TPTP Problem File: SLH0759^1.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_00333_012710__14057774_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1427 ( 805 unt; 148 typ; 0 def)
% Number of atoms : 2800 (2065 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 8642 ( 400 ~; 72 |; 115 &;7322 @)
% ( 0 <=>; 733 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 376 ( 376 >; 0 *; 0 +; 0 <<)
% Number of symbols : 134 ( 131 usr; 23 con; 0-3 aty)
% Number of variables : 2778 ( 186 ^;2537 !; 55 ?;2778 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:58:33.550
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
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__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
formal_Power_fps_nat: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
formal_Power_fps_int: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Complex__Ocomplex_J,type,
filter_complex: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
filter_real: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $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 (131)
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_Filter_Oat__top_001t__Nat__Onat,type,
at_top_nat: filter_nat ).
thf(sy_c_Filter_Ofilterlim_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
filter8330067395343389202omplex: ( complex > complex ) > filter_complex > filter_complex > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Complex__Ocomplex_001t__Nat__Onat,type,
filter1319825749481401652ex_nat: ( complex > nat ) > filter_nat > filter_complex > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Complex__Ocomplex_001t__Real__Oreal,type,
filter8559879285478333968x_real: ( complex > real ) > filter_real > filter_complex > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Complex__Ocomplex,type,
filter6923414461901439796omplex: ( nat > complex ) > filter_complex > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Complex__Ocomplex,type,
filter8506290784974013328omplex: ( real > complex ) > filter_complex > filter_real > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Nat__Onat,type,
filterlim_real_nat: ( real > nat ) > filter_nat > filter_real > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Complex__Ocomplex,type,
formal3666518339620930912omplex: formal670952693614245302omplex > nat > complex ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Int__Oint,type,
formal3717847055265219294th_int: formal_Power_fps_int > nat > int ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Nat__Onat,type,
formal3720337525774269570th_nat: formal_Power_fps_nat > nat > nat ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Real__Oreal,type,
formal2580924720334399070h_real: formal3361831859752904756s_real > nat > real ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Complex__Ocomplex,type,
formal1655152611307539683omplex: formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Int__Oint,type,
formal812433016830480481XD_int: formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Nat__Onat,type,
formal814923487339530757XD_nat: formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Real__Oreal,type,
formal4292469010823635553D_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__XDp_001t__Complex__Ocomplex,type,
formal5989188765539143467omplex: complex > formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__XDp_001t__Int__Oint,type,
formal9195297484582036137Dp_int: int > formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__XDp_001t__Nat__Onat,type,
formal9197787955091086413Dp_nat: nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Formal__Power__Series_Ofps__XDp_001t__Real__Oreal,type,
formal2839450981996073129p_real: real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__hypergeo_001t__Complex__Ocomplex,type,
formal7789817089755916220omplex: list_complex > list_complex > complex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__hypergeo_001t__Real__Oreal,type,
formal6618874005373735610o_real: list_real > list_real > real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__ln_001t__Complex__Ocomplex,type,
formal6928690614366948857omplex: complex > formal670952693614245302omplex ).
thf(sy_c_Formal__Power__Series_Ofps__ln_001t__Real__Oreal,type,
formal8688746759596762231n_real: real > formal3361831859752904756s_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_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__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
one_on1590755018477040891omplex: formal670952693614245302omplex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
one_on8395608022581818233ps_int: formal_Power_fps_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
one_on3350087005236239133ps_nat: formal_Power_fps_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
one_on8598947968683843321s_real: formal3361831859752904756s_real ).
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__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
plus_p8472957120637115327omplex: formal670952693614245302omplex > formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
plus_p1865620787042574909ps_int: formal_Power_fps_int > formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
plus_p6043471806551771617ps_nat: formal_Power_fps_nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
plus_p6008488439947570109s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
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_Oplus__class_Oplus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
plus_p7052360327008956141omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Int__Oint_J,type,
plus_plus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
times_1444617028055533883omplex: formal670952693614245302omplex > formal670952693614245302omplex > formal670952693614245302omplex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
times_3091854549176928185ps_int: formal_Power_fps_int > formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
times_7269705568686124893ps_nat: formal_Power_fps_nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
times_7561426564079326009s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Complex__Ocomplex_J,type,
times_6048082448287401577omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Int__Oint_J,type,
times_times_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_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_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__Int__Oint_J,type,
zero_z4353722679246354365ps_int: formal_Power_fps_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
zero_z8531573698755551073ps_nat: formal_Power_fps_nat ).
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__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
ring_1_Ints_complex: set_complex ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
ring_1_Ints_int: set_int ).
thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
ring_1_Ints_real: set_real ).
thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Complex__Ocomplex,type,
bfun_nat_complex: ( nat > complex ) > filter_nat > $o ).
thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
semiri8948773824294531479omplex: nat > formal670952693614245302omplex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
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thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
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_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > 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_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Formal____Power____Series__Ofps_It__Complex__Ocomplex_J,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Complex__Ocomplex,type,
topolo2444363109189100025omplex: complex > filter_complex ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Nat__Onat,type,
topolo8926549440605965083ds_nat: nat > filter_nat ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_x,type,
x: complex ).
% Relevant facts (1269)
thf(fact_0__092_060open_062x_A_092_060noteq_062_A0_092_060close_062,axiom,
x != zero_zero_complex ).
% \<open>x \<noteq> 0\<close>
thf(fact_1__092_060open_062x_A_L_A1_A_092_060noteq_062_A0_092_060close_062,axiom,
( ( plus_plus_complex @ x @ one_one_complex )
!= zero_zero_complex ) ).
% \<open>x + 1 \<noteq> 0\<close>
thf(fact_2_assms,axiom,
~ ( member_complex @ x @ ring_1_Ints_complex ) ).
% assms
thf(fact_3_tendsto__zero__divide__iff,axiom,
! [C: complex,A: nat > complex] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ ( A @ N ) @ C )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat )
= ( filter6923414461901439796omplex @ A @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ at_top_nat ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_4_tendsto__zero__divide__iff,axiom,
! [C: real,A: nat > real] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ ( A @ N ) @ C )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
= ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_5_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_6_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% of_nat_Suc
thf(fact_7_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% of_nat_Suc
thf(fact_8_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_9_LIMSEQ__Suc__n__over__n,axiom,
( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri8010041392384452111omplex @ N ) )
@ ( topolo2444363109189100025omplex @ one_one_complex )
@ at_top_nat ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_10_LIMSEQ__Suc__n__over__n,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri5074537144036343181t_real @ N ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_11_LIMSEQ__n__over__Suc__n,axiom,
( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) )
@ ( topolo2444363109189100025omplex @ one_one_complex )
@ at_top_nat ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_12_LIMSEQ__n__over__Suc__n,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_13_lim__1__over__n,axiom,
( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ one_one_complex @ ( semiri8010041392384452111omplex @ N ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat ) ).
% lim_1_over_n
thf(fact_14_lim__1__over__n,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_1_over_n
thf(fact_15_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_16_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_17_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_18_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_19_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_20_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_21_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A @ B ) )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_22_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_23_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_24_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_25_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_26_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_27_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_28_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_29__C_K_K_C,axiom,
! [X: complex,N2: nat] :
( ~ ( member_complex @ X @ ring_1_Ints_complex )
=> ( ( plus_plus_complex @ X @ ( semiri8010041392384452111omplex @ N2 ) )
!= zero_zero_complex ) ) ).
% "**"
thf(fact_30_add__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ M @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc_right
thf(fact_31_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_32_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_33_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= ( semiri8010041392384452111omplex @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_34_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_35_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_36_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_37_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_38_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_39_divide__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_40_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_41_divide__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ C @ A )
= ( divide1717551699836669952omplex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_42_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_43_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_44_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_45_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_46_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_47_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_48_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_49_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_50_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_51_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_52_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_53_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= zero_zero_complex )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_54_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_55_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_56_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_57_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_58_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_59_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_60_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_61_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_62_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_63_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_64_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_65_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_66_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_67_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_68_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_69_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri8010041392384452111omplex @ N2 )
= one_one_complex )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_70_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_71_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_72_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_73_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_74_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_75_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_76_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_77_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_78_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_79_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_80_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_81_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% of_nat_add
thf(fact_82_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_add
thf(fact_83_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_add
thf(fact_84_of__nat__add,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_add
thf(fact_85_Ints__add__iff2,axiom,
! [Y: complex,X: complex] :
( ( member_complex @ Y @ ring_1_Ints_complex )
=> ( ( member_complex @ ( plus_plus_complex @ X @ Y ) @ ring_1_Ints_complex )
= ( member_complex @ X @ ring_1_Ints_complex ) ) ) ).
% Ints_add_iff2
thf(fact_86_Ints__add__iff2,axiom,
! [Y: int,X: int] :
( ( member_int @ Y @ ring_1_Ints_int )
=> ( ( member_int @ ( plus_plus_int @ X @ Y ) @ ring_1_Ints_int )
= ( member_int @ X @ ring_1_Ints_int ) ) ) ).
% Ints_add_iff2
thf(fact_87_Ints__add__iff2,axiom,
! [Y: real,X: real] :
( ( member_real @ Y @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X @ Y ) @ ring_1_Ints_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff2
thf(fact_88_Ints__add__iff1,axiom,
! [X: complex,Y: complex] :
( ( member_complex @ X @ ring_1_Ints_complex )
=> ( ( member_complex @ ( plus_plus_complex @ X @ Y ) @ ring_1_Ints_complex )
= ( member_complex @ Y @ ring_1_Ints_complex ) ) ) ).
% Ints_add_iff1
thf(fact_89_Ints__add__iff1,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ring_1_Ints_int )
=> ( ( member_int @ ( plus_plus_int @ X @ Y ) @ ring_1_Ints_int )
= ( member_int @ Y @ ring_1_Ints_int ) ) ) ).
% Ints_add_iff1
thf(fact_90_Ints__add__iff1,axiom,
! [X: real,Y: real] :
( ( member_real @ X @ ring_1_Ints_real )
=> ( ( member_real @ ( plus_plus_real @ X @ Y ) @ ring_1_Ints_real )
= ( member_real @ Y @ ring_1_Ints_real ) ) ) ).
% Ints_add_iff1
thf(fact_91_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_92_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_93_add__is__1,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_94_one__is__add,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N2 ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_95_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_96_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_97_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_98_lim__inverse__n_H,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_inverse_n'
thf(fact_99_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X3: complex] : ( member_complex @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_101_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_102_add__Suc__shift,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_103_add__Suc,axiom,
! [M: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% add_Suc
thf(fact_104_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_105_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_106_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_107_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_108_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_109_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_110_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_111_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_112_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_113_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_114_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_115_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_116_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_117_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_118_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_119_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_120_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_121_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_122_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_123_add__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_124_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_125_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_126_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_127_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
!= zero_zero_complex ) ).
% of_nat_neq_0
thf(fact_128_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_129_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_130_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_131_tendsto__add__const__iff,axiom,
! [C: complex,F: nat > complex,D: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( plus_plus_complex @ C @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( plus_plus_complex @ C @ D ) )
@ F2 )
= ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ D ) @ F2 ) ) ).
% tendsto_add_const_iff
thf(fact_132_tendsto__add__const__iff,axiom,
! [C: real,F: nat > real,D: real,F2: filter_nat] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( plus_plus_real @ C @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ C @ D ) )
@ F2 )
= ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ D ) @ F2 ) ) ).
% tendsto_add_const_iff
thf(fact_133_tendsto__add,axiom,
! [F: nat > nat,A: nat,F2: filter_nat,G: nat > nat,B: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ A ) @ F2 )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ B ) @ F2 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( plus_plus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo8926549440605965083ds_nat @ ( plus_plus_nat @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_add
thf(fact_134_tendsto__add,axiom,
! [F: nat > complex,A: complex,F2: filter_nat,G: nat > complex,B: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ B ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( plus_plus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( plus_plus_complex @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_add
thf(fact_135_tendsto__add,axiom,
! [F: nat > real,A: real,F2: filter_nat,G: nat > real,B: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( plus_plus_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( plus_plus_real @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_add
thf(fact_136_tendsto__add__zero,axiom,
! [F: nat > nat,F2: filter_nat,G: nat > nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ zero_zero_nat ) @ F2 )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ zero_zero_nat ) @ F2 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( plus_plus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo8926549440605965083ds_nat @ zero_zero_nat )
@ F2 ) ) ) ).
% tendsto_add_zero
thf(fact_137_tendsto__add__zero,axiom,
! [F: nat > complex,F2: filter_nat,G: nat > complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( plus_plus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 ) ) ) ).
% tendsto_add_zero
thf(fact_138_tendsto__add__zero,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( plus_plus_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ) ).
% tendsto_add_zero
thf(fact_139_tendsto__divide__zero,axiom,
! [F: nat > complex,F2: filter_nat,C: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( divide1717551699836669952omplex @ ( F @ X3 ) @ C )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 ) ) ).
% tendsto_divide_zero
thf(fact_140_tendsto__divide__zero,axiom,
! [F: nat > real,F2: filter_nat,C: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( divide_divide_real @ ( F @ X3 ) @ C )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% tendsto_divide_zero
thf(fact_141_tendsto__divide,axiom,
! [F: nat > complex,A: complex,F2: filter_nat,G: nat > complex,B: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ B ) @ F2 )
=> ( ( B != zero_zero_complex )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( divide1717551699836669952omplex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( divide1717551699836669952omplex @ A @ B ) )
@ F2 ) ) ) ) ).
