TPTP Problem File: SLH0221^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Actuarial_Mathematics/0001_Interest/prob_00218_010494__12858186_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1407 ( 810 unt; 125 typ; 0 def)
% Number of atoms : 2791 (2095 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 9432 ( 319 ~; 57 |; 117 &;8229 @)
% ( 0 <=>; 710 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 4 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 187 ( 187 >; 0 *; 0 +; 0 <<)
% Number of symbols : 118 ( 115 usr; 19 con; 0-3 aty)
% Number of variables : 2796 ( 66 ^;2689 !; 41 ?;2796 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:12:04.895
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
formal_Power_fps_nat: $tType ).
thf(ty_n_t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
formal_Power_fps_int: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (115)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
invers68952373231134600s_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Int__Oint,type,
formal3717847055265219294th_int: formal_Power_fps_int > nat > int ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Nat__Onat,type,
formal3720337525774269570th_nat: formal_Power_fps_nat > nat > nat ).
thf(sy_c_Formal__Power__Series_Ofps_Ofps__nth_001t__Real__Oreal,type,
formal2580924720334399070h_real: formal3361831859752904756s_real > nat > real ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Int__Oint,type,
formal812433016830480481XD_int: formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Nat__Onat,type,
formal814923487339530757XD_nat: formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Formal__Power__Series_Ofps__XD_001t__Real__Oreal,type,
formal4292469010823635553D_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__XDp_001t__Int__Oint,type,
formal9195297484582036137Dp_int: int > formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__XDp_001t__Nat__Onat,type,
formal9197787955091086413Dp_nat: nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Formal__Power__Series_Ofps__XDp_001t__Real__Oreal,type,
formal2839450981996073129p_real: real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__X_001t__Int__Oint,type,
formal1741671657928595837_X_int: formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__X_001t__Nat__Onat,type,
formal1744162128437646113_X_nat: formal_Power_fps_nat ).
thf(sy_c_Formal__Power__Series_Ofps__X_001t__Real__Oreal,type,
formal4708490801539276157X_real: formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__compose_001t__Int__Oint,type,
formal7318879853629353975se_int: formal_Power_fps_int > formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__compose_001t__Real__Oreal,type,
formal8268054683415598839e_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__const_001t__Int__Oint,type,
formal5284259319228341128st_int: int > formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__const_001t__Nat__Onat,type,
formal5286749789737391404st_nat: nat > formal_Power_fps_nat ).
thf(sy_c_Formal__Power__Series_Ofps__const_001t__Real__Oreal,type,
formal2098867297714113032t_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__cos_001t__Real__Oreal,type,
formal461277676486907980s_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__deriv_001t__Int__Oint,type,
formal4461971871990784675iv_int: formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Formal__Power__Series_Ofps__deriv_001t__Nat__Onat,type,
formal4464462342499834951iv_nat: formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Formal__Power__Series_Ofps__deriv_001t__Real__Oreal,type,
formal4557910837323084707v_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__exp_001t__Real__Oreal,type,
formal3452214891061569154p_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__ginv_001t__Real__Oreal,type,
formal1301361369515107775v_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__hypergeo_001t__Real__Oreal,type,
formal6618874005373735610o_real: list_real > list_real > real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__integral_001t__Real__Oreal,type,
formal8984515926053063617l_real: formal3361831859752904756s_real > real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__inv_001t__Real__Oreal,type,
formal2886580842492807190v_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__ln_001t__Real__Oreal,type,
formal8688746759596762231n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__radical_001t__Real__Oreal,type,
formal8604817403481219167l_real: ( nat > real > real ) > nat > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__sin_001t__Real__Oreal,type,
formal6437758938379178589n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
minus_1563896255634514737ps_nat: formal_Power_fps_nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
minus_6791916864952032525s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
one_on8395608022581818233ps_int: formal_Power_fps_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
one_on3350087005236239133ps_nat: formal_Power_fps_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
one_on8598947968683843321s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
plus_p6043471806551771617ps_nat: formal_Power_fps_nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
plus_p6008488439947570109s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
times_3091854549176928185ps_int: formal_Power_fps_int > formal_Power_fps_int > formal_Power_fps_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
times_7269705568686124893ps_nat: formal_Power_fps_nat > formal_Power_fps_nat > formal_Power_fps_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
times_7561426564079326009s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Int__Oint_J,type,
times_times_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
times_times_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
uminus8389970968385878141s_real: formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
zero_z4353722679246354365ps_int: formal_Power_fps_int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
zero_z8531573698755551073ps_nat: formal_Power_fps_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Interest_Oi__force,type,
i_force: real > real ).
thf(sy_c_Interest_Oi__nom,type,
i_nom: real > nat > real ).
thf(sy_c_Interest_Ointerest,type,
interest: real > $o ).
thf(sy_c_Interest_Operp,type,
perp: real > nat > real ).
thf(sy_c_Linear__Algebra_Oinfnorm_001t__Real__Oreal,type,
linear_infnorm_real: real > real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Int__Oint_J,type,
semiri6570152736363784213ps_int: nat > formal_Power_fps_int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Nat__Onat_J,type,
semiri1524631719018205113ps_nat: nat > formal_Power_fps_nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
semiri2475410149736220053s_real: nat > formal3361831859752904756s_real ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
power_1846127563762588094s_real: formal3361831859752904756s_real > nat > formal3361831859752904756s_real ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Real__Vector__Spaces_Orepresentation_001t__Real__Oreal,type,
real_V2383402355066202452n_real: set_real > real > real > real ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V1485227260804924795R_real: real > real > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
divide1155267253282662278s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
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__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
dvd_dv1093944294739598810s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > $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__Real__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
modulo6537955338513192006s_real: formal3361831859752904756s_real > formal3361831859752904756s_real > formal3361831859752904756s_real ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_i,type,
i: real ).
thf(sy_v_m,type,
m: nat ).
% Relevant facts (1274)
thf(fact_0__092_060open_062i_A_061_A0_092_060close_062,axiom,
i = zero_zero_real ).
% \<open>i = 0\<close>
thf(fact_1_interest__axioms,axiom,
interest @ i ).
% interest_axioms
thf(fact_2_that,axiom,
m != zero_zero_nat ).
% that
thf(fact_3_i__nom__1,axiom,
( ( i_nom @ i @ one_one_nat )
= i ) ).
% i_nom_1
thf(fact_4_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_5_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_6_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_7_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_8_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_9_representation__zero,axiom,
! [Basis: set_real] :
( ( real_V2383402355066202452n_real @ Basis @ zero_zero_real )
= ( ^ [B: real] : zero_zero_real ) ) ).
% representation_zero
thf(fact_10_infnorm__0,axiom,
( ( linear_infnorm_real @ zero_zero_real )
= zero_zero_real ) ).
% infnorm_0
thf(fact_11_infnorm__eq__0,axiom,
! [X: real] :
( ( ( linear_infnorm_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% infnorm_eq_0
thf(fact_12_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_13_scaleR__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% scaleR_eq_0_iff
thf(fact_14_scaleR__zero__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% scaleR_zero_left
thf(fact_15_scaleR__cancel__right,axiom,
! [A: real,X: real,B2: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ B2 @ X ) )
= ( ( A = B2 )
| ( X = zero_zero_real ) ) ) ).
% scaleR_cancel_right
thf(fact_16_scaleR__zero__right,axiom,
! [A: real] :
( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% scaleR_zero_right
thf(fact_17_scaleR__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% scaleR_cancel_left
thf(fact_18_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_19_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_20_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_21_scaleR__left__commute,axiom,
! [A: real,B2: real,X: real] :
( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B2 @ X ) )
= ( real_V1485227260804924795R_real @ B2 @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ).
% scaleR_left_commute
thf(fact_22_scaleR__right__imp__eq,axiom,
! [X: real,A: real,B2: real] :
( ( X != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ B2 @ X ) )
=> ( A = B2 ) ) ) ).
% scaleR_right_imp_eq
thf(fact_23_scaleR__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A @ X )
= ( real_V1485227260804924795R_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% scaleR_left_imp_eq
thf(fact_24_representation__ne__zero,axiom,
! [Basis: set_real,V: real,B2: real] :
( ( ( real_V2383402355066202452n_real @ Basis @ V @ B2 )
!= zero_zero_real )
=> ( member_real @ B2 @ Basis ) ) ).
% representation_ne_zero
thf(fact_25_interest_Oi__nom__1,axiom,
! [I: real] :
( ( interest @ I )
=> ( ( i_nom @ I @ one_one_nat )
= I ) ) ).
% interest.i_nom_1
thf(fact_26_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_27_pth__4_I1_J,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% pth_4(1)
thf(fact_28_scaleR__cong__right,axiom,
! [X: real,R: real,P: real] :
( ( ( X != zero_zero_real )
=> ( R = P ) )
=> ( ( real_V1485227260804924795R_real @ R @ X )
= ( real_V1485227260804924795R_real @ P @ X ) ) ) ).
% scaleR_cong_right
thf(fact_29_pth__4_I2_J,axiom,
! [C: real] :
( ( real_V1485227260804924795R_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% pth_4(2)
thf(fact_30_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_31_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_32_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_33_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_34_powr__zero__eq__one,axiom,
! [X: real] :
( ( ( X = zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( ( powr_real @ X @ zero_zero_real )
= one_one_real ) ) ) ).
% powr_zero_eq_one
thf(fact_35_sin__coeff__0,axiom,
( ( sin_coeff @ zero_zero_nat )
= zero_zero_real ) ).
% sin_coeff_0
thf(fact_36_fps__sin__0,axiom,
( ( formal6437758938379178589n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_sin_0
thf(fact_37_fps__one__nth,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ one_on8598947968683843321s_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ one_on8598947968683843321s_real @ N )
= zero_zero_real ) ) ) ).
% fps_one_nth
thf(fact_38_fps__one__nth,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ one_on3350087005236239133ps_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ one_on3350087005236239133ps_nat @ N )
= zero_zero_nat ) ) ) ).
% fps_one_nth
thf(fact_39_fps__one__nth,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ one_on8395608022581818233ps_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ one_on8395608022581818233ps_int @ N )
= zero_zero_int ) ) ) ).
% fps_one_nth
thf(fact_40_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_41_scaleR__one,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ one_one_real @ X )
= X ) ).
% scaleR_one
thf(fact_42_powr__0,axiom,
! [Z: real] :
( ( powr_real @ zero_zero_real @ Z )
= zero_zero_real ) ).
% powr_0
thf(fact_43_powr__eq__0__iff,axiom,
! [W: real,Z: real] :
( ( ( powr_real @ W @ Z )
= zero_zero_real )
= ( W = zero_zero_real ) ) ).
% powr_eq_0_iff
thf(fact_44_powr__one__eq__one,axiom,
! [A: real] :
( ( powr_real @ one_one_real @ A )
= one_one_real ) ).
% powr_one_eq_one
thf(fact_45_fps__zero__nth,axiom,
! [N: nat] :
( ( formal2580924720334399070h_real @ zero_z7760665558314615101s_real @ N )
= zero_zero_real ) ).
% fps_zero_nth
thf(fact_46_fps__zero__nth,axiom,
! [N: nat] :
( ( formal3720337525774269570th_nat @ zero_z8531573698755551073ps_nat @ N )
= zero_zero_nat ) ).
% fps_zero_nth
thf(fact_47_fps__zero__nth,axiom,
! [N: nat] :
( ( formal3717847055265219294th_int @ zero_z4353722679246354365ps_int @ N )
= zero_zero_int ) ).
% fps_zero_nth
thf(fact_48_fps__sin__nth__0,axiom,
! [C: real] :
( ( formal2580924720334399070h_real @ ( formal6437758938379178589n_real @ C ) @ zero_zero_nat )
= zero_zero_real ) ).
% fps_sin_nth_0
thf(fact_49_pth__1,axiom,
! [X: real] :
( X
= ( real_V1485227260804924795R_real @ one_one_real @ X ) ) ).
% pth_1
thf(fact_50_fps__nonzeroI,axiom,
! [F: formal3361831859752904756s_real,N: nat] :
( ( ( formal2580924720334399070h_real @ F @ N )
!= zero_zero_real )
=> ( F != zero_z7760665558314615101s_real ) ) ).
% fps_nonzeroI
thf(fact_51_fps__nonzeroI,axiom,
! [F: formal_Power_fps_nat,N: nat] :
( ( ( formal3720337525774269570th_nat @ F @ N )
!= zero_zero_nat )
=> ( F != zero_z8531573698755551073ps_nat ) ) ).
% fps_nonzeroI
thf(fact_52_fps__nonzeroI,axiom,
! [F: formal_Power_fps_int,N: nat] :
( ( ( formal3717847055265219294th_int @ F @ N )
!= zero_zero_int )
=> ( F != zero_z4353722679246354365ps_int ) ) ).
% fps_nonzeroI
thf(fact_53_fps__nonzero__nth,axiom,
! [F: formal3361831859752904756s_real] :
( ( F != zero_z7760665558314615101s_real )
= ( ? [N2: nat] :
( ( formal2580924720334399070h_real @ F @ N2 )
!= zero_zero_real ) ) ) ).
% fps_nonzero_nth
thf(fact_54_fps__nonzero__nth,axiom,
! [F: formal_Power_fps_nat] :
( ( F != zero_z8531573698755551073ps_nat )
= ( ? [N2: nat] :
( ( formal3720337525774269570th_nat @ F @ N2 )
!= zero_zero_nat ) ) ) ).
% fps_nonzero_nth
thf(fact_55_fps__nonzero__nth,axiom,
! [F: formal_Power_fps_int] :
( ( F != zero_z4353722679246354365ps_int )
= ( ? [N2: nat] :
( ( formal3717847055265219294th_int @ F @ N2 )
!= zero_zero_int ) ) ) ).
% fps_nonzero_nth
thf(fact_56_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_57_fps__hypergeo__0,axiom,
! [As: list_real,Bs: list_real,C: real] :
( ( formal2580924720334399070h_real @ ( formal6618874005373735610o_real @ As @ Bs @ C ) @ zero_zero_nat )
= one_one_real ) ).
% fps_hypergeo_0
thf(fact_58_fps__ln__0,axiom,
! [C: real] :
( ( formal2580924720334399070h_real @ ( formal8688746759596762231n_real @ C ) @ zero_zero_nat )
= zero_zero_real ) ).
% fps_ln_0
thf(fact_59_fps__inv__idempotent,axiom,
! [A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
!= zero_zero_real )
=> ( ( formal2886580842492807190v_real @ ( formal2886580842492807190v_real @ A ) )
= A ) ) ) ).
% fps_inv_idempotent
thf(fact_60_fps__XD__0th,axiom,
! [A: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( formal4292469010823635553D_real @ A ) @ zero_zero_nat )
= zero_zero_real ) ).
% fps_XD_0th
thf(fact_61_fps__XD__0th,axiom,
! [A: formal_Power_fps_nat] :
( ( formal3720337525774269570th_nat @ ( formal814923487339530757XD_nat @ A ) @ zero_zero_nat )
= zero_zero_nat ) ).
% fps_XD_0th
thf(fact_62_fps__XD__0th,axiom,
! [A: formal_Power_fps_int] :
( ( formal3717847055265219294th_int @ ( formal812433016830480481XD_int @ A ) @ zero_zero_nat )
= zero_zero_int ) ).
% fps_XD_0th
thf(fact_63_fps__cos__nth__0,axiom,
! [C: real] :
( ( formal2580924720334399070h_real @ ( formal461277676486907980s_real @ C ) @ zero_zero_nat )
= one_one_real ) ).
% fps_cos_nth_0
thf(fact_64_fps__is__unit__iff,axiom,
! [F: formal3361831859752904756s_real] :
( ( dvd_dv1093944294739598810s_real @ F @ one_on8598947968683843321s_real )
= ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real ) ) ).
% fps_is_unit_iff
thf(fact_65_fps__radical__nth__0,axiom,
! [N: nat,R: nat > real > real,A: formal3361831859752904756s_real] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( formal8604817403481219167l_real @ R @ N @ A ) @ zero_zero_nat )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( formal8604817403481219167l_real @ R @ N @ A ) @ zero_zero_nat )
= ( R @ N @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) ) ) ) ).
% fps_radical_nth_0
thf(fact_66_fps__inverse__eq__0__iff,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real )
= ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real ) ) ).
% fps_inverse_eq_0_iff
thf(fact_67_fps__X__nth,axiom,
! [N: nat] :
( ( ( N = one_one_nat )
=> ( ( formal2580924720334399070h_real @ formal4708490801539276157X_real @ N )
= one_one_real ) )
& ( ( N != one_one_nat )
=> ( ( formal2580924720334399070h_real @ formal4708490801539276157X_real @ N )
= zero_zero_real ) ) ) ).
% fps_X_nth
thf(fact_68_fps__X__nth,axiom,
! [N: nat] :
( ( ( N = one_one_nat )
=> ( ( formal3720337525774269570th_nat @ formal1744162128437646113_X_nat @ N )
= one_one_nat ) )
& ( ( N != one_one_nat )
=> ( ( formal3720337525774269570th_nat @ formal1744162128437646113_X_nat @ N )
= zero_zero_nat ) ) ) ).
% fps_X_nth
thf(fact_69_fps__X__nth,axiom,
! [N: nat] :
( ( ( N = one_one_nat )
=> ( ( formal3717847055265219294th_int @ formal1741671657928595837_X_int @ N )
= one_one_int ) )
& ( ( N != one_one_nat )
=> ( ( formal3717847055265219294th_int @ formal1741671657928595837_X_int @ N )
= zero_zero_int ) ) ) ).
% fps_X_nth
thf(fact_70_fps__mod__unit,axiom,
! [G: formal3361831859752904756s_real,F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ G @ zero_zero_nat )
!= zero_zero_real )
=> ( ( modulo6537955338513192006s_real @ F @ G )
= zero_z7760665558314615101s_real ) ) ).
% fps_mod_unit
thf(fact_71_dvd__0__left__iff,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
= ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
thf(fact_72_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_73_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_74_dvd__0__right,axiom,
! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% dvd_0_right
thf(fact_75_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_76_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_77_mod__self,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ A )
= zero_zero_nat ) ).
% mod_self
thf(fact_78_mod__self,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ A )
= zero_zero_int ) ).
% mod_self
thf(fact_79_mod__by__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ zero_zero_nat )
= A ) ).
% mod_by_0
thf(fact_80_mod__by__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ zero_zero_int )
= A ) ).
% mod_by_0
thf(fact_81_mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mod_0
thf(fact_82_mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mod_0
thf(fact_83_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_84_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_85_dvd__imp__mod__0,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( modulo_modulo_nat @ B2 @ A )
= zero_zero_nat ) ) ).
% dvd_imp_mod_0
thf(fact_86_dvd__imp__mod__0,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( modulo_modulo_int @ B2 @ A )
= zero_zero_int ) ) ).
% dvd_imp_mod_0
thf(fact_87_fps__cos__0,axiom,
( ( formal461277676486907980s_real @ zero_zero_real )
= one_on8598947968683843321s_real ) ).
% fps_cos_0
thf(fact_88_fps__inverse__0__iff,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= zero_zero_real )
= ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real ) ) ).
% fps_inverse_0_iff
thf(fact_89_fps__inverse__idempotent,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( invers68952373231134600s_real @ ( invers68952373231134600s_real @ F ) )
= F ) ) ).
% fps_inverse_idempotent
thf(fact_90_fps__inverse__nth__0,axiom,
! [F: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) ) ) ).
% fps_inverse_nth_0
thf(fact_91_dvd__mod__imp__dvd,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( dvd_dvd_nat @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_92_dvd__mod__imp__dvd,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( dvd_dvd_int @ C @ A ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_93_dvd__mod__iff,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
= ( dvd_dvd_nat @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_94_dvd__mod__iff,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
= ( dvd_dvd_int @ C @ A ) ) ) ).
% dvd_mod_iff
thf(fact_95_dvd__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_96_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_97_mod__eq__0__iff__dvd,axiom,
! [A: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ B2 )
= zero_zero_nat )
= ( dvd_dvd_nat @ B2 @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_98_mod__eq__0__iff__dvd,axiom,
! [A: int,B2: int] :
( ( ( modulo_modulo_int @ A @ B2 )
= zero_zero_int )
= ( dvd_dvd_int @ B2 @ A ) ) ).
