TPTP Problem File: SLH0760^1.p
View Solutions
- 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_00484_019475__14104050_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1442 ( 805 unt; 155 typ; 0 def)
% Number of atoms : 2767 (1817 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 9293 ( 294 ~; 72 |; 66 &;8129 @)
% ( 0 <=>; 732 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 406 ( 406 >; 0 *; 0 +; 0 <<)
% Number of symbols : 141 ( 138 usr; 21 con; 0-3 aty)
% Number of variables : 2842 ( 242 ^;2576 !; 24 ?;2842 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:59:42.955
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_n_t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2417102609627094330l_num1: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Extended____Real__Oereal_J,type,
filter5793248256798695124_ereal: $tType ).
thf(ty_n_t__Filter__Ofilter_It__Complex__Ocomplex_J,type,
filter_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__Filter__Ofilter_It__Int__Oint_J,type,
filter_int: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $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__Extended____Real__Oereal,type,
extended_ereal: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (138)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(sy_c_Cotangent__PFD__Formula_Ocot__pfd_001t__Complex__Ocomplex,type,
cotang8298477626502807258omplex: complex > complex ).
thf(sy_c_Cotangent__PFD__Formula_Ocot__pfd_001t__Real__Oreal,type,
cotang1502006655779026648d_real: real > real ).
thf(sy_c_Filter_Ofilterlim_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
filter8330067395343389202omplex: ( complex > complex ) > filter_complex > filter_complex > $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__Extended____Real__Oereal,type,
filter2787094286561295468_ereal: ( real > extended_ereal ) > filter5793248256798695124_ereal > filter_real > $o ).
thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Int__Oint,type,
filterlim_real_int: ( real > int ) > filter_int > 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_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Real__Oereal,type,
minus_2816186181549245109_ereal: extended_ereal > extended_ereal > extended_ereal ).
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__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
minus_838314146864362899l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
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__Extended____Real__Oereal,type,
one_on4623092294121504201_ereal: extended_ereal ).
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__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
one_on3868389512446148991l_num1: numera2417102609627094330l_num1 ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Real__Oereal,type,
plus_p7876563987511257093_ereal: extended_ereal > extended_ereal > extended_ereal ).
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__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Real__Oereal,type,
times_7703590493115627913_ereal: extended_ereal > extended_ereal > extended_ereal ).
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__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
times_8498157372700349887l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
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__Num__Onum_J,type,
times_times_set_num: set_num > set_num > set_num ).
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_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Real__Oereal,type,
zero_z2744965634713055877_ereal: extended_ereal ).
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__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
zero_z5982384998485459395l_num1: numera2417102609627094330l_num1 ).
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__Extended____Real__Oereal,type,
if_Extended_ereal: $o > extended_ereal > extended_ereal > extended_ereal ).
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__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
if_Num9196306924077011444l_num1: $o > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > numera2417102609627094330l_num1 ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Limits_Oat__infinity_001t__Complex__Ocomplex,type,
at_infinity_complex: filter_complex ).
thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
at_infinity_real: filter_real ).
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__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__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
semiri1795386414920522267l_num1: nat > numera2417102609627094330l_num1 ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set_nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
numera6690914467698888265omplex: num > complex ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Real__Oereal,type,
numera1204434989813589363_ereal: num > extended_ereal ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
numera2161328050825114965l_num1: num > numera2417102609627094330l_num1 ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le1083603963089353582_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
top_top_set_complex: set_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Parity_Oadjust__mod,type,
adjust_mod: num > int > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Extended____Real__Oereal,type,
power_1054015426188190660_ereal: extended_ereal > nat > extended_ereal ).
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__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
power_7402600760894073284l_num1: numera2417102609627094330l_num1 > nat > numera2417102609627094330l_num1 ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Extended____Real__Oereal,type,
divide8893690120176169980_ereal: extended_ereal > extended_ereal > extended_ereal ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
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_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
dvd_dvd_complex: complex > complex > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Extended____Real__Oereal,type,
dvd_dv547123082783338664_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J,type,
dvd_dv2285863382094241760l_num1: numera2417102609627094330l_num1 > numera2417102609627094330l_num1 > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
suminf_complex: ( nat > complex ) > complex ).
thf(sy_c_Series_Osuminf_001t__Extended____Real__Oereal,type,
suminf4411151127299490740_ereal: ( nat > extended_ereal ) > extended_ereal ).
thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
suminf_int: ( nat > int ) > int ).
thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
suminf_nat: ( nat > nat ) > nat ).
thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
suminf_real: ( nat > real ) > real ).
thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
summable_complex: ( nat > complex ) > $o ).
thf(sy_c_Series_Osummable_001t__Extended____Real__Oereal,type,
summab1499615597823217518_ereal: ( nat > extended_ereal ) > $o ).
thf(sy_c_Series_Osummable_001t__Int__Oint,type,
summable_int: ( nat > int ) > $o ).
thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
summable_nat: ( nat > nat ) > $o ).
thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
summable_real: ( nat > real ) > $o ).
thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(sy_c_Series_Osums_001t__Extended____Real__Oereal,type,
sums_Extended_ereal: ( nat > extended_ereal ) > extended_ereal > $o ).
thf(sy_c_Series_Osums_001t__Int__Oint,type,
sums_int: ( nat > int ) > int > $o ).
thf(sy_c_Series_Osums_001t__Nat__Onat,type,
sums_nat: ( nat > nat ) > nat > $o ).
thf(sy_c_Series_Osums_001t__Real__Oreal,type,
sums_real: ( nat > real ) > real > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Complex__Ocomplex,type,
topolo3898190780742966766omplex: complex > set_complex > filter_complex ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Nat__Onat,type,
topolo4659099751122792720in_nat: nat > set_nat > filter_nat ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
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__Extended____Real__Oereal,type,
topolo4481849189275076035_ereal: extended_ereal > filter5793248256798695124_ereal ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Int__Oint,type,
topolo8924058970096914807ds_int: int > filter_int ).
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_Transcendental_Oarsinh_001t__Complex__Ocomplex,type,
arsinh_complex: complex > complex ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Complex__Ocomplex,type,
artanh_complex: complex > complex ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
diffs_complex: ( nat > complex ) > nat > complex ).
thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
diffs_real: ( nat > real ) > nat > 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__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_n____,type,
n: nat ).
% Relevant facts (1273)
thf(fact_0_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_1_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_2_zero__eq__power2,axiom,
! [A: complex] :
( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% zero_eq_power2
thf(fact_3_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_4_tendsto__mult__left__iff,axiom,
! [C: real,F: real > real,L: real,F2: filter_real] :
( ( C != zero_zero_real )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ C @ L ) )
@ F2 )
= ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ) ).
% tendsto_mult_left_iff
thf(fact_5_tendsto__mult__right__iff,axiom,
! [C: real,F: real > real,L: real,F2: filter_real] :
( ( C != zero_zero_real )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ L @ C ) )
@ F2 )
= ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ) ).
% tendsto_mult_right_iff
thf(fact_6_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= ( semiri1316708129612266289at_nat @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_7_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N )
= ( semiri5074537144036343181t_real @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_8_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N )
= ( semiri1314217659103216013at_int @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_9_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y: nat] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N )
= ( semiri8010041392384452111omplex @ Y ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N )
= Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_10_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X2: num,N: nat] :
( ( ( semiri1316708129612266289at_nat @ Y )
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_11_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X2: num,N: nat] :
( ( ( semiri5074537144036343181t_real @ Y )
= ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_12_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X2: num,N: nat] :
( ( ( semiri1314217659103216013at_int @ Y )
= ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_13_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X2: num,N: nat] :
( ( ( semiri8010041392384452111omplex @ Y )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N ) )
= ( Y
= ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_14_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
= A )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_15_divide__eq__eq__numeral1_I1_J,axiom,
! [B: complex,W: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
= A )
= ( ( ( ( numera6690914467698888265omplex @ W )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
& ( ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_16_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
= ( ( ( ( numeral_numeral_real @ W )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
= B ) )
& ( ( ( numeral_numeral_real @ W )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_17_eq__divide__eq__numeral1_I1_J,axiom,
! [A: complex,B: complex,W: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
= ( ( ( ( numera6690914467698888265omplex @ W )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
= B ) )
& ( ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_18_tendsto__ident__at,axiom,
! [A: real,S: set_real] :
( filterlim_real_real
@ ^ [X: real] : X
@ ( topolo2815343760600316023s_real @ A )
@ ( topolo2177554685111907308n_real @ A @ S ) ) ).
% tendsto_ident_at
thf(fact_19_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1795386414920522267l_num1 @ ( numeral_numeral_nat @ N ) )
= ( numera2161328050825114965l_num1 @ N ) ) ).
% of_nat_numeral
thf(fact_20_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_21_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% of_nat_numeral
thf(fact_22_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_23_of__nat__numeral,axiom,
! [N: num] :
( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
= ( numera6690914467698888265omplex @ N ) ) ).
% of_nat_numeral
thf(fact_24_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_25_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_26_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
= zero_zero_complex ) ).
% power_zero_numeral
thf(fact_27_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_28_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_29_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_30_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_31_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_32_left__diff__distrib__numeral,axiom,
! [A: numera2417102609627094330l_num1,B: numera2417102609627094330l_num1,V: num] :
( ( times_8498157372700349887l_num1 @ ( minus_838314146864362899l_num1 @ A @ B ) @ ( numera2161328050825114965l_num1 @ V ) )
= ( minus_838314146864362899l_num1 @ ( times_8498157372700349887l_num1 @ A @ ( numera2161328050825114965l_num1 @ V ) ) @ ( times_8498157372700349887l_num1 @ B @ ( numera2161328050825114965l_num1 @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_33_left__diff__distrib__numeral,axiom,
! [A: complex,B: complex,V: num] :
( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_34_left__diff__distrib__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_35_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_36_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera6690914467698888265omplex @ M )
= ( numera6690914467698888265omplex @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_37_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_38_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_39_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_40_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ W ) @ Z ) )
= ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_41_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_42_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_43_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_44_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_45_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ M ) @ ( numera2161328050825114965l_num1 @ N ) )
= ( numera2161328050825114965l_num1 @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_46_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_47_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_48_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_49_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_50_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_51_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_52_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_53_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_54_power__Suc0__right,axiom,
! [A: extended_ereal] :
( ( power_1054015426188190660_ereal @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_55_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_56_power__mult__numeral,axiom,
! [A: real,M: num,N: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_57_power__mult__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_58_power__mult__numeral,axiom,
! [A: complex,M: num,N: num] :
( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_59_power__mult__numeral,axiom,
! [A: extended_ereal,M: num,N: num] :
( ( power_1054015426188190660_ereal @ ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_60_power__mult__numeral,axiom,
! [A: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_61_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_62_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M: nat] :
( ( ( power_power_nat @ X2 @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_63_tendsto__const,axiom,
! [K: real,F2: filter_real] :
( filterlim_real_real
@ ^ [X: real] : K
@ ( topolo2815343760600316023s_real @ K )
@ F2 ) ).
% tendsto_const
thf(fact_64_right__diff__distrib__numeral,axiom,
! [V: num,B: numera2417102609627094330l_num1,C: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ ( minus_838314146864362899l_num1 @ B @ C ) )
= ( minus_838314146864362899l_num1 @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ B ) @ ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_65_right__diff__distrib__numeral,axiom,
! [V: num,B: complex,C: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_66_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_67_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_68_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X2 )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_69_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X2 )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_70_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri8010041392384452111omplex @ X2 )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_71_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X2: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X2 )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X2
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_72_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_73_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_74_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
= ( semiri8010041392384452111omplex @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_75_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X2: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X2 ) )
= ( ( power_power_nat @ B @ W )
= X2 ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_76_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_77_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_78_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% of_nat_power
thf(fact_79_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_80_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1795386414920522267l_num1 @ ( power_power_nat @ M @ N ) )
= ( power_7402600760894073284l_num1 @ ( semiri1795386414920522267l_num1 @ M ) @ N ) ) ).
