ~ ! [Y4: complex] :
( ( member_complex @ Y4 @ T2 )
=> ( ( G2 @ Y4 )
!= Y4 ) ) ) ) ) ) ) ) ) ) ).
% invertible_fixpoint_property
thf(fact_595_scaleR__cong__right,axiom,
! [X: complex,R: real,P2: real] :
( ( ( X != zero_zero_complex )
=> ( R = P2 ) )
=> ( ( real_V2046097035970521341omplex @ R @ X )
= ( real_V2046097035970521341omplex @ P2 @ X ) ) ) ).
% scaleR_cong_right
thf(fact_596_scaleR__cong__right,axiom,
! [X: real,R: real,P2: real] :
( ( ( X != zero_zero_real )
=> ( R = P2 ) )
=> ( ( real_V1485227260804924795R_real @ R @ X )
= ( real_V1485227260804924795R_real @ P2 @ X ) ) ) ).
% scaleR_cong_right
thf(fact_597_Multiseries__Expansion__Bounds_Oeq__zero__imp__nonneg,axiom,
! [X: real] :
( ( X = zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% Multiseries_Expansion_Bounds.eq_zero_imp_nonneg
thf(fact_598_subsetI,axiom,
! [A2: set_complex,B2: set_complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A2 )
=> ( member_complex @ X4 @ B2 ) )
=> ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).
% subsetI
thf(fact_599_in__mono,axiom,
! [A2: set_complex,B2: set_complex,X: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( member_complex @ X @ A2 )
=> ( member_complex @ X @ B2 ) ) ) ).
% in_mono
thf(fact_600_subsetD,axiom,
! [A2: set_complex,B2: set_complex,C: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B2 ) ) ) ).
% subsetD
thf(fact_601_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B3: set_complex] :
! [X2: complex] :
( ( member_complex @ X2 @ A3 )
=> ( member_complex @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_602_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B3: set_complex] :
! [T3: complex] :
( ( member_complex @ T3 @ A3 )
=> ( member_complex @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_603_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_604_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_605_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_606_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_607_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_608_Im__Arctan__of__real,axiom,
! [X: real] :
( ( im @ ( complex_Arctan @ ( real_V4546457046886955230omplex @ X ) ) )
= zero_zero_real ) ).
% Im_Arctan_of_real
thf(fact_609_cmod__le,axiom,
! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% cmod_le
thf(fact_610_norm__inverse__le__norm,axiom,
! [R: real,X: complex] :
( ( ord_less_eq_real @ R @ ( real_V1022390504157884413omplex @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ R )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X ) ) @ ( inverse_inverse_real @ R ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_611_norm__inverse__le__norm,axiom,
! [R: real,X: real] :
( ( ord_less_eq_real @ R @ ( real_V7735802525324610683m_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ R )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X ) ) @ ( inverse_inverse_real @ R ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_612_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_613_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_614_add__right__cancel,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_615_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_616_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_617_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_618_add__left__cancel,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_619_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_620_ComplI,axiom,
! [C: complex,A2: set_complex] :
( ~ ( member_complex @ C @ A2 )
=> ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) ) ) ).
% ComplI
thf(fact_621_Compl__iff,axiom,
! [C: complex,A2: set_complex] :
( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
= ( ~ ( member_complex @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_622_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_623_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_624_add__le__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
= ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_625_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_626_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_627_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_628_add__le__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
= ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_629_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_630_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_631_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_632_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_633_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_634_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_635_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_636_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_637_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_638_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_639_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_640_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_641_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_642_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_643_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_644_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_645_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_646_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_647_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_648_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_649_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_650_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_651_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_652_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_653_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_654_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_655_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_656_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_657_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_658_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_659_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_660_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_661_add__less__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
= ( ord_less_complex @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_662_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_663_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_664_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_665_add__less__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
= ( ord_less_complex @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_666_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_667_neg__less__iff__less,axiom,
! [B: complex,A: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) )
= ( ord_less_complex @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_668_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_669_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_670_add__minus__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_671_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_672_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_673_minus__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_674_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_675_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_676_minus__add__distrib,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_add_distrib
thf(fact_677_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_678_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_679_abs__add__abs,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_add_abs
thf(fact_680_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_681_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% exp_less_mono
thf(fact_682_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_683_Arctan__0,axiom,
( ( complex_Arctan @ zero_zero_complex )
= zero_zero_complex ) ).
% Arctan_0
thf(fact_684_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_685_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_686_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_687_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_688_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_689_le__add__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( plus_plus_complex @ B @ A ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B ) ) ).
% le_add_same_cancel2
thf(fact_690_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_691_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_692_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_693_le__add__same__cancel1,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( ord_less_eq_complex @ zero_zero_complex @ B ) ) ).
% le_add_same_cancel1
thf(fact_694_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_695_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_696_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_697_add__le__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ B ) @ B )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% add_le_same_cancel2
thf(fact_698_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_699_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_700_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_701_add__le__same__cancel1,axiom,
! [B: complex,A: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ B @ A ) @ B )
= ( ord_less_eq_complex @ A @ zero_zero_complex ) ) ).
% add_le_same_cancel1
thf(fact_702_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_703_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_704_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_705_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_706_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_707_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_708_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_709_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_710_less__add__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( plus_plus_complex @ B @ A ) )
= ( ord_less_complex @ zero_zero_complex @ B ) ) ).
% less_add_same_cancel2
thf(fact_711_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_712_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_713_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_714_less__add__same__cancel1,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( ord_less_complex @ zero_zero_complex @ B ) ) ).
% less_add_same_cancel1
thf(fact_715_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_716_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_717_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_718_add__less__same__cancel2,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ B )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% add_less_same_cancel2
thf(fact_719_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_720_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_721_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_722_add__less__same__cancel1,axiom,
! [B: complex,A: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ B @ A ) @ B )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% add_less_same_cancel1
thf(fact_723_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_724_neg__less__0__iff__less,axiom,
! [A: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A ) @ zero_zero_complex )
= ( ord_less_complex @ zero_zero_complex @ A ) ) ).
% neg_less_0_iff_less
thf(fact_725_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_726_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_727_neg__0__less__iff__less,axiom,
! [A: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( uminus1482373934393186551omplex @ A ) )
= ( ord_less_complex @ A @ zero_zero_complex ) ) ).