% mod_eq_0_iff_dvd
thf(fact_99_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_nat
= ( ^ [A2: nat,B: nat] :
( ( modulo_modulo_nat @ B @ A2 )
= zero_zero_nat ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_100_dvd__eq__mod__eq__0,axiom,
( dvd_dvd_int
= ( ^ [A2: int,B: int] :
( ( modulo_modulo_int @ B @ A2 )
= zero_zero_int ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_101_mod__0__imp__dvd,axiom,
! [A: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ B2 )
= zero_zero_nat )
=> ( dvd_dvd_nat @ B2 @ A ) ) ).
% mod_0_imp_dvd
thf(fact_102_mod__0__imp__dvd,axiom,
! [A: int,B2: int] :
( ( ( modulo_modulo_int @ A @ B2 )
= zero_zero_int )
=> ( dvd_dvd_int @ B2 @ A ) ) ).
% mod_0_imp_dvd
thf(fact_103_inverse__scaleR__distrib,axiom,
! [A: real,X: real] :
( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
= ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ).
% inverse_scaleR_distrib
thf(fact_104_fps__inverse__zero_H,axiom,
( ( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ zero_z7760665558314615101s_real )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_zero'
thf(fact_105_fps__inverse__one_H,axiom,
( ( ( inverse_inverse_real @ one_one_real )
= one_one_real )
=> ( ( invers68952373231134600s_real @ one_on8598947968683843321s_real )
= one_on8598947968683843321s_real ) ) ).
% fps_inverse_one'
thf(fact_106_dvd__0__left,axiom,
! [A: real] :
( ( dvd_dvd_real @ zero_zero_real @ A )
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
thf(fact_107_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_108_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_109_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_110_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_111_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_112_unit__imp__dvd,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ B2 @ A ) ) ).
% unit_imp_dvd
thf(fact_113_unit__imp__dvd,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ B2 @ A ) ) ).
% unit_imp_dvd
thf(fact_114_dvd__unit__imp__unit,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_115_dvd__unit__imp__unit,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_116_nonzero__inverse__scaleR__distrib,axiom,
! [A: real,X: real] :
( ( A != zero_zero_real )
=> ( ( X != zero_zero_real )
=> ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
= ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_117_fps__inverse__0__iff_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ ( invers68952373231134600s_real @ F ) @ zero_zero_nat )
= zero_zero_real )
= ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real ) ) ).
% fps_inverse_0_iff'
thf(fact_118_fps__inverse__eq__0_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_eq_0'
thf(fact_119_fps__inverse__eq__0__iff_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real )
= ( ( inverse_inverse_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) )
= zero_zero_real ) ) ).
% fps_inverse_eq_0_iff'
thf(fact_120_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_121_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_122_fps__unit__dvd,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( dvd_dv1093944294739598810s_real @ F @ G ) ) ).
% fps_unit_dvd
thf(fact_123_fps__inverse__eq__0,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
= zero_zero_real )
=> ( ( invers68952373231134600s_real @ F )
= zero_z7760665558314615101s_real ) ) ).
% fps_inverse_eq_0
thf(fact_124_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_125_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_126_inverse__eq__1__iff,axiom,
! [X: real] :
( ( ( inverse_inverse_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_127_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_128_bits__mod__0,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_129_bits__mod__0,axiom,
! [A: int] :
( ( modulo_modulo_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_130_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_131_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_132_unit__imp__mod__eq__0,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( modulo_modulo_nat @ A @ B2 )
= zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
thf(fact_133_unit__imp__mod__eq__0,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( modulo_modulo_int @ A @ B2 )
= zero_zero_int ) ) ).
% unit_imp_mod_eq_0
thf(fact_134_mod__mod__trivial,axiom,
! [A: nat,B2: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mod_trivial
thf(fact_135_mod__mod__trivial,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mod_trivial
thf(fact_136_inverse__eq__iff__eq,axiom,
! [A: real,B2: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B2 ) )
= ( A = B2 ) ) ).
% inverse_eq_iff_eq
thf(fact_137_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_138_inverse__eq__imp__eq,axiom,
! [A: real,B2: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B2 ) )
=> ( A = B2 ) ) ).
% inverse_eq_imp_eq
thf(fact_139_dvd__field__iff,axiom,
( dvd_dvd_real
= ( ^ [A2: real,B: real] :
( ( A2 = zero_zero_real )
=> ( B = zero_zero_real ) ) ) ) ).
% dvd_field_iff
thf(fact_140_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_141_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_142_nonzero__inverse__eq__imp__eq,axiom,
! [A: real,B2: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B2 ) )
=> ( ( A != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( A = B2 ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_143_nonzero__inverse__inverse__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_144_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_145_mod__mod__cancel,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C )
= ( modulo_modulo_nat @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_146_mod__mod__cancel,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ C )
= ( modulo_modulo_int @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_147_dvd__mod,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_148_dvd__mod,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ M )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_149_divideR__right,axiom,
! [R: real,Y: real,X: real] :
( ( R != zero_zero_real )
=> ( ( Y
= ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ R ) @ X ) )
= ( ( real_V1485227260804924795R_real @ R @ Y )
= X ) ) ) ).
% divideR_right
thf(fact_150_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_151_eucl__induct,axiom,
! [P2: nat > nat > $o,A: nat,B2: nat] :
( ! [B3: nat] : ( P2 @ B3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( P2 @ B3 @ ( modulo_modulo_nat @ A3 @ B3 ) )
=> ( P2 @ A3 @ B3 ) ) )
=> ( P2 @ A @ B2 ) ) ) ).
% eucl_induct
thf(fact_152_eucl__induct,axiom,
! [P2: int > int > $o,A: int,B2: int] :
( ! [B3: int] : ( P2 @ B3 @ zero_zero_int )
=> ( ! [A3: int,B3: int] :
( ( B3 != zero_zero_int )
=> ( ( P2 @ B3 @ ( modulo_modulo_int @ A3 @ B3 ) )
=> ( P2 @ A3 @ B3 ) ) )
=> ( P2 @ A @ B2 ) ) ) ).
% eucl_induct
thf(fact_153_inverse__mult__eq__1,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( times_7561426564079326009s_real @ ( invers68952373231134600s_real @ F ) @ F )
= one_on8598947968683843321s_real ) ) ).
% inverse_mult_eq_1
thf(fact_154_fps__inv__right,axiom,
! [A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
!= zero_zero_real )
=> ( ( formal8268054683415598839e_real @ A @ ( formal2886580842492807190v_real @ A ) )
= formal4708490801539276157X_real ) ) ) ).
% fps_inv_right
thf(fact_155_fps__inv,axiom,
! [A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
!= zero_zero_real )
=> ( ( formal8268054683415598839e_real @ ( formal2886580842492807190v_real @ A ) @ A )
= formal4708490801539276157X_real ) ) ) ).
% fps_inv
thf(fact_156_fps__XDp0,axiom,
( ( formal2839450981996073129p_real @ zero_zero_real )
= formal4292469010823635553D_real ) ).
% fps_XDp0
thf(fact_157_fps__XDp0,axiom,
( ( formal9197787955091086413Dp_nat @ zero_zero_nat )
= formal814923487339530757XD_nat ) ).
% fps_XDp0
thf(fact_158_fps__XDp0,axiom,
( ( formal9195297484582036137Dp_int @ zero_zero_int )
= formal812433016830480481XD_int ) ).
% fps_XDp0
thf(fact_159_mult__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_160_mult__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_161_mult__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_162_mult__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_163_mult__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_164_mult__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_165_mult__eq__0__iff,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_166_mult__eq__0__iff,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_167_mult__eq__0__iff,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_168_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_169_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_170_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_171_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_172_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_173_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_174_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_175_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_176_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_177_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_178_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_179_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_180_mult__scaleR__left,axiom,
! [A: real,X: real,Y: real] :
( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X ) @ Y )
= ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y ) ) ) ).
% mult_scaleR_left
thf(fact_181_mult__scaleR__right,axiom,
! [X: real,A: real,Y: real] :
( ( times_times_real @ X @ ( real_V1485227260804924795R_real @ A @ Y ) )
= ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y ) ) ) ).
% mult_scaleR_right
thf(fact_182_inverse__mult__distrib,axiom,
! [A: real,B2: real] :
( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
= ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) ) ) ).
% inverse_mult_distrib
thf(fact_183_mult__cancel__left1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_184_mult__cancel__left1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_185_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_186_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_187_mult__cancel__right1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_188_mult__cancel__right1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_189_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_190_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_191_dvd__times__right__cancel__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_192_dvd__times__right__cancel__iff,axiom,
! [A: int,B2: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_193_dvd__times__left__cancel__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_194_dvd__times__left__cancel__iff,axiom,
! [A: int,B2: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_195_dvd__mult__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B2 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_196_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B2 ) ) ) ).
% dvd_mult_cancel_right
thf(fact_197_dvd__mult__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B2 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_198_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B2 ) ) ) ).
% dvd_mult_cancel_left
thf(fact_199_unit__prod,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_200_unit__prod,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_201_mod__mult__self2__is__0,axiom,
! [A: nat,B2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_202_mod__mult__self2__is__0,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ B2 )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_203_mod__mult__self1__is__0,axiom,
! [B2: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A ) @ B2 )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_204_mod__mult__self1__is__0,axiom,
! [B2: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B2 @ A ) @ B2 )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_205_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_206_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_207_fps__mult__nth__0,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ zero_zero_nat )
= ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_208_fps__mult__nth__0,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G ) @ zero_zero_nat )
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_209_fps__mult__nth__0,axiom,
! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ zero_zero_nat )
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ).
% fps_mult_nth_0
thf(fact_210_fps__X__fps__compose__startby0,axiom,
! [A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( formal8268054683415598839e_real @ formal4708490801539276157X_real @ A )
= A ) ) ).
% fps_X_fps_compose_startby0
thf(fact_211_fps__X__fps__compose__startby0,axiom,
! [A: formal_Power_fps_int] :
( ( ( formal3717847055265219294th_int @ A @ zero_zero_nat )
= zero_zero_int )
=> ( ( formal7318879853629353975se_int @ formal1741671657928595837_X_int @ A )
= A ) ) ).
% fps_X_fps_compose_startby0
thf(fact_212_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_213_mult_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_214_mult_Oleft__commute,axiom,
! [B2: real,A: real,C: real] :
( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_215_mult_Oleft__commute,axiom,
! [B2: int,A: int,C: int] :
( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_216_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B: nat] : ( times_times_nat @ B @ A2 ) ) ) ).
% mult.commute
thf(fact_217_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B: real] : ( times_times_real @ B @ A2 ) ) ) ).
% mult.commute
thf(fact_218_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B: int] : ( times_times_int @ B @ A2 ) ) ) ).
% mult.commute
thf(fact_219_mult_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_220_mult_Oassoc,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_221_mult_Oassoc,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_222_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_223_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_224_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_225_mult__delta__right,axiom,
! [B2: $o,X: nat,Y: nat] :
( ( B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_226_mult__delta__right,axiom,
! [B2: $o,X: real,Y: real] :
( ( B2
=> ( ( times_times_real @ X @ ( if_real @ B2 @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_real @ X @ ( if_real @ B2 @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_227_mult__delta__right,axiom,
! [B2: $o,X: int,Y: int] :
( ( B2
=> ( ( times_times_int @ X @ ( if_int @ B2 @ Y @ zero_zero_int ) )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_int @ X @ ( if_int @ B2 @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_228_mult__delta__left,axiom,
! [B2: $o,X: nat,Y: nat] :
( ( B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_229_mult__delta__left,axiom,
! [B2: $o,X: real,Y: real] :
( ( B2
=> ( ( times_times_real @ ( if_real @ B2 @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_real @ ( if_real @ B2 @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_230_mult__delta__left,axiom,
! [B2: $o,X: int,Y: int] :
( ( B2
=> ( ( times_times_int @ ( if_int @ B2 @ X @ zero_zero_int ) @ Y )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_int @ ( if_int @ B2 @ X @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_231_mult__right__cancel,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_232_mult__right__cancel,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_233_mult__right__cancel,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_234_mult__left__cancel,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_235_mult__left__cancel,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_236_mult__left__cancel,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_237_no__zero__divisors,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_238_no__zero__divisors,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( times_times_real @ A @ B2 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_239_no__zero__divisors,axiom,
! [A: int,B2: int] :
( ( A != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_240_divisors__zero,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_241_divisors__zero,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_242_divisors__zero,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_243_mult__not__zero,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_244_mult__not__zero,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B2 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_245_mult__not__zero,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_246_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_247_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_248_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_249_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_250_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_251_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_252_dvd__triv__right,axiom,
! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ A ) ) ).
% dvd_triv_right
thf(fact_253_dvd__triv__right,axiom,
! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ A ) ) ).
% dvd_triv_right
thf(fact_254_dvd__triv__right,axiom,
! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ A ) ) ).
% dvd_triv_right
thf(fact_255_dvd__mult__right,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
=> ( dvd_dvd_nat @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_256_dvd__mult__right,axiom,
! [A: real,B2: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
=> ( dvd_dvd_real @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_257_dvd__mult__right,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
=> ( dvd_dvd_int @ B2 @ C ) ) ).
% dvd_mult_right
thf(fact_258_mult__dvd__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_259_mult__dvd__mono,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( dvd_dvd_real @ A @ B2 )
=> ( ( dvd_dvd_real @ C @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_260_mult__dvd__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_261_dvd__triv__left,axiom,
! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B2 ) ) ).
% dvd_triv_left
thf(fact_262_dvd__triv__left,axiom,
! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B2 ) ) ).
% dvd_triv_left
thf(fact_263_dvd__triv__left,axiom,
! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B2 ) ) ).
% dvd_triv_left
thf(fact_264_dvd__mult__left,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_265_dvd__mult__left,axiom,
! [A: real,B2: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
=> ( dvd_dvd_real @ A @ C ) ) ).
% dvd_mult_left
thf(fact_266_dvd__mult__left,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_267_dvd__mult2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_268_dvd__mult2,axiom,
! [A: real,B2: real,C: real] :
( ( dvd_dvd_real @ A @ B2 )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_269_dvd__mult2,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% dvd_mult2
thf(fact_270_dvd__mult,axiom,
! [A: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_271_dvd__mult,axiom,
! [A: real,C: real,B2: real] :
( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_272_dvd__mult,axiom,
! [A: int,C: int,B2: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% dvd_mult
thf(fact_273_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B: nat,A2: nat] :
? [K2: nat] :
( A2
= ( times_times_nat @ B @ K2 ) ) ) ) ).
% dvd_def
thf(fact_274_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B: real,A2: real] :
? [K2: real] :
( A2
= ( times_times_real @ B @ K2 ) ) ) ) ).
% dvd_def
thf(fact_275_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B: int,A2: int] :
? [K2: int] :
( A2
= ( times_times_int @ B @ K2 ) ) ) ) ).
% dvd_def
thf(fact_276_dvdI,axiom,
! [A: nat,B2: nat,K: nat] :
( ( A
= ( times_times_nat @ B2 @ K ) )
=> ( dvd_dvd_nat @ B2 @ A ) ) ).
% dvdI
thf(fact_277_dvdI,axiom,
! [A: real,B2: real,K: real] :
( ( A
= ( times_times_real @ B2 @ K ) )
=> ( dvd_dvd_real @ B2 @ A ) ) ).
% dvdI
thf(fact_278_dvdI,axiom,
! [A: int,B2: int,K: int] :
( ( A
= ( times_times_int @ B2 @ K ) )
=> ( dvd_dvd_int @ B2 @ A ) ) ).
% dvdI
thf(fact_279_dvdE,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ A )
=> ~ ! [K3: nat] :
( A
!= ( times_times_nat @ B2 @ K3 ) ) ) ).
% dvdE
thf(fact_280_dvdE,axiom,
! [B2: real,A: real] :
( ( dvd_dvd_real @ B2 @ A )
=> ~ ! [K3: real] :
( A
!= ( times_times_real @ B2 @ K3 ) ) ) ).
% dvdE
thf(fact_281_dvdE,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ A )
=> ~ ! [K3: int] :
( A
!= ( times_times_int @ B2 @ K3 ) ) ) ).
% dvdE
thf(fact_282_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: real,X: real] :
( ( ( times_times_real @ Y @ X )
= ( times_times_real @ X @ Y ) )
=> ( ( times_times_real @ ( inverse_inverse_real @ Y ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_283_mod__mult__eq,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% mod_mult_eq
thf(fact_284_mod__mult__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% mod_mult_eq
thf(fact_285_mod__mult__cong,axiom,
! [A: nat,C: nat,A4: nat,B2: nat,B4: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A4 @ C ) )
=> ( ( ( modulo_modulo_nat @ B2 @ C )
= ( modulo_modulo_nat @ B4 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_286_mod__mult__cong,axiom,
! [A: int,C: int,A4: int,B2: int,B4: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A4 @ C ) )
=> ( ( ( modulo_modulo_int @ B2 @ C )
= ( modulo_modulo_int @ B4 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_287_mod__mult__mult2,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C ) ) ).
% mod_mult_mult2
thf(fact_288_mod__mult__mult2,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ B2 ) @ C ) ) ).
% mod_mult_mult2
thf(fact_289_mult__mod__right,axiom,
! [C: nat,A: nat,B2: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ).
% mult_mod_right
thf(fact_290_mult__mod__right,axiom,
! [C: int,A: int,B2: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ).
% mult_mod_right
thf(fact_291_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_292_mod__mult__left__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_293_mod__mult__right__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_294_mod__mult__right__eq,axiom,
! [A: int,B2: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_295_fps__compose__mult__distrib,axiom,
! [C: formal3361831859752904756s_real,A: formal3361831859752904756s_real,B2: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ C @ zero_zero_nat )
= zero_zero_real )
=> ( ( formal8268054683415598839e_real @ ( times_7561426564079326009s_real @ A @ B2 ) @ C )
= ( times_7561426564079326009s_real @ ( formal8268054683415598839e_real @ A @ C ) @ ( formal8268054683415598839e_real @ B2 @ C ) ) ) ) ).
% fps_compose_mult_distrib
thf(fact_296_fps__compose__mult__distrib,axiom,
! [C: formal_Power_fps_int,A: formal_Power_fps_int,B2: formal_Power_fps_int] :
( ( ( formal3717847055265219294th_int @ C @ zero_zero_nat )
= zero_zero_int )
=> ( ( formal7318879853629353975se_int @ ( times_3091854549176928185ps_int @ A @ B2 ) @ C )
= ( times_3091854549176928185ps_int @ ( formal7318879853629353975se_int @ A @ C ) @ ( formal7318879853629353975se_int @ B2 @ C ) ) ) ) ).
% fps_compose_mult_distrib
thf(fact_297_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
! [F: formal3361831859752904756s_real] :
( ( ? [G2: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ F @ G2 ) ) )
= ( ? [K2: real] :
( one_one_real
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ K2 ) ) ) ) ).
% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_298_fps__is__left__unit__iff__zeroth__is__left__unit,axiom,
! [F: formal_Power_fps_int] :
( ( ? [G2: formal_Power_fps_int] :
( one_on8395608022581818233ps_int
= ( times_3091854549176928185ps_int @ F @ G2 ) ) )
= ( ? [K2: int] :
( one_one_int
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ K2 ) ) ) ) ).
% fps_is_left_unit_iff_zeroth_is_left_unit
thf(fact_299_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
! [F: formal3361831859752904756s_real] :
( ( ? [G2: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ G2 @ F ) ) )
= ( ? [K2: real] :
( one_one_real
= ( times_times_real @ K2 @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) ) ) ) ) ).
% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_300_fps__is__right__unit__iff__zeroth__is__right__unit,axiom,
! [F: formal_Power_fps_int] :
( ( ? [G2: formal_Power_fps_int] :
( one_on8395608022581818233ps_int
= ( times_3091854549176928185ps_int @ G2 @ F ) ) )
= ( ? [K2: int] :
( one_one_int
= ( times_times_int @ K2 @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) ) ) ) ) ).
