TPTP Problem File: SLH0880^1.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 : Risk_Free_Lending/0000_Risk_Free_Lending/prob_00595_018552__5839830_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1414 ( 557 unt; 142 typ; 0 def)
% Number of atoms : 3814 (1208 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10424 ( 346 ~; 88 |; 215 &;8026 @)
% ( 0 <=>;1749 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 950 ( 950 >; 0 *; 0 +; 0 <<)
% Number of symbols : 133 ( 130 usr; 18 con; 0-3 aty)
% Number of variables : 3631 ( 357 ^;3145 !; 129 ?;3631 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:57:01.679
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
set_Ri1641125681238393385ccount: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
set_nat_real: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Risk____Free____Lending__Oaccount,type,
risk_Free_account: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (130)
thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
abs_abs_complex: complex > complex ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_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_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Risk____Free____Lending__Oaccount,type,
uminus3377898441596595772ccount: risk_Free_account > risk_Free_account ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
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_Groups_Ozero__class_Ozero_001t__Risk____Free____Lending__Oaccount,type,
zero_z1425366712893667068ccount: risk_Free_account ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_It__Nat__Onat_Mt__Real__Oreal_J_001t__Complex__Ocomplex,type,
groups2834896168486368005omplex: ( ( nat > real ) > complex ) > set_nat_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_It__Nat__Onat_Mt__Real__Oreal_J_001t__Real__Oreal,type,
groups4253619806861319043l_real: ( ( nat > real ) > real ) > set_nat_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Real__Oreal,type,
groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Risk____Free____Lending__Oaccount,type,
groups6146125110001601245ccount: ( complex > risk_Free_account ) > set_complex > risk_Free_account ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Risk____Free____Lending__Oaccount,type,
groups6033208628184776703ccount: ( nat > risk_Free_account ) > set_nat > risk_Free_account ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Risk____Free____Lending__Oaccount,type,
groups8516999891779824987ccount: ( real > risk_Free_account ) > set_real > risk_Free_account ).
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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
ord_less_nat_real_o: ( ( nat > real ) > $o ) > ( ( nat > real ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Complex__Ocomplex_M_Eo_J,type,
ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Risk____Free____Lending__Oaccount,type,
ord_le2131251472502387783ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
ord_le3527643927072297637t_real: set_nat_real > set_nat_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
ord_le7676461544873280788real_o: ( ( nat > real ) > $o ) > ( ( nat > real ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Complex__Ocomplex_M_Eo_J,type,
ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
ord_less_eq_nat_real: ( nat > real ) > ( nat > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Risk____Free____Lending__Oaccount,type,
ord_le4245800335709223507ccount: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
ord_le2908806416726583473t_real: set_nat_real > set_nat_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Risk____Free____Lending__Oaccount_J,type,
ord_le4487465848215339657ccount: set_Ri1641125681238393385ccount > set_Ri1641125681238393385ccount > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
top_top_nat_real_o: ( nat > real ) > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Complex__Ocomplex_M_Eo_J,type,
top_top_complex_o: complex > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
top_top_real_o: real > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
top_top_set_nat_real: set_nat_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Complex__Ocomplex_J,type,
top_top_set_complex: set_complex ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
top_top_set_real: set_real ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Ordinal__Arithmetic_Ofin__support_001t__Real__Oreal_001t__Nat__Onat,type,
ordina1579063754167848977al_nat: real > set_nat > set_nat_real ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
real_V1022390504157884413omplex: complex > real ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
real_V7735802525324610683m_real: real > real ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
real_V4546457046886955230omplex: real > complex ).
thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
real_V1803761363581548252l_real: real > real ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V1485227260804924795R_real: real > real > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Risk__Free__Lending_Oaccount_OAbs__account,type,
risk_F5458100604530014700ccount: ( nat > real ) > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Oaccount_ORep__account,type,
risk_F170160801229183585ccount: risk_Free_account > nat > real ).
thf(sy_c_Risk__Free__Lending_Ocash__reserve,type,
risk_F1914734008469130493eserve: risk_Free_account > real ).
thf(sy_c_Risk__Free__Lending_Ojust__cash,type,
risk_Free_just_cash: real > risk_Free_account ).
thf(sy_c_Risk__Free__Lending_Ostrictly__solvent,type,
risk_F1636578016437888323olvent: risk_Free_account > $o ).
thf(sy_c_Risk__Free__Lending_Ovalid__transfer,type,
risk_F1023690899723030139ansfer: risk_Free_account > risk_Free_account > $o ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
collect_nat_real: ( ( nat > real ) > $o ) > set_nat_real ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Risk____Free____Lending__Oaccount,type,
collec1856553087948576712ccount: ( risk_Free_account > $o ) > set_Ri1641125681238393385ccount ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
set_or1122926678442080148t_real: ( nat > real ) > set_nat_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Risk____Free____Lending__Oaccount,type,
set_or3854930313887350124ccount: risk_Free_account > set_Ri1641125681238393385ccount ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: 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_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Typedef_Otype__definition_001t__Risk____Free____Lending__Oaccount_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
type_d8982087200295354172t_real: ( risk_Free_account > nat > real ) > ( ( nat > real ) > risk_Free_account ) > set_nat_real > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
member_nat_real: ( nat > real ) > set_nat_real > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Risk____Free____Lending__Oaccount,type,
member5612106785598075018ccount: risk_Free_account > set_Ri1641125681238393385ccount > $o ).
thf(sy_v_a,type,
a: real ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1264)
thf(fact_0_Rep__account__inject,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ( risk_F170160801229183585ccount @ X )
= ( risk_F170160801229183585ccount @ Y ) )
= ( X = Y ) ) ).
% Rep_account_inject
thf(fact_1_just__cash__embed,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A: real,B: real] :
( ( risk_Free_just_cash @ A )
= ( risk_Free_just_cash @ B ) ) ) ) ).
% just_cash_embed
thf(fact_2_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_3_sum_Oswap,axiom,
! [G: nat > nat > real,B2: set_nat,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups6591440286371151544t_real @ ( G @ I ) @ B2 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_4_sum_Oswap,axiom,
! [G: complex > complex > complex,B2: set_complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups7754918857620584856omplex @ ( G @ I ) @ B2 )
@ A2 )
= ( groups7754918857620584856omplex
@ ^ [J: complex] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_5_sum_Oswap,axiom,
! [G: nat > real > real,B2: set_real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups8097168146408367636l_real @ ( G @ I ) @ B2 )
@ A2 )
= ( groups8097168146408367636l_real
@ ^ [J: real] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_6_sum_Oswap,axiom,
! [G: complex > real > complex,B2: set_real,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups5754745047067104278omplex @ ( G @ I ) @ B2 )
@ A2 )
= ( groups5754745047067104278omplex
@ ^ [J: real] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_7_sum_Oswap,axiom,
! [G: real > complex > complex,B2: set_complex,A2: set_real] :
( ( groups5754745047067104278omplex
@ ^ [I: real] : ( groups7754918857620584856omplex @ ( G @ I ) @ B2 )
@ A2 )
= ( groups7754918857620584856omplex
@ ^ [J: complex] :
( groups5754745047067104278omplex
@ ^ [I: real] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_8_sum_Oswap,axiom,
! [G: real > real > complex,B2: set_real,A2: set_real] :
( ( groups5754745047067104278omplex
@ ^ [I: real] : ( groups5754745047067104278omplex @ ( G @ I ) @ B2 )
@ A2 )
= ( groups5754745047067104278omplex
@ ^ [J: real] :
( groups5754745047067104278omplex
@ ^ [I: real] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_9_sum_Oswap,axiom,
! [G: real > nat > real,B2: set_nat,A2: set_real] :
( ( groups8097168146408367636l_real
@ ^ [I: real] : ( groups6591440286371151544t_real @ ( G @ I ) @ B2 )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [J: nat] :
( groups8097168146408367636l_real
@ ^ [I: real] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_10_sum_Oswap,axiom,
! [G: real > real > real,B2: set_real,A2: set_real] :
( ( groups8097168146408367636l_real
@ ^ [I: real] : ( groups8097168146408367636l_real @ ( G @ I ) @ B2 )
@ A2 )
= ( groups8097168146408367636l_real
@ ^ [J: real] :
( groups8097168146408367636l_real
@ ^ [I: real] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_11_sum_Oswap,axiom,
! [G: nat > ( nat > real ) > real,B2: set_nat_real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I: nat] : ( groups4253619806861319043l_real @ ( G @ I ) @ B2 )
@ A2 )
= ( groups4253619806861319043l_real
@ ^ [J: nat > real] :
( groups6591440286371151544t_real
@ ^ [I: nat] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_12_sum_Oswap,axiom,
! [G: complex > ( nat > real ) > complex,B2: set_nat_real,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups2834896168486368005omplex @ ( G @ I ) @ B2 )
@ A2 )
= ( groups2834896168486368005omplex
@ ^ [J: nat > real] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ J )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_13_sum_Ocong,axiom,
! [A2: set_nat_real,B2: set_nat_real,G: ( nat > real ) > complex,H: ( nat > real ) > complex] :
( ( A2 = B2 )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups2834896168486368005omplex @ G @ A2 )
= ( groups2834896168486368005omplex @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_14_sum_Ocong,axiom,
! [A2: set_real,B2: set_real,G: real > complex,H: real > complex] :
( ( A2 = B2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ A2 )
= ( groups5754745047067104278omplex @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_15_sum_Ocong,axiom,
! [A2: set_nat_real,B2: set_nat_real,G: ( nat > real ) > real,H: ( nat > real ) > real] :
( ( A2 = B2 )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4253619806861319043l_real @ G @ A2 )
= ( groups4253619806861319043l_real @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_16_sum_Ocong,axiom,
! [A2: set_real,B2: set_real,G: real > real,H: real > real] :
( ( A2 = B2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ A2 )
= ( groups8097168146408367636l_real @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_17_sum_Ocong,axiom,
! [A2: set_nat,B2: set_nat,G: nat > real,H: nat > real] :
( ( A2 = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= ( groups6591440286371151544t_real @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_18_sum_Ocong,axiom,
! [A2: set_complex,B2: set_complex,G: complex > complex,H: complex > complex] :
( ( A2 = B2 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ B2 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= ( groups7754918857620584856omplex @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_19_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_20_sum_Oeq__general,axiom,
! [B2: set_complex,A2: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_21_sum_Oeq__general,axiom,
! [B2: set_real,A2: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_real @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_22_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_23_sum_Oeq__general,axiom,
! [B2: set_real,A2: set_complex,H: complex > real,Gamma: real > complex,Phi: complex > complex] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B2 )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_real @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups5754745047067104278omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_24_sum_Oeq__general,axiom,
! [B2: set_complex,A2: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B2 )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_25_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_nat_real,H: ( nat > real ) > nat,Gamma: nat > real,Phi: ( nat > real ) > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ? [X3: nat > real] :
( ( member_nat_real @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: nat > real] :
( ( ( member_nat_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4253619806861319043l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_26_sum_Oeq__general,axiom,
! [B2: set_complex,A2: set_nat_real,H: ( nat > real ) > complex,Gamma: complex > complex,Phi: ( nat > real ) > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B2 )
=> ? [X3: nat > real] :
( ( member_nat_real @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: nat > real] :
( ( ( member_nat_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2834896168486368005omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_27_sum_Oeq__general,axiom,
! [B2: set_nat_real,A2: set_nat,H: nat > nat > real,Gamma: ( nat > real ) > real,Phi: nat > real] :
( ! [Y3: nat > real] :
( ( member_nat_real @ Y3 @ B2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat_real @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups4253619806861319043l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_28_sum_Oeq__general,axiom,
! [B2: set_nat_real,A2: set_complex,H: complex > nat > real,Gamma: ( nat > real ) > complex,Phi: complex > complex] :
( ! [Y3: nat > real] :
( ( member_nat_real @ Y3 @ B2 )
=> ? [X3: complex] :
( ( member_complex @ X3 @ A2 )
& ( ( H @ X3 )
= Y3 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y3 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_nat_real @ ( H @ X2 ) @ B2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups2834896168486368005omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_29_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > real,A2: set_real,H: real > nat,Gamma: nat > real,Phi: real > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_real @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_30_sum_Oeq__general__inverses,axiom,
! [B2: set_complex,K: complex > real,A2: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B2 )
=> ( ( member_real @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_31_sum_Oeq__general__inverses,axiom,
! [B2: set_real,K: real > nat,A2: set_nat,H: nat > real,Gamma: real > real,Phi: nat > real] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B2 )
=> ( ( member_nat @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_real @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_32_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > real,Phi: nat > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_nat @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_33_sum_Oeq__general__inverses,axiom,
! [B2: set_real,K: real > complex,A2: set_complex,H: complex > real,Gamma: real > complex,Phi: complex > complex] :
( ! [Y3: real] :
( ( member_real @ Y3 @ B2 )
=> ( ( member_complex @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_real @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups5754745047067104278omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_34_sum_Oeq__general__inverses,axiom,
! [B2: set_complex,K: complex > complex,A2: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B2 )
=> ( ( member_complex @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_35_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K: nat > nat > real,A2: set_nat_real,H: ( nat > real ) > nat,Gamma: nat > real,Phi: ( nat > real ) > real] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B2 )
=> ( ( member_nat_real @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A2 )
=> ( ( member_nat @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4253619806861319043l_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_36_sum_Oeq__general__inverses,axiom,
! [B2: set_complex,K: complex > nat > real,A2: set_nat_real,H: ( nat > real ) > complex,Gamma: complex > complex,Phi: ( nat > real ) > complex] :
( ! [Y3: complex] :
( ( member_complex @ Y3 @ B2 )
=> ( ( member_nat_real @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A2 )
=> ( ( member_complex @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups2834896168486368005omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_37_sum_Oeq__general__inverses,axiom,
! [B2: set_nat_real,K: ( nat > real ) > nat,A2: set_nat,H: nat > nat > real,Gamma: ( nat > real ) > real,Phi: nat > real] :
( ! [Y3: nat > real] :
( ( member_nat_real @ Y3 @ B2 )
=> ( ( member_nat @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat_real @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups4253619806861319043l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_38_sum_Oeq__general__inverses,axiom,
! [B2: set_nat_real,K: ( nat > real ) > complex,A2: set_complex,H: complex > nat > real,Gamma: ( nat > real ) > complex,Phi: complex > complex] :
( ! [Y3: nat > real] :
( ( member_nat_real @ Y3 @ B2 )
=> ( ( member_complex @ ( K @ Y3 ) @ A2 )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( member_nat_real @ ( H @ X2 ) @ B2 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups2834896168486368005omplex @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_39_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: nat > real,J2: real > nat,T: set_nat,H: nat > real,G: real > real] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_40_sum_Oreindex__bij__witness,axiom,
! [S: set_real,I2: complex > real,J2: real > complex,T: set_complex,H: complex > complex,G: real > complex] :
( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( member_complex @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( member_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_41_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: real > nat,J2: nat > real,T: set_real,H: real > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_real @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8097168146408367636l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_42_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: nat > nat,J2: nat > nat,T: set_nat,H: nat > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_43_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: real > complex,J2: complex > real,T: set_real,H: real > complex,G: complex > complex] :
( ! [A3: complex] :
( ( member_complex @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S )
=> ( member_real @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T )
=> ( member_complex @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S )
= ( groups5754745047067104278omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_44_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: complex > complex,J2: complex > complex,T: set_complex,H: complex > complex,G: complex > complex] :
( ! [A3: complex] :
( ( member_complex @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S )
=> ( member_complex @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( member_complex @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_45_sum_Oreindex__bij__witness,axiom,
! [S: set_nat_real,I2: nat > nat > real,J2: ( nat > real ) > nat,T: set_nat,H: nat > real,G: ( nat > real ) > real] :
( ! [A3: nat > real] :
( ( member_nat_real @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat > real] :
( ( member_nat_real @ A3 @ S )
=> ( member_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T )
=> ( member_nat_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: nat > real] :
( ( member_nat_real @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4253619806861319043l_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_46_sum_Oreindex__bij__witness,axiom,
! [S: set_nat_real,I2: complex > nat > real,J2: ( nat > real ) > complex,T: set_complex,H: complex > complex,G: ( nat > real ) > complex] :
( ! [A3: nat > real] :
( ( member_nat_real @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat > real] :
( ( member_nat_real @ A3 @ S )
=> ( member_complex @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: complex] :
( ( member_complex @ B3 @ T )
=> ( member_nat_real @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: nat > real] :
( ( member_nat_real @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2834896168486368005omplex @ G @ S )
= ( groups7754918857620584856omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_47_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I2: ( nat > real ) > nat,J2: nat > nat > real,T: set_nat_real,H: ( nat > real ) > real,G: nat > real] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat_real @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: nat > real] :
( ( member_nat_real @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat > real] :
( ( member_nat_real @ B3 @ T )
=> ( member_nat @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups4253619806861319043l_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_48_sum_Oreindex__bij__witness,axiom,
! [S: set_complex,I2: ( nat > real ) > complex,J2: complex > nat > real,T: set_nat_real,H: ( nat > real ) > complex,G: complex > complex] :
( ! [A3: complex] :
( ( member_complex @ A3 @ S )
=> ( ( I2 @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S )
=> ( member_nat_real @ ( J2 @ A3 ) @ T ) )
=> ( ! [B3: nat > real] :
( ( member_nat_real @ B3 @ T )
=> ( ( J2 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: nat > real] :
( ( member_nat_real @ B3 @ T )
=> ( member_complex @ ( I2 @ B3 ) @ S ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S )
= ( groups2834896168486368005omplex @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_49_Rep__account__inverse,axiom,
! [X: risk_Free_account] :
( ( risk_F5458100604530014700ccount @ ( risk_F170160801229183585ccount @ X ) )
= X ) ).