% of_nat_power
thf(fact_81_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_82_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_83_power__mult,axiom,
! [A: complex,M: nat,N: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_84_power__mult,axiom,
! [A: extended_ereal,M: nat,N: nat] :
( ( power_1054015426188190660_ereal @ A @ ( times_times_nat @ M @ N ) )
= ( power_1054015426188190660_ereal @ ( power_1054015426188190660_ereal @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_85_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_86_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_87_power__even__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_88_power__even__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_89_power__even__eq,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_90_power__even__eq,axiom,
! [A: extended_ereal,N: nat] :
( ( power_1054015426188190660_ereal @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_1054015426188190660_ereal @ ( power_1054015426188190660_ereal @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_91_power__even__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_92_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_93_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( numera6690914467698888265omplex @ N ) ) ).
% zero_neq_numeral
thf(fact_94_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_95_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_96_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_97_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_98_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_99_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_100_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_101_power__commuting__commutes,axiom,
! [X2: extended_ereal,Y: extended_ereal,N: nat] :
( ( ( times_7703590493115627913_ereal @ X2 @ Y )
= ( times_7703590493115627913_ereal @ Y @ X2 ) )
=> ( ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ X2 @ N ) @ Y )
= ( times_7703590493115627913_ereal @ Y @ ( power_1054015426188190660_ereal @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_102_power__commuting__commutes,axiom,
! [X2: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X2 @ Y )
= ( times_times_complex @ Y @ X2 ) )
=> ( ( times_times_complex @ ( power_power_complex @ X2 @ N ) @ Y )
= ( times_times_complex @ Y @ ( power_power_complex @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_103_power__commuting__commutes,axiom,
! [X2: real,Y: real,N: nat] :
( ( ( times_times_real @ X2 @ Y )
= ( times_times_real @ Y @ X2 ) )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_104_power__commuting__commutes,axiom,
! [X2: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y )
= ( times_times_nat @ Y @ X2 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_105_power__commuting__commutes,axiom,
! [X2: int,Y: int,N: nat] :
( ( ( times_times_int @ X2 @ Y )
= ( times_times_int @ Y @ X2 ) )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X2 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_106_power__mult__distrib,axiom,
! [A: extended_ereal,B: extended_ereal,N: nat] :
( ( power_1054015426188190660_ereal @ ( times_7703590493115627913_ereal @ A @ B ) @ N )
= ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ N ) @ ( power_1054015426188190660_ereal @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_107_power__mult__distrib,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_108_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_109_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_110_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_111_power__commutes,axiom,
! [A: extended_ereal,N: nat] :
( ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ N ) @ A )
= ( times_7703590493115627913_ereal @ A @ ( power_1054015426188190660_ereal @ A @ N ) ) ) ).
% power_commutes
thf(fact_112_power__commutes,axiom,
! [A: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_commutes
thf(fact_113_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_114_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_115_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_116_power__divide,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
= ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_divide
thf(fact_117_power__divide,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_divide
thf(fact_118_tendsto__cong__limit,axiom,
! [F: real > real,L: real,F2: filter_real,K: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( ( K = L )
=> ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ K ) @ F2 ) ) ) ).
% tendsto_cong_limit
thf(fact_119_tendsto__eq__rhs,axiom,
! [F: real > real,X2: real,F2: filter_real,Y: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ X2 ) @ F2 )
=> ( ( X2 = Y )
=> ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ F2 ) ) ) ).
% tendsto_eq_rhs
thf(fact_120_mult__numeral__1__right,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ ( numera2161328050825114965l_num1 @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_121_mult__numeral__1__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_122_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_123_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_124_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_125_mult__numeral__1,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( numera2161328050825114965l_num1 @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_126_mult__numeral__1,axiom,
! [A: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_127_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_128_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_129_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_130_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_131_divide__numeral__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_132_power__Suc2,axiom,
! [A: extended_ereal,N: nat] :
( ( power_1054015426188190660_ereal @ A @ ( suc @ N ) )
= ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_133_power__Suc2,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_134_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_135_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_136_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_137_power__Suc,axiom,
! [A: extended_ereal,N: nat] :
( ( power_1054015426188190660_ereal @ A @ ( suc @ N ) )
= ( times_7703590493115627913_ereal @ A @ ( power_1054015426188190660_ereal @ A @ N ) ) ) ).
% power_Suc
thf(fact_138_power__Suc,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_Suc
thf(fact_139_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_140_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_141_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_142_Lim__transform__away__within,axiom,
! [A: real,B: real,S2: set_real,F: real > real,G: real > real,L: real] :
( ( A != B )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( ( X3 != A )
& ( X3 != B ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A @ S2 ) )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A @ S2 ) ) ) ) ) ).
% Lim_transform_away_within
thf(fact_143_Lim__cong__within,axiom,
! [A: nat,B: nat,X2: real,Y: real,S2: set_nat,T: set_nat,F: nat > real,G: nat > real] :
( ( A = B )
=> ( ( X2 = Y )
=> ( ( S2 = T )
=> ( ! [X3: nat] :
( ( X3 != B )
=> ( ( member_nat @ X3 @ T )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) )
=> ( ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ X2 ) @ ( topolo4659099751122792720in_nat @ A @ S2 ) )
= ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ Y ) @ ( topolo4659099751122792720in_nat @ B @ T ) ) ) ) ) ) ) ).
% Lim_cong_within
thf(fact_144_Lim__cong__within,axiom,
! [A: real,B: real,X2: real,Y: real,S2: set_real,T: set_real,F: real > real,G: real > real] :
( ( A = B )
=> ( ( X2 = Y )
=> ( ( S2 = T )
=> ( ! [X3: real] :
( ( X3 != B )
=> ( ( member_real @ X3 @ T )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ X2 ) @ ( topolo2177554685111907308n_real @ A @ S2 ) )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ Y ) @ ( topolo2177554685111907308n_real @ B @ T ) ) ) ) ) ) ) ).
% Lim_cong_within
thf(fact_145_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_146_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_147_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_148_filterlim__at__If,axiom,
! [F: real > real,G2: filter_real,X2: real,P: real > $o,G: real > real] :
( ( filterlim_real_real @ F @ G2 @ ( topolo2177554685111907308n_real @ X2 @ ( collect_real @ P ) ) )
=> ( ( filterlim_real_real @ G @ G2
@ ( topolo2177554685111907308n_real @ X2
@ ( collect_real
@ ^ [X: real] :
~ ( P @ X ) ) ) )
=> ( filterlim_real_real
@ ^ [X: real] : ( if_real @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ G2
@ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% filterlim_at_If
thf(fact_149_tendsto__mult__right,axiom,
! [F: real > real,L: real,F2: filter_real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ L @ C ) )
@ F2 ) ) ).
% tendsto_mult_right
thf(fact_150_tendsto__mult__left,axiom,
! [F: real > real,L: real,F2: filter_real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ C @ L ) )
@ F2 ) ) ).
% tendsto_mult_left
thf(fact_151_tendsto__mult,axiom,
! [F: real > real,A: real,F2: filter_real,G: real > real,B: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ ( times_times_real @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_mult
thf(fact_152_tendsto__diff,axiom,
! [F: real > real,A: real,F2: filter_real,G: real > real,B: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ ( minus_minus_real @ A @ B ) )
@ F2 ) ) ) ).
% tendsto_diff
thf(fact_153_tendsto__power,axiom,
! [F: real > real,A: real,F2: filter_real,N: nat] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
@ ( topolo2815343760600316023s_real @ ( power_power_real @ A @ N ) )
@ F2 ) ) ).
% tendsto_power
thf(fact_154_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_155_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: complex,C: complex] :
( ( ( numera6690914467698888265omplex @ W )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_156_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_157_divide__eq__eq__numeral_I1_J,axiom,
! [B: complex,C: complex,W: num] :
( ( ( divide1717551699836669952omplex @ B @ C )
= ( numera6690914467698888265omplex @ W ) )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_158_tendsto__at__iff__tendsto__nhds,axiom,
! [G: real > real,L: real] :
( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo2815343760600316023s_real @ L ) ) ) ).
% tendsto_at_iff_tendsto_nhds
thf(fact_159_LIM__unique,axiom,
! [F: real > real,L2: real,A: real,M2: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ M2 ) @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( L2 = M2 ) ) ) ).
% LIM_unique
thf(fact_160_LIM__equal,axiom,
! [A: real,F: real > real,G: real > real,L: real] :
( ! [X3: real] :
( ( X3 != A )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ).
% LIM_equal
thf(fact_161_Lim__transform__away__at,axiom,
! [A: real,B: real,F: real > real,G: real > real,L: real] :
( ( A != B )
=> ( ! [X3: real] :
( ( ( X3 != A )
& ( X3 != B ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ).
% Lim_transform_away_at
thf(fact_162_LIM__cong,axiom,
! [A: real,B: real,F: real > real,G: real > real,L: real,M: real] :
( ( A = B )
=> ( ! [X3: real] :
( ( X3 != B )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( L = M )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ M ) @ ( topolo2177554685111907308n_real @ B @ top_top_set_real ) ) ) ) ) ) ).
% LIM_cong
thf(fact_163_tendsto__mult__right__zero,axiom,
! [F: real > real,F2: filter_real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ C @ ( F @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% tendsto_mult_right_zero
thf(fact_164_tendsto__mult__left__zero,axiom,
! [F: real > real,F2: filter_real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ C )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% tendsto_mult_left_zero
thf(fact_165_tendsto__mult__zero,axiom,
! [F: real > real,F2: filter_real,G: real > real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ) ).
% tendsto_mult_zero
thf(fact_166_Lim__transform__eq,axiom,
! [F: real > real,G: real > real,F2: filter_real,A: real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
= ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ A ) @ F2 ) ) ) ).
% Lim_transform_eq
thf(fact_167_LIM__zero__cancel,axiom,
! [F: real > real,L: real,F2: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ).
% LIM_zero_cancel
thf(fact_168_Lim__transform2,axiom,
! [F: real > real,A: real,F2: filter_real,G: real > real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ A ) @ F2 ) ) ) ).
% Lim_transform2
thf(fact_169_Lim__transform,axiom,
! [G: real > real,A: real,F2: filter_real,F: real > real] :
( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
=> ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 ) ) ) ).
% Lim_transform
thf(fact_170_LIM__zero__iff,axiom,
! [F: real > real,L: real,F2: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 )
= ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 ) ) ).
% LIM_zero_iff
thf(fact_171_LIM__zero,axiom,
! [F: real > real,L: real,F2: filter_real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% LIM_zero
thf(fact_172_power__numeral__even,axiom,
! [Z: extended_ereal,W: num] :
( ( power_1054015426188190660_ereal @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_1054015426188190660_ereal @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_173_power__numeral__even,axiom,
! [Z: complex,W: num] :
( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_174_power__numeral__even,axiom,
! [Z: real,W: num] :
( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_175_power__numeral__even,axiom,
! [Z: nat,W: num] :
( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_176_power__numeral__even,axiom,
! [Z: int,W: num] :
( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% power_numeral_even
thf(fact_177_tendsto__divide__zero,axiom,
! [F: real > real,F2: filter_real,C: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ C )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ).
% tendsto_divide_zero
thf(fact_178_tendsto__divide,axiom,
! [F: real > real,A: real,F2: filter_real,G: real > real,B: real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ A ) @ F2 )
=> ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ B ) @ F2 )
=> ( ( B != zero_zero_real )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ ( divide_divide_real @ A @ B ) )
@ F2 ) ) ) ) ).