% fps_is_right_unit_iff_zeroth_is_right_unit
thf(fact_301_unit__mult__right__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B2 @ A )
= ( times_times_nat @ C @ A ) )
= ( B2 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_302_unit__mult__right__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B2 @ A )
= ( times_times_int @ C @ A ) )
= ( B2 = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_303_unit__mult__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B2 )
= ( times_times_nat @ A @ C ) )
= ( B2 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_304_unit__mult__left__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B2 )
= ( times_times_int @ A @ C ) )
= ( B2 = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_305_mult__unit__dvd__iff_H,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_306_mult__unit__dvd__iff_H,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
= ( dvd_dvd_int @ B2 @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_307_dvd__mult__unit__iff_H,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_308_dvd__mult__unit__iff_H,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_309_mult__unit__dvd__iff,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_310_mult__unit__dvd__iff,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_311_dvd__mult__unit__iff,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_312_dvd__mult__unit__iff,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_313_is__unit__mult__iff,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_314_is__unit__mult__iff,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_315_nonzero__inverse__mult__distrib,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
= ( times_times_real @ ( inverse_inverse_real @ B2 ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_316_inverse__unique,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
= one_one_real )
=> ( ( inverse_inverse_real @ A )
= B2 ) ) ).
% inverse_unique
thf(fact_317_fps__compose__assoc,axiom,
! [C: formal3361831859752904756s_real,B2: formal3361831859752904756s_real,A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ C @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ B2 @ zero_zero_nat )
= zero_zero_real )
=> ( ( formal8268054683415598839e_real @ A @ ( formal8268054683415598839e_real @ B2 @ C ) )
= ( formal8268054683415598839e_real @ ( formal8268054683415598839e_real @ A @ B2 ) @ C ) ) ) ) ).
% fps_compose_assoc
thf(fact_318_fps__compose__assoc,axiom,
! [C: formal_Power_fps_int,B2: formal_Power_fps_int,A: formal_Power_fps_int] :
( ( ( formal3717847055265219294th_int @ C @ zero_zero_nat )
= zero_zero_int )
=> ( ( ( formal3717847055265219294th_int @ B2 @ zero_zero_nat )
= zero_zero_int )
=> ( ( formal7318879853629353975se_int @ A @ ( formal7318879853629353975se_int @ B2 @ C ) )
= ( formal7318879853629353975se_int @ ( formal7318879853629353975se_int @ A @ B2 ) @ C ) ) ) ) ).
% fps_compose_assoc
thf(fact_319_unit__dvdE,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C2: nat] :
( B2
!= ( times_times_nat @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_320_unit__dvdE,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C2: int] :
( B2
!= ( times_times_int @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_321_field__class_Ofield__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% field_class.field_inverse
thf(fact_322_fps__compose__inj__right,axiom,
! [A: formal3361831859752904756s_real,B2: formal3361831859752904756s_real,C: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
!= zero_zero_real )
=> ( ( ( formal8268054683415598839e_real @ B2 @ A )
= ( formal8268054683415598839e_real @ C @ A ) )
= ( B2 = C ) ) ) ) ).
% fps_compose_inj_right
thf(fact_323_fps__compose__inj__right,axiom,
! [A: formal_Power_fps_int,B2: formal_Power_fps_int,C: formal_Power_fps_int] :
( ( ( formal3717847055265219294th_int @ A @ zero_zero_nat )
= zero_zero_int )
=> ( ( ( formal3717847055265219294th_int @ A @ one_one_nat )
!= zero_zero_int )
=> ( ( ( formal7318879853629353975se_int @ B2 @ A )
= ( formal7318879853629353975se_int @ C @ A ) )
= ( B2 = C ) ) ) ) ).
% fps_compose_inj_right
thf(fact_324_fps__inverse__compose,axiom,
! [B2: formal3361831859752904756s_real,A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ B2 @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
!= zero_zero_real )
=> ( ( formal8268054683415598839e_real @ ( invers68952373231134600s_real @ A ) @ B2 )
= ( invers68952373231134600s_real @ ( formal8268054683415598839e_real @ A @ B2 ) ) ) ) ) ).
% fps_inverse_compose
thf(fact_325_fps__unit__dvd__left,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ? [G3: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ F @ G3 ) ) ) ).
% fps_unit_dvd_left
thf(fact_326_fps__unit__dvd__right,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ? [G3: formal3361831859752904756s_real] :
( one_on8598947968683843321s_real
= ( times_7561426564079326009s_real @ G3 @ F ) ) ) ).
% fps_unit_dvd_right
thf(fact_327_inverse__mult__eq__1_H,axiom,
! [F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ F @ zero_zero_nat )
!= zero_zero_real )
=> ( ( times_7561426564079326009s_real @ F @ ( invers68952373231134600s_real @ F ) )
= one_on8598947968683843321s_real ) ) ).
% inverse_mult_eq_1'
thf(fact_328_vector__space__over__itself_Oscale__one,axiom,
! [X: real] :
( ( times_times_real @ one_one_real @ X )
= X ) ).
% vector_space_over_itself.scale_one
thf(fact_329_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X: real,B2: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ B2 @ X ) )
= ( ( A = B2 )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_330_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_331_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_332_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_333_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( times_times_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_334_fps__ginv__ginv,axiom,
! [A: formal3361831859752904756s_real,C: formal3361831859752904756s_real,B2: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
!= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ C @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ C @ one_one_nat )
!= zero_zero_real )
=> ( ( formal1301361369515107775v_real @ B2 @ ( formal1301361369515107775v_real @ C @ A ) )
= ( formal8268054683415598839e_real @ ( formal8268054683415598839e_real @ B2 @ A ) @ ( formal2886580842492807190v_real @ C ) ) ) ) ) ) ) ).
% fps_ginv_ginv
thf(fact_335_fps__ginv,axiom,
! [A: formal3361831859752904756s_real,B2: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
!= zero_zero_real )
=> ( ( formal8268054683415598839e_real @ ( formal1301361369515107775v_real @ B2 @ A ) @ A )
= B2 ) ) ) ).
% fps_ginv
thf(fact_336_mult__if__delta,axiom,
! [P2: $o,Q: nat] :
( ( P2
=> ( ( times_times_nat @ ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) @ Q )
= Q ) )
& ( ~ P2
=> ( ( times_times_nat @ ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) @ Q )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_337_mult__if__delta,axiom,
! [P2: $o,Q: real] :
( ( P2
=> ( ( times_times_real @ ( if_real @ P2 @ one_one_real @ zero_zero_real ) @ Q )
= Q ) )
& ( ~ P2
=> ( ( times_times_real @ ( if_real @ P2 @ one_one_real @ zero_zero_real ) @ Q )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_338_mult__if__delta,axiom,
! [P2: $o,Q: int] :
( ( P2
=> ( ( times_times_int @ ( if_int @ P2 @ one_one_int @ zero_zero_int ) @ Q )
= Q ) )
& ( ~ P2
=> ( ( times_times_int @ ( if_int @ P2 @ one_one_int @ zero_zero_int ) @ Q )
= zero_zero_int ) ) ) ).
% mult_if_delta
thf(fact_339_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_340_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_341_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_342_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_343_scaleR__scaleR,axiom,
! [A: real,B2: real,X: real] :
( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B2 @ X ) )
= ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B2 ) @ X ) ) ).
% scaleR_scaleR
thf(fact_344_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_345_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_346_ln__powr,axiom,
! [X: real,Y: real] :
( ( X != zero_zero_real )
=> ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
= ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).
% ln_powr
thf(fact_347_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_348_pth__5,axiom,
! [C: real,D: real,X: real] :
( ( real_V1485227260804924795R_real @ C @ ( real_V1485227260804924795R_real @ D @ X ) )
= ( real_V1485227260804924795R_real @ ( times_times_real @ C @ D ) @ X ) ) ).
% pth_5
thf(fact_349_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_350_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_351_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_352_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B2: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B2 @ X ) )
= ( times_times_real @ B2 @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_353_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B2: real,X: real] :
( ( times_times_real @ A @ ( times_times_real @ B2 @ X ) )
= ( times_times_real @ ( times_times_real @ A @ B2 ) @ X ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_354_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A: real,B2: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ B2 @ X ) )
=> ( A = B2 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_355_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_356_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_357_set__times__intro,axiom,
! [A: nat,C3: set_nat,B2: nat,D2: set_nat] :
( ( member_nat @ A @ C3 )
=> ( ( member_nat @ B2 @ D2 )
=> ( member_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_set_nat @ C3 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_358_set__times__intro,axiom,
! [A: real,C3: set_real,B2: real,D2: set_real] :
( ( member_real @ A @ C3 )
=> ( ( member_real @ B2 @ D2 )
=> ( member_real @ ( times_times_real @ A @ B2 ) @ ( times_times_set_real @ C3 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_359_set__times__intro,axiom,
! [A: int,C3: set_int,B2: int,D2: set_int] :
( ( member_int @ A @ C3 )
=> ( ( member_int @ B2 @ D2 )
=> ( member_int @ ( times_times_int @ A @ B2 ) @ ( times_times_set_int @ C3 @ D2 ) ) ) ) ).
% set_times_intro
thf(fact_360_fps__X__mult__right__nth,axiom,
! [N: nat,A: formal3361831859752904756s_real] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ A @ formal4708490801539276157X_real ) @ N )
= zero_zero_real ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ A @ formal4708490801539276157X_real ) @ N )
= ( formal2580924720334399070h_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fps_X_mult_right_nth
thf(fact_361_fps__X__mult__right__nth,axiom,
! [N: nat,A: formal_Power_fps_nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ A @ formal1744162128437646113_X_nat ) @ N )
= zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ A @ formal1744162128437646113_X_nat ) @ N )
= ( formal3720337525774269570th_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fps_X_mult_right_nth
thf(fact_362_fps__X__mult__right__nth,axiom,
! [N: nat,A: formal_Power_fps_int] :
( ( ( N = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ A @ formal1741671657928595837_X_int ) @ N )
= zero_zero_int ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ A @ formal1741671657928595837_X_int ) @ N )
= ( formal3717847055265219294th_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fps_X_mult_right_nth
thf(fact_363_fps__X__mult__nth,axiom,
! [N: nat,F: formal3361831859752904756s_real] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ formal4708490801539276157X_real @ F ) @ N )
= zero_zero_real ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ formal4708490801539276157X_real @ F ) @ N )
= ( formal2580924720334399070h_real @ F @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fps_X_mult_nth
thf(fact_364_fps__X__mult__nth,axiom,
! [N: nat,F: formal_Power_fps_nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ formal1744162128437646113_X_nat @ F ) @ N )
= zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ formal1744162128437646113_X_nat @ F ) @ N )
= ( formal3720337525774269570th_nat @ F @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fps_X_mult_nth
thf(fact_365_fps__X__mult__nth,axiom,
! [N: nat,F: formal_Power_fps_int] :
( ( ( N = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ formal1741671657928595837_X_int @ F ) @ N )
= zero_zero_int ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ formal1741671657928595837_X_int @ F ) @ N )
= ( formal3717847055265219294th_int @ F @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fps_X_mult_nth
thf(fact_366_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_367_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_368_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_369_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_370_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_371_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_372_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_373_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_374_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_375_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_376_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_377_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_378_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_379_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_380_minus__mod__self2,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% minus_mod_self2
thf(fact_381_fps__sub__nth,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( minus_1563896255634514737ps_nat @ F @ G ) @ N )
= ( minus_minus_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ ( formal3720337525774269570th_nat @ G @ N ) ) ) ).
% fps_sub_nth
thf(fact_382_fps__sub__nth,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( minus_6791916864952032525s_real @ F @ G ) @ N )
= ( minus_minus_real @ ( formal2580924720334399070h_real @ F @ N ) @ ( formal2580924720334399070h_real @ G @ N ) ) ) ).
% fps_sub_nth
thf(fact_383_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_384_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_385_bezout1__nat,axiom,
! [A: nat,B2: nat] :
? [D3: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B2 )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X2 ) @ ( times_times_nat @ B2 @ Y2 ) )
= D3 )
| ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X2 ) @ ( times_times_nat @ A @ Y2 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_386_powr__powr,axiom,
! [X: real,A: real,B2: real] :
( ( powr_real @ ( powr_real @ X @ A ) @ B2 )
= ( powr_real @ X @ ( times_times_real @ A @ B2 ) ) ) ).
% powr_powr
thf(fact_387_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_388_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_389_real__scaleR__def,axiom,
real_V1485227260804924795R_real = times_times_real ).
% real_scaleR_def
thf(fact_390_scaleR__left__diff__distrib,axiom,
! [A: real,B2: real,X: real] :
( ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B2 ) @ X )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B2 @ X ) ) ) ).
% scaleR_left_diff_distrib
thf(fact_391_scaleR__left_Odiff,axiom,
! [X: real,Y: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X @ Y ) @ Xa )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% scaleR_left.diff
thf(fact_392_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_393_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
= ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_394_diff__eq__diff__eq,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_395_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_396_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A2: int,B: int] :
( ( minus_minus_int @ A2 @ B )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_397_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A2: real,B: real] :
( ( minus_minus_real @ A2 @ B )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_398_right__diff__distrib_H,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B2 @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_399_right__diff__distrib_H,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_400_right__diff__distrib_H,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_401_left__diff__distrib_H,axiom,
! [B2: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_402_left__diff__distrib_H,axiom,
! [B2: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B2 @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_403_left__diff__distrib_H,axiom,
! [B2: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_404_right__diff__distrib,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_405_right__diff__distrib,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_406_left__diff__distrib,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% left_diff_distrib
thf(fact_407_left__diff__distrib,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% left_diff_distrib
thf(fact_408_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( times_times_real @ A @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_409_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A: real,B2: real,X: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ X )
= ( minus_minus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ B2 @ X ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_410_dvd__diff,axiom,
! [X: real,Y: real,Z: real] :
( ( dvd_dvd_real @ X @ Y )
=> ( ( dvd_dvd_real @ X @ Z )
=> ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_411_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_412_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_413_scaleR__right__diff__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( real_V1485227260804924795R_real @ A @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).
% scaleR_right_diff_distrib
thf(fact_414_mod__diff__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% mod_diff_eq
thf(fact_415_mod__diff__cong,axiom,
! [A: int,C: int,A4: int,B2: int,B4: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A4 @ C ) )
=> ( ( ( modulo_modulo_int @ B2 @ C )
= ( modulo_modulo_int @ B4 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_416_mod__diff__left__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_417_mod__diff__right__eq,axiom,
! [A: int,B2: int,C: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_418_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_419_infnorm__sub,axiom,
! [X: real,Y: real] :
( ( linear_infnorm_real @ ( minus_minus_real @ X @ Y ) )
= ( linear_infnorm_real @ ( minus_minus_real @ Y @ X ) ) ) ).
% infnorm_sub
thf(fact_420_dvd__minus__mod,axiom,
! [B2: nat,A: nat] : ( dvd_dvd_nat @ B2 @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% dvd_minus_mod
thf(fact_421_dvd__minus__mod,axiom,
! [B2: int,A: int] : ( dvd_dvd_int @ B2 @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% dvd_minus_mod
thf(fact_422_mod__eq__dvd__iff,axiom,
! [A: int,C: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B2 @ C ) )
= ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B2 ) ) ) ).
% mod_eq_dvd_iff
thf(fact_423_division__ring__inverse__diff,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B2 @ A ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_424_set__times__elim,axiom,
! [X: nat,A5: set_nat,B5: set_nat] :
( ( member_nat @ X @ ( times_times_set_nat @ A5 @ B5 ) )
=> ~ ! [A3: nat,B3: nat] :
( ( X
= ( times_times_nat @ A3 @ B3 ) )
=> ( ( member_nat @ A3 @ A5 )
=> ~ ( member_nat @ B3 @ B5 ) ) ) ) ).
% set_times_elim
thf(fact_425_set__times__elim,axiom,
! [X: real,A5: set_real,B5: set_real] :
( ( member_real @ X @ ( times_times_set_real @ A5 @ B5 ) )
=> ~ ! [A3: real,B3: real] :
( ( X
= ( times_times_real @ A3 @ B3 ) )
=> ( ( member_real @ A3 @ A5 )
=> ~ ( member_real @ B3 @ B5 ) ) ) ) ).
% set_times_elim
thf(fact_426_set__times__elim,axiom,
! [X: int,A5: set_int,B5: set_int] :
( ( member_int @ X @ ( times_times_set_int @ A5 @ B5 ) )
=> ~ ! [A3: int,B3: int] :
( ( X
= ( times_times_int @ A3 @ B3 ) )
=> ( ( member_int @ A3 @ A5 )
=> ~ ( member_int @ B3 @ B5 ) ) ) ) ).
% set_times_elim
thf(fact_427_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_428_gcd__nat_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ~ ( ( dvd_dvd_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% gcd_nat.asym
thf(fact_429_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_430_gcd__nat_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.trans
thf(fact_431_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A2: nat,B: nat] :
( ( dvd_dvd_nat @ A2 @ B )
& ( dvd_dvd_nat @ B @ A2 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_432_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_433_gcd__nat_Oantisym,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% gcd_nat.antisym
thf(fact_434_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ( ( ( dvd_dvd_nat @ B2 @ C )
& ( B2 != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_435_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( ( dvd_dvd_nat @ B2 @ C )
& ( B2 != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_436_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ( ( dvd_dvd_nat @ B2 @ C )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_437_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
= ( ( dvd_dvd_nat @ A @ B2 )
& ~ ( dvd_dvd_nat @ B2 @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_438_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A2: nat,B: nat] :
( ( ( dvd_dvd_nat @ A2 @ B )
& ( A2 != B ) )
| ( A2 = B ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_439_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
= ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_440_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ( dvd_dvd_nat @ A @ B2 ) ) ).
% gcd_nat.strict_implies_order
thf(fact_441_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) )
=> ( A != B2 ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_442_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ A @ B2 )
& ( A != B2 ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_443_dvd__productE,axiom,
! [P: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ P @ ( times_times_nat @ A @ B2 ) )
=> ~ ! [X2: nat,Y2: nat] :
( ( P
= ( times_times_nat @ X2 @ Y2 ) )
=> ( ( dvd_dvd_nat @ X2 @ A )
=> ~ ( dvd_dvd_nat @ Y2 @ B2 ) ) ) ) ).
% dvd_productE
thf(fact_444_dvd__productE,axiom,
! [P: int,A: int,B2: int] :
( ( dvd_dvd_int @ P @ ( times_times_int @ A @ B2 ) )
=> ~ ! [X2: int,Y2: int] :
( ( P
= ( times_times_int @ X2 @ Y2 ) )
=> ( ( dvd_dvd_int @ X2 @ A )
=> ~ ( dvd_dvd_int @ Y2 @ B2 ) ) ) ) ).
% dvd_productE
thf(fact_445_division__decomp,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
=> ? [B6: nat,C4: nat] :
( ( A
= ( times_times_nat @ B6 @ C4 ) )
& ( dvd_dvd_nat @ B6 @ B2 )
& ( dvd_dvd_nat @ C4 @ C ) ) ) ).
% division_decomp
thf(fact_446_division__decomp,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
=> ? [B6: int,C4: int] :
( ( A
= ( times_times_int @ B6 @ C4 ) )
& ( dvd_dvd_int @ B6 @ B2 )
& ( dvd_dvd_int @ C4 @ C ) ) ) ).
% division_decomp
thf(fact_447_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_448_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_449_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_450_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_451_fps__inv__deriv,axiom,
! [A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
!= zero_zero_real )
=> ( ( formal4557910837323084707v_real @ ( formal2886580842492807190v_real @ A ) )
= ( invers68952373231134600s_real @ ( formal8268054683415598839e_real @ ( formal4557910837323084707v_real @ A ) @ ( formal2886580842492807190v_real @ A ) ) ) ) ) ) ).
% fps_inv_deriv
thf(fact_452_fps__compose__divide__distrib,axiom,
! [G: formal3361831859752904756s_real,F: formal3361831859752904756s_real,H: formal3361831859752904756s_real] :
( ( dvd_dv1093944294739598810s_real @ G @ F )
=> ( ( ( formal2580924720334399070h_real @ H @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal8268054683415598839e_real @ G @ H )
!= zero_z7760665558314615101s_real )
=> ( ( formal8268054683415598839e_real @ ( divide1155267253282662278s_real @ F @ G ) @ H )
= ( divide1155267253282662278s_real @ ( formal8268054683415598839e_real @ F @ H ) @ ( formal8268054683415598839e_real @ G @ H ) ) ) ) ) ) ).