% neg_0_less_iff_less
thf(fact_728_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_729_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_730_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_731_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_732_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_733_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_734_image__add__0,axiom,
! [S: set_real] :
( ( image_real_real @ ( plus_plus_real @ zero_zero_real ) @ S )
= S ) ).
% image_add_0
thf(fact_735_image__add__0,axiom,
! [S: set_complex] :
( ( image_1468599708987790691omplex @ ( plus_plus_complex @ zero_zero_complex ) @ S )
= S ) ).
% image_add_0
thf(fact_736_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_737_image__add__0,axiom,
! [S: set_int] :
( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S )
= S ) ).
% image_add_0
thf(fact_738_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_739_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_740_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_741_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_742_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_743_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_744_zero__less__abs__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
= ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_745_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_746_inverse__positive__iff__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_747_inverse__negative__iff__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_748_inverse__less__iff__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_749_inverse__less__iff__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_750_of__real__add,axiom,
! [X: real,Y: real] :
( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).
% of_real_add
thf(fact_751_of__real__add,axiom,
! [X: real,Y: real] :
( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).
% of_real_add
thf(fact_752_inverse__le__iff__le,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_753_inverse__le__iff__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_754_zero__less__norm__iff,axiom,
! [X: complex] :
( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
= ( X != zero_zero_complex ) ) ).
% zero_less_norm_iff
thf(fact_755_zero__less__norm__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
= ( X != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_756_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_757_add__mono__thms__linordered__field_I4_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I @ J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_758_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_759_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_760_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_761_add__mono__thms__linordered__field_I3_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I @ J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_762_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_763_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_764_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_765_add__le__less__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ( ord_less_complex @ C @ D )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_766_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_767_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_768_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_769_add__less__le__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_complex @ A @ B )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_770_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_771_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_772_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_773_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_774_add__less__imp__less__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
=> ( ord_less_complex @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_775_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_776_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_777_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_778_add__less__imp__less__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
=> ( ord_less_complex @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_779_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_780_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_781_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_782_add__strict__right__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ A @ B )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_783_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_784_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_785_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_786_add__strict__left__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ A @ B )
=> ( ord_less_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_787_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_788_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_789_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_790_add__strict__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_complex @ A @ B )
=> ( ( ord_less_complex @ C @ D )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_791_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_792_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_793_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% linordered_field_no_lb
thf(fact_794_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_795_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_796_add__right__imp__eq,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_797_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_798_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_799_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_800_add__left__imp__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_801_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_802_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_803_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_804_add_Oleft__commute,axiom,
! [B: complex,A: complex,C: complex] :
( ( plus_plus_complex @ B @ ( plus_plus_complex @ A @ C ) )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% add.left_commute
thf(fact_805_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_806_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_807_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_808_add_Ocommute,axiom,
( plus_plus_complex
= ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_809_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_810_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_811_add_Oright__cancel,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_812_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_813_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_814_add_Oleft__cancel,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_815_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_816_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_817_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_818_add_Oassoc,axiom,
! [A: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% add.assoc
thf(fact_819_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_820_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_821_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_822_group__cancel_Oadd2,axiom,
! [B2: complex,K: complex,B: complex,A: complex] :
( ( B2
= ( plus_plus_complex @ K @ B ) )
=> ( ( plus_plus_complex @ A @ B2 )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_823_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_824_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_825_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_826_group__cancel_Oadd1,axiom,
! [A2: complex,K: complex,A: complex,B: complex] :
( ( A2
= ( plus_plus_complex @ K @ A ) )
=> ( ( plus_plus_complex @ A2 @ B )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_827_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_828_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_829_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_830_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_complex @ I @ K )
= ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_831_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_832_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_833_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_834_add__mono__thms__linordered__field_I1_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I @ J )
& ( K = L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_835_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_836_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_837_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_838_add__mono__thms__linordered__field_I2_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( I = J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_839_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_840_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_841_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_842_add__mono__thms__linordered__field_I5_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_complex @ I @ J )
& ( ord_less_complex @ K @ L ) )
=> ( ord_less_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_843_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_844_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_845_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_846_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_847_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_848_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_849_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_850_verit__comp__simplify1_I1_J,axiom,
! [A: complex] :
~ ( ord_less_complex @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_851_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_852_ComplD,axiom,
! [C: complex,A2: set_complex] :
( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
=> ~ ( member_complex @ C @ A2 ) ) ).
% ComplD
thf(fact_853_is__num__normalize_I8_J,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_854_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_855_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_856_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_857_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_858_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_859_scaleR__left__distrib,axiom,
! [A: real,B: real,X: complex] :
( ( real_V2046097035970521341omplex @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).
% scaleR_left_distrib
thf(fact_860_scaleR__left__distrib,axiom,
! [A: real,B: real,X: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).
% scaleR_left_distrib
thf(fact_861_scaleR__left_Oadd,axiom,
! [X: real,Y: real,Xa: complex] :
( ( real_V2046097035970521341omplex @ ( plus_plus_real @ X @ Y ) @ Xa )
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ X @ Xa ) @ ( real_V2046097035970521341omplex @ Y @ Xa ) ) ) ).
% scaleR_left.add
thf(fact_862_scaleR__left_Oadd,axiom,
! [X: real,Y: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y ) @ Xa )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% scaleR_left.add
thf(fact_863_norm__add__less,axiom,
! [X: complex,R: real,Y: complex,S2: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R @ S2 ) ) ) ) ).
% norm_add_less
thf(fact_864_norm__add__less,axiom,
! [X: real,R: real,Y: real,S2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R @ S2 ) ) ) ) ).