% Rep_account_inverse
thf(fact_50_less__eq__account__def,axiom,
( ord_le4245800335709223507ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
! [N: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_1 ) @ ( set_ord_atMost_nat @ N ) ) @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha_2 ) @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% less_eq_account_def
thf(fact_51_of__real__sum,axiom,
! [F: complex > real,S2: set_complex] :
( ( real_V4546457046886955230omplex @ ( groups5808333547571424918x_real @ F @ S2 ) )
= ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( real_V4546457046886955230omplex @ ( F @ X4 ) )
@ S2 ) ) ).
% of_real_sum
thf(fact_52_of__real__sum,axiom,
! [F: nat > real,S2: set_nat] :
( ( real_V1803761363581548252l_real @ ( groups6591440286371151544t_real @ F @ S2 ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( real_V1803761363581548252l_real @ ( F @ X4 ) )
@ S2 ) ) ).
% of_real_sum
thf(fact_53_of__real__sum,axiom,
! [F: ( nat > real ) > real,S2: set_nat_real] :
( ( real_V4546457046886955230omplex @ ( groups4253619806861319043l_real @ F @ S2 ) )
= ( groups2834896168486368005omplex
@ ^ [X4: nat > real] : ( real_V4546457046886955230omplex @ ( F @ X4 ) )
@ S2 ) ) ).
% of_real_sum
thf(fact_54_of__real__sum,axiom,
! [F: ( nat > real ) > real,S2: set_nat_real] :
( ( real_V1803761363581548252l_real @ ( groups4253619806861319043l_real @ F @ S2 ) )
= ( groups4253619806861319043l_real
@ ^ [X4: nat > real] : ( real_V1803761363581548252l_real @ ( F @ X4 ) )
@ S2 ) ) ).
% of_real_sum
thf(fact_55_of__real__sum,axiom,
! [F: real > real,S2: set_real] :
( ( real_V4546457046886955230omplex @ ( groups8097168146408367636l_real @ F @ S2 ) )
= ( groups5754745047067104278omplex
@ ^ [X4: real] : ( real_V4546457046886955230omplex @ ( F @ X4 ) )
@ S2 ) ) ).
% of_real_sum
thf(fact_56_of__real__sum,axiom,
! [F: real > real,S2: set_real] :
( ( real_V1803761363581548252l_real @ ( groups8097168146408367636l_real @ F @ S2 ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1803761363581548252l_real @ ( F @ X4 ) )
@ S2 ) ) ).
% of_real_sum
thf(fact_57_scaleR__left_Osum,axiom,
! [G: complex > real,A2: set_complex,X: complex] :
( ( real_V2046097035970521341omplex @ ( groups5808333547571424918x_real @ G @ A2 ) @ X )
= ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( real_V2046097035970521341omplex @ ( G @ X4 ) @ X )
@ A2 ) ) ).
% scaleR_left.sum
thf(fact_58_scaleR__left_Osum,axiom,
! [G: ( nat > real ) > real,A2: set_nat_real,X: complex] :
( ( real_V2046097035970521341omplex @ ( groups4253619806861319043l_real @ G @ A2 ) @ X )
= ( groups2834896168486368005omplex
@ ^ [X4: nat > real] : ( real_V2046097035970521341omplex @ ( G @ X4 ) @ X )
@ A2 ) ) ).
% scaleR_left.sum
thf(fact_59_scaleR__left_Osum,axiom,
! [G: real > real,A2: set_real,X: complex] :
( ( real_V2046097035970521341omplex @ ( groups8097168146408367636l_real @ G @ A2 ) @ X )
= ( groups5754745047067104278omplex
@ ^ [X4: real] : ( real_V2046097035970521341omplex @ ( G @ X4 ) @ X )
@ A2 ) ) ).
% scaleR_left.sum
thf(fact_60_scaleR__left_Osum,axiom,
! [G: nat > real,A2: set_nat,X: real] :
( ( real_V1485227260804924795R_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ X )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( real_V1485227260804924795R_real @ ( G @ X4 ) @ X )
@ A2 ) ) ).
% scaleR_left.sum
thf(fact_61_scaleR__left_Osum,axiom,
! [G: ( nat > real ) > real,A2: set_nat_real,X: real] :
( ( real_V1485227260804924795R_real @ ( groups4253619806861319043l_real @ G @ A2 ) @ X )
= ( groups4253619806861319043l_real
@ ^ [X4: nat > real] : ( real_V1485227260804924795R_real @ ( G @ X4 ) @ X )
@ A2 ) ) ).
% scaleR_left.sum
thf(fact_62_scaleR__left_Osum,axiom,
! [G: real > real,A2: set_real,X: real] :
( ( real_V1485227260804924795R_real @ ( groups8097168146408367636l_real @ G @ A2 ) @ X )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ ( G @ X4 ) @ X )
@ A2 ) ) ).
% scaleR_left.sum
thf(fact_63_scaleR__sum__left,axiom,
! [F: complex > real,A2: set_complex,X: complex] :
( ( real_V2046097035970521341omplex @ ( groups5808333547571424918x_real @ F @ A2 ) @ X )
= ( groups7754918857620584856omplex
@ ^ [A: complex] : ( real_V2046097035970521341omplex @ ( F @ A ) @ X )
@ A2 ) ) ).
% scaleR_sum_left
thf(fact_64_scaleR__sum__left,axiom,
! [F: ( nat > real ) > real,A2: set_nat_real,X: complex] :
( ( real_V2046097035970521341omplex @ ( groups4253619806861319043l_real @ F @ A2 ) @ X )
= ( groups2834896168486368005omplex
@ ^ [A: nat > real] : ( real_V2046097035970521341omplex @ ( F @ A ) @ X )
@ A2 ) ) ).
% scaleR_sum_left
thf(fact_65_scaleR__sum__left,axiom,
! [F: real > real,A2: set_real,X: complex] :
( ( real_V2046097035970521341omplex @ ( groups8097168146408367636l_real @ F @ A2 ) @ X )
= ( groups5754745047067104278omplex
@ ^ [A: real] : ( real_V2046097035970521341omplex @ ( F @ A ) @ X )
@ A2 ) ) ).
% scaleR_sum_left
thf(fact_66_scaleR__sum__left,axiom,
! [F: nat > real,A2: set_nat,X: real] :
( ( real_V1485227260804924795R_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ X )
= ( groups6591440286371151544t_real
@ ^ [A: nat] : ( real_V1485227260804924795R_real @ ( F @ A ) @ X )
@ A2 ) ) ).
% scaleR_sum_left
thf(fact_67_scaleR__sum__left,axiom,
! [F: ( nat > real ) > real,A2: set_nat_real,X: real] :
( ( real_V1485227260804924795R_real @ ( groups4253619806861319043l_real @ F @ A2 ) @ X )
= ( groups4253619806861319043l_real
@ ^ [A: nat > real] : ( real_V1485227260804924795R_real @ ( F @ A ) @ X )
@ A2 ) ) ).
% scaleR_sum_left
thf(fact_68_scaleR__sum__left,axiom,
! [F: real > real,A2: set_real,X: real] :
( ( real_V1485227260804924795R_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ X )
= ( groups8097168146408367636l_real
@ ^ [A: real] : ( real_V1485227260804924795R_real @ ( F @ A ) @ X )
@ A2 ) ) ).
% scaleR_sum_left
thf(fact_69_atMost__subset__iff,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4487465848215339657ccount @ ( set_or3854930313887350124ccount @ X ) @ ( set_or3854930313887350124ccount @ Y ) )
= ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_70_atMost__subset__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X ) @ ( set_ord_atMost_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_71_atMost__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_72_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_73_atMost__iff,axiom,
! [I2: nat > real,K: nat > real] :
( ( member_nat_real @ I2 @ ( set_or1122926678442080148t_real @ K ) )
= ( ord_less_eq_nat_real @ I2 @ K ) ) ).
% atMost_iff
thf(fact_74_atMost__iff,axiom,
! [I2: risk_Free_account,K: risk_Free_account] :
( ( member5612106785598075018ccount @ I2 @ ( set_or3854930313887350124ccount @ K ) )
= ( ord_le4245800335709223507ccount @ I2 @ K ) ) ).
% atMost_iff
thf(fact_75_atMost__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I2 @ K ) ) ).
% atMost_iff
thf(fact_76_atMost__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I2 @ K ) ) ).
% atMost_iff
thf(fact_77_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_78_scaleR__left__commute,axiom,
! [A4: real,B4: real,X: real] :
( ( real_V1485227260804924795R_real @ A4 @ ( real_V1485227260804924795R_real @ B4 @ X ) )
= ( real_V1485227260804924795R_real @ B4 @ ( real_V1485227260804924795R_real @ A4 @ X ) ) ) ).
% scaleR_left_commute
thf(fact_79_scaleR__sum__right,axiom,
! [A4: real,F: complex > complex,A2: set_complex] :
( ( real_V2046097035970521341omplex @ A4 @ ( groups7754918857620584856omplex @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( real_V2046097035970521341omplex @ A4 @ ( F @ X4 ) )
@ A2 ) ) ).
% scaleR_sum_right
thf(fact_80_scaleR__sum__right,axiom,
! [A4: real,F: ( nat > real ) > complex,A2: set_nat_real] :
( ( real_V2046097035970521341omplex @ A4 @ ( groups2834896168486368005omplex @ F @ A2 ) )
= ( groups2834896168486368005omplex
@ ^ [X4: nat > real] : ( real_V2046097035970521341omplex @ A4 @ ( F @ X4 ) )
@ A2 ) ) ).
% scaleR_sum_right
thf(fact_81_scaleR__sum__right,axiom,
! [A4: real,F: real > complex,A2: set_real] :
( ( real_V2046097035970521341omplex @ A4 @ ( groups5754745047067104278omplex @ F @ A2 ) )
= ( groups5754745047067104278omplex
@ ^ [X4: real] : ( real_V2046097035970521341omplex @ A4 @ ( F @ X4 ) )
@ A2 ) ) ).
% scaleR_sum_right
thf(fact_82_scaleR__sum__right,axiom,
! [A4: real,F: nat > real,A2: set_nat] :
( ( real_V1485227260804924795R_real @ A4 @ ( groups6591440286371151544t_real @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( real_V1485227260804924795R_real @ A4 @ ( F @ X4 ) )
@ A2 ) ) ).
% scaleR_sum_right
thf(fact_83_scaleR__sum__right,axiom,
! [A4: real,F: ( nat > real ) > real,A2: set_nat_real] :
( ( real_V1485227260804924795R_real @ A4 @ ( groups4253619806861319043l_real @ F @ A2 ) )
= ( groups4253619806861319043l_real
@ ^ [X4: nat > real] : ( real_V1485227260804924795R_real @ A4 @ ( F @ X4 ) )
@ A2 ) ) ).
% scaleR_sum_right
thf(fact_84_scaleR__sum__right,axiom,
! [A4: real,F: real > real,A2: set_real] :
( ( real_V1485227260804924795R_real @ A4 @ ( groups8097168146408367636l_real @ F @ A2 ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ A4 @ ( F @ X4 ) )
@ A2 ) ) ).
% scaleR_sum_right
thf(fact_85_scaleR__right_Osum,axiom,
! [A4: real,G: complex > complex,A2: set_complex] :
( ( real_V2046097035970521341omplex @ A4 @ ( groups7754918857620584856omplex @ G @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( real_V2046097035970521341omplex @ A4 @ ( G @ X4 ) )
@ A2 ) ) ).
% scaleR_right.sum
thf(fact_86_scaleR__right_Osum,axiom,
! [A4: real,G: ( nat > real ) > complex,A2: set_nat_real] :
( ( real_V2046097035970521341omplex @ A4 @ ( groups2834896168486368005omplex @ G @ A2 ) )
= ( groups2834896168486368005omplex
@ ^ [X4: nat > real] : ( real_V2046097035970521341omplex @ A4 @ ( G @ X4 ) )
@ A2 ) ) ).
% scaleR_right.sum
thf(fact_87_scaleR__right_Osum,axiom,
! [A4: real,G: real > complex,A2: set_real] :
( ( real_V2046097035970521341omplex @ A4 @ ( groups5754745047067104278omplex @ G @ A2 ) )
= ( groups5754745047067104278omplex
@ ^ [X4: real] : ( real_V2046097035970521341omplex @ A4 @ ( G @ X4 ) )
@ A2 ) ) ).
% scaleR_right.sum
thf(fact_88_scaleR__right_Osum,axiom,
! [A4: real,G: nat > real,A2: set_nat] :
( ( real_V1485227260804924795R_real @ A4 @ ( groups6591440286371151544t_real @ G @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( real_V1485227260804924795R_real @ A4 @ ( G @ X4 ) )
@ A2 ) ) ).
% scaleR_right.sum
thf(fact_89_scaleR__right_Osum,axiom,
! [A4: real,G: ( nat > real ) > real,A2: set_nat_real] :
( ( real_V1485227260804924795R_real @ A4 @ ( groups4253619806861319043l_real @ G @ A2 ) )
= ( groups4253619806861319043l_real
@ ^ [X4: nat > real] : ( real_V1485227260804924795R_real @ A4 @ ( G @ X4 ) )
@ A2 ) ) ).
% scaleR_right.sum
thf(fact_90_scaleR__right_Osum,axiom,
! [A4: real,G: real > real,A2: set_real] :
( ( real_V1485227260804924795R_real @ A4 @ ( groups8097168146408367636l_real @ G @ A2 ) )
= ( groups8097168146408367636l_real
@ ^ [X4: real] : ( real_V1485227260804924795R_real @ A4 @ ( G @ X4 ) )
@ A2 ) ) ).
% scaleR_right.sum
thf(fact_91_sum__mono,axiom,
! [K2: set_real,F: real > risk_Free_account,G: real > risk_Free_account] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ K2 ) @ ( groups8516999891779824987ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_92_sum__mono,axiom,
! [K2: set_complex,F: complex > risk_Free_account,G: complex > risk_Free_account] :
( ! [I3: complex] :
( ( member_complex @ I3 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups6146125110001601245ccount @ F @ K2 ) @ ( groups6146125110001601245ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_93_sum__mono,axiom,
! [K2: set_nat,F: nat > risk_Free_account,G: nat > risk_Free_account] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_le4245800335709223507ccount @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( groups6033208628184776703ccount @ F @ K2 ) @ ( groups6033208628184776703ccount @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_94_sum__mono,axiom,
! [K2: set_complex,F: complex > real,G: complex > real] :
( ! [I3: complex] :
( ( member_complex @ I3 @ K2 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ K2 ) @ ( groups5808333547571424918x_real @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_95_sum__mono,axiom,
! [K2: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K2 ) @ ( groups1935376822645274424al_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_96_sum__mono,axiom,
! [K2: set_complex,F: complex > nat,G: complex > nat] :
( ! [I3: complex] :
( ( member_complex @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K2 ) @ ( groups5693394587270226106ex_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_97_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K2 ) @ ( groups3542108847815614940at_nat @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_98_sum__mono,axiom,
! [K2: set_real,F: real > int,G: real > int] :
( ! [I3: real] :
( ( member_real @ I3 @ K2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K2 ) @ ( groups1932886352136224148al_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_99_sum__mono,axiom,
! [K2: set_complex,F: complex > int,G: complex > int] :
( ! [I3: complex] :
( ( member_complex @ I3 @ K2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K2 ) @ ( groups5690904116761175830ex_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_100_sum__mono,axiom,
! [K2: set_nat,F: nat > int,G: nat > int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K2 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K2 ) @ ( groups3539618377306564664at_int @ G @ K2 ) ) ) ).