% tendsto_divide
thf(fact_142_tendsto__divide,axiom,
! [F: nat > real,A: real,F2: filter_nat,G: nat > real,B: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F2 )
=> ( ( B != zero_zero_real )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ A @ B ) )
@ F2 ) ) ) ) ).
% tendsto_divide
thf(fact_143_lim__const__over__n,axiom,
! [A: complex] :
( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ N ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat ) ).
% lim_const_over_n
thf(fact_144_lim__const__over__n,axiom,
! [A: real] :
( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_const_over_n
thf(fact_145_tendsto__const,axiom,
! [K: nat,F2: filter_nat] :
( filterlim_nat_nat
@ ^ [X3: nat] : K
@ ( topolo8926549440605965083ds_nat @ K )
@ F2 ) ).
% tendsto_const
thf(fact_146_tendsto__const,axiom,
! [K: complex,F2: filter_nat] :
( filter6923414461901439796omplex
@ ^ [X3: nat] : K
@ ( topolo2444363109189100025omplex @ K )
@ F2 ) ).
% tendsto_const
thf(fact_147_tendsto__const,axiom,
! [K: real,F2: filter_nat] :
( filterlim_nat_real
@ ^ [X3: nat] : K
@ ( topolo2815343760600316023s_real @ K )
@ F2 ) ).
% tendsto_const
thf(fact_148_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_149_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_150_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_151_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_152_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_153_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_154_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_155_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_156_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_157_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_158_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_159_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_187_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_188_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_189_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_190_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_191_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_192_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_193_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_194_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_195_zadd__int__left,axiom,
! [M: nat,N2: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_196_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_197_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_198_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_199_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_200_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_201_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_202_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_203_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_204_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_add_Ocommute,axiom,
( plus_plus_complex
= ( ^ [A3: complex,B3: complex] : ( plus_plus_complex @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_232_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_233_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_234_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_252_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_253_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_254_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_255_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_256_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_257_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_258_tendsto__eq__rhs,axiom,
! [F: nat > nat,X: nat,F2: filter_nat,Y: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ X ) @ F2 )
=> ( ( X = Y )
=> ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ Y ) @ F2 ) ) ) ).
% tendsto_eq_rhs
thf(fact_259_tendsto__eq__rhs,axiom,
! [F: nat > complex,X: complex,F2: filter_nat,Y: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ X ) @ F2 )
=> ( ( X = Y )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ Y ) @ F2 ) ) ) ).
% tendsto_eq_rhs
thf(fact_260_tendsto__eq__rhs,axiom,
! [F: nat > real,X: real,F2: filter_nat,Y: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ X ) @ F2 )
=> ( ( X = Y )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ F2 ) ) ) ).
% tendsto_eq_rhs
thf(fact_261_tendsto__cong__limit,axiom,
! [F: nat > nat,L: nat,F2: filter_nat,K: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ L ) @ F2 )
=> ( ( K = L )
=> ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ K ) @ F2 ) ) ) ).
% tendsto_cong_limit
thf(fact_262_tendsto__cong__limit,axiom,
! [F: nat > complex,L: complex,F2: filter_nat,K: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 )
=> ( ( K = L )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ K ) @ F2 ) ) ) ).
% tendsto_cong_limit
thf(fact_263_tendsto__cong__limit,axiom,
! [F: nat > real,L: real,F2: filter_nat,K: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( ( K = L )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ K ) @ F2 ) ) ) ).
% tendsto_cong_limit
thf(fact_264_Ints__0,axiom,
member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
% Ints_0
thf(fact_265_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
% Ints_0
thf(fact_266_Ints__0,axiom,
member_int @ zero_zero_int @ ring_1_Ints_int ).
% Ints_0
thf(fact_267_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
% Ints_1
thf(fact_268_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
% Ints_1
thf(fact_269_Ints__1,axiom,
member_complex @ one_one_complex @ ring_1_Ints_complex ).
% Ints_1
thf(fact_270_Ints__add,axiom,
! [A: complex,B: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( member_complex @ B @ ring_1_Ints_complex )
=> ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% Ints_add
thf(fact_271_Ints__add,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_add
thf(fact_272_Ints__add,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_add
thf(fact_273_Ints__of__nat,axiom,
! [N2: nat] : ( member_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ring_1_Ints_complex ) ).
% Ints_of_nat
thf(fact_274_Ints__of__nat,axiom,
! [N2: nat] : ( member_real @ ( semiri5074537144036343181t_real @ N2 ) @ ring_1_Ints_real ) ).
% Ints_of_nat
thf(fact_275_Ints__of__nat,axiom,
! [N2: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N2 ) @ ring_1_Ints_int ) ).
% Ints_of_nat
thf(fact_276_Ints__double__eq__0__iff,axiom,
! [A: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( ( plus_plus_complex @ A @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_277_Ints__double__eq__0__iff,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_278_Ints__double__eq__0__iff,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_279_LIMSEQ__unique,axiom,
! [X5: nat > nat,A: nat,B: nat] :
( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ A ) @ at_top_nat )
=> ( ( filterlim_nat_nat @ X5 @ ( topolo8926549440605965083ds_nat @ B ) @ at_top_nat )
=> ( A = B ) ) ) ).
% LIMSEQ_unique
thf(fact_280_LIMSEQ__unique,axiom,
! [X5: nat > complex,A: complex,B: complex] :
( ( filter6923414461901439796omplex @ X5 @ ( topolo2444363109189100025omplex @ A ) @ at_top_nat )
=> ( ( filter6923414461901439796omplex @ X5 @ ( topolo2444363109189100025omplex @ B ) @ at_top_nat )
=> ( A = B ) ) ) ).
% LIMSEQ_unique
thf(fact_281_LIMSEQ__unique,axiom,
! [X5: nat > real,A: real,B: real] :
( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ A ) @ at_top_nat )
=> ( ( filterlim_nat_real @ X5 @ ( topolo2815343760600316023s_real @ B ) @ at_top_nat )
=> ( A = B ) ) ) ).
% LIMSEQ_unique
thf(fact_282_LIMSEQ__const__iff,axiom,
! [K: nat,L: nat] :
( ( filterlim_nat_nat
@ ^ [N: nat] : K
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_283_LIMSEQ__const__iff,axiom,
! [K: complex,L: complex] :
( ( filter6923414461901439796omplex
@ ^ [N: nat] : K
@ ( topolo2444363109189100025omplex @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_284_LIMSEQ__const__iff,axiom,
! [K: real,L: real] :
( ( filterlim_nat_real
@ ^ [N: nat] : K
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat )
= ( K = L ) ) ).
% LIMSEQ_const_iff
thf(fact_285_LIMSEQ__offset,axiom,
! [F: nat > nat,K: nat,A: nat] :
( ( filterlim_nat_nat
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo8926549440605965083ds_nat @ A )
@ at_top_nat )
=> ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ A ) @ at_top_nat ) ) ).
% LIMSEQ_offset
thf(fact_286_LIMSEQ__offset,axiom,
! [F: nat > complex,K: nat,A: complex] :
( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2444363109189100025omplex @ A )
@ at_top_nat )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ at_top_nat ) ) ).
% LIMSEQ_offset
thf(fact_287_LIMSEQ__offset,axiom,
! [F: nat > real,K: nat,A: real] :
( ( filterlim_nat_real
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2815343760600316023s_real @ A )
@ at_top_nat )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ at_top_nat ) ) ).
% LIMSEQ_offset
thf(fact_288_LIMSEQ__ignore__initial__segment,axiom,
! [F: nat > nat,A: nat,K: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ A ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo8926549440605965083ds_nat @ A )
@ at_top_nat ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_289_LIMSEQ__ignore__initial__segment,axiom,
! [F: nat > complex,A: complex,K: nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ at_top_nat )
=> ( filter6923414461901439796omplex
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2444363109189100025omplex @ A )
@ at_top_nat ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_290_LIMSEQ__ignore__initial__segment,axiom,
! [F: nat > real,A: real,K: nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) )
@ ( topolo2815343760600316023s_real @ A )
@ at_top_nat ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_291_Ints__odd__nonzero,axiom,
! [A: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
!= zero_zero_complex ) ) ).
% Ints_odd_nonzero
thf(fact_292_Ints__odd__nonzero,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
!= zero_zero_real ) ) ).
% Ints_odd_nonzero
thf(fact_293_Ints__odd__nonzero,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
!= zero_zero_int ) ) ).
% Ints_odd_nonzero
thf(fact_294_LIMSEQ__Suc,axiom,
! [F: nat > nat,L: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat )
=> ( filterlim_nat_nat
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat ) ) ).
% LIMSEQ_Suc
thf(fact_295_LIMSEQ__Suc,axiom,
! [F: nat > complex,L: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ at_top_nat )
=> ( filter6923414461901439796omplex
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo2444363109189100025omplex @ L )
@ at_top_nat ) ) ).
% LIMSEQ_Suc
thf(fact_296_LIMSEQ__Suc,axiom,
! [F: nat > real,L: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat ) ) ).
% LIMSEQ_Suc
thf(fact_297_LIMSEQ__imp__Suc,axiom,
! [F: nat > nat,L: nat] :
( ( filterlim_nat_nat
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo8926549440605965083ds_nat @ L )
@ at_top_nat )
=> ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ L ) @ at_top_nat ) ) ).
% LIMSEQ_imp_Suc
thf(fact_298_LIMSEQ__imp__Suc,axiom,
! [F: nat > complex,L: complex] :
( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo2444363109189100025omplex @ L )
@ at_top_nat )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ at_top_nat ) ) ).
% LIMSEQ_imp_Suc
thf(fact_299_LIMSEQ__imp__Suc,axiom,
! [F: nat > real,L: real] :
( ( filterlim_nat_real
@ ^ [N: nat] : ( F @ ( suc @ N ) )
@ ( topolo2815343760600316023s_real @ L )
@ at_top_nat )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ).
% LIMSEQ_imp_Suc
thf(fact_300_tendsto__inverse__real,axiom,
! [U: nat > real,L: real,F2: filter_nat] :
( ( filterlim_nat_real @ U @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( ( L != zero_zero_real )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( divide_divide_real @ one_one_real @ ( U @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ one_one_real @ L ) )
@ F2 ) ) ) ).
% tendsto_inverse_real
thf(fact_301_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_302_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_303_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_304_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_305_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_306_filterlim__sequentially__Suc,axiom,
! [F: nat > complex,F2: filter_complex] :
( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( F @ ( suc @ X3 ) )
@ F2
@ at_top_nat )
= ( filter6923414461901439796omplex @ F @ F2 @ at_top_nat ) ) ).
% filterlim_sequentially_Suc
thf(fact_307_filterlim__sequentially__Suc,axiom,
! [F: nat > real,F2: filter_real] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( F @ ( suc @ X3 ) )
@ F2
@ at_top_nat )
= ( filterlim_nat_real @ F @ F2 @ at_top_nat ) ) ).
% filterlim_sequentially_Suc
thf(fact_308_filterlim__sequentially__Suc,axiom,
! [F: nat > nat,F2: filter_nat] :
( ( filterlim_nat_nat
@ ^ [X3: nat] : ( F @ ( suc @ X3 ) )
@ F2
@ at_top_nat )
= ( filterlim_nat_nat @ F @ F2 @ at_top_nat ) ) ).
% filterlim_sequentially_Suc
thf(fact_309_of__nat__code,axiom,
( semiri8010041392384452111omplex
= ( ^ [N: nat] :
( semiri2816024913162550771omplex
@ ^ [I2: complex] : ( plus_plus_complex @ I2 @ one_one_complex )
@ N
@ zero_zero_complex ) ) ) ).
% of_nat_code
thf(fact_310_of__nat__code,axiom,
( semiri5074537144036343181t_real
= ( ^ [N: nat] :
( semiri7260567687927622513x_real
@ ^ [I2: real] : ( plus_plus_real @ I2 @ one_one_real )
@ N
@ zero_zero_real ) ) ) ).
% of_nat_code
thf(fact_311_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N: nat] :
( semiri8420488043553186161ux_int
@ ^ [I2: int] : ( plus_plus_int @ I2 @ one_one_int )
@ N
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_312_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I2: nat] : ( plus_plus_nat @ I2 @ one_one_nat )
@ N
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_313_Multiseries__Expansion_Ointyness__1,axiom,
( one_one_real
= ( semiri5074537144036343181t_real @ one_one_nat ) ) ).
% Multiseries_Expansion.intyness_1
thf(fact_314_Multiseries__Expansion_Ointyness__0,axiom,
( zero_zero_real
= ( semiri5074537144036343181t_real @ zero_zero_nat ) ) ).
% Multiseries_Expansion.intyness_0
thf(fact_315_filterlim__add__const__nat__at__top,axiom,
! [C: nat] :
( filterlim_nat_nat
@ ^ [N: nat] : ( plus_plus_nat @ N @ C )
@ at_top_nat
@ at_top_nat ) ).