% tendsto_divide
thf(fact_179_LIM__const__not__eq,axiom,
! [K: real,L2: real,A: real] :
( ( K != L2 )
=> ~ ( filterlim_real_real
@ ^ [X: real] : K
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ).
% LIM_const_not_eq
thf(fact_180_tendsto__compose,axiom,
! [G: real > real,L: real,F: real > real,F2: filter_real] :
( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ ( G @ L ) ) @ ( topolo2177554685111907308n_real @ L @ top_top_set_real ) )
=> ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( G @ ( F @ X ) )
@ ( topolo2815343760600316023s_real @ ( G @ L ) )
@ F2 ) ) ) ).
% tendsto_compose
thf(fact_181_LIM__const__eq,axiom,
! [K: real,L2: real,A: real] :
( ( filterlim_real_real
@ ^ [X: real] : K
@ ( topolo2815343760600316023s_real @ L2 )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
=> ( K = L2 ) ) ).
% LIM_const_eq
thf(fact_182_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_183_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_184_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex ) ).
% zero_power2
thf(fact_185_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_186_power2__eq__square,axiom,
! [A: extended_ereal] :
( ( power_1054015426188190660_ereal @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_7703590493115627913_ereal @ A @ A ) ) ).
% power2_eq_square
thf(fact_187_power2__eq__square,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_complex @ A @ A ) ) ).
% power2_eq_square
thf(fact_188_power2__eq__square,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A @ A ) ) ).
% power2_eq_square
thf(fact_189_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_190_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_191_power4__eq__xxxx,axiom,
! [X2: extended_ereal] :
( ( power_1054015426188190660_ereal @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_7703590493115627913_ereal @ ( times_7703590493115627913_ereal @ ( times_7703590493115627913_ereal @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_192_power4__eq__xxxx,axiom,
! [X2: complex] :
( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_193_power4__eq__xxxx,axiom,
! [X2: real] :
( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_194_power4__eq__xxxx,axiom,
! [X2: nat] :
( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_195_power4__eq__xxxx,axiom,
! [X2: int] :
( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).
% power4_eq_xxxx
thf(fact_196_power2__commute,axiom,
! [X2: real,Y: real] :
( ( power_power_real @ ( minus_minus_real @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ ( minus_minus_real @ Y @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_197_power2__commute,axiom,
! [X2: int,Y: int] :
( ( power_power_int @ ( minus_minus_int @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_198_power2__commute,axiom,
! [X2: complex,Y: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X2 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ ( minus_minus_complex @ Y @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_199_LIM__not__zero,axiom,
! [K: nat,A: real] :
( ( K != zero_zero_nat )
=> ~ ( filterlim_real_nat
@ ^ [X: real] : K
@ ( topolo8926549440605965083ds_nat @ zero_zero_nat )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ).
% LIM_not_zero
thf(fact_200_LIM__not__zero,axiom,
! [K: int,A: real] :
( ( K != zero_zero_int )
=> ~ ( filterlim_real_int
@ ^ [X: real] : K
@ ( topolo8924058970096914807ds_int @ zero_zero_int )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ).
% LIM_not_zero
thf(fact_201_LIM__not__zero,axiom,
! [K: extended_ereal,A: real] :
( ( K != zero_z2744965634713055877_ereal )
=> ~ ( filter2787094286561295468_ereal
@ ^ [X: real] : K
@ ( topolo4481849189275076035_ereal @ zero_z2744965634713055877_ereal )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ).
% LIM_not_zero
thf(fact_202_LIM__not__zero,axiom,
! [K: complex,A: real] :
( ( K != zero_zero_complex )
=> ~ ( filter8506290784974013328omplex
@ ^ [X: real] : K
@ ( topolo2444363109189100025omplex @ zero_zero_complex )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ).
% LIM_not_zero
thf(fact_203_LIM__not__zero,axiom,
! [K: real,A: real] :
( ( K != zero_zero_real )
=> ~ ( filterlim_real_real
@ ^ [X: real] : K
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ).
% LIM_not_zero
thf(fact_204_power__odd__eq,axiom,
! [A: extended_ereal,N: nat] :
( ( power_1054015426188190660_ereal @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_7703590493115627913_ereal @ A @ ( power_1054015426188190660_ereal @ ( power_1054015426188190660_ereal @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_205_power__odd__eq,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_206_power__odd__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_207_power__odd__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_208_power__odd__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_209_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_210_Suc__0__div__numeral_I2_J,axiom,
! [N: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(2)
thf(fact_211_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_212_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_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_zdiv__numeral__Bit0,axiom,
! [V: num,W: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_236_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_237_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_238_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_239_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_240_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_241_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_242_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_243_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_257_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_258_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_259_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_260_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_261_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_262_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_263_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_264_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_265_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_266_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_267_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_268_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_269_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_270_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_271_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_281_zdiv__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zdiv_int
thf(fact_282_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_283_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_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_left__diff__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C )
= ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_304_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_305_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_306_right__diff__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_307_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_308_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_309_left__diff__distrib_H,axiom,
! [B: complex,C: complex,A: complex] :
( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A )
= ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_310_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_311_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_312_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_313_right__diff__distrib_H,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_314_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_315_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_316_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_317_times__divide__times__eq,axiom,
! [X2: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_318_times__divide__times__eq,axiom,
! [X2: complex,Y: complex,Z: complex,W: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X2 @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_319_divide__divide__times__eq,axiom,
! [X2: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_320_divide__divide__times__eq,axiom,
! [X2: complex,Y: complex,Z: complex,W: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X2 @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X2 @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_321_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_322_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_323_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_324_diff__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
= ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_325_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_326_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L: num] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_327_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_328_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_329_lambda__zero,axiom,
( ( ^ [H: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_330_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_331_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_332_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_333_frac__eq__eq,axiom,
! [Y: real,Z: real,X2: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X2 @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X2 @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_334_frac__eq__eq,axiom,
! [Y: complex,Z: complex,X2: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X2 @ Y )
= ( divide1717551699836669952omplex @ W @ Z ) )
= ( ( times_times_complex @ X2 @ Z )
= ( times_times_complex @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_335_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_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_348_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_349_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_350_add__divide__eq__if__simps_I4_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_351_diff__frac__eq,axiom,
! [Y: real,Z: real,X2: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_352_diff__frac__eq,axiom,
! [Y: complex,Z: complex,X2: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_353_diff__divide__eq__iff,axiom,
! [Z: real,X2: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_354_diff__divide__eq__iff,axiom,
! [Z: complex,X2: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ X2 @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_355_divide__diff__eq__iff,axiom,
! [Z: real,X2: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_356_divide__diff__eq__iff,axiom,
! [Z: complex,X2: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_357_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_358_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_359_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_360_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_361_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_362_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_363_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_364_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_365_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% of_nat_mult
thf(fact_366_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_367_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1795386414920522267l_num1 @ ( times_times_nat @ M @ N ) )
= ( times_8498157372700349887l_num1 @ ( semiri1795386414920522267l_num1 @ M ) @ ( semiri1795386414920522267l_num1 @ N ) ) ) ).
% of_nat_mult
thf(fact_368_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_369_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_370_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_371_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_372_of__nat__0,axiom,
( ( semiri1795386414920522267l_num1 @ zero_zero_nat )
= zero_z5982384998485459395l_num1 ) ).
% of_nat_0
thf(fact_373_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_374_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_375_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_376_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_377_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_378_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_379_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= zero_zero_complex )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_380_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_381_four__x__squared,axiom,
! [X2: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_382_filterlim__atI_H,axiom,
! [F: real > real,C: real,F2: filter_real] :
( ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ C )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real )
@ F2 )
=> ( filterlim_real_real @ F @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) @ F2 ) ) ).
% filterlim_atI'
thf(fact_383_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_384_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_385_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_386_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_387_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_388_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_389_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_390_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_391_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_392_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_393_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_394_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= ( semiri8010041392384452111omplex @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_395_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_396_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_397_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_398_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_399_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_400_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_401_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_402_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_403_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_404_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_405_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_406_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_407_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_408_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_409_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_410_Multiseries__Expansion_Ointyness__of__nat,axiom,
! [N: nat] :
( ( N = N )
=> ( ( semiri5074537144036343181t_real @ N )
= ( semiri5074537144036343181t_real @ N ) ) ) ).
% Multiseries_Expansion.intyness_of_nat
thf(fact_411_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_412_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_413_mult__of__nat__commute,axiom,
! [X2: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_414_mult__of__nat__commute,axiom,
! [X2: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_415_mult__of__nat__commute,axiom,
! [X2: nat,Y: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ X2 ) @ Y )
= ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_416_mult__of__nat__commute,axiom,
! [X2: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_417_mult__of__nat__commute,axiom,
! [X2: nat,Y: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ ( semiri1795386414920522267l_num1 @ X2 ) @ Y )
= ( times_8498157372700349887l_num1 @ Y @ ( semiri1795386414920522267l_num1 @ X2 ) ) ) ).
% mult_of_nat_commute
thf(fact_418_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_419_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_420_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_421_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_422_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_423_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_424_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_425_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_426_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_427_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_428_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_429_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_430_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_431_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_432_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_433_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_434_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_435_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_436_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ N ) )
!= zero_zero_complex ) ).
% of_nat_neq_0
thf(fact_437_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_438_Multiseries__Expansion_Ointyness__0,axiom,
( zero_zero_real
= ( semiri5074537144036343181t_real @ zero_zero_nat ) ) ).
% Multiseries_Expansion.intyness_0
thf(fact_439_Multiseries__Expansion_Ointyness__numeral,axiom,
! [Num: num] :
( ( Num = Num )
=> ( ( numeral_numeral_real @ Num )
= ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ Num ) ) ) ) ).
% Multiseries_Expansion.intyness_numeral
thf(fact_440_Multiseries__Expansion_Ointyness__simps_I6_J,axiom,
( numeral_numeral_real
= ( ^ [Num2: num] : ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ Num2 ) ) ) ) ).
% Multiseries_Expansion.intyness_simps(6)
thf(fact_441_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_442_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_443_Multiseries__Expansion_Ointyness__simps_I3_J,axiom,
! [A: nat,N: nat] :
( ( power_power_real @ ( semiri5074537144036343181t_real @ A ) @ N )
= ( semiri5074537144036343181t_real @ ( power_power_nat @ A @ N ) ) ) ).
% Multiseries_Expansion.intyness_simps(3)
thf(fact_444_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_445_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_446_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_447_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_448_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_449_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_450_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_451_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_452_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_453_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_454_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_455_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_456_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_457_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_458_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_459_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_460_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_461_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_462_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_463_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_464_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_465_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_466_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_467_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_468_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_469_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_470_zero__reorient,axiom,
! [X2: extended_ereal] :
( ( zero_z2744965634713055877_ereal = X2 )
= ( X2 = zero_z2744965634713055877_ereal ) ) ).