% norm_add_less
thf(fact_865_norm__triangle__lt,axiom,
! [X: complex,Y: complex,E: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% norm_triangle_lt
thf(fact_866_norm__triangle__lt,axiom,
! [X: real,Y: real,E: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% norm_triangle_lt
thf(fact_867_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_868_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_869_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_870_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_871_add__neg__neg,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ zero_zero_complex )
=> ( ( ord_less_complex @ B @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_neg_neg
thf(fact_872_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_873_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_874_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_875_add__pos__pos,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ zero_zero_complex @ B )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_876_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_877_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_878_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_879_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_880_pos__add__strict,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ B @ C )
=> ( ord_less_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_881_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_882_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_883_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_884_add__neg__nonpos,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_neg_nonpos
thf(fact_885_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_886_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_887_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_888_add__nonneg__pos,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ zero_zero_complex @ B )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_889_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_890_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_891_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_892_add__nonpos__neg,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ zero_zero_complex )
=> ( ( ord_less_complex @ B @ zero_zero_complex )
=> ( ord_less_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_nonpos_neg
thf(fact_893_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_894_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_895_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_896_add__pos__nonneg,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B )
=> ( ord_less_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_897_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_898_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_899_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_900_add__strict__increasing,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ B @ C )
=> ( ord_less_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_901_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_902_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_903_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_904_add__strict__increasing2,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_complex @ B @ C )
=> ( ord_less_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_905_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_906_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_907_norm__add__rule__thm,axiom,
! [X1: complex,B1: real,X22: complex,B22: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X1 ) @ B1 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X22 ) @ B22 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X1 @ X22 ) ) @ ( plus_plus_real @ B1 @ B22 ) ) ) ) ).
% norm_add_rule_thm
thf(fact_908_norm__add__rule__thm,axiom,
! [X1: real,B1: real,X22: real,B22: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X1 ) @ B1 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X22 ) @ B22 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X1 @ X22 ) ) @ ( plus_plus_real @ B1 @ B22 ) ) ) ) ).
% norm_add_rule_thm
thf(fact_909_norm__add__leD,axiom,
! [A: complex,B: complex,C: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_910_norm__add__leD,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_911_norm__triangle__le,axiom,
! [X: complex,Y: complex,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% norm_triangle_le
thf(fact_912_norm__triangle__le,axiom,
! [X: real,Y: real,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% norm_triangle_le
thf(fact_913_norm__triangle__ineq,axiom,
! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_914_norm__triangle__ineq,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% norm_triangle_ineq
thf(fact_915_norm__triangle__mono,axiom,
! [A: complex,R: real,B: complex,S2: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S2 ) ) ) ) ).
% norm_triangle_mono
thf(fact_916_norm__triangle__mono,axiom,
! [A: real,R: real,B: real,S2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S2 ) ) ) ) ).
% norm_triangle_mono
thf(fact_917_verit__comp__simplify1_I3_J,axiom,
! [B5: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B5 @ A5 ) )
= ( ord_less_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_918_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_919_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_920_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_921_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_922_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_923_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_924_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_925_add__le__imp__le__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) )
=> ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_926_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_927_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_928_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_929_add__le__imp__le__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) )
=> ( ord_less_eq_complex @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_930_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_931_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_932_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_933_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_934_add__right__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ C ) ) ) ).
% add_right_mono
thf(fact_935_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_936_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_937_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_938_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_939_add__left__mono,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B ) ) ) ).
% add_left_mono
thf(fact_940_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_941_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_942_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_943_add__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ord_less_eq_complex @ A @ B )
=> ( ( ord_less_eq_complex @ C @ D )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B @ D ) ) ) ) ).
% add_mono
thf(fact_944_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_945_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_946_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_947_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I @ J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_948_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_949_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_950_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_951_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( I = J )
& ( ord_less_eq_complex @ K @ L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_952_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_953_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_954_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_955_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: complex,J: complex,K: complex,L: complex] :
( ( ( ord_less_eq_complex @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ I @ K ) @ ( plus_plus_complex @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_956_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_957_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_958_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_959_pth__d,axiom,
! [X: complex] :
( ( plus_plus_complex @ X @ zero_zero_complex )
= X ) ).
% pth_d
thf(fact_960_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_961_pth__7_I1_J,axiom,
! [X: complex] :
( ( plus_plus_complex @ zero_zero_complex @ X )
= X ) ).
% pth_7(1)
thf(fact_962_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_963_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_964_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_965_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_966_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_967_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_968_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_969_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_970_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_971_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_972_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_973_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_974_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_975_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_976_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_977_verit__negate__coefficient_I2_J,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ B )
=> ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_978_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_979_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_980_less__minus__iff,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( ord_less_complex @ B @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% less_minus_iff
thf(fact_981_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_982_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_983_minus__less__iff,axiom,
! [A: complex,B: complex] :
( ( ord_less_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_984_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_985_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_986_translation__Compl,axiom,
! [A: real,T: set_real] :
( ( image_real_real @ ( plus_plus_real @ A ) @ ( uminus612125837232591019t_real @ T ) )
= ( uminus612125837232591019t_real @ ( image_real_real @ ( plus_plus_real @ A ) @ T ) ) ) ).
% translation_Compl
thf(fact_987_translation__Compl,axiom,
! [A: complex,T: set_complex] :
( ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ ( uminus8566677241136511917omplex @ T ) )
= ( uminus8566677241136511917omplex @ ( image_1468599708987790691omplex @ ( plus_plus_complex @ A ) @ T ) ) ) ).
% translation_Compl
thf(fact_988_translation__Compl,axiom,
! [A: int,T: set_int] :
( ( image_int_int @ ( plus_plus_int @ A ) @ ( uminus1532241313380277803et_int @ T ) )
= ( uminus1532241313380277803et_int @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).
% translation_Compl
thf(fact_989_group__cancel_Oneg1,axiom,
! [A2: complex,K: complex,A: complex] :
( ( A2
= ( plus_plus_complex @ K @ A ) )
=> ( ( uminus1482373934393186551omplex @ A2 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_990_group__cancel_Oneg1,axiom,
! [A2: real,K: real,A: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_991_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_992_add_Oinverse__distrib__swap,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_993_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_994_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_995_scaleR__right__distrib,axiom,
! [A: real,X: complex,Y: complex] :
( ( real_V2046097035970521341omplex @ A @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y ) ) ) ).
% scaleR_right_distrib
thf(fact_996_scaleR__right__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).
% scaleR_right_distrib
thf(fact_997_Reals__add,axiom,
! [A: real,B: real] :
( ( member_real @ A @ real_V470468836141973256s_real )
=> ( ( member_real @ B @ real_V470468836141973256s_real )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).