% fps_compose_divide_distrib
thf(fact_453_fps__compose__divide,axiom,
! [G: formal3361831859752904756s_real,F: formal3361831859752904756s_real,H: formal3361831859752904756s_real] :
( ( dvd_dv1093944294739598810s_real @ G @ F )
=> ( ( ( formal2580924720334399070h_real @ H @ zero_zero_nat )
= zero_zero_real )
=> ( ( formal8268054683415598839e_real @ F @ H )
= ( times_7561426564079326009s_real @ ( formal8268054683415598839e_real @ ( divide1155267253282662278s_real @ F @ G ) @ H ) @ ( formal8268054683415598839e_real @ G @ H ) ) ) ) ) ).
% fps_compose_divide
thf(fact_454_inf__period_I2_J,axiom,
! [P2: real > $o,D2: real,Q2: real > $o] :
( ! [X2: real,K3: real] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D2 ) ) ) )
=> ( ! [X2: real,K3: real] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D2 ) ) ) )
=> ! [X3: real,K4: real] :
( ( ( P2 @ X3 )
| ( Q2 @ X3 ) )
= ( ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) )
| ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_455_inf__period_I2_J,axiom,
! [P2: int > $o,D2: int,Q2: int > $o] :
( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ! [X3: int,K4: int] :
( ( ( P2 @ X3 )
| ( Q2 @ X3 ) )
= ( ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) )
| ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_456_inf__period_I1_J,axiom,
! [P2: real > $o,D2: real,Q2: real > $o] :
( ! [X2: real,K3: real] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D2 ) ) ) )
=> ( ! [X2: real,K3: real] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus_real @ X2 @ ( times_times_real @ K3 @ D2 ) ) ) )
=> ! [X3: real,K4: real] :
( ( ( P2 @ X3 )
& ( Q2 @ X3 ) )
= ( ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) )
& ( Q2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_457_inf__period_I1_J,axiom,
! [P2: int > $o,D2: int,Q2: int > $o] :
( ! [X2: int,K3: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ( ! [X2: int,K3: int] :
( ( Q2 @ X2 )
= ( Q2 @ ( minus_minus_int @ X2 @ ( times_times_int @ K3 @ D2 ) ) ) )
=> ! [X3: int,K4: int] :
( ( ( P2 @ X3 )
& ( Q2 @ X3 ) )
= ( ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) )
& ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_458_fps__XDp__fps__integral,axiom,
! [A: formal3361831859752904756s_real,C: real] :
( ( formal2839450981996073129p_real @ zero_zero_real @ ( formal8984515926053063617l_real @ A @ C ) )
= ( times_7561426564079326009s_real @ formal4708490801539276157X_real @ A ) ) ).
% fps_XDp_fps_integral
thf(fact_459_divide__eq__0__iff,axiom,
! [A: real,B2: real] :
( ( ( divide_divide_real @ A @ B2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_460_divide__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% divide_cancel_left
thf(fact_461_divide__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% divide_cancel_right
thf(fact_462_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_463_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_464_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_465_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_466_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_467_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_468_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_469_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_470_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_471_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_472_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_473_times__divide__eq__left,axiom,
! [B2: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B2 @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B2 @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_474_divide__divide__eq__left,axiom,
! [A: real,B2: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_475_divide__divide__eq__right,axiom,
! [A: real,B2: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B2 @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B2 ) ) ).
% divide_divide_eq_right
thf(fact_476_times__divide__eq__right,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B2 @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ C ) ) ).
% times_divide_eq_right
thf(fact_477_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_478_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_479_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_480_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_481_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_482_div__dvd__div,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A ) @ ( divide_divide_nat @ C @ A ) )
= ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_483_div__dvd__div,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A ) @ ( divide_divide_int @ C @ A ) )
= ( dvd_dvd_int @ B2 @ C ) ) ) ) ).
% div_dvd_div
thf(fact_484_inverse__divide,axiom,
! [A: real,B2: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A @ B2 ) )
= ( divide_divide_real @ B2 @ A ) ) ).
% inverse_divide
thf(fact_485_nonzero__mult__div__cancel__right,axiom,
! [B2: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_486_nonzero__mult__div__cancel__right,axiom,
! [B2: real,A: real] :
( ( B2 != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ B2 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_487_nonzero__mult__div__cancel__right,axiom,
! [B2: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ B2 )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_488_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ A )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_489_nonzero__mult__div__cancel__left,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ A )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_490_nonzero__mult__div__cancel__left,axiom,
! [A: int,B2: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ A )
= B2 ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_491_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
= ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_492_div__mult__mult1__if,axiom,
! [C: int,A: int,B2: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_mult1_if
thf(fact_493_div__mult__mult2,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ A @ B2 ) ) ) ).
% div_mult_mult2
thf(fact_494_div__mult__mult2,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ A @ B2 ) ) ) ).
% div_mult_mult2
thf(fact_495_div__mult__mult1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
= ( divide_divide_nat @ A @ B2 ) ) ) ).
% div_mult_mult1
thf(fact_496_div__mult__mult1,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( divide_divide_int @ A @ B2 ) ) ) ).
% div_mult_mult1
thf(fact_497_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B2 ) )
= ( divide_divide_real @ A @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_498_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
= ( divide_divide_real @ A @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_499_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B2 @ C ) )
= ( divide_divide_real @ A @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_500_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( divide_divide_real @ A @ B2 ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_501_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B2: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( divide_divide_real @ A @ B2 ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_502_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_503_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_504_eq__divide__eq__1,axiom,
! [B2: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B2 @ A ) )
= ( ( A != zero_zero_real )
& ( A = B2 ) ) ) ).
% eq_divide_eq_1
thf(fact_505_divide__eq__eq__1,axiom,
! [B2: real,A: real] :
( ( ( divide_divide_real @ B2 @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B2 ) ) ) ).
% divide_eq_eq_1
thf(fact_506_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_507_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_508_one__eq__divide__iff,axiom,
! [A: real,B2: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B2 ) )
= ( ( B2 != zero_zero_real )
& ( A = B2 ) ) ) ).
% one_eq_divide_iff
thf(fact_509_divide__eq__1__iff,axiom,
! [A: real,B2: real] :
( ( ( divide_divide_real @ A @ B2 )
= one_one_real )
= ( ( B2 != zero_zero_real )
& ( A = B2 ) ) ) ).
% divide_eq_1_iff
thf(fact_510_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_511_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_512_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_513_dvd__mult__div__cancel,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ A ) )
= B2 ) ) ).
% dvd_mult_div_cancel
thf(fact_514_dvd__mult__div__cancel,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ A ) )
= B2 ) ) ).
% dvd_mult_div_cancel
thf(fact_515_dvd__div__mult__self,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
= B2 ) ) ).
% dvd_div_mult_self
thf(fact_516_dvd__div__mult__self,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ B2 )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
= B2 ) ) ).
% dvd_div_mult_self
thf(fact_517_unit__div,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_518_unit__div,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_519_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_520_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_521_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_522_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_523_div__diff,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( minus_minus_int @ A @ B2 ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% div_diff
thf(fact_524_bits__mod__div__trivial,axiom,
! [A: nat,B2: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_525_bits__mod__div__trivial,axiom,
! [A: int,B2: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_526_mod__div__trivial,axiom,
! [A: nat,B2: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_527_mod__div__trivial,axiom,
! [A: int,B2: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_528_nonzero__divide__mult__cancel__right,axiom,
! [B2: real,A: real] :
( ( B2 != zero_zero_real )
=> ( ( divide_divide_real @ B2 @ ( times_times_real @ A @ B2 ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_529_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B2 ) )
= ( divide_divide_real @ one_one_real @ B2 ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_530_unit__div__mult__self,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
= B2 ) ) ).
% unit_div_mult_self
thf(fact_531_unit__div__mult__self,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
= B2 ) ) ).
% unit_div_mult_self
thf(fact_532_unit__mult__div__div,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A ) )
= ( divide_divide_nat @ B2 @ A ) ) ) ).
% unit_mult_div_div
thf(fact_533_unit__mult__div__div,axiom,
! [A: int,B2: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A ) )
= ( divide_divide_int @ B2 @ A ) ) ) ).
% unit_mult_div_div
thf(fact_534_fps__divide__nth__0,axiom,
! [G: formal3361831859752904756s_real,F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ G @ zero_zero_nat )
!= zero_zero_real )
=> ( ( formal2580924720334399070h_real @ ( divide1155267253282662278s_real @ F @ G ) @ zero_zero_nat )
= ( divide_divide_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ) ).
% fps_divide_nth_0
thf(fact_535_fps__integral0__by__parts,axiom,
! [A: formal3361831859752904756s_real,B2: formal3361831859752904756s_real] :
( ( formal8984515926053063617l_real @ ( times_7561426564079326009s_real @ A @ B2 ) @ zero_zero_real )
= ( minus_6791916864952032525s_real @ ( times_7561426564079326009s_real @ A @ ( formal8984515926053063617l_real @ B2 @ zero_zero_real ) ) @ ( formal8984515926053063617l_real @ ( times_7561426564079326009s_real @ ( formal4557910837323084707v_real @ A ) @ ( formal8984515926053063617l_real @ B2 @ zero_zero_real ) ) @ zero_zero_real ) ) ) ).
% fps_integral0_by_parts
thf(fact_536_fps__integral0__sub,axiom,
! [A: formal3361831859752904756s_real,B2: formal3361831859752904756s_real] :
( ( formal8984515926053063617l_real @ ( minus_6791916864952032525s_real @ A @ B2 ) @ zero_zero_real )
= ( minus_6791916864952032525s_real @ ( formal8984515926053063617l_real @ A @ zero_zero_real ) @ ( formal8984515926053063617l_real @ B2 @ zero_zero_real ) ) ) ).
% fps_integral0_sub
thf(fact_537_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_538_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_539_divide__divide__eq__left_H,axiom,
! [A: real,B2: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B2 ) ) ) ).
% divide_divide_eq_left'
thf(fact_540_diff__divide__distrib,axiom,
! [A: real,B2: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% diff_divide_distrib
thf(fact_541_dvd__div__eq__iff,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B2 @ C ) )
= ( A = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_542_dvd__div__eq__iff,axiom,
! [C: real,A: real,B2: real] :
( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B2 )
=> ( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B2 @ C ) )
= ( A = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_543_dvd__div__eq__iff,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B2 @ C ) )
= ( A = B2 ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_544_dvd__div__eq__cancel,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B2 @ C ) )
=> ( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( A = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_545_dvd__div__eq__cancel,axiom,
! [A: real,C: real,B2: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B2 @ C ) )
=> ( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B2 )
=> ( A = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_546_dvd__div__eq__cancel,axiom,
! [A: int,C: int,B2: int] :
( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B2 @ C ) )
=> ( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( A = B2 ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_547_div__div__div__same,axiom,
! [D: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ D @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B2 @ D ) )
= ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_div_div_same
thf(fact_548_div__div__div__same,axiom,
! [D: int,B2: int,A: int] :
( ( dvd_dvd_int @ D @ B2 )
=> ( ( dvd_dvd_int @ B2 @ A )
=> ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B2 @ D ) )
= ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_div_div_same
thf(fact_549_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B2 @ C ) )
= ( ( times_times_real @ A @ C )
= B2 ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_550_nonzero__divide__eq__eq,axiom,
! [C: real,B2: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B2 @ C )
= A )
= ( B2
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_551_eq__divide__imp,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B2 )
=> ( A
= ( divide_divide_real @ B2 @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_552_divide__eq__imp,axiom,
! [C: real,B2: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B2
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B2 @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_553_eq__divide__eq,axiom,
! [A: real,B2: real,C: real] :
( ( A
= ( divide_divide_real @ B2 @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B2 ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_554_divide__eq__eq,axiom,
! [B2: real,C: real,A: real] :
( ( ( divide_divide_real @ B2 @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B2
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_555_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_556_right__inverse__eq,axiom,
! [B2: real,A: real] :
( ( B2 != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B2 )
= one_one_real )
= ( A = B2 ) ) ) ).
% right_inverse_eq
thf(fact_557_dvd__div__eq__0__iff,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ A )
=> ( ( ( divide_divide_nat @ A @ B2 )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_558_dvd__div__eq__0__iff,axiom,
! [B2: real,A: real] :
( ( dvd_dvd_real @ B2 @ A )
=> ( ( ( divide_divide_real @ A @ B2 )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_559_dvd__div__eq__0__iff,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ A )
=> ( ( ( divide_divide_int @ A @ B2 )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_560_div__mult__div__if__dvd,axiom,
! [B2: nat,A: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ A )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_561_div__mult__div__if__dvd,axiom,
! [B2: int,A: int,D: int,C: int] :
( ( dvd_dvd_int @ B2 @ A )
=> ( ( dvd_dvd_int @ D @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ C @ D ) )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_562_dvd__mult__imp__div,axiom,
! [A: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 )
=> ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_563_dvd__mult__imp__div,axiom,
! [A: int,C: int,B2: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 )
=> ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_564_dvd__div__mult2__eq,axiom,
! [B2: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_565_dvd__div__mult2__eq,axiom,
! [B2: int,C: int,A: int] :
( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_566_div__div__eq__right,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_567_div__div__eq__right,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ B2 @ A )
=> ( ( divide_divide_int @ A @ ( divide_divide_int @ B2 @ C ) )
= ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_568_div__mult__swap,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_569_div__mult__swap,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% div_mult_swap
thf(fact_570_dvd__div__mult,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A )
= ( divide_divide_nat @ ( times_times_nat @ B2 @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_571_dvd__div__mult,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A )
= ( divide_divide_int @ ( times_times_int @ B2 @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_572_unit__div__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B2 @ A )
= ( divide_divide_nat @ C @ A ) )
= ( B2 = C ) ) ) ).
% unit_div_cancel
thf(fact_573_unit__div__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B2 @ A )
= ( divide_divide_int @ C @ A ) )
= ( B2 = C ) ) ) ).
% unit_div_cancel
thf(fact_574_div__unit__dvd__iff,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_575_div__unit__dvd__iff,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_576_dvd__div__unit__iff,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B2 ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_577_dvd__div__unit__iff,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B2 ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_578_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B2: nat] :
( ( ( modulo_modulo_nat @ A @ B2 )
= A )
= ( ( divide_divide_nat @ A @ B2 )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_579_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B2: int] :
( ( ( modulo_modulo_int @ A @ B2 )
= A )
= ( ( divide_divide_int @ A @ B2 )
= zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_580_divide__inverse__commute,axiom,
( divide_divide_real
= ( ^ [A2: real,B: real] : ( times_times_real @ ( inverse_inverse_real @ B ) @ A2 ) ) ) ).
% divide_inverse_commute
thf(fact_581_divide__inverse,axiom,
( divide_divide_real
= ( ^ [A2: real,B: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B ) ) ) ) ).
% divide_inverse
thf(fact_582_field__class_Ofield__divide__inverse,axiom,
( divide_divide_real
= ( ^ [A2: real,B: real] : ( times_times_real @ A2 @ ( inverse_inverse_real @ B ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_583_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_584_powr__diff,axiom,
! [W: real,Z1: real,Z22: real] :
( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
= ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).
% powr_diff
thf(fact_585_fps__integral0__zero,axiom,
( ( formal8984515926053063617l_real @ zero_z7760665558314615101s_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_integral0_zero
thf(fact_586_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_587_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_588_diff__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_589_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B2: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_590_dvd__div__div__eq__mult,axiom,
! [A: nat,C: nat,B2: nat,D: nat] :
( ( A != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( ( ( divide_divide_nat @ B2 @ A )
= ( divide_divide_nat @ D @ C ) )
= ( ( times_times_nat @ B2 @ C )
= ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_591_dvd__div__div__eq__mult,axiom,
! [A: int,C: int,B2: int,D: int] :
( ( A != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B2 )
=> ( ( dvd_dvd_int @ C @ D )
=> ( ( ( divide_divide_int @ B2 @ A )
= ( divide_divide_int @ D @ C ) )
= ( ( times_times_int @ B2 @ C )
= ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_592_dvd__div__iff__mult,axiom,
! [C: nat,B2: nat,A: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_593_dvd__div__iff__mult,axiom,
! [C: int,B2: int,A: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_594_div__dvd__iff__mult,axiom,
! [B2: nat,A: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
= ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_595_div__dvd__iff__mult,axiom,
! [B2: int,A: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ A )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
= ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_596_dvd__div__eq__mult,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B2 )
=> ( ( ( divide_divide_nat @ B2 @ A )
= C )
= ( B2
= ( times_times_nat @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_597_dvd__div__eq__mult,axiom,
! [A: int,B2: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B2 )
=> ( ( ( divide_divide_int @ B2 @ A )
= C )
= ( B2
= ( times_times_int @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_598_unit__div__eq__0__iff,axiom,
! [B2: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B2 )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_599_unit__div__eq__0__iff,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B2 )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_600_is__unit__div__mult2__eq,axiom,
! [B2: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_601_is__unit__div__mult2__eq,axiom,
! [B2: int,C: int,A: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_602_unit__div__mult__swap,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_603_unit__div__mult__swap,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_604_unit__div__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ).
% unit_div_commute
thf(fact_605_unit__div__commute,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ).
% unit_div_commute
thf(fact_606_div__mult__unit2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( dvd_dvd_nat @ B2 @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_607_div__mult__unit2,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( dvd_dvd_int @ B2 @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_608_unit__eq__div2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( A
= ( divide_divide_nat @ C @ B2 ) )
= ( ( times_times_nat @ A @ B2 )
= C ) ) ) ).
% unit_eq_div2
thf(fact_609_unit__eq__div2,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( A
= ( divide_divide_int @ C @ B2 ) )
= ( ( times_times_int @ A @ B2 )
= C ) ) ) ).
% unit_eq_div2
thf(fact_610_unit__eq__div1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B2 )
= C )
= ( A
= ( times_times_nat @ C @ B2 ) ) ) ) ).
% unit_eq_div1
thf(fact_611_unit__eq__div1,axiom,
! [B2: int,A: int,C: int] :
( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B2 )
= C )
= ( A
= ( times_times_int @ C @ B2 ) ) ) ) ).
% unit_eq_div1
thf(fact_612_nonzero__inverse__eq__divide,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_613_minus__div__mult__eq__mod,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% minus_div_mult_eq_mod
thf(fact_614_minus__div__mult__eq__mod,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% minus_div_mult_eq_mod
thf(fact_615_minus__mod__eq__div__mult,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) ) ).
% minus_mod_eq_div_mult
thf(fact_616_minus__mod__eq__div__mult,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
= ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) ) ).
% minus_mod_eq_div_mult
thf(fact_617_minus__mod__eq__mult__div,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
= ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_618_minus__mod__eq__mult__div,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
= ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_619_minus__mult__div__eq__mod,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% minus_mult_div_eq_mod
thf(fact_620_minus__mult__div__eq__mod,axiom,
! [A: int,B2: int] :
( ( minus_minus_int @ A @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% minus_mult_div_eq_mod
thf(fact_621_fps__div__by__zero_H,axiom,
! [G: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real )
=> ( ( divide1155267253282662278s_real @ G @ zero_z7760665558314615101s_real )
= zero_z7760665558314615101s_real ) ) ).
% fps_div_by_zero'
thf(fact_622_fps__divide__1_H,axiom,
! [A: formal3361831859752904756s_real] :
( ( ( inverse_inverse_real @ one_one_real )
= one_one_real )
=> ( ( divide1155267253282662278s_real @ A @ one_on8598947968683843321s_real )
= A ) ) ).
% fps_divide_1'
thf(fact_623_fps__integral0__one,axiom,
( ( formal8984515926053063617l_real @ one_on8598947968683843321s_real @ zero_zero_real )
= formal4708490801539276157X_real ) ).
% fps_integral0_one
thf(fact_624_fps__ginv__deriv,axiom,
! [A: formal3361831859752904756s_real,B2: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
!= zero_zero_real )
=> ( ( formal4557910837323084707v_real @ ( formal1301361369515107775v_real @ B2 @ A ) )
= ( formal8268054683415598839e_real @ ( divide1155267253282662278s_real @ ( formal4557910837323084707v_real @ B2 ) @ ( formal4557910837323084707v_real @ A ) ) @ ( formal1301361369515107775v_real @ formal4708490801539276157X_real @ A ) ) ) ) ) ).