% zero_reorient
thf(fact_471_zero__reorient,axiom,
! [X2: complex] :
( ( zero_zero_complex = X2 )
= ( X2 = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_472_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_473_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_474_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_475_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_476_mult_Oassoc,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% mult.assoc
thf(fact_477_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_478_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_479_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_480_mult_Ocommute,axiom,
( times_times_complex
= ( ^ [A3: complex,B2: complex] : ( times_times_complex @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_481_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_482_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_483_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_484_mult_Oleft__commute,axiom,
! [B: complex,A: complex,C: complex] :
( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_485_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_486_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_487_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_488_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_489_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_490_diff__eq__diff__eq,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ( minus_minus_complex @ A @ B )
= ( minus_minus_complex @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_491_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_492_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_493_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_494_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: complex,C: complex,B: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B )
= ( minus_minus_complex @ ( minus_minus_complex @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_495_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_496_int__diff__cases,axiom,
! [Z: int] :
~ ! [M3: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_497_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_498_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_499_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: complex,Z2: complex] : ( Y4 = Z2 ) )
= ( ^ [A3: complex,B2: complex] :
( ( minus_minus_complex @ A3 @ B2 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_500_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: complex,X2: complex] :
( ( ( times_times_complex @ A @ X2 )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( X2 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_501_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X2: real] :
( ( ( times_times_real @ A @ X2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_502_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: complex] :
( ( times_times_complex @ zero_zero_complex @ X2 )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_503_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_504_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_505_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_506_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: complex,X2: complex,Y: complex] :
( ( ( times_times_complex @ A @ X2 )
= ( times_times_complex @ A @ Y ) )
= ( ( X2 = Y )
| ( A = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_507_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X2: real,Y: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y ) )
= ( ( X2 = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_508_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: complex,X2: complex,B: complex] :
( ( ( times_times_complex @ A @ X2 )
= ( times_times_complex @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_509_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X2: real,B: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_510_unset__bit__0,axiom,
! [A: nat] :
( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_511_unset__bit__0,axiom,
! [A: int] :
( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_512_Lim__null,axiom,
! [F: real > real,L: real,Net: filter_real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ Net )
= ( filterlim_real_real
@ ^ [X: real] : ( minus_minus_real @ ( F @ X ) @ L )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ Net ) ) ).
% Lim_null
thf(fact_513_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_514_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_515_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_516_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: complex,B: complex,X2: complex] :
( ( times_times_complex @ A @ ( times_times_complex @ B @ X2 ) )
= ( times_times_complex @ B @ ( times_times_complex @ A @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_517_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_518_vector__space__over__itself_Oscale__scale,axiom,
! [A: complex,B: complex,X2: complex] :
( ( times_times_complex @ A @ ( times_times_complex @ B @ X2 ) )
= ( times_times_complex @ ( times_times_complex @ A @ B ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_519_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_520_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: complex,A: complex,B: complex] :
( ( X2 != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ X2 )
= ( times_times_complex @ B @ X2 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_521_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_522_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: complex,X2: complex,Y: complex] :
( ( A != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ X2 )
= ( times_times_complex @ A @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_523_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X2: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y ) )
=> ( X2 = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_524_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A: complex,X2: complex,Y: complex] :
( ( times_times_complex @ A @ ( minus_minus_complex @ X2 @ Y ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ X2 ) @ ( times_times_complex @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_525_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A: real,X2: real,Y: real] :
( ( times_times_real @ A @ ( minus_minus_real @ X2 @ Y ) )
= ( minus_minus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_526_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A: complex,B: complex,X2: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ X2 )
= ( minus_minus_complex @ ( times_times_complex @ A @ X2 ) @ ( times_times_complex @ B @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_527_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ X2 )
= ( minus_minus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_528_Lim__at__imp__Lim__at__within,axiom,
! [F: real > real,L: real,X2: real,S2: set_real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
=> ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ) ).
% Lim_at_imp_Lim_at_within
thf(fact_529_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_530_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_531_power__numeral,axiom,
! [K: num,L: num] :
( ( power_7402600760894073284l_num1 @ ( numera2161328050825114965l_num1 @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numera2161328050825114965l_num1 @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_532_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_533_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_534_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_535_power__numeral,axiom,
! [K: num,L: num] :
( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
= ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).
% power_numeral
thf(fact_536_summable__cot__pfd__real,axiom,
! [X2: real] :
( summable_real
@ ^ [N3: nat] : ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% summable_cot_pfd_real
thf(fact_537_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y2: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y2 ) )
= ( X22 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_538_iso__tuple__UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_539_iso__tuple__UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_540_pow_Osimps_I1_J,axiom,
! [X2: num] :
( ( pow @ X2 @ one )
= X2 ) ).
% pow.simps(1)
thf(fact_541_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_542_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_543_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_544_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_545_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_546_summable__divide__iff,axiom,
! [F: nat > real,C: real] :
( ( summable_real
@ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_divide_iff
thf(fact_547_summable__divide__iff,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
= ( ( C = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_divide_iff
thf(fact_548_summable__cmult__iff,axiom,
! [C: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
= ( ( C = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_cmult_iff
thf(fact_549_summable__cmult__iff,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_cmult_iff
thf(fact_550_summable__zero,axiom,
( summable_real
@ ^ [N3: nat] : zero_zero_real ) ).
% summable_zero
thf(fact_551_summable__zero,axiom,
( summable_nat
@ ^ [N3: nat] : zero_zero_nat ) ).
% summable_zero
thf(fact_552_summable__zero,axiom,
( summable_int
@ ^ [N3: nat] : zero_zero_int ) ).
% summable_zero
thf(fact_553_summable__zero,axiom,
( summab1499615597823217518_ereal
@ ^ [N3: nat] : zero_z2744965634713055877_ereal ) ).
% summable_zero
thf(fact_554_summable__zero,axiom,
( summable_complex
@ ^ [N3: nat] : zero_zero_complex ) ).
% summable_zero
thf(fact_555_summable__single,axiom,
! [I: nat,F: nat > real] :
( summable_real
@ ^ [R2: nat] : ( if_real @ ( R2 = I ) @ ( F @ R2 ) @ zero_zero_real ) ) ).
% summable_single
thf(fact_556_summable__single,axiom,
! [I: nat,F: nat > nat] :
( summable_nat
@ ^ [R2: nat] : ( if_nat @ ( R2 = I ) @ ( F @ R2 ) @ zero_zero_nat ) ) ).
% summable_single
thf(fact_557_summable__single,axiom,
! [I: nat,F: nat > int] :
( summable_int
@ ^ [R2: nat] : ( if_int @ ( R2 = I ) @ ( F @ R2 ) @ zero_zero_int ) ) ).
% summable_single
thf(fact_558_summable__single,axiom,
! [I: nat,F: nat > extended_ereal] :
( summab1499615597823217518_ereal
@ ^ [R2: nat] : ( if_Extended_ereal @ ( R2 = I ) @ ( F @ R2 ) @ zero_z2744965634713055877_ereal ) ) ).
% summable_single
thf(fact_559_summable__single,axiom,
! [I: nat,F: nat > complex] :
( summable_complex
@ ^ [R2: nat] : ( if_complex @ ( R2 = I ) @ ( F @ R2 ) @ zero_zero_complex ) ) ).
% summable_single
thf(fact_560_powser__split__head_I3_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
=> ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% powser_split_head(3)
thf(fact_561_powser__split__head_I3_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
=> ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% powser_split_head(3)
thf(fact_562_summable__powser__split__head,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
= ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).
% summable_powser_split_head
thf(fact_563_summable__powser__split__head,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
= ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).
% summable_powser_split_head
thf(fact_564_summable__0__powser,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).
% summable_0_powser
thf(fact_565_summable__0__powser,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).
% summable_0_powser
thf(fact_566_summable__cot__pfd__complex,axiom,
! [X2: complex] :
( summable_complex
@ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% summable_cot_pfd_complex
thf(fact_567_summable__const__iff,axiom,
! [C: real] :
( ( summable_real
@ ^ [Uu: nat] : C )
= ( C = zero_zero_real ) ) ).
% summable_const_iff
thf(fact_568_summable__const__iff,axiom,
! [C: complex] :
( ( summable_complex
@ ^ [Uu: nat] : C )
= ( C = zero_zero_complex ) ) ).
% summable_const_iff
thf(fact_569_summable__mult,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) ) ) ).
% summable_mult
thf(fact_570_summable__mult,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) ) ) ).
% summable_mult
thf(fact_571_summable__mult2,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C ) ) ) ).
% summable_mult2
thf(fact_572_summable__mult2,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ).
% summable_mult2
thf(fact_573_summable__diff,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% summable_diff
thf(fact_574_summable__diff,axiom,
! [F: nat > complex,G: nat > complex] :
( ( summable_complex @ F )
=> ( ( summable_complex @ G )
=> ( summable_complex
@ ^ [N3: nat] : ( minus_minus_complex @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).
% summable_diff
thf(fact_575_summable__divide,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) ) ) ).
% summable_divide
thf(fact_576_summable__divide,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( summable_complex
@ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) ) ) ).
% summable_divide
thf(fact_577_summable__Suc__iff,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
= ( summable_real @ F ) ) ).
% summable_Suc_iff
thf(fact_578_summable__Suc__iff,axiom,
! [F: nat > complex] :
( ( summable_complex
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
= ( summable_complex @ F ) ) ).
% summable_Suc_iff
thf(fact_579_summable__mult__D,axiom,
! [C: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
=> ( ( C != zero_zero_complex )
=> ( summable_complex @ F ) ) ) ).
% summable_mult_D
thf(fact_580_summable__mult__D,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
=> ( ( C != zero_zero_real )
=> ( summable_real @ F ) ) ) ).
% summable_mult_D
thf(fact_581_summable__zero__power,axiom,
summable_real @ ( power_power_real @ zero_zero_real ) ).
% summable_zero_power
thf(fact_582_summable__zero__power,axiom,
summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% summable_zero_power
thf(fact_583_summable__zero__power,axiom,
summable_int @ ( power_power_int @ zero_zero_int ) ).
% summable_zero_power
thf(fact_584_summable__zero__power_H,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).
% summable_zero_power'
thf(fact_585_summable__zero__power_H,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).
% summable_zero_power'
thf(fact_586_summable__zero__power_H,axiom,
! [F: nat > int] :
( summable_int
@ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ ( power_power_int @ zero_zero_int @ N3 ) ) ) ).
% summable_zero_power'
thf(fact_587_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_588_UNIV__I,axiom,
! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_I
thf(fact_589_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_590_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_591_set__times__intro,axiom,
! [A: num,C2: set_num,B: num,D2: set_num] :
( ( member_num @ A @ C2 )
=> ( ( member_num @ B @ D2 )
=> ( member_num @ ( times_times_num @ A @ B ) @ ( times_times_set_num @ C2 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_592_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_593_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_594_Parity_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L3: num,R2: int] : ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ ( numeral_numeral_int @ L3 ) @ R2 ) ) ) ) ).
% Parity.adjust_mod_def
thf(fact_595_cot__pfd__def,axiom,
( cotang8298477626502807258omplex
= ( ^ [X: complex] :
( suminf_complex
@ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cot_pfd_def
thf(fact_596_cot__pfd__def,axiom,
( cotang1502006655779026648d_real
= ( ^ [X: real] :
( suminf_real
@ ^ [N3: nat] : ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cot_pfd_def
thf(fact_597_triangle__def,axiom,
( nat_triangle
= ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% triangle_def
thf(fact_598_suminf__zero,axiom,
( ( suminf_real
@ ^ [N3: nat] : zero_zero_real )
= zero_zero_real ) ).
% suminf_zero
thf(fact_599_suminf__zero,axiom,
( ( suminf_nat
@ ^ [N3: nat] : zero_zero_nat )
= zero_zero_nat ) ).
% suminf_zero
thf(fact_600_suminf__zero,axiom,
( ( suminf_int
@ ^ [N3: nat] : zero_zero_int )
= zero_zero_int ) ).
% suminf_zero
thf(fact_601_suminf__zero,axiom,
( ( suminf4411151127299490740_ereal
@ ^ [N3: nat] : zero_z2744965634713055877_ereal )
= zero_z2744965634713055877_ereal ) ).
% suminf_zero
thf(fact_602_suminf__zero,axiom,
( ( suminf_complex
@ ^ [N3: nat] : zero_zero_complex )
= zero_zero_complex ) ).
% suminf_zero
thf(fact_603_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_604_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_605_top__set__def,axiom,
( top_top_set_real
= ( collect_real @ top_top_real_o ) ) ).