% sum_mono
thf(fact_101_atMost__def,axiom,
( set_or3854930313887350124ccount
= ( ^ [U: risk_Free_account] :
( collec1856553087948576712ccount
@ ^ [X4: risk_Free_account] : ( ord_le4245800335709223507ccount @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_102_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U: real] :
( collect_real
@ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_103_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U: int] :
( collect_int
@ ^ [X4: int] : ( ord_less_eq_int @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_104_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U ) ) ) ) ).
% atMost_def
thf(fact_105_order__refl,axiom,
! [X: risk_Free_account] : ( ord_le4245800335709223507ccount @ X @ X ) ).
% order_refl
thf(fact_106_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_107_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_108_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_109_dual__order_Orefl,axiom,
! [A4: risk_Free_account] : ( ord_le4245800335709223507ccount @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_110_dual__order_Orefl,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_111_dual__order_Orefl,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_112_dual__order_Orefl,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_113_norm__sum,axiom,
! [F: nat > complex,A2: set_nat] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( real_V1022390504157884413omplex @ ( F @ I ) )
@ A2 ) ) ).
% norm_sum
thf(fact_114_norm__sum,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( real_V7735802525324610683m_real @ ( F @ I ) )
@ A2 ) ) ).
% norm_sum
thf(fact_115_norm__sum,axiom,
! [F: complex > complex,A2: set_complex] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
@ ( groups5808333547571424918x_real
@ ^ [I: complex] : ( real_V1022390504157884413omplex @ ( F @ I ) )
@ A2 ) ) ).
% norm_sum
thf(fact_116_norm__sum,axiom,
! [F: ( nat > real ) > complex,A2: set_nat_real] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2834896168486368005omplex @ F @ A2 ) )
@ ( groups4253619806861319043l_real
@ ^ [I: nat > real] : ( real_V1022390504157884413omplex @ ( F @ I ) )
@ A2 ) ) ).
% norm_sum
thf(fact_117_norm__sum,axiom,
! [F: real > complex,A2: set_real] :
( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ A2 ) )
@ ( groups8097168146408367636l_real
@ ^ [I: real] : ( real_V1022390504157884413omplex @ ( F @ I ) )
@ A2 ) ) ).
% norm_sum
thf(fact_118_norm__sum,axiom,
! [F: ( nat > real ) > real,A2: set_nat_real] :
( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups4253619806861319043l_real @ F @ A2 ) )
@ ( groups4253619806861319043l_real
@ ^ [I: nat > real] : ( real_V7735802525324610683m_real @ ( F @ I ) )
@ A2 ) ) ).
% norm_sum
thf(fact_119_norm__sum,axiom,
! [F: real > real,A2: set_real] :
( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups8097168146408367636l_real @ F @ A2 ) )
@ ( groups8097168146408367636l_real
@ ^ [I: real] : ( real_V7735802525324610683m_real @ ( F @ I ) )
@ A2 ) ) ).
% norm_sum
thf(fact_120_scaleR__left__mono,axiom,
! [X: real,Y: real,A4: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ X ) @ ( real_V1485227260804924795R_real @ A4 @ Y ) ) ) ) ).
% scaleR_left_mono
thf(fact_121_scaleR__left__mono__neg,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_eq_real @ B4 @ A4 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A4 ) @ ( real_V1485227260804924795R_real @ C @ B4 ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_122_scaleR__right__mono,axiom,
! [A4: real,B4: real,X: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ X ) @ ( real_V1485227260804924795R_real @ B4 @ X ) ) ) ) ).
% scaleR_right_mono
thf(fact_123_scaleR__right__mono__neg,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_eq_real @ B4 @ A4 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ C ) @ ( real_V1485227260804924795R_real @ B4 @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_124_complete__real,axiom,
! [S: set_real] :
( ? [X3: real] : ( member_real @ X3 @ S )
=> ( ? [Z2: real] :
! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ? [Y3: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Y3 ) )
& ! [Z2: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ) ) ).
% complete_real
thf(fact_125_verit__comp__simplify1_I2_J,axiom,
! [A4: risk_Free_account] : ( ord_le4245800335709223507ccount @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_126_verit__comp__simplify1_I2_J,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_127_verit__comp__simplify1_I2_J,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_128_verit__comp__simplify1_I2_J,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% verit_comp_simplify1(2)
thf(fact_129_nle__le,axiom,
! [A4: real,B4: real] :
( ( ~ ( ord_less_eq_real @ A4 @ B4 ) )
= ( ( ord_less_eq_real @ B4 @ A4 )
& ( B4 != A4 ) ) ) ).
% nle_le
thf(fact_130_nle__le,axiom,
! [A4: nat,B4: nat] :
( ( ~ ( ord_less_eq_nat @ A4 @ B4 ) )
= ( ( ord_less_eq_nat @ B4 @ A4 )
& ( B4 != A4 ) ) ) ).
% nle_le
thf(fact_131_nle__le,axiom,
! [A4: int,B4: int] :
( ( ~ ( ord_less_eq_int @ A4 @ B4 ) )
= ( ( ord_less_eq_int @ B4 @ A4 )
& ( B4 != A4 ) ) ) ).
% nle_le
thf(fact_132_le__cases3,axiom,
! [X: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_133_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_134_le__cases3,axiom,
! [X: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_135_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y4 )
& ( ord_le4245800335709223507ccount @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_136_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_137_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_138_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_139_scaleR__cancel__right,axiom,
! [A4: real,X: complex,B4: real] :
( ( ( real_V2046097035970521341omplex @ A4 @ X )
= ( real_V2046097035970521341omplex @ B4 @ X ) )
= ( ( A4 = B4 )
| ( X = zero_zero_complex ) ) ) ).
% scaleR_cancel_right
thf(fact_140_scaleR__cancel__right,axiom,
! [A4: real,X: real,B4: real] :
( ( ( real_V1485227260804924795R_real @ A4 @ X )
= ( real_V1485227260804924795R_real @ B4 @ X ) )
= ( ( A4 = B4 )
| ( X = zero_zero_real ) ) ) ).
% scaleR_cancel_right
thf(fact_141_scaleR__zero__right,axiom,
! [A4: real] :
( ( real_V2046097035970521341omplex @ A4 @ zero_zero_complex )
= zero_zero_complex ) ).
% scaleR_zero_right
thf(fact_142_scaleR__zero__right,axiom,
! [A4: real] :
( ( real_V1485227260804924795R_real @ A4 @ zero_zero_real )
= zero_zero_real ) ).
% scaleR_zero_right
thf(fact_143_scaleR__cancel__left,axiom,
! [A4: real,X: real,Y: real] :
( ( ( real_V1485227260804924795R_real @ A4 @ X )
= ( real_V1485227260804924795R_real @ A4 @ Y ) )
= ( ( X = Y )
| ( A4 = zero_zero_real ) ) ) ).
% scaleR_cancel_left
thf(fact_144_Rep__account__zero,axiom,
( ( risk_F170160801229183585ccount @ zero_z1425366712893667068ccount )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% Rep_account_zero
thf(fact_145_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [Uu: nat] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_146_sum_Oneutral__const,axiom,
! [A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu: complex] : zero_zero_complex
@ A2 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_147_sum_Oneutral__const,axiom,
! [A2: set_nat_real] :
( ( groups2834896168486368005omplex
@ ^ [Uu: nat > real] : zero_zero_complex
@ A2 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_148_sum_Oneutral__const,axiom,
! [A2: set_real] :
( ( groups5754745047067104278omplex
@ ^ [Uu: real] : zero_zero_complex
@ A2 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_149_sum_Oneutral__const,axiom,
! [A2: set_nat_real] :
( ( groups4253619806861319043l_real
@ ^ [Uu: nat > real] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_150_sum_Oneutral__const,axiom,
! [A2: set_real] :
( ( groups8097168146408367636l_real
@ ^ [Uu: real] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_151_norm__zero,axiom,
( ( real_V7735802525324610683m_real @ zero_zero_real )
= zero_zero_real ) ).
% norm_zero
thf(fact_152_norm__zero,axiom,
( ( real_V1022390504157884413omplex @ zero_zero_complex )
= zero_zero_real ) ).
% norm_zero
thf(fact_153_norm__eq__zero,axiom,
! [X: real] :
( ( ( real_V7735802525324610683m_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_eq_zero
thf(fact_154_norm__eq__zero,axiom,
! [X: complex] :
( ( ( real_V1022390504157884413omplex @ X )
= zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_eq_zero
thf(fact_155_scaleR__zero__left,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
= zero_zero_complex ) ).
% scaleR_zero_left
thf(fact_156_scaleR__zero__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% scaleR_zero_left
thf(fact_157_scaleR__eq__0__iff,axiom,
! [A4: real,X: complex] :
( ( ( real_V2046097035970521341omplex @ A4 @ X )
= zero_zero_complex )
= ( ( A4 = zero_zero_real )
| ( X = zero_zero_complex ) ) ) ).
% scaleR_eq_0_iff
thf(fact_158_scaleR__eq__0__iff,axiom,
! [A4: real,X: real] :
( ( ( real_V1485227260804924795R_real @ A4 @ X )
= zero_zero_real )
= ( ( A4 = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% scaleR_eq_0_iff
thf(fact_159_of__real__eq__0__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_160_of__real__eq__0__iff,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= zero_zero_complex )
= ( X = zero_zero_real ) ) ).
% of_real_eq_0_iff
thf(fact_161_of__real__0,axiom,
( ( real_V1803761363581548252l_real @ zero_zero_real )
= zero_zero_real ) ).
% of_real_0
thf(fact_162_of__real__0,axiom,
( ( real_V4546457046886955230omplex @ zero_zero_real )
= zero_zero_complex ) ).
% of_real_0
thf(fact_163_mem__Collect__eq,axiom,
! [A4: real,P: real > $o] :
( ( member_real @ A4 @ ( collect_real @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_164_mem__Collect__eq,axiom,
! [A4: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ A4 @ ( collect_nat_real @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_165_mem__Collect__eq,axiom,
! [A4: nat,P: nat > $o] :
( ( member_nat @ A4 @ ( collect_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_166_mem__Collect__eq,axiom,
! [A4: complex,P: complex > $o] :
( ( member_complex @ A4 @ ( collect_complex @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_167_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_168_Collect__mem__eq,axiom,
! [A2: set_nat_real] :
( ( collect_nat_real
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X4: complex] : ( member_complex @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_171_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X2: complex] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_172_Rep__account__just__cash,axiom,
! [C: real] :
( ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ C ) )
= ( ^ [N: nat] : ( if_real @ ( N = zero_zero_nat ) @ C @ zero_zero_real ) ) ) ).
% Rep_account_just_cash
thf(fact_173_norm__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
= ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
thf(fact_174_norm__le__zero__iff,axiom,
! [X: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
= ( X = zero_zero_complex ) ) ).
% norm_le_zero_iff
thf(fact_175_norm__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) ) ).
% norm_ge_zero
thf(fact_176_norm__ge__zero,axiom,
! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% norm_ge_zero
thf(fact_177_zero__account__alt__def,axiom,
( ( risk_Free_just_cash @ zero_zero_real )
= zero_z1425366712893667068ccount ) ).
% zero_account_alt_def
thf(fact_178_zero__account__def,axiom,
( zero_z1425366712893667068ccount
= ( risk_F5458100604530014700ccount
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_account_def
thf(fact_179_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_180_sum_Oneutral,axiom,
! [A2: set_complex,G: complex > complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_181_sum_Oneutral,axiom,
! [A2: set_nat_real,G: ( nat > real ) > complex] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_complex ) )
=> ( ( groups2834896168486368005omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_182_sum_Oneutral,axiom,
! [A2: set_real,G: real > complex] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_complex ) )
=> ( ( groups5754745047067104278omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_183_sum_Oneutral,axiom,
! [A2: set_nat_real,G: ( nat > real ) > real] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups4253619806861319043l_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_184_sum_Oneutral,axiom,
! [A2: set_real,G: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( G @ X2 )
= zero_zero_real ) )
=> ( ( groups8097168146408367636l_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_185_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > risk_Free_account,A2: set_real] :
( ( ( groups8516999891779824987ccount @ G @ A2 )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_186_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > risk_Free_account,A2: set_complex] :
( ( ( groups6146125110001601245ccount @ G @ A2 )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_187_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > risk_Free_account,A2: set_nat] :
( ( ( groups6033208628184776703ccount @ G @ A2 )
!= zero_z1425366712893667068ccount )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_z1425366712893667068ccount ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_188_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > real,A2: set_complex] :
( ( ( groups5808333547571424918x_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_189_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_190_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > nat,A2: set_complex] :
( ( ( groups5693394587270226106ex_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_191_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_192_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A2: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_193_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > int,A2: set_complex] :
( ( ( groups5690904116761175830ex_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_194_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A2: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_195_scaleR__right__imp__eq,axiom,
! [X: complex,A4: real,B4: real] :
( ( X != zero_zero_complex )
=> ( ( ( real_V2046097035970521341omplex @ A4 @ X )
= ( real_V2046097035970521341omplex @ B4 @ X ) )
=> ( A4 = B4 ) ) ) ).
% scaleR_right_imp_eq
thf(fact_196_scaleR__right__imp__eq,axiom,
! [X: real,A4: real,B4: real] :
( ( X != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A4 @ X )
= ( real_V1485227260804924795R_real @ B4 @ X ) )
=> ( A4 = B4 ) ) ) ).
% scaleR_right_imp_eq
thf(fact_197_scaleR__left__imp__eq,axiom,
! [A4: real,X: real,Y: real] :
( ( A4 != zero_zero_real )
=> ( ( ( real_V1485227260804924795R_real @ A4 @ X )
= ( real_V1485227260804924795R_real @ A4 @ Y ) )
=> ( X = Y ) ) ) ).
% scaleR_left_imp_eq
thf(fact_198_scaleR__nonpos__nonpos,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A4 @ B4 ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_199_scaleR__nonpos__nonneg,axiom,
! [A4: real,X: real] :
( ( ord_less_eq_real @ A4 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_200_scaleR__nonneg__nonpos,axiom,
! [A4: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_201_scaleR__nonneg__nonneg,axiom,
! [A4: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A4 @ X ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_202_split__scaleR__pos__le,axiom,
! [A4: real,B4: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_eq_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A4 @ B4 ) ) ) ).
% split_scaleR_pos_le
thf(fact_203_split__scaleR__neg__le,axiom,
! [A4: real,X: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ X @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ X ) ) )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ X ) @ zero_zero_real ) ) ).
% split_scaleR_neg_le
thf(fact_204_scaleR__mono_H,axiom,
! [A4: real,B4: real,C: real,D: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ C ) @ ( real_V1485227260804924795R_real @ B4 @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_205_scaleR__mono,axiom,
! [A4: real,B4: real,X: real,Y: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ B4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ X ) @ ( real_V1485227260804924795R_real @ B4 @ Y ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_206_just__cash__def,axiom,
( risk_Free_just_cash
= ( ^ [C2: real] :
( risk_F5458100604530014700ccount
@ ^ [N: nat] : ( if_real @ ( N = zero_zero_nat ) @ C2 @ zero_zero_real ) ) ) ) ).
% just_cash_def
thf(fact_207_sum__nonpos,axiom,
! [A2: set_real,F: real > risk_Free_account] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups8516999891779824987ccount @ F @ A2 ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_208_sum__nonpos,axiom,
! [A2: set_complex,F: complex > risk_Free_account] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups6146125110001601245ccount @ F @ A2 ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_209_sum__nonpos,axiom,
! [A2: set_nat,F: nat > risk_Free_account] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ zero_z1425366712893667068ccount ) )
=> ( ord_le4245800335709223507ccount @ ( groups6033208628184776703ccount @ F @ A2 ) @ zero_z1425366712893667068ccount ) ) ).
% sum_nonpos
thf(fact_210_sum__nonpos,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_211_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_212_sum__nonpos,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_213_sum__nonpos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_214_sum__nonpos,axiom,
! [A2: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_215_sum__nonpos,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_216_sum__nonpos,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ zero_zero_int ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).