% fps_ginv_deriv
thf(fact_625_is__unitE,axiom,
! [A: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [B3: nat] :
( ( B3 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B3 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A )
= B3 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B3 )
= A )
=> ( ( ( times_times_nat @ A @ B3 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A )
!= ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_626_is__unitE,axiom,
! [A: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [B3: int] :
( ( B3 != zero_zero_int )
=> ( ( dvd_dvd_int @ B3 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A )
= B3 )
=> ( ( ( divide_divide_int @ one_one_int @ B3 )
= A )
=> ( ( ( times_times_int @ A @ B3 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A )
!= ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_627_is__unit__div__mult__cancel__left,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B2 ) )
= ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_628_is__unit__div__mult__cancel__left,axiom,
! [A: int,B2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ A @ B2 ) )
= ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_629_is__unit__div__mult__cancel__right,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ A ) )
= ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_630_is__unit__div__mult__cancel__right,axiom,
! [A: int,B2: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ A ) )
= ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_631_fps__divide__compose,axiom,
! [C: formal3361831859752904756s_real,B2: formal3361831859752904756s_real,A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ C @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ B2 @ zero_zero_nat )
!= zero_zero_real )
=> ( ( formal8268054683415598839e_real @ ( divide1155267253282662278s_real @ A @ B2 ) @ C )
= ( divide1155267253282662278s_real @ ( formal8268054683415598839e_real @ A @ C ) @ ( formal8268054683415598839e_real @ B2 @ C ) ) ) ) ) ).
% fps_divide_compose
thf(fact_632_fps__compose__deriv,axiom,
! [B2: formal3361831859752904756s_real,A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ B2 @ zero_zero_nat )
= zero_zero_real )
=> ( ( formal4557910837323084707v_real @ ( formal8268054683415598839e_real @ A @ B2 ) )
= ( times_7561426564079326009s_real @ ( formal8268054683415598839e_real @ ( formal4557910837323084707v_real @ A ) @ B2 ) @ ( formal4557910837323084707v_real @ B2 ) ) ) ) ).
% fps_compose_deriv
thf(fact_633_fps__compose__deriv,axiom,
! [B2: formal_Power_fps_int,A: formal_Power_fps_int] :
( ( ( formal3717847055265219294th_int @ B2 @ zero_zero_nat )
= zero_zero_int )
=> ( ( formal4461971871990784675iv_int @ ( formal7318879853629353975se_int @ A @ B2 ) )
= ( times_3091854549176928185ps_int @ ( formal7318879853629353975se_int @ ( formal4461971871990784675iv_int @ A ) @ B2 ) @ ( formal4461971871990784675iv_int @ B2 ) ) ) ) ).
% fps_compose_deriv
thf(fact_634_fps__divide__unit,axiom,
! [G: formal3361831859752904756s_real,F: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ G @ zero_zero_nat )
!= zero_zero_real )
=> ( ( divide1155267253282662278s_real @ F @ G )
= ( times_7561426564079326009s_real @ F @ ( invers68952373231134600s_real @ G ) ) ) ) ).
% fps_divide_unit
thf(fact_635_fps__integral0__one_H,axiom,
( ( ( inverse_inverse_real @ one_one_real )
= one_one_real )
=> ( ( formal8984515926053063617l_real @ one_on8598947968683843321s_real @ zero_zero_real )
= formal4708490801539276157X_real ) ) ).
% fps_integral0_one'
thf(fact_636_divide__mult__cancel,axiom,
! [B2: real,A: real] :
( ( B2 != zero_zero_real )
=> ( ( times_times_real @ ( divide_divide_real @ A @ B2 ) @ B2 )
= A ) ) ).
% divide_mult_cancel
thf(fact_637_fps__inv__fps__exp__compose_I1_J,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( formal8268054683415598839e_real @ ( formal2886580842492807190v_real @ ( minus_6791916864952032525s_real @ ( formal3452214891061569154p_real @ A ) @ one_on8598947968683843321s_real ) ) @ ( minus_6791916864952032525s_real @ ( formal3452214891061569154p_real @ A ) @ one_on8598947968683843321s_real ) )
= formal4708490801539276157X_real ) ) ).
% fps_inv_fps_exp_compose(1)
thf(fact_638_fps__inv__fps__exp__compose_I2_J,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( formal8268054683415598839e_real @ ( minus_6791916864952032525s_real @ ( formal3452214891061569154p_real @ A ) @ one_on8598947968683843321s_real ) @ ( formal2886580842492807190v_real @ ( minus_6791916864952032525s_real @ ( formal3452214891061569154p_real @ A ) @ one_on8598947968683843321s_real ) ) )
= formal4708490801539276157X_real ) ) ).
% fps_inv_fps_exp_compose(2)
thf(fact_639_fps__ln__fps__exp__inv,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( formal8688746759596762231n_real @ A )
= ( formal2886580842492807190v_real @ ( minus_6791916864952032525s_real @ ( formal3452214891061569154p_real @ A ) @ one_on8598947968683843321s_real ) ) ) ) ).
% fps_ln_fps_exp_inv
thf(fact_640_fps__inverse__deriv__divring,axiom,
! [A: formal3361831859752904756s_real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
!= zero_zero_real )
=> ( ( formal4557910837323084707v_real @ ( invers68952373231134600s_real @ A ) )
= ( times_7561426564079326009s_real @ ( times_7561426564079326009s_real @ ( uminus8389970968385878141s_real @ ( invers68952373231134600s_real @ A ) ) @ ( formal4557910837323084707v_real @ A ) ) @ ( invers68952373231134600s_real @ A ) ) ) ) ).
% fps_inverse_deriv_divring
thf(fact_641_neg__equal__iff__equal,axiom,
! [A: real,B2: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B2 ) )
= ( A = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_642_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_643_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_644_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_645_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_646_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_647_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_648_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_649_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_650_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_651_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_652_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_653_mult__minus__right,axiom,
! [A: int,B2: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% mult_minus_right
thf(fact_654_mult__minus__right,axiom,
! [A: real,B2: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% mult_minus_right
thf(fact_655_minus__mult__minus,axiom,
! [A: int,B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
= ( times_times_int @ A @ B2 ) ) ).
% minus_mult_minus
thf(fact_656_minus__mult__minus,axiom,
! [A: real,B2: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
= ( times_times_real @ A @ B2 ) ) ).
% minus_mult_minus
thf(fact_657_mult__minus__left,axiom,
! [A: int,B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% mult_minus_left
thf(fact_658_mult__minus__left,axiom,
! [A: real,B2: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% mult_minus_left
thf(fact_659_vector__space__over__itself_Oscale__minus__left,axiom,
! [A: real,X: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ X )
= ( uminus_uminus_real @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_minus_left
thf(fact_660_vector__space__over__itself_Oscale__minus__right,axiom,
! [A: real,X: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ X ) ) ) ).
% vector_space_over_itself.scale_minus_right
thf(fact_661_minus__diff__eq,axiom,
! [A: real,B2: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) )
= ( minus_minus_real @ B2 @ A ) ) ).
% minus_diff_eq
thf(fact_662_div__minus__minus,axiom,
! [A: int,B2: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
= ( divide_divide_int @ A @ B2 ) ) ).
% div_minus_minus
thf(fact_663_minus__dvd__iff,axiom,
! [X: real,Y: real] :
( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
= ( dvd_dvd_real @ X @ Y ) ) ).
% minus_dvd_iff
thf(fact_664_dvd__minus__iff,axiom,
! [X: real,Y: real] :
( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
= ( dvd_dvd_real @ X @ Y ) ) ).
% dvd_minus_iff
thf(fact_665_scaleR__left_Ominus,axiom,
! [X: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ X ) @ Xa )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) ) ) ).
% scaleR_left.minus
thf(fact_666_scaleR__minus__left,axiom,
! [A: real,X: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ A ) @ X )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ).
% scaleR_minus_left
thf(fact_667_scaleR__minus__right,axiom,
! [A: real,X: real] :
( ( real_V1485227260804924795R_real @ A @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ).
% scaleR_minus_right
thf(fact_668_inverse__minus__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).
% inverse_minus_eq
thf(fact_669_mod__minus__minus,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% mod_minus_minus
thf(fact_670_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_671_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_672_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_673_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_674_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_675_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_676_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_677_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_678_scaleR__minus1__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X )
= ( uminus_uminus_real @ X ) ) ).
% scaleR_minus1_left
thf(fact_679_minus__mod__self1,axiom,
! [B2: int,A: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ B2 @ A ) @ B2 )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% minus_mod_self1
thf(fact_680_fps__neg__nth,axiom,
! [F: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( uminus8389970968385878141s_real @ F ) @ N )
= ( uminus_uminus_real @ ( formal2580924720334399070h_real @ F @ N ) ) ) ).
% fps_neg_nth
thf(fact_681_fps__exp__0,axiom,
( ( formal3452214891061569154p_real @ zero_zero_real )
= one_on8598947968683843321s_real ) ).
% fps_exp_0
thf(fact_682_fps__exp__eq__1__iff,axiom,
! [C: real] :
( ( ( formal3452214891061569154p_real @ C )
= one_on8598947968683843321s_real )
= ( C = zero_zero_real ) ) ).
% fps_exp_eq_1_iff
thf(fact_683_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_684_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_685_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_686_fps__fps__exp__compose__minus,axiom,
! [C: real] :
( ( formal8268054683415598839e_real @ ( formal3452214891061569154p_real @ C ) @ ( uminus8389970968385878141s_real @ formal4708490801539276157X_real ) )
= ( formal3452214891061569154p_real @ ( uminus_uminus_real @ C ) ) ) ).
% fps_fps_exp_compose_minus
thf(fact_687_pth__3,axiom,
( uminus_uminus_real
= ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% pth_3
thf(fact_688_fps__exp__neg,axiom,
! [A: real] :
( ( formal3452214891061569154p_real @ ( uminus_uminus_real @ A ) )
= ( invers68952373231134600s_real @ ( formal3452214891061569154p_real @ A ) ) ) ).
% fps_exp_neg
thf(fact_689_fps__sin__even,axiom,
! [C: real] :
( ( formal6437758938379178589n_real @ ( uminus_uminus_real @ C ) )
= ( uminus8389970968385878141s_real @ ( formal6437758938379178589n_real @ C ) ) ) ).
% fps_sin_even
thf(fact_690_minus__equation__iff,axiom,
! [A: real,B2: real] :
( ( ( uminus_uminus_real @ A )
= B2 )
= ( ( uminus_uminus_real @ B2 )
= A ) ) ).
% minus_equation_iff
thf(fact_691_equation__minus__iff,axiom,
! [A: real,B2: real] :
( ( A
= ( uminus_uminus_real @ B2 ) )
= ( B2
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_692_minus__mult__commute,axiom,
! [A: int,B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) ) ) ).
% minus_mult_commute
thf(fact_693_minus__mult__commute,axiom,
! [A: real,B2: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% minus_mult_commute
thf(fact_694_square__eq__iff,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B2 @ B2 ) )
= ( ( A = B2 )
| ( A
= ( uminus_uminus_int @ B2 ) ) ) ) ).
% square_eq_iff
thf(fact_695_square__eq__iff,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ A )
= ( times_times_real @ B2 @ B2 ) )
= ( ( A = B2 )
| ( A
= ( uminus_uminus_real @ B2 ) ) ) ) ).
% square_eq_iff
thf(fact_696_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_697_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_698_minus__diff__commute,axiom,
! [B2: real,A: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A )
= ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% minus_diff_commute
thf(fact_699_div__minus__right,axiom,
! [A: int,B2: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% div_minus_right
thf(fact_700_minus__divide__left,axiom,
! [A: real,B2: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% minus_divide_left
thf(fact_701_minus__divide__divide,axiom,
! [A: real,B2: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
= ( divide_divide_real @ A @ B2 ) ) ).
% minus_divide_divide
thf(fact_702_minus__divide__right,axiom,
! [A: real,B2: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% minus_divide_right
thf(fact_703_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_704_mod__minus__eq,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) @ B2 )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% mod_minus_eq
thf(fact_705_mod__minus__cong,axiom,
! [A: int,B2: int,A4: int] :
( ( ( modulo_modulo_int @ A @ B2 )
= ( modulo_modulo_int @ A4 @ B2 ) )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B2 ) ) ) ).
% mod_minus_cong
thf(fact_706_mod__minus__right,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% mod_minus_right
thf(fact_707_infnorm__neg,axiom,
! [X: real] :
( ( linear_infnorm_real @ ( uminus_uminus_real @ X ) )
= ( linear_infnorm_real @ X ) ) ).
% infnorm_neg
thf(fact_708_fps__cos__odd,axiom,
! [C: real] :
( ( formal461277676486907980s_real @ ( uminus_uminus_real @ C ) )
= ( formal461277676486907980s_real @ C ) ) ).
% fps_cos_odd
thf(fact_709_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_710_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_711_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_712_square__eq__1__iff,axiom,
! [X: real] :
( ( ( times_times_real @ X @ X )
= one_one_real )
= ( ( X = one_one_real )
| ( X
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_713_nonzero__minus__divide__right,axiom,
! [B2: real,A: real] :
( ( B2 != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_714_nonzero__minus__divide__divide,axiom,
! [B2: real,A: real] :
( ( B2 != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
= ( divide_divide_real @ A @ B2 ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_715_nonzero__inverse__minus__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
= ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_716_dvd__neg__div,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ A )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% dvd_neg_div
thf(fact_717_dvd__neg__div,axiom,
! [B2: real,A: real] :
( ( dvd_dvd_real @ B2 @ A )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% dvd_neg_div
thf(fact_718_dvd__div__neg,axiom,
! [B2: int,A: int] :
( ( dvd_dvd_int @ B2 @ A )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% dvd_div_neg
thf(fact_719_dvd__div__neg,axiom,
! [B2: real,A: real] :
( ( dvd_dvd_real @ B2 @ A )
=> ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% dvd_div_neg
thf(fact_720_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_721_powr__minus,axiom,
! [X: real,A: real] :
( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
= ( inverse_inverse_real @ ( powr_real @ X @ A ) ) ) ).
% powr_minus
thf(fact_722_fps__integral0__neg,axiom,
! [A: formal3361831859752904756s_real] :
( ( formal8984515926053063617l_real @ ( uminus8389970968385878141s_real @ A ) @ zero_zero_real )
= ( uminus8389970968385878141s_real @ ( formal8984515926053063617l_real @ A @ zero_zero_real ) ) ) ).
% fps_integral0_neg
thf(fact_723_eq__minus__divide__eq,axiom,
! [A: real,B2: real,C: real] :
( ( A
= ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= ( uminus_uminus_real @ B2 ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_724_minus__divide__eq__eq,axiom,
! [B2: real,C: real,A: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) )
= A )
= ( ( ( C != zero_zero_real )
=> ( ( uminus_uminus_real @ B2 )
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_725_nonzero__neg__divide__eq__eq,axiom,
! [B2: real,A: real,C: real] :
( ( B2 != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
= C )
= ( ( uminus_uminus_real @ A )
= ( times_times_real @ C @ B2 ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_726_nonzero__neg__divide__eq__eq2,axiom,
! [B2: real,C: real,A: real] :
( ( B2 != zero_zero_real )
=> ( ( C
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) )
= ( ( times_times_real @ C @ B2 )
= ( uminus_uminus_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_727_divide__eq__minus__1__iff,axiom,
! [A: real,B2: real] :
( ( ( divide_divide_real @ A @ B2 )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B2 != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B2 ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_728_vector__fraction__eq__iff,axiom,
! [U: real,V: real,A: real,X: real] :
( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
= X )
= ( ( ( V = zero_zero_real )
=> ( X = zero_zero_real ) )
& ( ( V != zero_zero_real )
=> ( ( real_V1485227260804924795R_real @ U @ A )
= ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_729_eq__vector__fraction__iff,axiom,
! [X: real,U: real,V: real,A: real] :
( ( X
= ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
= ( ( ( V = zero_zero_real )
=> ( X = zero_zero_real ) )
& ( ( V != zero_zero_real )
=> ( ( real_V1485227260804924795R_real @ V @ X )
= ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_730_powr__minus__divide,axiom,
! [X: real,A: real] :
( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
= ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).
% powr_minus_divide
thf(fact_731_modulo__nat__def,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N2: nat] : ( minus_minus_nat @ M2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% modulo_nat_def
thf(fact_732_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A: real,B2: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
= ( uminus_uminus_real @ B2 ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_733_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A: real,B2: real] :
( ( ( Z = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
= ( uminus_uminus_real @ B2 ) ) )
& ( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
= ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_734_minus__divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_735_verit__minus__simplify_I3_J,axiom,
! [B2: int] :
( ( minus_minus_int @ zero_zero_int @ B2 )
= ( uminus_uminus_int @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_736_verit__minus__simplify_I3_J,axiom,
! [B2: real] :
( ( minus_minus_real @ zero_zero_real @ B2 )
= ( uminus_uminus_real @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_737_inverse__diff__inverse,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
= ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B2 ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_738_perp__def,axiom,
( perp
= ( ^ [I2: real,M2: nat] : ( divide_divide_real @ one_one_real @ ( i_nom @ I2 @ M2 ) ) ) ) ).
% perp_def
thf(fact_739_eq__fps__cos,axiom,
! [A: formal3361831859752904756s_real,C: real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= one_one_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
= zero_zero_real )
=> ( ( ( formal4557910837323084707v_real @ ( formal4557910837323084707v_real @ A ) )
= ( uminus8389970968385878141s_real @ ( times_7561426564079326009s_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ ( formal2098867297714113032t_real @ C ) ) @ A ) ) )
=> ( ( formal461277676486907980s_real @ C )
= A ) ) ) ) ).
% eq_fps_cos
thf(fact_740_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arsinh_real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_741_fps__const__mult,axiom,
! [C: nat,D: nat] :
( ( times_7269705568686124893ps_nat @ ( formal5286749789737391404st_nat @ C ) @ ( formal5286749789737391404st_nat @ D ) )
= ( formal5286749789737391404st_nat @ ( times_times_nat @ C @ D ) ) ) ).
% fps_const_mult
thf(fact_742_fps__const__mult,axiom,
! [C: real,D: real] :
( ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ ( formal2098867297714113032t_real @ D ) )
= ( formal2098867297714113032t_real @ ( times_times_real @ C @ D ) ) ) ).
% fps_const_mult
thf(fact_743_fps__const__mult,axiom,
! [C: int,D: int] :
( ( times_3091854549176928185ps_int @ ( formal5284259319228341128st_int @ C ) @ ( formal5284259319228341128st_int @ D ) )
= ( formal5284259319228341128st_int @ ( times_times_int @ C @ D ) ) ) ).
% fps_const_mult
thf(fact_744_fps__const__0__eq__0,axiom,
( ( formal2098867297714113032t_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_const_0_eq_0
thf(fact_745_fps__const__0__eq__0,axiom,
( ( formal5286749789737391404st_nat @ zero_zero_nat )
= zero_z8531573698755551073ps_nat ) ).
% fps_const_0_eq_0
thf(fact_746_fps__const__0__eq__0,axiom,
( ( formal5284259319228341128st_int @ zero_zero_int )
= zero_z4353722679246354365ps_int ) ).
% fps_const_0_eq_0
thf(fact_747_fps__const__1__eq__1,axiom,
( ( formal5286749789737391404st_nat @ one_one_nat )
= one_on3350087005236239133ps_nat ) ).
% fps_const_1_eq_1
thf(fact_748_fps__const__1__eq__1,axiom,
( ( formal2098867297714113032t_real @ one_one_real )
= one_on8598947968683843321s_real ) ).
% fps_const_1_eq_1
thf(fact_749_fps__const__1__eq__1,axiom,
( ( formal5284259319228341128st_int @ one_one_int )
= one_on8395608022581818233ps_int ) ).