% top_set_def
thf(fact_606_suminf__mult2,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( ( times_times_complex @ ( suminf_complex @ F ) @ C )
= ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C ) ) ) ) ).
% suminf_mult2
thf(fact_607_suminf__mult2,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( times_times_real @ ( suminf_real @ F ) @ C )
= ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ) ).
% suminf_mult2
thf(fact_608_suminf__mult,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
= ( times_times_complex @ C @ ( suminf_complex @ F ) ) ) ) ).
% suminf_mult
thf(fact_609_suminf__mult,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
= ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% suminf_mult
thf(fact_610_suminf__diff,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
= ( suminf_real
@ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% suminf_diff
thf(fact_611_suminf__diff,axiom,
! [F: nat > complex,G: nat > complex] :
( ( summable_complex @ F )
=> ( ( summable_complex @ G )
=> ( ( minus_minus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
= ( suminf_complex
@ ^ [N3: nat] : ( minus_minus_complex @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).
% suminf_diff
thf(fact_612_suminf__divide,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
= ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% suminf_divide
thf(fact_613_suminf__divide,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( ( suminf_complex
@ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
= ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% suminf_divide
thf(fact_614_set__times__elim,axiom,
! [X2: complex,A2: set_complex,B3: set_complex] :
( ( member_complex @ X2 @ ( times_6048082448287401577omplex @ A2 @ B3 ) )
=> ~ ! [A4: complex,B4: complex] :
( ( X2
= ( times_times_complex @ A4 @ B4 ) )
=> ( ( member_complex @ A4 @ A2 )
=> ~ ( member_complex @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_615_set__times__elim,axiom,
! [X2: real,A2: set_real,B3: set_real] :
( ( member_real @ X2 @ ( times_times_set_real @ A2 @ B3 ) )
=> ~ ! [A4: real,B4: real] :
( ( X2
= ( times_times_real @ A4 @ B4 ) )
=> ( ( member_real @ A4 @ A2 )
=> ~ ( member_real @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_616_set__times__elim,axiom,
! [X2: num,A2: set_num,B3: set_num] :
( ( member_num @ X2 @ ( times_times_set_num @ A2 @ B3 ) )
=> ~ ! [A4: num,B4: num] :
( ( X2
= ( times_times_num @ A4 @ B4 ) )
=> ( ( member_num @ A4 @ A2 )
=> ~ ( member_num @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_617_set__times__elim,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ X2 @ ( times_times_set_nat @ A2 @ B3 ) )
=> ~ ! [A4: nat,B4: nat] :
( ( X2
= ( times_times_nat @ A4 @ B4 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_618_set__times__elim,axiom,
! [X2: int,A2: set_int,B3: set_int] :
( ( member_int @ X2 @ ( times_times_set_int @ A2 @ B3 ) )
=> ~ ! [A4: int,B4: int] :
( ( X2
= ( times_times_int @ A4 @ B4 ) )
=> ( ( member_int @ A4 @ A2 )
=> ~ ( member_int @ B4 @ B3 ) ) ) ) ).
% set_times_elim
thf(fact_619_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_620_UNIV__eq__I,axiom,
! [A2: set_real] :
( ! [X3: real] : ( member_real @ X3 @ A2 )
=> ( top_top_set_real = A2 ) ) ).
% UNIV_eq_I
thf(fact_621_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_622_UNIV__witness,axiom,
? [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_623_suminf__split__head,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% suminf_split_head
thf(fact_624_suminf__split__head,axiom,
! [F: nat > complex] :
( ( summable_complex @ F )
=> ( ( suminf_complex
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
= ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% suminf_split_head
thf(fact_625_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X: nat] : $true ) ) ).
% UNIV_def
thf(fact_626_UNIV__def,axiom,
( top_top_set_real
= ( collect_real
@ ^ [X: real] : $true ) ) ).
% UNIV_def
thf(fact_627_powser__split__head_I2_J,axiom,
! [F: nat > complex,Z: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
=> ( ( times_times_complex
@ ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
@ Z )
= ( minus_minus_complex
@ ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_628_powser__split__head_I2_J,axiom,
! [F: nat > real,Z: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
=> ( ( times_times_real
@ ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
@ Z )
= ( minus_minus_real
@ ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_629_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_630_powser__zero,axiom,
! [F: nat > complex] :
( ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_631_powser__zero,axiom,
! [F: nat > real] :
( ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_632_set__decode__Suc,axiom,
! [N: nat,X2: nat] :
( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X2 ) )
= ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_633_sums__cot__pfd__complex,axiom,
! [X2: complex] :
( sums_complex
@ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( cotang8298477626502807258omplex @ X2 ) ) ).
% sums_cot_pfd_complex
thf(fact_634_sums__cot__pfd__real,axiom,
! [X2: real] :
( sums_real
@ ^ [N3: nat] : ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( cotang1502006655779026648d_real @ X2 ) ) ).
% sums_cot_pfd_real
thf(fact_635_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_636_filterlim__divide__at__infinity,axiom,
! [F: complex > complex,C: complex,F2: filter_complex,G: complex > complex] :
( ( filter8330067395343389202omplex @ F @ ( topolo2444363109189100025omplex @ C ) @ F2 )
=> ( ( filter8330067395343389202omplex @ G @ ( topolo3898190780742966766omplex @ zero_zero_complex @ top_top_set_complex ) @ F2 )
=> ( ( C != zero_zero_complex )
=> ( filter8330067395343389202omplex
@ ^ [X: complex] : ( divide1717551699836669952omplex @ ( F @ X ) @ ( G @ X ) )
@ at_infinity_complex
@ F2 ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_637_filterlim__divide__at__infinity,axiom,
! [F: real > real,C: real,F2: filter_real,G: real > real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ C ) @ F2 )
=> ( ( filterlim_real_real @ G @ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) @ F2 )
=> ( ( C != zero_zero_real )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
@ at_infinity_real
@ F2 ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_638_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_639_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_640_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_641_dvd__0__left__iff,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
= ( A = zero_zero_complex ) ) ).
% dvd_0_left_iff
thf(fact_642_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_643_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_644_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_645_dvd__0__right,axiom,
! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% dvd_0_right
thf(fact_646_div__dvd__div,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_647_div__dvd__div,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_648_sums__zero,axiom,
( sums_real
@ ^ [N3: nat] : zero_zero_real
@ zero_zero_real ) ).
% sums_zero
thf(fact_649_sums__zero,axiom,
( sums_nat
@ ^ [N3: nat] : zero_zero_nat
@ zero_zero_nat ) ).
% sums_zero
thf(fact_650_sums__zero,axiom,
( sums_int
@ ^ [N3: nat] : zero_zero_int
@ zero_zero_int ) ).
% sums_zero
thf(fact_651_sums__zero,axiom,
( sums_Extended_ereal
@ ^ [N3: nat] : zero_z2744965634713055877_ereal
@ zero_z2744965634713055877_ereal ) ).
% sums_zero
thf(fact_652_sums__zero,axiom,
( sums_complex
@ ^ [N3: nat] : zero_zero_complex
@ zero_zero_complex ) ).
% sums_zero
thf(fact_653_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_654_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_655_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_656_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_657_dvd__mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_658_dvd__mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_659_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_660_dvd__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_661_dvd__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_662_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_663_dvd__mult__div__cancel,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_664_dvd__mult__div__cancel,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_665_dvd__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_666_dvd__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_667_div__diff,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_diff
thf(fact_668_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_669_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_670_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_671_even__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_672_even__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_673_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_674_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_675_powser__sums__zero__iff,axiom,
! [A: nat > complex,X2: complex] :
( ( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
@ X2 )
= ( ( A @ zero_zero_nat )
= X2 ) ) ).
% powser_sums_zero_iff
thf(fact_676_powser__sums__zero__iff,axiom,
! [A: nat > real,X2: real] :
( ( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
@ X2 )
= ( ( A @ zero_zero_nat )
= X2 ) ) ).
% powser_sums_zero_iff
thf(fact_677_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_678_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_679_set__decode__0,axiom,
! [X2: nat] :
( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X2 ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) ) ).
% set_decode_0
thf(fact_680_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_681_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_682_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_683_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_684_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_685_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_686_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_687_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_688_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_689_sums__0,axiom,
! [F: nat > real] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_real )
=> ( sums_real @ F @ zero_zero_real ) ) ).
% sums_0
thf(fact_690_sums__0,axiom,
! [F: nat > nat] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_nat )
=> ( sums_nat @ F @ zero_zero_nat ) ) ).
% sums_0
thf(fact_691_sums__0,axiom,
! [F: nat > int] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_int )
=> ( sums_int @ F @ zero_zero_int ) ) ).
% sums_0
thf(fact_692_sums__0,axiom,
! [F: nat > extended_ereal] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_z2744965634713055877_ereal )
=> ( sums_Extended_ereal @ F @ zero_z2744965634713055877_ereal ) ) ).
% sums_0
thf(fact_693_sums__0,axiom,
! [F: nat > complex] :
( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_complex )
=> ( sums_complex @ F @ zero_zero_complex ) ) ).
% sums_0
thf(fact_694_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_695_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_696_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_697_dvd__0__left,axiom,
! [A: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A )
=> ( A = zero_zero_complex ) ) ).
% dvd_0_left
thf(fact_698_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A3: real,B2: real] :
( ( A3 = zero_zero_real )
=> ( B2 = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_699_dvd__field__iff,axiom,
( dvd_dvd_complex
= ( ^ [A3: complex,B2: complex] :
( ( A3 = zero_zero_complex )
=> ( B2 = zero_zero_complex ) ) ) ) ).
% dvd_field_iff
thf(fact_700_dvd__triv__right,axiom,
! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ A ) ) ).
% dvd_triv_right
thf(fact_701_dvd__triv__right,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_702_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_703_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_704_dvd__mult__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
=> ( dvd_dvd_complex @ B @ C ) ) ).
% dvd_mult_right
thf(fact_705_dvd__mult__right,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ B @ C ) ) ).
% dvd_mult_right
thf(fact_706_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_707_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_708_mult__dvd__mono,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( dvd_dvd_complex @ A @ B )
=> ( ( dvd_dvd_complex @ C @ D )
=> ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_709_mult__dvd__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ C @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_710_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_711_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_712_dvd__triv__left,axiom,
! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ A @ B ) ) ).
% dvd_triv_left
thf(fact_713_dvd__triv__left,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_714_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_715_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_716_dvd__mult__left,axiom,
! [A: complex,B: complex,C: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
=> ( dvd_dvd_complex @ A @ C ) ) ).
% dvd_mult_left
thf(fact_717_dvd__mult__left,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ A @ C ) ) ).
% dvd_mult_left
thf(fact_718_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_719_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_720_dvd__mult2,axiom,
! [A: complex,B: complex,C: complex] :
( ( dvd_dvd_complex @ A @ B )
=> ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_721_dvd__mult2,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_722_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_723_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_724_dvd__mult,axiom,
! [A: complex,C: complex,B: complex] :
( ( dvd_dvd_complex @ A @ C )
=> ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% dvd_mult
thf(fact_725_dvd__mult,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult
thf(fact_726_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_727_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_728_dvd__def,axiom,
( dvd_dvd_complex
= ( ^ [B2: complex,A3: complex] :
? [K2: complex] :
( A3
= ( times_times_complex @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_729_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B2: real,A3: real] :
? [K2: real] :
( A3
= ( times_times_real @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_730_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B2: nat,A3: nat] :
? [K2: nat] :
( A3
= ( times_times_nat @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_731_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B2: int,A3: int] :
? [K2: int] :
( A3
= ( times_times_int @ B2 @ K2 ) ) ) ) ).
% dvd_def
thf(fact_732_dvdI,axiom,
! [A: complex,B: complex,K: complex] :
( ( A
= ( times_times_complex @ B @ K ) )
=> ( dvd_dvd_complex @ B @ A ) ) ).