% Reals_add
thf(fact_998_Reals__add,axiom,
! [A: complex,B: complex] :
( ( member_complex @ A @ real_V2521375963428798218omplex )
=> ( ( member_complex @ B @ real_V2521375963428798218omplex )
=> ( member_complex @ ( plus_plus_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).
% Reals_add
thf(fact_999_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_1000_neg__less__divideR__eq,axiom,
! [C: real,A: complex,B: complex] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_complex @ A @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ C ) @ B ) )
= ( ord_less_complex @ B @ ( real_V2046097035970521341omplex @ C @ A ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_1001_neg__less__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
= ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_1002_neg__divideR__less__eq,axiom,
! [C: real,B: complex,A: complex] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ C ) @ B ) @ A )
= ( ord_less_complex @ ( real_V2046097035970521341omplex @ C @ A ) @ B ) ) ) ).
% neg_divideR_less_eq
thf(fact_1003_neg__divideR__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% neg_divideR_less_eq
thf(fact_1004_pos__less__divideR__eq,axiom,
! [C: real,A: complex,B: complex] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_complex @ A @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ C ) @ B ) )
= ( ord_less_complex @ ( real_V2046097035970521341omplex @ C @ A ) @ B ) ) ) ).
% pos_less_divideR_eq
thf(fact_1005_pos__less__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% pos_less_divideR_eq
thf(fact_1006_pos__divideR__less__eq,axiom,
! [C: real,B: complex,A: complex] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ C ) @ B ) @ A )
= ( ord_less_complex @ B @ ( real_V2046097035970521341omplex @ C @ A ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_1007_pos__divideR__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
= ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_1008_Arctan__in__Reals,axiom,
! [Z: complex] :
( ( member_complex @ Z @ real_V2521375963428798218omplex )
=> ( member_complex @ ( complex_Arctan @ Z ) @ real_V2521375963428798218omplex ) ) ).
% Arctan_in_Reals
thf(fact_1009_neg__minus__divideR__less__eq,axiom,
! [C: real,B: complex,A: complex] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_complex @ ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
= ( ord_less_complex @ ( real_V2046097035970521341omplex @ C @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_1010_neg__minus__divideR__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_1011_neg__less__minus__divideR__eq,axiom,
! [C: real,A: complex,B: complex] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_complex @ A @ ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ C ) @ B ) ) )
= ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ ( real_V2046097035970521341omplex @ C @ A ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_1012_neg__less__minus__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_1013_pos__minus__divideR__less__eq,axiom,
! [C: real,B: complex,A: complex] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_complex @ ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
= ( ord_less_complex @ ( uminus1482373934393186551omplex @ B ) @ ( real_V2046097035970521341omplex @ C @ A ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_1014_pos__minus__divideR__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_1015_pos__less__minus__divideR__eq,axiom,
! [C: real,A: complex,B: complex] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_complex @ A @ ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ C ) @ B ) ) )
= ( ord_less_complex @ ( real_V2046097035970521341omplex @ C @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_1016_pos__less__minus__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_1017_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1018_add__decreasing,axiom,
! [A: complex,C: complex,B: complex] :
( ( ord_less_eq_complex @ A @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ C @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1019_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1020_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1021_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1022_add__increasing,axiom,
! [A: complex,B: complex,C: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ B @ C )
=> ( ord_less_eq_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1023_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1024_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1025_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1026_add__decreasing2,axiom,
! [C: complex,A: complex,B: complex] :
( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ A @ B )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1027_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1028_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1029_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1030_add__increasing2,axiom,
! [C: complex,B: complex,A: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ( ord_less_eq_complex @ B @ A )
=> ( ord_less_eq_complex @ B @ ( plus_plus_complex @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1031_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1032_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1033_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1034_add__nonneg__nonneg,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1035_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1036_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1037_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_1038_add__nonpos__nonpos,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_complex @ A @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( plus_plus_complex @ A @ B ) @ zero_zero_complex ) ) ) ).
% add_nonpos_nonpos
thf(fact_1039_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1040_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1041_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1042_add__nonneg__eq__0__iff,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ Y )
=> ( ( ( plus_plus_complex @ X @ Y )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1043_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1044_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1045_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1046_add__nonpos__eq__0__iff,axiom,
! [X: complex,Y: complex] :
( ( ord_less_eq_complex @ X @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ Y @ zero_zero_complex )
=> ( ( ( plus_plus_complex @ X @ Y )
= zero_zero_complex )
= ( ( X = zero_zero_complex )
& ( Y = zero_zero_complex ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1047_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1048_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1049_neg__eq__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1050_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1051_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1052_eq__neg__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1053_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1054_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1055_add_Oinverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_1056_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_1057_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_1058_ab__group__add__class_Oab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1059_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1060_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1061_add__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% add_eq_0_iff
thf(fact_1062_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_1063_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_1064_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1065_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1066_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_1067_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1068_positive__imp__inverse__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_1069_Multiseries__Expansion__Bounds_Oneg__imp__inverse__neg,axiom,
! [F2: real > real,X5: real] :
( ( ord_less_real @ ( F2 @ X5 ) @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ ( F2 @ X5 ) ) @ zero_zero_real ) ) ).
% Multiseries_Expansion_Bounds.neg_imp_inverse_neg
thf(fact_1070_Multiseries__Expansion__Bounds_Opos__imp__inverse__pos,axiom,
! [F2: real > real,X5: real] :
( ( ord_less_real @ zero_zero_real @ ( F2 @ X5 ) )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( F2 @ X5 ) ) ) ) ).
% Multiseries_Expansion_Bounds.pos_imp_inverse_pos
thf(fact_1071_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% not_exp_less_zero
thf(fact_1072_exp__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% exp_gt_zero
thf(fact_1073_exp__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X4: real] :
( ( exp_real @ X4 )
= Y ) ) ).