% sum_nonpos
thf(fact_217_sum__nonneg,axiom,
! [A2: set_real,F: real > risk_Free_account] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups8516999891779824987ccount @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_218_sum__nonneg,axiom,
! [A2: set_complex,F: complex > risk_Free_account] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups6146125110001601245ccount @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_219_sum__nonneg,axiom,
! [A2: set_nat,F: nat > risk_Free_account] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( F @ X2 ) ) )
=> ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ ( groups6033208628184776703ccount @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_220_sum__nonneg,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_221_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_222_sum__nonneg,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_223_sum__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_224_sum__nonneg,axiom,
! [A2: set_real,F: real > int] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_225_sum__nonneg,axiom,
! [A2: set_complex,F: complex > int] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_226_sum__nonneg,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X2 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_227_sum__norm__le,axiom,
! [S: set_complex,F: complex > real,G: complex > real] :
( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups5808333547571424918x_real @ F @ S ) ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_228_sum__norm__le,axiom,
! [S: set_nat,F: nat > complex,G: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_229_sum__norm__le,axiom,
! [S: set_nat,F: nat > real,G: nat > real] :
( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_230_sum__norm__le,axiom,
! [S: set_complex,F: complex > complex,G: complex > real] :
( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_231_sum__norm__le,axiom,
! [S: set_nat_real,F: ( nat > real ) > complex,G: ( nat > real ) > real] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2834896168486368005omplex @ F @ S ) ) @ ( groups4253619806861319043l_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_232_sum__norm__le,axiom,
! [S: set_real,F: real > complex,G: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( groups8097168146408367636l_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_233_sum__norm__le,axiom,
! [S: set_nat_real,F: ( nat > real ) > real,G: ( nat > real ) > real] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups4253619806861319043l_real @ F @ S ) ) @ ( groups4253619806861319043l_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_234_sum__norm__le,axiom,
! [S: set_real,F: real > real,G: real > real] :
( ! [X2: real] :
( ( member_real @ X2 @ S )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X2 ) ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups8097168146408367636l_real @ F @ S ) ) @ ( groups8097168146408367636l_real @ G @ S ) ) ) ).
% sum_norm_le
thf(fact_235_order__antisym__conv,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ( ( ord_le4245800335709223507ccount @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_236_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_237_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_238_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_239_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_240_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_241_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_242_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_243_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_244_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_245_ord__le__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_246_ord__le__eq__subst,axiom,
! [A4: real,B4: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_247_ord__le__eq__subst,axiom,
! [A4: real,B4: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_248_ord__le__eq__subst,axiom,
! [A4: real,B4: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_249_ord__le__eq__subst,axiom,
! [A4: real,B4: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_250_ord__le__eq__subst,axiom,
! [A4: nat,B4: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_251_ord__le__eq__subst,axiom,
! [A4: nat,B4: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_252_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_253_ord__eq__le__subst,axiom,
! [A4: real,F: risk_Free_account > real,B4: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_ord__eq__le__subst,axiom,
! [A4: nat,F: risk_Free_account > nat,B4: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_255_ord__eq__le__subst,axiom,
! [A4: int,F: risk_Free_account > int,B4: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_256_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B4: real,C: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_257_ord__eq__le__subst,axiom,
! [A4: real,F: real > real,B4: real,C: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_258_ord__eq__le__subst,axiom,
! [A4: nat,F: real > nat,B4: real,C: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_259_ord__eq__le__subst,axiom,
! [A4: int,F: real > int,B4: real,C: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_260_ord__eq__le__subst,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B4: nat,C: nat] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_261_ord__eq__le__subst,axiom,
! [A4: real,F: nat > real,B4: nat,C: nat] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_262_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_263_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_264_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_265_verit__la__disequality,axiom,
! [A4: real,B4: real] :
( ( A4 = B4 )
| ~ ( ord_less_eq_real @ A4 @ B4 )
| ~ ( ord_less_eq_real @ B4 @ A4 ) ) ).
% verit_la_disequality
thf(fact_266_verit__la__disequality,axiom,
! [A4: nat,B4: nat] :
( ( A4 = B4 )
| ~ ( ord_less_eq_nat @ A4 @ B4 )
| ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ).
% verit_la_disequality
thf(fact_267_verit__la__disequality,axiom,
! [A4: int,B4: int] :
( ( A4 = B4 )
| ~ ( ord_less_eq_int @ A4 @ B4 )
| ~ ( ord_less_eq_int @ B4 @ A4 ) ) ).
% verit_la_disequality
thf(fact_268_order__eq__refl,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( X = Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_eq_refl
thf(fact_269_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_270_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_271_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_272_order__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_273_order__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_274_order__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_275_order__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_less_eq_int @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_276_order__subst2,axiom,
! [A4: real,B4: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_277_order__subst2,axiom,
! [A4: real,B4: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_278_order__subst2,axiom,
! [A4: real,B4: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_279_order__subst2,axiom,
! [A4: real,B4: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_int @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_280_order__subst2,axiom,
! [A4: nat,B4: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_281_order__subst2,axiom,
! [A4: nat,B4: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_282_order__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_283_order__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B4: real,C: real] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_284_order__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B4: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_285_order__subst1,axiom,
! [A4: risk_Free_account,F: int > risk_Free_account,B4: int,C: int] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_int @ B4 @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le4245800335709223507ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_286_order__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_287_order__subst1,axiom,
! [A4: real,F: real > real,B4: real,C: real] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_288_order__subst1,axiom,
! [A4: real,F: nat > real,B4: nat,C: nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_289_order__subst1,axiom,
! [A4: real,F: int > real,B4: int,C: int] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_int @ B4 @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_290_order__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_291_order__subst1,axiom,
! [A4: nat,F: real > nat,B4: real,C: real] :
( ( ord_less_eq_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_292_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
& ( ord_le4245800335709223507ccount @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_293_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_294_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_295_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ B @ A ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_296_antisym,axiom,
! [A4: risk_Free_account,B4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_297_antisym,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_298_antisym,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_299_antisym,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_eq_int @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% antisym
thf(fact_300_dual__order_Otrans,axiom,
! [B4: risk_Free_account,A4: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B4 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ C @ B4 )
=> ( ord_le4245800335709223507ccount @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_301_dual__order_Otrans,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_eq_real @ B4 @ A4 )
=> ( ( ord_less_eq_real @ C @ B4 )
=> ( ord_less_eq_real @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_302_dual__order_Otrans,axiom,
! [B4: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
=> ( ( ord_less_eq_nat @ C @ B4 )
=> ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_303_dual__order_Otrans,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_eq_int @ B4 @ A4 )
=> ( ( ord_less_eq_int @ C @ B4 )
=> ( ord_less_eq_int @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_304_dual__order_Oantisym,axiom,
! [B4: risk_Free_account,A4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B4 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_305_dual__order_Oantisym,axiom,
! [B4: real,A4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_306_dual__order_Oantisym,axiom,
! [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_307_dual__order_Oantisym,axiom,
! [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
=> ( ( ord_less_eq_int @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ).
% dual_order.antisym
thf(fact_308_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: risk_Free_account,Z: risk_Free_account] : ( Y2 = Z ) )
= ( ^ [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
& ( ord_le4245800335709223507ccount @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_309_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: real,Z: real] : ( Y2 = Z ) )
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_310_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_311_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_312_linorder__wlog,axiom,
! [P: real > real > $o,A4: real,B4: real] :
( ! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_313_linorder__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B4: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_314_linorder__wlog,axiom,
! [P: int > int > $o,A4: int,B4: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ).
% linorder_wlog
thf(fact_315_order__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z3 )
=> ( ord_le4245800335709223507ccount @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_316_order__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_317_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_318_order__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_319_order_Otrans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ord_le4245800335709223507ccount @ A4 @ C ) ) ) ).
% order.trans
thf(fact_320_order_Otrans,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ord_less_eq_real @ A4 @ C ) ) ) ).
% order.trans
thf(fact_321_order_Otrans,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% order.trans
thf(fact_322_order_Otrans,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_eq_int @ B4 @ C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% order.trans
thf(fact_323_order__antisym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_324_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_325_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_326_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_327_ord__le__eq__trans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_le4245800335709223507ccount @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_328_ord__le__eq__trans,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq_real @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_329_ord__le__eq__trans,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_330_ord__le__eq__trans,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_331_ord__eq__le__trans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( A4 = B4 )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ord_le4245800335709223507ccount @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_332_ord__eq__le__trans,axiom,
! [A4: real,B4: real,C: real] :
( ( A4 = B4 )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ord_less_eq_real @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_333_ord__eq__le__trans,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( A4 = B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_334_ord__eq__le__trans,axiom,
! [A4: int,B4: int,C: int] :
( ( A4 = B4 )
=> ( ( ord_less_eq_int @ B4 @ C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_335_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_336_strictly__solvent__def,axiom,
( risk_F1636578016437888323olvent
= ( ^ [Alpha: risk_Free_account] :
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ Alpha ) @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% strictly_solvent_def
thf(fact_337_subsetI,axiom,
! [A2: set_real,B2: set_real] :
( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B2 ) )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% subsetI
thf(fact_338_subsetI,axiom,
! [A2: set_nat_real,B2: set_nat_real] :
( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ A2 )
=> ( member_nat_real @ X2 @ B2 ) )
=> ( ord_le2908806416726583473t_real @ A2 @ B2 ) ) ).
% subsetI
thf(fact_339_subsetI,axiom,
! [A2: set_complex,B2: set_complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_complex @ X2 @ B2 ) )
=> ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).
% subsetI
thf(fact_340_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_341_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_342_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_343_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_344_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_345_strictly__solvent__just__cash__equiv,axiom,
! [C: real] :
( ( risk_F1636578016437888323olvent @ ( risk_Free_just_cash @ C ) )
= ( ord_less_eq_real @ zero_zero_real @ C ) ) ).
% strictly_solvent_just_cash_equiv
thf(fact_346_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_347_Collect__subset,axiom,
! [A2: set_nat_real,P: ( nat > real ) > $o] :
( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_348_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_349_Collect__subset,axiom,
! [A2: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_350_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A5 )
@ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_351_less__eq__set__def,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B5: set_nat_real] :
( ord_le7676461544873280788real_o
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ A5 )
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_352_less__eq__set__def,axiom,
( ord_le211207098394363844omplex
= ( ^ [A5: set_complex,B5: set_complex] :
( ord_le4573692005234683329plex_o
@ ^ [X4: complex] : ( member_complex @ X4 @ A5 )
@ ^ [X4: complex] : ( member_complex @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_353_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_354_pred__subset__eq,axiom,
! [R: set_real,S: set_real] :
( ( ord_less_eq_real_o
@ ^ [X4: real] : ( member_real @ X4 @ R )
@ ^ [X4: real] : ( member_real @ X4 @ S ) )
= ( ord_less_eq_set_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_355_pred__subset__eq,axiom,
! [R: set_nat_real,S: set_nat_real] :
( ( ord_le7676461544873280788real_o
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ R )
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ S ) )
= ( ord_le2908806416726583473t_real @ R @ S ) ) ).
% pred_subset_eq
thf(fact_356_pred__subset__eq,axiom,
! [R: set_complex,S: set_complex] :
( ( ord_le4573692005234683329plex_o
@ ^ [X4: complex] : ( member_complex @ X4 @ R )
@ ^ [X4: complex] : ( member_complex @ X4 @ S ) )
= ( ord_le211207098394363844omplex @ R @ S ) ) ).
% pred_subset_eq
thf(fact_357_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ R )
@ ^ [X4: nat] : ( member_nat @ X4 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_358_strictly__solvent__alt__def,axiom,
( risk_F1636578016437888323olvent
= ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount ) ) ).
% strictly_solvent_alt_def
thf(fact_359_zero__reorient,axiom,
! [X: risk_Free_account] :
( ( zero_z1425366712893667068ccount = X )
= ( X = zero_z1425366712893667068ccount ) ) ).
% zero_reorient
thf(fact_360_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_361_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_362_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_363_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_364_Collect__mono__iff,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
= ( ! [X4: complex] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_365_Collect__mono,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X2: complex] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_366_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [T2: real] :
( ( member_real @ T2 @ A5 )
=> ( member_real @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_367_subset__iff,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B5: set_nat_real] :
! [T2: nat > real] :
( ( member_nat_real @ T2 @ A5 )
=> ( member_nat_real @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_368_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A5: set_complex,B5: set_complex] :
! [T2: complex] :
( ( member_complex @ T2 @ A5 )
=> ( member_complex @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_369_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_370_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B5: set_real] :
! [X4: real] :
( ( member_real @ X4 @ A5 )
=> ( member_real @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_371_subset__eq,axiom,
( ord_le2908806416726583473t_real
= ( ^ [A5: set_nat_real,B5: set_nat_real] :
! [X4: nat > real] :
( ( member_nat_real @ X4 @ A5 )
=> ( member_nat_real @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_372_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A5: set_complex,B5: set_complex] :
! [X4: complex] :
( ( member_complex @ X4 @ A5 )
=> ( member_complex @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_373_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A5 )
=> ( member_nat @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_374_subsetD,axiom,
! [A2: set_real,B2: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% subsetD
thf(fact_375_subsetD,axiom,
! [A2: set_nat_real,B2: set_nat_real,C: nat > real] :
( ( ord_le2908806416726583473t_real @ A2 @ B2 )
=> ( ( member_nat_real @ C @ A2 )
=> ( member_nat_real @ C @ B2 ) ) ) ).
% subsetD
thf(fact_376_subsetD,axiom,
! [A2: set_complex,B2: set_complex,C: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B2 ) ) ) ).
% subsetD
thf(fact_377_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_378_in__mono,axiom,
! [A2: set_real,B2: set_real,X: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ X @ A2 )
=> ( member_real @ X @ B2 ) ) ) ).
% in_mono
thf(fact_379_in__mono,axiom,
! [A2: set_nat_real,B2: set_nat_real,X: nat > real] :
( ( ord_le2908806416726583473t_real @ A2 @ B2 )
=> ( ( member_nat_real @ X @ A2 )
=> ( member_nat_real @ X @ B2 ) ) ) ).
% in_mono
thf(fact_380_in__mono,axiom,
! [A2: set_complex,B2: set_complex,X: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( member_complex @ X @ A2 )
=> ( member_complex @ X @ B2 ) ) ) ).
% in_mono
thf(fact_381_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_382_strictly__solvent__non__negative__cash,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F1636578016437888323olvent @ Alpha2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( risk_F1914734008469130493eserve @ Alpha2 ) ) ) ).
% strictly_solvent_non_negative_cash
thf(fact_383_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_384_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_385_valid__transfer__alt__def,axiom,
( risk_F1023690899723030139ansfer
= ( ^ [Alpha: risk_Free_account,Tau: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ zero_z1425366712893667068ccount @ Tau )
& ( ord_le4245800335709223507ccount @ Tau @ Alpha ) ) ) ) ).
% valid_transfer_alt_def
thf(fact_386_cash__reserve__def,axiom,
( risk_F1914734008469130493eserve
= ( ^ [Alpha: risk_Free_account] : ( risk_F170160801229183585ccount @ Alpha @ zero_zero_nat ) ) ) ).