% filterlim_add_const_nat_at_top
thf(fact_316_set__plus__intro,axiom,
! [A: complex,C2: set_complex,B: complex,D2: set_complex] :
( ( member_complex @ A @ C2 )
=> ( ( member_complex @ B @ D2 )
=> ( member_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_p7052360327008956141omplex @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_317_set__plus__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_set_nat @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_318_set__plus__intro,axiom,
! [A: int,C2: set_int,B: int,D2: set_int] :
( ( member_int @ A @ C2 )
=> ( ( member_int @ B @ D2 )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_set_int @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_319_set__plus__intro,axiom,
! [A: real,C2: set_real,B: real,D2: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_set_real @ C2 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_320_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_321_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_322_int__int__eq,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% int_int_eq
thf(fact_323_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_324_zdiv__int,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zdiv_int
thf(fact_325_set__plus__elim,axiom,
! [X: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ X @ ( plus_p7052360327008956141omplex @ A2 @ B2 ) )
=> ~ ! [A4: complex,B4: complex] :
( ( X
= ( plus_plus_complex @ A4 @ B4 ) )
=> ( ( member_complex @ A4 @ A2 )
=> ~ ( member_complex @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_326_set__plus__elim,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( plus_plus_set_nat @ A2 @ B2 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X
= ( plus_plus_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_327_set__plus__elim,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ( member_int @ X @ ( plus_plus_set_int @ A2 @ B2 ) )
=> ~ ! [A4: int,B4: int] :
( ( X
= ( plus_plus_int @ A4 @ B4 ) )
=> ( ( member_int @ A4 @ A2 )
=> ~ ( member_int @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_328_set__plus__elim,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( member_real @ X @ ( plus_plus_set_real @ A2 @ B2 ) )
=> ~ ! [A4: real,B4: real] :
( ( X
= ( plus_plus_real @ A4 @ B4 ) )
=> ( ( member_real @ A4 @ A2 )
=> ~ ( member_real @ B4 @ B2 ) ) ) ) ).
% set_plus_elim
thf(fact_329_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_330_Multiseries__Expansion_Ointyness__of__nat,axiom,
! [N2: nat] :
( ( N2 = N2 )
=> ( ( semiri5074537144036343181t_real @ N2 )
= ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% Multiseries_Expansion.intyness_of_nat
thf(fact_331_filterlim__compose,axiom,
! [G: complex > complex,F3: filter_complex,F22: filter_complex,F: nat > complex,F1: filter_nat] :
( ( filter8330067395343389202omplex @ G @ F3 @ F22 )
=> ( ( filter6923414461901439796omplex @ F @ F22 @ F1 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( G @ ( F @ X3 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_332_filterlim__compose,axiom,
! [G: complex > real,F3: filter_real,F22: filter_complex,F: nat > complex,F1: filter_nat] :
( ( filter8559879285478333968x_real @ G @ F3 @ F22 )
=> ( ( filter6923414461901439796omplex @ F @ F22 @ F1 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( G @ ( F @ X3 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_333_filterlim__compose,axiom,
! [G: complex > nat,F3: filter_nat,F22: filter_complex,F: nat > complex,F1: filter_nat] :
( ( filter1319825749481401652ex_nat @ G @ F3 @ F22 )
=> ( ( filter6923414461901439796omplex @ F @ F22 @ F1 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( G @ ( F @ X3 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_334_filterlim__compose,axiom,
! [G: real > complex,F3: filter_complex,F22: filter_real,F: nat > real,F1: filter_nat] :
( ( filter8506290784974013328omplex @ G @ F3 @ F22 )
=> ( ( filterlim_nat_real @ F @ F22 @ F1 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( G @ ( F @ X3 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_335_filterlim__compose,axiom,
! [G: real > real,F3: filter_real,F22: filter_real,F: nat > real,F1: filter_nat] :
( ( filterlim_real_real @ G @ F3 @ F22 )
=> ( ( filterlim_nat_real @ F @ F22 @ F1 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( G @ ( F @ X3 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_336_filterlim__compose,axiom,
! [G: real > nat,F3: filter_nat,F22: filter_real,F: nat > real,F1: filter_nat] :
( ( filterlim_real_nat @ G @ F3 @ F22 )
=> ( ( filterlim_nat_real @ F @ F22 @ F1 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( G @ ( F @ X3 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_337_filterlim__compose,axiom,
! [G: nat > complex,F3: filter_complex,F22: filter_nat,F: nat > nat,F1: filter_nat] :
( ( filter6923414461901439796omplex @ G @ F3 @ F22 )
=> ( ( filterlim_nat_nat @ F @ F22 @ F1 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( G @ ( F @ X3 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_338_filterlim__compose,axiom,
! [G: nat > real,F3: filter_real,F22: filter_nat,F: nat > nat,F1: filter_nat] :
( ( filterlim_nat_real @ G @ F3 @ F22 )
=> ( ( filterlim_nat_nat @ F @ F22 @ F1 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( G @ ( F @ X3 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_339_filterlim__compose,axiom,
! [G: nat > nat,F3: filter_nat,F22: filter_nat,F: nat > nat,F1: filter_nat] :
( ( filterlim_nat_nat @ G @ F3 @ F22 )
=> ( ( filterlim_nat_nat @ F @ F22 @ F1 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( G @ ( F @ X3 ) )
@ F3
@ F1 ) ) ) ).
% filterlim_compose
thf(fact_340_filterlim__ident,axiom,
! [F2: filter_nat] :
( filterlim_nat_nat
@ ^ [X3: nat] : X3
@ F2
@ F2 ) ).
% filterlim_ident
thf(fact_341_filterlim__Suc,axiom,
filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% filterlim_Suc
thf(fact_342_Multiseries__Expansion_Ointyness__simps_I1_J,axiom,
! [A: nat,B: nat] :
( ( plus_plus_real @ ( semiri5074537144036343181t_real @ A ) @ ( semiri5074537144036343181t_real @ B ) )
= ( semiri5074537144036343181t_real @ ( plus_plus_nat @ A @ B ) ) ) ).
% Multiseries_Expansion.intyness_simps(1)
thf(fact_343_int__plus,axiom,
! [N2: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_344_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_345_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_346_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_347_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_348_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N2 ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_349_int__ops_I8_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(8)
thf(fact_350_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_351_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_352_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_353_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_354_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_355_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_356_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_357_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_358_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_359_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_360_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_361_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R: real] :
( filterlim_nat_real
@ ^ [N: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
@ ( topolo2815343760600316023s_real @ R )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_362_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_363_pth__7_I1_J,axiom,
! [X: complex] :
( ( plus_plus_complex @ zero_zero_complex @ X )
= X ) ).
% pth_7(1)
thf(fact_364_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_365_eq__add__iff,axiom,
! [X: complex,Y: complex] :
( ( X
= ( plus_plus_complex @ X @ Y ) )
= ( Y = zero_zero_complex ) ) ).
% eq_add_iff
thf(fact_366_eq__add__iff,axiom,
! [X: real,Y: real] :
( ( X
= ( plus_plus_real @ X @ Y ) )
= ( Y = zero_zero_real ) ) ).
% eq_add_iff
thf(fact_367_eq__add__iff,axiom,
! [X: int,Y: int] :
( ( X
= ( plus_plus_int @ X @ Y ) )
= ( Y = zero_zero_int ) ) ).
% eq_add_iff
thf(fact_368_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_369_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_370_inverse__nonzero__iff__nonzero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_371_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_372_inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% inverse_zero
thf(fact_373_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_374_inverse__1,axiom,
( ( invers8013647133539491842omplex @ one_one_complex )
= one_one_complex ) ).
% inverse_1
thf(fact_375_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_376_inverse__eq__1__iff,axiom,
! [X: complex] :
( ( ( invers8013647133539491842omplex @ X )
= one_one_complex )
= ( X = one_one_complex ) ) ).
% inverse_eq_1_iff
thf(fact_377_inverse__eq__1__iff,axiom,
! [X: real] :
( ( ( inverse_inverse_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_378_inverse__divide,axiom,
! [A: complex,B: complex] :
( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ B @ A ) ) ).
% inverse_divide
thf(fact_379_inverse__divide,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ B @ A ) ) ).
% inverse_divide
thf(fact_380_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_381_nonzero__imp__inverse__nonzero,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
!= zero_zero_complex ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_382_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_383_nonzero__inverse__inverse__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_384_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_385_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_386_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_387_inverse__zero__imp__zero,axiom,
! [A: complex] :
( ( ( invers8013647133539491842omplex @ A )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ).
% inverse_zero_imp_zero
thf(fact_388_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_389_field__class_Ofield__inverse__zero,axiom,
( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% field_class.field_inverse_zero
thf(fact_390_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_391_inverse__eq__divide,axiom,
( invers8013647133539491842omplex
= ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% inverse_eq_divide
thf(fact_392_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_393_nonzero__inverse__eq__divide,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_394_nonzero__inverse__eq__divide,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_395_tendsto__inverse,axiom,
! [F: nat > complex,A: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( A != zero_zero_complex )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( invers8013647133539491842omplex @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( invers8013647133539491842omplex @ A ) )
@ F2 ) ) ) ).
% tendsto_inverse
thf(fact_396_tendsto__inverse,axiom,
! [F: nat > real,A: real,F2: filter_nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( A != zero_zero_real )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( inverse_inverse_real @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( inverse_inverse_real @ A ) )
@ F2 ) ) ) ).
% tendsto_inverse
thf(fact_397_lim__inverse__n,axiom,
( filter6923414461901439796omplex
@ ^ [N: nat] : ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ N ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat ) ).
% lim_inverse_n
thf(fact_398_lim__inverse__n,axiom,
( filterlim_nat_real
@ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_inverse_n
thf(fact_399_pth__d,axiom,
! [X: complex] :
( ( plus_plus_complex @ X @ zero_zero_complex )
= X ) ).
% pth_d
thf(fact_400_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_401_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R: real] :
( filterlim_nat_real
@ ^ [N: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_402_Bfun__inverse,axiom,
! [F: nat > complex,A: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( A != zero_zero_complex )
=> ( bfun_nat_complex
@ ^ [X3: nat] : ( invers8013647133539491842omplex @ ( F @ X3 ) )
@ F2 ) ) ) ).
% Bfun_inverse
thf(fact_403_Bfun__inverse,axiom,
! [F: nat > real,A: real,F2: filter_nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( A != zero_zero_real )
=> ( bfun_nat_real
@ ^ [X3: nat] : ( inverse_inverse_real @ ( F @ X3 ) )
@ F2 ) ) ) ).
% Bfun_inverse
thf(fact_404_triangle__Suc,axiom,
! [N2: nat] :
( ( nat_triangle @ ( suc @ N2 ) )
= ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% triangle_Suc
thf(fact_405_fps__divide__1_H,axiom,
! [A: formal670952693614245302omplex] :
( ( ( invers8013647133539491842omplex @ one_one_complex )
= one_one_complex )
=> ( ( divide1348722040316500488omplex @ A @ one_on1590755018477040891omplex )
= A ) ) ).
% fps_divide_1'
thf(fact_406_fps__divide__1_H,axiom,
! [A: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ one_one_real )
= one_one_real )
=> ( ( divide1155267253282662278s_real @ A @ one_on8598947968683843321s_real )
= A ) ) ).
% fps_divide_1'
thf(fact_407_fps__inverse__one_H,axiom,
( ( ( invers8013647133539491842omplex @ one_one_complex )
= one_one_complex )
=> ( ( invers8029835185024579338omplex @ one_on1590755018477040891omplex )
= one_on1590755018477040891omplex ) ) ).
% fps_inverse_one'
thf(fact_408_fps__inverse__one_H,axiom,
( ( ( inverse_inverse_real @ one_one_real )
= one_one_real )
=> ( ( invers68952373231134600s_real @ one_on8598947968683843321s_real )
= one_on8598947968683843321s_real ) ) ).
% fps_inverse_one'
thf(fact_409_fps__div__by__zero_H,axiom,
! [G: formal670952693614245302omplex] :
( ( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex )
=> ( ( divide1348722040316500488omplex @ G @ zero_z1877163951443063103omplex )
= zero_z1877163951443063103omplex ) ) ).
% fps_div_by_zero'
thf(fact_410_fps__div__by__zero_H,axiom,
! [G: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real )
=> ( ( divide1155267253282662278s_real @ G @ zero_z7760665558314615101s_real )
= zero_z7760665558314615101s_real ) ) ).
% fps_div_by_zero'
thf(fact_411_fps__inverse__zero_H,axiom,
( ( ( invers8013647133539491842omplex @ zero_zero_complex )
= zero_zero_complex )
=> ( ( invers8029835185024579338omplex @ zero_z1877163951443063103omplex )
= zero_z1877163951443063103omplex ) ) ).
% fps_inverse_zero'
thf(fact_412_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_413_add__0__iff,axiom,
! [B: complex,A: complex] :
( ( B
= ( plus_plus_complex @ B @ A ) )
= ( A = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_414_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_415_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_416_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_417_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_418_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_419_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_420_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_421_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_422_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_423_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_424_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_425_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_426_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_427_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_428_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_429_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_430_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_431_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_432_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_433_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_434_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_435_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_436_add__minus__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_437_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_438_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_439_minus__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_440_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_441_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_442_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_443_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_444_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_445_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_446_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_447_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_448_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_449_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_450_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_451_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_452_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_453_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_454_divide__minus1,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% divide_minus1
thf(fact_455_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_456_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_457_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_458_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_459_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_460_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_461_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_462_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_463_Bseq__minus__iff,axiom,
! [X5: nat > real] :
( ( bfun_nat_real
@ ^ [N: nat] : ( uminus_uminus_real @ ( X5 @ N ) )
@ at_top_nat )
= ( bfun_nat_real @ X5 @ at_top_nat ) ) ).