% exp_total
thf(fact_1074_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1075_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1076_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1077_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1078_cmod__plus__Re__le__0__iff,axiom,
! [Z: complex] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
= ( ( re @ Z )
= ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_1079_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_1080_abs__real__def,axiom,
( abs_abs_real
= ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% abs_real_def
thf(fact_1081_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1082_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1083_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1084_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1085_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1086_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1087_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1088_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1089_complex__cnj__add,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% complex_cnj_add
thf(fact_1090_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1091_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1092_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1093_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1094_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1095_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1096_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1097_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1098_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1099_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1100_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1101_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1102_plus__complex_Osimps_I1_J,axiom,
! [X: complex,Y: complex] :
( ( re @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% plus_complex.simps(1)
thf(fact_1103_plus__complex_Osimps_I2_J,axiom,
! [X: complex,Y: complex] :
( ( im @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% plus_complex.simps(2)
thf(fact_1104_complex__add,axiom,
! [A: real,B: real,C: real,D: real] :
( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
= ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% complex_add
thf(fact_1105_less__complex__def,axiom,
( ord_less_complex
= ( ^ [X2: complex,Y2: complex] :
( ( ord_less_real @ ( re @ X2 ) @ ( re @ Y2 ) )
& ( ( im @ X2 )
= ( im @ Y2 ) ) ) ) ) ).
% less_complex_def
thf(fact_1106_Complex__add__complex__of__real,axiom,
! [X: real,Y: real,R: real] :
( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
= ( complex2 @ ( plus_plus_real @ X @ R ) @ Y ) ) ).
% Complex_add_complex_of_real
thf(fact_1107_complex__of__real__add__Complex,axiom,
! [R: real,X: real,Y: real] :
( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
= ( complex2 @ ( plus_plus_real @ R @ X ) @ Y ) ) ).
% complex_of_real_add_Complex
thf(fact_1108_plus__complex_Ocode,axiom,
( plus_plus_complex
= ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ) ) ).
% plus_complex.code
thf(fact_1109_cmod__add__real__less,axiom,
! [Z: complex,R: real] :
( ( ( im @ Z )
!= zero_zero_real )
=> ( ( R != zero_zero_real )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ ( real_V4546457046886955230omplex @ R ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( abs_abs_real @ R ) ) ) ) ) ).
% cmod_add_real_less
thf(fact_1110_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
= ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_1111_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% real_0_le_add_iff
thf(fact_1112_cmod__diff__real__less,axiom,
! [Z: complex,X: real] :
( ( ( im @ Z )
!= zero_zero_real )
=> ( ( X != zero_zero_real )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ ( real_V4546457046886955230omplex @ X ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( abs_abs_real @ X ) ) ) ) ) ).
% cmod_diff_real_less
thf(fact_1113_continuous__on__Arctan,axiom,
! [S2: set_complex] :
( ! [Z4: complex] :
( ( member_complex @ Z4 @ S2 )
=> ( ( ( re @ Z4 )
= zero_zero_real )
=> ( ord_less_real @ ( abs_abs_real @ ( im @ Z4 ) ) @ one_one_real ) ) )
=> ( topolo9015423870875150044omplex @ S2 @ complex_Arctan ) ) ).
% continuous_on_Arctan
thf(fact_1114_complex__cnj__diff,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% complex_cnj_diff
thf(fact_1115_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp_real @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ).
% exp_eq_one_iff
thf(fact_1116_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% exp_le_one_iff
thf(fact_1117_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% one_le_exp_iff
thf(fact_1118_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% exp_less_one_iff
thf(fact_1119_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_less_exp_iff
thf(fact_1120_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( artanh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_1121_exp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% exp_gt_one
thf(fact_1122_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_1123_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_1124_norm__exp__imaginary,axiom,
! [Z: complex] :
( ( ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) )
= one_one_real )
=> ( ( re @ Z )
= zero_zero_real ) ) ).
% norm_exp_imaginary
thf(fact_1125_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1126_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1127_complex__cnj__one,axiom,
( ( cnj @ one_one_complex )
= one_one_complex ) ).
% complex_cnj_one
thf(fact_1128_complex__cnj__one__iff,axiom,
! [Z: complex] :
( ( ( cnj @ Z )
= one_one_complex )
= ( Z = one_one_complex ) ) ).
% complex_cnj_one_iff
thf(fact_1129_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1130_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1131_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1132_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1133_nat0__intermed__int__val,axiom,
! [N2: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N2 ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1134_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1135_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1136_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1137_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1138_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X2: real,Y2: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).
% minus_real_def
thf(fact_1139_minus__complex_Osimps_I1_J,axiom,
! [X: complex,Y: complex] :
( ( re @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% minus_complex.simps(1)
thf(fact_1140_minus__complex_Osimps_I2_J,axiom,
! [X: complex,Y: complex] :
( ( im @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% minus_complex.simps(2)
thf(fact_1141_one__complex_Osimps_I2_J,axiom,
( ( im @ one_one_complex )
= zero_zero_real ) ).
% one_complex.simps(2)
thf(fact_1142_one__complex_Osimps_I1_J,axiom,
( ( re @ one_one_complex )
= one_one_real ) ).
% one_complex.simps(1)
thf(fact_1143_complex__diff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
= ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% complex_diff
thf(fact_1144_Bolzano,axiom,
! [A: real,B: real,P: real > real > $o] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [A6: real,B6: real,C2: real] :
( ( P @ A6 @ B6 )
=> ( ( P @ B6 @ C2 )
=> ( ( ord_less_eq_real @ A6 @ B6 )
=> ( ( ord_less_eq_real @ B6 @ C2 )
=> ( P @ A6 @ C2 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq_real @ A @ X4 )
=> ( ( ord_less_eq_real @ X4 @ B )
=> ? [D2: real] :
( ( ord_less_real @ zero_zero_real @ D2 )
& ! [A6: real,B6: real] :
( ( ( ord_less_eq_real @ A6 @ X4 )
& ( ord_less_eq_real @ X4 @ B6 )
& ( ord_less_real @ ( minus_minus_real @ B6 @ A6 ) @ D2 ) )
=> ( P @ A6 @ B6 ) ) ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Bolzano
thf(fact_1145_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1146_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1147_complex__mod__triangle__ineq2,axiom,
! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% complex_mod_triangle_ineq2
thf(fact_1148_one__complex_Ocode,axiom,
( one_one_complex
= ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% one_complex.code
thf(fact_1149_Complex__eq__1,axiom,
! [A: real,B: real] :
( ( ( complex2 @ A @ B )
= one_one_complex )
= ( ( A = one_one_real )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_1
thf(fact_1150_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ Y )
=> ? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
& ( ( exp_real @ X4 )
= Y ) ) ) ).