% cash_reserve_def
thf(fact_387_subset__Collect__iff,axiom,
! [B2: set_real,A2: set_real,P: real > $o] :
( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ( ord_less_eq_set_real @ B2
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( ! [X4: real] :
( ( member_real @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_388_subset__Collect__iff,axiom,
! [B2: set_nat_real,A2: set_nat_real,P: ( nat > real ) > $o] :
( ( ord_le2908806416726583473t_real @ B2 @ A2 )
=> ( ( ord_le2908806416726583473t_real @ B2
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( ! [X4: nat > real] :
( ( member_nat_real @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_389_subset__Collect__iff,axiom,
! [B2: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ B2
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_390_subset__Collect__iff,axiom,
! [B2: set_complex,A2: set_complex,P: complex > $o] :
( ( ord_le211207098394363844omplex @ B2 @ A2 )
=> ( ( ord_le211207098394363844omplex @ B2
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( P @ X4 ) ) ) )
= ( ! [X4: complex] :
( ( member_complex @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_391_subset__CollectI,axiom,
! [B2: set_real,A2: set_real,Q: real > $o,P: real > $o] :
( ( ord_less_eq_set_real @ B2 @ A2 )
=> ( ! [X2: real] :
( ( member_real @ X2 @ B2 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ B2 )
& ( Q @ X4 ) ) )
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ A2 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_392_subset__CollectI,axiom,
! [B2: set_nat_real,A2: set_nat_real,Q: ( nat > real ) > $o,P: ( nat > real ) > $o] :
( ( ord_le2908806416726583473t_real @ B2 @ A2 )
=> ( ! [X2: nat > real] :
( ( member_nat_real @ X2 @ B2 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ B2 )
& ( Q @ X4 ) ) )
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ A2 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_393_subset__CollectI,axiom,
! [B2: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( Q @ X4 ) ) )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_394_subset__CollectI,axiom,
! [B2: set_complex,A2: set_complex,Q: complex > $o,P: complex > $o] :
( ( ord_le211207098394363844omplex @ B2 @ A2 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ B2 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ B2 )
& ( Q @ X4 ) ) )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_395_Collect__restrict,axiom,
! [X5: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_396_Collect__restrict,axiom,
! [X5: set_nat_real,P: ( nat > real ) > $o] :
( ord_le2908806416726583473t_real
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_397_Collect__restrict,axiom,
! [X5: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_398_Collect__restrict,axiom,
! [X5: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ X5 )
& ( P @ X4 ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_399_prop__restrict,axiom,
! [X: real,Z4: set_real,X5: set_real,P: real > $o] :
( ( member_real @ X @ Z4 )
=> ( ( ord_less_eq_set_real @ Z4
@ ( collect_real
@ ^ [X4: real] :
( ( member_real @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_400_prop__restrict,axiom,
! [X: nat > real,Z4: set_nat_real,X5: set_nat_real,P: ( nat > real ) > $o] :
( ( member_nat_real @ X @ Z4 )
=> ( ( ord_le2908806416726583473t_real @ Z4
@ ( collect_nat_real
@ ^ [X4: nat > real] :
( ( member_nat_real @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_401_prop__restrict,axiom,
! [X: nat,Z4: set_nat,X5: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_402_prop__restrict,axiom,
! [X: complex,Z4: set_complex,X5: set_complex,P: complex > $o] :
( ( member_complex @ X @ Z4 )
=> ( ( ord_le211207098394363844omplex @ Z4
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ X5 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_403_conj__subset__def,axiom,
! [A2: set_complex,P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ A2
@ ( collect_complex
@ ^ [X4: complex] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_le211207098394363844omplex @ A2 @ ( collect_complex @ P ) )
& ( ord_le211207098394363844omplex @ A2 @ ( collect_complex @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_404_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_405_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_406_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_407_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_408_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_409_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B4: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B4 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_410_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M2: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_411_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_412_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_413_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_414_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_415_only__strictly__solvent__accounts__can__transfer,axiom,
! [Alpha2: risk_Free_account,Tau2: risk_Free_account] :
( ( risk_F1023690899723030139ansfer @ Alpha2 @ Tau2 )
=> ( risk_F1636578016437888323olvent @ Alpha2 ) ) ).
% only_strictly_solvent_accounts_can_transfer
thf(fact_416_scaleR__left__le__one__le,axiom,
! [X: real,A4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ A4 @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ X ) @ X ) ) ) ).
% scaleR_left_le_one_le
thf(fact_417_zero__le__scaleR__iff,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A4 @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( A4 = zero_zero_real ) ) ) ).
% zero_le_scaleR_iff
thf(fact_418_scaleR__le__0__iff,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A4 @ B4 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A4 )
& ( ord_less_eq_real @ B4 @ zero_zero_real ) )
| ( ( ord_less_real @ A4 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B4 ) )
| ( A4 = zero_zero_real ) ) ) ).
% scaleR_le_0_iff
thf(fact_419_zero__less__norm__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
= ( X != zero_zero_real ) ) ).
% zero_less_norm_iff
thf(fact_420_zero__less__norm__iff,axiom,
! [X: complex] :
( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
= ( X != zero_zero_complex ) ) ).
% zero_less_norm_iff
thf(fact_421_sum__abs__ge__zero,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( abs_abs_real @ ( F @ I ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_422_sum__abs__ge__zero,axiom,
! [F: ( nat > real ) > real,A2: set_nat_real] :
( ord_less_eq_real @ zero_zero_real
@ ( groups4253619806861319043l_real
@ ^ [I: nat > real] : ( abs_abs_real @ ( F @ I ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_423_sum__abs__ge__zero,axiom,
! [F: real > real,A2: set_real] :
( ord_less_eq_real @ zero_zero_real
@ ( groups8097168146408367636l_real
@ ^ [I: real] : ( abs_abs_real @ ( F @ I ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_424_atMost__UNIV__triv,axiom,
( ( set_or4236626031148496127et_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% atMost_UNIV_triv
thf(fact_425_UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% UNIV_I
thf(fact_426_UNIV__I,axiom,
! [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).
% UNIV_I
thf(fact_427_UNIV__I,axiom,
! [X: complex] : ( member_complex @ X @ top_top_set_complex ) ).
% UNIV_I
thf(fact_428_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_429_abs__idempotent,axiom,
! [A4: real] :
( ( abs_abs_real @ ( abs_abs_real @ A4 ) )
= ( abs_abs_real @ A4 ) ) ).
% abs_idempotent
thf(fact_430_abs__idempotent,axiom,
! [A4: int] :
( ( abs_abs_int @ ( abs_abs_int @ A4 ) )
= ( abs_abs_int @ A4 ) ) ).
% abs_idempotent
thf(fact_431_abs__norm__cancel,axiom,
! [A4: real] :
( ( abs_abs_real @ ( real_V7735802525324610683m_real @ A4 ) )
= ( real_V7735802525324610683m_real @ A4 ) ) ).
% abs_norm_cancel
thf(fact_432_abs__norm__cancel,axiom,
! [A4: complex] :
( ( abs_abs_real @ ( real_V1022390504157884413omplex @ A4 ) )
= ( real_V1022390504157884413omplex @ A4 ) ) ).
% abs_norm_cancel
thf(fact_433_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_434_abs__0__eq,axiom,
! [A4: real] :
( ( zero_zero_real
= ( abs_abs_real @ A4 ) )
= ( A4 = zero_zero_real ) ) ).
% abs_0_eq
thf(fact_435_abs__0__eq,axiom,
! [A4: int] :
( ( zero_zero_int
= ( abs_abs_int @ A4 ) )
= ( A4 = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_436_abs__eq__0,axiom,
! [A4: real] :
( ( ( abs_abs_real @ A4 )
= zero_zero_real )
= ( A4 = zero_zero_real ) ) ).
% abs_eq_0
thf(fact_437_abs__eq__0,axiom,
! [A4: int] :
( ( ( abs_abs_int @ A4 )
= zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_438_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_zero
thf(fact_439_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_440_scaleR__one,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ one_one_real @ X )
= X ) ).
% scaleR_one
thf(fact_441_norm__of__real,axiom,
! [R2: real] :
( ( real_V7735802525324610683m_real @ ( real_V1803761363581548252l_real @ R2 ) )
= ( abs_abs_real @ R2 ) ) ).
% norm_of_real
thf(fact_442_norm__of__real,axiom,
! [R2: real] :
( ( real_V1022390504157884413omplex @ ( real_V4546457046886955230omplex @ R2 ) )
= ( abs_abs_real @ R2 ) ) ).
% norm_of_real
thf(fact_443_abs__sum__abs,axiom,
! [F: nat > real,A2: set_nat] :
( ( abs_abs_real
@ ( groups6591440286371151544t_real
@ ^ [A: nat] : ( abs_abs_real @ ( F @ A ) )
@ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [A: nat] : ( abs_abs_real @ ( F @ A ) )
@ A2 ) ) ).
% abs_sum_abs
thf(fact_444_abs__sum__abs,axiom,
! [F: ( nat > real ) > real,A2: set_nat_real] :
( ( abs_abs_real
@ ( groups4253619806861319043l_real
@ ^ [A: nat > real] : ( abs_abs_real @ ( F @ A ) )
@ A2 ) )
= ( groups4253619806861319043l_real
@ ^ [A: nat > real] : ( abs_abs_real @ ( F @ A ) )
@ A2 ) ) ).
% abs_sum_abs
thf(fact_445_abs__sum__abs,axiom,
! [F: real > real,A2: set_real] :
( ( abs_abs_real
@ ( groups8097168146408367636l_real
@ ^ [A: real] : ( abs_abs_real @ ( F @ A ) )
@ A2 ) )
= ( groups8097168146408367636l_real
@ ^ [A: real] : ( abs_abs_real @ ( F @ A ) )
@ A2 ) ) ).
% abs_sum_abs
thf(fact_446_abs__le__zero__iff,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A4 ) @ zero_zero_real )
= ( A4 = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_447_abs__le__zero__iff,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A4 ) @ zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_448_abs__le__self__iff,axiom,
! [A4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A4 ) @ A4 )
= ( ord_less_eq_real @ zero_zero_real @ A4 ) ) ).
% abs_le_self_iff
thf(fact_449_abs__le__self__iff,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A4 ) @ A4 )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% abs_le_self_iff
thf(fact_450_abs__of__nonneg,axiom,
! [A4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A4 )
=> ( ( abs_abs_real @ A4 )
= A4 ) ) ).
% abs_of_nonneg
thf(fact_451_abs__of__nonneg,axiom,
! [A4: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( abs_abs_int @ A4 )
= A4 ) ) ).
% abs_of_nonneg
thf(fact_452_zero__less__abs__iff,axiom,
! [A4: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A4 ) )
= ( A4 != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_453_zero__less__abs__iff,axiom,
! [A4: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A4 ) )
= ( A4 != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_454_norm__one,axiom,
( ( real_V7735802525324610683m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_455_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_456_of__real__1,axiom,
( ( real_V1803761363581548252l_real @ one_one_real )
= one_one_real ) ).
% of_real_1
thf(fact_457_of__real__1,axiom,
( ( real_V4546457046886955230omplex @ one_one_real )
= one_one_complex ) ).
% of_real_1
thf(fact_458_of__real__eq__1__iff,axiom,
! [X: real] :
( ( ( real_V1803761363581548252l_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% of_real_eq_1_iff
thf(fact_459_of__real__eq__1__iff,axiom,
! [X: real] :
( ( ( real_V4546457046886955230omplex @ X )
= one_one_complex )
= ( X = one_one_real ) ) ).
% of_real_eq_1_iff
thf(fact_460_sum__abs,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I: nat] : ( abs_abs_real @ ( F @ I ) )
@ A2 ) ) ).
% sum_abs
thf(fact_461_sum__abs,axiom,
! [F: ( nat > real ) > real,A2: set_nat_real] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups4253619806861319043l_real @ F @ A2 ) )
@ ( groups4253619806861319043l_real
@ ^ [I: nat > real] : ( abs_abs_real @ ( F @ I ) )
@ A2 ) ) ).
% sum_abs
thf(fact_462_sum__abs,axiom,
! [F: real > real,A2: set_real] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups8097168146408367636l_real @ F @ A2 ) )
@ ( groups8097168146408367636l_real
@ ^ [I: real] : ( abs_abs_real @ ( F @ I ) )
@ A2 ) ) ).
% sum_abs
thf(fact_463_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_464_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_465_order__less__imp__not__less,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_466_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_467_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_468_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_469_order__less__imp__not__eq2,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_470_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_471_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_472_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_473_order__less__imp__not__eq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_474_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_475_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_476_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_477_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_478_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_479_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_480_order__less__imp__triv,axiom,
! [X: risk_Free_account,Y: risk_Free_account,P: $o] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_481_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_482_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_483_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_484_order__less__not__sym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_485_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_486_order__less__subst2,axiom,
! [A4: real,B4: real,F: real > real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_487_order__less__subst2,axiom,
! [A4: real,B4: real,F: real > nat,C: nat] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_488_order__less__subst2,axiom,
! [A4: real,B4: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_489_order__less__subst2,axiom,
! [A4: real,B4: real,F: real > int,C: int] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_int @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_490_order__less__subst2,axiom,
! [A4: nat,B4: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_491_order__less__subst2,axiom,
! [A4: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_492_order__less__subst2,axiom,
! [A4: nat,B4: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_493_order__less__subst2,axiom,
! [A4: nat,B4: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_int @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_494_order__less__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_495_order__less__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_496_order__less__subst1,axiom,
! [A4: real,F: real > real,B4: real,C: real] :
( ( ord_less_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_497_order__less__subst1,axiom,
! [A4: real,F: nat > real,B4: nat,C: nat] :
( ( ord_less_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_498_order__less__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_le2131251472502387783ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_499_order__less__subst1,axiom,
! [A4: real,F: int > real,B4: int,C: int] :
( ( ord_less_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_500_order__less__subst1,axiom,
! [A4: nat,F: real > nat,B4: real,C: real] :
( ( ord_less_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_501_order__less__subst1,axiom,
! [A4: nat,F: nat > nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_502_order__less__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_le2131251472502387783ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_503_order__less__subst1,axiom,
! [A4: nat,F: int > nat,B4: int,C: int] :
( ( ord_less_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_504_order__less__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B4: real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_505_order__less__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B4: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_506_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_507_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_508_order__less__irrefl,axiom,
! [X: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ X @ X ) ).
% order_less_irrefl
thf(fact_509_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_510_ord__less__eq__subst,axiom,
! [A4: real,B4: real,F: real > real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_511_ord__less__eq__subst,axiom,
! [A4: real,B4: real,F: real > nat,C: nat] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_512_ord__less__eq__subst,axiom,
! [A4: real,B4: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_513_ord__less__eq__subst,axiom,
! [A4: real,B4: real,F: real > int,C: int] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_514_ord__less__eq__subst,axiom,
! [A4: nat,B4: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_515_ord__less__eq__subst,axiom,
! [A4: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_516_ord__less__eq__subst,axiom,
! [A4: nat,B4: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_517_ord__less__eq__subst,axiom,
! [A4: nat,B4: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_518_ord__less__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_519_ord__less__eq__subst,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_520_ord__eq__less__subst,axiom,
! [A4: real,F: real > real,B4: real,C: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_521_ord__eq__less__subst,axiom,
! [A4: nat,F: real > nat,B4: real,C: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_522_ord__eq__less__subst,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B4: real,C: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_523_ord__eq__less__subst,axiom,
! [A4: int,F: real > int,B4: real,C: real] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_524_ord__eq__less__subst,axiom,
! [A4: real,F: nat > real,B4: nat,C: nat] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_525_ord__eq__less__subst,axiom,
! [A4: nat,F: nat > nat,B4: nat,C: nat] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_526_ord__eq__less__subst,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B4: nat,C: nat] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_527_ord__eq__less__subst,axiom,
! [A4: int,F: nat > int,B4: nat,C: nat] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_528_ord__eq__less__subst,axiom,
! [A4: real,F: risk_Free_account > real,B4: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le2131251472502387783ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_529_ord__eq__less__subst,axiom,
! [A4: nat,F: risk_Free_account > nat,B4: risk_Free_account,C: risk_Free_account] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_le2131251472502387783ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_530_order__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_531_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_532_order__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z3 )
=> ( ord_le2131251472502387783ccount @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_533_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_534_order__less__asym_H,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ~ ( ord_less_real @ B4 @ A4 ) ) ).
% order_less_asym'
thf(fact_535_order__less__asym_H,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ~ ( ord_less_nat @ B4 @ A4 ) ) ).
% order_less_asym'
thf(fact_536_order__less__asym_H,axiom,
! [A4: risk_Free_account,B4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ~ ( ord_le2131251472502387783ccount @ B4 @ A4 ) ) ).
% order_less_asym'
thf(fact_537_order__less__asym_H,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ~ ( ord_less_int @ B4 @ A4 ) ) ).
% order_less_asym'
thf(fact_538_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_539_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_540_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_541_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_542_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_543_order__less__asym,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ~ ( ord_le2131251472502387783ccount @ Y @ X ) ) ).
% order_less_asym
thf(fact_544_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_545_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_546_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_547_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_548_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( A4 != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_549_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ( A4 != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_550_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B4 @ A4 )
=> ( A4 != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_551_dual__order_Ostrict__implies__not__eq,axiom,
! [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( A4 != B4 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_552_order_Ostrict__implies__not__eq,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( A4 != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_553_order_Ostrict__implies__not__eq,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( A4 != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_554_order_Ostrict__implies__not__eq,axiom,
! [A4: risk_Free_account,B4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( A4 != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_555_order_Ostrict__implies__not__eq,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( A4 != B4 ) ) ).