% Bseq_minus_iff
thf(fact_464_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_465_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_466_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_467_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_468_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_469_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_470_minus__divide__right,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_divide_right
thf(fact_471_minus__divide__right,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_472_minus__divide__divide,axiom,
! [A: complex,B: complex] :
( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ).
% minus_divide_divide
thf(fact_473_minus__divide__divide,axiom,
! [A: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_474_minus__divide__left,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_475_minus__divide__left,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_476_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_477_Multiseries__Expansion_Ointyness__uminus,axiom,
! [X: real,N2: nat] :
( ( X
= ( semiri5074537144036343181t_real @ N2 ) )
=> ( ( uminus_uminus_real @ X )
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% Multiseries_Expansion.intyness_uminus
thf(fact_478_minus__in__Ints__iff,axiom,
! [X: complex] :
( ( member_complex @ ( uminus1482373934393186551omplex @ X ) @ ring_1_Ints_complex )
= ( member_complex @ X @ ring_1_Ints_complex ) ) ).
% minus_in_Ints_iff
thf(fact_479_minus__in__Ints__iff,axiom,
! [X: real] :
( ( member_real @ ( uminus_uminus_real @ X ) @ ring_1_Ints_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ).
% minus_in_Ints_iff
thf(fact_480_minus__in__Ints__iff,axiom,
! [X: int] :
( ( member_int @ ( uminus_uminus_int @ X ) @ ring_1_Ints_int )
= ( member_int @ X @ ring_1_Ints_int ) ) ).
% minus_in_Ints_iff
thf(fact_481_Ints__minus,axiom,
! [A: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( member_complex @ ( uminus1482373934393186551omplex @ A ) @ ring_1_Ints_complex ) ) ).
% Ints_minus
thf(fact_482_Ints__minus,axiom,
! [A: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( member_real @ ( uminus_uminus_real @ A ) @ ring_1_Ints_real ) ) ).
% Ints_minus
thf(fact_483_Ints__minus,axiom,
! [A: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( member_int @ ( uminus_uminus_int @ A ) @ ring_1_Ints_int ) ) ).
% Ints_minus
thf(fact_484_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_485_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_486_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_487_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_488_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_489_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_490_add_Oinverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_491_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_492_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_493_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_494_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_495_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_496_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_497_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_498_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_499_nonzero__minus__divide__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_500_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_501_nonzero__minus__divide__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_502_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_503_nonzero__inverse__minus__eq,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
= ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_504_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_505_tendsto__uminus__nhds,axiom,
! [A: complex] : ( filter8330067395343389202omplex @ uminus1482373934393186551omplex @ ( topolo2444363109189100025omplex @ ( uminus1482373934393186551omplex @ A ) ) @ ( topolo2444363109189100025omplex @ A ) ) ).
% tendsto_uminus_nhds
thf(fact_506_tendsto__uminus__nhds,axiom,
! [A: real] : ( filterlim_real_real @ uminus_uminus_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ A ) ) @ ( topolo2815343760600316023s_real @ A ) ) ).
% tendsto_uminus_nhds
thf(fact_507_tendsto__minus,axiom,
! [F: nat > complex,A: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( uminus1482373934393186551omplex @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( uminus1482373934393186551omplex @ A ) )
@ F2 ) ) ).
% tendsto_minus
thf(fact_508_tendsto__minus,axiom,
! [F: nat > real,A: real,F2: filter_nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( uminus_uminus_real @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ A ) )
@ F2 ) ) ).
% tendsto_minus
thf(fact_509_tendsto__minus__cancel,axiom,
! [F: nat > complex,A: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( uminus1482373934393186551omplex @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( uminus1482373934393186551omplex @ A ) )
@ F2 )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 ) ) ).
% tendsto_minus_cancel
thf(fact_510_tendsto__minus__cancel,axiom,
! [F: nat > real,A: real,F2: filter_nat] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( uminus_uminus_real @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ A ) )
@ F2 )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 ) ) ).
% tendsto_minus_cancel
thf(fact_511_tendsto__minus__cancel__left,axiom,
! [F: nat > complex,Y: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ ( uminus1482373934393186551omplex @ Y ) ) @ F2 )
= ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( uminus1482373934393186551omplex @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ Y )
@ F2 ) ) ).
% tendsto_minus_cancel_left
thf(fact_512_tendsto__minus__cancel__left,axiom,
! [F: nat > real,Y: real,F2: filter_nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ Y ) ) @ F2 )
= ( filterlim_nat_real
@ ^ [X3: nat] : ( uminus_uminus_real @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ Y )
@ F2 ) ) ).
% tendsto_minus_cancel_left
thf(fact_513_Bseq__add,axiom,
! [F: nat > complex,C: complex] :
( ( bfun_nat_complex @ F @ at_top_nat )
=> ( bfun_nat_complex
@ ^ [X3: nat] : ( plus_plus_complex @ ( F @ X3 ) @ C )
@ at_top_nat ) ) ).
% Bseq_add
thf(fact_514_Bseq__add,axiom,
! [F: nat > real,C: real] :
( ( bfun_nat_real @ F @ at_top_nat )
=> ( bfun_nat_real
@ ^ [X3: nat] : ( plus_plus_real @ ( F @ X3 ) @ C )
@ at_top_nat ) ) ).
% Bseq_add
thf(fact_515_Bseq__add__iff,axiom,
! [F: nat > complex,C: complex] :
( ( bfun_nat_complex
@ ^ [X3: nat] : ( plus_plus_complex @ ( F @ X3 ) @ C )
@ at_top_nat )
= ( bfun_nat_complex @ F @ at_top_nat ) ) ).
% Bseq_add_iff
thf(fact_516_Bseq__add__iff,axiom,
! [F: nat > real,C: real] :
( ( bfun_nat_real
@ ^ [X3: nat] : ( plus_plus_real @ ( F @ X3 ) @ C )
@ at_top_nat )
= ( bfun_nat_real @ F @ at_top_nat ) ) ).
% Bseq_add_iff
thf(fact_517_divide__eq__minus__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( B != zero_zero_complex )
& ( A
= ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_518_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_519_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= zero_zero_complex ) ).
% add_neg_numeral_special(8)
thf(fact_520_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_521_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_522_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% add_neg_numeral_special(7)
thf(fact_523_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_524_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_525_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_526_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R: real] :
( filterlim_nat_real
@ ^ [N: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_527_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_528_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_529_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_530_fps__tan__0,axiom,
( ( formal6482914284900457064omplex @ zero_zero_complex )
= zero_z1877163951443063103omplex ) ).
% fps_tan_0
thf(fact_531_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_532_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_533_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_534_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_535_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_536_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_537_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_538_set__times__intro,axiom,
! [A: complex,C2: set_complex,B: complex,D2: set_complex] :
( ( member_complex @ A @ C2 )
=> ( ( member_complex @ B @ D2 )
=> ( member_complex @ ( times_times_complex @ A @ B ) @ ( times_6048082448287401577omplex @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_539_set__times__intro,axiom,
! [A: real,C2: set_real,B: real,D2: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D2 )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_540_set__times__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D2 )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_541_set__times__intro,axiom,
! [A: int,C2: set_int,B: int,D2: set_int] :
( ( member_int @ A @ C2 )
=> ( ( member_int @ B @ D2 )
=> ( member_int @ ( times_times_int @ A @ B ) @ ( times_times_set_int @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_542_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_543_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_544_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_545_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_546_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_547_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_548_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_549_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_550_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_551_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_552_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_553_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_554_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_555_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_556_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_557_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_558_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_559_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_560_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_561_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_562_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_563_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_564_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_565_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_566_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_567_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_568_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_569_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_570_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_571_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_572_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_573_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_574_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_575_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_576_times__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_577_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_578_divide__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_579_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_580_divide__divide__eq__left,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_581_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_582_times__divide__eq__left,axiom,
! [B: complex,C: complex,A: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
= ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_583_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_584_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% of_nat_mult
thf(fact_585_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_586_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_587_of__nat__mult,axiom,
! [M: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_588_inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_589_negative__eq__positive,axiom,
! [N2: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N2 = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_590_mult__cancel__right2,axiom,
! [A: complex,C: complex] :
( ( ( times_times_complex @ A @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_591_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_592_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_593_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_594_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_595_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_596_mult__cancel__left2,axiom,
! [C: complex,A: complex] :
( ( ( times_times_complex @ C @ A )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_597_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_598_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_599_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_600_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_601_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_602_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_603_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_604_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_605_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_606_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_607_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_608_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_609_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_610_nonzero__mult__div__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_611_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_612_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_613_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_614_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_615_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_616_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_617_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_618_nonzero__mult__div__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_619_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_620_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_621_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_622_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_623_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_624_mult__divide__mult__cancel__left__if,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( C = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= zero_zero_complex ) )
& ( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_625_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_626_mult__minus1__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1_right
thf(fact_627_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_628_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_629_mult__minus1,axiom,
! [Z: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1
thf(fact_630_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_631_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_632_nonzero__divide__mult__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_633_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_634_nonzero__divide__mult__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_635_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_636_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_637_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_638_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_639_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_640_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_641_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_642_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_643_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_644_right__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
= one_one_complex ) ) ).
% right_inverse
thf(fact_645_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_646_left__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
= one_one_complex ) ) ).
% left_inverse
thf(fact_647_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_648_tendsto__mult__right__iff,axiom,
! [C: complex,F: nat > complex,L: complex,F2: filter_nat] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ ( F @ X3 ) @ C )
@ ( topolo2444363109189100025omplex @ ( times_times_complex @ L @ C ) )
@ F2 )
= ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 ) ) ) ).
% tendsto_mult_right_iff
thf(fact_649_tendsto__mult__right__iff,axiom,
! [C: real,F: nat > real,L: real,F2: filter_nat] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ ( F @ X3 ) @ C )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ L @ C ) )
@ F2 )
= ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ) ).
% tendsto_mult_right_iff
thf(fact_650_tendsto__mult__left__iff,axiom,
! [C: complex,F: nat > complex,L: complex,F2: filter_nat] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ C @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( times_times_complex @ C @ L ) )
@ F2 )
= ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 ) ) ) ).
% tendsto_mult_left_iff
thf(fact_651_tendsto__mult__left__iff,axiom,
! [C: real,F: nat > real,L: real,F2: filter_nat] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ C @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ C @ L ) )
@ F2 )
= ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ) ).
% tendsto_mult_left_iff
thf(fact_652_tendsto__zero__mult__right__iff,axiom,
! [C: complex,A: nat > complex] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ C )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat )
= ( filter6923414461901439796omplex @ A @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ at_top_nat ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_653_tendsto__zero__mult__right__iff,axiom,
! [C: real,A: nat > real] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ C )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
= ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_654_tendsto__zero__mult__left__iff,axiom,
! [C: complex,A: nat > complex] :
( ( C != zero_zero_complex )
=> ( ( filter6923414461901439796omplex
@ ^ [N: nat] : ( times_times_complex @ C @ ( A @ N ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat )
= ( filter6923414461901439796omplex @ A @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ at_top_nat ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_655_tendsto__zero__mult__left__iff,axiom,
! [C: real,A: nat > real] :
( ( C != zero_zero_real )
=> ( ( filterlim_nat_real
@ ^ [N: nat] : ( times_times_real @ C @ ( A @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
= ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_656_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_657_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_658_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_659_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_660_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_661_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_662_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_663_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_664_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_665_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_666_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_667_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_668_set__times__elim,axiom,
! [X: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ X @ ( times_6048082448287401577omplex @ A2 @ B2 ) )
=> ~ ! [A4: complex,B4: complex] :
( ( X
= ( times_times_complex @ A4 @ B4 ) )
=> ( ( member_complex @ A4 @ A2 )
=> ~ ( member_complex @ B4 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_669_set__times__elim,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ( member_real @ X @ ( times_times_set_real @ A2 @ B2 ) )
=> ~ ! [A4: real,B4: real] :
( ( X
= ( times_times_real @ A4 @ B4 ) )
=> ( ( member_real @ A4 @ A2 )
=> ~ ( member_real @ B4 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_670_set__times__elim,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A2 @ B2 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X
= ( times_times_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B4 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_671_set__times__elim,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ( member_int @ X @ ( times_times_set_int @ A2 @ B2 ) )
=> ~ ! [A4: int,B4: int] :
( ( X
= ( times_times_int @ A4 @ B4 ) )
=> ( ( member_int @ A4 @ A2 )
=> ~ ( member_int @ B4 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_672_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_673_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_674_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_675_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_676_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_677_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_678_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_679_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_680_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_681_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_682_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_683_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_684_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_685_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_686_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_687_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_688_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_689_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_690_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_691_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_692_mult__delta__right,axiom,
! [B: $o,X: complex,Y: complex] :
( ( B
=> ( ( times_times_complex @ X @ ( if_complex @ B @ Y @ zero_zero_complex ) )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_complex @ X @ ( if_complex @ B @ Y @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_693_mult__delta__right,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_694_mult__delta__right,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_695_mult__delta__right,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_696_mult__delta__left,axiom,
! [B: $o,X: complex,Y: complex] :
( ( B
=> ( ( times_times_complex @ ( if_complex @ B @ X @ zero_zero_complex ) @ Y )
= ( times_times_complex @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_complex @ ( if_complex @ B @ X @ zero_zero_complex ) @ Y )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_697_mult__delta__left,axiom,
! [B: $o,X: real,Y: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_698_mult__delta__left,axiom,
! [B: $o,X: nat,Y: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_699_mult__delta__left,axiom,
! [B: $o,X: int,Y: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_700_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_701_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_702_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_703_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_704_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_705_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_706_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_707_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_708_combine__common__factor,axiom,
! [A: complex,E: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_709_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_710_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_711_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_712_distrib__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% distrib_right
thf(fact_713_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_714_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_715_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_716_distrib__left,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% distrib_left
thf(fact_717_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_718_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_719_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_720_comm__semiring__class_Odistrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_721_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_722_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_723_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_724_ring__class_Oring__distribs_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_725_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_726_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_727_ring__class_Oring__distribs_I2_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_728_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_729_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_730_crossproduct__noteq,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
!= ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_731_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_732_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_733_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_734_crossproduct__eq,axiom,
! [W: complex,Y: complex,X: complex,Z: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y ) @ ( times_times_complex @ X @ Z ) )
= ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_735_crossproduct__eq,axiom,
! [W: real,Y: real,X: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_736_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_737_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_738_minus__mult__commute,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_739_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_740_square__eq__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ A )
= ( times_times_real @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_741_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_742_divide__divide__eq__left_H,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_743_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_744_divide__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_745_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_746_times__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_747_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_748_mult__of__nat__commute,axiom,
! [X: nat,Y: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
= ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_749_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_750_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_751_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_752_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_753_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: real,X: real] :
( ( ( times_times_real @ Y @ X )
= ( times_times_real @ X @ Y ) )
=> ( ( times_times_real @ ( inverse_inverse_real @ Y ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_754_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_755_Ints__mult,axiom,
! [A: complex,B: complex] :
( ( member_complex @ A @ ring_1_Ints_complex )
=> ( ( member_complex @ B @ ring_1_Ints_complex )
=> ( member_complex @ ( times_times_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).