% fps_const_1_eq_1
thf(fact_750_fps__const__minus,axiom,
! [C: real,D: real] :
( ( minus_6791916864952032525s_real @ ( formal2098867297714113032t_real @ C ) @ ( formal2098867297714113032t_real @ D ) )
= ( formal2098867297714113032t_real @ ( minus_minus_real @ C @ D ) ) ) ).
% fps_const_minus
thf(fact_751_fps__const__neg,axiom,
! [C: real] :
( ( uminus8389970968385878141s_real @ ( formal2098867297714113032t_real @ C ) )
= ( formal2098867297714113032t_real @ ( uminus_uminus_real @ C ) ) ) ).
% fps_const_neg
thf(fact_752_fps__nth__fps__const,axiom,
! [N: nat,C: real] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( formal2098867297714113032t_real @ C ) @ N )
= C ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( formal2098867297714113032t_real @ C ) @ N )
= zero_zero_real ) ) ) ).
% fps_nth_fps_const
thf(fact_753_fps__nth__fps__const,axiom,
! [N: nat,C: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( formal5286749789737391404st_nat @ C ) @ N )
= C ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( formal5286749789737391404st_nat @ C ) @ N )
= zero_zero_nat ) ) ) ).
% fps_nth_fps_const
thf(fact_754_fps__nth__fps__const,axiom,
! [N: nat,C: int] :
( ( ( N = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( formal5284259319228341128st_int @ C ) @ N )
= C ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( formal5284259319228341128st_int @ C ) @ N )
= zero_zero_int ) ) ) ).
% fps_nth_fps_const
thf(fact_755_fps__mult__right__const__nth,axiom,
! [F: formal_Power_fps_nat,C: nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ ( formal5286749789737391404st_nat @ C ) ) @ N )
= ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ C ) ) ).
% fps_mult_right_const_nth
thf(fact_756_fps__mult__right__const__nth,axiom,
! [F: formal3361831859752904756s_real,C: real,N: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ ( formal2098867297714113032t_real @ C ) ) @ N )
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ N ) @ C ) ) ).
% fps_mult_right_const_nth
thf(fact_757_fps__mult__right__const__nth,axiom,
! [F: formal_Power_fps_int,C: int,N: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ ( formal5284259319228341128st_int @ C ) ) @ N )
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ N ) @ C ) ) ).
% fps_mult_right_const_nth
thf(fact_758_fps__mult__left__const__nth,axiom,
! [C: nat,F: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( formal5286749789737391404st_nat @ C ) @ F ) @ N )
= ( times_times_nat @ C @ ( formal3720337525774269570th_nat @ F @ N ) ) ) ).
% fps_mult_left_const_nth
thf(fact_759_fps__mult__left__const__nth,axiom,
! [C: real,F: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ F ) @ N )
= ( times_times_real @ C @ ( formal2580924720334399070h_real @ F @ N ) ) ) ).
% fps_mult_left_const_nth
thf(fact_760_fps__mult__left__const__nth,axiom,
! [C: int,F: formal_Power_fps_int,N: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ ( formal5284259319228341128st_int @ C ) @ F ) @ N )
= ( times_times_int @ C @ ( formal3717847055265219294th_int @ F @ N ) ) ) ).
% fps_mult_left_const_nth
thf(fact_761_fps__exp__eq__fps__const__iff,axiom,
! [C: real,C5: real] :
( ( ( formal3452214891061569154p_real @ C )
= ( formal2098867297714113032t_real @ C5 ) )
= ( ( C = zero_zero_real )
& ( C5 = one_one_real ) ) ) ).
% fps_exp_eq_fps_const_iff
thf(fact_762_divide__fps__const,axiom,
! [F: formal3361831859752904756s_real,C: real] :
( ( divide1155267253282662278s_real @ F @ ( formal2098867297714113032t_real @ C ) )
= ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ ( inverse_inverse_real @ C ) ) @ F ) ) ).
% divide_fps_const
thf(fact_763_fps__exp__compose__linear,axiom,
! [D: real,C: real] :
( ( formal8268054683415598839e_real @ ( formal3452214891061569154p_real @ D ) @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ formal4708490801539276157X_real ) )
= ( formal3452214891061569154p_real @ ( times_times_real @ C @ D ) ) ) ).
% fps_exp_compose_linear
thf(fact_764_fps__const__nonzero__eq__nonzero,axiom,
! [C: real] :
( ( C != zero_zero_real )
=> ( ( formal2098867297714113032t_real @ C )
!= zero_z7760665558314615101s_real ) ) ).
% fps_const_nonzero_eq_nonzero
thf(fact_765_fps__const__nonzero__eq__nonzero,axiom,
! [C: nat] :
( ( C != zero_zero_nat )
=> ( ( formal5286749789737391404st_nat @ C )
!= zero_z8531573698755551073ps_nat ) ) ).
% fps_const_nonzero_eq_nonzero
thf(fact_766_fps__const__nonzero__eq__nonzero,axiom,
! [C: int] :
( ( C != zero_zero_int )
=> ( ( formal5284259319228341128st_int @ C )
!= zero_z4353722679246354365ps_int ) ) ).
% fps_const_nonzero_eq_nonzero
thf(fact_767_fps__const__divide,axiom,
! [X: real,Y: real] :
( ( divide1155267253282662278s_real @ ( formal2098867297714113032t_real @ X ) @ ( formal2098867297714113032t_real @ Y ) )
= ( formal2098867297714113032t_real @ ( divide_divide_real @ X @ Y ) ) ) ).
% fps_const_divide
thf(fact_768_fps__const__inverse,axiom,
! [A: real] :
( ( invers68952373231134600s_real @ ( formal2098867297714113032t_real @ A ) )
= ( formal2098867297714113032t_real @ ( inverse_inverse_real @ A ) ) ) ).
% fps_const_inverse
thf(fact_769_divide__powr__uminus,axiom,
! [A: real,B2: real,C: real] :
( ( divide_divide_real @ A @ ( powr_real @ B2 @ C ) )
= ( times_times_real @ A @ ( powr_real @ B2 @ ( uminus_uminus_real @ C ) ) ) ) ).
% divide_powr_uminus
thf(fact_770_divide__fps__const_H,axiom,
! [F: formal3361831859752904756s_real,C: real] :
( ( divide1155267253282662278s_real @ F @ ( formal2098867297714113032t_real @ C ) )
= ( times_7561426564079326009s_real @ F @ ( formal2098867297714113032t_real @ ( inverse_inverse_real @ C ) ) ) ) ).
% divide_fps_const'
thf(fact_771_fps__integral0__fps__const__mult__left,axiom,
! [C: real,A: formal3361831859752904756s_real] :
( ( formal8984515926053063617l_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ A ) @ zero_zero_real )
= ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ ( formal8984515926053063617l_real @ A @ zero_zero_real ) ) ) ).
% fps_integral0_fps_const_mult_left
thf(fact_772_fps__integral0__fps__const__mult__right,axiom,
! [A: formal3361831859752904756s_real,C: real] :
( ( formal8984515926053063617l_real @ ( times_7561426564079326009s_real @ A @ ( formal2098867297714113032t_real @ C ) ) @ zero_zero_real )
= ( times_7561426564079326009s_real @ ( formal8984515926053063617l_real @ A @ zero_zero_real ) @ ( formal2098867297714113032t_real @ C ) ) ) ).
% fps_integral0_fps_const_mult_right
thf(fact_773_fps__inverse__zero__conv__fps__const,axiom,
( ( invers68952373231134600s_real @ zero_z7760665558314615101s_real )
= ( formal2098867297714113032t_real @ ( inverse_inverse_real @ zero_zero_real ) ) ) ).
% fps_inverse_zero_conv_fps_const
thf(fact_774_fps__integral0__fps__const,axiom,
! [C: real] :
( ( formal8984515926053063617l_real @ ( formal2098867297714113032t_real @ C ) @ zero_zero_real )
= ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ formal4708490801539276157X_real ) ) ).
% fps_integral0_fps_const
thf(fact_775_fps__cos__deriv,axiom,
! [C: real] :
( ( formal4557910837323084707v_real @ ( formal461277676486907980s_real @ C ) )
= ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ ( uminus_uminus_real @ C ) ) @ ( formal6437758938379178589n_real @ C ) ) ) ).
% fps_cos_deriv
thf(fact_776_fps__integral0__fps__const_H,axiom,
! [C: real] :
( ( ( inverse_inverse_real @ one_one_real )
= one_one_real )
=> ( ( formal8984515926053063617l_real @ ( formal2098867297714113032t_real @ C ) @ zero_zero_real )
= ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ formal4708490801539276157X_real ) ) ) ).
% fps_integral0_fps_const'
thf(fact_777_fps__integral0__deriv,axiom,
! [A: formal3361831859752904756s_real] :
( ( formal8984515926053063617l_real @ ( formal4557910837323084707v_real @ A ) @ zero_zero_real )
= ( minus_6791916864952032525s_real @ A @ ( formal2098867297714113032t_real @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) ) ) ).
% fps_integral0_deriv
thf(fact_778_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X4: real,Y4: real] : ( times_times_real @ X4 @ ( inverse_inverse_real @ Y4 ) ) ) ) ).
% divide_real_def
thf(fact_779_eq__fps__sin,axiom,
! [A: formal3361831859752904756s_real,C: real] :
( ( ( formal2580924720334399070h_real @ A @ zero_zero_nat )
= zero_zero_real )
=> ( ( ( formal2580924720334399070h_real @ A @ one_one_nat )
= C )
=> ( ( ( formal4557910837323084707v_real @ ( formal4557910837323084707v_real @ A ) )
= ( uminus8389970968385878141s_real @ ( times_7561426564079326009s_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ C ) @ ( formal2098867297714113032t_real @ C ) ) @ A ) ) )
=> ( ( formal6437758938379178589n_real @ C )
= A ) ) ) ) ).
% eq_fps_sin
thf(fact_780_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_781_fps__ln__deriv,axiom,
! [C: real] :
( ( formal4557910837323084707v_real @ ( formal8688746759596762231n_real @ C ) )
= ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ ( divide_divide_real @ one_one_real @ C ) ) @ ( invers68952373231134600s_real @ ( plus_p6008488439947570109s_real @ one_on8598947968683843321s_real @ formal4708490801539276157X_real ) ) ) ) ).
% fps_ln_deriv
thf(fact_782_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_783_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_784_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_785_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_786_add__right__cancel,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_787_add__right__cancel,axiom,
! [B2: real,A: real,C: real] :
( ( ( plus_plus_real @ B2 @ A )
= ( plus_plus_real @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_788_add__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_789_add__left__cancel,axiom,
! [A: real,B2: real,C: real] :
( ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_790_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_791_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_792_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_793_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_794_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_795_add__cancel__left__left,axiom,
! [B2: real,A: real] :
( ( ( plus_plus_real @ B2 @ A )
= A )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_796_add__cancel__left__left,axiom,
! [B2: nat,A: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_797_add__cancel__left__left,axiom,
! [B2: int,A: int] :
( ( ( plus_plus_int @ B2 @ A )
= A )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_798_add__cancel__left__right,axiom,
! [A: real,B2: real] :
( ( ( plus_plus_real @ A @ B2 )
= A )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_799_add__cancel__left__right,axiom,
! [A: nat,B2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_800_add__cancel__left__right,axiom,
! [A: int,B2: int] :
( ( ( plus_plus_int @ A @ B2 )
= A )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_801_add__cancel__right__left,axiom,
! [A: real,B2: real] :
( ( A
= ( plus_plus_real @ B2 @ A ) )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_802_add__cancel__right__left,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ B2 @ A ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_803_add__cancel__right__left,axiom,
! [A: int,B2: int] :
( ( A
= ( plus_plus_int @ B2 @ A ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_804_add__cancel__right__right,axiom,
! [A: real,B2: real] :
( ( A
= ( plus_plus_real @ A @ B2 ) )
= ( B2 = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_805_add__cancel__right__right,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ A @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_806_add__cancel__right__right,axiom,
! [A: int,B2: int] :
( ( A
= ( plus_plus_int @ A @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_807_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_808_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_809_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_810_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_811_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_812_add__diff__cancel,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel
thf(fact_813_diff__add__cancel,axiom,
! [A: real,B2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
= A ) ).
% diff_add_cancel
thf(fact_814_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_815_add__diff__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
= ( minus_minus_real @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_816_add__diff__cancel__left_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_817_add__diff__cancel__left_H,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_818_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_819_add__diff__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( minus_minus_real @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_820_add__diff__cancel__right_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_821_add__diff__cancel__right_H,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_822_add__minus__cancel,axiom,
! [A: real,B2: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 ) )
= B2 ) ).
% add_minus_cancel
thf(fact_823_minus__add__cancel,axiom,
! [A: real,B2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B2 ) )
= B2 ) ).
% minus_add_cancel
thf(fact_824_minus__add__distrib,axiom,
! [A: real,B2: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) ) ) ).
% minus_add_distrib
thf(fact_825_dvd__add__triv__left__iff,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_826_dvd__add__triv__left__iff,axiom,
! [A: real,B2: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B2 ) )
= ( dvd_dvd_real @ A @ B2 ) ) ).
% dvd_add_triv_left_iff
thf(fact_827_dvd__add__triv__right__iff,axiom,
! [A: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_828_dvd__add__triv__right__iff,axiom,
! [A: real,B2: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ A ) )
= ( dvd_dvd_real @ A @ B2 ) ) ).
% dvd_add_triv_right_iff
thf(fact_829_mod__add__self2,axiom,
! [A: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_add_self2
thf(fact_830_mod__add__self2,axiom,
! [A: int,B2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_add_self2
thf(fact_831_mod__add__self1,axiom,
! [B2: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_add_self1
thf(fact_832_mod__add__self1,axiom,
! [B2: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_add_self1
thf(fact_833_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_834_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_835_diff__add__zero,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_836_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_837_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_838_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_839_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_840_uminus__add__conv__diff,axiom,
! [A: real,B2: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( minus_minus_real @ B2 @ A ) ) ).
% uminus_add_conv_diff
thf(fact_841_diff__minus__eq__add,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( plus_plus_real @ A @ B2 ) ) ).
% diff_minus_eq_add
thf(fact_842_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_843_dvd__add__times__triv__right__iff,axiom,
! [A: real,B2: real,C: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A ) ) )
= ( dvd_dvd_real @ A @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_844_dvd__add__times__triv__right__iff,axiom,
! [A: int,B2: int,C: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A ) ) )
= ( dvd_dvd_int @ A @ B2 ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_845_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B2 ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_846_dvd__add__times__triv__left__iff,axiom,
! [A: real,C: real,B2: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B2 ) )
= ( dvd_dvd_real @ A @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_847_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B2: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B2 ) )
= ( dvd_dvd_int @ A @ B2 ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_848_div__add,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).
% div_add
thf(fact_849_div__add,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% div_add
thf(fact_850_mod__mult__self4,axiom,
! [B2: nat,C: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mult_self4
thf(fact_851_mod__mult__self4,axiom,
! [B2: int,C: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mult_self4
thf(fact_852_mod__mult__self3,axiom,
! [C: nat,B2: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mult_self3
thf(fact_853_mod__mult__self3,axiom,
! [C: int,B2: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mult_self3
thf(fact_854_mod__mult__self2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mult_self2
thf(fact_855_mod__mult__self2,axiom,
! [A: int,B2: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mult_self2
thf(fact_856_mod__mult__self1,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
= ( modulo_modulo_nat @ A @ B2 ) ) ).
% mod_mult_self1
thf(fact_857_mod__mult__self1,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
= ( modulo_modulo_int @ A @ B2 ) ) ).
% mod_mult_self1
thf(fact_858_scaleR__eq__iff,axiom,
! [B2: real,U: real,A: real] :
( ( ( plus_plus_real @ B2 @ ( real_V1485227260804924795R_real @ U @ A ) )
= ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B2 ) ) )
= ( ( A = B2 )
| ( U = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_859_fps__add__nth,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( plus_p6043471806551771617ps_nat @ F @ G ) @ N )
= ( plus_plus_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ ( formal3720337525774269570th_nat @ G @ N ) ) ) ).
% fps_add_nth
thf(fact_860_fps__add__nth,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( plus_p6008488439947570109s_real @ F @ G ) @ N )
= ( plus_plus_real @ ( formal2580924720334399070h_real @ F @ N ) @ ( formal2580924720334399070h_real @ G @ N ) ) ) ).
% fps_add_nth
thf(fact_861_fps__const__add,axiom,
! [C: nat,D: nat] :
( ( plus_p6043471806551771617ps_nat @ ( formal5286749789737391404st_nat @ C ) @ ( formal5286749789737391404st_nat @ D ) )
= ( formal5286749789737391404st_nat @ ( plus_plus_nat @ C @ D ) ) ) ).
% fps_const_add
thf(fact_862_fps__const__add,axiom,
! [C: real,D: real] :
( ( plus_p6008488439947570109s_real @ ( formal2098867297714113032t_real @ C ) @ ( formal2098867297714113032t_real @ D ) )
= ( formal2098867297714113032t_real @ ( plus_plus_real @ C @ D ) ) ) ).
% fps_const_add
thf(fact_863_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_864_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_865_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_866_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_867_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_868_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_869_div__mult__self1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_self1
thf(fact_870_div__mult__self1,axiom,
! [B2: int,A: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_self1
thf(fact_871_div__mult__self2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_self2
thf(fact_872_div__mult__self2,axiom,
! [B2: int,A: int,C: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_self2
thf(fact_873_div__mult__self3,axiom,
! [B2: nat,C: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_self3
thf(fact_874_div__mult__self3,axiom,
! [B2: int,C: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_self3
thf(fact_875_div__mult__self4,axiom,
! [B2: nat,C: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% div_mult_self4
thf(fact_876_div__mult__self4,axiom,
! [B2: int,C: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% div_mult_self4
thf(fact_877_scaleR__collapse,axiom,
! [U: real,A: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
= A ) ).
% scaleR_collapse
thf(fact_878_add_Oinverse__distrib__swap,axiom,
! [A: real,B2: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_879_group__cancel_Oneg1,axiom,
! [A5: real,K: real,A: real] :
( ( A5
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A5 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_880_add__divide__distrib,axiom,
! [A: real,B2: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% add_divide_distrib
thf(fact_881_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_882_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_883_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_884_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_885_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X4: real] : ( plus_plus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_886_scaleR__left_Oadd,axiom,
! [X: real,Y: real,Xa: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y ) @ Xa )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).
% scaleR_left.add
thf(fact_887_scaleR__left__distrib,axiom,
! [A: real,B2: real,X: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B2 ) @ X )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B2 @ X ) ) ) ).
% scaleR_left_distrib
thf(fact_888_pth__8,axiom,
! [C: real,X: real,D: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( real_V1485227260804924795R_real @ D @ X ) )
= ( real_V1485227260804924795R_real @ ( plus_plus_real @ C @ D ) @ X ) ) ).
% pth_8
thf(fact_889_pth__9_I1_J,axiom,
! [C: real,X: real,Z: real,D: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ Z ) @ ( real_V1485227260804924795R_real @ D @ X ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( plus_plus_real @ C @ D ) @ X ) @ Z ) ) ).
% pth_9(1)
thf(fact_890_pth__9_I2_J,axiom,
! [C: real,X: real,D: real,Z: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ X ) @ Z ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( plus_plus_real @ C @ D ) @ X ) @ Z ) ) ).
% pth_9(2)
thf(fact_891_pth__9_I3_J,axiom,
! [C: real,X: real,W: real,D: real,Z: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ X ) @ Z ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( plus_plus_real @ C @ D ) @ X ) @ ( plus_plus_real @ W @ Z ) ) ) ).
% pth_9(3)
thf(fact_892_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_893_add__right__imp__eq,axiom,
! [B2: real,A: real,C: real] :
( ( ( plus_plus_real @ B2 @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_894_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_895_add__left__imp__eq,axiom,
! [A: real,B2: real,C: real] :
( ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_896_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_897_add_Oleft__commute,axiom,
! [B2: real,A: real,C: real] :
( ( plus_plus_real @ B2 @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_898_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B: nat] : ( plus_plus_nat @ B @ A2 ) ) ) ).
% add.commute
thf(fact_899_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A2: real,B: real] : ( plus_plus_real @ B @ A2 ) ) ) ).