% order.strict_implies_not_eq
thf(fact_556_dual__order_Ostrict__trans,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( ( ord_less_real @ C @ B4 )
=> ( ord_less_real @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_557_dual__order_Ostrict__trans,axiom,
! [B4: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ( ( ord_less_nat @ C @ B4 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_558_dual__order_Ostrict__trans,axiom,
! [B4: risk_Free_account,A4: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B4 @ A4 )
=> ( ( ord_le2131251472502387783ccount @ C @ B4 )
=> ( ord_le2131251472502387783ccount @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_559_dual__order_Ostrict__trans,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( ( ord_less_int @ C @ B4 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_560_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_561_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_562_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_563_top_Onot__eq__extremum,axiom,
! [A4: set_nat] :
( ( A4 != top_top_set_nat )
= ( ord_less_set_nat @ A4 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_564_top_Oextremum__strict,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A4 ) ).
% top.extremum_strict
thf(fact_565_order_Ostrict__trans,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_real @ B4 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_566_order_Ostrict__trans,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_567_order_Ostrict__trans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ( ord_le2131251472502387783ccount @ B4 @ C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_568_order_Ostrict__trans,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_int @ B4 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_569_linorder__less__wlog,axiom,
! [P: real > real > $o,A4: real,B4: real] :
( ! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B3: real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_570_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B4: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_571_linorder__less__wlog,axiom,
! [P: int > int > $o,A4: int,B4: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A4 @ B4 ) ) ) ) ).
% linorder_less_wlog
thf(fact_572_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_573_dual__order_Oirrefl,axiom,
! [A4: real] :
~ ( ord_less_real @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_574_dual__order_Oirrefl,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_575_dual__order_Oirrefl,axiom,
! [A4: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_576_dual__order_Oirrefl,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_577_dual__order_Oasym,axiom,
! [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
=> ~ ( ord_less_real @ A4 @ B4 ) ) ).
% dual_order.asym
thf(fact_578_dual__order_Oasym,axiom,
! [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ~ ( ord_less_nat @ A4 @ B4 ) ) ).
% dual_order.asym
thf(fact_579_dual__order_Oasym,axiom,
! [B4: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B4 @ A4 )
=> ~ ( ord_le2131251472502387783ccount @ A4 @ B4 ) ) ).
% dual_order.asym
thf(fact_580_dual__order_Oasym,axiom,
! [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
=> ~ ( ord_less_int @ A4 @ B4 ) ) ).
% dual_order.asym
thf(fact_581_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_582_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_583_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_584_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_585_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_586_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_587_less__induct,axiom,
! [P: nat > $o,A4: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A4 ) ) ).
% less_induct
thf(fact_588_ord__less__eq__trans,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_589_ord__less__eq__trans,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_590_ord__less__eq__trans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_591_ord__less__eq__trans,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_592_ord__eq__less__trans,axiom,
! [A4: real,B4: real,C: real] :
( ( A4 = B4 )
=> ( ( ord_less_real @ B4 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_593_ord__eq__less__trans,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( A4 = B4 )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_594_ord__eq__less__trans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( A4 = B4 )
=> ( ( ord_le2131251472502387783ccount @ B4 @ C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_595_ord__eq__less__trans,axiom,
! [A4: int,B4: int,C: int] :
( ( A4 = B4 )
=> ( ( ord_less_int @ B4 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_596_order_Oasym,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ~ ( ord_less_real @ B4 @ A4 ) ) ).
% order.asym
thf(fact_597_order_Oasym,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ~ ( ord_less_nat @ B4 @ A4 ) ) ).
% order.asym
thf(fact_598_order_Oasym,axiom,
! [A4: risk_Free_account,B4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ~ ( ord_le2131251472502387783ccount @ B4 @ A4 ) ) ).
% order.asym
thf(fact_599_order_Oasym,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ~ ( ord_less_int @ B4 @ A4 ) ) ).
% order.asym
thf(fact_600_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_601_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_602_less__imp__neq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_603_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_604_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z5: real] :
( ( ord_less_real @ X @ Z5 )
& ( ord_less_real @ Z5 @ Y ) ) ) ).
% dense
thf(fact_605_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_606_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_607_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_608_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_609_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_610_verit__comp__simplify1_I1_J,axiom,
! [A4: real] :
~ ( ord_less_real @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_611_verit__comp__simplify1_I1_J,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_612_verit__comp__simplify1_I1_J,axiom,
! [A4: risk_Free_account] :
~ ( ord_le2131251472502387783ccount @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_613_verit__comp__simplify1_I1_J,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% verit_comp_simplify1(1)
thf(fact_614_atMost__eq__UNIV__iff,axiom,
! [X: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= top_top_set_set_nat )
= ( X = top_top_set_nat ) ) ).
% atMost_eq_UNIV_iff
thf(fact_615_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_616_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_617_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_618_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_619_UNIV__witness,axiom,
? [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).
% UNIV_witness
thf(fact_620_UNIV__witness,axiom,
? [X2: nat > real] : ( member_nat_real @ X2 @ top_top_set_nat_real ) ).
% UNIV_witness
thf(fact_621_UNIV__witness,axiom,
? [X2: complex] : ( member_complex @ X2 @ top_top_set_complex ) ).
% UNIV_witness
thf(fact_622_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_623_UNIV__eq__I,axiom,
! [A2: set_real] :
( ! [X2: real] : ( member_real @ X2 @ A2 )
=> ( top_top_set_real = A2 ) ) ).
% UNIV_eq_I
thf(fact_624_UNIV__eq__I,axiom,
! [A2: set_nat_real] :
( ! [X2: nat > real] : ( member_nat_real @ X2 @ A2 )
=> ( top_top_set_nat_real = A2 ) ) ).
% UNIV_eq_I
thf(fact_625_UNIV__eq__I,axiom,
! [A2: set_complex] :
( ! [X2: complex] : ( member_complex @ X2 @ A2 )
=> ( top_top_set_complex = A2 ) ) ).
% UNIV_eq_I
thf(fact_626_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_627_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_628_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_629_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_630_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_631_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_632_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_633_abs__of__pos,axiom,
! [A4: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( abs_abs_real @ A4 )
= A4 ) ) ).
% abs_of_pos
thf(fact_634_abs__of__pos,axiom,
! [A4: int] :
( ( ord_less_int @ zero_zero_int @ A4 )
=> ( ( abs_abs_int @ A4 )
= A4 ) ) ).
% abs_of_pos
thf(fact_635_abs__not__less__zero,axiom,
! [A4: real] :
~ ( ord_less_real @ ( abs_abs_real @ A4 ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_636_abs__not__less__zero,axiom,
! [A4: int] :
~ ( ord_less_int @ ( abs_abs_int @ A4 ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_637_UNIV__def,axiom,
( top_top_set_complex
= ( collect_complex
@ ^ [X4: complex] : $true ) ) ).
% UNIV_def
thf(fact_638_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X4: nat] : $true ) ) ).
% UNIV_def
thf(fact_639_dense__eq0__I,axiom,
! [X: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
=> ( X = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_640_abs__le__D1,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A4 ) @ B4 )
=> ( ord_less_eq_real @ A4 @ B4 ) ) ).
% abs_le_D1
thf(fact_641_abs__le__D1,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A4 ) @ B4 )
=> ( ord_less_eq_int @ A4 @ B4 ) ) ).
% abs_le_D1
thf(fact_642_abs__ge__self,axiom,
! [A4: real] : ( ord_less_eq_real @ A4 @ ( abs_abs_real @ A4 ) ) ).
% abs_ge_self
thf(fact_643_abs__ge__self,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ ( abs_abs_int @ A4 ) ) ).
% abs_ge_self
thf(fact_644_top__greatest,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% top_greatest
thf(fact_645_top_Oextremum__unique,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
= ( A4 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_646_top_Oextremum__uniqueI,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A4 )
=> ( A4 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_647_verit__comp__simplify1_I3_J,axiom,
! [B6: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
= ( ord_less_real @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_648_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_649_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_650_leD,axiom,
! [Y: risk_Free_account,X: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Y @ X )
=> ~ ( ord_le2131251472502387783ccount @ X @ Y ) ) ).
% leD
thf(fact_651_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_652_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_653_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_654_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_655_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_656_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_657_nless__le,axiom,
! [A4: risk_Free_account,B4: risk_Free_account] :
( ( ~ ( ord_le2131251472502387783ccount @ A4 @ B4 ) )
= ( ~ ( ord_le4245800335709223507ccount @ A4 @ B4 )
| ( A4 = B4 ) ) ) ).
% nless_le
thf(fact_658_nless__le,axiom,
! [A4: real,B4: real] :
( ( ~ ( ord_less_real @ A4 @ B4 ) )
= ( ~ ( ord_less_eq_real @ A4 @ B4 )
| ( A4 = B4 ) ) ) ).
% nless_le
thf(fact_659_nless__le,axiom,
! [A4: nat,B4: nat] :
( ( ~ ( ord_less_nat @ A4 @ B4 ) )
= ( ~ ( ord_less_eq_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ).
% nless_le
thf(fact_660_nless__le,axiom,
! [A4: int,B4: int] :
( ( ~ ( ord_less_int @ A4 @ B4 ) )
= ( ~ ( ord_less_eq_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ).
% nless_le
thf(fact_661_antisym__conv1,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ~ ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_662_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_663_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_664_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_665_antisym__conv2,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ~ ( ord_le2131251472502387783ccount @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_666_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_667_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_668_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_669_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X2: real] :
( ( ord_less_real @ Z3 @ X2 )
=> ( ord_less_eq_real @ Y @ X2 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_670_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_671_less__le__not__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y4 )
& ~ ( ord_le4245800335709223507ccount @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_672_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_673_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_674_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_675_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_676_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_677_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_678_order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_679_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A: real,B: real] :
( ( ord_less_real @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_680_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_681_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] :
( ( ord_less_int @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_682_order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_683_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_684_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_685_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_686_order_Ostrict__trans1,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_le2131251472502387783ccount @ B4 @ C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_687_order_Ostrict__trans1,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_real @ B4 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_688_order_Ostrict__trans1,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_689_order_Ostrict__trans1,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( ord_less_int @ B4 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_690_order_Ostrict__trans2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ord_le2131251472502387783ccount @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_691_order_Ostrict__trans2,axiom,
! [A4: real,B4: real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ord_less_real @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_692_order_Ostrict__trans2,axiom,
! [A4: nat,B4: nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_693_order_Ostrict__trans2,axiom,
! [A4: int,B4: int,C: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_eq_int @ B4 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_694_order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [A: risk_Free_account,B: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A @ B )
& ~ ( ord_le4245800335709223507ccount @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_695_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
& ~ ( ord_less_eq_real @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_696_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
& ~ ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_697_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ~ ( ord_less_eq_int @ B @ A ) ) ) ) ).
% order.strict_iff_not
thf(fact_698_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_699_dense__le__bounded,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_700_dual__order_Oorder__iff__strict,axiom,
( ord_le4245800335709223507ccount
= ( ^ [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_701_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B: real,A: real] :
( ( ord_less_real @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_702_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_703_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( ( ord_less_int @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_704_dual__order_Ostrict__iff__order,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_705_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_706_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_707_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_708_dual__order_Ostrict__trans1,axiom,
! [B4: risk_Free_account,A4: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B4 @ A4 )
=> ( ( ord_le2131251472502387783ccount @ C @ B4 )
=> ( ord_le2131251472502387783ccount @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_709_dual__order_Ostrict__trans1,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_eq_real @ B4 @ A4 )
=> ( ( ord_less_real @ C @ B4 )
=> ( ord_less_real @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_710_dual__order_Ostrict__trans1,axiom,
! [B4: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
=> ( ( ord_less_nat @ C @ B4 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_711_dual__order_Ostrict__trans1,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_eq_int @ B4 @ A4 )
=> ( ( ord_less_int @ C @ B4 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_712_dual__order_Ostrict__trans2,axiom,
! [B4: risk_Free_account,A4: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B4 @ A4 )
=> ( ( ord_le4245800335709223507ccount @ C @ B4 )
=> ( ord_le2131251472502387783ccount @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_713_dual__order_Ostrict__trans2,axiom,
! [B4: real,A4: real,C: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( ( ord_less_eq_real @ C @ B4 )
=> ( ord_less_real @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_714_dual__order_Ostrict__trans2,axiom,
! [B4: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ( ( ord_less_eq_nat @ C @ B4 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_715_dual__order_Ostrict__trans2,axiom,
! [B4: int,A4: int,C: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( ( ord_less_eq_int @ C @ B4 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_716_dual__order_Ostrict__iff__not,axiom,
( ord_le2131251472502387783ccount
= ( ^ [B: risk_Free_account,A: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ B @ A )
& ~ ( ord_le4245800335709223507ccount @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_717_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
& ~ ( ord_less_eq_real @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_718_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
& ~ ( ord_less_eq_nat @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_719_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ~ ( ord_less_eq_int @ A @ B ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_720_order_Ostrict__implies__order,axiom,
! [A4: risk_Free_account,B4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ord_le4245800335709223507ccount @ A4 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_721_order_Ostrict__implies__order,axiom,
! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ord_less_eq_real @ A4 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_722_order_Ostrict__implies__order,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ord_less_eq_nat @ A4 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_723_order_Ostrict__implies__order,axiom,
! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( ord_less_eq_int @ A4 @ B4 ) ) ).
% order.strict_implies_order
thf(fact_724_dual__order_Ostrict__implies__order,axiom,
! [B4: risk_Free_account,A4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ B4 @ A4 )
=> ( ord_le4245800335709223507ccount @ B4 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_725_dual__order_Ostrict__implies__order,axiom,
! [B4: real,A4: real] :
( ( ord_less_real @ B4 @ A4 )
=> ( ord_less_eq_real @ B4 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_726_dual__order_Ostrict__implies__order,axiom,
! [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
=> ( ord_less_eq_nat @ B4 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_727_dual__order_Ostrict__implies__order,axiom,
! [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
=> ( ord_less_eq_int @ B4 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_728_order__le__less,axiom,
( ord_le4245800335709223507ccount
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_729_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_730_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_731_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_732_order__less__le,axiom,
( ord_le2131251472502387783ccount
= ( ^ [X4: risk_Free_account,Y4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_733_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_734_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_735_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_736_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_737_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_738_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_739_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_740_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_741_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_742_order__less__imp__le,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ord_le4245800335709223507ccount @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_743_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_744_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_745_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_746_order__le__neq__trans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_le2131251472502387783ccount @ A4 @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_747_order__le__neq__trans,axiom,
! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_real @ A4 @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_748_order__le__neq__trans,axiom,
! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_nat @ A4 @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_749_order__le__neq__trans,axiom,
! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_int @ A4 @ B4 ) ) ) ).
% order_le_neq_trans
thf(fact_750_order__neq__le__trans,axiom,
! [A4: risk_Free_account,B4: risk_Free_account] :
( ( A4 != B4 )
=> ( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ord_le2131251472502387783ccount @ A4 @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_751_order__neq__le__trans,axiom,
! [A4: real,B4: real] :
( ( A4 != B4 )
=> ( ( ord_less_eq_real @ A4 @ B4 )
=> ( ord_less_real @ A4 @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_752_order__neq__le__trans,axiom,
! [A4: nat,B4: nat] :
( ( A4 != B4 )
=> ( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ord_less_nat @ A4 @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_753_order__neq__le__trans,axiom,
! [A4: int,B4: int] :
( ( A4 != B4 )
=> ( ( ord_less_eq_int @ A4 @ B4 )
=> ( ord_less_int @ A4 @ B4 ) ) ) ).