% lemma_exp_total
thf(fact_1151_minus__complex_Ocode,axiom,
( minus_minus_complex
= ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( minus_minus_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ) ) ).
% minus_complex.code
thf(fact_1152_Complex__eq__neg__1,axiom,
! [A: real,B: real] :
( ( ( complex2 @ A @ B )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( A
= ( uminus_uminus_real @ one_one_real ) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_neg_1
thf(fact_1153_lemma__interval,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
=> ( ( ord_less_eq_real @ A @ Y5 )
& ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1154_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F2: ( nat > real ) > nat > real,Q3: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F2 @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I3: nat] :
( ( Q3 @ I3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I3 ) )
& ( ord_less_eq_real @ ( X4 @ I3 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X5: nat > real,I2: nat] : ( ord_less_eq_nat @ ( L3 @ X5 @ I2 ) @ one_one_nat )
& ! [X5: nat > real,I2: nat] :
( ( ( P @ X5 )
& ( Q3 @ I2 )
& ( ( X5 @ I2 )
= zero_zero_real ) )
=> ( ( L3 @ X5 @ I2 )
= zero_zero_nat ) )
& ! [X5: nat > real,I2: nat] :
( ( ( P @ X5 )
& ( Q3 @ I2 )
& ( ( X5 @ I2 )
= one_one_real ) )
=> ( ( L3 @ X5 @ I2 )
= one_one_nat ) )
& ! [X5: nat > real,I2: nat] :
( ( ( P @ X5 )
& ( Q3 @ I2 )
& ( ( L3 @ X5 @ I2 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X5 @ I2 ) @ ( F2 @ X5 @ I2 ) ) )
& ! [X5: nat > real,I2: nat] :
( ( ( P @ X5 )
& ( Q3 @ I2 )
& ( ( L3 @ X5 @ I2 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F2 @ X5 @ I2 ) @ ( X5 @ I2 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1155_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1156_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1157_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1158_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1159_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1160_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1161_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1162_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1163_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1164_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1165_kuhn__lemma,axiom,
! [P2: nat,N2: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ! [X4: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I2 ) @ P2 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( Label @ X4 @ I3 )
= zero_zero_nat )
| ( ( Label @ X4 @ I3 )
= one_one_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I2 ) @ P2 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( X4 @ I3 )
= zero_zero_nat )
=> ( ( Label @ X4 @ I3 )
= zero_zero_nat ) ) ) )
=> ( ! [X4: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_nat @ ( X4 @ I2 ) @ P2 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ( X4 @ I3 )
= P2 )
=> ( ( Label @ X4 @ I3 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_nat @ ( Q2 @ I2 ) @ P2 ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ? [R3: nat > nat] :
( ! [J2: nat] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J2 ) @ ( R3 @ J2 ) )
& ( ord_less_eq_nat @ ( R3 @ J2 ) @ ( plus_plus_nat @ ( Q2 @ J2 ) @ one_one_nat ) ) ) )
& ? [S3: nat > nat] :
( ! [J2: nat] :
( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_eq_nat @ ( Q2 @ J2 ) @ ( S3 @ J2 ) )
& ( ord_less_eq_nat @ ( S3 @ J2 ) @ ( plus_plus_nat @ ( Q2 @ J2 ) @ one_one_nat ) ) ) )
& ( ( Label @ R3 @ I2 )
!= ( Label @ S3 @ I2 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1166_lemma__interval__lt,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D4: real] :
( ( ord_less_real @ zero_zero_real @ D4 )
& ! [Y5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
=> ( ( ord_less_real @ A @ Y5 )
& ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1167_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I1_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_real @ A @ B ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(1)
thf(fact_1168_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1169_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1170_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1171_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1172_gbinomial__series__aux_Oexhaust,axiom,
! [Abort: $o,Acc: real] :
( ( Abort
=> ( Acc != zero_zero_real ) )
=> ( ~ Abort
| ( Acc != zero_zero_real ) ) ) ).
% gbinomial_series_aux.exhaust
thf(fact_1173_Multiseries__Expansion_Oreal__eqI,axiom,
! [A: real,B: real] :
( ( ( minus_minus_real @ A @ B )
= zero_zero_real )
=> ( A = B ) ) ).
% Multiseries_Expansion.real_eqI
thf(fact_1174_Multiseries__Expansion_Ocompare__reals__diff__sgnD_I3_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
=> ( ord_less_real @ B @ A ) ) ).
% Multiseries_Expansion.compare_reals_diff_sgnD(3)
thf(fact_1175_continuous__on__Arcsin,axiom,
! [S2: set_complex] :
( ! [Z4: complex] :
( ( member_complex @ Z4 @ S2 )
=> ( ( ( im @ Z4 )
= zero_zero_real )
=> ( ord_less_real @ ( abs_abs_real @ ( re @ Z4 ) ) @ one_one_real ) ) )
=> ( topolo9015423870875150044omplex @ S2 @ complex_Arcsin ) ) ).
% continuous_on_Arcsin
thf(fact_1176_Arcsin__0,axiom,
( ( complex_Arcsin @ zero_zero_complex )
= zero_zero_complex ) ).
% Arcsin_0
thf(fact_1177_Arcsin__in__Reals,axiom,
! [Z: complex] :
( ( member_complex @ Z @ real_V2521375963428798218omplex )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ ( re @ Z ) ) @ one_one_real )
=> ( member_complex @ ( complex_Arcsin @ Z ) @ real_V2521375963428798218omplex ) ) ) ).
% Arcsin_in_Reals
thf(fact_1178_Im__Arcsin__of__real,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( im @ ( complex_Arcsin @ ( real_V4546457046886955230omplex @ X ) ) )
= zero_zero_real ) ) ).
% Im_Arcsin_of_real
thf(fact_1179_continuous__on__Arccos,axiom,
! [S2: set_complex] :
( ! [Z4: complex] :
( ( member_complex @ Z4 @ S2 )
=> ( ( ( im @ Z4 )
= zero_zero_real )
=> ( ord_less_real @ ( abs_abs_real @ ( re @ Z4 ) ) @ one_one_real ) ) )
=> ( topolo9015423870875150044omplex @ S2 @ complex_Arccos ) ) ).