% add.commute
thf(fact_900_add_Oright__cancel,axiom,
! [B2: real,A: real,C: real] :
( ( ( plus_plus_real @ B2 @ A )
= ( plus_plus_real @ C @ A ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_901_add_Oleft__cancel,axiom,
! [A: real,B2: real,C: real] :
( ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ A @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_902_add_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_903_add_Oassoc,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_904_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B2: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_905_group__cancel_Oadd2,axiom,
! [B5: real,K: real,B2: real,A: real] :
( ( B5
= ( plus_plus_real @ K @ B2 ) )
=> ( ( plus_plus_real @ A @ B5 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_906_group__cancel_Oadd1,axiom,
! [A5: nat,K: nat,A: nat,B2: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A5 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_907_group__cancel_Oadd1,axiom,
! [A5: real,K: real,A: real,B2: real] :
( ( A5
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A5 @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_908_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_909_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_910_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_911_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_912_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_913_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X4: real] : ( minus_minus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_914_mod__add__right__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% mod_add_right_eq
thf(fact_915_mod__add__right__eq,axiom,
! [A: int,B2: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% mod_add_right_eq
thf(fact_916_mod__add__left__eq,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% mod_add_left_eq
thf(fact_917_mod__add__left__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% mod_add_left_eq
thf(fact_918_mod__add__cong,axiom,
! [A: nat,C: nat,A4: nat,B2: nat,B4: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A4 @ C ) )
=> ( ( ( modulo_modulo_nat @ B2 @ C )
= ( modulo_modulo_nat @ B4 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_919_mod__add__cong,axiom,
! [A: int,C: int,A4: int,B2: int,B4: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A4 @ C ) )
=> ( ( ( modulo_modulo_int @ B2 @ C )
= ( modulo_modulo_int @ B4 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B4 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_920_mod__add__eq,axiom,
! [A: nat,C: nat,B2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% mod_add_eq
thf(fact_921_mod__add__eq,axiom,
! [A: int,C: int,B2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% mod_add_eq
thf(fact_922_pth__6,axiom,
! [C: real,X: real,Y: real] :
( ( real_V1485227260804924795R_real @ C @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( real_V1485227260804924795R_real @ C @ Y ) ) ) ).
% pth_6
thf(fact_923_pth__c_I1_J,axiom,
! [C: real,X: real,D: real,Y: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( real_V1485227260804924795R_real @ D @ Y ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ ( real_V1485227260804924795R_real @ C @ X ) ) ) ).
% pth_c(1)
thf(fact_924_pth__c_I2_J,axiom,
! [C: real,X: real,Z: real,D: real,Y: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ Z ) @ ( real_V1485227260804924795R_real @ D @ Y ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ Z ) ) ) ).
% pth_c(2)
thf(fact_925_pth__c_I3_J,axiom,
! [C: real,X: real,D: real,Y: real,Z: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ Z ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ Z ) ) ) ).
% pth_c(3)
thf(fact_926_pth__c_I4_J,axiom,
! [C: real,X: real,W: real,D: real,Y: real,Z: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ Z ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ ( plus_plus_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ W ) @ Z ) ) ) ).
% pth_c(4)
thf(fact_927_pth__b_I1_J,axiom,
! [C: real,X: real,D: real,Y: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( real_V1485227260804924795R_real @ D @ Y ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( real_V1485227260804924795R_real @ D @ Y ) ) ) ).
% pth_b(1)
thf(fact_928_pth__b_I2_J,axiom,
! [C: real,X: real,Z: real,D: real,Y: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ Z ) @ ( real_V1485227260804924795R_real @ D @ Y ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( plus_plus_real @ Z @ ( real_V1485227260804924795R_real @ D @ Y ) ) ) ) ).
% pth_b(2)
thf(fact_929_pth__b_I3_J,axiom,
! [C: real,X: real,D: real,Y: real,Z: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ Z ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ Z ) ) ) ).
% pth_b(3)
thf(fact_930_pth__b_I4_J,axiom,
! [C: real,X: real,W: real,D: real,Y: real,Z: real] :
( ( plus_plus_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ W ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ Z ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ C @ X ) @ ( plus_plus_real @ W @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ D @ Y ) @ Z ) ) ) ) ).
% pth_b(4)
thf(fact_931_scaleR__right__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).
% scaleR_right_distrib
thf(fact_932_dvd__add,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_933_dvd__add,axiom,
! [A: real,B2: real,C: real] :
( ( dvd_dvd_real @ A @ B2 )
=> ( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ) ).
% dvd_add
thf(fact_934_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B2: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_935_dvd__add__left__iff,axiom,
! [A: real,C: real,B2: real] :
( ( dvd_dvd_real @ A @ C )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
= ( dvd_dvd_real @ A @ B2 ) ) ) ).
% dvd_add_left_iff
thf(fact_936_dvd__add__right__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B2 )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_937_dvd__add__right__iff,axiom,
! [A: real,B2: real,C: real] :
( ( dvd_dvd_real @ A @ B2 )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
= ( dvd_dvd_real @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_938_group__cancel_Osub1,axiom,
! [A5: real,K: real,A: real,B2: real] :
( ( A5
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A5 @ B2 )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_939_diff__eq__eq,axiom,
! [A: real,B2: real,C: real] :
( ( ( minus_minus_real @ A @ B2 )
= C )
= ( A
= ( plus_plus_real @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_940_eq__diff__eq,axiom,
! [A: real,C: real,B2: real] :
( ( A
= ( minus_minus_real @ C @ B2 ) )
= ( ( plus_plus_real @ A @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_941_add__diff__eq,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_942_diff__diff__eq2,axiom,
! [A: real,B2: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_943_diff__add__eq,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_944_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B2: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_945_add__implies__diff,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A )
=> ( C
= ( minus_minus_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_946_add__implies__diff,axiom,
! [C: real,B2: real,A: real] :
( ( ( plus_plus_real @ C @ B2 )
= A )
=> ( C
= ( minus_minus_real @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_947_diff__diff__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_948_diff__diff__eq,axiom,
! [A: real,B2: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_949_vector__space__over__itself_Oscale__left__distrib,axiom,
! [A: real,B2: real,X: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ X )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ B2 @ X ) ) ) ).
% vector_space_over_itself.scale_left_distrib
thf(fact_950_vector__space__over__itself_Oscale__right__distrib,axiom,
! [A: real,X: real,Y: real] :
( ( times_times_real @ A @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ A @ X ) @ ( times_times_real @ A @ Y ) ) ) ).
% vector_space_over_itself.scale_right_distrib
thf(fact_951_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_952_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_953_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_954_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_955_comm__semiring__class_Odistrib,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_956_comm__semiring__class_Odistrib,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_957_comm__semiring__class_Odistrib,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_958_distrib__left,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_959_distrib__left,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_960_distrib__left,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_961_distrib__right,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_962_distrib__right,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_963_distrib__right,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_964_combine__common__factor,axiom,
! [A: nat,E: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_965_combine__common__factor,axiom,
! [A: real,E: real,B2: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_966_combine__common__factor,axiom,
! [A: int,E: int,B2: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_967_pth__7_I1_J,axiom,
! [X: real] :
( ( plus_plus_real @ zero_zero_real @ X )
= X ) ).
% pth_7(1)
thf(fact_968_pth__d,axiom,
! [X: real] :
( ( plus_plus_real @ X @ zero_zero_real )
= X ) ).
% pth_d
thf(fact_969_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_970_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_971_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_972_eq__add__iff,axiom,
! [X: real,Y: real] :
( ( X
= ( plus_plus_real @ X @ Y ) )
= ( Y = zero_zero_real ) ) ).
% eq_add_iff
thf(fact_973_eq__add__iff,axiom,
! [X: int,Y: int] :
( ( X
= ( plus_plus_int @ X @ Y ) )
= ( Y = zero_zero_int ) ) ).
% eq_add_iff
thf(fact_974_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_975_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_976_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_977_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_978_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_979_fps__sin__add,axiom,
! [A: real,B2: real] :
( ( formal6437758938379178589n_real @ ( plus_plus_real @ A @ B2 ) )
= ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ ( formal6437758938379178589n_real @ A ) @ ( formal461277676486907980s_real @ B2 ) ) @ ( times_7561426564079326009s_real @ ( formal461277676486907980s_real @ A ) @ ( formal6437758938379178589n_real @ B2 ) ) ) ) ).
% fps_sin_add
thf(fact_980_neg__eq__iff__add__eq__0,axiom,
! [A: int,B2: int] :
( ( ( uminus_uminus_int @ A )
= B2 )
= ( ( plus_plus_int @ A @ B2 )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_981_neg__eq__iff__add__eq__0,axiom,
! [A: real,B2: real] :
( ( ( uminus_uminus_real @ A )
= B2 )
= ( ( plus_plus_real @ A @ B2 )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_982_eq__neg__iff__add__eq__0,axiom,
! [A: int,B2: int] :
( ( A
= ( uminus_uminus_int @ B2 ) )
= ( ( plus_plus_int @ A @ B2 )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_983_eq__neg__iff__add__eq__0,axiom,
! [A: real,B2: real] :
( ( A
= ( uminus_uminus_real @ B2 ) )
= ( ( plus_plus_real @ A @ B2 )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_984_add_Oinverse__unique,axiom,
! [A: int,B2: int] :
( ( ( plus_plus_int @ A @ B2 )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B2 ) ) ).
% add.inverse_unique
thf(fact_985_add_Oinverse__unique,axiom,
! [A: real,B2: real] :
( ( ( plus_plus_real @ A @ B2 )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B2 ) ) ).
% add.inverse_unique
thf(fact_986_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_987_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_988_add__eq__0__iff,axiom,
! [A: int,B2: int] :
( ( ( plus_plus_int @ A @ B2 )
= zero_zero_int )
= ( B2
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_989_add__eq__0__iff,axiom,
! [A: real,B2: real] :
( ( ( plus_plus_real @ A @ B2 )
= zero_zero_real )
= ( B2
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_990_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_991_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_992_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B2: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_993_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B2: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_994_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B2: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_995_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B2: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_996_mult__diff__mult,axiom,
! [X: real,Y: real,A: real,B2: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B2 ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B2 ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B2 ) ) ) ).
% mult_diff_mult
thf(fact_997_mult__diff__mult,axiom,
! [X: int,Y: int,A: int,B2: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B2 ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B2 ) ) ) ).
% mult_diff_mult
thf(fact_998_pth__2,axiom,
( minus_minus_real
= ( ^ [X4: real,Y4: real] : ( plus_plus_real @ X4 @ ( uminus_uminus_real @ Y4 ) ) ) ) ).
% pth_2
thf(fact_999_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A2: real,B: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1000_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A2: real,B: real] : ( plus_plus_real @ A2 @ ( uminus_uminus_real @ B ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1001_group__cancel_Osub2,axiom,
! [B5: real,K: real,B2: real,A: real] :
( ( B5
= ( plus_plus_real @ K @ B2 ) )
=> ( ( minus_minus_real @ A @ B5 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% group_cancel.sub2
thf(fact_1002_fps__integral__linear,axiom,
! [A: real,F: formal3361831859752904756s_real,B2: real,G: formal3361831859752904756s_real,A0: real,B0: real] :
( ( formal8984515926053063617l_real @ ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ A ) @ F ) @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ B2 ) @ G ) ) @ ( plus_plus_real @ ( times_times_real @ A @ A0 ) @ ( times_times_real @ B2 @ B0 ) ) )
= ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ A ) @ ( formal8984515926053063617l_real @ F @ A0 ) ) @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ B2 ) @ ( formal8984515926053063617l_real @ G @ B0 ) ) ) ) ).
% fps_integral_linear
thf(fact_1003_pth__a,axiom,
! [X: real,Y: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ zero_zero_real @ X ) @ Y )
= Y ) ).
% pth_a
thf(fact_1004_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B2: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B2 )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1005_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B2: int,A: int] :
( ( dvd_dvd_int @ C @ B2 )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1006_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1007_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A: int,B2: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1008_mod__eqE,axiom,
! [A: int,C: int,B2: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B2 @ C ) )
=> ~ ! [D3: int] :
( B2
!= ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% mod_eqE
thf(fact_1009_div__add1__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_1010_div__add1__eq,axiom,
! [A: int,B2: int,C: int] :
( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_1011_powr__add,axiom,
! [X: real,A: real,B2: real] :
( ( powr_real @ X @ ( plus_plus_real @ A @ B2 ) )
= ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) ) ) ).
% powr_add
thf(fact_1012_fps__integral0__add,axiom,
! [A: formal3361831859752904756s_real,B2: formal3361831859752904756s_real] :
( ( formal8984515926053063617l_real @ ( plus_p6008488439947570109s_real @ A @ B2 ) @ zero_zero_real )
= ( plus_p6008488439947570109s_real @ ( formal8984515926053063617l_real @ A @ zero_zero_real ) @ ( formal8984515926053063617l_real @ B2 @ zero_zero_real ) ) ) ).
% fps_integral0_add
thf(fact_1013_fps__exp__add__mult,axiom,
! [A: real,B2: real] :
( ( formal3452214891061569154p_real @ ( plus_plus_real @ A @ B2 ) )
= ( times_7561426564079326009s_real @ ( formal3452214891061569154p_real @ A ) @ ( formal3452214891061569154p_real @ B2 ) ) ) ).
% fps_exp_add_mult
thf(fact_1014_fps__ln__mult__add,axiom,
! [C: real,D: real] :
( ( C != zero_zero_real )
=> ( ( D != zero_zero_real )
=> ( ( plus_p6008488439947570109s_real @ ( formal8688746759596762231n_real @ C ) @ ( formal8688746759596762231n_real @ D ) )
= ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ ( plus_plus_real @ C @ D ) ) @ ( formal8688746759596762231n_real @ ( times_times_real @ C @ D ) ) ) ) ) ) ).
% fps_ln_mult_add
thf(fact_1015_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B2: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
= B2 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1016_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B2: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1017_add__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_1018_add__frac__num,axiom,
! [Y: real,X: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_1019_add__num__frac,axiom,
! [Y: real,Z: real,X: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_1020_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_1021_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_1022_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_1023_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_1024_div__add__self1,axiom,
! [B2: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1025_div__add__self1,axiom,
! [B2: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_1026_div__add__self2,axiom,
! [B2: nat,A: nat] :
( ( B2 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1027_div__add__self2,axiom,
! [B2: int,A: int] :
( ( B2 != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_1028_unity__coeff__ex,axiom,
! [P2: nat > $o,L: nat] :
( ( ? [X4: nat] : ( P2 @ ( times_times_nat @ L @ X4 ) ) )
= ( ? [X4: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
& ( P2 @ X4 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1029_unity__coeff__ex,axiom,
! [P2: real > $o,L: real] :
( ( ? [X4: real] : ( P2 @ ( times_times_real @ L @ X4 ) ) )
= ( ? [X4: real] :
( ( dvd_dvd_real @ L @ ( plus_plus_real @ X4 @ zero_zero_real ) )
& ( P2 @ X4 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1030_unity__coeff__ex,axiom,
! [P2: int > $o,L: int] :
( ( ? [X4: int] : ( P2 @ ( times_times_int @ L @ X4 ) ) )
= ( ? [X4: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
& ( P2 @ X4 ) ) ) ) ).
% unity_coeff_ex
thf(fact_1031_inf__period_I3_J,axiom,
! [D: real,D2: real,T: real] :
( ( dvd_dvd_real @ D @ D2 )
=> ! [X3: real,K4: real] :
( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_1032_inf__period_I3_J,axiom,
! [D: int,D2: int,T: int] :
( ( dvd_dvd_int @ D @ D2 )
=> ! [X3: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_1033_inf__period_I4_J,axiom,
! [D: real,D2: real,T: real] :
( ( dvd_dvd_real @ D @ D2 )
=> ! [X3: real,K4: real] :
( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D2 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_1034_inf__period_I4_J,axiom,
! [D: int,D2: int,T: int] :
( ( dvd_dvd_int @ D @ D2 )
=> ! [X3: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D2 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_1035_division__ring__inverse__add,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B2 ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_1036_inverse__add,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
= ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B2 ) ) ) ) ) ).
% inverse_add
thf(fact_1037_mult__div__mod__eq,axiom,
! [B2: nat,A: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) )
= A ) ).
% mult_div_mod_eq
thf(fact_1038_mult__div__mod__eq,axiom,
! [B2: int,A: int] :
( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) )
= A ) ).
% mult_div_mod_eq
thf(fact_1039_mod__mult__div__eq,axiom,
! [A: nat,B2: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_1040_mod__mult__div__eq,axiom,
! [A: int,B2: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_1041_mod__div__mult__eq,axiom,
! [A: nat,B2: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
= A ) ).
% mod_div_mult_eq
thf(fact_1042_mod__div__mult__eq,axiom,
! [A: int,B2: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
= A ) ).
% mod_div_mult_eq
thf(fact_1043_div__mult__mod__eq,axiom,
! [A: nat,B2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) )
= A ) ).
% div_mult_mod_eq
thf(fact_1044_div__mult__mod__eq,axiom,
! [A: int,B2: int] :
( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) )
= A ) ).
% div_mult_mod_eq
thf(fact_1045_mod__div__decomp,axiom,
! [A: nat,B2: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% mod_div_decomp
thf(fact_1046_mod__div__decomp,axiom,
! [A: int,B2: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% mod_div_decomp
thf(fact_1047_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1048_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1049_cancel__div__mod__rules_I2_J,axiom,
! [B2: nat,A: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1050_cancel__div__mod__rules_I2_J,axiom,
! [B2: int,A: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1051_div__mult1__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_1052_div__mult1__eq,axiom,
! [A: int,B2: int,C: int] :
( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_1053_fps__integral__conv__plus__const,axiom,
( formal8984515926053063617l_real
= ( ^ [A2: formal3361831859752904756s_real,A02: real] : ( plus_p6008488439947570109s_real @ ( formal8984515926053063617l_real @ A2 @ zero_zero_real ) @ ( formal2098867297714113032t_real @ A02 ) ) ) ) ).
% fps_integral_conv_plus_const
thf(fact_1054_fps__mult__right__fps__X__plus__1__nth,axiom,
! [N: nat,A: formal_Power_fps_nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ A @ ( plus_p6043471806551771617ps_nat @ one_on3350087005236239133ps_nat @ formal1744162128437646113_X_nat ) ) @ N )
= ( formal3720337525774269570th_nat @ A @ N ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ A @ ( plus_p6043471806551771617ps_nat @ one_on3350087005236239133ps_nat @ formal1744162128437646113_X_nat ) ) @ N )
= ( plus_plus_nat @ ( formal3720337525774269570th_nat @ A @ N ) @ ( formal3720337525774269570th_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% fps_mult_right_fps_X_plus_1_nth
thf(fact_1055_fps__mult__right__fps__X__plus__1__nth,axiom,
! [N: nat,A: formal3361831859752904756s_real] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ A @ ( plus_p6008488439947570109s_real @ one_on8598947968683843321s_real @ formal4708490801539276157X_real ) ) @ N )
= ( formal2580924720334399070h_real @ A @ N ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ A @ ( plus_p6008488439947570109s_real @ one_on8598947968683843321s_real @ formal4708490801539276157X_real ) ) @ N )
= ( plus_plus_real @ ( formal2580924720334399070h_real @ A @ N ) @ ( formal2580924720334399070h_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% fps_mult_right_fps_X_plus_1_nth
thf(fact_1056_fps__mult__fps__X__plus__1__nth,axiom,
! [N: nat,A: formal_Power_fps_nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( plus_p6043471806551771617ps_nat @ one_on3350087005236239133ps_nat @ formal1744162128437646113_X_nat ) @ A ) @ N )
= ( formal3720337525774269570th_nat @ A @ N ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( plus_p6043471806551771617ps_nat @ one_on3350087005236239133ps_nat @ formal1744162128437646113_X_nat ) @ A ) @ N )
= ( plus_plus_nat @ ( formal3720337525774269570th_nat @ A @ N ) @ ( formal3720337525774269570th_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% fps_mult_fps_X_plus_1_nth
thf(fact_1057_fps__mult__fps__X__plus__1__nth,axiom,
! [N: nat,A: formal3361831859752904756s_real] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( plus_p6008488439947570109s_real @ one_on8598947968683843321s_real @ formal4708490801539276157X_real ) @ A ) @ N )
= ( formal2580924720334399070h_real @ A @ N ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( plus_p6008488439947570109s_real @ one_on8598947968683843321s_real @ formal4708490801539276157X_real ) @ A ) @ N )
= ( plus_plus_real @ ( formal2580924720334399070h_real @ A @ N ) @ ( formal2580924720334399070h_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% fps_mult_fps_X_plus_1_nth
thf(fact_1058_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1059_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A: real,B2: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
= B2 ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1060_real__vector__affinity__eq,axiom,
! [M: real,X: real,C: real,Y: real] :
( ( M != zero_zero_real )
=> ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C )
= Y )
= ( X
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_1061_real__vector__eq__affinity,axiom,
! [M: real,Y: real,X: real,C: real] :
( ( M != zero_zero_real )
=> ( ( Y
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C ) )
= ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) )
= X ) ) ) ).