% order_neq_le_trans
thf(fact_754_order__le__less__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ Y @ Z3 )
=> ( ord_le2131251472502387783ccount @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_755_order__le__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_756_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_757_order__le__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_758_order__less__le__trans,axiom,
! [X: risk_Free_account,Y: risk_Free_account,Z3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X @ Y )
=> ( ( ord_le4245800335709223507ccount @ Y @ Z3 )
=> ( ord_le2131251472502387783ccount @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_759_order__less__le__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_760_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_761_order__less__le__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_762_order__le__less__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B4: real,C: real] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_763_order__le__less__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B4: nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_764_order__le__less__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_le2131251472502387783ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_765_order__le__less__subst1,axiom,
! [A4: risk_Free_account,F: int > risk_Free_account,B4: int,C: int] :
( ( ord_le4245800335709223507ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_766_order__le__less__subst1,axiom,
! [A4: real,F: real > real,B4: real,C: real] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_767_order__le__less__subst1,axiom,
! [A4: real,F: nat > real,B4: nat,C: nat] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_768_order__le__less__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_le2131251472502387783ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_769_order__le__less__subst1,axiom,
! [A4: real,F: int > real,B4: int,C: int] :
( ( ord_less_eq_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_int @ B4 @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_770_order__le__less__subst1,axiom,
! [A4: nat,F: real > nat,B4: real,C: real] :
( ( ord_less_eq_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_771_order__le__less__subst1,axiom,
! [A4: nat,F: nat > nat,B4: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_772_order__le__less__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_773_order__le__less__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_774_order__le__less__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > nat,C: nat] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_775_order__le__less__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > int,C: int] :
( ( ord_le4245800335709223507ccount @ A4 @ B4 )
=> ( ( ord_less_int @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_776_order__le__less__subst2,axiom,
! [A4: real,B4: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_777_order__le__less__subst2,axiom,
! [A4: real,B4: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_778_order__le__less__subst2,axiom,
! [A4: real,B4: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_779_order__le__less__subst2,axiom,
! [A4: real,B4: real,F: real > int,C: int] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( ( ord_less_int @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_780_order__le__less__subst2,axiom,
! [A4: nat,B4: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_le2131251472502387783ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_781_order__le__less__subst2,axiom,
! [A4: nat,B4: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( ( ord_less_real @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_782_order__less__le__subst1,axiom,
! [A4: risk_Free_account,F: risk_Free_account > risk_Free_account,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_783_order__less__le__subst1,axiom,
! [A4: real,F: risk_Free_account > real,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_784_order__less__le__subst1,axiom,
! [A4: nat,F: risk_Free_account > nat,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_785_order__less__le__subst1,axiom,
! [A4: int,F: risk_Free_account > int,B4: risk_Free_account,C: risk_Free_account] :
( ( ord_less_int @ A4 @ ( F @ B4 ) )
=> ( ( ord_le4245800335709223507ccount @ B4 @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_786_order__less__le__subst1,axiom,
! [A4: risk_Free_account,F: real > risk_Free_account,B4: real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_787_order__less__le__subst1,axiom,
! [A4: real,F: real > real,B4: real,C: real] :
( ( ord_less_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_788_order__less__le__subst1,axiom,
! [A4: nat,F: real > nat,B4: real,C: real] :
( ( ord_less_nat @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_789_order__less__le__subst1,axiom,
! [A4: int,F: real > int,B4: real,C: real] :
( ( ord_less_int @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_real @ B4 @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_790_order__less__le__subst1,axiom,
! [A4: risk_Free_account,F: nat > risk_Free_account,B4: nat,C: nat] :
( ( ord_le2131251472502387783ccount @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_le4245800335709223507ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_791_order__less__le__subst1,axiom,
! [A4: real,F: nat > real,B4: nat,C: nat] :
( ( ord_less_real @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq_nat @ B4 @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A4 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_792_order__less__le__subst2,axiom,
! [A4: real,B4: real,F: real > risk_Free_account,C: risk_Free_account] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_793_order__less__le__subst2,axiom,
! [A4: nat,B4: nat,F: nat > risk_Free_account,C: risk_Free_account] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_794_order__less__le__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > risk_Free_account,C: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_795_order__less__le__subst2,axiom,
! [A4: int,B4: int,F: int > risk_Free_account,C: risk_Free_account] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_le4245800335709223507ccount @ ( F @ B4 ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_le2131251472502387783ccount @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_le2131251472502387783ccount @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_796_order__less__le__subst2,axiom,
! [A4: real,B4: real,F: real > real,C: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_797_order__less__le__subst2,axiom,
! [A4: nat,B4: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_798_order__less__le__subst2,axiom,
! [A4: risk_Free_account,B4: risk_Free_account,F: risk_Free_account > real,C: real] :
( ( ord_le2131251472502387783ccount @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C )
=> ( ! [X2: risk_Free_account,Y3: risk_Free_account] :
( ( ord_le2131251472502387783ccount @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_799_order__less__le__subst2,axiom,
! [A4: int,B4: int,F: int > real,C: real] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( ord_less_eq_real @ ( F @ B4 ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_800_order__less__le__subst2,axiom,
! [A4: real,B4: real,F: real > nat,C: nat] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_801_order__less__le__subst2,axiom,
! [A4: nat,B4: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( F @ B4 ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_802_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_803_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_804_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_805_order__le__imp__less__or__eq,axiom,
! [X: risk_Free_account,Y: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ X @ Y )
=> ( ( ord_le2131251472502387783ccount @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_806_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_807_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_808_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_809_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_810_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_811_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_812_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_813_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_814_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_815_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_816_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_817_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_818_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_819_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_820_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_821_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_822_not__UNIV__eq__Iic,axiom,
! [H2: nat] :
( top_top_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_UNIV_eq_Iic
thf(fact_823_abs__ge__zero,axiom,
! [A4: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A4 ) ) ).
% abs_ge_zero
thf(fact_824_abs__ge__zero,axiom,
! [A4: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A4 ) ) ).
% abs_ge_zero
thf(fact_825_norm__not__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_826_norm__not__less__zero,axiom,
! [X: complex] :
~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% norm_not_less_zero
thf(fact_827_not__UNIV__le__Iic,axiom,
! [H: nat] :
~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_ord_atMost_nat @ H ) ) ).
% not_UNIV_le_Iic
thf(fact_828_of__real__def,axiom,
( real_V4546457046886955230omplex
= ( ^ [R3: real] : ( real_V2046097035970521341omplex @ R3 @ one_one_complex ) ) ) ).
% of_real_def
thf(fact_829_of__real__def,axiom,
( real_V1803761363581548252l_real
= ( ^ [R3: real] : ( real_V1485227260804924795R_real @ R3 @ one_one_real ) ) ) ).
% of_real_def
thf(fact_830_scaleR__le__cancel__left,axiom,
! [C: real,A4: real,B4: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A4 ) @ ( real_V1485227260804924795R_real @ C @ B4 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A4 @ B4 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_831_scaleR__le__cancel__left__neg,axiom,
! [C: real,A4: real,B4: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A4 ) @ ( real_V1485227260804924795R_real @ C @ B4 ) )
= ( ord_less_eq_real @ B4 @ A4 ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_832_scaleR__le__cancel__left__pos,axiom,
! [C: real,A4: real,B4: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A4 ) @ ( real_V1485227260804924795R_real @ C @ B4 ) )
= ( ord_less_eq_real @ A4 @ B4 ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_833_abs__1,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_1
thf(fact_834_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_835_abs__1,axiom,
( ( abs_abs_complex @ one_one_complex )
= one_one_complex ) ).
% abs_1
thf(fact_836_abs__0,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
% abs_0
thf(fact_837_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_838_abs__0,axiom,
( ( abs_abs_complex @ zero_zero_complex )
= zero_zero_complex ) ).
% abs_0
thf(fact_839_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_840_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_841_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_842_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_843_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_844_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_845_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_846_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_847_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_848_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_849_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_850_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_851_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_852_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_853_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_854_abs__abs,axiom,
! [A4: real] :
( ( abs_abs_real @ ( abs_abs_real @ A4 ) )
= ( abs_abs_real @ A4 ) ) ).
% abs_abs
thf(fact_855_abs__abs,axiom,
! [A4: int] :
( ( abs_abs_int @ ( abs_abs_int @ A4 ) )
= ( abs_abs_int @ A4 ) ) ).
% abs_abs
thf(fact_856_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_857_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_858_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_859_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_860_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_861_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_862_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_863_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_864_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_865_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_866_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_867_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_868_real__norm__def,axiom,
real_V7735802525324610683m_real = abs_abs_real ).
% real_norm_def
thf(fact_869_top__set__def,axiom,
( top_top_set_complex
= ( collect_complex @ top_top_complex_o ) ) ).
% top_set_def
thf(fact_870_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_871_top__empty__eq,axiom,
( top_top_real_o
= ( ^ [X4: real] : ( member_real @ X4 @ top_top_set_real ) ) ) ).
% top_empty_eq
thf(fact_872_top__empty__eq,axiom,
( top_top_nat_real_o
= ( ^ [X4: nat > real] : ( member_nat_real @ X4 @ top_top_set_nat_real ) ) ) ).
% top_empty_eq
thf(fact_873_top__empty__eq,axiom,
( top_top_complex_o
= ( ^ [X4: complex] : ( member_complex @ X4 @ top_top_set_complex ) ) ) ).
% top_empty_eq
thf(fact_874_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_875_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_876_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_877_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_878_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_879_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_880_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_881_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_882_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
& ( M4 != N ) ) ) ) ).
% nat_less_le
thf(fact_883_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_884_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
| ( M4 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_885_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_886_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_887_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_888_less__account__def,axiom,
( ord_le2131251472502387783ccount
= ( ^ [Alpha_1: risk_Free_account,Alpha_2: risk_Free_account] :
( ( ord_le4245800335709223507ccount @ Alpha_1 @ Alpha_2 )
& ~ ( ord_le4245800335709223507ccount @ Alpha_2 @ Alpha_1 ) ) ) ) ).
% less_account_def
thf(fact_889_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_890_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_891_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_892_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_893_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_894_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_895_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_896_abs__eq__0__iff,axiom,
! [A4: real] :
( ( ( abs_abs_real @ A4 )
= zero_zero_real )
= ( A4 = zero_zero_real ) ) ).
% abs_eq_0_iff
thf(fact_897_abs__eq__0__iff,axiom,
! [A4: int] :
( ( ( abs_abs_int @ A4 )
= zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_898_abs__eq__0__iff,axiom,
! [A4: complex] :
( ( ( abs_abs_complex @ A4 )
= zero_zero_complex )
= ( A4 = zero_zero_complex ) ) ).
% abs_eq_0_iff
thf(fact_899_abs__one,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_one
thf(fact_900_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_901_iso__tuple__UNIV__I,axiom,
! [X: real] : ( member_real @ X @ top_top_set_real ) ).
% iso_tuple_UNIV_I
thf(fact_902_iso__tuple__UNIV__I,axiom,
! [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).
% iso_tuple_UNIV_I
thf(fact_903_iso__tuple__UNIV__I,axiom,
! [X: complex] : ( member_complex @ X @ top_top_set_complex ) ).
% iso_tuple_UNIV_I
thf(fact_904_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_905_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_906_minf_I8_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ~ ( ord_less_eq_real @ T3 @ X3 ) ) ).
% minf(8)
thf(fact_907_minf_I8_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ~ ( ord_less_eq_nat @ T3 @ X3 ) ) ).
% minf(8)
thf(fact_908_minf_I8_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ~ ( ord_less_eq_int @ T3 @ X3 ) ) ).
% minf(8)
thf(fact_909_minf_I6_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ord_less_eq_real @ X3 @ T3 ) ) ).
% minf(6)
thf(fact_910_minf_I6_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ord_less_eq_nat @ X3 @ T3 ) ) ).
% minf(6)
thf(fact_911_minf_I6_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ord_less_eq_int @ X3 @ T3 ) ) ).
% minf(6)
thf(fact_912_pinf_I8_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ord_less_eq_real @ T3 @ X3 ) ) ).
% pinf(8)
thf(fact_913_pinf_I8_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ord_less_eq_nat @ T3 @ X3 ) ) ).
% pinf(8)
thf(fact_914_pinf_I8_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ord_less_eq_int @ T3 @ X3 ) ) ).
% pinf(8)
thf(fact_915_less__set__def,axiom,
( ord_less_set_real
= ( ^ [A5: set_real,B5: set_real] :
( ord_less_real_o
@ ^ [X4: real] : ( member_real @ X4 @ A5 )
@ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_916_less__set__def,axiom,
( ord_le3527643927072297637t_real
= ( ^ [A5: set_nat_real,B5: set_nat_real] :
( ord_less_nat_real_o
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ A5 )
@ ^ [X4: nat > real] : ( member_nat_real @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_917_less__set__def,axiom,
( ord_less_set_complex
= ( ^ [A5: set_complex,B5: set_complex] :
( ord_less_complex_o
@ ^ [X4: complex] : ( member_complex @ X4 @ A5 )
@ ^ [X4: complex] : ( member_complex @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_918_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_919_psubsetD,axiom,
! [A2: set_real,B2: set_real,C: real] :
( ( ord_less_set_real @ A2 @ B2 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_920_psubsetD,axiom,
! [A2: set_nat_real,B2: set_nat_real,C: nat > real] :
( ( ord_le3527643927072297637t_real @ A2 @ B2 )
=> ( ( member_nat_real @ C @ A2 )
=> ( member_nat_real @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_921_psubsetD,axiom,
! [A2: set_complex,B2: set_complex,C: complex] :
( ( ord_less_set_complex @ A2 @ B2 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_922_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_923_minf_I7_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ~ ( ord_less_real @ T3 @ X3 ) ) ).
% minf(7)
thf(fact_924_minf_I7_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ~ ( ord_less_nat @ T3 @ X3 ) ) ).
% minf(7)
thf(fact_925_minf_I7_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ~ ( ord_less_int @ T3 @ X3 ) ) ).
% minf(7)
thf(fact_926_minf_I5_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ord_less_real @ X3 @ T3 ) ) ).
% minf(5)
thf(fact_927_minf_I5_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ord_less_nat @ X3 @ T3 ) ) ).
% minf(5)
thf(fact_928_minf_I5_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ord_less_int @ X3 @ T3 ) ) ).
% minf(5)
thf(fact_929_minf_I4_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(4)
thf(fact_930_minf_I4_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(4)
thf(fact_931_minf_I4_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(4)
thf(fact_932_minf_I3_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(3)
thf(fact_933_minf_I3_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(3)
thf(fact_934_minf_I3_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( X3 != T3 ) ) ).
% minf(3)
thf(fact_935_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_936_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_937_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_938_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_939_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_940_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_941_pinf_I7_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ord_less_real @ T3 @ X3 ) ) ).
% pinf(7)
thf(fact_942_pinf_I7_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ord_less_nat @ T3 @ X3 ) ) ).
% pinf(7)
thf(fact_943_pinf_I7_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ord_less_int @ T3 @ X3 ) ) ).
% pinf(7)
thf(fact_944_pinf_I5_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ~ ( ord_less_real @ X3 @ T3 ) ) ).
% pinf(5)
thf(fact_945_pinf_I5_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ~ ( ord_less_nat @ X3 @ T3 ) ) ).
% pinf(5)
thf(fact_946_pinf_I5_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ~ ( ord_less_int @ X3 @ T3 ) ) ).
% pinf(5)
thf(fact_947_pinf_I4_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(4)
thf(fact_948_pinf_I4_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(4)
thf(fact_949_pinf_I4_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(4)
thf(fact_950_pinf_I3_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(3)
thf(fact_951_pinf_I3_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(3)
thf(fact_952_pinf_I3_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( X3 != T3 ) ) ).
% pinf(3)
thf(fact_953_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_954_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_955_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P4 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_956_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: real] :
! [X2: real] :
( ( ord_less_real @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_957_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_958_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P4 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_959_pinf_I6_J,axiom,
! [T3: real] :
? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ~ ( ord_less_eq_real @ X3 @ T3 ) ) ).
% pinf(6)
thf(fact_960_pinf_I6_J,axiom,
! [T3: nat] :
? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T3 ) ) ).
% pinf(6)
thf(fact_961_pinf_I6_J,axiom,
! [T3: int] :
? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ T3 ) ) ).
% pinf(6)
thf(fact_962_complete__interval,axiom,
! [A4: real,B4: real,P: real > $o] :
( ( ord_less_real @ A4 @ B4 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B4 )
=> ? [C3: real] :
( ( ord_less_eq_real @ A4 @ C3 )
& ( ord_less_eq_real @ C3 @ B4 )
& ! [X3: real] :
( ( ( ord_less_eq_real @ A4 @ X3 )
& ( ord_less_real @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D3: real] :
( ! [X2: real] :
( ( ( ord_less_eq_real @ A4 @ X2 )
& ( ord_less_real @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_963_complete__interval,axiom,
! [A4: nat,B4: nat,P: nat > $o] :
( ( ord_less_nat @ A4 @ B4 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B4 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A4 @ C3 )
& ( ord_less_eq_nat @ C3 @ B4 )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A4 @ X3 )
& ( ord_less_nat @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A4 @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_964_complete__interval,axiom,
! [A4: int,B4: int,P: int > $o] :
( ( ord_less_int @ A4 @ B4 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B4 )
=> ? [C3: int] :
( ( ord_less_eq_int @ A4 @ C3 )
& ( ord_less_eq_int @ C3 @ B4 )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A4 @ X3 )
& ( ord_less_int @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D3: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A4 @ X2 )
& ( ord_less_int @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_965_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_966_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_967_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_968_Abs__account__inverse,axiom,
! [Y: nat > real] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( risk_F170160801229183585ccount @ ( risk_F5458100604530014700ccount @ Y ) )
= Y ) ) ).