% continuous_on_Arccos
thf(fact_1180_cmod__cos__le__exp,axiom,
! [U: real,Z: complex] :
( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( cos_complex @ ( real_V2046097035970521341omplex @ U @ Z ) ) ) @ ( exp_real @ ( abs_abs_real @ ( im @ Z ) ) ) ) ) ) ).
% cmod_cos_le_exp
thf(fact_1181_Arccos__1,axiom,
( ( complex_Arccos @ one_one_complex )
= zero_zero_complex ) ).
% Arccos_1
thf(fact_1182_norm__cos__le,axiom,
! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( cos_complex @ Z ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% norm_cos_le
thf(fact_1183_Im__Arccos__bound,axiom,
! [W: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ ( complex_Arccos @ W ) ) ) @ ( real_V1022390504157884413omplex @ W ) ) ).
% Im_Arccos_bound
thf(fact_1184_Arccos__in__Reals,axiom,
! [Z: complex] :
( ( member_complex @ Z @ real_V2521375963428798218omplex )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ ( re @ Z ) ) @ one_one_real )
=> ( member_complex @ ( complex_Arccos @ Z ) @ real_V2521375963428798218omplex ) ) ) ).
% Arccos_in_Reals
thf(fact_1185_Im__Arccos__of__real,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( im @ ( complex_Arccos @ ( real_V4546457046886955230omplex @ X ) ) )
= zero_zero_real ) ) ).
% Im_Arccos_of_real
thf(fact_1186_cmod__sin__le__exp,axiom,
! [U: real,Z: complex] :
( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( sin_complex @ ( real_V2046097035970521341omplex @ U @ Z ) ) ) @ ( exp_real @ ( abs_abs_real @ ( im @ Z ) ) ) ) ) ) ).
% cmod_sin_le_exp
thf(fact_1187_Arccos__unique,axiom,
! [Z: complex,W: complex] :
( ( ( cos_complex @ Z )
= W )
=> ( ( ( ( ord_less_real @ zero_zero_real @ ( re @ Z ) )
& ( ord_less_real @ ( re @ Z ) @ pi ) )
| ( ( ( re @ Z )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ Z ) ) )
| ( ( ( re @ Z )
= pi )
& ( ord_less_eq_real @ ( im @ Z ) @ zero_zero_real ) ) )
=> ( ( complex_Arccos @ W )
= Z ) ) ) ).
% Arccos_unique
thf(fact_1188_cos__pi,axiom,
( ( cos_real @ pi )
= ( uminus_uminus_real @ one_one_real ) ) ).
% cos_pi
thf(fact_1189_cos__periodic__pi2,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi2
thf(fact_1190_cos__periodic__pi,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi
thf(fact_1191_cos__pi__minus,axiom,
! [X: real] :
( ( cos_real @ ( minus_minus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_pi_minus
thf(fact_1192_cos__minus__pi,axiom,
! [X: real] :
( ( cos_real @ ( minus_minus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_minus_pi
thf(fact_1193_real__cos__eq,axiom,
! [X: real] :
( ( re @ ( cos_complex @ ( real_V4546457046886955230omplex @ X ) ) )
= ( cos_real @ X ) ) ).
% real_cos_eq
thf(fact_1194_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_1195_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X4 )
& ( ord_less_eq_real @ X4 @ pi )
& ( ( cos_real @ X4 )
= Y )
& ! [Y5: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
& ( ord_less_eq_real @ Y5 @ pi )
& ( ( cos_real @ Y5 )
= Y ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% cos_total
thf(fact_1196_cos__monotone__0__pi__le,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_1197_cos__mono__le__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_1198_cos__inj__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ( cos_real @ X )
= ( cos_real @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_1199_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_1200_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_1201_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ pi )
=> ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
= ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_1202_cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% cos_le_one
thf(fact_1203_pi__neq__zero,axiom,
pi != zero_zero_real ).
% pi_neq_zero
thf(fact_1204_Re__sin__pos,axiom,
! [Z: complex] :
( ( ord_less_real @ zero_zero_real @ ( re @ Z ) )
=> ( ( ord_less_real @ ( re @ Z ) @ pi )
=> ( ord_less_real @ zero_zero_real @ ( re @ ( sin_complex @ Z ) ) ) ) ) ).
% Re_sin_pos
thf(fact_1205_pi__not__less__zero,axiom,
~ ( ord_less_real @ pi @ zero_zero_real ) ).
% pi_not_less_zero
thf(fact_1206_pi__gt__zero,axiom,
ord_less_real @ zero_zero_real @ pi ).
% pi_gt_zero
thf(fact_1207_pi__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ pi ).
% pi_ge_zero
thf(fact_1208_Im__sin__nonneg2,axiom,
! [Z: complex] :
( ( ( re @ Z )
= pi )
=> ( ( ord_less_eq_real @ ( im @ Z ) @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( im @ ( sin_complex @ Z ) ) ) ) ) ).
% Im_sin_nonneg2
thf(fact_1209_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% cos_ge_minus_one
thf(fact_1210_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% abs_cos_le_one
thf(fact_1211_Im__sin__nonneg,axiom,
! [Z: complex] :
( ( ( re @ Z )
= zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( im @ Z ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( im @ ( sin_complex @ Z ) ) ) ) ) ).
% Im_sin_nonneg
thf(fact_1212_Re__Arccos__bound,axiom,
! [Z: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ ( complex_Arccos @ Z ) ) ) @ pi ) ).
% Re_Arccos_bound
thf(fact_1213_Re__Arcsin__bound,axiom,
! [Z: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ ( complex_Arcsin @ Z ) ) ) @ pi ) ).
% Re_Arcsin_bound
thf(fact_1214_norm__Arccos__bounded,axiom,
! [W: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( complex_Arccos @ W ) ) @ ( plus_plus_real @ pi @ ( real_V1022390504157884413omplex @ W ) ) ) ).
% norm_Arccos_bounded
thf(fact_1215_Arccos__minus1,axiom,
( ( complex_Arccos @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( real_V4546457046886955230omplex @ pi ) ) ).
% Arccos_minus1
thf(fact_1216_Re__Arccos__bounds,axiom,
! [Z: complex] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( re @ ( complex_Arccos @ Z ) ) )
& ( ord_less_eq_real @ ( re @ ( complex_Arccos @ Z ) ) @ pi ) ) ).