% Ints_mult
thf(fact_756_Ints__mult,axiom,
! [A: real,B: real] :
( ( member_real @ A @ ring_1_Ints_real )
=> ( ( member_real @ B @ ring_1_Ints_real )
=> ( member_real @ ( times_times_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).
% Ints_mult
thf(fact_757_Ints__mult,axiom,
! [A: int,B: int] :
( ( member_int @ A @ ring_1_Ints_int )
=> ( ( member_int @ B @ ring_1_Ints_int )
=> ( member_int @ ( times_times_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).
% Ints_mult
thf(fact_758_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X3: real,Y5: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y5 ) ) ) ) ).
% divide_real_def
thf(fact_759_lambda__zero,axiom,
( ( ^ [H: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_760_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_761_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_762_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_763_lambda__one,axiom,
( ( ^ [X3: complex] : X3 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_764_lambda__one,axiom,
( ( ^ [X3: real] : X3 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_765_lambda__one,axiom,
( ( ^ [X3: nat] : X3 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_766_lambda__one,axiom,
( ( ^ [X3: int] : X3 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_767_Bseq__mult,axiom,
! [F: nat > real,G: nat > real] :
( ( bfun_nat_real @ F @ at_top_nat )
=> ( ( bfun_nat_real @ G @ at_top_nat )
=> ( bfun_nat_real
@ ^ [X3: nat] : ( times_times_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ at_top_nat ) ) ) ).
% Bseq_mult
thf(fact_768_add__scale__eq__noteq,axiom,
! [R: complex,A: complex,B: complex,C: complex,D: complex] :
( ( R != zero_zero_complex )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
!= ( plus_plus_complex @ B @ ( times_times_complex @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_769_add__scale__eq__noteq,axiom,
! [R: real,A: real,B: real,C: real,D: real] :
( ( R != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_770_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_771_add__scale__eq__noteq,axiom,
! [R: int,A: int,B: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_772_nonzero__eq__divide__eq,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( times_times_complex @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_773_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_774_nonzero__divide__eq__eq,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( B
= ( times_times_complex @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_775_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_776_eq__divide__imp,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= B )
=> ( A
= ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_777_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_778_divide__eq__imp,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( B
= ( times_times_complex @ A @ C ) )
=> ( ( divide1717551699836669952omplex @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_779_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_780_eq__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ A @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_781_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_782_divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_783_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_784_frac__eq__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X @ Y )
= ( divide1717551699836669952omplex @ W @ Z ) )
= ( ( times_times_complex @ X @ Z )
= ( times_times_complex @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_785_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_786_square__eq__1__iff,axiom,
! [X: complex] :
( ( ( times_times_complex @ X @ X )
= one_one_complex )
= ( ( X = one_one_complex )
| ( X
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% square_eq_1_iff
thf(fact_787_square__eq__1__iff,axiom,
! [X: real] :
( ( ( times_times_real @ X @ X )
= one_one_real )
= ( ( X = one_one_real )
| ( X
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_788_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_789_nonzero__inverse__mult__distrib,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
= ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_790_nonzero__inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_791_div__mult2__eq_H,axiom,
! [A: int,M: nat,N2: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% div_mult2_eq'
thf(fact_792_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N2: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% div_mult2_eq'
thf(fact_793_inverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= one_one_complex )
=> ( ( invers8013647133539491842omplex @ A )
= B ) ) ).
% inverse_unique
thf(fact_794_inverse__unique,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= one_one_real )
=> ( ( inverse_inverse_real @ A )
= B ) ) ).
% inverse_unique
thf(fact_795_divide__inverse__commute,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B3: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B3 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_796_divide__inverse__commute,axiom,
( divide_divide_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ ( inverse_inverse_real @ B3 ) @ A3 ) ) ) ).
% divide_inverse_commute
thf(fact_797_divide__inverse,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B3: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_798_divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_799_field__class_Ofield__divide__inverse,axiom,
( divide1717551699836669952omplex
= ( ^ [A3: complex,B3: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_800_field__class_Ofield__divide__inverse,axiom,
( divide_divide_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_801_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_802_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_803_mult__inverse__of__nat__commute,axiom,
! [Xa: nat,X: complex] :
( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_804_mult__inverse__of__nat__commute,axiom,
! [Xa: nat,X: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_805_Bseq__cmult__iff,axiom,
! [C: complex,F: nat > complex] :
( ( C != zero_zero_complex )
=> ( ( bfun_nat_complex
@ ^ [X3: nat] : ( times_times_complex @ C @ ( F @ X3 ) )
@ at_top_nat )
= ( bfun_nat_complex @ F @ at_top_nat ) ) ) ).
% Bseq_cmult_iff
thf(fact_806_Bseq__cmult__iff,axiom,
! [C: real,F: nat > real] :
( ( C != zero_zero_real )
=> ( ( bfun_nat_real
@ ^ [X3: nat] : ( times_times_real @ C @ ( F @ X3 ) )
@ at_top_nat )
= ( bfun_nat_real @ F @ at_top_nat ) ) ) ).
% Bseq_cmult_iff
thf(fact_807_tendsto__mult__right,axiom,
! [F: nat > nat,L: nat,F2: filter_nat,C: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ L ) @ F2 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( times_times_nat @ ( F @ X3 ) @ C )
@ ( topolo8926549440605965083ds_nat @ ( times_times_nat @ L @ C ) )
@ F2 ) ) ).
% tendsto_mult_right
thf(fact_808_tendsto__mult__right,axiom,
! [F: nat > complex,L: complex,F2: filter_nat,C: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ ( F @ X3 ) @ C )
@ ( topolo2444363109189100025omplex @ ( times_times_complex @ L @ C ) )
@ F2 ) ) ).
% tendsto_mult_right
thf(fact_809_tendsto__mult__right,axiom,
! [F: nat > real,L: real,F2: filter_nat,C: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ ( F @ X3 ) @ C )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ L @ C ) )
@ F2 ) ) ).
% tendsto_mult_right
thf(fact_810_tendsto__mult__left,axiom,
! [F: nat > nat,L: nat,F2: filter_nat,C: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ L ) @ F2 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( times_times_nat @ C @ ( F @ X3 ) )
@ ( topolo8926549440605965083ds_nat @ ( times_times_nat @ C @ L ) )
@ F2 ) ) ).
% tendsto_mult_left
thf(fact_811_tendsto__mult__left,axiom,
! [F: nat > complex,L: complex,F2: filter_nat,C: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ L ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ C @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( times_times_complex @ C @ L ) )
@ F2 ) ) ).
% tendsto_mult_left
thf(fact_812_tendsto__mult__left,axiom,
! [F: nat > real,L: real,F2: filter_nat,C: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ C @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ C @ L ) )
@ F2 ) ) ).
% tendsto_mult_left
thf(fact_813_tendsto__mult,axiom,
! [F: nat > nat,A: nat,F2: filter_nat,G: nat > nat,B: nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ A ) @ F2 )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ B ) @ F2 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( times_times_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo8926549440605965083ds_nat @ ( times_times_nat @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_mult
thf(fact_814_tendsto__mult,axiom,
! [F: nat > complex,A: complex,F2: filter_nat,G: nat > complex,B: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ B ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ ( times_times_complex @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_mult
thf(fact_815_tendsto__mult,axiom,
! [F: nat > real,A: real,F2: filter_nat,G: nat > real,B: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_mult
thf(fact_816_divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_817_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_818_add__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_819_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_820_add__num__frac,axiom,
! [Y: complex,Z: complex,X: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_821_add__num__frac,axiom,
! [Y: real,Z: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_822_add__frac__num,axiom,
! [Y: complex,X: complex,Z: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_823_add__frac__num,axiom,
! [Y: real,X: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_824_add__frac__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_825_add__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_826_add__divide__eq__if__simps_I1_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_827_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_828_add__divide__eq__if__simps_I2_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_829_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_830_nonzero__neg__divide__eq__eq2,axiom,
! [B: complex,C: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( C
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ( times_times_complex @ C @ B )
= ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_831_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C: real,A: real] :
( ( B != zero_zero_real )
=> ( ( C
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
= ( ( times_times_real @ C @ B )
= ( uminus_uminus_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_832_nonzero__neg__divide__eq__eq,axiom,
! [B: complex,A: complex,C: complex] :
( ( B != zero_zero_complex )
=> ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= C )
= ( ( uminus1482373934393186551omplex @ A )
= ( times_times_complex @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_833_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A: real,C: real] :
( ( B != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= C )
= ( ( uminus_uminus_real @ A )
= ( times_times_real @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_834_minus__divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
= A )
= ( ( ( C != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ B )
= ( times_times_complex @ A @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_835_minus__divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
= A )
= ( ( ( C != zero_zero_real )
=> ( ( uminus_uminus_real @ B )
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_836_eq__minus__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ A @ C )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_837_eq__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= ( uminus_uminus_real @ B ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_838_field__class_Ofield__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
= one_one_complex ) ) ).
% field_class.field_inverse
thf(fact_839_field__class_Ofield__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% field_class.field_inverse
thf(fact_840_division__ring__inverse__add,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_841_division__ring__inverse__add,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_842_inverse__add,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% inverse_add
thf(fact_843_inverse__add,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% inverse_add
thf(fact_844_is__num__normalize_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 ) ) ) ).
% is_num_normalize(1)
thf(fact_845_is__num__normalize_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 ) ) ) ).
% is_num_normalize(1)
thf(fact_846_is__num__normalize_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 ) ) ) ).
% is_num_normalize(1)
thf(fact_847_tendsto__mult__right__zero,axiom,
! [F: nat > complex,F2: filter_nat,C: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ C @ ( F @ X3 ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 ) ) ).
% tendsto_mult_right_zero
thf(fact_848_tendsto__mult__right__zero,axiom,
! [F: nat > real,F2: filter_nat,C: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ C @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% tendsto_mult_right_zero
thf(fact_849_tendsto__mult__left__zero,axiom,
! [F: nat > complex,F2: filter_nat,C: complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ ( F @ X3 ) @ C )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 ) ) ).
% tendsto_mult_left_zero
thf(fact_850_tendsto__mult__left__zero,axiom,
! [F: nat > real,F2: filter_nat,C: real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ ( F @ X3 ) @ C )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% tendsto_mult_left_zero
thf(fact_851_tendsto__mult__zero,axiom,
! [F: nat > complex,F2: filter_nat,G: nat > complex] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( ( filter6923414461901439796omplex @ G @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( filter6923414461901439796omplex
@ ^ [X3: nat] : ( times_times_complex @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ F2 ) ) ) ).
% tendsto_mult_zero
thf(fact_852_tendsto__mult__zero,axiom,
! [F: nat > real,F2: filter_nat,G: nat > real] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( times_times_real @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ) ).
% tendsto_mult_zero
thf(fact_853_tendsto__mult__one,axiom,
! [F: nat > nat,F2: filter_nat,G: nat > nat] :
( ( filterlim_nat_nat @ F @ ( topolo8926549440605965083ds_nat @ one_one_nat ) @ F2 )
=> ( ( filterlim_nat_nat @ G @ ( topolo8926549440605965083ds_nat @ one_one_nat ) @ F2 )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( times_times_nat @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( topolo8926549440605965083ds_nat @ one_one_nat )
@ F2 ) ) ) ).