% Abs_account_inverse
thf(fact_969_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
@ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_970_top_Oordering__top__axioms,axiom,
ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% top.ordering_top_axioms
thf(fact_971_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_972_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A4: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A4 @ Top ) ) ).
% ordering_top.extremum
thf(fact_973_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A4: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A4 ) ) ).
% ordering_top.extremum_strict
thf(fact_974_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A4: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
= ( A4 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_975_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A4: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A4 != Top )
= ( Less @ A4 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_976_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A4: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
=> ( A4 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_977_ex__gt__or__lt,axiom,
! [A4: real] :
? [B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( ord_less_real @ B3 @ A4 ) ) ).
% ex_gt_or_lt
thf(fact_978_Rep__account,axiom,
! [X: risk_Free_account] : ( member_nat_real @ ( risk_F170160801229183585ccount @ X ) @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ) ).
% Rep_account
thf(fact_979_Rep__account__cases,axiom,
! [Y: nat > real] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ~ ! [X2: risk_Free_account] :
( Y
!= ( risk_F170160801229183585ccount @ X2 ) ) ) ).
% Rep_account_cases
thf(fact_980_Rep__account__induct,axiom,
! [Y: nat > real,P: ( nat > real ) > $o] :
( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ! [X2: risk_Free_account] : ( P @ ( risk_F170160801229183585ccount @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_account_induct
thf(fact_981_Abs__account__inject,axiom,
! [X: nat > real,Y: nat > real] :
( ( member_nat_real @ X @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( member_nat_real @ Y @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( ( ( risk_F5458100604530014700ccount @ X )
= ( risk_F5458100604530014700ccount @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_account_inject
thf(fact_982_Abs__account__induct,axiom,
! [P: risk_Free_account > $o,X: risk_Free_account] :
( ! [Y3: nat > real] :
( ( member_nat_real @ Y3 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) )
=> ( P @ ( risk_F5458100604530014700ccount @ Y3 ) ) )
=> ( P @ X ) ) ).
% Abs_account_induct
thf(fact_983_Abs__account__cases,axiom,
! [X: risk_Free_account] :
~ ! [Y3: nat > real] :
( ( X
= ( risk_F5458100604530014700ccount @ Y3 ) )
=> ~ ( member_nat_real @ Y3 @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ) ) ).
% Abs_account_cases
thf(fact_984_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_985_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_986_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_987_type__definition__account,axiom,
type_d8982087200295354172t_real @ risk_F170160801229183585ccount @ risk_F5458100604530014700ccount @ ( ordina1579063754167848977al_nat @ zero_zero_real @ top_top_set_nat ) ).
% type_definition_account
thf(fact_988_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_989_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_990_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_991_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_992_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_993_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_994_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_995_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_996_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_997_type__copy__ex__RepI,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,F2: ( nat > real ) > $o] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ top_top_set_nat_real )
=> ( ( ? [X7: nat > real] : ( F2 @ X7 ) )
= ( ? [B: risk_Free_account] : ( F2 @ ( Rep @ B ) ) ) ) ) ).
% type_copy_ex_RepI
thf(fact_998_type__copy__obj__one__point__absE,axiom,
! [Rep: risk_Free_account > nat > real,Abs: ( nat > real ) > risk_Free_account,S2: risk_Free_account] :
( ( type_d8982087200295354172t_real @ Rep @ Abs @ top_top_set_nat_real )
=> ~ ! [X2: nat > real] :
( S2
!= ( Abs @ X2 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_999_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1000_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_1001_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1002_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1003_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1004_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_1005_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1006_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_1007_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_1008_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_1009_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_1010_zero__le__log__cancel__iff,axiom,
! [A4: real,X: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A4 @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_1011_log__le__zero__cancel__iff,axiom,
! [A4: real,X: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A4 @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_1012_one__le__log__cancel__iff,axiom,
! [A4: real,X: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A4 @ X ) )
= ( ord_less_eq_real @ A4 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_1013_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_1014_of__nat__eq__iff,axiom,
! [M: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M = N2 ) ) ).
% of_nat_eq_iff
thf(fact_1015_abs__of__nat,axiom,
! [N2: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ N2 ) ) ).
% abs_of_nat
thf(fact_1016_abs__of__nat,axiom,
! [N2: nat] :
( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% abs_of_nat
thf(fact_1017_norm__of__nat,axiom,
! [N2: nat] :
( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% norm_of_nat
thf(fact_1018_norm__of__nat,axiom,
! [N2: nat] :
( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% norm_of_nat
thf(fact_1019_of__real__of__nat__eq,axiom,
! [N2: nat] :
( ( real_V1803761363581548252l_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_real_of_nat_eq
thf(fact_1020_arcsin__0,axiom,
( ( arcsin @ zero_zero_real )
= zero_zero_real ) ).
% arcsin_0
thf(fact_1021_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_1022_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= zero_zero_complex )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1023_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_1024_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_1025_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1026_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1027_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1028_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1029_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_1030_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_1031_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_1032_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_1033_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_1034_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_1035_of__nat__le__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% of_nat_le_iff
thf(fact_1036_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1037_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1038_of__nat__less__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1039_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1040_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_1041_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1042_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_1043_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1044_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1045_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1046_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1047_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1048_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri8010041392384452111omplex @ N2 )
= one_one_complex )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1049_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1050_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1051_log__one,axiom,
! [A4: real] :
( ( log @ A4 @ one_one_real )
= zero_zero_real ) ).
% log_one
thf(fact_1052_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1053_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1054_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_1055_zero__less__log__cancel__iff,axiom,
! [A4: real,X: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( log @ A4 @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_1056_log__less__zero__cancel__iff,axiom,
! [A4: real,X: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A4 @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_1057_one__less__log__cancel__iff,axiom,
! [A4: real,X: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ ( log @ A4 @ X ) )
= ( ord_less_real @ A4 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_1058_log__less__one__cancel__iff,axiom,
! [A4: real,X: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A4 @ X ) @ one_one_real )
= ( ord_less_real @ X @ A4 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_1059_log__less__cancel__iff,axiom,
! [A4: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A4 @ X ) @ ( log @ A4 @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_1060_log__eq__one,axiom,
! [A4: real] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( A4 != one_one_real )
=> ( ( log @ A4 @ A4 )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_1061_log__le__cancel__iff,axiom,
! [A4: real,X: real,Y: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A4 @ X ) @ ( log @ A4 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_1062_log__le__one__cancel__iff,axiom,
! [A4: real,X: real] :
( ( ord_less_real @ one_one_real @ A4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A4 @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ A4 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_1063_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1064_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1065_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1066_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1067_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A: nat,B: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_less_as_int
thf(fact_1068_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A: nat,B: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_leq_as_int
thf(fact_1069_arcsin__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% arcsin_le_mono
thf(fact_1070_arcsin__eq__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y ) )
= ( X = Y ) ) ) ) ).
% arcsin_eq_iff
thf(fact_1071_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1072_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1073_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( abs_abs_int @ Z3 ) @ one_one_int )
= ( Z3 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1074_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ A )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1075_int__if,axiom,
! [P: $o,A4: nat,B4: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A4 @ B4 ) )
= ( semiri1314217659103216013at_int @ A4 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A4 @ B4 ) )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% int_if
thf(fact_1076_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1077_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1078_zle__int,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% zle_int
thf(fact_1079_log__of__power__le,axiom,
! [M: nat,B4: real,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B4 @ N2 ) )
=> ( ( ord_less_real @ one_one_real @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( log @ B4 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% log_of_power_le
thf(fact_1080_log__pow__cancel,axiom,
! [A4: real,B4: nat] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( A4 != one_one_real )
=> ( ( log @ A4 @ ( power_power_real @ A4 @ B4 ) )
= ( semiri5074537144036343181t_real @ B4 ) ) ) ) ).
% log_pow_cancel
thf(fact_1081_log__of__power__less,axiom,
! [M: nat,B4: real,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B4 @ N2 ) )
=> ( ( ord_less_real @ one_one_real @ B4 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log @ B4 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% log_of_power_less
thf(fact_1082_le__log__of__power,axiom,
! [B4: real,N2: nat,M: real] :
( ( ord_less_eq_real @ ( power_power_real @ B4 @ N2 ) @ M )
=> ( ( ord_less_real @ one_one_real @ B4 )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B4 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_1083_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_1084_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_1085_less__log__of__power,axiom,
! [B4: real,N2: nat,M: real] :
( ( ord_less_real @ ( power_power_real @ B4 @ N2 ) @ M )
=> ( ( ord_less_real @ one_one_real @ B4 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B4 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_1086_log__of__power__eq,axiom,
! [M: nat,B4: real,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( power_power_real @ B4 @ N2 ) )
=> ( ( ord_less_real @ one_one_real @ B4 )
=> ( ( semiri5074537144036343181t_real @ N2 )
= ( log @ B4 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_1087_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1088_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1089_realpow__pos__nth,axiom,
! [N2: nat,A4: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ? [R4: real] :
( ( ord_less_real @ zero_zero_real @ R4 )
& ( ( power_power_real @ R4 @ N2 )
= A4 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1090_realpow__pos__nth__unique,axiom,
! [N2: nat,A4: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ? [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
& ( ( power_power_real @ X2 @ N2 )
= A4 )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N2 )
= A4 ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1091_sum__roots__unity,axiom,
! [N2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] : X4
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N2 )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_1092_sum__nth__roots,axiom,
! [N2: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] : X4
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N2 )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_1093_real__root__increasing,axiom,
! [N2: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_1094_real__root__pow__pos2,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_1095_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1096_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1097_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1098_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1099_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1100_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N2 ) )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1101_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_1102_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1103_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1104_real__root__zero,axiom,
! [N2: nat] :
( ( root @ N2 @ zero_zero_real )
= zero_zero_real ) ).
% real_root_zero
thf(fact_1105_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1106_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_1107_real__root__eq__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= ( root @ N2 @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_1108_real__root__eq__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_1109_real__root__less__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_1110_real__root__le__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_1111_real__root__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_1112_real__root__eq__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_1113_real__root__gt__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_1114_real__root__lt__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_1115_real__root__le__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_1116_real__root__ge__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_1117_real__root__gt__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_1118_real__root__lt__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_1119_real__root__ge__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_1120_real__root__le__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_1121_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1122_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1123_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1124_mult__le__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1125_mult__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1126_mult__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1127_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1128_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1129_int__ops_I7_J,axiom,
! [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A4 @ B4 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ).
% int_ops(7)
thf(fact_1130_real__root__pos__pos__le,axiom,
! [X: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_1131_mult__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1132_mult__less__mono2,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1133_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times_nat @ M @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1134_zmult__zless__mono2,axiom,
! [I2: int,J2: int,K: int] :
( ( ord_less_int @ I2 @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1135_abs__zmult__eq__1,axiom,
! [M: int,N2: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1136_real__root__less__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_1137_real__root__le__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_1138_real__root__power,axiom,
! [N2: nat,X: real,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( root @ N2 @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_1139_real__root__abs,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ ( abs_abs_real @ X ) )
= ( abs_abs_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_abs
thf(fact_1140_log__base__root,axiom,
! [N2: nat,B4: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ B4 )
=> ( ( log @ ( root @ N2 @ B4 ) @ X )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B4 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_1141_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1142_pos__zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N2 )
= one_one_int )
= ( ( M = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1143_real__root__gt__zero,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_1144_real__root__strict__decreasing,axiom,
! [N2: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N4 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_1145_root__abs__power,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
= ( abs_abs_real @ Y ) ) ) ).
% root_abs_power
thf(fact_1146_zmult__zless__mono2__lemma,axiom,
! [I2: int,J2: int,K: nat] :
( ( ord_less_int @ I2 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1147_real__root__pos__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_1148_real__root__strict__increasing,axiom,
! [N2: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_1149_real__root__decreasing,axiom,
! [N2: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N4 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_1150_real__root__pow__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_1151_real__root__pos__unique,axiom,
! [N2: nat,Y: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ Y @ N2 )
= X )
=> ( ( root @ N2 @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_1152_real__root__power__cancel,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_1153_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1154_ln__realpow,axiom,
! [X: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_1155_log__nat__power,axiom,
! [X: real,B4: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B4 @ ( power_power_real @ X @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B4 @ X ) ) ) ) ).
% log_nat_power
thf(fact_1156_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1157_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1158_real__scaleR__def,axiom,
real_V1485227260804924795R_real = times_times_real ).
% real_scaleR_def
thf(fact_1159_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1160_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1161_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1162_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1163_log__root,axiom,
! [N2: nat,A4: real,B4: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A4 )
=> ( ( log @ B4 @ ( root @ N2 @ A4 ) )
= ( divide_divide_real @ ( log @ B4 @ A4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% log_root
thf(fact_1164_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1165_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1166_Rep__account__uminus,axiom,
! [Alpha2: risk_Free_account] :
( ( risk_F170160801229183585ccount @ ( uminus3377898441596595772ccount @ Alpha2 ) )
= ( ^ [N: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha2 @ N ) ) ) ) ).
% Rep_account_uminus
thf(fact_1167_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_1168_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1169_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1170_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_1171_negative__eq__positive,axiom,
! [N2: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N2 = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1172_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1173_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1174_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1175_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N2 ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1176_negative__zless,axiom,
! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1177_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( artanh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_1178_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_1179_power__Suc__0,axiom,
! [N2: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1180_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1181_one__le__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1182_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_1183_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1184_negative__zless__0,axiom,
! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1185_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1186_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1187_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1188_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1189_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1190_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1191_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1192_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1193_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1194_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1195_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1196_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1197_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1198_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_1199_uminus__account__def,axiom,
( uminus3377898441596595772ccount
= ( ^ [Alpha: risk_Free_account] :
( risk_F5458100604530014700ccount
@ ^ [N: nat] : ( uminus_uminus_real @ ( risk_F170160801229183585ccount @ Alpha @ N ) ) ) ) ) ).
% uminus_account_def
thf(fact_1200_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1201_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1202_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z5: nat] :
( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z5 )
=> ( R @ X2 @ Z5 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1203_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1204_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1205_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1206_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1207_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1208_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1209_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1210_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1211_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_1212_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1213_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_1214_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1215_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1216_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_1217_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1218_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ N2 )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1219_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1220_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1221_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ N2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1222_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1223_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_1224_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1225_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1226_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_1227_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1228_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1229_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1230_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1231_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N2 ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N2 )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1232_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1233_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1234_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1235_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1236_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1237_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1238_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1239_inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1240_dec__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1241_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_1242_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_1243_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1244_real__minus__mult__self__le,axiom,
! [U2: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U2 @ U2 ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1245_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1246_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1247_zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ( times_times_int @ M @ N2 )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N2 = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N2
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1248_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N2: int] :
( ( ( times_times_int @ M @ N2 )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1249_not__int__zless__negative,axiom,
! [N2: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1250_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1251_one__less__mult,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1252_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1253_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1254_realpow__pos__nth2,axiom,
! [A4: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ A4 )
=> ? [R4: real] :
( ( ord_less_real @ zero_zero_real @ R4 )
& ( ( power_power_real @ R4 @ ( suc @ N2 ) )
= A4 ) ) ) ).
% realpow_pos_nth2
thf(fact_1255_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1256_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1257_abs__real__def,axiom,
( abs_abs_real
= ( ^ [A: real] : ( if_real @ ( ord_less_real @ A @ zero_zero_real ) @ ( uminus_uminus_real @ A ) @ A ) ) ) ).
% abs_real_def
thf(fact_1258_int__zle__neg,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N2 = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1259_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_1260_nat__one__le__power,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_1261_zabs__def,axiom,
( abs_abs_int
= ( ^ [I: int] : ( if_int @ ( ord_less_int @ I @ zero_zero_int ) @ ( uminus_uminus_int @ I ) @ I ) ) ) ).
% zabs_def
thf(fact_1262_arcsin__le__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_1263_arcsin__minus,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( arcsin @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% arcsin_minus
% 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,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( groups6591440286371151544t_real @ ( risk_F170160801229183585ccount @ ( risk_Free_just_cash @ a ) ) @ ( set_ord_atMost_nat @ n ) )
= a ) ).
%------------------------------------------------------------------------------