% real_vector_eq_affinity
thf(fact_1062_fps__integral0__linear,axiom,
! [A: real,F: formal3361831859752904756s_real,B2: real,G: formal3361831859752904756s_real] :
( ( formal8984515926053063617l_real @ ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ A ) @ F ) @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ B2 ) @ G ) ) @ zero_zero_real )
= ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ A ) @ ( formal8984515926053063617l_real @ F @ zero_zero_real ) ) @ ( times_7561426564079326009s_real @ ( formal2098867297714113032t_real @ B2 ) @ ( formal8984515926053063617l_real @ G @ zero_zero_real ) ) ) ) ).
% fps_integral0_linear
thf(fact_1063_fps__integral0__linear2,axiom,
! [F: formal3361831859752904756s_real,A: real,G: formal3361831859752904756s_real,B2: real] :
( ( formal8984515926053063617l_real @ ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ F @ ( formal2098867297714113032t_real @ A ) ) @ ( times_7561426564079326009s_real @ G @ ( formal2098867297714113032t_real @ B2 ) ) ) @ zero_zero_real )
= ( plus_p6008488439947570109s_real @ ( times_7561426564079326009s_real @ ( formal8984515926053063617l_real @ F @ zero_zero_real ) @ ( formal2098867297714113032t_real @ A ) ) @ ( times_7561426564079326009s_real @ ( formal8984515926053063617l_real @ G @ zero_zero_real ) @ ( formal2098867297714113032t_real @ B2 ) ) ) ) ).
% fps_integral0_linear2
thf(fact_1064_fps__cos__add,axiom,
! [A: real,B2: real] :
( ( formal461277676486907980s_real @ ( plus_plus_real @ A @ B2 ) )
= ( minus_6791916864952032525s_real @ ( times_7561426564079326009s_real @ ( formal461277676486907980s_real @ A ) @ ( formal461277676486907980s_real @ B2 ) ) @ ( times_7561426564079326009s_real @ ( formal6437758938379178589n_real @ A ) @ ( formal6437758938379178589n_real @ B2 ) ) ) ) ).
% fps_cos_add
thf(fact_1065_fps__mult__nth__1,axiom,
! [F: formal_Power_fps_nat,G: formal_Power_fps_nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ G ) @ one_one_nat )
= ( plus_plus_nat @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ zero_zero_nat ) @ ( formal3720337525774269570th_nat @ G @ one_one_nat ) ) @ ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ one_one_nat ) @ ( formal3720337525774269570th_nat @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1
thf(fact_1066_fps__mult__nth__1,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ G ) @ one_one_nat )
= ( plus_plus_real @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ one_one_nat ) ) @ ( times_times_real @ ( formal2580924720334399070h_real @ F @ one_one_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1
thf(fact_1067_fps__mult__nth__1,axiom,
! [F: formal_Power_fps_int,G: formal_Power_fps_int] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ G ) @ one_one_nat )
= ( plus_plus_int @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ zero_zero_nat ) @ ( formal3717847055265219294th_int @ G @ one_one_nat ) ) @ ( times_times_int @ ( formal3717847055265219294th_int @ F @ one_one_nat ) @ ( formal3717847055265219294th_int @ G @ zero_zero_nat ) ) ) ) ).
% fps_mult_nth_1
thf(fact_1068_fps__deriv__eq__iff,axiom,
! [F: formal3361831859752904756s_real,G: formal3361831859752904756s_real] :
( ( ( formal4557910837323084707v_real @ F )
= ( formal4557910837323084707v_real @ G ) )
= ( F
= ( plus_p6008488439947570109s_real @ ( formal2098867297714113032t_real @ ( minus_minus_real @ ( formal2580924720334399070h_real @ F @ zero_zero_nat ) @ ( formal2580924720334399070h_real @ G @ zero_zero_nat ) ) ) @ G ) ) ) ).
% fps_deriv_eq_iff
thf(fact_1069_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1070_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1071_real__eq__affinity,axiom,
! [M: real,Y: real,X: real,C: real] :
( ( M != zero_zero_real )
=> ( ( Y
= ( plus_plus_real @ ( times_times_real @ M @ X ) @ C ) )
= ( ( plus_plus_real @ ( times_times_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( uminus_uminus_real @ ( divide_divide_real @ C @ M ) ) )
= X ) ) ) ).
% real_eq_affinity
thf(fact_1072_real__affinity__eq,axiom,
! [M: real,X: real,C: real,Y: real] :
( ( M != zero_zero_real )
=> ( ( ( plus_plus_real @ ( times_times_real @ M @ X ) @ C )
= Y )
= ( X
= ( plus_plus_real @ ( times_times_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( uminus_uminus_real @ ( divide_divide_real @ C @ M ) ) ) ) ) ) ).
% real_affinity_eq
thf(fact_1073_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1074_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_1075_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_1076_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_1077_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_1078_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_1079_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_1080_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_1081_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_1082_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1083_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1084_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_1085_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1086_Euclid__induct,axiom,
! [P2: nat > nat > $o,A: nat,B2: nat] :
( ! [A3: nat,B3: nat] :
( ( P2 @ A3 @ B3 )
= ( P2 @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ A3 @ B3 )
=> ( P2 @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P2 @ A @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1087_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_1088_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B2: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B2 )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y ) @ D ) )
| ( ( times_times_nat @ B2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
=> ? [X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B2 ) )
& ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ Y2 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1089_bezout__add__nat,axiom,
! [A: nat,B2: nat] :
? [D3: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B2 )
& ( ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y2 ) @ D3 ) )
| ( ( times_times_nat @ B2 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y2 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_1090_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1091_bezout__add__strong__nat,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ? [D3: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B2 )
& ( ( times_times_nat @ A @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B2 @ Y2 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1092_div__mod__decomp,axiom,
! [A5: nat,N: nat] :
( A5
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A5 @ N ) @ N ) @ ( modulo_modulo_nat @ A5 @ N ) ) ) ).
% div_mod_decomp
thf(fact_1093_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1094_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1095_i__nom__i,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ i @ M ) @ ( semiri5074537144036343181t_real @ M ) ) )
= ( powr_real @ ( plus_plus_real @ one_one_real @ i ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% i_nom_i
thf(fact_1096_i__force__def,axiom,
( i_force
= ( ^ [I2: real] : ( ln_ln_real @ ( plus_plus_real @ one_one_real @ I2 ) ) ) ) ).
% i_force_def
thf(fact_1097_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_1098_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_1106_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_1107_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_1108_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_1109_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_1110_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_1111_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_1112_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_1113_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_1114_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1115_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_1116_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_fps__nth__of__nat,axiom,
! [N: nat,C: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C ) @ N )
= ( semiri1316708129612266289at_nat @ C ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3720337525774269570th_nat @ ( semiri1524631719018205113ps_nat @ C ) @ N )
= zero_zero_nat ) ) ) ).
% fps_nth_of_nat
thf(fact_1124_fps__nth__of__nat,axiom,
! [N: nat,C: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C ) @ N )
= ( semiri5074537144036343181t_real @ C ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( semiri2475410149736220053s_real @ C ) @ N )
= zero_zero_real ) ) ) ).
% fps_nth_of_nat
thf(fact_1125_fps__nth__of__nat,axiom,
! [N: nat,C: nat] :
( ( ( N = zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C ) @ N )
= ( semiri1314217659103216013at_int @ C ) ) )
& ( ( N != zero_zero_nat )
=> ( ( formal3717847055265219294th_int @ ( semiri6570152736363784213ps_int @ C ) @ N )
= zero_zero_int ) ) ) ).
% fps_nth_of_nat
thf(fact_1126_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ ( semiri1524631719018205113ps_nat @ K ) @ F ) @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ K ) @ ( formal3720337525774269570th_nat @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_1127_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ ( semiri2475410149736220053s_real @ K ) @ F ) @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( formal2580924720334399070h_real @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_1128_fps__mult__of__nat__nth_I1_J,axiom,
! [K: nat,F: formal_Power_fps_int,N: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ ( semiri6570152736363784213ps_int @ K ) @ F ) @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ ( formal3717847055265219294th_int @ F @ N ) ) ) ).
% fps_mult_of_nat_nth(1)
thf(fact_1129_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal_Power_fps_nat,K: nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( times_7269705568686124893ps_nat @ F @ ( semiri1524631719018205113ps_nat @ K ) ) @ N )
= ( times_times_nat @ ( formal3720337525774269570th_nat @ F @ N ) @ ( semiri1316708129612266289at_nat @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_1130_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal3361831859752904756s_real,K: nat,N: nat] :
( ( formal2580924720334399070h_real @ ( times_7561426564079326009s_real @ F @ ( semiri2475410149736220053s_real @ K ) ) @ N )
= ( times_times_real @ ( formal2580924720334399070h_real @ F @ N ) @ ( semiri5074537144036343181t_real @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_1131_fps__mult__of__nat__nth_I2_J,axiom,
! [F: formal_Power_fps_int,K: nat,N: nat] :
( ( formal3717847055265219294th_int @ ( times_3091854549176928185ps_int @ F @ ( semiri6570152736363784213ps_int @ K ) ) @ N )
= ( times_times_int @ ( formal3717847055265219294th_int @ F @ N ) @ ( semiri1314217659103216013at_int @ K ) ) ) ).
% fps_mult_of_nat_nth(2)
thf(fact_1132_fps__XD__nth,axiom,
! [A: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( formal814923487339530757XD_nat @ A ) @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( formal3720337525774269570th_nat @ A @ N ) ) ) ).
% fps_XD_nth
thf(fact_1133_fps__XD__nth,axiom,
! [A: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( formal4292469010823635553D_real @ A ) @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( formal2580924720334399070h_real @ A @ N ) ) ) ).
% fps_XD_nth
thf(fact_1134_fps__XD__nth,axiom,
! [A: formal_Power_fps_int,N: nat] :
( ( formal3717847055265219294th_int @ ( formal812433016830480481XD_int @ A ) @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( formal3717847055265219294th_int @ A @ N ) ) ) ).
% fps_XD_nth
thf(fact_1135_fps__XDp__nth,axiom,
! [C: nat,A: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( formal9197787955091086413Dp_nat @ C @ A ) @ N )
= ( times_times_nat @ ( plus_plus_nat @ C @ ( semiri1316708129612266289at_nat @ N ) ) @ ( formal3720337525774269570th_nat @ A @ N ) ) ) ).
% fps_XDp_nth
thf(fact_1136_fps__XDp__nth,axiom,
! [C: real,A: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( formal2839450981996073129p_real @ C @ A ) @ N )
= ( times_times_real @ ( plus_plus_real @ C @ ( semiri5074537144036343181t_real @ N ) ) @ ( formal2580924720334399070h_real @ A @ N ) ) ) ).
% fps_XDp_nth
thf(fact_1137_fps__XDp__nth,axiom,
! [C: int,A: formal_Power_fps_int,N: nat] :
( ( formal3717847055265219294th_int @ ( formal9195297484582036137Dp_int @ C @ A ) @ N )
= ( times_times_int @ ( plus_plus_int @ C @ ( semiri1314217659103216013at_int @ N ) ) @ ( formal3717847055265219294th_int @ A @ N ) ) ) ).
% fps_XDp_nth
thf(fact_1138_fps__deriv__nth,axiom,
! [F: formal_Power_fps_nat,N: nat] :
( ( formal3720337525774269570th_nat @ ( formal4464462342499834951iv_nat @ F ) @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( formal3720337525774269570th_nat @ F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ).
% fps_deriv_nth
thf(fact_1139_fps__deriv__nth,axiom,
! [F: formal3361831859752904756s_real,N: nat] :
( ( formal2580924720334399070h_real @ ( formal4557910837323084707v_real @ F ) @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( formal2580924720334399070h_real @ F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ).
% fps_deriv_nth
thf(fact_1140_fps__deriv__nth,axiom,
! [F: formal_Power_fps_int,N: nat] :
( ( formal3717847055265219294th_int @ ( formal4461971871990784675iv_int @ F ) @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( formal3717847055265219294th_int @ F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ).
% fps_deriv_nth
thf(fact_1141_mult__inverse__of__nat__commute,axiom,
! [Xa: nat,X: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_1142_fps__of__nat,axiom,
! [C: nat] :
( ( formal2098867297714113032t_real @ ( semiri5074537144036343181t_real @ C ) )
= ( semiri2475410149736220053s_real @ C ) ) ).
% fps_of_nat
thf(fact_1143_fps__of__nat,axiom,
! [C: nat] :
( ( formal5284259319228341128st_int @ ( semiri1314217659103216013at_int @ C ) )
= ( semiri6570152736363784213ps_int @ C ) ) ).
% fps_of_nat
thf(fact_1144_inverse__fps__of__nat,axiom,
! [N: nat] :
( ( invers68952373231134600s_real @ ( semiri2475410149736220053s_real @ N ) )
= ( formal2098867297714113032t_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% inverse_fps_of_nat
thf(fact_1145_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_1146_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_1147_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_1148_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_1149_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_1150_i__nom__def,axiom,
( i_nom
= ( ^ [I2: real,M2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( minus_minus_real @ ( powr_real @ ( plus_plus_real @ one_one_real @ I2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) @ one_one_real ) ) ) ) ).
% i_nom_def
thf(fact_1151_interest_Oi__nom__i,axiom,
! [I: real,M: nat] :
( ( interest @ I )
=> ( ( M != zero_zero_nat )
=> ( ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ I @ M ) @ ( semiri5074537144036343181t_real @ M ) ) )
= ( powr_real @ ( plus_plus_real @ one_one_real @ I ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ).
% interest.i_nom_i
thf(fact_1152_v__futr__m__pos,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ i @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% v_futr_m_pos
thf(fact_1153_i__nom__eff,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ ( i_nom @ i @ M ) @ ( semiri5074537144036343181t_real @ M ) ) ) @ M )
= ( plus_plus_real @ one_one_real @ i ) ) ) ).
% i_nom_eff
thf(fact_1154_v__futr__pos,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ i ) ).
% v_futr_pos
thf(fact_1155_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1156_add__less__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1157_add__less__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( ord_less_real @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1158_add__less__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1159_add__less__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
= ( ord_less_real @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1160_neg__less__iff__less,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B2 ) ) ).
% neg_less_iff_less
thf(fact_1161_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_1162_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_1163_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_1164_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_1165_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_1166_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1167_fps__const__power,axiom,
! [C: real,N: nat] :
( ( power_1846127563762588094s_real @ ( formal2098867297714113032t_real @ C ) @ N )
= ( formal2098867297714113032t_real @ ( power_power_real @ C @ N ) ) ) ).
% fps_const_power
thf(fact_1168_add__less__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1169_add__less__same__cancel1,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1170_add__less__same__cancel1,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_1171_add__less__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1172_add__less__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1173_add__less__same__cancel2,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_1174_less__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1175_less__add__same__cancel1,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1176_less__add__same__cancel1,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B2 ) )
= ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1177_less__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1178_less__add__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1179_less__add__same__cancel2,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B2 @ A ) )
= ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1180_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1181_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1182_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1183_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1184_diff__gt__0__iff__gt,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
= ( ord_less_int @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1185_diff__gt__0__iff__gt,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
= ( ord_less_real @ B2 @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1186_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_1187_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_1188_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_1189_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_1190_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_1191_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_1192_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1193_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1194_power__strict__increasing__iff,axiom,
! [B2: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B2 )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1195_power__strict__increasing__iff,axiom,
! [B2: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B2 )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1196_power__strict__increasing__iff,axiom,
! [B2: real,X: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B2 )
=> ( ( ord_less_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1197_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1198_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1199_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1200_inverse__less__iff__less,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
= ( ord_less_real @ B2 @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_1201_inverse__less__iff__less__neg,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
= ( ord_less_real @ B2 @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_1202_inverse__negative__iff__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_1203_inverse__positive__iff__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_1204_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1205_scaleR__power,axiom,
! [X: real,Y: real,N: nat] :
( ( power_power_real @ ( real_V1485227260804924795R_real @ X @ Y ) @ N )
= ( real_V1485227260804924795R_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).
% scaleR_power
thf(fact_1206_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1207_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1208_powr__gt__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
= ( X != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_1209_powr__less__cancel__iff,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) )
= ( ord_less_real @ A @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_1210_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_1211_less__divide__eq__1__pos,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
= ( ord_less_real @ A @ B2 ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1212_less__divide__eq__1__neg,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
= ( ord_less_real @ B2 @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1213_divide__less__eq__1__pos,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
= ( ord_less_real @ B2 @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1214_divide__less__eq__1__neg,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
= ( ord_less_real @ A @ B2 ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1215_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_1216_power__strict__decreasing__iff,axiom,
! [B2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_nat @ B2 @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1217_power__strict__decreasing__iff,axiom,
! [B2: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ B2 @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1218_power__strict__decreasing__iff,axiom,
! [B2: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_real @ B2 @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1219_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_1220_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_1221_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_1222_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_1223_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_1224_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_1225_left__minus__one__mult__self,axiom,
! [N: nat,A: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_1226_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1227_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_1228_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1229_powr__eq__one__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( powr_real @ A @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_1230_powr__eq__one__iff__gen,axiom,
! [A: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ( powr_real @ A @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ) ) ).
% powr_eq_one_iff_gen
thf(fact_1231_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1232_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1233_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1234_fps__radical__power__nth,axiom,
! [R: nat > real > real,K: nat,A: formal3361831859752904756s_real] :
( ( ( power_power_real @ ( R @ K @ ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) @ K )
= ( formal2580924720334399070h_real @ A @ zero_zero_nat ) )
=> ( ( ( K = zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ ( formal8604817403481219167l_real @ R @ K @ A ) @ K ) @ zero_zero_nat )
= one_one_real ) )
& ( ( K != zero_zero_nat )
=> ( ( formal2580924720334399070h_real @ ( power_1846127563762588094s_real @ ( formal8604817403481219167l_real @ R @ K @ A ) @ K ) @ zero_zero_nat )
= ( formal2580924720334399070h_real @ A @ zero_zero_nat ) ) ) ) ) ).
% fps_radical_power_nth
thf(fact_1235_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1236_add__less__imp__less__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
=> ( ord_less_real @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1237_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1238_add__less__imp__less__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
=> ( ord_less_real @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1239_add__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1240_add__strict__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1241_add__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1242_add__strict__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1243_add__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1244_add__strict__mono,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1245_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1246_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1247_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1248_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1249_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1250_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1251_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_1252_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_1253_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_1254_add__neg__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1255_add__neg__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1256_add__neg__neg,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_1257_add__pos__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_1258_add__pos__pos,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_1259_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_1260_int__ops_I7_J,axiom,
! [A: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1261_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1262_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1263_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_1264_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
= ( power_power_real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_1265_zmod__int,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zmod_int
thf(fact_1266_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1267_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1268_powr__less__mono,axiom,
! [A: real,B2: real,X: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_1269_powr__less__cancel,axiom,
! [X: real,A: real,B2: real] :
( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B2 ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_1270_powr__less__cancel2,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) )
=> ( ord_less_real @ X @ Y ) ) ) ) ) ).
% powr_less_cancel2
thf(fact_1271_powr__non__neg,axiom,
! [A: real,X: real] :
~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_1272_powr__less__mono2__neg,axiom,
! [A: real,X: real,Y: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_1273_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( i_nom @ i @ m )
= zero_zero_real ) ).
%------------------------------------------------------------------------------