% tendsto_mult_one
thf(fact_854_minus__divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_855_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_856_add__divide__eq__if__simps_I3_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_857_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_858_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_859_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_860_real__eq__affinity,axiom,
! [M: real,Y: real,X: real,C: real] :
( ( M != zero_zero_real )
=> ( ( Y
= ( plus_plus_real @ ( times_times_real @ M @ X ) @ C ) )
= ( ( plus_plus_real @ ( times_times_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( uminus_uminus_real @ ( divide_divide_real @ C @ M ) ) )
= X ) ) ) ).
% real_eq_affinity
thf(fact_861_real__affinity__eq,axiom,
! [M: real,X: real,C: real,Y: real] :
( ( M != zero_zero_real )
=> ( ( ( plus_plus_real @ ( times_times_real @ M @ X ) @ C )
= Y )
= ( X
= ( plus_plus_real @ ( times_times_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( uminus_uminus_real @ ( divide_divide_real @ C @ M ) ) ) ) ) ) ).
% real_affinity_eq
thf(fact_862_real__divide__square__eq,axiom,
! [R: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
= ( divide_divide_real @ A @ R ) ) ).
% real_divide_square_eq
thf(fact_863_vector__space__over__itself_Oscale__one,axiom,
! [X: complex] :
( ( times_times_complex @ one_one_complex @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_864_vector__space__over__itself_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_865_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: complex,X: complex,B: complex] :
( ( ( times_times_complex @ A @ X )
= ( times_times_complex @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_866_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X: real,B: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
= ( ( A = B )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_867_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: complex,X: complex,Y: complex] :
( ( ( times_times_complex @ A @ X )
= ( times_times_complex @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_868_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_869_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: complex,X: complex] :
( ( ( times_times_complex @ A @ X )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_870_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( times_times_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_871_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: complex] :
( ( times_times_complex @ zero_zero_complex @ X )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_872_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_873_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_874_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_875_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_876_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_877_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_878_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_879_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N2 ) )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_880_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_881_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N2 ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_882_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_883_mult__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ M @ ( suc @ N2 ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_884_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_885_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_886_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_887_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_888_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_889_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_890_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_891_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_892_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_893_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N2: int] :
( ( ( times_times_int @ M @ N2 )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_894_zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ( times_times_int @ M @ N2 )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N2 = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_895_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_896_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_897_div__mult2__eq,axiom,
! [M: nat,N2: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q ) ) ).
% div_mult2_eq
thf(fact_898_mult__Suc,axiom,
! [M: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).
% mult_Suc
thf(fact_899_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times_nat @ M @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_900_Multiseries__Expansion_Ointyness__simps_I2_J,axiom,
! [A: nat,B: nat] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ A ) @ ( semiri5074537144036343181t_real @ B ) )
= ( semiri5074537144036343181t_real @ ( times_times_nat @ A @ B ) ) ) ).
% Multiseries_Expansion.intyness_simps(2)
thf(fact_901_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: complex,A: complex,B: complex] :
( ( X != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ X )
= ( times_times_complex @ B @ X ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_902_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A: real,B: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ B @ X ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_903_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: complex,X: complex,Y: complex] :
( ( A != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ X )
= ( times_times_complex @ A @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_904_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_905_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: complex,B: complex,X: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ X )
= ( plus_plus_complex @ ( times_times_complex @ A @ X ) @ ( times_times_complex @ B @ X ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_906_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: real,B: real,X: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ B @ X ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_907_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: complex,X: complex,Y: complex] :
( ( times_times_complex @ A @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ X ) @ ( times_times_complex @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_908_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_909_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_910_mult__if__delta,axiom,
! [P: $o,Q: complex] :
( ( P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
= zero_zero_complex ) ) ) ).
% mult_if_delta
thf(fact_911_mult__if__delta,axiom,
! [P: $o,Q: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_912_mult__if__delta,axiom,
! [P: $o,Q: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_913_mult__if__delta,axiom,
! [P: $o,Q: int] :
( ( P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_int @ ( if_int @ P @ one_one_int @ zero_zero_int ) @ Q )
= zero_zero_int ) ) ) ).
% mult_if_delta
thf(fact_914_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_dec_simps(2)
thf(fact_915_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_916_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_917_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_918_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_919_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6511756317524482435omplex @ one_one_complex )
= one_one_complex ) ).
% dbl_dec_simps(3)
thf(fact_920_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_921_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_922_tendsto__norm__zero__cancel,axiom,
! [F: nat > complex,F2: filter_nat] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( real_V1022390504157884413omplex @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 ) ) ).
% tendsto_norm_zero_cancel
thf(fact_923_tendsto__norm__zero__cancel,axiom,
! [F: nat > real,F2: filter_nat] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( real_V7735802525324610683m_real @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 ) ) ).
% tendsto_norm_zero_cancel
thf(fact_924_tendsto__norm__zero__iff,axiom,
! [F: nat > complex,F2: filter_nat] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( real_V1022390504157884413omplex @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
= ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 ) ) ).
% tendsto_norm_zero_iff
thf(fact_925_tendsto__norm__zero__iff,axiom,
! [F: nat > real,F2: filter_nat] :
( ( filterlim_nat_real
@ ^ [X3: nat] : ( real_V7735802525324610683m_real @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
= ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 ) ) ).
% tendsto_norm_zero_iff
thf(fact_926_tendsto__norm__zero,axiom,
! [F: nat > complex,F2: filter_nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( real_V1022390504157884413omplex @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% tendsto_norm_zero
thf(fact_927_tendsto__norm__zero,axiom,
! [F: nat > real,F2: filter_nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( real_V7735802525324610683m_real @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% tendsto_norm_zero
thf(fact_928_inverse__mult__eq__1,axiom,
! [F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ F @ zero_zero_nat )
!= zero_zero_complex )
=> ( ( times_1444617028055533883omplex @ ( invers8029835185024579338omplex @ F ) @ F )
= one_on1590755018477040891omplex ) ) ).
% inverse_mult_eq_1
thf(fact_929_inverse__mult__eq__1,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( times_7561426564079326009s_real @ ( invers68952373231134600s_real @ F ) @ F )
= one_on8598947968683843321s_real ) ) ).
% inverse_mult_eq_1
thf(fact_930_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_inc_simps(4)
thf(fact_931_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_932_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_933_norm__minus__cancel,axiom,
! [X: real] :
( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ X ) )
= ( real_V7735802525324610683m_real @ X ) ) ).
% norm_minus_cancel
thf(fact_934_fps__add__nth,axiom,
! [F: formal670952693614245302omplex,G: formal670952693614245302omplex,N2: nat] :
( ( formal3666518339620930912omplex @ ( plus_p8472957120637115327omplex @ F @ G ) @ N2 )
= ( plus_plus_complex @ ( formal3666518339620930912omplex @ F @ N2 ) @ ( formal3666518339620930912omplex @ G @ N2 ) ) ) ).
% fps_add_nth
thf(fact_935_fps__add__nth,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat,N2: nat] :
( ( formal3720337525774269570th_nat @ ( plus_p6043471806551771617ps_nat @ F @ G ) @ N2 )
= ( plus_plus_nat @ ( formal3720337525774269570th_nat @ F @ N2 ) @ ( formal3720337525774269570th_nat @ G @ N2 ) ) ) ).
% fps_add_nth
thf(fact_936_fps__add__nth,axiom,
! [F: formal_Power_fps_int,G: formal_Power_fps_int,N2: nat] :
( ( formal3717847055265219294th_int @ ( plus_p1865620787042574909ps_int @ F @ G ) @ N2 )
= ( plus_plus_int @ ( formal3717847055265219294th_int @ F @ N2 ) @ ( formal3717847055265219294th_int @ G @ N2 ) ) ) ).
% fps_add_nth
thf(fact_937_fps__add__nth,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real,N2: nat] :
( ( formal2580924720334399070h_real @ ( plus_p6008488439947570109s_real @ F @ G ) @ N2 )
= ( plus_plus_real @ ( formal2580924720334399070h_real @ F @ N2 ) @ ( formal2580924720334399070h_real @ G @ N2 ) ) ) ).
% fps_add_nth
thf(fact_938_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_939_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_940_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_941_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_942_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_943_norm__one,axiom,
( ( real_V7735802525324610683m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_944_norm__of__nat,axiom,
! [N2: nat] :
( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% norm_of_nat
thf(fact_945_norm__of__nat,axiom,
! [N2: nat] :
( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% norm_of_nat
thf(fact_946_fps__zero__nth,axiom,
! [N2: nat] :
( ( formal3666518339620930912omplex @ zero_z1877163951443063103omplex @ N2 )
= zero_zero_complex ) ).
% fps_zero_nth
thf(fact_947_fps__zero__nth,axiom,
! [N2: nat] :
( ( formal3720337525774269570th_nat @ zero_z8531573698755551073ps_nat @ N2 )
= zero_zero_nat ) ).
% fps_zero_nth
thf(fact_948_fps__zero__nth,axiom,
! [N2: nat] :
( ( formal2580924720334399070h_real @ zero_z7760665558314615101s_real @ N2 )
= zero_zero_real ) ).
% fps_zero_nth
thf(fact_949_fps__zero__nth,axiom,
! [N2: nat] :
( ( formal3717847055265219294th_int @ zero_z4353722679246354365ps_int @ N2 )
= zero_zero_int ) ).
% fps_zero_nth
thf(fact_950_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_951_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_952_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_953_fps__nth__of__nat,axiom,
! [N2: nat,C: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( formal3666518339620930912omplex @ ( semiri8948773824294531479omplex @ C ) @ N2 )
= ( semiri8010041392384452111omplex @ C ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( formal3666518339620930912omplex @ ( semiri8948773824294531479omplex @ C ) @ N2 )
= zero_zero_complex ) ) ) ).
% fps_nth_of_nat
thf(fact_954_fps__nth__of__nat,axiom,
! [N2: nat,C: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C ) @ N2 )
= ( semiri5074537144036343181t_real @ C ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C ) @ N2 )
= zero_zero_real ) ) ) ).
% fps_nth_of_nat
thf(fact_955_fps__nth__of__nat,axiom,
! [N2: nat,C: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C ) @ N2 )
= ( semiri1314217659103216013at_int @ C ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C ) @ N2 )
= zero_zero_int ) ) ) ).
% fps_nth_of_nat
thf(fact_956_fps__nth__of__nat,axiom,
! [N2: nat,C: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C ) @ N2 )
= ( semiri1316708129612266289at_nat @ C ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C ) @ N2 )
= zero_zero_nat ) ) ) ).
% fps_nth_of_nat
thf(fact_957_fps__mult__nth__0,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G ) @ zero_zero_nat )
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_958_fps__mult__nth__0,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ zero_zero_nat )
= ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_959_fps__mult__nth__0,axiom,
! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ zero_zero_nat )
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_960_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal670952693614245302omplex,N2: nat] :
( ( formal3666518339620930912omplex @ ( times_1444617028055533883omplex @ ( semiri8948773824294531479omplex @ K ) @ F ) @ N2 )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( formal3666518339620930912omplex @ F @ N2 ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_961_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal3361831859752904756s_real,N2: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( semiri2475410149736220053s_real @ K ) @ F ) @ N2 )
= ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( formal2580924720334399070h_real @ F @ N2 ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_962_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal_Power_fps_int,N2: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ ( semiri6570152736363784213ps_int @ K ) @ F ) @ N2 )
= ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ ( formal3717847055265219294th_int @ F @ N2 ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_963_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal_Power_fps_nat,N2: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( semiri1524631719018205113ps_nat @ K ) @ F ) @ N2 )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ K ) @ ( formal3720337525774269570th_nat @ F @ N2 ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_964_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal670952693614245302omplex,K: nat,N2: nat] :
( ( formal3666518339620930912omplex @ ( times_1444617028055533883omplex @ F @ ( semiri8948773824294531479omplex @ K ) ) @ N2 )
= ( times_times_complex @ ( formal3666518339620930912omplex @ F @ N2 ) @ ( semiri8010041392384452111omplex @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_965_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal3361831859752904756s_real,K: nat,N2: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ ( semiri2475410149736220053s_real @ K ) ) @ N2 )
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ N2 ) @ ( semiri5074537144036343181t_real @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_966_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal_Power_fps_int,K: nat,N2: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ ( semiri6570152736363784213ps_int @ K ) ) @ N2 )
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ N2 ) @ ( semiri1314217659103216013at_int @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_967_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal_Power_fps_nat,K: nat,N2: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ ( semiri1524631719018205113ps_nat @ K ) ) @ N2 )
= ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ N2 ) @ ( semiri1316708129612266289at_nat @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_968_fps__inverse__0__iff,axiom,
! [F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ ( invers8029835185024579338omplex @ F ) @ zero_zero_nat )
= zero_zero_complex )
= ( ( formal3666518339620930912omplex @ F @ zero_zero_nat )
= zero_zero_complex ) ) ).
% fps_inverse_0_iff
thf(fact_969_fps__inverse__0__iff,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= zero_zero_real )
= ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real ) ) ).
% fps_inverse_0_iff
thf(fact_970_fps__inverse__idempotent,axiom,
! [F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ F @ zero_zero_nat )
!= zero_zero_complex )
=> ( ( invers8029835185024579338omplex @ ( invers8029835185024579338omplex @ F ) )
= F ) ) ).