% dvdI
thf(fact_733_dvdI,axiom,
! [A: real,B: real,K: real] :
( ( A
= ( times_times_real @ B @ K ) )
=> ( dvd_dvd_real @ B @ A ) ) ).
% dvdI
thf(fact_734_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_735_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_736_dvdE,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ~ ! [K3: complex] :
( A
!= ( times_times_complex @ B @ K3 ) ) ) ).
% dvdE
thf(fact_737_dvdE,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ~ ! [K3: real] :
( A
!= ( times_times_real @ B @ K3 ) ) ) ).
% dvdE
thf(fact_738_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K3: nat] :
( A
!= ( times_times_nat @ B @ K3 ) ) ) ).
% dvdE
thf(fact_739_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K3: int] :
( A
!= ( times_times_int @ B @ K3 ) ) ) ).
% dvdE
thf(fact_740_dvd__diff__commute,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
= ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% dvd_diff_commute
thf(fact_741_dvd__diff,axiom,
! [X2: real,Y: real,Z: real] :
( ( dvd_dvd_real @ X2 @ Y )
=> ( ( dvd_dvd_real @ X2 @ Z )
=> ( dvd_dvd_real @ X2 @ ( minus_minus_real @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_742_dvd__diff,axiom,
! [X2: int,Y: int,Z: int] :
( ( dvd_dvd_int @ X2 @ Y )
=> ( ( dvd_dvd_int @ X2 @ Z )
=> ( dvd_dvd_int @ X2 @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_743_dvd__diff,axiom,
! [X2: complex,Y: complex,Z: complex] :
( ( dvd_dvd_complex @ X2 @ Y )
=> ( ( dvd_dvd_complex @ X2 @ Z )
=> ( dvd_dvd_complex @ X2 @ ( minus_minus_complex @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_744_div__div__div__same,axiom,
! [D: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ D @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_745_div__div__div__same,axiom,
! [D: int,B: int,A: int] :
( ( dvd_dvd_int @ D @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_746_dvd__div__eq__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
=> ( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_747_dvd__div__eq__cancel,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B @ C ) )
=> ( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_748_dvd__div__eq__cancel,axiom,
! [A: int,C: int,B: int] :
( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
=> ( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_749_dvd__div__eq__cancel,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
=> ( ( dvd_dvd_complex @ C @ A )
=> ( ( dvd_dvd_complex @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_750_dvd__div__eq__iff,axiom,
! [C: real,A: real,B: real] :
( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B )
=> ( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_751_dvd__div__eq__iff,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_752_dvd__div__eq__iff,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_753_dvd__div__eq__iff,axiom,
! [C: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ C @ A )
=> ( ( dvd_dvd_complex @ C @ B )
=> ( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_754_dvd__power__same,axiom,
! [X2: real,Y: real,N: nat] :
( ( dvd_dvd_real @ X2 @ Y )
=> ( dvd_dvd_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_755_dvd__power__same,axiom,
! [X2: nat,Y: nat,N: nat] :
( ( dvd_dvd_nat @ X2 @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_756_dvd__power__same,axiom,
! [X2: complex,Y: complex,N: nat] :
( ( dvd_dvd_complex @ X2 @ Y )
=> ( dvd_dvd_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_757_dvd__power__same,axiom,
! [X2: int,Y: int,N: nat] :
( ( dvd_dvd_int @ X2 @ Y )
=> ( dvd_dvd_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_758_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_759_sums__single,axiom,
! [I: nat,F: nat > real] :
( sums_real
@ ^ [R2: nat] : ( if_real @ ( R2 = I ) @ ( F @ R2 ) @ zero_zero_real )
@ ( F @ I ) ) ).
% sums_single
thf(fact_760_sums__single,axiom,
! [I: nat,F: nat > nat] :
( sums_nat
@ ^ [R2: nat] : ( if_nat @ ( R2 = I ) @ ( F @ R2 ) @ zero_zero_nat )
@ ( F @ I ) ) ).
% sums_single
thf(fact_761_sums__single,axiom,
! [I: nat,F: nat > int] :
( sums_int
@ ^ [R2: nat] : ( if_int @ ( R2 = I ) @ ( F @ R2 ) @ zero_zero_int )
@ ( F @ I ) ) ).
% sums_single
thf(fact_762_sums__single,axiom,
! [I: nat,F: nat > extended_ereal] :
( sums_Extended_ereal
@ ^ [R2: nat] : ( if_Extended_ereal @ ( R2 = I ) @ ( F @ R2 ) @ zero_z2744965634713055877_ereal )
@ ( F @ I ) ) ).
% sums_single
thf(fact_763_sums__single,axiom,
! [I: nat,F: nat > complex] :
( sums_complex
@ ^ [R2: nat] : ( if_complex @ ( R2 = I ) @ ( F @ R2 ) @ zero_zero_complex )
@ ( F @ I ) ) ).
% sums_single
thf(fact_764_sums__mult2,axiom,
! [F: nat > complex,A: complex,C: complex] :
( ( sums_complex @ F @ A )
=> ( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C )
@ ( times_times_complex @ A @ C ) ) ) ).
% sums_mult2
thf(fact_765_sums__mult2,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
@ ( times_times_real @ A @ C ) ) ) ).
% sums_mult2
thf(fact_766_sums__mult,axiom,
! [F: nat > complex,A: complex,C: complex] :
( ( sums_complex @ F @ A )
=> ( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
@ ( times_times_complex @ C @ A ) ) ) ).
% sums_mult
thf(fact_767_sums__mult,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
@ ( times_times_real @ C @ A ) ) ) ).
% sums_mult
thf(fact_768_sums__diff,axiom,
! [F: nat > real,A: real,G: nat > real,B: real] :
( ( sums_real @ F @ A )
=> ( ( sums_real @ G @ B )
=> ( sums_real
@ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
@ ( minus_minus_real @ A @ B ) ) ) ) ).
% sums_diff
thf(fact_769_sums__diff,axiom,
! [F: nat > complex,A: complex,G: nat > complex,B: complex] :
( ( sums_complex @ F @ A )
=> ( ( sums_complex @ G @ B )
=> ( sums_complex
@ ^ [N3: nat] : ( minus_minus_complex @ ( F @ N3 ) @ ( G @ N3 ) )
@ ( minus_minus_complex @ A @ B ) ) ) ) ).
% sums_diff
thf(fact_770_sums__divide,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C )
@ ( divide_divide_real @ A @ C ) ) ) ).
% sums_divide
thf(fact_771_sums__divide,axiom,
! [F: nat > complex,A: complex,C: complex] :
( ( sums_complex @ F @ A )
=> ( sums_complex
@ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C )
@ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% sums_divide
thf(fact_772_dvd__div__eq__0__iff,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_773_dvd__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_774_dvd__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_775_dvd__div__eq__0__iff,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_776_div__mult__div__if__dvd,axiom,
! [B: nat,A: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_777_div__mult__div__if__dvd,axiom,
! [B: int,A: int,D: int,C: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ D @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_778_dvd__mult__imp__div,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
=> ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_779_dvd__mult__imp__div,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
=> ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_780_dvd__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_781_dvd__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_782_div__div__eq__right,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_783_div__div__eq__right,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_784_div__mult__swap,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_785_div__mult__swap,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_786_dvd__div__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
= ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_787_dvd__div__mult,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
= ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_788_sums__mult2__iff,axiom,
! [C: complex,F: nat > complex,D: complex] :
( ( C != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C )
@ ( times_times_complex @ D @ C ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_789_sums__mult2__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
@ ( times_times_real @ D @ C ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_790_sums__mult__iff,axiom,
! [C: complex,F: nat > complex,D: complex] :
( ( C != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
@ ( times_times_complex @ C @ D ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_791_sums__mult__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
@ ( times_times_real @ C @ D ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_792_div__power,axiom,
! [B: nat,A: nat,N: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
= ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% div_power
thf(fact_793_div__power,axiom,
! [B: int,A: int,N: nat] :
( ( dvd_dvd_int @ B @ A )
=> ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
= ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% div_power
thf(fact_794_powser__sums__zero,axiom,
! [A: nat > complex] :
( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_795_powser__sums__zero,axiom,
! [A: nat > real] :
( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_796_sums__mult__D,axiom,
! [C: real,F: nat > real,A: real] :
( ( sums_real
@ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
@ A )
=> ( ( C != zero_zero_real )
=> ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% sums_mult_D
thf(fact_797_sums__mult__D,axiom,
! [C: complex,F: nat > complex,A: complex] :
( ( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
@ A )
=> ( ( C != zero_zero_complex )
=> ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% sums_mult_D
thf(fact_798_sums__Suc__imp,axiom,
! [F: nat > real,S: real] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( sums_real
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
@ S )
=> ( sums_real @ F @ S ) ) ) ).
% sums_Suc_imp
thf(fact_799_sums__Suc__imp,axiom,
! [F: nat > complex,S: complex] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( sums_complex
@ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
@ S )
=> ( sums_complex @ F @ S ) ) ) ).
% sums_Suc_imp
thf(fact_800_dvd__div__div__eq__mult,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( A != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ D @ C ) )
= ( ( times_times_nat @ B @ C )
= ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_801_dvd__div__div__eq__mult,axiom,
! [A: int,C: int,B: int,D: int] :
( ( A != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ D @ C ) )
= ( ( times_times_int @ B @ C )
= ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_802_dvd__div__iff__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_803_dvd__div__iff__mult,axiom,
! [C: int,B: int,A: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_804_div__dvd__iff__mult,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_805_div__dvd__iff__mult,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_806_dvd__div__eq__mult,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( ( divide_divide_nat @ B @ A )
= C )
= ( B
= ( times_times_nat @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_807_dvd__div__eq__mult,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( ( divide_divide_int @ B @ A )
= C )
= ( B
= ( times_times_int @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_808_odd__Numeral1,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ one ) ) ).
% odd_Numeral1
thf(fact_809_odd__Numeral1,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ one ) ) ).
% odd_Numeral1
thf(fact_810_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_811_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_812_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X: nat] :
( collect_nat
@ ^ [N3: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_813_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_814_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_815_odd__multI,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) ) ) ) ).
% odd_multI
thf(fact_816_odd__multI,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) ) ) ) ).
% odd_multI
thf(fact_817_even__multI_I1_J,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) ) ) ).
% even_multI(1)
thf(fact_818_even__multI_I1_J,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) ) ) ).
% even_multI(1)
thf(fact_819_even__multI_I2_J,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) ) ) ).
% even_multI(2)
thf(fact_820_even__multI_I2_J,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) ) ) ).
% even_multI(2)
thf(fact_821_evenE,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B4: nat] :
( A
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% evenE
thf(fact_822_evenE,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B4: int] :
( A
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) ) ) ).
% evenE
thf(fact_823_even__diffI_I1_J,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% even_diffI(1)
thf(fact_824_even__diffI_I2_J,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% even_diffI(2)
thf(fact_825_odd__diffI_I1_J,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% odd_diffI(1)
thf(fact_826_odd__diffI_I2_J,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% odd_diffI(2)
thf(fact_827_bit__eq__rec,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
& ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_828_bit__eq__rec,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A3: int,B2: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
& ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_829_odd__powerI,axiom,
! [A: nat,N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) ) ) ).
% odd_powerI
thf(fact_830_odd__powerI,axiom,
! [A: int,N: nat] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) ) ) ).