% Re_Arccos_bounds
thf(fact_1217_Re__Arcsin__bounds,axiom,
! [Z: complex] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( re @ ( complex_Arcsin @ Z ) ) )
& ( ord_less_eq_real @ ( re @ ( complex_Arcsin @ Z ) ) @ pi ) ) ).
% Re_Arcsin_bounds
thf(fact_1218_Arccos__bounds,axiom,
! [Z: complex] :
( ( ord_less_real @ ( abs_abs_real @ ( re @ Z ) ) @ one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( re @ ( complex_Arccos @ Z ) ) )
& ( ord_less_real @ ( re @ ( complex_Arccos @ Z ) ) @ pi ) ) ) ).
% Arccos_bounds
thf(fact_1219_Arccos__cos,axiom,
! [Z: complex] :
( ( ( ( ord_less_real @ zero_zero_real @ ( re @ Z ) )
& ( ord_less_real @ ( re @ Z ) @ pi ) )
| ( ( ( re @ Z )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ Z ) ) )
| ( ( ( re @ Z )
= pi )
& ( ord_less_eq_real @ ( im @ Z ) @ zero_zero_real ) ) )
=> ( ( complex_Arccos @ ( cos_complex @ Z ) )
= Z ) ) ).
% Arccos_cos
thf(fact_1220_sin__pi,axiom,
( ( sin_real @ pi )
= zero_zero_real ) ).
% sin_pi
thf(fact_1221_sin__pi__minus,axiom,
! [X: real] :
( ( sin_real @ ( minus_minus_real @ pi @ X ) )
= ( sin_real @ X ) ) ).
% sin_pi_minus
thf(fact_1222_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi2
thf(fact_1223_sin__periodic__pi,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi
thf(fact_1224_sin__minus__pi,axiom,
! [X: real] :
( ( sin_real @ ( minus_minus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_minus_pi
thf(fact_1225_real__sin__eq,axiom,
! [X: real] :
( ( re @ ( sin_complex @ ( real_V4546457046886955230omplex @ X ) ) )
= ( sin_real @ X ) ) ).
% real_sin_eq
thf(fact_1226_norm__cos__sin,axiom,
! [T: real] :
( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
= one_one_real ) ).
% norm_cos_sin
thf(fact_1227_sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% sin_le_one
thf(fact_1228_abs__sin__x__le__abs__x,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% abs_sin_x_le_abs_x
thf(fact_1229_sin__x__le__x,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% sin_x_le_x
thf(fact_1230_sin__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_ge_zero
thf(fact_1231_sin__x__ge__neg__x,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% sin_x_ge_neg_x
thf(fact_1232_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ pi )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_1233_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% sin_ge_minus_one
thf(fact_1234_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% abs_sin_le_one
thf(fact_1235_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( ord_less_real @ X @ pi )
=> ( ( ( sin_real @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ) ) ).
% sin_eq_0_pi
thf(fact_1236_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
=> ( ( ( sin_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% sin_zero_pi_iff
thf(fact_1237_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
=> ( ( abs_abs_real @ ( cos_real @ X ) )
= one_one_real ) ) ).
% sin_zero_abs_cos_one
thf(fact_1238_sincos__principal__value,axiom,
! [X: real] :
? [Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
& ( ord_less_eq_real @ Y4 @ pi )
& ( ( sin_real @ Y4 )
= ( sin_real @ X ) )
& ( ( cos_real @ Y4 )
= ( cos_real @ X ) ) ) ).
% sincos_principal_value
thf(fact_1239_Re__Im__le__cmod,axiom,
! [W: complex,Phi: real] : ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( im @ W ) @ ( sin_real @ Phi ) ) @ ( times_times_real @ ( re @ W ) @ ( cos_real @ Phi ) ) ) @ ( real_V1022390504157884413omplex @ W ) ) ).
% Re_Im_le_cmod
thf(fact_1240_nat__ivt__aux,axiom,
! [N2: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F2 @ N2 ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1241_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1242_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1243_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1244_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1245_polar__Ex,axiom,
! [X: real,Y: real] :
? [R3: real,A6: real] :
( ( X
= ( times_times_real @ R3 @ ( cos_real @ A6 ) ) )
& ( Y
= ( times_times_real @ R3 @ ( sin_real @ A6 ) ) ) ) ).
% polar_Ex
thf(fact_1246_scaleR__complex_Osimps_I1_J,axiom,
! [R: real,X: complex] :
( ( re @ ( real_V2046097035970521341omplex @ R @ X ) )
= ( times_times_real @ R @ ( re @ X ) ) ) ).
% scaleR_complex.simps(1)
thf(fact_1247_scaleR__complex_Osimps_I2_J,axiom,
! [R: real,X: complex] :
( ( im @ ( real_V2046097035970521341omplex @ R @ X ) )
= ( times_times_real @ R @ ( im @ X ) ) ) ).
% scaleR_complex.simps(2)
thf(fact_1248_real__scaleR__def,axiom,
real_V1485227260804924795R_real = times_times_real ).
% real_scaleR_def
thf(fact_1249_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1250_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1251_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1252_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1253_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1254_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1255_old_Onat_Odistinct_I2_J,axiom,
! [Nat: nat] :
( ( suc @ Nat )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1256_old_Onat_Odistinct_I1_J,axiom,
! [Nat: nat] :
( zero_zero_nat
!= ( suc @ Nat ) ) ).
% old.nat.distinct(1)
thf(fact_1257_nat_OdiscI,axiom,
! [Nat2: nat,X22: nat] :
( ( Nat2
= ( suc @ X22 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1258_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1259_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1260_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1261_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1262_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1263_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_1264_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_1265_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1266_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1267_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1268_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1269_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1270_complex__scaleR,axiom,
! [R: real,A: real,B: real] :
( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B ) )
= ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B ) ) ) ).
% complex_scaleR
thf(fact_1271_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1272_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1273_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( comp_c3333796836230738283l_real @ ( comp_c2063761206571265261omplex @ re @ cotang8298477626502807258omplex ) @ real_V4546457046886955230omplex )
= cotang1502006655779026648d_real ) ).
%------------------------------------------------------------------------------