% fps_inverse_idempotent
thf(fact_971_fps__inverse__idempotent,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( invers68952373231134600s_real @ ( invers68952373231134600s_real @ F ) )
= F ) ) ).
% fps_inverse_idempotent
thf(fact_972_fps__inverse__nth__0,axiom,
! [F: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) ) ) ).
% fps_inverse_nth_0
thf(fact_973_fps__one__nth,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( formal3666518339620930912omplex @ one_on1590755018477040891omplex @ N2 )
= one_one_complex ) )
& ( ( N2 != zero_zero_nat )
=> ( ( formal3666518339620930912omplex @ one_on1590755018477040891omplex @ N2 )
= zero_zero_complex ) ) ) ).
% fps_one_nth
thf(fact_974_fps__one__nth,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ one_on3350087005236239133ps_nat @ N2 )
= one_one_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ one_on3350087005236239133ps_nat @ N2 )
= zero_zero_nat ) ) ) ).
% fps_one_nth
thf(fact_975_fps__one__nth,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ one_on8598947968683843321s_real @ N2 )
= one_one_real ) )
& ( ( N2 != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ one_on8598947968683843321s_real @ N2 )
= zero_zero_real ) ) ) ).
% fps_one_nth
thf(fact_976_fps__one__nth,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ one_on8395608022581818233ps_int @ N2 )
= one_one_int ) )
& ( ( N2 != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ one_on8395608022581818233ps_int @ N2 )
= zero_zero_int ) ) ) ).
% fps_one_nth
thf(fact_977_fps__divide__nth__0,axiom,
! [G: formal670952693614245302omplex,F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ G @ zero_zero_nat )
!= zero_zero_complex )
=> ( ( formal3666518339620930912omplex @ ( divide1348722040316500488omplex @ F @ G ) @ zero_zero_nat )
= ( divide1717551699836669952omplex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) @ ( formal3666518339620930912omplex @ G @ zero_zero_nat ) ) ) ) ).
% fps_divide_nth_0
thf(fact_978_fps__divide__nth__0,axiom,
! [G: formal3361831859752904756s_real,F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ G @ zero_zero_nat )
!= zero_zero_real )
=> ( ( formal2580924720334399070h_real @ ( divide1155267253282662278s_real @ F @ G ) @ zero_zero_nat )
= ( divide_divide_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ) ).
% fps_divide_nth_0
thf(fact_979_fps__inverse__eq__0__iff,axiom,
! [F: formal670952693614245302omplex] :
( ( ( invers8029835185024579338omplex @ F )
= zero_z1877163951443063103omplex )
= ( ( formal3666518339620930912omplex @ F @ zero_zero_nat )
= zero_zero_complex ) ) ).
% fps_inverse_eq_0_iff
thf(fact_980_fps__inverse__eq__0__iff,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real )
= ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real ) ) ).
% fps_inverse_eq_0_iff
thf(fact_981_fps__mult__nth__1_H,axiom,
! [F: formal670952693614245302omplex,G: formal670952693614245302omplex] :
( ( formal3666518339620930912omplex @ ( times_1444617028055533883omplex @ F @ G ) @ ( suc @ zero_zero_nat ) )
= ( plus_plus_complex @ ( times_times_complex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) @ ( formal3666518339620930912omplex @ G @ ( suc @ zero_zero_nat ) ) ) @ ( times_times_complex @ ( formal3666518339620930912omplex @ F @ ( suc @ zero_zero_nat ) ) @ ( formal3666518339620930912omplex @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1'
thf(fact_982_fps__mult__nth__1_H,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G ) @ ( suc @ zero_zero_nat ) )
= ( plus_plus_real @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ ( suc @ zero_zero_nat ) ) ) @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ ( suc @ zero_zero_nat ) ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1'
thf(fact_983_fps__mult__nth__1_H,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ ( suc @ zero_zero_nat ) )
= ( plus_plus_nat @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ ( suc @ zero_zero_nat ) ) ) @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ ( suc @ zero_zero_nat ) ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1'
thf(fact_984_fps__mult__nth__1_H,axiom,
! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ ( suc @ zero_zero_nat ) )
= ( plus_plus_int @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ ( suc @ zero_zero_nat ) ) ) @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ ( suc @ zero_zero_nat ) ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1'
thf(fact_985_norm__mult,axiom,
! [X: real,Y: real] :
( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% norm_mult
thf(fact_986_fps__nonzeroI,axiom,
! [F: formal670952693614245302omplex,N2: nat] :
( ( ( formal3666518339620930912omplex @ F @ N2 )
!= zero_zero_complex )
=> ( F != zero_z1877163951443063103omplex ) ) ).
% fps_nonzeroI
thf(fact_987_fps__nonzeroI,axiom,
! [F: formal_Power_fps_nat,N2: nat] :
( ( ( formal3720337525774269570th_nat @ F @ N2 )
!= zero_zero_nat )
=> ( F != zero_z8531573698755551073ps_nat ) ) ).
% fps_nonzeroI
thf(fact_988_fps__nonzeroI,axiom,
! [F: formal3361831859752904756s_real,N2: nat] :
( ( ( formal2580924720334399070h_real @ F @ N2 )
!= zero_zero_real )
=> ( F != zero_z7760665558314615101s_real ) ) ).
% fps_nonzeroI
thf(fact_989_fps__nonzeroI,axiom,
! [F: formal_Power_fps_int,N2: nat] :
( ( ( formal3717847055265219294th_int @ F @ N2 )
!= zero_zero_int )
=> ( F != zero_z4353722679246354365ps_int ) ) ).
% fps_nonzeroI
thf(fact_990_fps__nonzero__nth,axiom,
! [F: formal670952693614245302omplex] :
( ( F != zero_z1877163951443063103omplex )
= ( ? [N: nat] :
( ( formal3666518339620930912omplex @ F @ N )
!= zero_zero_complex ) ) ) ).
% fps_nonzero_nth
thf(fact_991_fps__nonzero__nth,axiom,
! [F: formal_Power_fps_nat] :
( ( F != zero_z8531573698755551073ps_nat )
= ( ? [N: nat] :
( ( formal3720337525774269570th_nat @ F @ N )
!= zero_zero_nat ) ) ) ).
% fps_nonzero_nth
thf(fact_992_fps__nonzero__nth,axiom,
! [F: formal3361831859752904756s_real] :
( ( F != zero_z7760665558314615101s_real )
= ( ? [N: nat] :
( ( formal2580924720334399070h_real @ F @ N )
!= zero_zero_real ) ) ) ).
% fps_nonzero_nth
thf(fact_993_fps__nonzero__nth,axiom,
! [F: formal_Power_fps_int] :
( ( F != zero_z4353722679246354365ps_int )
= ( ? [N: nat] :
( ( formal3717847055265219294th_int @ F @ N )
!= zero_zero_int ) ) ) ).
% fps_nonzero_nth
thf(fact_994_norm__divide,axiom,
! [A: complex,B: complex] :
( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% norm_divide
thf(fact_995_norm__divide,axiom,
! [A: real,B: real] :
( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% norm_divide
thf(fact_996_norm__inverse,axiom,
! [A: real] :
( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
= ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% norm_inverse
thf(fact_997_nonzero__norm__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_998_nonzero__norm__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_999_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_1000_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_1001_dbl__inc__def,axiom,
( neg_nu8557863876264182079omplex
= ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).
% dbl_inc_def
thf(fact_1002_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_1003_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_1004_fps__inverse__0__iff_H,axiom,
! [F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ ( invers8029835185024579338omplex @ F ) @ zero_zero_nat )
= zero_zero_complex )
= ( ( invers8013647133539491842omplex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) )
= zero_zero_complex ) ) ).
% fps_inverse_0_iff'
thf(fact_1005_fps__inverse__0__iff_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= zero_zero_real )
= ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real ) ) ).
% fps_inverse_0_iff'
thf(fact_1006_fps__unit__dvd__right,axiom,
! [F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ F @ zero_zero_nat )
!= zero_zero_complex )
=> ? [G2: formal670952693614245302omplex] :
( one_on1590755018477040891omplex
= ( times_1444617028055533883omplex @ G2 @ F ) ) ) ).
% fps_unit_dvd_right
thf(fact_1007_fps__unit__dvd__right,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ? [G2: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ G2 @ F ) ) ) ).
% fps_unit_dvd_right
thf(fact_1008_fps__unit__dvd__left,axiom,
! [F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ F @ zero_zero_nat )
!= zero_zero_complex )
=> ? [G2: formal670952693614245302omplex] :
( one_on1590755018477040891omplex
= ( times_1444617028055533883omplex @ F @ G2 ) ) ) ).
% fps_unit_dvd_left
thf(fact_1009_fps__unit__dvd__left,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ? [G2: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ F @ G2 ) ) ) ).
% fps_unit_dvd_left
thf(fact_1010_fps__inverse__eq__0,axiom,
! [F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( invers8029835185024579338omplex @ F )
= zero_z1877163951443063103omplex ) ) ).
% fps_inverse_eq_0
thf(fact_1011_fps__inverse__eq__0,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_eq_0
thf(fact_1012_tendsto__norm,axiom,
! [F: nat > complex,A: complex,F2: filter_nat] :
( ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ A ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( real_V1022390504157884413omplex @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( real_V1022390504157884413omplex @ A ) )
@ F2 ) ) ).
% tendsto_norm
thf(fact_1013_tendsto__norm,axiom,
! [F: nat > real,A: real,F2: filter_nat] :
( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( filterlim_nat_real
@ ^ [X3: nat] : ( real_V7735802525324610683m_real @ ( F @ X3 ) )
@ ( topolo2815343760600316023s_real @ ( real_V7735802525324610683m_real @ A ) )
@ F2 ) ) ).
% tendsto_norm
thf(fact_1014_fps__mult__nth__1,axiom,
! [F: formal670952693614245302omplex,G: formal670952693614245302omplex] :
( ( formal3666518339620930912omplex @ ( times_1444617028055533883omplex @ F @ G ) @ one_one_nat )
= ( plus_plus_complex @ ( times_times_complex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) @ ( formal3666518339620930912omplex @ G @ one_one_nat ) ) @ ( times_times_complex @ ( formal3666518339620930912omplex @ F @ one_one_nat ) @ ( formal3666518339620930912omplex @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1
thf(fact_1015_fps__mult__nth__1,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G ) @ one_one_nat )
= ( plus_plus_real @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ one_one_nat ) ) @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ one_one_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1
thf(fact_1016_fps__mult__nth__1,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ one_one_nat )
= ( plus_plus_nat @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ one_one_nat ) ) @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ one_one_nat ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1
thf(fact_1017_fps__mult__nth__1,axiom,
! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ one_one_nat )
= ( plus_plus_int @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ one_one_nat ) ) @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ one_one_nat ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1
thf(fact_1018_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
! [F: formal670952693614245302omplex] :
( ( ? [G3: formal670952693614245302omplex] :
( one_on1590755018477040891omplex
= ( times_1444617028055533883omplex @ F @ G3 ) ) )
= ( ? [K2: complex] :
( one_one_complex
= ( times_times_complex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) @ K2 ) ) ) ) ).
% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_1019_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
! [F: formal3361831859752904756s_real] :
( ( ? [G3: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ F @ G3 ) ) )
= ( ? [K2: real] :
( one_one_real
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ K2 ) ) ) ) ).
% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_1020_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
! [F: formal_Power_fps_int] :
( ( ? [G3: formal_Power_fps_int] :
( one_on8395608022581818233ps_int
= ( times_3091854549176928185ps_int @ F @ G3 ) ) )
= ( ? [K2: int] :
( one_one_int
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ K2 ) ) ) ) ).
% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_1021_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
! [F: formal670952693614245302omplex] :
( ( ? [G3: formal670952693614245302omplex] :
( one_on1590755018477040891omplex
= ( times_1444617028055533883omplex @ G3 @ F ) ) )
= ( ? [K2: complex] :
( one_one_complex
= ( times_times_complex @ K2 @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) ) ) ) ) ).
% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_1022_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
! [F: formal3361831859752904756s_real] :
( ( ? [G3: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ G3 @ F ) ) )
= ( ? [K2: real] :
( one_one_real
= ( times_times_real @ K2 @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) ) ) ) ) ).
% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_1023_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
! [F: formal_Power_fps_int] :
( ( ? [G3: formal_Power_fps_int] :
( one_on8395608022581818233ps_int
= ( times_3091854549176928185ps_int @ G3 @ F ) ) )
= ( ? [K2: int] :
( one_one_int
= ( times_times_int @ K2 @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) ) ) ) ) ).
% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_1024_fps__inverse__eq__0_H,axiom,
! [F: formal670952693614245302omplex] :
( ( ( invers8013647133539491842omplex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) )
= zero_zero_complex )
=> ( ( invers8029835185024579338omplex @ F )
= zero_z1877163951443063103omplex ) ) ).
% fps_inverse_eq_0'
thf(fact_1025_fps__inverse__eq__0_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_eq_0'
thf(fact_1026_fps__inverse__eq__0__iff_H,axiom,
! [F: formal670952693614245302omplex] :
( ( ( invers8029835185024579338omplex @ F )
= zero_z1877163951443063103omplex )
= ( ( invers8013647133539491842omplex @ ( formal3666518339620930912omplex @ F @ zero_zero_nat ) )
= zero_zero_complex ) ) ).