% odd_powerI
thf(fact_831_div2__even__ext__nat,axiom,
! [X2: nat,Y: nat] :
( ( ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
=> ( X2 = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_832_even__two__times__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_833_even__two__times__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_834_tendsto__mult__filterlim__at__infinity,axiom,
! [F: real > real,C: real,F2: filter_real,G: real > real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ C ) @ F2 )
=> ( ( C != zero_zero_real )
=> ( ( filterlim_real_real @ G @ at_infinity_real @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ at_infinity_real
@ F2 ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_835_tendsto__divide__0,axiom,
! [F: real > real,C: real,F2: filter_real,G: real > real] :
( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ C ) @ F2 )
=> ( ( filterlim_real_real @ G @ at_infinity_real @ F2 )
=> ( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ F2 ) ) ) ).
% tendsto_divide_0
thf(fact_836_even__unset__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_837_even__unset__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_838_real__of__nat__div,axiom,
! [D: nat,N: nat] :
( ( dvd_dvd_nat @ D @ N )
=> ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_839_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_840_artanh__0,axiom,
( ( artanh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% artanh_0
thf(fact_841_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_842_arsinh__0,axiom,
( ( arsinh_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% arsinh_0
thf(fact_843_diffs__equiv,axiom,
! [C: nat > real,X2: real] :
( ( summable_real
@ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) )
=> ( sums_real
@ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( C @ N3 ) ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_real
@ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X2 @ N3 ) ) ) ) ) ).
% diffs_equiv
thf(fact_844_diffs__equiv,axiom,
! [C: nat > complex,X2: complex] :
( ( summable_complex
@ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) )
=> ( sums_complex
@ ^ [N3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( C @ N3 ) ) @ ( power_power_complex @ X2 @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_complex
@ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X2 @ N3 ) ) ) ) ) ).
% diffs_equiv
thf(fact_845_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
= D3 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_846_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_847_mult_Oright__neutral,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ one_on3868389512446148991l_num1 )
= A ) ).
% mult.right_neutral
thf(fact_848_mult_Oright__neutral,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ one_on4623092294121504201_ereal )
= A ) ).
% mult.right_neutral
thf(fact_849_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_850_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_851_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_852_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_853_mult__1,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 @ A )
= A ) ).
% mult_1
thf(fact_854_mult__1,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ one_on4623092294121504201_ereal @ A )
= A ) ).
% mult_1
thf(fact_855_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_856_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_857_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_858_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_859_vector__space__over__itself_Oscale__one,axiom,
! [X2: complex] :
( ( times_times_complex @ one_one_complex @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_860_vector__space__over__itself_Oscale__one,axiom,
! [X2: real] :
( ( times_times_real @ one_one_real @ X2 )
= X2 ) ).
% vector_space_over_itself.scale_one
thf(fact_861_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_862_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_863_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_864_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_865_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_866_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_867_power__one,axiom,
! [N: nat] :
( ( power_7402600760894073284l_num1 @ one_on3868389512446148991l_num1 @ N )
= one_on3868389512446148991l_num1 ) ).
% power_one
thf(fact_868_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_869_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_870_power__one,axiom,
! [N: nat] :
( ( power_power_complex @ one_one_complex @ N )
= one_one_complex ) ).
% power_one
thf(fact_871_power__one,axiom,
! [N: nat] :
( ( power_1054015426188190660_ereal @ one_on4623092294121504201_ereal @ N )
= one_on4623092294121504201_ereal ) ).
% power_one
thf(fact_872_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_873_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_874_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_875_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_876_power__one__right,axiom,
! [A: extended_ereal] :
( ( power_1054015426188190660_ereal @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_877_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_878_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_879_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_880_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_881_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_diff__numeral__special_I9_J,axiom,
( ( minus_838314146864362899l_num1 @ one_on3868389512446148991l_num1 @ one_on3868389512446148991l_num1 )
= zero_z5982384998485459395l_num1 ) ).
% diff_numeral_special(9)
thf(fact_895_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_896_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_897_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_898_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_899_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_900_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_901_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_902_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_903_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_904_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_905_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_906_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_907_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_908_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_909_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_910_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_911_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_912_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_913_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_914_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera6690914467698888265omplex @ N )
= one_one_complex )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_915_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_916_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_917_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_918_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_complex
= ( numera6690914467698888265omplex @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_919_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_920_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_921_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_922_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_923_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_924_unit__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_925_unit__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_926_unit__div__1__unit,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_927_unit__div__1__unit,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_928_unit__div__1__div__1,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_929_unit__div__1__div__1,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_930_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_931_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_932_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_933_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_934_of__nat__1,axiom,
( ( semiri1795386414920522267l_num1 @ one_one_nat )
= one_on3868389512446148991l_num1 ) ).
% of_nat_1
thf(fact_935_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_936_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_937_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_938_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_939_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_940_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_941_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri8010041392384452111omplex @ N )
= one_one_complex )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_942_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_943_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_944_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_945_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_946_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_947_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_948_unit__mult__div__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
= ( divide_divide_nat @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_949_unit__mult__div__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_950_unit__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_951_unit__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_952_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_953_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_954_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_955_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_956_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_957_Suc__0__div__numeral_I1_J,axiom,
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= one_one_nat ) ).
% Suc_0_div_numeral(1)
thf(fact_958_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_959_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_960_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_961_one__reorient,axiom,
! [X2: numera2417102609627094330l_num1] :
( ( one_on3868389512446148991l_num1 = X2 )
= ( X2 = one_on3868389512446148991l_num1 ) ) ).
% one_reorient
thf(fact_962_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_963_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_964_one__reorient,axiom,
! [X2: extended_ereal] :
( ( one_on4623092294121504201_ereal = X2 )
= ( X2 = one_on4623092294121504201_ereal ) ) ).
% one_reorient
thf(fact_965_zero__neq__one,axiom,
zero_z5982384998485459395l_num1 != one_on3868389512446148991l_num1 ).
% zero_neq_one
thf(fact_966_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_967_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_968_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_969_zero__neq__one,axiom,
zero_z2744965634713055877_ereal != one_on4623092294121504201_ereal ).
% zero_neq_one
thf(fact_970_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_971_mult_Ocomm__neutral,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ A @ one_on3868389512446148991l_num1 )
= A ) ).
% mult.comm_neutral
thf(fact_972_mult_Ocomm__neutral,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ A @ one_on4623092294121504201_ereal )
= A ) ).
% mult.comm_neutral
thf(fact_973_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_974_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_975_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_976_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_977_comm__monoid__mult__class_Omult__1,axiom,
! [A: numera2417102609627094330l_num1] :
( ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_978_comm__monoid__mult__class_Omult__1,axiom,
! [A: extended_ereal] :
( ( times_7703590493115627913_ereal @ one_on4623092294121504201_ereal @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_979_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_980_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_981_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_982_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_983_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_984_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_985_one__dvd,axiom,
! [A: numera2417102609627094330l_num1] : ( dvd_dv2285863382094241760l_num1 @ one_on3868389512446148991l_num1 @ A ) ).
% one_dvd
thf(fact_986_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_987_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_988_one__dvd,axiom,
! [A: extended_ereal] : ( dvd_dv547123082783338664_ereal @ one_on4623092294121504201_ereal @ A ) ).
% one_dvd
thf(fact_989_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_990_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_991_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_992_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_993_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_994_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_995_lambda__one,axiom,
( ( ^ [X: numera2417102609627094330l_num1] : X )
= ( times_8498157372700349887l_num1 @ one_on3868389512446148991l_num1 ) ) ).
% lambda_one
thf(fact_996_lambda__one,axiom,
( ( ^ [X: extended_ereal] : X )
= ( times_7703590493115627913_ereal @ one_on4623092294121504201_ereal ) ) ).
% lambda_one
thf(fact_997_lambda__one,axiom,
( ( ^ [X: complex] : X )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_998_lambda__one,axiom,
( ( ^ [X: real] : X )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_999_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1000_lambda__one,axiom,
( ( ^ [X: int] : X )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_1001_power__eq__if,axiom,
( power_7402600760894073284l_num1
= ( ^ [P2: numera2417102609627094330l_num1,M4: nat] : ( if_Num9196306924077011444l_num1 @ ( M4 = zero_zero_nat ) @ one_on3868389512446148991l_num1 @ ( times_8498157372700349887l_num1 @ P2 @ ( power_7402600760894073284l_num1 @ P2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1002_power__eq__if,axiom,
( power_1054015426188190660_ereal
= ( ^ [P2: extended_ereal,M4: nat] : ( if_Extended_ereal @ ( M4 = zero_zero_nat ) @ one_on4623092294121504201_ereal @ ( times_7703590493115627913_ereal @ P2 @ ( power_1054015426188190660_ereal @ P2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1003_power__eq__if,axiom,
( power_power_complex
= ( ^ [P2: complex,M4: nat] : ( if_complex @ ( M4 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P2 @ ( power_power_complex @ P2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1004_power__eq__if,axiom,
( power_power_real
= ( ^ [P2: real,M4: nat] : ( if_real @ ( M4 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P2 @ ( power_power_real @ P2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1005_power__eq__if,axiom,
( power_power_nat
= ( ^ [P2: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P2 @ ( power_power_nat @ P2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1006_power__eq__if,axiom,
( power_power_int
= ( ^ [P2: int,M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P2 @ ( power_power_int @ P2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1007_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1008_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_1009_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_1010_numeral__One,axiom,
( ( numera1204434989813589363_ereal @ one )
= one_on4623092294121504201_ereal ) ).
% numeral_One
thf(fact_1011_numeral__One,axiom,
( ( numera2161328050825114965l_num1 @ one )
= one_on3868389512446148991l_num1 ) ).
% numeral_One
thf(fact_1012_numeral__One,axiom,
( ( numera6690914467698888265omplex @ one )
= one_one_complex ) ).
% numeral_One
thf(fact_1013_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_1014_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_1015_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_1016_left__right__inverse__power,axiom,
! [X2: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1,N: nat] :
( ( ( times_8498157372700349887l_num1 @ X2 @ Y )
= one_on3868389512446148991l_num1 )
=> ( ( times_8498157372700349887l_num1 @ ( power_7402600760894073284l_num1 @ X2 @ N ) @ ( power_7402600760894073284l_num1 @ Y @ N ) )
= one_on3868389512446148991l_num1 ) ) ).
% left_right_inverse_power
thf(fact_1017_left__right__inverse__power,axiom,
! [X2: extended_ereal,Y: extended_ereal,N: nat] :
( ( ( times_7703590493115627913_ereal @ X2 @ Y )
= one_on4623092294121504201_ereal )
=> ( ( times_7703590493115627913_ereal @ ( power_1054015426188190660_ereal @ X2 @ N ) @ ( power_1054015426188190660_ereal @ Y @ N ) )
= one_on4623092294121504201_ereal ) ) ).
% left_right_inverse_power
thf(fact_1018_left__right__inverse__power,axiom,
! [X2: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X2 @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y @ N ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_1019_left__right__inverse__power,axiom,
! [X2: real,Y: real,N: nat] :
( ( ( times_times_real @ X2 @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_1020_left__right__inverse__power,axiom,
! [X2: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X2 @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X2 @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1021_left__right__inverse__power,axiom,
! [X2: int,Y: int,N: nat] :
( ( ( times_times_int @ X2 @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_1022_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_1023_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_1024_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_1025_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_1026_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_1027_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_1028_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1029_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1030_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1031_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1032_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1033_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1034_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1035_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1036_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_1037_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_1038_power__one__over,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% power_one_over
thf(fact_1039_power__one__over,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% power_one_over
thf(fact_1040_power__0,axiom,
! [A: numera2417102609627094330l_num1] :
( ( power_7402600760894073284l_num1 @ A @ zero_zero_nat )
= one_on3868389512446148991l_num1 ) ).
% power_0
thf(fact_1041_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_1042_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1043_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_1044_power__0,axiom,
! [A: extended_ereal] :
( ( power_1054015426188190660_ereal @ A @ zero_zero_nat )
= one_on4623092294121504201_ereal ) ).