% fps_inverse_eq_0_iff'
thf(fact_1027_fps__inverse__eq__0__iff_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real )
= ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real ) ) ).
% fps_inverse_eq_0_iff'
thf(fact_1028_fps__divide__unit,axiom,
! [G: formal670952693614245302omplex,F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ G @ zero_zero_nat )
!= zero_zero_complex )
=> ( ( divide1348722040316500488omplex @ F @ G )
= ( times_1444617028055533883omplex @ F @ ( invers8029835185024579338omplex @ G ) ) ) ) ).
% fps_divide_unit
thf(fact_1029_fps__divide__unit,axiom,
! [G: formal3361831859752904756s_real,F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ G @ zero_zero_nat )
!= zero_zero_real )
=> ( ( divide1155267253282662278s_real @ F @ G )
= ( times_7561426564079326009s_real @ F @ ( invers68952373231134600s_real @ G ) ) ) ) ).
% fps_divide_unit
thf(fact_1030_inverse__mult__eq__1_H,axiom,
! [F: formal670952693614245302omplex] :
( ( ( formal3666518339620930912omplex @ F @ zero_zero_nat )
!= zero_zero_complex )
=> ( ( times_1444617028055533883omplex @ F @ ( invers8029835185024579338omplex @ F ) )
= one_on1590755018477040891omplex ) ) ).
% inverse_mult_eq_1'
thf(fact_1031_inverse__mult__eq__1_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( times_7561426564079326009s_real @ F @ ( invers68952373231134600s_real @ F ) )
= one_on8598947968683843321s_real ) ) ).
% inverse_mult_eq_1'
thf(fact_1032_fps__XDp__nth,axiom,
! [C: complex,A: formal670952693614245302omplex,N2: nat] :
( ( formal3666518339620930912omplex @ ( formal5989188765539143467omplex @ C @ A ) @ N2 )
= ( times_times_complex @ ( plus_plus_complex @ C @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( formal3666518339620930912omplex @ A @ N2 ) ) ) ).
% fps_XDp_nth
thf(fact_1033_fps__XDp__nth,axiom,
! [C: real,A: formal3361831859752904756s_real,N2: nat] :
( ( formal2580924720334399070h_real @ ( formal2839450981996073129p_real @ C @ A ) @ N2 )
= ( times_times_real @ ( plus_plus_real @ C @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( formal2580924720334399070h_real @ A @ N2 ) ) ) ).
% fps_XDp_nth
thf(fact_1034_fps__XDp__nth,axiom,
! [C: int,A: formal_Power_fps_int,N2: nat] :
( ( formal3717847055265219294th_int @ ( formal9195297484582036137Dp_int @ C @ A ) @ N2 )
= ( times_times_int @ ( plus_plus_int @ C @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( formal3717847055265219294th_int @ A @ N2 ) ) ) ).
% fps_XDp_nth
thf(fact_1035_fps__XDp__nth,axiom,
! [C: nat,A: formal_Power_fps_nat,N2: nat] :
( ( formal3720337525774269570th_nat @ ( formal9197787955091086413Dp_nat @ C @ A ) @ N2 )
= ( times_times_nat @ ( plus_plus_nat @ C @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( formal3720337525774269570th_nat @ A @ N2 ) ) ) ).
% fps_XDp_nth
thf(fact_1036_LIMSEQ__norm__0,axiom,
! [F: nat > complex] :
( ! [N3: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( filter6923414461901439796omplex @ F @ ( topolo2444363109189100025omplex @ zero_zero_complex ) @ at_top_nat ) ) ).
% LIMSEQ_norm_0
thf(fact_1037_LIMSEQ__norm__0,axiom,
! [F: nat > real] :
( ! [N3: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% LIMSEQ_norm_0
thf(fact_1038_fps__XD__Suc,axiom,
! [A: formal670952693614245302omplex,N2: nat] :
( ( formal3666518339620930912omplex @ ( formal1655152611307539683omplex @ A ) @ ( suc @ N2 ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( formal3666518339620930912omplex @ A @ ( suc @ N2 ) ) ) ) ).
% fps_XD_Suc
thf(fact_1039_fps__XD__Suc,axiom,
! [A: formal3361831859752904756s_real,N2: nat] :
( ( formal2580924720334399070h_real @ ( formal4292469010823635553D_real @ A ) @ ( suc @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( formal2580924720334399070h_real @ A @ ( suc @ N2 ) ) ) ) ).
% fps_XD_Suc
thf(fact_1040_fps__XD__Suc,axiom,
! [A: formal_Power_fps_int,N2: nat] :
( ( formal3717847055265219294th_int @ ( formal812433016830480481XD_int @ A ) @ ( suc @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( formal3717847055265219294th_int @ A @ ( suc @ N2 ) ) ) ) ).
% fps_XD_Suc
thf(fact_1041_fps__XD__Suc,axiom,
! [A: formal_Power_fps_nat,N2: nat] :
( ( formal3720337525774269570th_nat @ ( formal814923487339530757XD_nat @ A ) @ ( suc @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( formal3720337525774269570th_nat @ A @ ( suc @ N2 ) ) ) ) ).
% fps_XD_Suc
thf(fact_1042_fps__hypergeo__0,axiom,
! [As: list_complex,Bs: list_complex,C: complex] :
( ( formal3666518339620930912omplex @ ( formal7789817089755916220omplex @ As @ Bs @ C ) @ zero_zero_nat )
= one_one_complex ) ).
% fps_hypergeo_0
thf(fact_1043_fps__hypergeo__0,axiom,
! [As: list_real,Bs: list_real,C: real] :
( ( formal2580924720334399070h_real @ ( formal6618874005373735610o_real @ As @ Bs @ C ) @ zero_zero_nat )
= one_one_real ) ).
% fps_hypergeo_0
thf(fact_1044_fps__ln__0,axiom,
! [C: complex] :
( ( formal3666518339620930912omplex @ ( formal6928690614366948857omplex @ C ) @ zero_zero_nat )
= zero_zero_complex ) ).
% fps_ln_0
thf(fact_1045_fps__ln__0,axiom,
! [C: real] :
( ( formal2580924720334399070h_real @ ( formal8688746759596762231n_real @ C ) @ zero_zero_nat )
= zero_zero_real ) ).
% fps_ln_0
thf(fact_1046_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1047_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_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1058_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1059_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1082_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y6: real] :
? [N3: nat] : ( ord_less_real @ Y6 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1083_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_1084_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_1085_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
= ( ? [N: nat] :
( ( N != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_1086_forall__pos__mono,axiom,
! [P: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less_real @ D3 @ E2 )
=> ( ( P @ D3 )
=> ( P @ E2 ) ) )
=> ( ! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_1087_forall__pos__mono__1,axiom,
! [P: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less_real @ D3 @ E2 )
=> ( ( P @ D3 )
=> ( P @ E2 ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono_1
thf(fact_1088_real__arch__invD,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E ) ) ) ).
% real_arch_invD
thf(fact_1089_square__bound__lemma,axiom,
! [X: real] : ( ord_less_real @ X @ ( times_times_real @ ( plus_plus_real @ one_one_real @ X ) @ ( plus_plus_real @ one_one_real @ X ) ) ) ).
% square_bound_lemma
thf(fact_1090_Multiseries__Expansion__Bounds_Opos__imp__inverse__pos,axiom,
! [F: real > real,X6: real] :
( ( ord_less_real @ zero_zero_real @ ( F @ X6 ) )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( F @ X6 ) ) ) ) ).
% Multiseries_Expansion_Bounds.pos_imp_inverse_pos
thf(fact_1091_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_1092_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1093_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1094_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1095_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1096_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_1097_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1098_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1099_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1100_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1101_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1102_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1103_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1104_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1105_div__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1106_negative__zless,axiom,
! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1107_div__mult__self1__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1108_div__mult__self__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1109_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1110_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1111_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1112_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1113_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1114_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_1115_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_1116_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1117_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_1118_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_1119_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1120_add__less__mono,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_less_mono
thf(fact_1121_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1122_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1123_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1124_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1125_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1126_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1127_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1128_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1129_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_1130_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1131_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1132_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_1133_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1134_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
& ( P @ I2 ) ) )
= ( ( P @ N2 )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1135_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1136_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1137_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
=> ( P @ I2 ) ) )
= ( ( P @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_1138_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1139_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_1140_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1141_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1142_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1143_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1144_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1145_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1146_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1147_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1148_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1149_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1150_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1151_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_1152_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1153_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1154_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1155_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1156_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1157_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_1158_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1159_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1160_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1161_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1162_less__imp__Suc__add,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1163_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N: nat] :
? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M5 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1164_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1165_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1166_less__natE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ! [Q2: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1167_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1168_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1169_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1170_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1171_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1172_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1173_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_1174_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1175_not__int__zless__negative,axiom,
! [N2: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1176_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N2: nat] :
( ( ( divide_divide_nat @ M @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1177_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1178_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1179_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1180_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1181_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
thf(fact_1182_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1183_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1184_one__less__mult,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1185_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z3: int] :
? [N: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1186_pos__zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N2 )
= one_one_int )
= ( ( M = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1187_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_1188_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1189_div__less__dividend,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1190_div__eq__dividend__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N2 )
= M )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1191_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q ) @ N2 )
= ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1192_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1193_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1194_negative__zless__0,axiom,
! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1195_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1196_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1197_split__div,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N2 != zero_zero_nat )
=> ! [I2: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N2 )
& ( M
= ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1198_dividend__less__div__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1199_dividend__less__times__div,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1200_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1201_mult__nat__right__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat
@ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
@ at_top_nat
@ at_top_nat ) ) ).
% mult_nat_right_at_top
thf(fact_1202_mult__nat__left__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% mult_nat_left_at_top
thf(fact_1203_Multiseries__Expansion__Bounds_Oneg__imp__inverse__neg,axiom,
! [F: real > real,X6: real] :
( ( ord_less_real @ ( F @ X6 ) @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ ( F @ X6 ) ) @ zero_zero_real ) ) ).
% Multiseries_Expansion_Bounds.neg_imp_inverse_neg
thf(fact_1204_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_1205_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1206_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1207_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1208_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1209_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1210_Multiseries__Expansion_Ointyness__simps_I3_J,axiom,
! [A: nat,N2: nat] :
( ( power_power_real @ ( semiri5074537144036343181t_real @ A ) @ N2 )
= ( semiri5074537144036343181t_real @ ( power_power_nat @ A @ N2 ) ) ) ).
% Multiseries_Expansion.intyness_simps(3)
thf(fact_1211_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_1212_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1213_realpow__pos__nth2,axiom,
! [A: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ ( suc @ N2 ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1214_realpow__pos__nth,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R2: real] :
( ( ord_less_real @ zero_zero_real @ R2 )
& ( ( power_power_real @ R2 @ N2 )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1215_realpow__pos__nth__unique,axiom,
! [N2: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N2 )
= A )
& ! [Y6: real] :
( ( ( ord_less_real @ zero_zero_real @ Y6 )
& ( ( power_power_real @ Y6 @ N2 )
= A ) )
=> ( Y6 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1216_reals__power__lt__ex,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ Y )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_real @ ( power_power_real @ ( divide_divide_real @ one_one_real @ Y ) @ K3 ) @ X ) ) ) ) ).
% reals_power_lt_ex
thf(fact_1217_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_1218_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1219_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1220_Suc__diff__diff,axiom,
! [M: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1221_diff__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1222_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1223_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_1224_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1225_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_1226_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_1227_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1228_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1229_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_1230_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1231_diff__less__Suc,axiom,
! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1232_diff__add__0,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1233_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1234_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less_nat @ M @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1235_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ M @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1236_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1237_diff__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_cancel2
thf(fact_1238_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_1239_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_1240_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1241_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1242_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1243_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1244_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1245_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1246_filterlim__minus__const__nat__at__top,axiom,
! [C: nat] :
( filterlim_nat_nat
@ ^ [N: nat] : ( minus_minus_nat @ N @ C )
@ at_top_nat
@ at_top_nat ) ).
% filterlim_minus_const_nat_at_top
thf(fact_1247_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_1248_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 ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1249_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 ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1250_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1251_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1252_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1253_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N ) ) ) ) ) ).
% mult_eq_if
thf(fact_1254_div__if,axiom,
( divide_divide_nat
= ( ^ [M5: nat,N: nat] :
( if_nat
@ ( ( ord_less_nat @ M5 @ N )
| ( N = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N ) @ N ) ) ) ) ) ).
% div_if
thf(fact_1255_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1256_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1257_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_1258_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_1259_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X3: real,Y5: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_1260_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_1261_int__diff__cases,axiom,
! [Z: int] :
~ ! [M2: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_1262_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1263_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1264_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_1265_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1266_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1267_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K3: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1268_plusinfinity,axiom,
! [D: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int,K3: int] :
( ( P2 @ X4 )
= ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [X_1: int] : ( P2 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
% Helper facts (9)
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_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
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 ) ).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( if_complex @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( filter6923414461901439796omplex
@ ^ [N: nat] : ( divide1717551699836669952omplex @ one_one_complex @ ( plus_plus_complex @ x @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) ) )
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ at_top_nat ) ).
%------------------------------------------------------------------------------