% power_0
thf(fact_1045_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_1046_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_1047_dvd__div__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_1048_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_1049_div__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_1050_unit__div__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_1051_unit__div__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_1052_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1053_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1054_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1055_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1056_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C3: nat] :
( B
!= ( times_times_nat @ A @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_1057_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C3: int] :
( B
!= ( times_times_int @ A @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_1058_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_7402600760894073284l_num1 @ zero_z5982384998485459395l_num1 @ N )
= one_on3868389512446148991l_num1 ) )
& ( ( N != zero_zero_nat )
=> ( ( power_7402600760894073284l_num1 @ zero_z5982384998485459395l_num1 @ N )
= zero_z5982384998485459395l_num1 ) ) ) ).
% power_0_left
thf(fact_1059_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_1060_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1061_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_1062_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_1063_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1064_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1065_unit__eq__div1,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= C )
= ( A
= ( times_times_nat @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_1066_unit__eq__div1,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= C )
= ( A
= ( times_times_int @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_1067_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1068_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1069_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1070_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1071_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1072_sums__if_H,axiom,
! [G: nat > real,X2: real] :
( ( sums_real @ G @ X2 )
=> ( sums_real
@ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ X2 ) ) ).
% sums_if'
thf(fact_1073_exhaust__2,axiom,
! [X2: numera2417102609627094330l_num1] :
( ( X2 = one_on3868389512446148991l_num1 )
| ( X2
= ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ).
% exhaust_2
thf(fact_1074_forall__2,axiom,
( ( ^ [P3: numera2417102609627094330l_num1 > $o] :
! [X4: numera2417102609627094330l_num1] : ( P3 @ X4 ) )
= ( ^ [P4: numera2417102609627094330l_num1 > $o] :
( ( P4 @ one_on3868389512446148991l_num1 )
& ( P4 @ ( numera2161328050825114965l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).
% forall_2
thf(fact_1075_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1076_Multiseries__Expansion_Ointyness__1,axiom,
( one_one_real
= ( semiri5074537144036343181t_real @ one_one_nat ) ) ).
% Multiseries_Expansion.intyness_1
thf(fact_1077_power__half__series,axiom,
( sums_real
@ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
@ one_one_real ) ).
% power_half_series
thf(fact_1078_suminf__half__series__ereal,axiom,
( ( suminf4411151127299490740_ereal
@ ^ [N3: nat] : ( power_1054015426188190660_ereal @ ( divide8893690120176169980_ereal @ one_on4623092294121504201_ereal @ ( numera1204434989813589363_ereal @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) ) )
= one_on4623092294121504201_ereal ) ).
% suminf_half_series_ereal
thf(fact_1079_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_1080_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1081_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1082_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1083_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1084_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_1085_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_1086_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1087_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1088_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1089_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1090_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1091_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1092_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1093_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1094_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1095_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1096_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1097_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1098_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1099_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1100_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1101_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1102_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1103_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_1104_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1105_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1106_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1107_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1108_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_1109_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1110_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1111_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_1112_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1113_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1114_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1115_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1116_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1117_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1118_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1119_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1120_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1121_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1122_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1123_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1124_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1125_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1126_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1127_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1128_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1129_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1130_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1131_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1132_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_1133_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1134_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1135_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1136_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1137_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1138_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1139_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1140_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1141_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1142_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1143_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1144_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1145_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1146_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1147_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_1148_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_1149_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1150_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1151_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1152_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1153_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1154_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1155_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1156_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1157_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1158_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1159_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1160_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1161_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1162_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1163_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1164_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1165_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1166_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1167_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1168_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1169_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_1170_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1171_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_1172_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1173_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1174_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1175_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1176_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1177_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1178_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1179_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1180_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1181_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1182_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1183_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1184_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R: nat,M: nat] :
( ( ord_less_eq_nat @ Q2 @ N )
=> ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
=> ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
= ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1185_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1186_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1187_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1188_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1189_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1190_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1191_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1192_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_1193_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1194_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1195_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1196_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X2: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
| ( ( times_times_nat @ B @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1197_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_1198_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1199_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1200_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1201_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1202_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_1203_zdvd__period,axiom,
! [A: int,D: int,X2: int,T2: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X2 @ T2 ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X2 @ ( times_times_int @ C @ D ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1204_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1205_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1206_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1207_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1208_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1209_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1210_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1211_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_1212_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_1213_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_1214_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1215_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_1216_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1217_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_1218_sums__if,axiom,
! [G: nat > real,X2: real,F: nat > real,Y: real] :
( ( sums_real @ G @ X2 )
=> ( ( sums_real @ F @ Y )
=> ( sums_real
@ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ X2 @ Y ) ) ) ) ).
% sums_if
thf(fact_1219_real__average__minus__first,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_first
thf(fact_1220_real__average__minus__second,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_second
thf(fact_1221_iff__4k,axiom,
! [R: real,K: nat,M: nat,N: nat,M6: nat,N4: nat] :
( ( R
= ( semiri5074537144036343181t_real @ K ) )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ R ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ M6 ) ) @ R ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N4 ) ) ) )
= ( ( M = M6 )
& ( N = N4 ) ) ) ) ) ).
% iff_4k
thf(fact_1222_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1223_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_1224_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_1225_ereal__minus_I7_J,axiom,
! [X2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X2 @ zero_z2744965634713055877_ereal )
= X2 ) ).
% ereal_minus(7)
thf(fact_1226_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_1227_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1228_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1229_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_1230_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1231_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1232_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_1233_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1234_ereal__diff__le__mono__left,axiom,
! [X2: extended_ereal,Z: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Z )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ Z ) ) ) ).
% ereal_diff_le_mono_left
thf(fact_1235_ereal__diff__positive,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( minus_2816186181549245109_ereal @ B @ A ) ) ) ).
% ereal_diff_positive
thf(fact_1236_ereal__diff__le__self,axiom,
! [Y: extended_ereal,X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ X2 ) ) ).
% ereal_diff_le_self
thf(fact_1237_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1238_suminf__0__le,axiom,
! [F: nat > extended_ereal] :
( ! [N2: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F @ N2 ) )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( suminf4411151127299490740_ereal @ F ) ) ) ).
% suminf_0_le
thf(fact_1239_suminf__add__ereal,axiom,
! [F: nat > extended_ereal,G: nat > extended_ereal] :
( ! [I2: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F @ I2 ) )
=> ( ! [I2: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( G @ I2 ) )
=> ( ( suminf4411151127299490740_ereal
@ ^ [I3: nat] : ( plus_p7876563987511257093_ereal @ ( F @ I3 ) @ ( G @ I3 ) ) )
= ( plus_p7876563987511257093_ereal @ ( suminf4411151127299490740_ereal @ F ) @ ( suminf4411151127299490740_ereal @ G ) ) ) ) ) ).
% suminf_add_ereal
thf(fact_1240_suminf__ereal__eq__0,axiom,
! [F: nat > extended_ereal] :
( ! [I2: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F @ I2 ) )
=> ( ( ( suminf4411151127299490740_ereal @ F )
= zero_z2744965634713055877_ereal )
= ( ! [I3: nat] :
( ( F @ I3 )
= zero_z2744965634713055877_ereal ) ) ) ) ).
% suminf_ereal_eq_0
thf(fact_1241_suminf__cmult__ereal,axiom,
! [F: nat > extended_ereal,A: extended_ereal] :
( ! [I2: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F @ I2 ) )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( suminf4411151127299490740_ereal
@ ^ [I3: nat] : ( times_7703590493115627913_ereal @ A @ ( F @ I3 ) ) )
= ( times_7703590493115627913_ereal @ A @ ( suminf4411151127299490740_ereal @ F ) ) ) ) ) ).
% suminf_cmult_ereal
thf(fact_1242_suminf__ereal__offset__le,axiom,
! [F: nat > extended_ereal,K: nat] :
( ! [I2: nat] : ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( F @ I2 ) )
=> ( ord_le1083603963089353582_ereal
@ ( suminf4411151127299490740_ereal
@ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ K ) ) )
@ ( suminf4411151127299490740_ereal @ F ) ) ) ).
% suminf_ereal_offset_le
thf(fact_1243_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1244_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1245_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1246_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1247_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1248_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1249_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1250_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1251_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1252_real__of__nat__div4,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% real_of_nat_div4
thf(fact_1253_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_1254_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_1255_linear__plus__1__le__power,axiom,
! [X2: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_1256_real__of__nat__div2,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) ) ).
% real_of_nat_div2
thf(fact_1257_real__of__nat__div3,axiom,
! [N: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_1258_L2__set__mult__ineq__lemma,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_1259_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1260_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1261_Bernoulli__inequality__even,axiom,
! [N: nat,X2: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_1262_triangle__lemma,axiom,
! [X2: real,Y: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y @ Z ) ) ) ) ) ) ).
% triangle_lemma
thf(fact_1263_half__bounded__equal,axiom,
! [X2: real] :
( ( ord_less_eq_real @ one_one_real @ ( times_times_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real )
= ( X2
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% half_bounded_equal
thf(fact_1264_set__bit__greater__eq,axiom,
! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% set_bit_greater_eq
thf(fact_1265_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_1266_ereal__diff__add__assoc2,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ Z )
= ( plus_p7876563987511257093_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Z ) @ Y ) ) ).
% ereal_diff_add_assoc2
thf(fact_1267_diff__add__eq__ereal,axiom,
! [A: extended_ereal,B: extended_ereal,C: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( minus_2816186181549245109_ereal @ A @ B ) @ C )
= ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ A @ C ) @ B ) ) ).
% diff_add_eq_ereal
thf(fact_1268_add__diff__eq__ereal,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ X2 @ ( minus_2816186181549245109_ereal @ Y @ Z ) )
= ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X2 @ Y ) @ Z ) ) ).
% add_diff_eq_ereal
thf(fact_1269_ereal__minus__mono,axiom,
! [A2: extended_ereal,B3: extended_ereal,D2: extended_ereal,C2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ B3 )
=> ( ( ord_le1083603963089353582_ereal @ D2 @ C2 )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ A2 @ C2 ) @ ( minus_2816186181549245109_ereal @ B3 @ D2 ) ) ) ) ).
% ereal_minus_mono
thf(fact_1270_diff__diff__commute__ereal,axiom,
! [X2: extended_ereal,Y: extended_ereal,Z: extended_ereal] :
( ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y ) @ Z )
= ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Z ) @ Y ) ) ).
% diff_diff_commute_ereal
thf(fact_1271_sum__le__prod1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1272_segment__bound__lemma,axiom,
! [B3: real,X2: real,Y: real,U: real] :
( ( ord_less_eq_real @ B3 @ X2 )
=> ( ( ord_less_eq_real @ B3 @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ U @ one_one_real )
=> ( ord_less_eq_real @ B3 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ one_one_real @ U ) @ X2 ) @ ( times_times_real @ U @ Y ) ) ) ) ) ) ) ).
% segment_bound_lemma
% Helper facts (13)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y: real] :
( ( if_real @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X2: complex,Y: complex] :
( ( if_complex @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X2: complex,Y: complex] :
( ( if_complex @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Extended____Real__Oereal_T,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( if_Extended_ereal @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Extended____Real__Oereal_T,axiom,
! [X2: extended_ereal,Y: extended_ereal] :
( ( if_Extended_ereal @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_T,axiom,
! [X2: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( if_Num9196306924077011444l_num1 @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_T,axiom,
! [X2: numera2417102609627094330l_num1,Y: numera2417102609627094330l_num1] :
( ( if_Num9196306924077011444l_num1 @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( filterlim_real_real
@ ^ [X: real] : ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ ( suc @ n ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ top_top_set_real ) ) ).
%------------------------------------------------------------------------------